text:
Putin: Western Model of Democracy Doesn't Exist
Russia » Politics
Vladimir Putin gave a traditional interview to French journalists prior to his visit to Paris. In the interview, the chairman of the Russian government said that the Western model of democracy does not exist per se. Below you can read several most intersecting excerpts from Putin's interview.
About democracy
"Can you tell me what it is – the Western model of democracy? France has one model of democracy, the USA has another one. A French politician told me once: one can not do anything at the elections in the USA without money – whether they are elections to the Senate or Congress, not to mention the presidential election. There is nothing to do there without a sack of money. So what is that democracy about? Is it democracy of those who have a lot of money? The USA is a presidential republic, Great Britain is a monarchy. All of that are elements of democracy, but what kind of democracy? There is no single model of " Western democracy ". It doesn't exist.
"Tony Blair is a good friend of mine. His party won the elections under his leadership. Afterwards , Laborites decided that he should step down . Mr. Brown immediately took his position and automatically became the prime minister, without elections. What is it ? Democracy ? Yes , it is . That is the way it is – good or bad.
Russia Today: Solution on Iranian nuke program should fit all sides - Putin
"I've had many discussions with American colleagues. I asked them how could it happen that the majority of US citizens voted for one person, but another one took office as president owing to the system of electoral delegates. The Americans answered: you do not need to interfere, we are used to it, and that's the way it will be. We do not interfere. Why do you think that you can interfere with us? We will solve our problems alone."
About Georgia
"Georgia has a very long ground border with Russia. People died as a result of the criminal action launched by President Saakashvili two years ago. Russia was forced to defend the lives of its peacemakers and the citizens of South Ossetia. We stopped 15-20 kilometers far from Tbilisi. We stopped there not because we could not enter Tbilisi – we stopped because we didn't want to do it. We didn't want any military action at all. I hope there will never be another armed conflict between Russia and Georgia, never."
About sport
"You know, I've been going in for judo since childhood. It would be strange if I was trying to conceal that. I have a judo black belt. It's one of the biggest lifeworks for me. When it attracted so much attention in the media, I was very surprised because I did not see anything unusual about it. It's routine for me – the things that I do every day. Well, not every day – every other day. There was a period when I was getting ready to take part in the judo championship of the Soviet Union. I had daily training sessions in the national judo team."
About personality cult
"First off, I just live and work the way I can. I do the things that I like to do. There's one life and one has to learn to enjoy it despite various limitations. Secondly, speaking of the personality cult, I think that our citizens know exactly that personality cult is not just attention to one person. It also includes countless violations of law and repressions. I can not imagine, not in my wildest dreams, that it can happen again in modern-day Russia. The Russian society today is mature enough not to let the 1930s, 40s and 50s repeat again."
About democracy in Russia
"Authorities always try to look better than they really are in the whole world. They try to somehow restrict opportunities of mass media. As a rule, authorities can not do it in a country with a well-developed civil society. A lot of things need to be changed and improved in the Russian political system, but this is a long process that takes place the society grows.
About next presidential election in Russia
"Me and President Medvedev think about it, of course, but we agreed not to pay much attention to this problem. The decision will depend on how we work on our positions."
About World Cup
When asked which country Putin will support during the Word Cup, the prime minister dropped a curtsey to the French and the French team, which, unlike Russia, will go to South Africa to enjoy the game of football.


Fractal is a clever online tool that allows you to validate your email CSS across 24 of the most popular email clients simply by uploading your email code (HTML/CSS). You'll receive a report that outlines the problems in your code and it will show you which email clients support your code. This is great as it allows email developers to design, build and test their emails more efficiently. Fractal will even suggest fixes to your code to ensure that it is cross client compatible, you can even contribute your own fixes if you're feeling particularly inspired.
Fractal validates your HTML email across all the major email clients and automagically fixes known email quirks. It's FREE to use so give it a try next time you're working on an email.


text:
module Data.Matrix (
  matrix, Matrix
) where

import Data.Matrix.Construct

redpajama_set_name: RedPajamaGithub


Vigor representatives approached Vancouver officials "recently," said City Manager Eric Holmes, one of three officials in Clark County privy to talks with the company as it made its decision.
Holmes and Vigor officials discounted the influence of financial incentives in the company's decision.
The city manager said Vigor received the same incentives that would be offered to any prospective industrial recruit, such as a cap on building permit fees and expedited permit processing. Also, the city would assist Vigor in trying to obtain the necessary approvals to build a dock in the future, Holmes said.
A Vigor spokeswoman said the company would seek an unspecified amount from Washington's Economic Development Strategic Reserve Account to assist with the expenses of moving work to Vancouver.
Vigor, which is privately held with a mix of 2,300 union and non-union workers, has employees in several locations, including an existing site in Vancouver, its Portland headquarters and a 60-acre shipyard on Swan Island.
The company has previously done business in Oregon as Vigor Industrial LLC and in Washington as Vigor Shipyards Inc.
Operating through its subsidiaries, Vigor builds and repairs ships and provides related vessel maintenance services including painting, blasting and coating. It also builds ocean and inland barges. Customers include cruise lines, ferry systems and oil companies.
Kvichak Marine Industries in Ballard laid the groundwork to acquire the huge Army contract in 2012, three years before it was acquired by Vigor, becoming Vigor Kvichak. On Friday, workers were informed their facility would close and its projects would move to Vancouver. The 70 workers affected at Kvichak — it's named after a river in Alaska — were offered the opportunity to move to the Vancouver location.
Also Friday, about 60 of the 500 workers at Vigor's factory in Clackamas, Ore., were told their aluminum fabrication jobs would move to Vancouver. They, too, were offered the opportunity to follow their job to Vancouver, Foti said.
The Vancouver site will begin construction of the Army landing craft prototype this summer. Once the prototype is completed and testing and refinements have occurred, another 36 vessels will be built.
In addition to the Army landing craft production, the site will also be used to continue Vigor's ongoing production of the Combatant Craft Medium for the U.S. Navy as well as for U.S. allies, the Response Boat-Medium for the U.S. Coast Guard and export market, Vigor Fast Interceptor, aluminum fast ferries and commercial workboats.
Vigor representatives contacted Vancouver officials at the end of November after learning of a potential site in Columbia Business Park.
Foti said Vigor's existing site in Vancouver, the former Oregon Iron Works nearby on the riverfront at the former World War II-era Kaiser Shipyards, did not meet the needs for the aluminum-fabrication projects.
The site at 4400 S.E. Columbia Way in the Columbia Business Park is the former home of Christensen Yachts. Vigor is in the process of purchasing the site and the transaction is expected to be completed this summer.
Foti noted the marine heritage of the Christensen building in Friday's news release.
"We are honored to carry forward the vision begun by (company founder) Dave" Christensen, Foti says in the release. "When I think of his alignment with the Vigor values of Truth, Responsibility, Evolution and Love, it's Responsibility and Love that shine through. He stayed responsible to what he built, and his Love for the people who built amazing boats here was proven by Dave's actions many times over.
"There's a sign in the entry that reads, 'Through these doors walk the finest boat builders in the world.' The craftspeople of Vigor intend to honor that vision as we walk through those same doors every day," Foti said.
Christensen died Oct. 15, 2018, years after having any connection with his namesake company, which has been embroiled in a lawsuit between members of the Christensen family and the company's current owner, Tennessee investor Henry Luken. The parties reached a settlement earlier this week, according to court records, although the details of the settlement were not available.
Luken said on Friday that the Christensen Yachts business will continue at a new shipyard location near Tellico Lake, Tenn. Plans for the Tennessee facility were first announced in 2006, but according to a 2011 Columbian story, the company was hit hard by the recession starting in 2008 and had to halt construction of the second location. It never opened, Luken said, but he now plans to finish the site and move the company's equipment over from the Vancouver shipyard.
Best remembered for the Kaiser Shipyards, Vancouver's Columbia riverfront has a long maritime history.
The World War II-era operation was one of the defining institutions in Vancouver's history, spawning a huge population and housing boom. The yard was along the Columbia River, just downstream from Marine Park; the old slipways are visible from an observation tower near the park's boat launch.
Kaiser produced baby flattops, tank landing craft and Liberty ships. At its peak in December 1944, it employed 38,000 and led to creation of the Vancouver Housing Authority and such developments as McLoughlin Heights, Fruit Valley and Fourth Plain Village.
The WWII deal between the U.S. Maritime Commission and the Kaiser Co., a heavy-construction firm controlled by Henry Kaiser and his son Edgar, was signed Jan. 9, 1942. The first ship, the S.S. George Vancouver, was launched on July 4, 1942. The last ship was outfitted in May 1946.
Vancouver had been the site of a World War I shipyard as well, called the Standifer Shipyard, at what is now the Port of Vancouver's Terminal 1. That shipyard went out of business by 1921. Today the site is home to a restaurant, WareHouse '23.
A company Vigor acquired, Oregon Iron Works, also took on marine projects in Vancouver. In 2002, Oregon Iron Works helped build an 85-foot Coast Guard craft under a $2.4 million subcontract.
The Columbia River Economic Development Council kicked off the new year by welcoming three members to its board of directors, along with the economic development organization's new president, Jennifer Baker.
Baker most recently served as the deputy district director for the Portland office of the U.S. Small Business Administration, and previously worked in multiple business and trade fields, including four years as an international trade specialist with the U.S. Department of Commerce.
Baker began her tenure Jan. 14, and the new board members were officially voted in at the council's annual investor event at the ilani casino on Wednesday. Alan Hwang of ACH Marketing, Andy Lowery of RealWear, Kara Fox-LaRose of ilani and Tim Price of Yaculta were elected to three-year terms on the board.
The council's board officers are: chair (2018-2019): Casey Wyckoff, LSW Architects; vice chair (2018-2019): Lisa Lowe, Schwabe Williamson & Wyatt; immediate past chair (2018-2019): Greg Seifert, Biggs Insurance Services; treasurer (2019): Lisa Dow, Columbia Bank; secretary (2019): Helen Devery, BergerABAM.
The event included a recap of the council's progress on the Clark County Comprehensive Economic Development Plan, a five-year strategic plan that the organization adopted in 2017, which focuses on three core goals: expand the existing business base, support people and create place.
In her closing remarks to CREDC investors, Baker said that 2019 would be a year of increased direct action, and she urged the members to reach out to her and share their stories, as well as reaching out to others in their networks and focus on asking the questions that will spur action.
Earlier in the day, The Columbian sat down for an interview with Baker at the council's downtown Vancouver office to learn more about her plans for the CREDC. Answers have been edited for length and clarity.
Coming from the U.S. Small Business Administration, what was appealing about the CREDC and Clark County? What kind of opportunities did you see there?
I fully subscribe to the vision of the Clark County Comprehensive Economic Development Plan, and that vision is to recognize Clark County as the most inclusive, healthy an amenity-rich community in the U.S.
Those values and the commitment and investment that I see from community members is palpable in the meetings I attended when I was an SBA employee, and certainly now that I'm president of the CREDC. Being a part of a community that's so invested allows us to do great things.
How much interaction did you have with the CREDC prior to seeking out this role?
The CREDC was part of a consortium of partners who were administering a Jobs and Innovation Accelerator Challenge grant several years back, and so I worked with Elizabeth (Scott), who is now our business development director, on the implementation of that grant.
What appealed to you about this position, and how will your background contribute to your job performance?
I would say that I'm resourceful and diplomatic, and I feel like I have a knack for asking the right kinds of questions to help bring awareness and drive decision making. The experience of having worked with big businesses on international trade issues and businesses as small as a food cart — that expanse of experiences is what I'll draw on to be particularly creative and resourceful as I liaise with the board and the investor community in realizing the next steps for the economic development plan.
One of the things in particular that fuels me is when I can drive by a warehouse or a business that was able to grow by virtue of the help that's been infused by an organization — in this case, CREDC — and know that there are people employed by virtue of the efforts of the community. That's very gratifying, to be able to see the results in our community and to know that employment opportunities are being created.
Do you see this transition as a continuation of your more recent work with small businesses, or more of a transition back towards the kind of trade and commerce work you did prior to the Small Business Administration, or both?
I hope it'll be both. Well, I know it will be both. Depending on the industry, the SBA defines a small business as anything with 500 employees or fewer. So 'small' tends to still be pretty large in the lens of the SBA. I'm not anticipating a huge vacillation, and harkening back to my commerce days, I certainly worked with some very large corporations, helping them with trade policy.
Clark County is doing well economically, but like the rest of the Portland region, there have been struggles with housing affordability and struggles for businesses to attract skilled labor. What can the CREDC do to help address these issues, and what are some of the other issues facing the county that you want to be able to play a hand in tackling?
Stemming from the CREDC vision, we've outlined three business goals. (The first is to) expand the existing business space, and that entails building out traded sector. (The second is) supporting people, so making sure that the Clark County workforce is gaining skills and capacity to be a collaborative partner with industry, and/or attracting workforce from outside of Clark County to bolster workforce where there happen to be specific needs.
The third prong of the plan is creating place, and we keep an eye on the lands available for development and coordinate with our city and county partners on that inventory. But another part of creating place is the quality of life factor; being able to reach out and share that Clark county has amazing natural amenities.
Whether that's kayaking up in Ridgefield or taking a hike on trails around La Center, or taking a stroll around the Vancouver Waterfront, or walking through Main Street in Camas, or the old historic district in Battle Ground, there are so many amazing amenities that we can share about the quality of life here, and I think that will continue to attract top talent.
Clark County has a growing technology sector. What is it going to take to keep it growing and make the county competitive in the tech field?
That relates directly to the second two goals of the strategic plan, which are support people — helping to build a workforce that is capable of meeting the needs of the tech industry as they look to move to town — and creating place.
Tech industry salaries tend to be higher-end salaries, averaging around $102,000 annually, and oftentimes employees in tech firms want to move somewhere where they can work hard and play hard, and here you have a fusion of both.
The diversity of towns that are relatively proximal means that you can have employees who live in different towns depending on whether or not they really subscribe to an urban experience or if they want to have a wooded lot somewhere.
As you said, there are different cities here that are very diverse and the county overall has a lot of variation to it — so how do you balance the competing business interests? How do you find solutions that are going to fit with everybody in Clark County?
I think that really correlates back to the cities and their uniqueness. Each city has something distinct to offer that makes Clark County desirable and accessible and inviting. So I think in some ways it's up to those who are exploring Clark County to see what really fits with their life philosophy, because I think we really offer a little bit of it all.
The CREDC has been very public about wanting to move forward on replacing the Interstate 5 Bridge. What role do you see the CREDC playing in trying to pick those discussions back up, and how can we avoid another stalemate?
CREDC has set up an I-5 corridor task force and we're inviting city representatives, Port (of Vancouver) representatives and regional transportation representatives to the table. (A new bridge) really will help connect Clark County to the rest of the West Coast more fluidly, and Clark County internationally. We're pleased that Gov. Jay Inslee proposed a $17.5 million budget line item for an I-5 project office.
Coalescing the voices of the business industry and emphasizing the importance of transportation infrastructure to traded sector will help inform the dialogs to come. I think we have a huge commitment on behalf of interested parties to help share our story and help share the benefits of an optimized I-5 project.
Are there any other large-scale projects that the business community or the CREDC see Clark County as needing right now?
Investment in people is one of the key investments. I know I'm transitioning from infrastructure to workforce, but one of the common themes we hear from business owners in Clark County is that they're really having to be effortful to find the right workforce contributors to grow their businesses, so I think continued investment in skilled labor and STEM — those will all be important ingredients to building out the most competitive economy.
Is there anything else you'd like to say about the job, or to introduce yourself to residents and business owners in Clark County?
The community has been very welcoming so far, and I'm so excited to be out and about and hearing people's stories; their experiences building a business or resourcing business components or trying to explore new markets and troubleshoot parts of the supply chain. I think hearing those stories and being able to translate them into opportunity for other businesses and for businesses outside of the community will be really valuable.
VANCOUVER, Wash. (January 24, 2019) – Yesterday, at its annual investor meeting, the Columbia River Economic Development Council (CREDC) elected their new members to serve on the 2019 Board of Directors. At a special Board meeting immediately following, the new Board moved to elect its 2019 officers and private sector Executive Committee members.
1 . Brian Wolfe, Brian H. Wolfe, P.C.
3. Bill Dudley, Landerholm, P.S.
4. Kevin Tapani, Tapani Inc.
The full list of the 2019 CREDC Board of Directors can be found HERE.
VANCOUVER, Wash. (December 13, 2018) – Today, Governor Jay Inslee released details of his 2019-2021 budget proposal, which includes $17.5 million in funding to establish an I-5 bridge project office tasked with helping define a new project scope, evaluate the previous Record of Decision and re-establish vital partnerships that will lay the groundwork to ensure long-term success.
With a growing population and a rising number of businesses seeking to expand and locate in Southwest Washington, the Columbia River Economic Development Council (CREDC) is mobilizing its network of more than 200 business leaders and strategic partners to support a bridge project that provides a long-term solution to the mobility needs of Clark County, the metropolitan region and the entire West Coast.
To this end, CREDC recently formed the I-5 Bridge Replacement Task Force made up of business and public sector leaders across the County who will contribute the voice of the business community as this critical work moves forward. Aligned with the Clark County Comprehensive Economic Development Plan, CREDC's vision for the County is to be one of the most inclusive, healthy, and amenity-rich communities in the country. To accomplish this vision, it is critical that our transportation infrastructure accommodate our immediate needs and set us up for success as we continue on a rapidly growing trajectory.
The Columbia River Economic Development Council ("CREDC") is seeking responses to this Request for Proposals ("RFP") for the purpose of evaluating and selecting a consultant or team of consultants to provide an update to its 2016 Employment Land Study of Clark County, Washington (the "Land Study"). Responses to this RFP will be evaluated by a selection committee who will independently score each proposal based on the criteria set forth in this solicitation. Finalists with the highest aggregate score will be considered for interviews. Following interviews, the CREDC anticipates the selection committee will invite a finalist to enter into an agreement with CREDC to update the Land Study consistent with the requirements set forth in this RFP. Composite teams of individuals and/or firms may be formed to bring together the various disciplines necessary to provide the services described in this RFP. Having access to consultants with flexibility and various skill sets provides significant advantages and benefits. Firms of all sizes, depth, and capabilities are encouraged to submit proposals.
CREDC is often the first point of contact for traded sector businesses looking to locate or expand in Clark County, Washington. The industrial/office/business park market segment is distinct from the commercial and residential markets. Not only is the price of land per square foot lower than for other segments, but these kind of businesses often have specific needs (e.g., access to rail, utilities, or freeway) and requirements that generally are not transferable from one site to another. While lands may be classified as "available," environmental conditions, regulations, or infrastructure constraints may reduce the effective size, availability, or affordability of those lands for development.
In 2016, CREDC conducted an employment land study that identified employment sites and their varying state of readiness, which is an important baseline to inform policymakers and service providers as they respond to challenges replenishing the inventory of sites that support significant high-impact job density. The study, guided by a project advisory group and a consultant team, framed consensus on key definitions and industry-specific profiles. The study identified a total of 56 employment sites arranged in three tiers based on the estimated time needed to make each site development ready. The project also prepared a detailed development analysis that determined market opportunities and the economic impact for five sites from the inventory.
• Update inventory of employment sites: determine site-specific changes to development readiness, sites that should be removed from the inventory due to absorption or zone changes, and identify any new sites that should be added to the inventory within the tier structure.
• Quantify the total supply and readiness of large industry, business park, and office-industrial (excluding commercial) sites in Clark County, Washington.
• Analyze development of identified sites and update their tier rating (e.g., determine Tier III sites that have been developed, and former Tier I sites that have increased development readiness).
Report on jurisdictions' initiatives to advance sites (e.g., capital improvement plans, policy development, etc.) by conducting stakeholder interviews and/or communicating directly with city and county planning and/or public works departments.
• Prepare recommendations that detail how sites could be made more attractive with further design, infrastructure investments, or regulatory amendments, as well as how public policy could inform investments to strategically plan and execute in order to yield best and earliest results, increasing the supply of development-ready sites.
• Identify strategies for aggregating sites with multiple owners.
• Consider sites otherwise development-ready or near development-ready yet located just outside of the Urban Growth Boundary, providing jurisdictions with strategic direction for UGB expansions.
• Pinpoint recommendations for highest and best uses for the Chelatchie Railroad.
• Provide a final report to inform the discussion on future tools and policies to maintain market-ready inventory of employment sites and provide information that can assist in future infrastructure planning, prioritization, and policy/regulatory rulemaking decisions.
analysis for each. Provide cost estimate per site.
o Utility gap analysis to determine infrastructure deficiencies surrounding employment sites throughout Clark County, Washington.
Funding for this project is expected come from multiple regional partners and the budget will depend on final scope of work and deliverables. Digital delivery of the final report is required and CREDC reserves the sole ownership rights of the deliverables.
5. Total cost including number of hours and hourly rate.
A response may not be withdrawn or canceled for a period of 90 days following the submission deadline.
Respondents are expected to examine the instructions, specifications, terms, and conditions prior to submitting their response. Failure to do so will be at the respondent's risk.
At CREDC's request, respondents may be selected for in-person presentations.
All responses and related materials become property of CREDC.
CREDC reserves the right to reject any or all responses received.
Proposals must be submitted electronically to Brittany Bagent at bbagent@credc.org no later than COB on December 7, 2018. If you have any questions, please contact Brittany at the email above no later than November 30, 2018.
This application period is now closed.
CREDC's second Grow Clark County event of the year featured a dynamic panel of women founders from across the region who shared stories about their journey starting and growing their businesses in Clark County – including the challenges they faced and what it took to innovate in a fast-changing environment.
Allie Magyar, founder and CEO of event management software company Hubb. In just six years, Allie has led the company to achieve impressive regional recognition, including this year's Technology Association of Oregon's Emerging Company of the Year, Vancouver Business Journal's Innovator of the Year and inclusion in Portland Business Journal's 100 Fastest-Growing Private Companies.
Callie Christensen, co-founder of children's toy company Slumberkins. Along with Kelly Oriard, they co-founded the company in 2015 and have reached great heights, including an appearance on Shark Tank and, most recently, landed a TV deal with The Jim Henson Company.
Miriam Kim, co-founder and COO of the leading developer and innovator of connected kitchen products Perfect Company. With 12 years at Intel under her belt, Miriam is a serial entrepreneur who has co-founded two other successful companies: Ensequence and Pure Imagination.
Lisa Schauer, president of leadership development firm PointNorth Consulting. A champion of women in leadership roles, Lisa became the first female partner and the first female Board Director at MacKay Sposito and helped found H-RoC Political Action Committee to advance women leaders in Southwest Washington.
Below are key pieces of insight from this impressive group of Women Founders, highlighting their motivation, entrepreneurial spirit and advice for others looking to embark on their own entrepreneurial journey. We hope they inspire and motivate you as much as they did us!
Special thank you to our sponsors Zenith Properties NW, Kaiser Permanente, iQ Credit Union and Opsahl Dawson for making the event possible and to Schwabe, Williamson & Wyatt Shareholder Lisa Lowe for being a fantastic panel moderator.
The next GROW Clark County event is coming up in September. Register for our mailing list here or follow us on facebook, twitter or LinkedIn for more details as soon as they become available.


text:
\section{Introduction}
The Mumford-Shah functional is one of the most important variational model for image segmentation. It was introduced in the late 80's by Mumford and Shah (\cite{BDB},\cite{OAB}) and it can be defined in its general form as
\begin{equation}\label{mumford}
J(u,K) = \int_{\Omega\setminus K} |\nabla u|^2 \, dx +  \beta \mathcal{H}^{n-1}(K) + \alpha \int_{\Omega\setminus K} |u-g|^2 \, dx,
\end{equation}
where $\Omega \in \mathbb{R}^n$ is open, $K\subset \Omega$ is closed and such that $\mathcal{H}^{n-1}(K)<\infty$, $g\in L^{\infty} (\Omega)$, $u\in W^{1,2}(\Omega\setminus K)$ and $\beta$ and $\alpha$ are tuning parameters. \\
The idea of the model is that given $g$ representing the level of gray of an image, it is possible to get a ``smoother'' version of it, ``close'' to the starting one in the $L^2$ norm, by finding a minimizer of \eqref{mumford}. The gain of smoothness for the minimizers comes from penalizing the oscillation of the competitors (i.e. the Dirichlet energy) and the length of the contour, in order to avoid fractal behaviour of the boundary of the processed image.\\
The existence of minimizers for \eqref{mumford} was proved in \cite{ETF2} introducing a weak formulation obtained considering $u\in SBV(\Omega)$ and replacing the set $K$ with $S_u$, i.e. the singular set of $u$:
\begin{equation}\label{mum}
F(u) = \int_{\Omega} |\nabla u|^2 \, dx +  \beta\mathcal{H}^{n-1}(S_u) + \alpha\int_{\Omega} |u-g|^2 \, dx.
\end{equation}
It is worth to remark that when $\alpha = 0$ and $\beta=1$, $F$ is called homogeneous Mumford-Shah functional.\\
In the following years there have been a huge effort in understanding the regularity properties of the functional defined above. We can cite some relevant papers in this direction like \cite{HIG}, \cite{OTE}, \cite{CFM}. However, despite all the effort, the main conjecture proposed by Mumford and Shah in their seminal paper still remains open in its full generality.
\begin{conj}[Mumford, Shah]
Let $(u,K)$ be a pair minimizing \eqref{mum}. Then $K$ is locally union of finitely many $C^{1,1}$ embedded arcs.
\end{conj}
As pointed out for the first time in \cite{OTE}, a blow up limit of appropriate sequences of minimizers of \eqref{mum} is a local minimizer of the homogeneous Mumford-Shah functional; for this reason the characterization of these minimizers is directly related to the solution of the conjecture stated above. For example it is known that harmonic functions are local minimizers of \eqref{mum} (for $\alpha=0$ and $\beta=1$) in small domains and that the same result holds for step functions and triple junctions (\cite{TCM}). Moreover the main achievement in this direction is contained in \cite{CIS} and it answers affirmatively to a conjecture proposed by De Giorgi in \cite{PCD}:
\begin{equation}\label{crack}
u(\rho,\theta) = \sqrt{\frac{2\rho}{\pi}} \sin\left(\frac{\theta}{2}\right) \quad \rho>0,\ -\pi<\theta<\pi
\end{equation}
is a global minimizer of the homogeneous Mumford-Shah functional. \eqref{crack} is usually called crack-tip.

In \cite{TCM} Alberti, Bouchitt\'e and Dal Maso introduced the notion of calibration for the Mumford-Shah functional that resembles closely the classical theory for minimal surfaces by Harvey and Lawson (\cite{CGE}). With this technique, in \cite{TCM}, they were able to prove the minimality of some candidates for the homogeneous Mumford-Shah functional like the triple junction or reproving the minimality of harmonic functions in a very elegant way. However it remains open the problem of finding a calibration for the crack-tip and for general minimums in higher dimensions.
It is therefore a relevant issue to understand if, given $u$ a minimum for the Mumford-Shah functional, then there exists a calibration for $u$.\\
This is the question we are going to address in this paper. Existence of calibration is a common issue also in the field of minimal surfaces and also there it is not completely solved. One can refer to the work of Federer \cite{RFC} for the classical results in this theory.\\
As for the Mumford-Shah functional the main result in this direction was obtained by Chambolle in \cite{CRF}. He proved the existence of a calibration in dimension one in a weak asymptotic sense using the following representation formula introduced in \cite{TCM}:
\begin{displaymath}
F(u)= \sup_{\phi\in K} \int_{\Gamma_u} \langle \phi, \nu_{\Gamma_u}\rangle \, d \mathcal{H}^{n}= \sup_{\phi\in K} \int_{\Omega\times \mathbb{R}} \langle\phi, D\textbf{1}_{\{u>t\}}\rangle,
\end{displaymath}
where $K$ is the set of Borel vector fields $\phi:\Omega \times \mathbb{R} \rightarrow \mathbb{R}^{n+1}$ such that
\begin{equation}
\left\{\begin{array}{ll}
\phi^t(x,t) \geq \dfrac{|\phi^x(x,t)|^2}{4} -\beta(t-g)^2 & \forall x,t\\
\left|\displaystyle\int_{t_1}^{t_2} \phi^x(x,s)\,ds\right|\leq\alpha\ &\forall x,t_1,t_2.
\end{array}
\right.
\end{equation}
More precisely this representation formula is the particular case of a general one for ``local'' functionals in BV presented by Bouchitt\'e in \cite{CAN}.\\
In particular one can lift $F$ to higher dimension to obtain a convex functional $\mathcal{F}$ defined as
\begin{displaymath}
\mathcal{F}(w) = \sup_{\phi\in K\cap C_0} \int_{\Omega\times \mathbb{R}} \langle\phi , Dw\rangle
\end{displaymath}
for $w \in SBV(\Omega \times \mathbb{R})$ decreasing in the last variable. If one is able to prove that given $u$ a minimizer of $F$, then $\textbf{1}_{\{u>t\}}$ is a minimizer of $\mathcal{F}$, then this would imply the existence of a calibration in a weak asymptotic sense by argument of convex analysis. Moreover, another important consequence is that one can compute the minimum of $F$ using the functional $\mathcal{F}$ that, being convex, allows for an efficient gradient descent method (\cite{AAF}).
Chambolle, in \cite{CRF}, was able to prove these facts in dimension one and he pointed out that the same results could be obtained building up a coarea-type formula for the previous functional generalising the classical coarea formula for functionals (\cite{CLD}, \cite{GCF}):
\begin{equation}\label{solast}
\mathcal{F}(w) = \int_0^1 \mathcal{F}(\textbf{1}_{\{w(x,t)>s\}})\, ds,
\end{equation}
that is false for $\mathcal{F}$ as the example below shows:
\begin{displaymath}
u_1(x)=\left\{
\begin{array}{ll}
0 & \mbox{ if } x\leq 1/2\\
x & \mbox{ if } x> 1/2,
\end{array}
\right.
\qquad
u_2(x)=\left\{
\begin{array}{ll}
x & \mbox{ if } x\leq 1/2\\
1 & \mbox{ if } x> 1/2
\end{array}
\right.
\end{displaymath}
and
$w(x,t) = (1/2)\textbf{1}_{\{u_1(x)> t\}} + (1/2)\textbf{1}_{\{u_2(x)> t\}}$.

\vspace{3mm}

In this article we use an alternative representation of the Mumford-Shah functional by rectifiable currents of the type
\begin{displaymath}
G(T)=\sup_{\phi\in K} \int_{\mathcal{M}} \theta\langle \phi , \nu_T \rangle d\mathcal{H}^{n},
\end{displaymath}
where $T=(\mathcal{M},\xi,\theta)$ is a rectifiable current and $\nu_T$ is the normal to $\mathcal{M}$, and we start to exploit the validity of a general coarea-type formula for the functional $G$. In Section \ref{mais} we study the structure of the functional and we prove the following convex decomposition formula for a finite linear combination of graphs in dimension one.
\begin{thmnonumber}[Convex decomposition formula]
Let $I$ be an open interval. Given $T= \sum_{i=1}^k \lambda_i\Gamma_{u_i}$ with $u_i\in SBV(I)$ and $\lambda_i>0$ such that $|\bigcup S_{u_i}| < +\infty$ there exists $k' \in \mathbb{N}$, $\{\mu_i\}_{i=1\ldots k'} > 0$ and $\{w_i\}_{i=1\ldots k'} \subset SBV(I)$ such that $T = \sum_{i=1}^{k'} \mu_i\Gamma_{w_i}$ and
\begin{displaymath}
G(T) =  \sum_{i=1}^{k'} \mu_i G(\Gamma_{w_i}).
\end{displaymath}
\end{thmnonumber}
The previous formula can be viewed as a variant of the generalized coarea formula in the sense of \eqref{solast}, when the latter is applied to finite linear combination of graphs. \\
The immediate consequence of this result is the following theorem that links the minimizers of \eqref{mum} with the minimizers of $G$:
\begin{thmnonumber}
Given $u\in SBV(I)$ a minimizer of the Mumford-Shah functional, $\Gamma_u$ (i.e. the graph associated to $u$) is a minimizer of $G$ among all the linear combinations of graphs of the form $T = \sum_{i=1}^k \lambda_i \Gamma_{u_i}$ with $\partial \Gamma_u = \partial T$.
\end{thmnonumber}
In Section \ref{existence}, we use this theorem to prove the existence of calibrations in a weak sense (see Definition \ref{calcur}) as a consequence of the Hahn-Banach theorem. The general idea of this proof follows closely Federer's approach to calibrations for minimal surfaces in \cite{RFC} and it suggests that, at least in dimension one, it would be possible to produce the analogue result and to extract an $L^\infty$ vector field playing the role of a calibration.\\
It is worth to notice that the convex decomposition formula presented in this paper relies on the one dimensional structure of the domain. In particular in Proposition \ref{coareasame} it is necessary that the singular points of an SBV function disconnect the domain; this is clearly peculiar of the dimension one, but it is likely that similar decomposition can be found in higher dimension and similar results could be obtained.\\
Moreover, even if all the proof of this paper are carried on for the functional \eqref{mum} the results can be extended with minor modifications to more general Mumford-Shah type functionals. We refer to Remark \ref{gener} for further details in this direction.
\begin{ackn}
The author is warmly grateful to Professor Bernd Kirchheim for the useful discussions about this problem. The author would also like to thank Professor Giovanni Alberti for the valuable suggestions, Professor Domenico Mucci for the careful reading of the manuscript and the referee for the detailed review.
\end{ackn}

\section{Preliminaries}
Throughout the paper we consider $\Omega$ and $\Omega'$ to be open, bounded, regular sets of $\mathbb{R}^n$ such that $\Omega' \subset\subset \Omega$. Given $g\in L^\infty(\Omega)$ we define the Mumford-Shah functional as stated in the introduction
\begin{equation}\label{mum2}
\mathfrak{F}(u) = \int_{\Omega} |\nabla u|^2 \, dx +  \mathcal{H}^{n-1}(S_u) + \int_{\Omega} |u-g|^2 \, dx
\end{equation}
and the homogeneous version
\begin{equation}\label{ho2}
F(u)= \int_{\Omega} |\nabla u|^2 \, dx +  \mathcal{H}^{n-1}(S_u),
\end{equation}
where $u\in SBV(\Omega)$ and $S_u$ is the singular set of $u$. We refer to \cite{CCC} for the basic properties of BV and SBV functions and to \cite{SSO} for a comprehensive treatise on the Mumford-Shah functional.
\vspace{2mm}\\
We deal with the following notions of minimizers:
\begin{defi}[Minimizer of $\mathfrak{F}$]
Given $g\in L^\infty(\Omega)$ we say that $u\in SBV(\Omega)$ is a minimizer of $\mathfrak{F}$ if $\mathfrak{F}(u) \leq \mathfrak{F}(v)$ for all $v\in SBV(\Omega)$.
\end{defi}
\begin{defi}[Dirichlet minimizers]
We say that $u\in SBV(\Omega)$ is a Dirichlet minimizer of $F$ (resp. $\mathfrak{F}$) in $\Omega'$ if
\begin{displaymath}
F(u) \leq F(v) \quad \ \forall v\in SBV(\Omega)\quad \mbox{s.t.}\quad  u=v \quad \mbox{in} \quad \Omega\setminus \Omega'
\end{displaymath}
\begin{displaymath}
(resp. \,\ \mathfrak{F}(u) \leq \mathfrak{F}(v) \quad \ \forall v\in SBV(\Omega)\quad \mbox{s.t.} \quad  u=v \quad \mbox{in} \quad \Omega\setminus \Omega').
\end{displaymath}
\end{defi}
Proving that a function $u \in SBV(\Omega)$ is a Dirichlet minimizer in $\Omega'$ is not an easy question (in general); this is one of the main reasons why a notion of calibration resembling the one of minimal surfaces by Harvey and Lawson (\cite{CGE}) has turned out to be very useful. It was proposed by Alberti, Bouchittè and Dal Maso in \cite{TCM} and developed among the others in \cite{LCF} and \cite{LCFM}. In the next section we will give a brief introduction on this topic.
\subsection{Calibration for the Mumford-Shah Functional}\label{sec}
Given $H:L^1(\Omega) \rightarrow \mathbb{R}$ let us define an abstract calibration in the following way:
\begin{defi}[Abstract calibration]\label{abstract}
Given $u\in L^1(\Omega)$, an abstract calibration for $u$ is a functional $G:L^1(\Omega) \rightarrow \mathbb{R}$ such that
\begin{equation}
(i) \quad H(u) = G(u), \quad \  (ii) \quad H(v) \geq G(v),\quad \  (iii) \quad G(u) = G(v)
\end{equation}
for all $v\in L^1(\Omega)$ such that $u=v$ in $\Omega\setminus \Omega'$.
\end{defi}
\begin{rmk}
If $G$ is a calibration for $u$, then $u$ is a Dirichlet minimizer in $\Omega'$ for $H$, indeed
\begin{displaymath}
H(u) \overset{(i)}{=} G(u) \overset{(iii)}{=} G(v) \overset{(ii)}{\leq} H(v)
\end{displaymath}
for all $v \in L^1(\Omega)$  such that $u=v$ in $\Omega\setminus \Omega'$.
\end{rmk}
In \cite{TCM} Alberti, Bouchitt\'e and Dal Maso introduced a stronger notion of calibration for the Mumford-Shah functional.
Given $v\in SBV(\Omega)$, we denote by $v^-(x)$ and $v^+(x)$ the lower and the upper traces of $v$. Moreover let $\Gamma_v$ be the extended graph of $v$ defined as
\begin{equation}\label{completegraph}
\Gamma_v = \{(x,t) \in \Omega \times \mathbb{R}: v^-(x) \leq t\leq v^+(x)\}.
\end{equation}
For standard theory on BV functions (\cite{CCC}) $\Gamma_v$ is rectifiable and then it admits a generalized normal that we are going to denote with $\nu_{\Gamma_v}$.\\
The calibration proposed in \cite{TCM} has the following form:
\begin{displaymath}
G(v) = \int_{\Gamma_v} \langle \phi , \nu_{\Gamma_v}\rangle \, d\mathcal{H}^n,
\end{displaymath}
where $\phi : \Omega \times \mathbb{R} \rightarrow \mathbb{R}^{n+1}$ is a vector field to be determined. The regularity asked on $\phi$ is the least that guarantees the validity of a divergence theorem on $\Omega \times \mathbb{R}$. To be more precise we refer to \cite{TCM} and for reader convenience we propose the definition of $\emph{approximately regular}$ vector field:
\begin{defi}[Approximately regular vector field]
Given $A\subset \mathbb{R}^{n+1}$, a vectorfield $\phi: A \rightarrow \mathbb{R}^{n+1}$ is approximately regular if it is bounded and for every Lipschitz hypersurface $M$ in $\mathbb{R}^{n+1}$ there holds
\begin{equation}
\lim_{r\rightarrow 0}\mint_{B_r(x_0)\cap A} |(\phi(x)-\phi(x_0))\cdot \nu_M(x_0)|\, dx = 0
\end{equation}
for $\mathcal{H}^{n}$-a.e. $x_0\in M\cap A$.
\end{defi}
Comparing the functional $G$ with $F$, it is possible to find sufficient conditions on $\phi$ such that $G$ satisfies properties (i), (ii) and (iii) with respect to $F$ for a given $u\in SBV(\Omega)$. Then the vector field satisfying these properties is called \emph{calibration} for $u$.
\begin{defi}[Calibration for the Mumford-Shah Functional, \cite{TCM}]
Let $\Omega \subset \mathbb{R}^n$ be open and bounded and $u\in SBV(\Omega)$. Given $\phi = (\phi^x,\phi^t) : \Omega \times \mathbb{R} \rightarrow \mathbb{R}^{n+1}$ an approximately regular vector field, we say that it is a calibration for $u$ if it is divergence free and
\begin{itemize}
\item[a)] $\phi^t(x,t) \geq \dfrac{|\phi^x(x,t)|^2}{4} \quad \mbox{for } \mathscr{L}^n$-a.e. $x\in \Omega$ and for all $t\in \mathbb{R}$,
\item[b)] $\left|\displaystyle\int_{t_1}^{t_2} \phi^x(x,t) \, dt\right| \leq 1 \quad \mbox{for } \mathcal{H}^{n-1}$-a.e. $x\in \Omega$ and for all $t_1,t_2\in \mathbb{R}$, \\
\item[c)]$\phi^x(x,u(x) ) = 2\nabla u(x), \qquad \phi^t(x,u(x)) = |\nabla u(x)|^2 \quad \mbox{for } \mathscr{L}^n$-a.e. $x\in \Omega$, \\
\item[d)] $\displaystyle\int_{u^-(x)}^{u^+(x)} \phi^x(x,t) \, dt = \nu_u(x) \quad \mbox{for } \mathcal{H}^{n-1}$-a.e. $x\in S_u$,
\end{itemize}
where $\nu_u$ is the approximate normal of $S_u$.
\end{defi}
As properties $(a),(b),(c),(d)$ imply $(i),(ii)$ and $(iii)$ for $G$ we have the following theorem:
\begin{thm}[\cite{TCM}]
Given $u \in SBV (\Omega)$, suppose that there exists $\phi: \Omega \times \mathbb{R} \rightarrow \mathbb{R}^{n+1}$ a calibration for $u$. Then $u$ is a Dirichlet minimizer in $\Omega'$ of the homogeneous Mumford-Shah functional \eqref{ho2}.
\end{thm}
In an analogous way a similar notion can be introduced in order to study minimizers of $\mathfrak{F}$. It is enough to replace conditions $(a)$ and $(c)$ with
\begin{itemize}
\item[a')]
$\phi^t(x,t) \geq \dfrac{|\phi^x(x,t)|^2}{4} - (t-g)^2 \quad \mbox{for } \mathscr{L}^n$-a.e. $x\in \Omega$ and for all $t\in \mathbb{R}$,\\
\item[c')] $\phi^x(x,u(x) ) = 2\nabla u(x) , \quad \phi^t(x,u(x)) = |\nabla u(x)|^2 - (u-g)^2\quad \mbox{for } \mathscr{L}^n$-a.e. $x\in \Omega$.
\end{itemize}
\begin{thm}[\cite{TCM}]
Given $u \in SBV (\Omega)$, suppose that there exists $\phi: \Omega \times \mathbb{R} \rightarrow \mathbb{R}^{n+1}$ a calibration for $u$ with $(a)$ and $(c)$ replaced with $(a')$ and $(c')$. Then $u$ is a Dirichlet minimizer in $\Omega'$ of the Mumford-Shah functional \eqref{mum2}.
\end{thm}
As a consequence, in \cite{TCM}, the authors proposed the following alternative formulation of the Mumford-Shah functional
\begin{equation}\label{formula1}
F(u)= \max_{\phi\in K} \int_{\Gamma_u} \langle \phi, \nu_{\Gamma_u}\rangle \, d \mathcal{H}^{n}= \max_{\phi\in K} \int_{\Omega\times \mathbb{R}} \langle \phi, D\textbf{1}_{\{u>t\}}\rangle ,
\end{equation}
\begin{equation}\label{formula2}
\mathfrak{F}(u)= \max_{\phi\in K'} \int_{\Gamma_u} \langle \phi , \nu_{\Gamma_u} \rangle \, d \mathcal{H}^{n}= \max_{\phi\in K'} \int_{\Omega\times \mathbb{R}} \langle \phi ,D\textbf{1}_{\{u>t\}}\rangle,
\end{equation}
where
\begin{equation}\label{kappa1}
K = \{\phi:\Omega\times\mathbb{R}\rightarrow \mathbb{R}^{n+1},\ Borel: (a) \mbox{ and }(b)\mbox{ hold pointwise} \}
\end{equation}
and
\begin{equation}\label{kappa2}
K' = \{\phi: \Omega\times\mathbb{R}\rightarrow \mathbb{R}^{n+1},\ Borel: (a') \mbox{ and }(b)\mbox{ hold pointwise}\}.
\end{equation}
\begin{rmk}
The previous representation formula is the starting point for the proof of existence of calibration in dimension one, due to Chambolle \cite{CRF}. In particular one can introduce the following convex functional also called $\emph{lift}$ of $F$
\begin{displaymath}
\mathcal{F}_K(w)= \sup_{\phi\in K \cap C_0(\Omega \times \mathbb{R}, \mathbb{R}^{n+1})} \int_{\Omega\times\mathbb{R}} \phi \cdot Dw,
\end{displaymath}
with $w:I\times \mathbb{R} \rightarrow [0,1]$ decreasing in the second variable and of bounded variation. In \cite{CRF} Chambolle proves that if $u\in SBV(I)$ is a minimizer of the Mumford-Shah functional then $\textbf{1}_{\{u(x)>t\}}$ is a minimizer of $\mathcal{F}_K$. Then by Hahn-Banach theorem it is possible to prove the existence of calibrations in a weak asymptotic sense.
\end{rmk}
\begin{rmk}\label{coareafalse}
It is interesting to notice that one can prove the same result in higher dimension if $\mathcal{F}_K$ satisfies a generalized coarea formula of the form
\begin{equation}\label{chacoa}
\mathcal{F}_K(w) = \int_0^1 \mathcal{F}_K(\textbf{1}_{\{w(x,t)>s\}})\, ds.
\end{equation}
Unfortunately this is false even in dimension one. Indeed it is enough to consider
\begin{displaymath}
u_1(x)=\left\{
\begin{array}{ll}
0 & \mbox{ if } x\leq 1/2\\
x & \mbox{ if } x> 1/2,
\end{array}
\right.
\qquad
u_2(x)=\left\{
\begin{array}{ll}
x & \mbox{ if } x\leq 1/2\\
1 & \mbox{ if } x> 1/2
\end{array}
\right.
\end{displaymath}
and
$w(x,t) = (1/2)\textbf{1}_{\{u_1(x)> t\}} + (1/2)\textbf{1}_{\{u_2(x)> t\}}$ to see that formula \eqref{chacoa} does not hold.
\end{rmk}

\subsection{A lifting of the Mumford-Shah functional in the space of rectifiable currents}
In this section we introduce a lifted functional that takes values in $\mathcal{R}_{n}(\Omega \times \mathbb{R})$ the $n$-dimensional rectifiable currents with real multiplicity. We briefly recall the basic theory of currents and we refer the reader to \cite{CCC} for a more detailed overview.\\
Let $U$ be an open subset of $\mathbb{R}^N$. A $k$-dimensional current on $U$ is a linear continuous (see \cite{CCC}) functional on the space of $k$-forms $\Lambda^k(U)$ with coefficients in $C_c^\infty(U)$. \\
In particular we define the space $\mathcal{R}_{k}(U)$ of $k$-dimensional rectifiable currents with real multiplicity as the triple $(\mathcal{M},\theta,\xi)$ where $\mathcal{M} \subset U$ is a $k$-rectifiable set, $\theta : \mathcal{M} \rightarrow \mathbb{R}_+$ is a function called multiplicity and $\xi$ is a map that associates to $\mathcal{H}^n$-a.e. $x$ in $\mathcal{M}$ a unit, simple $k$-vector orienting $\mathcal{M}$.
We define the current $(\mathcal{M},\theta,\xi)$ by its action on a $k$-diffential form $\omega \in \Lambda^k(U)$ in the following way:
\begin{displaymath}
(\mathcal{M},\theta,\xi)(\omega) = \int_{\mathcal{M}} \langle \omega, \xi\rangle \theta \, d\mathcal{H}^k,
\end{displaymath}
where $\langle\cdot , \cdot \rangle$ denote the duality product between vectors and covectors.
Moreover given $T=(\mathcal{M},\theta,\xi)$ we define the total variation measure associated to $T$ as
\begin{displaymath}
\|T\| (A) = \int_{\mathcal{M}\cap A} \theta \, d\mathcal{H}^k
\end{displaymath}
for every $A \subset U$ measurable. We call $\|T\|(U) = M(T)$ the mass of $T$.\\
We define the restriction of a rectifiable current $T=(\mathcal{M},\theta,\xi)$ on a measurable set as
\begin{displaymath}
T\res A(\omega) = \int_{\mathcal{M}\cap A} \langle \omega, \xi\rangle \theta \, d\mathcal{H}^k
\end{displaymath}
for every $A \subset U$ measurable. In addition given $\alpha\in \Lambda^h(U)$ with $h\leq k$, we define the restriction of $T \in \mathcal{R}_k(U)$ to $\alpha$ as the $(k-h)$-dimensional current $T\res \alpha$ defined as
\begin{displaymath}
T\res \alpha(\omega) = T(\alpha \wedge \omega)
\end{displaymath}
for every $\omega\in \Lambda^{k-h}(U)$.\\
Moreover, given $E$ a $k$-rectifiable set in $U$ we will denote by $[\![E]\!]$ the $k$-dimensional rectifiable current induced by $E$, that is defined as
\begin{displaymath}
[\![E]\!](\omega) := \int_{E} \langle\omega, \xi_{E}\rangle \, d\mathcal{H}^k,
\end{displaymath}
for $\omega \in \Lambda^k(U)$, where $\xi_{E}$ is the unit simple $k$-vector orienting $E$.
Therefore, as a consequence of \eqref{completegraph}, we can define $[\![\Gamma_u]\!]$, the $n$-rectifiable current associated to the complete graph of $u\in SBV(\mathbb{R}^n)$. From now on, with a little abuse of notation, we will denote it by $\Gamma_u$, instead of $[\![\Gamma_u]\!]$ (it will be clear by the context if we are dealing with the rectifiable set or with the current associated to it).
\vspace*{2mm}\\
We introduce the lifting of the Mumford-Shah functional on the space of rectifiable currents for the functionals $\mathfrak{F}$ and $F$.
\begin{defi}[Lifting to the space of rectifiable current]
Given $T=(\mathcal{M},\theta,\xi) \in \mathcal{R}_{n}(\Omega \times \mathbb{R})$ we define
\begin{equation}\label{func}
G_K(T) := \sup_{\phi\in K} \int_{\mathcal{M}} \langle \phi , \star (-\xi) \rangle d\|T\| =\sup_{\phi\in K} \int_{\mathcal{M}} \theta\langle \phi , \nu_T \rangle d\mathcal{H}^{n}
\end{equation}
and
\begin{equation}
G_{K'}(T) := \sup_{\phi\in K'} \int_{\mathcal{M}} \langle \phi , \star (-\xi) \rangle d\|T\| = \sup_{\phi\in K} \int_{\mathcal{M}} \theta\langle \phi , \nu_T \rangle d\mathcal{H}^{n}
\end{equation}
where $\nu_T := - (\star \xi)$, $\star$ is the Hodge star and $K$ and $K'$ are defined as in \eqref{kappa1} and in \eqref{kappa2}.
\end{defi}
\begin{prop}\label{convexlsc}
The functionals $G_K$ and $G_{K'}$ satisfy the following properties:
\begin{itemize}
\item [$(i)$] They are convex on $\mathcal{R}_{n}(\Omega \times \mathbb{R}).$
\item [$(ii)$] They are lower semicontinous with respect to the mass bounded convergence.
\item [$(iii)$] Given $v\in SBV(\Omega)$, $G_K(\Gamma_v) = F(v)$ and $G_{K'}(\Gamma_v) = \mathfrak{F}(v)$.
\end{itemize}
\end{prop}
\begin{dimo}
Statement $(i)$ follows from the definition and $(iii)$ is a consequence of the representation formulas \eqref{formula1} and \eqref{formula2}.
Moreover $(ii)$ can be proved with an easy modification of the argument in \cite{CCC} sec. $3.3.1$.
\end{dimo}

\section{A convex decomposition formula for the Mumford-Shah functional in dimension one}\label{mais}
We restrict our analysis to the case $n=1$. We also assume $\Omega = I$ and $\Omega' = I'$ to be open and bounded intervals such that $I' \subset \subset I$ and we consider the Mumford-Shah functional in its general form
\begin{equation}\label{generic}
F(u) := \int_I |u'(x)|^2\, dx + \beta\int_I|u-g|^2\, dx + \alpha\mathcal{H}^0(S_u),
\end{equation}
where $\alpha > 0$, $\beta \geq 0$, $g\in L^\infty(I)$ and $u\in SBV(I)$. Notice that when $\beta = 0$ and $\alpha = 1$, $F$ is the homogeneous version of the Mumford-Shah functional as defined in \eqref{ho2}. From now on we will denote by $u^l(x)$ (resp. $u^r(x)$) the left (resp. right) trace of $u$ in a point $x$.

\begin{rmk}\label{gener}
Even if we restrict our attention to \eqref{generic} it is important to remark that the results of this section and of the following one hold for a more general class of functionals with minor modification of the proofs. Functionals of the form
\begin{displaymath}
W(u) = \int_I f(u'(x),u(x),x)\, dx + \sum_{x\in S_u} \psi(x,u^l(x),u^r(x))
\end{displaymath}
with suitable hypothesis on $f$ and $\psi$ necessary to ensure the lower semicontinuity of $W$ and the existence of minimizers can be treated by this theory. We refer to \cite{ETF} for the precise assumptions and we stress the fact that in our setting $f$ need not to be assumed more regular as in \cite{CRF}. For example in the case of the Mumford-Shah functional $g$ can be taken in $L^\infty$ without affecting the proof, while in \cite{CRF} the function $g$ needs to have a l.s.c. and a u.s.c. representatives in $L^\infty$.
\end{rmk}
If we consider the functional $F$ as defined in \eqref{generic}, its convex lift defined in \eqref{func} on $\mathcal{R}_1(I\times\mathbb{R})$ reads
\begin{equation}\label{operator}
G(T) =  \sup_{\phi\in K} \int_{\mathcal{M}} \theta\langle \phi , \nu_T \rangle d\mathcal{H}^1
\end{equation}
for every $T=(\mathcal{M},\theta,\xi)$. \\
In particular $K$ is the set of $\phi :I\times \mathbb{R}\rightarrow \mathbb{R}^2$, Borel, such that
\begin{itemize}
\item[I)] $\phi^t(x,t) \geq \dfrac{|\phi^x(x,t)|^2}{4} - \beta(t-g)^2 \quad \mbox{for all }x\in I$ and for all $t\in \mathbb{R}$,\\
\item[II)] $\left|\displaystyle\int_{t_1}^{t_2} \phi^x(x,t) \, dt\right| \leq \alpha \quad \mbox{for all } x\in I$ and for all $t_1,t_2\in \mathbb{R}$.
\end{itemize}
We are going to consider as the domain of $G$ the cone $C \subset \mathcal{R}_1(I \times \mathbb{R})$ made by finite linear combinations of SBV graphs:
\begin{equation}\label{cone}
C:= \left\{T= \sum_{i=1}^{k} \lambda_i \Gamma_{u_i} : k\in \mathbb{N}, \lambda_i \in \mathbb{R}_+, u_i \in SBV(I)\right\}.
\end{equation}
For every $T\in C$ we will assume implicitly that, being a rectifiable current, it is defined by a triple $T=(\mathcal{M},\theta,\xi)$.

\subsection{Simplifying the cone $C$}
From the definition of the cone $C$ in \eqref{cone} one easily notices that for every current $T\in C$ there exists different combinations of SBV graphs $\{u_i\}$ that represent it. In particular there are some configurations we would like to avoid and this subsection is devoted to make this simplifications for $C$.
\begin{defi}
Given $\{u_i\}_{i=1\ldots k} \subset SBV(I)$. We say that the family $\{u_i\}_{i=1\ldots k}$ has \emph{cancellation on the jumps} if there exists $l_1,l_2$ and $x_0 \in S_{u_{l_1}} \cap S_{u_{l_2}}$ such that
\begin{displaymath}
u_{l_1}^l(x_0) \leq  u_{l_2}^r(x_0) < u_{l_1}^r(x_0) \leq u_{l_2}^l (x_0) \quad \mbox{or}
\end{displaymath}
\begin{displaymath}
u_{l_1}^r(x_0) \leq  u_{l_2}^l(x_0) < u_{l_1}^l(x_0) \leq u_{l_2}^r (x_0).
\end{displaymath}
\end{defi}
We need a lemma that ensures that we can rearrange the graphs in order not to have this cancellation.
\begin{lemma}\label{nocanc}
Given $T =\sum_{i=1}^k \lambda_i \Gamma_{u_i} \in C$ there exists $l \in \mathbb{N}$, $w_i \in SBV(I)$ and $\mu_i \in \mathbb{R}^+$ for $i=1\ldots l$ such that $T = \sum_{i=1}^l \mu_i \Gamma_{w_i}$ and
there is no cancellation on the jumps.
\end{lemma}
\begin{dimo}
Given $T= \sum_{i=1}^{k} \lambda_i \Gamma_{u_i}$ let us suppose that we have cancellation between $\Gamma_{u_1}$ and $\Gamma_{u_2}$ in $A\subset S_{u_1} \cap S_{u_2}$ and $\lambda_1 \geq \lambda_2$ (without loss of generality).
As $A$ is countable we will denote it by the sequence $\{x_1, x_2, \ldots \}$ possibly infinite.
Given $I=(a,b)$ consider the new sequence $\{a=x_0, x_1, x_2, \ldots \}$ and define two SBV functions in the following way:
\begin{displaymath}
w_1(x) = \left\{
\begin{array}{ll}
u_1(x) & \mbox{ for } x_{i-1}<x\leq x_{i},\ i\geq 1 \mbox{ and odd}\\
u_2(x) & \mbox{ for } x_{i-1}<x\leq x_{i},\ i\geq 1 \mbox{ and even}
\end{array}
\right.
\end{displaymath}
and
\begin{displaymath}
w_2(x) = \left\{
\begin{array}{ll}
u_2(x) & \mbox{ for } x_{i-1}<x\leq x_{i},\ i\geq 1 \mbox{ and odd }\\
u_1(x) & \mbox{ for } x_{i-1}<x\leq x_{i},\ i\geq 1 \mbox{ and even}.
\end{array}
\right.
\end{displaymath}
Then we have that $\lambda_2 \Gamma_{w_1} + \lambda_2\Gamma_{w_2} + (\lambda_1 - \lambda_2) \Gamma_{u_1} =  \lambda_1\Gamma_{u_1}+ \lambda_2\Gamma_{u_2}$. Hence we produce a decomposition of $\lambda_1\Gamma_{u_1}+ \lambda_2\Gamma_{u_2}$ that has no cancellation on the jumps. It is easy to check that one can repeat this operation for any pair of graphs that has cancellation on jumps and that this procedure ends in a finite number of steps.
\end{dimo}

\subsection{Properties of the regular part of $G(T)$}
\begin{defi}[Regular part and singular part of $T$]
We define the singular part of $T = \sum_{i=1}^k\lambda_i \Gamma_{u_i} \in C$ as
\begin{equation}
S_T := \bigcup_{i=1}^k S_{u_i}
\end{equation}
and the regular part as $R_T := I \setminus S_T$.
\end{defi}
\begin{rmk}
One can easily notice that if we assume that the graphs do not have cancellation according to Lemma \ref{nocanc}, $S_T$ is well defined, so it does not depend on the representation of $T$.
\end{rmk}
Given a measurable set $A\subset I$ we define the localized version of $G$ as
\begin{displaymath}
G(T,A) := \sup_{\phi\in K} \int_{\mathcal{M}\cap (A\times \mathbb{R})} \langle \phi , \nu_T \rangle d\|T\|.
\end{displaymath}
\begin{rmk}
It is clear that given $A_1$, $A_2$ disjoint measurable sets we have
\begin{displaymath}
G(T,A_1 \cup A_2) = G(T,A_1) + G(T,A_2)
\end{displaymath}
so in particular
\begin{equation}
G(T) = G(T,S_T) + G(T,R_T).
\end{equation}
Moreover when one computes the localized functional, it is possible to restrict the set $K$ accordingly:
\begin{displaymath}
G(T,A) = \sup_{\phi\in K_A} \int_{\mathcal{M}\cap (A\times \mathbb{R})} \langle \phi , \nu_T \rangle d\|T\|,
\end{displaymath}
where $K_A$ is the set of $\phi :I\times \mathbb{R}\rightarrow \mathbb{R}$, Borel, such that
\begin{itemize}
\item
$\phi^t(x,t) \geq \dfrac{|\phi^x(x,t)|^2}{4} - \beta(t-g)^2 \quad \forall x\in A$ and $\forall t\in \mathbb{R}$,\\
\item $\left|\displaystyle\int_{t_1}^{t_2} \phi^x(x,t) \, dt\right| \leq \alpha \quad \mbox{for every } x\in A$ and for all $t_1,t_2\in \mathbb{R}$.
\end{itemize}
\end{rmk}
We are presenting a proposition that allows us to split $G(T,R_T)$ as the sum of $\lambda_i G(\Gamma_{u_i},R_T)$.
\begin{prop}\label{decomposition}
Given $T = \sum_{i=1}^k \lambda_i \Gamma_{u_i} \in C$, then
\begin{equation}\label{regular}
G(T, R_T) = \sum_{i=1}^k \lambda_i G(\Gamma_{u_i}, R_T)=\sum_{i=1}^k \lambda_i\left(\alpha\int_I (u_i')^2 \, dx + \beta\int_I |u_i - g|^2 \,dx\right).
\end{equation}
\end{prop}
In order to give a proof of this fact we need some preliminary lemmas.
\begin{lemma}\label{first}
Given $T = \sum_{i=1}^k \lambda_i \Gamma_{u_i} \in C$ let $A \subset I$ be a measurable set such that $A \cap S_T = \emptyset$ and $\mathcal{H}^{1}(\Gamma_{u_i} \cap \Gamma_{u_j} \cap (A\times \mathbb{R})) = 0$ for every $i\neq j$. Then
\begin{displaymath}
G(T,A) = \sum_i \lambda_i G(\Gamma_{u_i},A).
\end{displaymath}
\end{lemma}
\begin{dimo}
By induction it is enough to show that given, $T_1 = \sum_{i=1}^{k-1}\lambda_i \Gamma_{u_i}$ and $T_2= \lambda_k \Gamma_{u_k}$ one has
\begin{displaymath}
G(T_1 + T_2 ,A) = G(T_1,A) + G(T_2,A).
\end{displaymath}
Fix $\varepsilon>0$. For $i=1,2$ there exist $\phi_i \in K_A$ such that
\begin{displaymath}
\int_{\mathcal{M}_i \cap (A\times \mathbb{R})} \langle \phi_i , \nu_{T_i} \rangle \, d\|T_i\| \geq  G(T_i, A) - \varepsilon,
\end{displaymath}
where  $T_i = (\mathcal{M}_i, \theta_{T_i} ,\xi_{T_i})$ and $\nu_{T_i} = -(\star \xi_{T_i})$.
Define then the following vector field
\begin{displaymath}
\tilde \phi = \left\{\begin{array}{ll}
\phi_1 & (x,t) \in \mathcal{M}_1 \setminus \mathcal{M}_2\\
\phi_2 & (x,t) \in \mathcal{M}_2\setminus \mathcal{M}_1 \\
 0 & \mbox{ otherwise.}
\end{array}
\right.
\end{displaymath}
Let us prove that $\tilde \phi \in K_A$. \\
For every $x \in A$ we have that $x\notin S_{T}$ by hypothesis, so that (II) is satisfied and (I) is trivial by definition.
Moreover, as $\mathcal{H}^1(\mathcal{M}_1\cap \mathcal{M}_2 \cap (A\times \mathbb{R})) = 0$, one has
\begin{eqnarray*}
\int_{(\mathcal{M}_1 \cup \mathcal{M}_2)\cap (A\times \mathbb{R})}  \langle \tilde \phi ,\nu_T \rangle \, d\mathcal{H}^1 = \int_{\mathcal{M}_1 \cap (A\times \mathbb{R})} \langle \phi_1 , \nu_{T_1} \rangle  \, d\mathcal{H}^1 + \int_{\mathcal{M}_2\cap (A\times \mathbb{R})} \langle \phi_2 , \nu_{T_2} \rangle  \, d\mathcal{H}^1.
\end{eqnarray*}
So
\begin{eqnarray*}
G(T_1,A) + G(T_2,A) &\leq &\int_{\mathcal{M}_1 \cap (A\times \mathbb{R})} \langle \phi_1 , \nu_{T_1} \rangle \theta_{T_1}  \, d\mathcal{H}^1 + \int_{\mathcal{M}_2\cap (A\times \mathbb{R})} \langle \phi_2 , \nu_{T_2} \rangle \theta_{T_2} \, d\mathcal{H}^1 + 2\varepsilon \\
&\leq & G(T_1 + T_2 ,A) + 2\varepsilon.
\end{eqnarray*}
Sending $\varepsilon$ to zero we obtain the first inequality.
The opposite one comes directly from the convexity of $G$.
\end{dimo}

\begin{figure}
\centering
\subfigure{
\begin{tikzpicture}[>=stealth, scale=0.9]
\draw [->](-1,0) -- (6,0) node [scale=0.7][below]{x};
\draw [->](0,-1) -- (0,6) node [scale=0.7][left]{t};
\draw[densely dashed] (5,0) -- (5,6);
\draw  (0,0) [thick][red]
to[out=10,in=180, looseness=1] (2,2)
to[out=90,in=-90, looseness=1] (2,2.5)
to[out=0,in=160, looseness=0.8] (5,5) node[black, scale=0.7] [below=18pt, left=4pt] {$\Gamma_{u_1}$} ;

\draw  (0,0)[thick][blue]
to[out=40,in=180, looseness=1] (1.5,3.5)
to[out=90,in=-90, looseness=1] (1.5,4)
to[out=0,in=160, looseness=0.8] (5,5)node[black, scale=0.7] [above=3pt,left=35pt]{$\Gamma_{u_2}$};

\draw  (2,0) node[black, scale=0.7] [below=0.5pt]{$S_{u_1}$} [dashed] -- (2,2) ;
\draw  (1.5,0) node[black, scale=0.7] [below=0.5pt]{$S_{u_2}$} [dashed] -- (1.5,3.5) ;

\draw  (0,2) node[black, scale=0.7] [left=0.5pt]{$u_1^l$} [dashed] -- (2,2) ;
\draw  (0,2.5) node[black, scale=0.7] [left=0.5pt]{$u_1^r$} [dashed] -- (2,2.5) ;
\draw  (0,3.5) node[black, scale=0.7] [left=0.5pt]{$u_2^l$} [dashed] -- (1.5,3.5) ;

\draw  (0,4) node[black, scale=0.7] [left=0.5pt]{$u_2^r$} [dashed] -- (1.5,4) ;

\end{tikzpicture}
}\qquad\subfigure{\begin{tikzpicture}[>=stealth, scale=0.9]
\draw [->](-1,0) -- (6,0) node [scale=0.7][below]{x};
\draw [->](0,-1) -- (0,6) node [scale=0.7][left]{t};
\draw[densely dashed] (5,0) -- (5,6);
\draw  (1,1) [thick][red]
to[out=10,in=180, looseness=1] (2,2)
to[out=90,in=-90, looseness=1] (2,2.5)
to[out=0,in=160, looseness=0.8] (5,5) node[black, scale=0.7] [below=14pt, left=4pt] {$\Gamma_{u_1}$} ;

\draw  (1,1)[thick][blue]
to[out=65,in=250, looseness=1] (1.5,3.5)
to[out=90,in=-90, looseness=1] (1.5,4)
to[out=0,in=160, looseness=0.8] (5,5)node[black, scale=0.7] [above=6pt,left=35pt]{$\Gamma_{u_2}$};

\draw  (0,0)[thick][green]
to[out=40,in=180, looseness=1] (1,1)node[black, scale=0.7] [left=7pt,below=5pt]{$\Gamma_{u_1} = \Gamma_{u_2}$};

\draw  (2,0) node[black, scale=0.7] [below=0.5pt]{$S_{u_1}$} [dashed] -- (2,2) ;
\draw  (1.5,0) node[black, scale=0.7] [below=0.5pt]{$S_{u_2}$} [dashed] -- (1.5,3.5) ;

\draw  (0,2) node[black, scale=0.7] [left=0.5pt]{$u_1^l$} [dashed] -- (2,2) ;
\draw  (0,2.5) node[black, scale=0.7] [left=0.5pt]{$u_1^r$} [dashed] -- (2,2.5) ;
\draw  (0,3.5) node[black, scale=0.7] [left=0.5pt]{$u_2^l$} [dashed] -- (1.5,3.5) ;

\draw  (0,4) node[black, scale=0.7] [left=0.5pt]{$u_2^r$} [dashed] -- (1.5,4) ;

\end{tikzpicture}
}
\caption{Configuration in Lemma \ref{first} and \ref{overlapping}}
\label{firstover}
\end{figure}

\begin{lemma}\label{overlapping}
Given $T = \sum_{i=1}^k \lambda_i \Gamma_{u_i} \in C$ let $A\subset I$ be a measurable set such that $A \cap S_T = \emptyset$. Then
\begin{displaymath}
G(T,A) = \sum_i \lambda_i G(\Gamma_{u_i},A).
\end{displaymath}
\end{lemma}
\begin{dimo}
Given $T\in C$, let $J$ be a set of indexes. Denote by $\Gamma = \bigcap_{i\in J} \Gamma_{u_i}$ an intersection of graphs and let $\theta = \sum_{i\in J} \lambda_i$ be the multiplicity on $\Gamma$. So
\begin{eqnarray*}
 \sup_{\phi\in K} \int_{\Gamma \cap (A\times \mathbb{R})} \langle \phi,\nu_T\rangle \, d\|T\|&=& \sup_{\phi\in K} \int_{\Gamma\cap (A\times \mathbb{R})}  \theta\langle \phi,\nu_T\rangle \, d\mathcal{H}^1 = \sup_{\phi\in K} \int_{\Gamma \cap (A\times \mathbb{R})} \sum_{i\in J} \lambda_i\langle \phi,\nu_T\rangle \, d\mathcal{H}^1 \\
&=& \sum_{i\in J} \lambda_i\sup_{\phi\in K} \int_{\Gamma \cap (A\times \mathbb{R})}  \langle \phi,\nu_T\rangle \, d\mathcal{H}^1.
\end{eqnarray*}
Clearly this can be repeated for every intersection of an arbitrary number of graphs. Combining this result with Lemma \ref{first} we have the thesis.
\end{dimo}
\emph{Proof of Proposition \ref{decomposition}}
\vspace{2mm}\\
Proposition \ref{decomposition} is a direct consequence of Lemma \ref{overlapping} choosing $A = R_T$ and the second equality in \eqref{regular} follows from Proposition \ref{convexlsc}.
\begin{flushright}
$\Box$
\end{flushright}
\subsection{Properties of the singular part of $G(T)$}
In this section we are going to study the properties of $\mathcal{G}(T):= G(T,S_T)$. \\
Given $T= \sum_{i=1}^k \lambda_i \Gamma_{u_i}\in C$ and calling $\nu_T= ((\nu_T)^x, (\nu_T)^t)$, by \eqref{operator} we have
\begin{displaymath}
\mathcal{G}(T) = \sup_{\phi \in K}\int_{\mathcal{M} \cap (S_T \times \mathbb{R})} \theta \phi^x (\nu_{T})^x \, d\mathcal{H}^1
\end{displaymath}
and it is easy to see that
\begin{displaymath}
(\nu_{T})^x(x,t) = \left\{\begin{array}{ll}
+1 & (x,t) \in S_{u_i} \times (u_i^l, u_i^r)\\
-1 & (x,t) \in S_{u_i} \times (u_i^r, u_i^l).
\end{array}
\right.
\end{displaymath}
Hence
\begin{displaymath}
\mathcal{G}(T) = \sup_{\phi \in K} \sum_{i=1}^k\int_{S_{u_i} \times (u_i^l, u_i^r)} \theta\phi^x\, d\mathcal{H}^1.
\end{displaymath}
From now on we will work with linear combinations of graphs with the same multiplicity. We will see later the reason why we can reduce to this situation. We want to prove that, given $T= \sum_i \Gamma_{u_i}$, $\mathcal{G}(T)$ can be written as the sum of $\mathcal{G}(\Gamma_{u_i})$ in all the configurations in which there is non-adjacency of the jumps of the graphs.
\begin{thm}\label{split}
Consider $T \in C$ such that $T = \sum_{i=1}^k \Gamma_{u_i}$. Suppose that for every $i,j = 1\ldots k$
\begin{displaymath}
\{x\in S_{u_i} \cap S_{u_{j}} : u^r_{i} (x) = u^l_{j}(x)\} = \emptyset.
\end{displaymath}
Then
\begin{displaymath}
\mathcal{G}\left(\sum_{i=1}^k \Gamma_{u_i}\right) = \sum_{i=1}^k \mathcal{G}\left(\Gamma_{u_i}\right).
\end{displaymath}
\end{thm}
\begin{rmk}
Notice that without loss of generality we can prove the previous statement restricting the functional $\mathcal{G}$ to every $x\in S_T$. So the lemmas needed to prove Theorem \ref{split} will be stated for a fixed point $x\in S_T$.
\end{rmk}
For sake of clarity we propose two lemmas (Lemma \ref{fillingone} and \ref{fillingtwo}) that deals with a simple situation that is enough to explain the general strategy (See Figure \ref{splitfig}). Then, in Proposition \ref{summ1} and \ref{summ2}, we generalize this procedure and finally we prove the theorem.
\begin{lemma}\label{fillingone}
Consider $T= \sum_{i=1}^k \Gamma_{u_i} \in C$ such that $u_i$ are ordered in an increasing way. Fix $x\in S_T$ and suppose that we have $u_i^l(x) \leq u_i^r(x)$ for every $i=1 \ldots k$. Suppose in addition that
\begin{displaymath}
u_i^r(x) < u_j^l(x)\quad \mbox{ for every }  \quad i<j.
\end{displaymath}
Then
\begin{displaymath}
\mathcal{G}(T,\{x\}) = \sum_{i=1}^k\mathcal{G}(\Gamma_{u_i},\{x\})= \alpha |\{i: x\in S_{u_i}\}|.
\end{displaymath}
In addition the maximum is achieved and letting $\phi_{T}$ be the vector field realizing the maximum for $T$
\begin{displaymath}
\phi^x_T(x,t) = \alpha/(u_i^r - u_i^l) \quad \mbox{ for every } \quad t\in (u_i^l,u_i^r)
\end{displaymath}
for every $i=1\ldots k$ such that $x\in S_{u_i}$.
\end{lemma}
\begin{dimo}
First of all notice that it is not restrictive to assume that $x	\in S_{u_i}$ for every $i=1,\ldots,k$. By induction it is enough to prove that for $T=T_1 + T_2$ where $T_1 = \sum_{i=1}^{k-1} \Gamma_{u_i}$ and $T_2 = \Gamma_{u_k}$ one has
\begin{displaymath}
\mathcal{G}(T_1 + T_2,\{x\}) = \mathcal{G}(T_1,\{x\}) + \mathcal{G}(T_2,\{x\})
\end{displaymath}
and
\begin{displaymath}
\phi^x_{T}(x,t) = \alpha/(u_k^r - u_k^l)  \quad \mbox{ for every }  \quad t\in (u_k^l,u_k^r).
\end{displaymath}
(We suppose $x\in S_{u_k}$ because if not, there is nothing to prove).\\
For the inductive hypothesis we have that for all $i=1\ldots k-1$
\begin{displaymath}
\phi^x_{T_1}(x,t) = \alpha/(u_i^r - u_i^l) \quad \mbox{for every }  \quad t\in (u_i^l,u_i^r).
\end{displaymath}
For the general theory of calibration we have that, calling $\phi_{T_2}$ the vector field realizing the maximum in $\mathcal{G}(T_2,\{x\})$,
\begin{displaymath}
\phi^x_{T_2}(x,t) = \alpha/(u_k^r - u_k^l)  \quad \mbox{for every }  \quad t\in (u_k^l,u_k^r),
\end{displaymath}
because
\begin{displaymath}
\int_{u_k^-}^{u_k^+} \phi^x_{T_2}(x) = \alpha \quad \mbox{for every } x \in S_{u_k}.
\end{displaymath}
Define the following vector field on $\{x\} \times \mathbb{R}$:
\begin{displaymath}
\tilde \phi = \left\{\begin{array}{ll}
\phi_{T_1} & (x,t) \in \{x\} \times (u_1^l, u_{k-1}^r),\\
\phi_{T_2} & (x,t) \in \{x\} \times (u_k^l, u_{k}^r),\\
\{-\alpha/(u_k^l - u_{k-1}^r),\frac{(\tilde\phi^x)^2}{4} - \beta(t-g)^2\}  & (x,t) \in \{x\} \times (u_{k-1}^r, u_{k}^l),\\
0 & \mbox{ otherwise}.
\end{array}
\right.
\end{displaymath}
Let us prove that $\tilde \phi \in K_{\{x\}}$. \\
\begin{eqnarray*}
\left|\int_{t_1}^{t_2} \tilde \phi(x,t) \, dt\right| &=&  \left|\int_{t_1}^{u^l_{k-1}}\phi^x_{T_1}(x,t)\, dt - \alpha + \int_{u^r_k}^{t_2}\phi^x_{T_2}(x,t)\, dt\right| \\
&=& \left|\alpha\frac{(u_{k-1}^l - t_1)}{(u_{k-1}^l - u_1^l)}  - \alpha  + \alpha\frac{(t_2 - u_{k}^r )}{(u_{k}^r - u_k^l)}\right| \leq \alpha
\end{eqnarray*}
for every $t_1 \leq u_1^l$, $t_2 \geq u_k^r$. As in all the other cases the computation is similar, then $\tilde \phi \in K_{\{x\}}$. Therefore
\begin{displaymath}
\mathcal{G}(T_1,\{x\}) + \mathcal{G}(T_2,\{x\}) = \int_{\mathcal{M} \cap (\{x\} \times \mathbb{R})} \langle \tilde \phi, \nu_T\rangle \theta \, d\mathcal{H}^1 \leq \mathcal{G}(T,\{x\}).
\end{displaymath}
On the other hand by convexity
\begin{displaymath}
\mathcal{G}(T,\{x\}) \leq \mathcal{G}(T_1,\{x\}) + \mathcal{G}(T_2,\{x\}) = \int_{\mathcal{M} \cap (\{x\} \times \mathbb{R})} \langle \tilde \phi, \nu_T\rangle \theta \, d\mathcal{H}^1.
\end{displaymath}
So the thesis follows.
\end{dimo}
We can prove the analogue:

\begin{figure}
\centering
\subfigure{
\begin{tikzpicture}[>=stealth, scale=0.9]
\draw [->](-1,0) -- (6,0) node [scale=0.7][below]{x};
\draw [->](0,-1) -- (0,6) node [scale=0.7][left]{t};
\draw[densely dashed] (5,0) -- (5,6);
\draw  (0,0) [thick][red]
to[out=10,in=180, looseness=1] (2,2)
to[out=90,in=90, looseness=1] (2,2.5)
to[out=0,in=160, looseness=0.8] (5,5) node[black, scale=0.7] [below=18pt, left=4pt] {$\Gamma_{u_1}$} ;

\draw  (0,0)[thick][blue]
to[out=40,in=180, looseness=1] (2,3.5)
to[out=90,in=90, looseness=1] (2,4)
to[out=0,in=160, looseness=0.8] (5,5)node[black, scale=0.7] [left=35pt]{$\Gamma_{u_2}$};

\draw  (2,0) node[black, scale=0.7] [below=0.5pt]{$S_{u_1}\cap S_{u_2}$} [dashed] -- (2,2) ;
\draw  (0,2) node[black, scale=0.7] [left=0.5pt]{$u_1^l$} [dashed] -- (2,2) ;
\draw  (0,2.5) node[black, scale=0.7] [left=0.5pt]{$u_1^r$} [dashed] -- (2,2.5) ;
\draw  (0,3.5) node[black, scale=0.7] [left=0.5pt]{$u_2^l$} [dashed] -- (2,3.5) ;

\draw  (0,4) node[black, scale=0.7] [left=0.5pt]{$u_2^r$} [dashed] -- (2,4) ;

\end{tikzpicture}
}\qquad
\subfigure{
\begin{tikzpicture}[>=stealth, scale=0.9]
\draw [->](-1,0) -- (6,0) node [scale=0.7][below]{x};
\draw [->](0,-1) -- (0,6) node [scale=0.7][left]{t};
\draw[densely dashed] (5,0) -- (5,6);
\draw  (0,5) [thick][red]
to[out=10,in=180, looseness=1] (2,2.5)
to[out=90,in=90, looseness=1]  (2,2)
to[out=0,in=160, looseness=0.8] (5,0) node[black, scale=0.7] [above=42pt, left=7pt] {$\Gamma_{u_2}$} ;

\draw  (0,5)[thick][blue]
to[out=40,in=180, looseness=1]  (2,4)
to[out=90,in=90, looseness=1] (2,3.5)
to[out=0,in=160, looseness=0.8] (5,0)node[black, scale=0.7] [left=35pt, above=8pt]{$\Gamma_{u_1}$};

\draw  (2,0) node[black, scale=0.7] [below=0.5pt]{$S_{u_1}\cap S_{u_2}$} [dashed] -- (2,2) ;
\draw  (0,2) node[black, scale=0.7] [left=0.5pt]{$u_1^r$} [dashed] -- (2,2) ;
\draw  (0,2.5) node[black, scale=0.7] [left=0.5pt]{$u_1^l$} [dashed] -- (2,2.5) ;
\draw  (0,3.5) node[black, scale=0.7] [left=0.5pt]{$u_2^r$} [dashed] -- (2,3.5) ;

\draw  (0,4) node[black, scale=0.7] [left=0.5pt]{$u_2^l$} [dashed] -- (2,4) ;

\end{tikzpicture}
}
\caption{Configuration in Lemma \ref{fillingone} and in Lemma \ref{fillingtwo}}
\label{splitfig}
\end{figure}

\begin{lemma}\label{fillingtwo}
Given $T= \sum_{i=1}^k \Gamma_{u_i}\in C$ such that $u_i$ are ordered in an increasing way. Fix $x\in S_T$ and suppose that we have $u_i^r(x) \leq u_i^l(x)$ for every $i=1 \ldots k$. Suppose in addition that
\begin{displaymath}
u_i^r(x) > u_j^l(x)\quad \mbox{ for every }  \quad i>j.
\end{displaymath}
Then
\begin{displaymath}
\mathcal{G}(T,\{x\}) = \sum_{i=1}^k\mathcal{G}(\Gamma_{u_i},\{x\})= \alpha |\{i: x\in S_{u_i}\}|.
\end{displaymath}
In addition the maximum is achieved and letting $\phi_{T}$ be the vector field realizing the maximum for $T$
\begin{displaymath}
\phi^x_T(x,t) = \alpha/(u_i^r - u_i^l) \quad \mbox{ for every } \quad t\in (u_i^l,u_i^r)
\end{displaymath}
for every $i=1\ldots k$ such that $x\in S_{u_i}$.
\end{lemma}
\begin{dimo}
See Lemma \ref{fillingone}.
\end{dimo}
We are now in position to prove two general statements that are generalizations of Lemmas \ref{fillingone} and \ref{fillingtwo}.
\begin{prop}\label{summ1}
Consider $T \in C$ such that $T = \sum_{i=1}^k \Gamma_{u_i}$. Fix $x\in S_T$ and suppose that we have $u_i^l(x) \leq u_i^r(x)$ for every $i=1 \ldots k$. Moreover assume that $u^r_{i} (x)  \neq  u^l_{j}(x)$ for every $i,j$ such that $x\in S_{u_i}\cap S_{u_j}$.\\
Then
\begin{displaymath}
\mathcal{G}(T, \{x\}) = \sum_{i=1}^k \mathcal{G}(\Gamma_{u_i}, \{x\}).
\end{displaymath}
\end{prop}
\begin{dimo}
We can assume without loss of generality that $x\in S_{u_i}$ for every $i=1 \ldots k$.\\
Then it is easy to see that
\begin{equation}\label{ah}
T\res(\{x\} \times \mathbb{R}) = \sum_{i=1}^{k'} \mu_i[\![\{x\} \times(a_i,a_{i+1})]\!]
\end{equation}
for some $\lambda_i \in \mathbb{N}$ and $a_i \in \mathbb{R}$ with $a_i < a_{i+1}$ and $\lambda_i \neq \lambda_{i+1}$ for every $i$. Let us denote by $\{\lambda_{M_j}\}$ the local maxima of the sequence $\{\lambda_i\}$ and let $\lambda_{m_j}$ be the minimum multiplicity in $\{\lambda_{M_j}, \lambda_{M_j + 1}, \ldots , \lambda_{M_{j+1} - 1}, \lambda_{M_{j+1}}\}$ for every $j$. \\
Thanks to the assumptions, we have that
\begin{equation}\label{maxmin}
k = \sum_j \lambda_{M_j} - \sum_j \lambda_{m_j}.
\end{equation}
Equation \eqref{maxmin} can be proved by induction. Consider $T=\sum_{i=1}^k \Gamma_{u_i}$ associated to a sequence of natural numbers $\{\lambda_i\}_{i=1,\ldots,k'}$ and intervals $(a_i,a_{i+1})$ according to \eqref{ah} with $\lambda_i \neq \lambda_{i+1}$ and $a_i < a_{i+1}$ for every $i$. Let $\Gamma_{w}$ be the graph composing $T$ such that
\begin{displaymath}
\Gamma_w \res (\{x\} \times \mathbb{R}) = [\![\{x\} \times (b,c)]\!]
\end{displaymath}
with $b \neq c$ and $c = a_{k'}$ and call $\hat T = T - \Gamma_w \in C$. We want to show that adding $\Gamma_w$ to $\hat T$ we are increasing the quantity $\sum_j \lambda_{M_j} - \sum_j \lambda_{m_j}$ by one. This fact can be verified considering separately the cases in which $ b\in (a_h,a_{h+1})$ where $\lambda_h$ is a local maximum, a local minimum and none of the two for the sequence $\{\lambda_i\}_{i=1,\ldots,k'}$.

Then the proof proceeds similarly to the proof of Lemma \ref{fillingone}.
One can define the following vector field on $\{x\} \times \mathbb{R}$:
\begin{displaymath}
\tilde \phi = \left\{\begin{array}{ll}
\Big(\alpha/(a_{M_{j} + 1}-a_{M_j}),\frac{(\tilde\phi^x)^2}{4} - \beta(t-g)^2\Big) & (x,t) \in \{x\} \times \bigcup_j (a_{M_j}, a_{M_j + 1}),\\
\Big(-\alpha/(a_{m_{j} + 1}-a_{m_{j}}), \frac{(\tilde\phi^x)^2}{4} - \beta(t-g)^2\Big) & (x,t) \in \{x\}\times \bigcup_j  (a_{m_{j}}, a_{m_{j}+1}), \\
0 & \mbox{ otherwise,}
\end{array}
\right.
\end{displaymath}
proving that $\tilde \phi \in K_{\{x\}}$, similarly as in Lemma \ref{fillingone}. Then, thanks to \eqref{maxmin}, one obtains
\begin{displaymath}
\mathcal{G}(T,\{x\}) = \alpha k = \sum_{i=1}^k \mathcal{G}(\Gamma_{u_i}, \{x\})
\end{displaymath}
as we wanted to prove.

\end{dimo}

\begin{prop}\label{summ2}
Consider $T \in C$ such that $T = \sum_{i=1}^k \Gamma_{u_i}$. Fix $x\in S_T$ and suppose that we have $u_i^r(x) \leq u_i^l(x)$ for every $i=1 \ldots k$. Moreover assume that $u^l_{i} (x)  \neq  u^r_{j}(x)$ for every $i,j$ such that $x\in S_{u_i}\cap S_{u_j}$.\\
Then
\begin{displaymath}
\mathcal{G}(T, \{x\}) = \sum_{i=1}^k \mathcal{G}(\Gamma_{u_i}, \{x\}).
\end{displaymath}
\end{prop}
\begin{dimo}
See Proposition \ref{summ1}.
\end{dimo}
Now Theorem \ref{split} is an immediate consequence of the previous propositions.
\vspace{2mm}\\
\emph{Proof of Theorem \ref{split}}
\vspace{2mm}\\
Fix $x\in S_T$ and define
\begin{displaymath}
\mathcal{I} = \{i=1\ldots k: u^l_{i} (x) \leq u^r_{i}(x)\} \qquad \mathcal{J} = \{i=1\ldots k: u^l_{i} (x) > u^r_{i}(x)\}
\end{displaymath}
and call $T_\mathcal{I} = \sum_{i\in \mathcal{I}}\Gamma_{u_i}$ and $T_\mathcal{J} = \sum_{i\in \mathcal{J}}\Gamma_{u_i}$.
Moreover let $\phi_{\mathcal{I}}$ ($\phi_\mathcal{J}$) be the vector field realizing the maximum in $\mathcal{G}(T_\mathcal{I},\{x\})$ ($\mathcal{G}(T_\mathcal{J}),\{x\})$. From Proposition \ref{summ1} and \ref{summ2} it is easy to see that $\phi^x_\mathcal{\mathcal{I}} \leq 0$ outside the support of $T_\mathcal{I}$ restricted to $\{x\} \times \mathbb{R}$ and $\phi^x_\mathcal{J} \geq 0$ outside the support of $T_\mathcal{I}$ restricted to $\{x\} \times \mathbb{R}$. Therefore defining $\tilde \phi =  \phi_\mathcal{I} + \phi_\mathcal{J}$, as we assumed that there is no cancellation on the jumps by Lemma \ref{nocanc}, we have that $\tilde \phi \in K_{\{x\}}$ and
\begin{displaymath}
\mathcal{G}(T_\mathcal{I},\{x\}) + \mathcal{G}(T_\mathcal{J},\{x\}) = \int_{\{x\} \times \mathbb{R}}  \langle\phi_\mathcal{I}^x + \phi_\mathcal{J}^x,\nu_T\rangle \, d\|T\|  \leq \mathcal{G}(T,\{x\}).
\end{displaymath}
So by convexity
\begin{displaymath}
\mathcal{G}(T_\mathcal{I},\{x\}) + \mathcal{G}(T_\mathcal{J},\{x\}) = \mathcal{G}(T,\{x\}).
\end{displaymath}
Finally we apply Proposition \ref{summ1} and \ref{summ2} to $T_\mathcal{I}$ and $T_\mathcal{J}$ to get the thesis.
\begin{flushright}
$\Box$
\end{flushright}
We conclude this section with a lemma that shows that we can reduce any combination of graphs belonging to $C$ to a combination of graphs, all with the same multiplicity. We are going to use this property in the proof of the convex decomposition formula in the next section.
\begin{lemma}\label{samemult}
Consider $T_1,T_2\in C$ and $x \in S_{T_1} \cap S_{T_2}$. Suppose that $T_1\res (\{x\} \times \mathbb{R}) = \sum_{i=1}^k \lambda_i [\![\{x\}\times (a_i,a_{i+1})]\!]$ with $a_i < a_{i+1}$ and let $\{M_j\}_{j\in J}$ be the indexes of the maximums of the multiplicities. Assume in addition that $T_2 \res (\{x\} \times \mathbb{R}) = \nu \sum_{j\in J} [\![\{x\}\times (a_{M_j}, a_{M_j+1})]\!]$ for some $\nu>0$. Then we have
\begin{equation}\label{kin}
\mathcal{G}(T_1 + T_2,\{x\}) = \mathcal{G}(T_1,\{x\}) + \mathcal{G}(T_2,\{x\}).
\end{equation}
\end{lemma}
\begin{dimo}
Consider the vector field $\tilde \phi \in K_{\{x\}}$ defined in Lemma \ref{summ1}:
\begin{displaymath}
\tilde \phi = \left\{\begin{array}{ll}
\Big(\alpha/(a_{M_{j} + 1}-a_{M_j}),\frac{(\tilde\phi^x)^2}{4} - \beta(t-g)^2\Big) & (x,t) \in \{x\} \times \bigcup_j (a_{M_j}, a_{M_j + 1}),\\
\Big(-\alpha/(a_{m_{j} + 1}-a_{m_{j}}), \frac{(\tilde\phi^x)^2}{4} - \beta(t-g)^2\Big) & (x,t) \in \{x\}\times \bigcup_j  (a_{m_{j}}, a_{m_{j}+1}), \\
0 & \mbox{ otherwise}.
\end{array}
\right.
\end{displaymath}
Thanks to \eqref{maxmin} we have
\begin{displaymath}
\mathcal{G}(T_1, \{x\}) = \int_{\mathcal{M}_1 \cap (\{x\} \times \mathbb{R})} \langle \tilde \phi^x, \nu_{T_1}\rangle \theta_{T_1}\, d\mathcal{H}^1 \quad \mbox{and} \quad \mathcal{G}(T_2, \{x\}) = \int_{\mathcal{M}_2 \cap (\{x\} \times \mathbb{R})} \langle \tilde \phi^x, \nu_{T_2}\rangle\theta_{T_2}\, d\mathcal{H}^1,
\end{displaymath}
where $T_i = (\mathcal{M}_i, \theta_{T_i}, \nu_{T_i})$ for $i=1,2$. Hence setting $T_1 + T_2 =(\mathcal{M},\theta,\nu)$ we have
\begin{eqnarray*}
\mathcal{G}(T_1, \{x\}) + \mathcal{G}(T_2, \{x\}) & = & \int_{\mathcal{M}_1 \cap (\{x\} \times \mathbb{R})} \langle \tilde{\phi}^x, \nu_{T_1}\rangle \theta_{T_1}\, d\mathcal{H}^1 + \int_{\mathcal{M}_2 \cap (\{x\} \times \mathbb{R})} \langle \tilde \phi^x, \nu_{T_2}\rangle\theta_{T_2}\, d\mathcal{H}^1\\
& = & \int_{\mathcal{M} \cap (\{x\} \times \mathbb{R})} \langle \tilde \phi^x, \nu \rangle \theta \, d\mathcal{H}^1 \leq G(T_1 + T_2).
\end{eqnarray*}
As the opposite inequality follows by convexity, we infer \eqref{kin}.

\end{dimo}

\begin{cor}\label{same}
Given $T_1 = \sum_{i=1}^{k} \lambda_i \Gamma_{u_i}$, let $\{M_j\}_{j\in J}$ be the indexes of the maximums of the multiplicities. Given $T_2 =  \nu\sum_{i \in J} \Gamma_{u_i}$ with $\nu > 0$ we have that $G(T_1 + T_2) = G(T_1) + G(T_2)$.
\end{cor}
\begin{dimo}
Notice that by Lemma \ref{overlapping} it is enough to prove the thesis for every $x\in S_{T_2}\cap S_{T_1}$. Thanks to Lemma \ref{samemult} one has
\begin{displaymath}
G(T_1 + T_2, \{x\}) =  G(T_1, \{x\}) + G(T_2, \{x\}).
\end{displaymath}
\end{dimo}
\subsection{Convex decomposition formula}
As anticipated in the introduction, this section is devoted to the proof of a decomposition formula for the Mumford-Shah functional in one dimension. This formula resembles closely a generalized coarea formula for functionals and it is performed for a finite combination of graphs with multiplicity. It is interesting to notice that the counterexample in the end of Remark \ref{coareafalse} is ``solved'' by this decomposition, but it is difficult to generalize it to the continuous case. However it gives a strong indication on how this decomposition should be performed at least in dimension one. The higher dimensional case is a completely different issue, as the convex decomposition formula we are going to present strongly relies on the one dimensional structure of the problem and cannot be extended in an easy way.
\begin{figure}
\centering
\subfigure{
\begin{tikzpicture}[>=stealth, scale=0.9]
\draw [->](-1,0) -- (6,0) node [scale=0.7][below]{x};
\draw [->](0,-1) -- (0,6) node [scale=0.7][left]{t};
\draw[densely dashed] (5,0) -- (5,6);
\draw  (0,0) [thick][red]
to[out=10,in=180, looseness=1] (2,2)
to[out=90,in=90, looseness=1] (2,3)
to[out=0,in=160, looseness=0.8] (5,5) node[black, scale=0.7] [below=18pt, left=4pt] {$\Gamma_{u_1}$} ;

\draw  (0,0)[thick][blue]
to[out=40,in=180, looseness=1] (2,3)
to[out=90,in=-90, looseness=1] (2,4)
to[out=0,in=160, looseness=0.8] (5,5)node[black, scale=0.7] [left=35pt]{$\Gamma_{u_2}$};

\draw  (2,0) node[black, scale=0.7] [below=0.5pt]{$S_{u_1}\cap S_{u_2}$} [dashed] -- (2,2) ;
\draw  (0,2) node[black, scale=0.7] [left=0.5pt]{$u_1^l$} [dashed] -- (2,2) ;
\draw  (0,3) node[black, scale=0.7] [left=0.5pt]{$u_1^r = u_2^l$} [dashed] -- (2,3) ;

\draw  (0,4) node[black, scale=0.7] [left=0.5pt]{$u_2^r$} [dashed] -- (2,4) ;

\end{tikzpicture}
} \qquad
\subfigure{
\begin{tikzpicture}[>=stealth, scale=0.9]
\draw [->](-1,0) -- (6,0) node [scale=0.7][below]{x};
\draw [->](0,-1) -- (0,6) node [scale=0.7][left]{t};
\draw[densely dashed] (5,0) -- (5,6);
\draw  (0,0) [thick][green]
to[out=10,in=180, looseness=1] (2,2)
to[out=90,in=-90, looseness=1] (2,4)
to[out=0,in=160, looseness=1] (5,5) node[black, scale=0.7] [below=18pt, left=4pt] {$\Gamma_{w_2}$} ;

\draw  (0,0)[thick][violet]
to[out=40,in=180, looseness=1] (2,3)
to[out=0,in=160, looseness=0.8] (5,5)node[black, scale=0.7] [left=35pt]{$\Gamma_{w_1}$};

\draw  (2,0) node[black, scale=0.7] [below=0.5pt]{$S_{w_1}$} [dashed] -- (2,2) ;
\draw  (0,2) node[black, scale=0.7] [left=0.5pt]{$w_1^l$} [dashed] -- (2,2) ;

\draw  (0,4) node[black, scale=0.7] [left=0.5pt]{$w_1^r$} [dashed] -- (2,4) ;

\end{tikzpicture}

}

\caption{Convex decomposition of two SBV graphs}
\label{space}
\end{figure}

\begin{prop}\label{coareasame}
Given $T= \sum_{i=1}^k \Gamma_{u_i} \in C$ such that $|S_T| <+\infty$ there exists $\{w_i\}_{i=1\ldots k} \subset SBV(I)$ such that $T = \sum_{i=1}^k\Gamma_{w_i}$ and
\begin{displaymath}
G(T) =  \sum_{i=1}^k G(\Gamma_{w_i}).
\end{displaymath}
\end{prop}
\begin{dimo}
Set $I=(a,b)$ and $S_T = \{p_1,\ldots,p_N\} \subset I$ (the cardinality of $S_T$ is finite by assumption).
Choose $N-1$ points $\{x_s\}$ so that $p_s < x_s < p_{s+1}$ and define a partition of $(a,b]$ as
\begin{displaymath}
(a,b]=\bigcup_{s=0}^{N-1} (x_s,x_{s+1}]
\end{displaymath}
where $x_0 = a$ and $x_N =b$.\\
Consider an interval $(x_s,x_{s+1})$ and define $\mathcal{I} \subset \{1,\ldots,k\}\times \{1,\ldots,k\}$ as
\begin{equation}
\mathcal{I} := \{(i,j) : p_s \in S_{u_i} \cap S_{u_j} \ \mbox{and} \  u^r_i(p_s) = u^l_j(p_s) \}.
\end{equation}

For a given $(i,j) \in \mathcal{I}$ define the following functions (see Figure \ref{space}):

\begin{displaymath}
w_i = \left\{
\begin{array}{ll}
u_{i} & \mbox{ for } x \in (x_s,p_s)\\
u_{j} & \mbox{ for } x \in (p_s,x_{s+1})
\end{array}
\right.
\end{displaymath}
and
\begin{displaymath}
w_j = \left\{
\begin{array}{ll}
u_{j} & \mbox{ for } x \in (x_s,p_s)\\
u_{i} & \mbox{ for } x \in (p_s,x_{s+1}).
\end{array}
\right.
\end{displaymath}
We have that $\Gamma_{u_i} + \Gamma_{u_j} = \Gamma_{w_i} + \Gamma_{w_j}$ in $(x_s,x_{s+1})$ and $w_i^r(p_s) \neq w_j^l(p_s)$. Then, considering the new collection of functions with $u_i,u_j$ substituted with $w_i$ and $w_j$, we can repeat this operation. It is easy to see that we can perform this procedure only a finite number of times (until the set $\mathcal{I}$ is empty), as at every step we are strictly decreasing the cardinality of $\mathcal{I}$ by at least one. In this way we produce a family of functions $\{w_i\} \subset SBV((x_s,x_{s+1}))$ such that
$\sum_{i=1}^k \Gamma_{w_i} = \sum_{i=1}^k \Gamma_{u_i}$ in $(x_s, x_{s+1})$ and $w^r_{i} (p_s) \neq w^l_{j}(p_s)$ for every $(i,j)$ such that $p_s \in S_{w_i}\cap S_{w_j}$. \\
Finally we can construct a family of functions $\{w_i\} \subset SBV(I)$ repeating this procedure inductively starting from the first interval $(a,x_1)$ and ending in $(x_{n-1},b)$. The family $\{w_i\} \subset SBV(I)$ has the following properties:
\begin{displaymath}
\sum_{i=1}^k \Gamma_{w_i} = \sum_{i=1}^k \Gamma_{u_i} \quad \mbox{and} \quad w^r_{i} (x) \neq w^l_{j}(x)\  \  \forall x,i,j \mbox{ such that } x \in S_{w_i}\cap S_{w_j}.
\end{displaymath}
Hence using Theorem \ref{split} and Proposition \ref{decomposition} one obtains the thesis.

\end{dimo}

\begin{thm}[Convex decomposition formula]\label{coarea}
Given $T= \sum_{i=1}^k \lambda_i\Gamma_{u_i}$ such that $|S_T| < +\infty$ there exists $k' \in \mathbb{N}$, $\{\mu_i\}_{i=1\ldots k'} \geq 0$ and $\{w_i\}_{i=1\ldots k'} \subset SBV(I)$ such that $T = \sum_{i=1}^{k'} \mu_i\Gamma_{w_i}$ and
\begin{equation}\label{coar}
G(T) =  \sum_{i=1}^{k'} \mu_i G(\Gamma_{w_i}).
\end{equation}
\end{thm}
\begin{dimo}
Consider $T= \sum_{i=1}^k \lambda_i \Gamma_{u_i} \in C$ and suppose without loss of generality that also $\lambda_i$ are ordered and $\lambda_k$ is the maximum (if the multiplicities are not ordered the proof is analogous). Then $T$ can be rewritten as
\begin{displaymath}
T= (\lambda_k-\lambda_{k-1}) \Gamma_{u_k} + \lambda_{k-1} \Gamma_{u_k} + \sum_{i=1}^{k-1} \lambda_i \Gamma_{u_i}.
\end{displaymath}
Hence by Corollary \ref{same}
\begin{displaymath}
G(T) = G((\lambda_k-\lambda_{k-1}) \Gamma_{u_k}) + G\left(\lambda_{k-1} \Gamma_{u_k} + \sum_{i=1}^{k-1} \lambda_i \Gamma_{u_i}\right).
\end{displaymath}
Then one can rewrite
\begin{displaymath}
\lambda_{k-1} \Gamma_{u_k} + \sum_{i=1}^{k-1} \lambda_i \Gamma_{u_i} = \lambda_{k-2} (\Gamma_{u_k} + \Gamma_{u_{k-1}}) +  (\lambda_{k-1} -\lambda_{k-2}) (\Gamma_{u_k} +  \Gamma_{u_{k-1}}) + \sum_{i=1}^{k-2} \lambda_i \Gamma_{u_i}
\end{displaymath}
and applying again Corollary \ref{same}
\begin{eqnarray}
G\left(\lambda_{k-1} \Gamma_{u_k} + \sum_{i=1}^{k-1} \lambda_i \Gamma_{u_i}\right) &=&
G((\lambda_{k-1} -\lambda_{k-2}) (\Gamma_{u_k} +  \Gamma_{u_{k-1}}) ) \nonumber\\
&+& G\left(\lambda_{k-2} (\Gamma_{u_k} + \Gamma_{u_{k-1}}) + \sum_{i=1}^{k-2} \lambda_i \Gamma_{u_i}\right).\label{tr}
\end{eqnarray}
By Proposition \ref{coareasame} there exists $u^2_k$ and $u^2_{k-1}$ SBV functions such that $\Gamma_{u^2_k} +  \Gamma_{u^2_{k-1}} = \Gamma_{u_k} + \Gamma_{u_{k-1}}$ and
\begin{displaymath}
\eqref{tr}= G((\lambda_{k-1} -\lambda_{k-2}) \Gamma_{u^2_k}) + G((\lambda_{k-1} -\lambda_{k-2}) \Gamma_{u^2_{k-1}})  +  G\left(\lambda_{k-2} (\Gamma_{u_k} + \Gamma_{u_{k-1}}) + \sum_{i=1}^{k-2} \lambda_i \Gamma_{u_i}\right)
\end{displaymath}
and so on. Repeating this procedure $k$ times one gets to
\begin{displaymath}
G(T) = \sum_{i=2}^{k}\sum_{j=i}^k (\lambda_{i} -\lambda_{i-1})G(\Gamma_{u^{k-i+1}_j}) + G\left(\sum_{i=1}^k \lambda_1 \Gamma_{u_i}\right).
\end{displaymath}
Hence, applying again Proposition \ref{coareasame} to the last term we obtain the desired decomposition \eqref{coar}.

\end{dimo}
\section{Existence of calibration as a functional defined on currents}\label{existence}
We now want to show an application of the previous convex decomposition formula to the existence of calibration for the Mumford-Shah type functionals. Firstly we set the Dirichlet problem in $I'$ associated to the previous functional $G$.
We recall that $G$ takes values in the convex cone defined as
\begin{equation}
C:= \left\{T= \sum_{i=1}^{k} \lambda_i \Gamma_{u_i} : k\in \mathbb{N}, \lambda_i \in \mathbb{R}_+, u_i \in SBV(I)\right\}.
\end{equation}
Consider $S\in C$ and setting $I_{\mathbb{R}} := (I\setminus I') \times \mathbb{R}$ define
\begin{displaymath}
\psi_{G}(S) = \inf\{G(T) : T\in C, \ T\res I_{\mathbb{R}} = S \res I_{\mathbb{R}}\}.
\end{displaymath}

\begin{prop}\label{conv}
The functional $\psi_{G}$ is convex in $C$.
\end{prop}
\begin{dimo}
As $G$ is convex and the constraint is linear the proof is straightforward.
\end{dimo}
It is easy to see that by the convex decomposition formula in Theorem \ref{coarea} we have the following theorem:
\begin{thm}\label{mainthm1}
If $u \in SBV(I)$ is a Dirichlet minimizer in $I'$ of $F$, then $\psi_{G}(\Gamma_u) = G(\Gamma_u) = F(u)$.
\end{thm}
\begin{dimo}
Consider $T = \sum_{i=1}^k \lambda_i \Gamma_{u_i} \in C$ such that $T\res I_{\mathbb{R}} = \Gamma_u \res I_{\mathbb{R}}$.
Without loss of generality we can suppose that $|S_T| <+\infty$.
Then by Theorem \ref{coarea} there exist $k'$ and $\{\mu_i\}_{i=1,\ldots, k'}>0$ such that
\begin{equation}\label{lastcor}
G(T) = G\left(\sum_{i=1}^k \lambda_i \Gamma_{u_i}\right) = \sum_{i=1}^{k'}\mu_i G(\Gamma_{w_i}) = \sum_{i=1}^{k'}\mu_i F(w_i)
\end{equation}
and $\sum_{i=1}^k \lambda_i \Gamma_{u_i} = \sum_{i=1}^{k'}\mu_i \Gamma_{w_i}$. As $T\res I_{\mathbb{R}} = \Gamma_u \res I_{\mathbb{R}}$, we infer that $\sum_{i=1}^{k'}\mu_i = 1$ and
$w_i = u$ in $I\setminus I'$ for every $i=1,\ldots,k'$. Finally, from Formula \eqref{lastcor}, using that $u$ is a Dirichlet minimizer of $F$ in $I'$ we obtain that $\psi_{G}(\Gamma_u) = G(\Gamma_u)$.

\end{dimo}

\begin{rmk}
Theorem \ref{mainthm1} can be obtained also with similar techniques as the ones presented in \cite{CRF}, provided that the functional $F$ satisfies stronger regularity assumptions (see Remark \ref{gener}).
\end{rmk}

The previous theorem allows us to state a weak existence results for calibrations as an application of Hahn-Banach theorem. Let
\begin{displaymath}
\hat C = \left\{T = \sum_{i=1}^k \lambda_i \Gamma_{u_i}: k\in \mathbb{N}, \lambda_i \in \mathbb{R}, u_i \in SBV(I)\right\}
\end{displaymath}
be the double cone and denote by $Hom(\hat C)$ the set of all the linear maps from $\hat C$ to $\mathbb{R}$.
We define the following notion of calibration for linear combinations of graphs:
\begin{defi}[Calibration for minimal graphs]\label{calcur}
Given $u \in SBV(I)$ and $\Gamma_u$ its associated graph, we say that $\xi\in \mbox{Hom}(\hat C)$ is a calibration for $\Gamma_u$ with respect to $G$ if
\begin{itemize}
\item[i)] $\xi(\Gamma_u) = G(\Gamma_u) = F(u)$,
\item[ii)] $\xi(T) = 0$ for every $T \in \hat C$ such that $T\res I_\mathbb{R} = 0$,
\item[iii)] $\xi (T) \leq \hat G(T)$ for every $T \in \hat C$,
\end{itemize}
where $\hat G : \hat C \rightarrow \mathbb{R}$ is the extension of $G$ to $\hat C$ according to Formula \eqref{operator}.
\end{defi}
\begin{thm}
Given $u\in SBV(I)$ a Dirichlet minimizer of $F$ in $I'$ there exists a calibration for $\Gamma_u$ with respect to $G$ according to Definition \ref{calcur}.
\end{thm}
\begin{dimo}
From Theorem \ref{mainthm1} follows that
\begin{displaymath}
G(\Gamma_{u}) = \psi_{G}(\Gamma_u).
\end{displaymath}
We firstly notice that, as a consequence of the definition of $K$, we have that
\begin{equation*}
\hat G(T) = \left\{
\begin{array}{ll}
G(T) & \mbox{if } T\in C\\
+\infty & \mbox{of }T\in \hat C \setminus C.
\end{array}
\right.
\end{equation*}
We define $\psi_{\hat G} : \hat C \rightarrow \mathbb{R}$ as
\begin{equation*}
\psi_{\hat G}(S) := \inf\{\hat G(T) : T\in \hat C, \ T\res I_{\mathbb{R}} = S \res I_{\mathbb{R}}\},
\end{equation*}
that is convex and such that $\psi_{\hat G}(\Gamma_{u}) = G(\Gamma_{u}) > 0$.\\
Consider the vector subspace $L = \{a\Gamma_u: a\in \mathbb{R}\}$ and define $\psi: L \rightarrow \mathbb{R}$ as $\psi(a\Gamma_u) = a\psi_{\hat G}(\Gamma_u)$ clearly linear. As we have that $\psi \leq \psi_{\hat G}$ on $L$, by Hahn-Banach theorem there exists $\xi\in Hom(\hat C, \mathbb{R})$ such that
\begin{equation}\label{too}
\xi(\Gamma_{u}) =  \psi(\Gamma_u) = \psi_{\hat G}(\Gamma_{u}) \quad \mbox{ and }\quad  \xi(T) \leq \psi_{\hat G}(T) \ \quad \forall T\in \hat C.
\end{equation}
We want to prove that $\xi$ is a calibration according to Definition \ref{calcur}. Let $T_0\in \hat C$ be such that $T_0\res I_\mathbb{R}  = 0$, then
\begin{displaymath}
\psi_{\hat G}(T_0)= \inf\{\hat G(S) : S\in \hat C, \  S\res I_\mathbb{R} = 0\} \leq G(0) = 0.
\end{displaymath}
In combination with \eqref{too} this implies $\xi(T)\leq 0$ for every $T_0 \in \hat C$ such that $T_0\res I_\mathbb{R}  = 0$.\\
So, as $\xi$ is an homeomorphism, one has also that $\xi(T_0)= 0$, so that $(ii)$ holds.
Moreover from \eqref{too}, $\xi(\Gamma_u) = \psi_{\hat G}(\Gamma_{u}) = F(u)$ that is $(i)$.\\
Let us show that also $(iii)$ is satisfied: if $T\in \hat C \setminus C$ then $G(T) = +\infty$ and so there is nothing to prove. On the other hand given $T = \sum_{i=1}^{k} \lambda_i \Gamma_{u_i} \in C$ with $\lambda_i \in \mathbb{R}_+$ by \eqref{too} and using the definition of $\psi_{\hat G}$
\begin{displaymath}
\xi(T) \leq \psi_{\hat G}(T) \leq \hat G(T).
\end{displaymath}
Hence $\xi$ is a calibration according to Definition \ref{calcur}.
\end{dimo}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv