text:
\section{A conjecture on rank-one convexity and polyconvexity}

Generalized notions of convexity play an important role in the multidimensional calculus of variations. In order to ensure the existence of minimizers for functionals of the form
\begin{align}
	I\col W^{1,p}(\Omega;\R^n)\to\R\,,\qquad \varphi\mapsto I(\varphi) = \int_{\Omega} W(\grad\varphi(x))\,\dx
\end{align}
in the Sobolev space $W^{1,p}(\Omega)$ with $\Omega\subset\R^n$ under various boundary conditions, weak lower semicontinuity (w.l.s.c.) of $I$ is usually required. While (classical) convexity and lower semicontinuity of the function $W\col\Rnn\to\R$ on the set of \nnmatrices\ are certainly sufficient for $I$ to be w.l.s.c., convexity is oftentimes too strong as requirement for practical applications. Therefore, weaker convexity properties have been introduced which in many cases suffice to ensure the weak lower semicontinuity and thus the existence of minimizers as well. In particular, Morrey \cite{morrey1952quasi} famously showed that, under appropriate conditions, $I$ is w.l.s.c.\ if and only if $W$ is \emph{quasiconvex}, i.e.\ if
\begin{equation}\label{eq:definitionQuasiconvexity}
	\int_{\Omega}W(F_0+\nabla \vartheta)\,\dx\geq \int_{\Omega}W(F_0)\,\dx=W(F_0)\cdot \abs{\Omega}
	\qquad\tforall F\in\Rnn
\end{equation}
for every bounded open set $\Omega\subset\R^n$ and all test functions $\vartheta\in W_0^{1,\infty} (\Omega;\R^n)$. Further generalizations of the convexity of functions have been subsequently introduced, with two of the most important ones being \emph{rank-one convexity}, i.e.
\begin{equation}\label{eq:definitionRankOneConvexity}
	W((1-t)F+t(F+H)) \;\leq\; (1-t)\.W(F) + t\.W(F+tH)
\end{equation}
for all $F\in\Rnn$, $t\in[0,1]$ and $H\in\Rnn$ with $\rank(H)=1$, and \emph{polyconvexity} of $W$ which, for $n=2$ and $n=3$, can be expressed as
\begin{align}
	W(F) = P(F,\det F) \qquad&\text{for a convex function }\;P\col\R^{2\times2}\times\R \;\cong\; \R^5\to\R_\infty\label{eq:definitionPolyconvexityTwoDimensional}\\[-0.7em]
\intertext{and}\notag\\[-2em]
	W(F) = P(F,\Cof F,\det F) \qquad&\text{for a convex function }\;P\col\R^{3\times3}\times\R^{3\times3}\times\R \;\cong\; \R^{19}\to\R_\infty\label{eq:definitionPolyconvexityThreeDimensional}
\end{align}
respectively, where $\Cof F$ denotes the cofactor of $F$ and $\R_\infty=\R\cup \{\infty\}$. Major interest in these two concepts stems from the observation that polyconvexity implies quasiconvexity, which in turn implies rank-one convexity \cite{Dacorogna08,rindler2018calculus}. Since quasiconvexity is difficult to verify or falsify for a given function $W$ on $\Rnn$, the availability of a sufficient and a necessary criterion has proven quite useful in the past. However, for $n\geq3$, these three notions of convexity are generally not equivalent \cite{sverak1992rank}. For $n=2$, on the other hand, it is still an open question whether rank-one convexity implies quasiconvexity \cite{morrey1952quasi,ball1987does,ball2002openProblems,conti2005rank,parry1995planar,parry2000rank,pedregal1998note}. In fact, many classes of functions on $\R^{2\times2}$ have been identified for which rank-one convexity even implies polyconvexity \cite{Ball84b,muller1999rank,agn_martin2015rank,agn_ghiba2018rank,agn_martin2019quasiconvex} and thus quasiconvexity.

\medskip

In a 2005 \emph{Journal of Convex Analysis} article \cite{mielke2005necessary}, Alexander Mielke put forward a further conjecture on the relation between rank-one convexity and polyconvexity in the planar case.

\begin{conjecture}[Mielke \cite{mielke2005necessary}]\label{conjecture:mielke}
	Let $W\col\GLp(2)\to \R_{\infty}$ be an objective, isotropic and rank-one convex function such that the sublevel sets
	\[
		S_c \colonequals \{F\in\R^{2\times2} \setvert \det F > 0\,,\; W(F) \leq c\}, \qquad c\in \R
	\]
	are connected and compact. Then $W$ is polyconvex.
\end{conjecture}

We note that although the requirement of compact sublevel sets is not explicitly included in the original statement of the conjecture \cite{mielke2005necessary}, it can be inferred from the context, especially since Mielke excludes an otherwise viable counterexample by Aubert despite the connectedness of the corresponding sublevel sets (cf.~\eqref{eq:aubertEnergy} and Appendix~\ref{appendix:aubertConnectedness}).

Conjecture \ref{conjecture:mielke}\footnote{%
	Mielke also later clarified his originally intended formulation \cite{mielkePersonalCommunication}. The original conjecture \cite{mielke2005necessary} is phrased as follows, with the additional requirement of compactness added here for clarity:
	\begin{quote}
	{\it%
		However, there is even further similarity between rank-one convexity and polyconvexity for isotropic functions which stems from the theory in \quoteescnobr{\cite{aubert1987counterexample,cardaliaguet2002equivalence1,cardaliaguet2002equivalence2}}. There it is shown that compact, connected, and isotropic subsets of $\{A\in \R^{2\times 2} \setvert \det A>0\}$ are rank-one convex if and only if they are polyconvex. So we conjecture that all isotropic, rank-one convex functions are in fact polyconvex, if all the sublevel sets $\{F\in \R^{2\times 2} \,\setvert\, \det F>0\,,\; \Phi(\Lambda(F))\leq t\}\,,\;t\in \R$, are \quoteesc{compact and} connected. The last condition rules out the famous counterexample by Aubert in \cite{aubert1987counterexample} given via
		\begin{equation*}
			\Phi(\nu)=\frac{1}{3} \left(\nu_1^4+\nu_2^4\right)+\frac{1}{2}\nu_1^2\nu_2^2-\frac{2}{3} \nu_1\nu_2 \left(\nu_1^2+\nu_2^2\right)\,.
		\end{equation*}%
	}
	\end{quote}
	Here, Mielke uses the notation $W(F)=\Phi(\Lambda(F))$ for the representation of an isotropic energy $W\col\GLp(2)\to\R$ in terms of the ordered singular values $\Lambda_1(F)\geq\Lambda_2(F)$ of $F$.
} is mainly motivated by analogous results relating the polyconvexity and rank-one convexity of sets (cf.~Proposition \ref{proposition:rankOnePolyEquivalenceCompactConnectedSetsGLp}). In Section \ref{section:counterExample}, however, we will show that the function (cf.~eq.~\eqref{eq:counterexampleDefinition})
\[
	W_0\col\GLp(2)\to\R\,,\qquad W_0(F) = \frac{\opnorm{F}^2}{\det F} - \log\biggl(\frac{\opnorm{F}^2}{\det F}\biggr) + \log(\det F) + \frac{1}{\det F}
	\,,
\]
where $\opnorm{\..\.}$ denotes the operator norm, is indeed rank-one convex and isotropic with compact and connected sublevel sets, but not polyconvex. We will thereby establish the following proposition.
\begin{proposition}
\label{prop:conjectureFalse}
	Conjecture \ref{conjecture:mielke} does not hold.
\end{proposition}

In the following, we will provide some of the background which originally motivated Conjecture \ref{conjecture:mielke} as well as some basic results required for demonstrating that $W_0$ has the claimed properties.
\section{Background and preliminaries}

While the generalized notions of convexity discussed above are defined for (and applied to), first and foremost, \emph{functions} on $\Rnn$, analogous definitions are sometimes considered for \emph{sets} of matrices as well.

\begin{definition}\label{def:generalizedConvexitySets}
	Let $\mathbb{M}\subset\Rnn$. Then $\mathbb{M}$ is called \emph{rank-one convex} if $(1-t)F_1+tF_2\in \mathbb{M}$ for all $t\in[0,1]$ and all $F_1,F_2\in \mathbb{M}$ such that $\rank(F_2-F_1)= 1$. For $n=2$, the set $\mathbb{M}\subset\R^{2\times2}$ is called \emph{polyconvex} if $\mathbb{M}=\{F\in\R^{2\times2} \setvert P(F,\det F)\leq0\}$ for some convex function $P\col\R^{2\times2}\times\R\cong\R^5\to\R_\infty$.
\end{definition}

Of course, the definition of polyconvex sets can easily be extended \cite{Dacorogna08,szekelyhidi2006local} to the case of arbitrary dimension $n$. Note that Definition \ref{def:generalizedConvexitySets} of rank-one convex subsets is used by Cardaliaguet and Tahraoui \cite{cardaliaguet2000equivalence,cardaliaguet2002equivalence1,cardaliaguet2002equivalence2} as well as by Dacorogna \cite{Dacorogna08}, while Conti et al.\ \cite{conti2005rank} call subsets with the stated properties \emph{laminated convex} subsets of $\R^{2\times 2}$.

The rather recent development of these notions of convexity for sets has, again, been primarily motivated by applications in the calculus of variations. In particular, it has been shown that if a compact set $\mathbb{M}\subset\R^{2\times2}$ is polyconvex and $\OO(2)$-invariant (see Remark \ref{remark:isotropyHemitropyOfSets}), then the set
\[
	\mathcal{M} = \{ \varphi\in W^{1,\infty}(\Omega;\R^n) \setvert \grad\varphi(x)\in \mathbb{M} \;\text{ for }x\in\Omega \;\text{ a.e.}\}
\]
is weakly-$\ast$ closed; furthermore, if $\mathcal{M}$ is weakly-$\ast$ closed, then $\mathbb{M}$ is rank-one convex \cite{cardaliaguet2000equivalence,cardaliaguet2002equivalence1}.

In the context of nonlinear hyperelasticity, where generalized convexity properties have an especially long and rich history \cite{ball1976convexity,chelminski2006new,knowles1976failure,knowles1978failure,rosakis1994,agn_schroder2010poly}, the material behaviour of an elastic solid is described by a potential energy function
\[
	W\col\GLp(n)\to\R\,,\quad F\mapsto W(F)
\]
defined on the group $\GLp(n)$ of invertible matrices with positive determinant; here, $F=\grad\varphi$ represents the \emph{deformation gradient} corresponding to a deformation $\varphi\col\Omega\to\R^n$ of an elastic body $\Omega\subset\R^n$, which is assumed to be non-singular and orientation preserving. While this restriction of $W$ to a subset of $\Rnn$ poses additional challenges for the application of variational methods, the above concepts of generalized convexity can still be utilized \cite{ball1976convexity}, partly due to the fact that the set $\GLpn$ is itself rank-one convex and polyconvex.

An elastic energy potential $W\col\GLp(n)\to\R$ is also often assumed to be \emph{objective} (or \emph{frame-indifferent}) as well as \emph{isotropic}, i.e.\ it satisfies
\begin{equation}
	W(Q_1F\.Q_2) = W(F)
	\qquad\text{for all }\;F\in\GLp(n)
	\quad\text{and all }\;\; Q_1,Q_2\in\SO(n)\,,
\end{equation}
where $\SO(n)=\{X\in \Rnn \setvert Q^T Q=\id\,,\;\det Q=1\}$ denotes the special orthogonal group. In terms of sets of matrices, the theory of nonlinear elasticity thereby directly motivates the consideration of the particular class of \emph{isotropic subsets} of $\GLpn$.

\begin{definition}
\label{definition:isotropicSetGLp}
	A set $\mathbb{M}\subset\GLpn$ is called \emph{isotropic} if
	\[
		Q_1F\.Q_2\in \mathbb{M}
		\qquad\text{for all }\;F\in M
		\quad\text{and all }\;\; Q_1,Q_2\in\SO(n)\,.
	\]
\end{definition}

\begin{remark}
\label{remark:isotropyHemitropyOfSets}
	While Definition \ref{definition:isotropicSetGLp} is in agreement with the notion of isotropic subsets of $\GLpn$ employed by Cardaliaguet and Tahraoui \cite{cardaliaguet2002equivalence2}, different notions of isotropy for arbitrary subsets of $\Rnn$ can be found in the literature as well. In particular, there is no clear consensus on whether isotropy of $\mathbb{M}\subset\Rnn$ encompasses only the invariance under multiplication with $Q\in\SOn$ or with any $Q\in\On$, where $\On=\{X\in\Rnn\setvert X^TX=\id\}$ denotes the orthogonal group.\footnote{%
		If the term \emph{isotropy} is used to denote $\On$-invariance, then $\SOn$-invariance is sometimes called \emph{hemitropy} instead.%
	}
	In the following, we will therefore use the terms \enquote{$\SOn$-invariant} and \enquote{$\On$-invariant} if there is any danger of ambiguity.
\end{remark}

Similar to the case of isotropic energy functions (cf.~Section \ref{sectionContains:energySVrepresentation}), any isotropic (in the sense of Definition \ref{definition:isotropicSetGLp}) set $\mathbb{M}\subset\GLpn$ can be expressed in terms of \emph{singular values}. More specifically, there exists a unique set $M\subset\Oset$ such that $F\in\mathbb{M}$ if and only if
\begin{align*}
	\mathbb{M} &= \mathbb{M}(M) \colonequals \{ F\in\GLpn \setvert \lambdahat(F)\in M \}\,,\\[-0.7em]
	\intertext{where}\\[-2em]
	\Oset &= \{ (\lambda_1,\dotsc,\lambda_n)\in\R^n \setvert \lambda_1\geq\ldots\geq\lambda_n>0 \}
\end{align*}
and $\lambdahat\col\GLpn\to\Oset$ denotes the mapping of any $F\in\GLp(2)$ to the vector $\lambdahat(F)$ of its singular values in descending order. Similarly, for any $M\subset\Oset$, the set $\mathbb{M}(M)$ is isotropic.

In the planar case $n=2$, a number of results concerning the generalized convexity properties have been established for isotropic sets of matrices. Cardaliaguet and Tahraoui \cite{cardaliaguet2002equivalence2} characterized the relation between rank-one convexity and polyconvexity of compact isotropic sets.

\begin{proposition}[{Cardaliaguet and Tahraoui \cite{cardaliaguet2002equivalence2}}]
\label{proposition:cardaliaguetMainResult}
	Let $M\subset\Oset[2]$ be compact. Then the set $\mathbb{M}(M)$ is rank-one convex if and only if the two following properties are satisfied:
	\begin{itemize}
		\item[i)] if $C$ is a connected component of $M$, then $\mathbb{M}(C)$ is polyconvex;
		\item[ii)] if $C_1$ and $C_2$ are two distinct connected components of $M$, then $C_1$ and $C_2$ can be strictly separated in the following sense:\footnote{%
			Note that Cardaliaguet and Tahraoui state their result in terms of singular values in \emph{ascending} order.%
		}
		either $\displaystyle\sup_{(x,y)\in C_1} x < \inf_{(x,y)\in C_2} y$\;\; or \;$\displaystyle\sup_{(x,y)\in C_2} x < \inf_{(x,y)\in C_1} y$.\directqed
	\end{itemize}
\end{proposition}

In particular, Proposition \ref{proposition:cardaliaguetMainResult} yields the equivalence of rank-one convexity and polyconvexity of $\mathbb{M}\subset\GLp(2)$ if the set $\mathbb{M}$, and thus its singular value representation $M\subset\Oset[2]$ with $\mathbb{M}=\mathbb{M}(M)$, is compact and connected.

\begin{proposition}[\cite{cardaliaguet2002equivalence2}]
\label{proposition:rankOnePolyEquivalenceCompactConnectedSetsGLp}
	Let $\mathbb{M}\subset\GLp(2)$ be isotropic, compact and connected. Then $\mathbb{M}$
	is rank-one convex if and only if $\mathbb{M}$ polyconvex.
	\directqed
\end{proposition}

By a counterexample \cite[p.~1225]{cardaliaguet2002equivalence2}, Cardaliaguet and Tahraoui also showed that the connectedness required in Proposition \ref{proposition:rankOnePolyEquivalenceCompactConnectedSetsGLp} cannot be simply omitted. In a companion paper \cite{cardaliaguet2002equivalence1}, they also considered the \emph{unconstrained} case of $\OO(2)$-invariant sets of matrices, i.e.\ subsets $\mathbb{M}\subset\R^{2\times2}$ of the form
\begin{equation}\label{eq:unconstrainedSetSingularValueRepresentation}
	\mathbb{M} = \{ X\in\R^{2\times2} \setvert (\lambdahat_1(X),\lambdahat_2(X))\in M \}
	\qquad\text{with }\;
	M\subset \{ (x,y)\in \R^2 \setvert x\geq y\geq 0 \}\,,
\end{equation}
where $\lambdahat_1(X)\geq\lambdahat_2(X)\geq0$ are the singular values of $X\in\R^{2\times2}$ in descending order, obtaining a result analogous to Proposition \ref{proposition:rankOnePolyEquivalenceCompactConnectedSetsGLp}.

\begin{proposition}[\cite{cardaliaguet2002equivalence1}]
\label{proposition:cardliaguetOinvariantSetsUnconstrained}
	Let $\mathbb{M}\subset\R^{2\times 2}$ be $\OO(2)-invariant$, compact, connected and rank-one convex. Then $\mathbb{M}$ is polyconvex.
	\directqed
\end{proposition}

An interesting generalization of Proposition \ref{proposition:cardliaguetOinvariantSetsUnconstrained} has been given by Conti et al.\ \cite{conti2003polyconvexity}.

\begin{proposition}[{Conti et al.\ \cite[Theorem 2.1]{conti2003polyconvexity}}]
\label{proposition:contiSOinvariantSets}
	Let $\mathbb{M}\subset\R^{2\times 2}$ be $\SO(2)$-invariant, compact, connected and rank-one convex. Then $\mathbb{M}$ is polyconvex.
	\directqed
\end{proposition}

Again, note carefully that $\mathbb{M}\subset\R^{2\times2}$ is of the form \eqref{eq:unconstrainedSetSingularValueRepresentation} if and only if it is $\OO(2)$-invariant, whereas the weaker requirement of $\SO(2)$-invariance in Proposition \ref{proposition:contiSOinvariantSets} does not allow for this particular single value representation in general.

More recently, it was also shown by Heinz that rank-one convexity of an $\OO(2)$-invariant compact set $\mathbb{M}\subset\R^{2\times2}$ implies the \emph{quasiconvexity} \cite{heinz2015quasiconvexity} of $\mathbb{M}$; note that Heinz' result does not require $\mathbb{M}$ to be connected.

\bigskip

In terms of energy functions on $\GLp(2)$, Proposition \ref{proposition:rankOnePolyEquivalenceCompactConnectedSetsGLp} yields an immediate consequence for the generalized notion of \emph{q-convexity}.\footnote{%
	In the literature, functions satisfying Definition \ref{definition:q-Convexity} are more commonly known as \emph{quasiconvex} functions; however, due to the obvious ambiguity accompanied by the term, we will exclusively refer to this property as q-convexity here and throughout.%
}

\begin{definition}\label{definition:q-Convexity}
	Let $W\col M\to\R_\infty$ be an extended real-valued function on a convex subset $M$ of a linear space. Then $W$ is called \emph{q-convex} if for every $c\in\R$, the sublevel set $S_c \colonequals \{x\in M \setvert W(x) \leq c\}$ is convex.
\end{definition}
While every convex function is q-convex, the reverse implication does not hold in general; for example, the mapping $t\mapsto \ln^2(t)$ is q-convex on $\Rp=(0,\infty)$ but not convex.

The concept of q-convexity can be generalized to weakened convexity conditions, including rank-one convexity and polyconvexity.

\begin{definition}\label{definition:q-ConvexityGeneralized}
	A function $W\col \Rnn\to\R_\infty$ is called \emph{q-rank-one-convex} [\emph{q-polyconvex}] if all sublevel sets of $W$ are rank-one convex [polyconvex].
\end{definition}

Since the sublevel sets of any isotropic function $W\col\GLp(2)\to\R$ are isotropic, the following corollary follows immediately from Proposition \ref{proposition:rankOnePolyEquivalenceCompactConnectedSetsGLp}; note the obvious analogy to Conjecture \ref{conjecture:mielke}.

\begin{corollary}
\label{corollary:qConvexityEquivalence}
	Let $W\col\GLp(2)\to\R_\infty$ be an isotropic function such that all sublevel sets of $W$ are compact and connected. Then $W$ is q-rank-one-convex if and only if $W$ is q-polyconvex.
	\directqed
\end{corollary}
\subsection{Polyconvexity criteria in terms of singular values}

In the aforementioned 2005 article \cite{mielke2005necessary}, Mielke obtained a number of results for generalized convexity properties of isotropic energy functions on $\GLpn$ in terms of their singular value representation, including a necessary and sufficient criterion for polyconvexity in the planar case suited for both numerical and analytical applications (cf.\ Section \ref{section:assemblingTheProof}).

Mielke considered the representation of an isotropic energy $W\col\GLp(2)\to\R$ in terms of \emph{ordered} singular values, i.e.\ the uniquely determined function
\label{sectionContains:energySVrepresentation}
\begin{equation}
	\ghat\col\Oset[2]\to\R
	\qquad\text{on the set }\;\;
	\Oset[2]
	= \{ (\lambda_1,\lambda_2)\in\R^2 \setvert \lambda_1\geq\lambda_2>0 \}
	\subset \Rp^2
\end{equation}
such that
\begin{equation}\label{eq:mielkeOrderedSingularValueRepresentation}
	W(F) = \ghat(\lambdahat(F))
	\qquad\tforall F\in\GLp(2)\,,
\end{equation}
where $\Rp=(0,\infty)$ and $\lambdahat\col\GLp(2)\to\Oset[2]$ is the mapping of any $F\in\GLp(2)$ to the vector $\lambdahat(F)$ of its singular values in descending order. After establishing general necessary and sufficient criteria for the polyconvexity\footnote{%
	It is important to note that the definition of polyconvexity of $W\col\GLpn\to\R$ employed by Mielke requires $W$ to satisfy the growth condition $W(F)\to+\infty$ for $\det F\to0$, which corresponds to the lower semicontinuity of the extension of $W$ to $\Rnn$ obtained by setting $W(F)=+\infty$ for all $F\in\Rnn\setminus\GLpn$, cf.\ Remark \ref{remark:compactnessGrowthCondition}.%
}
of functions, Mielke \cite[Theorem 4.1]{mielke2005necessary} also obtained a previous result by Miroslav \v{S}ilhav\'{y} for the differentiable planar case as a direct corollary.
\begin{theorem}[{\v{S}ilhav\'{y} \cite[Proposition 4.1]{vsilhavy2002convexity}}]\label{polycrit}
	Let $\ghat\in C^1(\Oset[2];\R)$. Then the function $W\col\GLp(2)\to\R$ represented by $\ghat$ in terms of ordered singular values as defined by \eqref{eq:mielkeOrderedSingularValueRepresentation} is polyconvex if and only if
	\begin{samepage}
		\begin{equation}
			\forall\;\gamma\in \Oset[2]
			\quad\exists\;c\in \left[
				-\frac{\frac{\partial \ghat}{\partial{\lambda_1}}(\gamma_1,\gamma_2)-\frac{\partial \ghat}{\partial{\lambda_2}}(\gamma_1,\gamma_2)}{\gamma_1-\gamma_2}
				\,,\;
				\frac{\frac{\partial \ghat}{\partial\lambda_1}(\gamma_1,\gamma_2)+\frac{\partial \ghat}{\partial\lambda_2}\ghat(\gamma_1,\gamma_2)}{\gamma_1+\gamma_2}
			\right]
			\quad\forall\;\nu\in \Oset[2]:
		\end{equation}
		\[
			\quad\ghat(\nu_1,\nu_2)
			\geq \ghat(\gamma_1,\gamma_2) + \frac{\partial\ghat}{\partial{\lambda_1}}(\gamma_1,\gamma_2)\cdot(\nu_1-\gamma_1) + \frac{\partial\ghat}{\partial{\lambda_2}}(\gamma_1,\gamma_2)\cdot(\nu_2-\gamma_2) + c\,(\nu_1-\gamma_1)\,(\nu_2-\gamma_2)
			\,.
			\directqed
		\]
	\end{samepage}
\end{theorem}
\subsection{Previous examples of rank-one convex, non-polyconvex energy functions}
\label{section:previousExamples}

A number of examples for rank-one convex, non-polyconvex functions have already been given in the literature \cite{aubert1987counterexample,alibert1992example,dacorogna1988counterexample}. The first such counterexample, a fourth-degree homogeneous polynomial expression, is due to Aubert \cite{aubert1986contribution,aubert1987counterexample}, who considered the function $W_{\mathrm{A}}$ given by
\begin{align}\label{eq:aubertEnergy}
	W_{\mathrm{A}}\col\GLp(2)\to\R\,,\quad
	W_{\mathrm{A}}(F)
	&=\frac13\.\norm{F}^4-\frac16\.(\det F)^2-\frac23\.\det F\cdot\norm{F}^2
	\\&= \frac13(\lambda_1^4+\lambda_2^4) + \frac12\lambda_1^2\lambda_2^2 - \frac23(\lambda_1^3\lambda_2+\lambda_1\lambda_2^3)\notag
\end{align}
for all $F\in\GLp(2)$ with (not necessarily ordered) singular values $\lambda_1,\lambda_2$.

\begin{figure}[h!]\begin{center}
		\begin{minipage}[t]{0.45\linewidth}
			\centering
			\includegraphics[height=.8\textwidth]{WA1}

			\caption{The sublevels of the energy $W_{\mathrm{A}}$ on $\GLp(2)$ are connected but not compact.}
			\label{Fig3}
		\end{minipage}
		\hfill
		\begin{minipage}[t]{0.450\linewidth}
			\centering
			\includegraphics[height=.8\textwidth]{WA2}

			\caption{The sublevels of the extension of $W_{\mathrm{A}}$ to $\R^{2\times 2}$ are neither connected nor compact.}
			\label{Fig4}
			\label{fig:firstDiagonalFigure}
		\end{minipage}
	\end{center}
\end{figure}

However, the function \eqref{eq:aubertEnergy} is not a counterexample to Conjecture \ref{conjecture:mielke}, since the sublevel sets are not compact;\footnote{In Appendix \ref{appendix:aubertConnectedness}, we show that the sublevel sets are indeed connected.} simply note that
\begin{equation}\label{eq:nonCompactExample}
	W_{\mathrm{A}}(\diag(t,t)) = -\frac{t^4}{6}\,,
\end{equation}
i.e.\ that each sublevel set contains an (unbounded) set of the form $\{a\cdot\id\setvert a>d\}$ for some $d\geq0$, as visualized\footnote{%
	Figs.~\ref{Fig3}, \ref{Fig4} and \ref{fig:firstDiagonalFigureInBlock}--\ref{fig:lastDiagonalFigure} show the graphs and sublevel set contours for different energies applied to \emph{diagonal} matrices, i.e.\ visualize the mappings $(x,y)\mapsto W(\diag(x,y))$ for some isotropic energy function $W$, with $(x,y)\in\R^2$ if $W$ is defined on $\R^{2\times2}$ or $(x,y)\in\Rp^2$ if $W$ is defined on the domain $\GLp(2)$. In the latter case, the figures can equivalently be interpreted as showing $W$ in terms of (unordered) singular values. Here and throughout, lower sublevel sets are represented by darker regions.%
}
in Figs.~\ref{Fig3} and \ref{Fig4}. Indeed, such negative growth conditions are often employed to simplify the otherwise cumbersome task of showing that a function is not polyconvex \cite{ball1976convexity,Dacorogna08,alibert1992example} (cf.~\cite[Remark (1)]{aubert1987counterexample}). Therefore, the examples of rank-one convex functions which are not polyconvex encountered in the literature are usually not suitable to falsify Conjecture \ref{conjecture:mielke}.

This observation is highlighted further by another important example (which is a homogeneous polynomial expression of grade 4 as well): for the family of isotropic energies
\begin{equation}\label{eq:dacorognaEnergy}
	W_{\mathrm{ADM}}^\gamma\col\R^{2\times2}\to\R\,,\qquad W_{\mathrm{ADM}}^\gamma(F) = \norm{F}^2\.(\norm{F}^2-2\gamma\.\det F)\,,\qquad \gamma\in\R\,,
\end{equation}
it was shown by Alibert, Dacorogna and Marcellini \cite{alibert1992example,dacorogna1988counterexample} that
\begin{align*}
	W_{\mathrm{ADM}}^\gamma \quad \text{is convex on } \R^{2\times2}\quad &\iff \quad \abs{\gamma}\leq \frac{2\sqrt{2}}{3}\approx 0.942809\,,\\[-0.3em]
	W_{\mathrm{ADM}}^\gamma\quad \text{is polyconvex on } \R^{2\times2}\quad &\iff \quad\abs{\gamma}\leq 1\,,\vphantom{\frac23}\\[-0.3em]
	W_{\mathrm{ADM}}^\gamma \quad \text{is quasiconvex on } \R^{2\times2}\quad &\iff \quad\abs{\gamma}\leq \gamma_q \ \ \text{with}\ \ \gamma_q>1\,,\vphantom{\frac23}\\[-0.3em]
	W_{\mathrm{ADM}}^\gamma \quad \text{is rank-one convex on } \R^{2\times2}\quad &\iff \quad \abs{\gamma}\leq \frac{2}{\sqrt{3}}\approx 1.1547
	\,.
\end{align*}
Again, in the non-polyconvex case $\abs{\gamma}>1$, the sublevel sets are not compact, since $W_{\mathrm{ADM}}(a\cdot\id)=4\.a^4(1-\gamma)<0$ for all $a>0$ if $\gamma>1$ and $W_{\mathrm{ADM}}(\diag(a,-a))=4\.a^4(1+\gamma)<0$ for all $a>0$ if $\gamma<-1$, where $\diag(x,y)\in\R^{2\times2}$ denotes the diagonal matrix with diagonal entries $x,y\in\R$.

\tikzset{
	state/.style={
	minimum height = 0.55cm,
	outer sep = 0,
	inner sep = 0,
	black
	}
}
\begin{figure}[h!]
\centering
\small
\begin{tikzpicture}
	\node[state](1){$0$};
	\node[right of=1, node distance = 4cm, state](2){$\frac{2\sqrt{2}}{3}$};
	\node[right of=1, node distance = 6.5cm, state](3){$1$};
	\node[right of=1, node distance = 9cm, state](4){$\gamma_q$};
	\node[right of=1, node distance = 12cm, state](5){$\frac{2}{\sqrt{3}}$};
	\node[right of=1, node distance = 14.5cm, state](6){$\abs\gamma$};
	\path[very thick, udedarkblue]
		(1) edge ++(0,-2cm)
		(2) edge ++(0,-1.1cm)
		(3) edge ++(0,-1.4cm)
		(4) edge ++(0,-1.7cm)
		(5) edge ++(0,-2cm)
		(1)++(0,-0.4cm) edge[->] ($(6)+(0.3cm,-0.4cm)$)
		(1)++(0,-1cm) edge[<->] node[black,pos=0.5,above]{convex} ($(2)-(0,1cm)$)
		(1)++(0,-1.3cm) edge[<->] node[black,pos=0.82,above]{polyconvex} ($(3)-(0,1.3cm)$)
		(1)++(0,-1.6cm) edge[<->] node[black,pos=0.86,above]{quasiconvex} ($(4)-(0,1.6cm)$)
		(1)++(0,-1.9cm) edge[<->] node[black,pos=0.88,above]{rank-one convex} ($(5)-(0,1.9cm)$)
	;
\end{tikzpicture}
\caption{A schematic overview for the Alibert-Dacorogna-Marcellini family of energies. It is currently unknown whether $\gamma_q<\frac{2}{\sqrt{3}}$.}\label{schita}
\end{figure}

Now, for $\gamma\in\R$, consider the restriction
\begin{equation}\label{eq:dacorognaEnergy2}
	W_{\mathrm{ADM}^+}^\gamma = W_{\mathrm{ADM}}^\gamma\big|_{\GLp(2)}\col\GLp(2)\to\R\,,\quad W_{\mathrm{ADM}}^\gamma(F) = \norm{F}^2\.(\norm{F}^2-2\gamma\.\det F)
\end{equation}
of $W_{\mathrm{ADM}}^\gamma$ to $\GLp(2)$. Then $W_{\mathrm{ADM}^+}^\gamma$ can be expressed in terms of (unordered) singular values $\lambda_1,\lambda_2$ of $F\in\GLp(2)$ via
\begin{equation}\label{eq:dacorognaEnergy2singularValues}
	W_{\mathrm{ADM}}^\gamma(F)=(\lambda_1^2+\lambda_2^2)^2-2 \,\gamma\,(\lambda_1^2+\lambda_2^2) \,\lambda_1 \lambda_2\,.
\end{equation}
In this case, the sublevel sets are not compact for any $\gamma\in\R$; observe that for every $c>0$, since
\[
	\lim_{n\to\infty}\,W_{\mathrm{ADM}^+}\Bigl(\frac1n\cdot\id\Bigr)
	= \lim_{n\to\infty}\,\frac{4\.(1-\gamma)}{n^4}
	= 0\,,
\]
we find $\frac1n\cdot\id\in S_c$ for all sufficiently large $n\in\N$, thus the sublevel set $S_c$ contains a sequence without a subsequence which converges in $\GLp(2)$.

\begin{remark}
\label{remark:compactnessGrowthCondition}
	From generalizing the above argument, it follows that the compactness of sublevel sets for an energy on $\GLp(2)$ requires $W$ to satisfy the growth condition $W(F)\to\infty$ for $\det F\to0$. In particular, this condition is never satisfied by any polynomial function $W$, which immediately excludes the class of polynomial energies as possible counterexamples to Conjecture \ref{conjecture:mielke}.
\end{remark}

\begin{figure}[h!]\centering
	\begin{minipage}[t]{0.45\linewidth}
		\centering
		\includegraphics[height=.8\textwidth]{WADM_1_1}

		\caption{The sublevels of $W_{\mathrm{ADM}^+}^{1.1}$ are connected, but unbounded and thus not compact.}
		\label{DM2}
		\label{fig:firstDiagonalFigureInBlock}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.45\linewidth}
		\centering
		\includegraphics[height=.8\textwidth]{WADM_0_97}

		\caption{The sublevels of $W_{\mathrm{ADM}^+}^{0.97}$, which are connected and bounded but not compact.}
		\label{DM1}
	\end{minipage}
\end{figure}

Another example of an $\OO(2)$-invariant energy which is rank-one convex but not polyconvex was introduced by \v{S}ilhav\'{y} \cite{silhavy2002invariant}, who proposed the function $W_{\mathrm{S}}\col\R^{2\times 2}\to \R$ given by
\[
	W_{\mathrm{S}}\col\R^{2\times 2}\to \R\,,\quad
	W_{\mathrm{S}}(F)
	=\ghat_{\mathrm{S}}(\lambdahat(F))
	\qquad\text{with}\qquad
	\ghat_{\mathrm{S}}\col\Oset[2]\to\R\,,\quad
	\ghat_{\mathrm{S}}(\lambdahat_1,\lambdahat_2)
	=\begin{cases}
		\lambdahat_1\lambdahat_2 &:\lambdahat_1\leq 1\\
		\lambdahat_1+\lambdahat_2 -1&: \lambdahat_1\geq 1\\
	\end{cases}
\]
in terms of ordered singular values $\lambdahat_1,\lambdahat_2$. Again, if we consider the restriction $W_{\mathrm{S}^+}=W_{\mathrm{S}}\restrict{\GLp(2)}$ of $W_{\mathrm{S}}$ to $\GLp(2)$, then the sublevel sets (cf.~Figs.~\ref{FigS3} and \ref{FigS4}) are generally not compact due to the boundedness of $W_{\mathrm{S}^+}(F)$ for $\det F\to0$.

\begin{figure}[h!]\begin{center}
		\begin{minipage}[t]{0.54\linewidth}
			\centering
			\includegraphics[width=1\textwidth]{WS1}

			\caption{The energy $W_{\mathrm{S}^+}(F)$ on $\GLp(2)$ in terms of singular values of $F$.}
			\label{FigS3}
		\end{minipage}
		\hfill
		\begin{minipage}[t]{0.41\linewidth}
			\centering
			\includegraphics[height=0.85\textwidth]{WS2}

			\caption{The sublevels of the energy $W_{\mathrm{S}^+}$ on $\GLp(2)$ are connected, but not compact.}
			\label{FigS4}
		\end{minipage}
	\end{center}
\end{figure}

\pagebreak
\section{The counterexample}\label{section:counterExample}

In the following, we will show that the energy function
\begin{align}
\label{eq:counterexampleDefinition}
	W_0\col\GLp(2)\to\R\,,\qquad W_0(F) &= \frac{\opnorm{F}^2}{\det F} - \log\biggl(\frac{\opnorm{F}^2}{\det F}\biggr) + \log(\det F) + \frac{1}{\det F}\\
	&= \frac{\lambdamax}{\lambdamin} - \log\left(\frac{\lambdamax}{\lambdamin}\right) + \log(\lambdamax\lambdamin) + \frac{1}{\lambdamax\lambdamin}\,\nonumber
\end{align}
is not polyconvex, although it satisfies all the conditions posed in Conjecture \ref{conjecture:mielke}, i.e.\ $W_0$ is isotropic and rank-one convex with connected, compact sublevel sets; here and in the following, we denote the singular values of $X\in\R^{2\times2}$ by $\lambdamax(X)\geq\lambdamin(X)$ and simply write $\lambdamax,\lambdamin$ for the ordered singular values of the energy's argument $F$, while $\opnorm{X}=\sup_{\norm{h}=1}\norm{X\, h}=\lambdamax(X)$ denotes the operator norm of any $X\in\R^{2\times2}$.

In order to establish that the desired properties hold for $W_0$, we will require a number of auxiliary results concerning the sublevel sets of isotropic functions with a so-called \emph{volumetric-isochoric split}, i.e.\ functions of the form
\begin{equation}\label{eq:volIsoSplitGeneralForm}
	W\col\GLp(2)\to\R\,,\qquad W(F) = W_{\textrm{\rm iso}}(F) + W_{\textrm{\rm vol}}(\det F)
\end{equation}
with $W_{\textrm{\rm iso}}\col\GLp(2)\to\R$ and $W_{\textrm{\rm vol}}\col\Rp=(0,\infty)\to\R$ such that $W_{\textrm{\rm iso}}$ is \emph{isochoric}, i.e.\ satisfies $W(aF)=W(F)$ for all $a>0$. If $W_{\textrm{\rm iso}}$, and therefore $W$, is objective and isotropic, then any function $W$ of the form \eqref{eq:volIsoSplitGeneralForm} can be expressed as \cite{agn_martin2015rank}
\begin{equation}\label{eq:volIsoSplitGeneralFormSingularValues}
	W(F) = \hhat(K(F)) + f(\det F) = \underbrace{\hhat\left(\frac{\lambdamax}{\lambdamin}\right)}_{=W_{\textrm{\rm iso}}(F)} + \underbrace{f(\lambdamax\lambdamin) \vphantom{\left(\frac{\lambdamax}{\lambdamin}\right)}}_{=W_{\textrm{\rm vol}}(\det F)}
\end{equation}
for all $F\in\GLp(2)$ with singular values $\lambdamax\geq\lambdamin>0$ and with uniquely defined functions $\hhat\col[1,\infty)\to\R$ and $f\col(0,\infty)\to\R$. Here,
\begin{equation}\label{eq:Kdefinition}
	K\col\GLp(2)\to\R\,,\qquad K(F) \colonequals \frac{\opnorm{F}^2}{\det F} = \frac{\lambdamax}{\lambdamin}
\end{equation}
is the \emph{linear distortion} (or \emph{dilation}) function, which plays an important role in the theory of conformal and quasiconformal mappings \cite{astala2008elliptic} as well as for the characterization of rank-one convexity, quasiconvexity and polyconvexity of isochoric planar energy functions \cite{agn_martin2015rank,agn_martin2019envelope}. More specifically, the isochoric part $W_{\textrm{\rm iso}}$ of an energy $W$ of the form \eqref{eq:volIsoSplitGeneralForm} is rank-one convex, quasiconvex and polyconvex if and only if the function $\hhat\col[1,\infty)\to\R$ given by \eqref{eq:volIsoSplitGeneralFormSingularValues} is nondecreasing and convex \cite[Theorem 3.3]{agn_martin2015rank}. This equivalence between polyconvexity and rank-one convexity does not hold for the full energy $W$ itself, as the example $W_0$ will show.

\begin{figure}[h!]\begin{center}
		\begin{minipage}[t]{0.54\linewidth}
			\centering
			\includegraphics[width=1\textwidth]{W01}

			\caption{The energy function $W_0$ in terms of singular values.}
			\label{FigW1}
		\end{minipage}
		\hfill
		\begin{minipage}[t]{0.35\linewidth}
			\centering
			\includegraphics[height=1\textwidth]{W02}

			\caption{The sublevels of the energy $W_0$ are compact and connected.}
			\label{FigW2}
			\label{fig:lastDiagonalFigure}
		\end{minipage}
	\end{center}
\end{figure}

It is clear that $W_0$ from \eqref{eq:counterexampleDefinition} can be expressed in the form \eqref{eq:volIsoSplitGeneralForm} with
\begin{equation}\label{eq:counterexampleIsoPart}
	 W_{\textrm{\rm iso}}(F) = K(F) - \log(K(F)) = \frac{\lambdamax}{\lambdamin} - \log\left(\frac{\lambdamax}{\lambdamin}\right)
\end{equation}
and
\begin{equation}\label{eq:counterexampleVolPart}
	 W_{\textrm{\rm vol}}(\det F) = \log(\det F) + \frac{1}{\det F} = \log(\lambdamax\lambdamin) + \frac{1}{\lambdamax\lambdamin}
\end{equation}
or, equivalently, in the form \eqref{eq:volIsoSplitGeneralFormSingularValues} with
\begin{equation}\label{eq:counterexampleVolIsoPartSingularValues}
	\hhat(t) = t-\log(t) \qquad\text{and}\qquad f(t)=\frac1t-\log\left(\frac{1}{t}\right)\,.
\end{equation}

As shown in Figs.\ \ref{FigW1} and \ref{FigW2}, the growth behaviour of $W_0(F)$ for $\det F\to0$ clearly distinguishes $W_0$ from the examples considered in Section \ref{section:previousExamples}.

\subsection{Auxiliary results for volumetric-isochorically split energy functions}

In order to establish $W_0$ as a counterexample to Conjecture \ref{conjecture:mielke}, we first state a sufficient criterion for the compactness of the sublevel sets for energy functions with a volumetric-isochoric split.

\begin{lemma}\label{lemma:compactness}
	Let $W\col\GLp(2)\to\R$ be of the form \eqref{eq:volIsoSplitGeneralFormSingularValues} such that $\hhat,f$ are lower semicontinuous and
	\begin{equation}\label{eq:compactnessGrowthConditions}
		\lim_{t\to\infty}\hhat(t)=\lim_{t\to\infty}f(t)=\lim_{t\to0}f(t)=\infty\,.
	\end{equation}
	Then every sublevel set of $W$ is compact.
\end{lemma}
\begin{proof}
	Since all sublevel sets of lower semicontinuous functions are closed (in this case relative to $\GLp(2)$), we only need to show that for each $c\in\R$, the corresponding sublevel set
	\[
		S_c \colonequals \{F\in\R^{2\times2} \setvert \det F > 0\,,\; W(F) \leq c\}
	\]
	is bounded and that $\dist(S_c,\partial\GLp(2))>0$. In order to simplify the computations, we will consider the distance on $\R^{2\times2}$ with respect to the operator norm, given by the largest singular value $\opnorm{X}=\lambdamax(X)$ of a matrix $X\in\R^{2\times2}$. First note that
	\begin{equation}\label{eq:opnormKdet}
		\opnorm{F}^2 = \frac{\lambdamax}{\lambdamin}\cdot\lambdamax\lambdamin = K(F)\cdot\det(F)
	\end{equation}
	and that the lower semicontinuity of $\hhat$ and $f$, together with \eqref{eq:compactnessGrowthConditions}, ensures that both $\hhat$ and $f$ are bounded below by some $d\in\R$.

	Now, let $c\in\R$. Then due to \eqref{eq:compactnessGrowthConditions}, there exists $r\geq1$ such that $\hhat(t)>c-d$ and $f(t)>c-d$ for any $t>r$. Since $\opnorm{F}^2>r^2$ implies that either $K(F)>r$ or $\det(F)>r$ by virtue of \eqref{eq:opnormKdet},
	\[
		W(F) = \hhat(K(F)) + f(\det(F)) > c-d+d = c
	\]
	for any $F\in\GLp(2)$ with $\opnorm{F}>r$. Thus $S_c$ is bounded.

	In order to establish a lower bound for $\dist(S_c,\partial\GLp(2))$, we first observe that, according to the Eckart-Young-Mirsky Theorem \cite{mirsky1960symmetric},\footnote{%
		Recall that the distance is taken with respect to the operator norm. Equality \eqref{eq:rankOneApproximation} can also easily be shown directly: on the one hand, for any $X\in\R^{2\times2}$ with $\rank(X)\leq1$ there exists $\xi\in\R^2$ with $X\xi=0$ and $\norm{\xi}=1$, thus $\opnorm{F-X}\geq\norm{(F-X)\xi}=\norm{F\xi}\geq\lambdamin$; on the other hand, $\opnorm{Q_1\diag(\lambdamax,\lambdamin)Q_2-Q_1\diag(\lambdamax,0)Q_2}=\lambdamin$ for $Q_1,Q_2\in\SO(2)$.%
	}
	\begin{align}
		\dist(F,\partial\GLp(2))
		&= \dist(F,\{X\in\R^{2\times2}\setvert\det(X)=0\}) \nonumber\\
		&= \dist(F,\{X\in\R^{2\times2}\setvert\rank(X)\leq1\})
		= \lambdamin(F)\,. \label{eq:rankOneApproximation}
	\end{align}
	Due to \eqref{eq:compactnessGrowthConditions}, there exists $r\leq1$ such that $\hhat(s)>c-d$ and $f(t)>c-d$ for all $t<r$ and all $s>\frac1r$. Then for any $F\in\GLp(2)$ with $\lambdamin<r$,
	\[
		r^2 > \lambdamin^2 = \frac{\lambdamax\lambdamin}{\frac{\lambdamax}{\lambdamin}} = \frac{\det(F)}{K(F)}
	\]
	and thus either $\det(F)>r$ or $K(F)<\frac1r$. Again, $W(F)>c$ in either case; therefore, $F\notin S_c$ for any $F\in\GLp(2)$ with $\dist(F,\partial\GLp(2)) = \lambdamin(F) < r$, which shows that $\dist(S_c,\partial\GLp(2))\geq r >0$.
\end{proof}

We also require a criterion for the sublevel sets of $W$ to be connected. Recall from Definition \ref{definition:q-Convexity} that a function is called q-convex if all its sublevel sets are convex.

\begin{lemma}\label{lemma:connectedness}
	Let $W\col\GLp(2)\to\R$ be of the form \eqref{eq:volIsoSplitGeneralFormSingularValues} such that $\hhat\col[1,\infty)\to \R$ is monotone and $f$ is q-convex. Then every sublevel set of $W$ is connected.
\end{lemma}
\begin{proof}
\newcommand{^{\raisebox{-.245em}{\!\text{\normalsize$s$}}}}{^{\raisebox{-.245em}{\!\text{\normalsize$s$}}}}
	In order to show that the sublevel set $S_c$ is (path) connected for any $c\in\R$ under the stated conditions, we will explicitly construct a curve connecting arbitrary $F,\Ftilde\in S_c$. The construction will be split into four parts, with the first and last one describing a continuous, orthogonal basis change. The second part will be constructed such that the linear distortion $K$ is decreasing and the determinant is constant along the curve, while for the third part, the linear distortion $K$ is kept constant. This will allow us to utilize the monotonicity of the isochoric part $\hhat$ and the q-convexity of the volumetric part $f$, respectively, to show that the curve remains in the sublevel set (cf.\ Figure \ref{fig:curveWithinSublevelsCounterexample}).

	For $c\in\R$, let $F,\Ftilde\in S_c$ and assume without loss of generality that $K(\Ftilde) \leq K(F)$. First, choose $Q_1,Q_2,\Qtilde_1,\Qtilde_2\in\SO(2)$ such that $\diag(\lambda_1,\lambda_2)\colonequals Q_1FQ_2$ and $\diag(\lambdatilde_1,\lambdatilde_2)\colonequals \Qtilde_1\Ftilde\Qtilde_2$ are diagonal with $\lambda_1\geq\lambda_2>0$ and $\lambdatilde_1\geq\lambdatilde_2>0$. Let
	\begin{equation}
		X_1\col[0,1]\to\GLp(2)\,,\qquad X_1(s) = Q_1^s FQ_2^s\,;
	\end{equation}
	here, $Q^s\colonequals \exp(s\log Q)$ for $Q\in\SO(2)$, where $\log$ denotes the principal matrix logarithm on $\SO(2)$ or, more explicitly,
	\begin{equation}
		Q^s = \matr{\cos(s\.\alpha)&\sin(s\.\alpha)\\-\sin(s\.\alpha)&\cos(s\.\alpha)} \qquad\text{for}\qquad Q = \matr{\cos(\alpha)&\sin(\alpha)\\-\sin(\alpha)&\cos(\alpha)} \quad\text{with }\;\alpha\in(-\pi,\pi]\,.
	\end{equation}
	In particular, $X_1$ is continuous with $X_1(0)=F$ and $X_1(1)=\diag(\lambda_1,\lambda_2)$. Furthermore, $s\mapsto W(X_1(s))$ is constant due to the objectivity and isotropy of $W$ and thus, in particular, $X_1(s)\in S_c$ for all $s\in[0,1]$.

	Now, for $\mu_1=\frac{\sqrt{\lambda_1\.\lambda_2}}{\sqrt{\lambdatilde_1\.\lambdatilde_2}}\,\lambdatilde_1$ and $\mu_2=\frac{\sqrt{\lambda_1\.\lambda_2}}{\sqrt{\lambdatilde_1\.\lambdatilde_2}}\,\lambdatilde_2$, let
	\begin{equation}
		X_2\col[0,1]\to\GLp(2)\,,\qquad X_2(s) = \diag(\lambda_1^{1-s}\.\mu_1^s,\lambda_2^{1-s}\.\mu_2^s)\,.
	\end{equation}
	Then $X_2$ is continuous with $X_2(0)=\diag(\lambda_1,\lambda_2)=X_1(1)$ and $X_2(1)=\diag(\mu_1,\mu_2)$. Furthermore,
	\[
		\det(X_2(s)) = \lambda_1^{1-s}\.\mu_1^s\.\lambda_2^{1-s}\.\mu_2^s = \lambda_1\lambda_2\cdot \biggl(\frac{\mu_1\mu_2}{\lambda_1\lambda_2}\biggr)^{\raisebox{-.245em}{\!\text{\normalsize$s$}}} = \lambda_1\lambda_2\cdot \left(\frac{\frac{\sqrt{\lambda_1\.\lambda_2}}{\sqrt{\lambdatilde_1\.\lambdatilde_2}}\,\lambdatilde_1\cdot\frac{\sqrt{\lambda_1\.\lambda_2}}{\sqrt{\lambdatilde_1\.\lambdatilde_2}}\,\lambdatilde_2}{\lambda_1\lambda_2}\right)^{\raisebox{-.245em}{\!\text{\normalsize$s$}}} = \lambda_1\lambda_2
	\]
	is independent of $s$ and, since $\frac{K(\Ftilde)}{K(F)}\leq1$ by assumption, the mapping
	\[
		s\mapsto K(X_2(s)) = \frac{\lambda_1^{1-s}\.\mu_1^s}{\lambda_2^{1-s}\.\mu_2^s} = \frac{\lambda_1}{\lambda_2}\cdot \biggl(\frac{\lambda_2\.\mu_1}{\lambda_1\.\mu_2}\biggr)^{\raisebox{-.245em}{\!\text{\normalsize$s$}}} = \frac{\lambda_1}{\lambda_2}\cdot \left(\frac{K(\Ftilde)}{K(F)}\right)^{\raisebox{-.245em}{\!\text{\normalsize$s$}}}
	\]
	is nonincreasing on $[0,1]$. Therefore, the assumed monotonicity of the function $\hhat$ implies that the mapping $s\mapsto W(X_2(s))=h(K(X_2(s)))+f(\det(X_2(s)))$ is nonincreasing, hence $X_2(s)\in S_c$ for all $s\in[0,1]$.

	Next, let
	\[
		X_3\col[0,1]\to\GLp(2)\,,\qquad X_3(s) = \left(\frac{\sqrt{\lambdatilde_1\.\lambdatilde_2}}{\sqrt{\lambda_1\.\lambda_2}}\right)^{\raisebox{-.245em}{\!\text{\normalsize$s$}}}\cdot\,\diag(\mu_1,\mu_2)\,.
	\]
	Then $X_3$ is continuous with $X_3(0)=\diag(\mu_1,\mu_2)=X_2(1)$ and $X_3(1)=\diag(\lambdatilde_1,\lambdatilde_2)$.
	We observe that $K$ is constant along $X_3$ and that
	\[
		\min\{\det(X_3(0))\,, \det(X_3(1))\}
		\leq\det(X_3(s))
		\leq \max\{\det(X_3(0))\,, \det(X_3(1))\}
	\]
	for all $s\in[0,1]$. Furthermore, the volumetric part $f\col(0,\infty)\to\R$ is q-convex by assumption which, in particular \cite{crouzeix1982criteria}, implies that $f(t)\leq \max\{f(a),\,f(b)\}$ for any $0<a\leq t \leq b$. We therefore find
	\[
		f(\det (X_3(s))) \leq \max\{f(\det(X_3(0) ))\,,\;f(\det(X_3(1)))\}
	\]
	and thus, since $s\mapsto K(X_3(s))$ is constant (i.e.\ $K(X_3(s))=K(X_3(0))=K(X_3(1))$ for all $s\in[0,1]$),
	\begin{align*}
		W(X_3(s)) &= f(\det (X_3(s))) + \hhat(K(X_3(s)))\\
		&\leq \max\{f(\det(X_3(0))) + \hhat(K(X_3(s)))\,,\; f(\det(X_3(1))) + \hhat(K(X_3(s)))\}\\
		&= \max\{f(\det(X_3(0))) + \hhat(K(X_3(0)))\,,\; f(\det(X_3(1))) + \hhat(K(X_3(1)))\}\\
		&= \max\{W(X_3(0))\,,\; W(X_3(1))\}
		\;\leq\; c
	\end{align*}
	for all $s\in[0,1]$.

	Finally, the curve
	\[
		X_4\col[0,1]\to\GLp(2)\,,\qquad X_4(s) = (\Qtilde_1^T)^s \diag(\lambdatilde_1,\lambdatilde_2)(\Qtilde_2^T)^s
	\]
	continuously connects $\diag(\lambdatilde_1,\lambdatilde_2)$ with $\Ftilde$ such that $W$ is constant along $X_4$. Therefore, the combined curve
	\[
		X\col[0,4]\to\GLp(2)\,,\qquad X(s) = X_i(s)
		\quad\text{ for }\; s\in[i-1,i]\,,
		\;\;i\in\{1,2,3,4\}
	\]
	is (well defined and) continuous with $X(0)=F$, $X(4)=\Ftilde$ and $X(s)\in S_c$ for all $s\in[0,4]$.
\end{proof}

\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
		\def3.5{3.5}
		\def1.26{.735}
			\begin{axis}[
				axis x line=middle,axis y line=middle,
				x label style={at={(current axis.right of origin)},anchor=north, below},
				xlabel={$\lambda_1$}, ylabel={$\lambda_2$},
				xmin=-0.07, xmax=4.9,
				ymin=-0.07, ymax=4.9,
				xtick={3.43},
				ytick={3.185},
				xticklabels={$\mu_1$},
				yticklabels={$\mu_2$},
				hide obscured x ticks=false,
				width=.49\linewidth,
				height=.49\linewidth
			]
				%
				\addplot[domain=0:4.41,dotted,thick] {x} node[pos=.63, above left] {$\lambda_1=\lambda_2$};
				%
				\addplot[domain=.343:sqrt(3.5/1.26),blue,dashed,thick] {3.5/x}
					node[pos=.7, above right] {$\lambda_1\lambda_2\equiv \det(F)$};
				\addplot[domain=sqrt(3.5/1.26):3.43,<-,ultra thick,blue] {3.5/x}
					node[pos=1.001,draw,circle,fill, inner sep=0.147em]{}
					node[pos=1.001, above right] {$F$}
					node[pos=.49, below left] {$X_2$};
				\addplot[domain=3.43:4.9,blue,dashed,thick] {3.5/x};
				%
				\addplot[domain=0:sqrt(3.5/1.26),red,dashed,thick] {1.26*x}
					node[pos=.392, below right] {$\frac{\lambda_1}{\lambda_2}\equiv K(\Ftilde)$};
				\addplot[domain=sqrt(3.5/1.26):4.312,->,ultra thick,red] {1.26*x}
					node[pos=1.029,draw,circle,fill, inner sep=0.147em]{}
					node[pos=1.029, below right] {$\Ftilde$}
					node[pos=.49, below right] {$X_3$};
				\addplot[domain=4.312:4.9,red,dashed,thick] {1.26*x};
				%
			\end{axis}
		\end{tikzpicture}
		%
		\hfill
		%
		\begin{tikzpicture}
				\def1.26{1.26}
		\def3.5{3.5}
			\begin{axis}[
				axis x line=middle,axis y line=middle,
				x label style={at={(current axis.right of origin)},anchor=north, below},
				xlabel={$s$}, ylabel={$W(X(s))$},
				xmin=-0.07, xmax=4.13,
				ymin=-.07, ymax=.847,
				xtick={1,2,3,4},
				ytick=\empty,
				hide obscured x ticks=false,
				width=.588\linewidth,
				height=.49\linewidth
			]
				%
				\addplot[domain=0:1.001, thick] {.49}
					node[pos=.49,below] {$X_1$};
				%
				\addplot[domain=1.001:2.002, thick,blue] {.49-0.343*(x-1.001)^2}
					node[pos=.63, below left] {$X_2$};
				%
				\addplot[domain=2.002:2.506, thick,red] {.49-0.343*(1.001)^2-.147*(x-2.002)};
				%
				\addplot[domain=2.506:3.003, thick,red] {.49-0.343*(1.001)^2-.147*(.504)+3.43*((x-2.1)^(1/7)-(2.506-2.1)^(1/7))}
					node[pos=.49, above left] {$X_3$};
				%
				\addplot[domain=3.003:3.997, thick] {.49-0.343*(1.001)^2-.147*(.504)+3.43*((3.003-2.1)^(1/7)-(2.506-2.1)^(1/7))}
					node[pos=.49, below] {$X_4$};
				%
			\end{axis}
		\end{tikzpicture}
		\vspace*{-1.47em}
	\par\end{center}
	\caption{%
	\label{fig:curveWithinSublevelsCounterexample}
		Along $X_2$, the linear distortion $K=\frac{\lambda_1}{\lambda_2}$ is decreasing and the determinant is constant, so the energy is nonincreasing; along $X_3$, $K$ is constant, so the curve does not leave the sublevel set due to the q-convexity of the energy with respect to the determinant.%
	}
\end{figure}

Of course, in order to show that $W_0$ given by \eqref{eq:counterexampleDefinition} is a counterexample to Conjecture \ref{conjecture:mielke}, the rank-one convexity of $W_0$ must be established as well. The following criterion, which is based on an earlier result by Knowles and Sternberg \cite{knowles1976failure,knowles1978failure} for twice differentiable functions (cf.\ Appendix \ref{section:knowlesSternberg}) and which is applicable to volumetric-isochorically split energies in the planar case, has been the subject of a recent contribution \cite{agn_voss2019volIsLog}.

\begin{lemma}[\cite{agn_voss2019volIsLog}]\label{lemma:rankOneConvexity}
	For $f,h\in C^2((0,\infty))$ with $h\left(\frac1t\right)=h(t)$, let $W(F)=h\bigl(\frac{\lambda_1}{\lambda_2}\bigr)+f(\lambda_1\lambda_2)$ for all $F\in\GLp(2)$ with singular values $\lambda_1,\lambda_2$. Then $W$ is rank-one convex on $\GLp(2)$ if and only if
	%
	\begin{itemize}
		\item[i)] \quad$\displaystyle h_0+f_0\geq 0$,
		\item[ii)] \quad$\displaystyle h'(t)\geq 0\qquad\text{for all }\; t\geq 1$,
		\item[iii)]\quad$\displaystyle \frac{2t}{t-1}\.h'(t)-t^2\.h''(t)+f_0\geq 0$\quad or\quad $\displaystyle a(t)+\left[b(t)-c(t)\right]f_0\geq 0\qquad\text{for all }\; t\in(0,\infty)\,,\;t\neq 1$,
		\item[iv)]\quad$\displaystyle \frac{2t}{t+1}\.h'(t)+t^2\.h''(t)-f_0\geq 0\quad\text{or}\quad a(t)+\left[b(t)+c(t)\right]f_0\geq 0\qquad\text{for all }\; t\in(0,\infty)$,
	\end{itemize}
	where
	\begin{align*}
		a(t)&=t^2(t^2-1)\.h'(t)h''(t)-2t\.h'(t)^2\,,\qquad b(t)=\left(t^2+3\right)h'(t)+2t\.(t^2+1)\.h''(t)\,,\\
		c(t)&=4t\left(h'(t)+t\.h''(t)\right)
	\end{align*}
	and
	\[
		h_0=\inf_{t\in(0,\infty)}t^2\.h''(t)\,,\qquad f_0=\inf_{t\in(0,\infty)}t^2\.f''(t)\,.\directqed
	\]
\end{lemma}
Note that the function $h\col(0,\infty)\to\R$, which represents the isochoric part of $W$ in terms of the ratio $\frac{\lambda_1}{\lambda_2}$ of the (not necessarily ordered) singular values, can be obtained from $\hhat\col[1,\infty)\to\R$ as given in \eqref{eq:volIsoSplitGeneralFormSingularValues} by setting
\begin{equation}\label{ht}
	h(t) \colonequals
	\begin{cases}
		\hhat(t) &: t\geq1\,,\\
		\hhat\left(\frac1t\right) &: t<1\,.
	\end{cases}
\end{equation}

\begin{remark}
	The energy $W_0$ can be written in the form
	\[
		W_0(F)=h\left(\frac{\lambdamax}{\lambdamin}\right)+h\left(\frac{1}{\lambdamax\lambdamin}\right)\,,
	\]
	where $h\col(0,\infty)\to\R$ is the function related to $\hhat\col[1,\infty)\to\R$ with $\hhat(t)=t-\log(t)$ by \eqref{ht}.
\end{remark}
\subsection{Main properties of the counterexample}%
\label{section:assemblingTheProof}

We can now prove Proposition \ref{prop:conjectureFalse}, i.e.\ demonstrate that Conjecture \ref{conjecture:mielke} does not hold, by showing that $W_0$ is indeed a counterexample.
\begin{proposition}
	The function
	\begin{equation}\label{eq:counterexampleRepetition}
		W_0(F) = \frac{\lambdamax}{\lambdamin} - \log\left(\frac{\lambdamax}{\lambdamin}\right) + \log(\lambdamax\lambdamin) + \frac{1}{\lambdamax\lambdamin} = \hhat(K(F)) + f(\det F)
	\end{equation}
	with
	\[
		\hhat(t) = t-\log(t) \qquad\text{and}\qquad f(t)=h\left(\frac{1}{t}\right)=\log(t)+\frac1t\,,
	\]
	where $\lambdamax\geq\lambdamin>0$ are the singular values of $F\in\GLp(2)$, is rank-one convex and isotropic with compact and connected sublevel sets, but not polyconvex.
\end{proposition}
\begin{corollary*}[Proposition \ref{prop:conjectureFalse}]
	Conjecture \ref{conjecture:mielke} does not hold.
\end{corollary*}
\begin{proof}
	The function $W_0$ is obviously objective and isotropic. Since $\hhat$ and $f$ are lower semicontinuous with $\lim_{t\to\infty}\hhat(t)=\lim_{t\to\infty}f(t)=\lim_{t\to0}f(t)=\infty$, Lemma \ref{lemma:compactness} shows that the sublevel sets of $W_0$ are compact. Furthermore, $\hhat$ is monotone increasing on $[1,\infty)$ and $f$ is q-convex on $(0,\infty)$, thus the sublevel sets are also connected due to Lemma \ref{lemma:connectedness}.

	\bigskip

	It remains to show that $W_0$ is rank-one convex\footnote{%
		An alternative proof of the rank-one convexity of $W_0$, based directly on the classical Knowles-Sternberg criterion, is given in Appendix \ref{section:knowlesSternberg}.%
	}
	but not polyconvex. In order to prove the rank-one convexity, we will apply Lemma \ref{lemma:rankOneConvexity}. Since\footnote{%
		Note that $h$ is two-times continuously differentiable.%
	}
	\[
		h(t)
		\;=\;
		\begin{cases}
			\hhat(t) &: t\geq1\\
			\hhat\left(\frac1t\right) &: t<1
		\end{cases}
		\quad=\quad
		\begin{cases}
			t-\log(t) &: t\geq1\\
			\frac1t+\log(t) &: t<1
		\end{cases}
	\]
	for $t>0$, we find
	\begin{align}
		h_0 &= \inf_{t\in(0,\infty)}t^2\.h''(t) = \min \left\{ \inf_{t\in[1,\infty)}t^2\.h''(t)\,,\; \inf_{t\in(0,1)}t^2\.h''(t) \right\} \nonumber\\
		&= \min \left\{ \inf_{t\in[1,\infty)} t^2\cdot\frac{1}{t^2}\,,\; \inf_{t\in(0,1)} t^2\cdot \left( \frac{2}{t^3} -\frac{1}{t^2} \right) \right\}
		= \min \left\{ 1\,,\; \inf_{t\in(0,1)} \left( \frac{2}{t} -1 \right) \right\}
		= 1
	\end{align}
	as well as
	\begin{align*}
		f_0 &= \inf_{t\in(0,\infty)}t^2\.f''(t) = \inf_{t\in(0,\infty)} t^2\cdot \left( -\frac{1}{t^2} + \frac{2}{t^3} \right) = \inf_{t\in(0,\infty)} -1+\frac{2}{t} = -1\,.
	\end{align*}
	Then $h_0+f_0=0$, thus condition i) in Lemma \ref{lemma:rankOneConvexity} is satisfied. Since $h'(t)=\hhat'(t)=1-\frac1t\geq0$ for all $t\geq1$, condition ii) is fulfilled as well. Next, we compute
	\begin{align*}
		\frac{2t}{t-1}\,h'(t)-t^2\.h''(t)+f_0
		&= \frac{2t}{t-1}\,\hhat'(t)-t^2\.\hhat''(t)+f_0\\
		&= \frac{2t}{t-1}\cdot \left( 1-\frac1t \right) - t^2\cdot \frac{1}{t^2} - 1
		= \frac{2t}{t-1}\cdot \frac{t-1}{t} - 2
		\;=\; 0
	\end{align*}
	for $t\geq1$ as well as
	\begin{align}
		\frac{2t}{t-1}\,h'(t)-t^2\.h''(t)+f_0
		&= \frac{2t}{t-1}\cdot \left( -\frac{1}{t^2}+\frac1t \right) - t^2\cdot \left( \frac{2}{t^3} - \frac{1}{t^2} \right) - 1 \nonumber\\
		&= \frac{2t}{t-1}\cdot \frac{t-1}{t^2} - t^2\cdot \frac{2-t}{t^3} - 1
		= \frac{2}{t} - \frac{2-t}{t} - 1
		\;=\; 0
	\end{align}
	for $t<1$, which shows that condition iii) is also satisfied. Finally, for condition iv), we find
	\[
		\frac{2t}{t+1}\.h'(t)+t^2\.h''(t)-f_0
		= \frac{2t}{t+1}\cdot\frac{t-1}{t} + t^2\cdot\frac{1}{t^2} + 1
		= 2\, \left( \frac{t-1}{t+1}+1 \right)
		\geq 0
	\]
	for all $t\geq1$ and
	\begin{align*}
		\frac{2t}{t+1}\.h'(t)+t^2\.h''(t)-f_0
		&= \frac{2t}{t+1}\cdot\frac{t-1}{t^2} + t^2\cdot\frac{2-t}{t^3} + 1\\
		&= \frac{2t-2}{t\.(t+1)} + \frac{2-t}{t} + 1
		= \frac{2t-2}{t\.(t+1)} + \frac2t
		= \frac{2t-2 + 2\.(t+1)}{t\.(t+1)}
		= \frac{4}{t+1}
		\;\geq\; 0
	\end{align*}
	for all $t<1$. Therefore, according to Lemma \ref{lemma:rankOneConvexity}, the function $W_0$ is rank-one convex.

	\bigskip

	Finally, in order to show that $W_0$ is not polyconvex, we will apply \v{S}ilhav\'{y}'s result given in Proposition \ref{polycrit}. The representation $\ghat$ of $W_0$ in terms of ordered singular values is given by
	\begin{equation}\label{eq:counterexampleSingularValueRepresentation}
		W_0(F) = \ghat(\lambdahat(F))
		\qquad\text{with}\qquad
		\ghat(\lambdahat_1,\lambdahat_2) = \frac{\lambdahat_1}{\lambdahat_2} - \log\left(\frac{\lambdahat_1}{\lambdahat_2}\right) + \log(\lambdahat_1\lambdahat_2) + \frac{1}{\lambdahat_1\lambdahat_2}\,.
	\end{equation}
	We compute
	\[
		\frac{\partial \ghat}{\partial \lambdahat_1}=\frac{\lambdahat_1^2-1}{\lambdahat_1^2 \lambdahat_2},\qquad
		\frac{\partial \ghat}{\partial \lambdahat_2}=-\frac{\lambdahat_1^2-2 \lambdahat_1 \lambdahat_2+1}{\lambdahat_1 \lambdahat_2^2}
	\]
	and choose
	\[
		\gamma_1 = e^4, \qquad \gamma_2 = e^3, \qquad \nu_1 = e, \qquad \nu_2 = 1\,.
	\]
	Then
	\begin{alignat}{2}
		\label{eq:nonPolyValues1}
		\frac{\partial \ghat}{\partial\lambdahat_1}(e^4,e^3) &= \frac{e^8-1}{e^{11}}\,,
		\qquad&
		\frac{\partial \ghat}{\partial\lambdahat_2}(e^4,e^3) &= -\frac{e^7 (e-2)+1}{e^{10}}\,,
		\\
		\notag
		-\frac{\frac{\partial \ghat}{\partial\lambdahat_1}(e^4,e^3)-\frac{\partial \ghat}{\partial\lambdahat_2}(e^4,e^3)}{e^4-e^3} &= -\frac{1+e^8}{e^{14}}\,,
		\qquad&
		\frac{\frac{\partial \ghat}{\partial\lambdahat_1}(e^4,e^3)+\frac{\partial \ghat}{\partial\lambdahat_2}(e^4,e^3)}{e^4+e^3} &= -\frac{1+e-3 e^8+e^9}{e^{14} (1+e)}
	\end{alignat}
	and therefore, if $W_0$ were polyconvex,
	\begin{equation}\label{eq:nonPolyMielkeCondition}
		\begin{aligned}
			&\exists\;c\in \left[ -\frac{1+e^8}{e^{14}}\,,\; -\frac{1+e-3 e^8+e^9}{e^{14} (1+e)} \right]
			\quad \text{ such that}
			\\
			&\quad \ghat(e,1)\geq \ghat(e^4,e^3)+\frac{\partial \ghat}{\partial\lambdahat_1}(e^4,e^3)\cdot(e-e^4)
			+\frac{\partial \ghat}{\partial\lambdahat_2}(e^4,e^3)\cdot(1-e^3)+c\, (e-e^4)\,(1-e^3)
		\end{aligned}
	\end{equation}
	according to Proposition \ref{polycrit}. Since
	\[
		\ghat(e^4,e^3) = e+6+\frac{1}{e^7}
		\qquad\text{and}\qquad
		\ghat(e,1) = \frac{1}{e}+e\,,
	\]
	combining \eqref{eq:nonPolyMielkeCondition} with \eqref{eq:nonPolyValues1} yields that polyconvexity of $W_0$ would imply
	\[
		\exists\; c\in \left[-\frac{1+e^8}{e^{14}},-\frac{1+e-3 e^8+e^9}{e^{14} (1+e)}\right]
		\qquad \text{such that}\quad
		c\leq \frac{2-3 e^3-2 e^7+e^9-4 e^{10}}{e^{11} \left(e^3-1\right)^2}\,,
	\]
	in contradiction to
	\[
		-0.00377147 \approx \frac{2-3 e^3-2 e^7+e^9-4 e^{10}}{e^{11} \left(e^3-1\right)^2}<-\frac{1+e^8}{e^{14}}\approx-0.00247958\,.
	\]
	Therefore, the energy $W_0$ cannot be polyconvex.
\end{proof}

\section{Conclusion}

We have provided yet another example of a rank-one convex isotropic energy function on $\GLp(2)$ which is not polyconvex. In contrast to classical examples encountered in the literature (cf.\ Section \ref{section:previousExamples}), however, the energy $W_0$ given in \eqref{eq:counterexampleDefinition} satisfies the growth condition $W_0(F)\to\infty$ for $\det F \to 0$. Moreover, all sublevel sets of $W_0$ are compact as well as connected, thus $W_0$ serves as a counterexample to an earlier conjecture by Mielke stating that rank-one convexity of an isotropic function with these properties implies its polyconvexity. As for many other examples of non-polyconvex planar energy functions, it is currently unknown whether the rank-one convex function $W_0$ is quasiconvex as well. In particular, it remains possible that the conditions of Conjecture \ref{conjecture:mielke} are indeed sufficient to ensure the quasiconvexity of rank-one convex planar functions in general.
\subsubsection*{Acknowledgements}
The work of I.D.\ Ghiba has been supported by a grant of the Romanian Ministry of Research and Innovation, CNCS--UEFISCDI, project number PN-III-P1-1.1-TE-2019-0397, within PNCDI III.

\section{References}
\footnotesize
\printbibliography[heading=none]

\begin{appendix}

\section{Connectedness of sublevels for Aubert's energy}
\label{appendix:aubertConnectedness}

Although Aubert's example \cite{aubert1987counterexample}
\begin{align*}
	W_{\mathrm{A}}\col\GLp(2)\to\R\,,\quad W_{\mathrm{A}}(F) &= \frac13(\lambda_1^4+\lambda_2^4) + \frac12\lambda_1^2\lambda_2^2 - \frac23(\lambda_1^3\lambda_2+\lambda_1\lambda_2^3)=\frac13\.\norm{F}^4-\frac16\.(\det F)^2-\frac23\.\det F\cdot\norm{F}^2
\end{align*}
of a rank-one convex, non-polyconvex energy function is not suitable as a counterexample to Conjecture \ref{conjecture:mielke} since the sublevel sets $S_c$ are not compact, we will show in the following that every set $S_c$ is indeed connected.

For $c\in\R$, let $F,\Ftilde\in S_c$. As in the proof of Lemma \ref{lemma:connectedness}, we will explicitly construct a continuous curve contained in $S_c$ which connects $F$ and $\Ftilde$. Without loss of generality,\footnote{%
	Since $W_{\mathrm{A}}$ is objective and isotropic, arbitrary $F$ and $\Ftilde$ can be continuously connected to diagonal matrices by curves within level sets of $W_{\mathrm{A}}$ as in the proof of Lemma \ref{lemma:connectedness}.%
}
assume that $F=\diag(\lambda_1,\lambda_2)$ and $\Ftilde=\diag(\lambdatilde_1,\lambdatilde_2)$ are diagonal with $\lambda_1\geq\lambda_2$, $\lambdatilde_1\geq\lambdatilde_2$ and $\lambdatilde_1\geq\lambda_1$. Then the curve
\begin{equation}
	X_1\col[\lambda_2,\lambda_1]\to\GLp(2)\,,\qquad X_1(s) = \diag(\lambda_1,s)
\end{equation}
is continuous with $X_1(\lambda_2)=F$ and, for all $s\in[\lambda_2,\lambda_1]$,
\begin{align}
	\dds\;W_{\mathrm{A}}(X_1(s))
	&= \frac43\,s^3 - 2\lambda_1\.s^2 + \lambda_1^2\.s - \frac23\,\lambda_1^3 \leq \frac13\,s^3 + \lambda_1\.s^2 - 2\lambda_1\.s^2 + \lambda_1^2\.s - \frac23\,\lambda_1^3\nonumber\\
	&= \frac13\,s^3 + \lambda_1\.s\,(\lambda_1-s) - \frac23\,\lambda_1^3 \leq \frac13\,s^3 + \frac12\,(\lambda_1^2+s^2)\,(\lambda_1-s) - \frac23\,\lambda_1^3 \label{eq:usedCauchyInequalityFirst}\\
	&= \frac13\,s^3 + \frac12\,(\lambda_1^3+\lambda_1\.s^2-\lambda_1^2\.s-s^3) - \frac23\,\lambda_1^3
	= -\frac16\,s^3 + \frac12\,\lambda_1\.s\,(s-\lambda_1) - \frac16\,\lambda_1^3 \nonumber
	\;<\;0\,, \nonumber
\end{align}
where \eqref{eq:usedCauchyInequalityFirst} holds due to \emph{Cauchy's inequality},
\begin{equation}\label{eq:cauchyInequality}
	x\,y\leq k\,x^2+\frac{y^2}{4\,k} \qquad\text{for all }\; x,y,k>0\,,
\end{equation}
with $x=\lambda_1$, $y=s$ and $k=\frac12$. Therefore, the mapping $s\mapsto W_{\mathrm{A}}(X_1(s))$ is monotone decreasing, which implies $W_{\mathrm{A}}(X_1(s))\leqW_{\mathrm{A}}(X_1(\lambda_2))=W_{\mathrm{A}}(F)\leq c$ for all $s\in[\lambda_2,\lambda_1]$.

Now, let
\[
	X_2\col[\lambda_1,\lambdatilde_1]\to\GLp(2)\,,\qquad X_2(s) = \diag(s,s)\,.
\]
Then $X_2$ is continuous with $X_2(\lambda_1)=X_1(\lambda_1)$. Furthermore, $\displaystyle W(X_2(s)) = -\frac{s^4}{6}\,,$ thus $W_{\mathrm{A}}$ is decreasing along $X_2$ as well. Finally, for the continuous curve
\[
	X_3\col[0,\lambdatilde_1-\lambdatilde_2]\to\GLp(2)\,,\quad X_3(s) = \diag(\lambdatilde_1,\lambdatilde_1-s)\,,
\]
we find $X_3(0)=X_2(\lambdatilde_1)$ and $X_3(\lambdatilde_1-\lambdatilde_2)=\diag(\lambdatilde_1,\lambdatilde_2)=\Ftilde$ as well as
\begin{align}
	\dds\;W_{\mathrm{A}}(X_3(s))
	&= \dds\;\left( \frac13(\lambdatilde_1^4+(\lambdatilde_1-s)^4) + \frac12\lambdatilde_1^2\.(\lambdatilde_1-s)^2 - \frac23(\lambdatilde_1^3\.(\lambdatilde_1-s)+\lambdatilde_1\.(\lambdatilde_1-s)^3) \right) \nonumber\\
	&= \frac13\,\lambdatilde_1^3 + \lambdatilde_1^2\.s - 2\lambdatilde_1\.s^2 + \frac43\.s^3 \geq \frac13\,\lambdatilde_1^3 - \lambdatilde_1\.s^2 + \frac43\.s^3 \nonumber\\
	&\geq \frac13\,\lambdatilde_1^3 - s\.\left( \frac13\.\lambdatilde^2 + \frac34\.s^2 \right) + \frac43\.s^3
	= \frac13\,\lambdatilde^2\.(\lambdatilde_1-s) + \left( \frac43-\frac34 \right)\.s^3
	\;>\; 0\,, \label{eq:usedCauchyInequalitySecond}
\end{align}
where Cauchy's inequality \eqref{eq:cauchyInequality} was again employed in \eqref{eq:usedCauchyInequalitySecond} with $x=\lambdatilde_1$, $y=s$ and $k=\frac13$. Thus $W_{\mathrm{A}}$ is nondecreasing along $X_3$, which implies
\[
	W_{\mathrm{A}}(X_3(s)) \leq W_{\mathrm{A}}(X_3(\lambdatilde_1-\lambdatilde_2)) = W_{\mathrm{A}}(\Ftilde) \leq c
\]
and hence $X_3(s)\in S_c$ for all $s\in[0,\lambdatilde_1-\lambdatilde_2]$. By concatenating $X_1$, $X_2$ and $X_3$, we therefore obtain a continuous curve contained in $S_c$ which connects $F$ and $\Ftilde$, cf.\ Figure \ref{fig:curveWithinSublevelsAubert}.

\begin{figure}
	\begin{center}
		\begin{tikzpicture}
			\begin{axis}[
				axis x line=middle,axis y line=middle,
				x label style={at={(current axis.right of origin)},anchor=north, below},
				xlabel={$\lambda_1$}, ylabel={$\lambda_2$},
				xmin=-0.07, xmax=4.9,
				ymin=-0.07, ymax=4.9,
				xtick=\empty,
				ytick=\empty,
				hide obscured x ticks=false,
				width=.49\linewidth,
				height=.39\linewidth
			]
				%
				\addplot[domain=0:4.41,dotted,thick] {x} node[pos=.49, above left] {$\lambda_1=\lambda_2$};
				%
				\addplot[domain=.49:1.47,samples=2,->,ultra thick,blue] ({1.47},{x})
					node[pos=0,draw,circle,fill, inner sep=0.147em]{}
					node[pos=0, below left] {$F$}
					node[pos=.49,right] {$X_1$};
				%
				\addplot[domain=1.47:3.43,samples=2,->,ultra thick,black] {x}
					node[pos=.49,below right] {$X_2$};
				%
				\addplot[domain=3.43:.98,samples=2,->,ultra thick,red] ({3.43},{x})
					node[pos=1.029,draw,circle,fill, inner sep=0.147em]{}
					node[pos=1.029, below right] {$\Ftilde$}
					node[pos=.49,right] {$X_3$};
				%
			\end{axis}
		\end{tikzpicture}
		\vspace*{-1.96em}
	\par\end{center}
	\caption{%
	\label{fig:curveWithinSublevelsAubert}
		\small The energy $W_{\mathrm{A}}$ is decreasing along $X_1$ and $X_2$ and increasing along $X_3$.%
	}
\end{figure}

\end{appendix}

\end{document}

redpajama_set_name: RedPajamaArXiv


Forced and arranged marriages in Asian tradition have created a furore in United Kingdom over the years. Now marriages of convenience are creating an upheaval that cannot be stopped. These two kinds of marriages, where plenty of money exchange hands mainly are entered into for circumventing immigration laws. Marriages have started taking place for such ludicrous reasons like getting a health plan for the partner. Breaking of the marriage too is equally quick. Usually as soon as the real intention is fulfilled, marriages are annulled and according to authorities, the bride and groom do not live together in most of these cases. We even come across marriages by proxy. Both the parties have to sign a contract and this is accepted in countries like El Salvador and Israel. The problem arises when the spouse applies for a UK Visa. UK being keen on cutting down as many illegal entries as possible would definitely discourage such marriages.
USA and other European countries too are facing a similar problem. In addition to these marriages, there are lesbian and gay partners. Once gay and lesbian marriages are accepted, ignoring them becomes a matter of denial of human rights and these partners could not be refused immigration. At least a few of these marriages are made only for the simple reason that the other partner should gain an entry to a more developed country. It is difficult to prove the sexual orientation of the partners, or if they are real partners at all.
It is believed that at least 10,000 marriages per year in London are marriages of convenience. A marriage of convenience could be a passport to reach another country, to escape from the cultural pressures, family squabbles, There are films like The Wedding Banquet and Green Card, and people are well aware of this situation. Mail order brides are available in all countries now and their parents do not have much problem about the situation either. UK and other Western countries have to think of many changes in the Immigration laws to deal with this situation. It is not very easy to separate legal from the illegal ones. Immigration and naturalization of aliens who are married to UK citizens had been going on for a long time, before this desperate situation arose.
There are also cases of feigning legitimate relationships when no such thing exists. They keep up the pretence as long as their immigration situation is cleared and go their own way afterwards. There are many examples of 'one-sided' marriages. In USA, Immigration of Marriage Fraud Amendment Acts, 1986 has effectively dealt with this matter to a large extent.
UK Home office has brought many rules to follow while assessing such applications. The careful assessment of marriage photographs and decide if they are genuine is one way of doing it. Whether the consummation of the marriage has taken place or not? Are there any children? Is the wife pregnant? Whether it is the first marriage or a subsequent marriage, especially for the sponsor? Nowadays almost all the Asian, especially Muslim marriages are looked at with suspicion that could be justified under the circumstances.
People could be deported from UK on the ground that their presence is 'not conducive to the public good' and marriage of convenience could be taken as one of the grounds. Such a deported person can request for the order to be revoked, through Home office, or British High Commission in his or her own country. The usual practice is that the order is not revoked at least for three years. Under certain circumstances, people who have entered without proper permission, or those who have come under the blanket suspicion of having made the marriage of convenience, could be treated as illegal entrants. Such people might have entered either by concealing many particulars, or by misleading the Immigration officers and by disclosing only partial information. Some people leave voluntarily and there is a provision of paying their fares to return to make it as less painful as possible. It is just a pleasant gesture and this includes people who are caught in this muddle for no fault of theirs.
The traditional registered marriages could be easily detected. Genuine marriages cannot be ignored as part of the Human Rights. Registered marriages make better impact on authorities. Authorities are looking for proof that the marriage was performed with the complete approval and co-operation of both the parties and entered into voluntarily, without any ulterior motive or pressure. Also it should withstand the verification that it was conducted with open consent of both the parties. Nowhere it should give the impression that the marriage was solely for the purpose of obtaining an entry into United Kingdom and if any of these fail, definitely the request would not be entertained.
There are many newly introduced rules now to discourage this evil. Both the partners should have attained majority. One of the spouses should hold a legal permission to stay in United Kingdom. A jobless or houseless spouse would not be able to offer much for the future life of the other person. Spouse's income and capability of financially looking after the other partner are crucial for an application and none of these should look temporary, ungenuine and dishonest. The education and employment history of the spouse residing in UK would be assessed. The capability of the spouse to support the alien partner would be weighed. Sometimes even after all the requirements are met, Immigration officer might get a doubt, just at the unbelievable perfection of the whole thing and might refuse the permission. With their vast experiences of dealing with hopeful UK citizens, they would have developed an acute sensitivity and sometimes, without any reason, and they feel sure that something is wrong somewhere and under those circumstances, they prefer to opt for more time, to ponder over things. Usually such cases end up in refusing the permission. Sometimes, very strong humanitarian considerations tip the scale in favour of the alien. Even if there is a child, if neither parent is a British citizen or legally settled in UK, the child cannot get the British citizenship.
Marriage of convenience is becoming a bigger problem among Asians and Third world countries. People in their desire to come to developed Western countries, make the mistake of getting into any kind of marriage alliance and get into trouble very often. It is not as easy as they think it is. Every Western country is discouraging this trend before it attains an enormous threatening form. This does not mean that the trend has fully stopped. It goes on in the face of discouraging laws and tightened immigration rules.
Foreign spouses, especially after the onslaught of terrorism, have been regarded with suspicion in all western countries for obvious reasons. Immigration authorities are also nervous that terrorists might be using the marriage of convenience as one of their tools to get access into the Western countries. There are enough reasons to be sure that many brides are willing to be part of the ongoing Jihad by helping the terrorists to gain entry. With that in mind, UK law had taken many measures to discourage the new and dangerous trend. Immigration authorities, while considering the foreign spouse applications are expected to mainly consider if the couple had shared residence earlier or not, and if so, for how long; whether they speak the same language, whether they come from the same country; is there any other common language other than English; the age difference between the partners; their personal background; is it an impossible union or a workable one? How much personal knowledge of each other they had before the marriage? How many times have they married earlier and are any children involved in the situation?
They also assess the overall family situation of both the partners, like the length and nature of partners' stays in the respective countries, both the partners' family ties with United Kingdom and mainly the alien person's country of birth. To fulfill the condition of maintenance, the UK partner is supposed to show his income and expenditure statement, an adequate place of dwelling, etc. Statements of income, self-employment income, public and private pensions benefit, sickness and even maternity benefit in some cases will be asked to be submitted.
Marriages are being scrutinized by all Immigration authorities in great detail for considering the family reunification request. "The examination of the role of marriage registrars in immigration co0ntrol enforcement is important for several reasons. First it is important in itself, not least as registrars have now imposed on them a statutory duty to report so called 'sham' marriages," says Cohen, (2003, p.128). Close relatives like cousins getting married to one another is viewed suspiciously, as forced or authoritatively arranged marriage could be detected. Details of even distant cousins, grandparents, siblings, uncles and aunts, relatives in various countries etc have been scrutinized. If partners could prove beyond doubt that the marriage has taken place according to their own desire, without any pressure from any one, usually the request for family reunification is granted. "The primacy of the right to family life has not been apparent historically in UK immigration law governing marriage-related applications….Marriage and child-rearing practices, concepts of the family and family duties are all raised by family settlement applications," says Clayton, (2004, p.206).
UK does not recognize proxy marriages. Nor does it recognize telephone marriages. In some countries like Israel, after signing certain forms, telephone marriage between the sponsor in UK and the other partner in Israel take place and are accepted under certain conditions. But they are unacceptable and regarded as marriages of convenience in UK. Commonwealth citizens used to receive visa easily if the other partner is in UK. This provision too has stopped recently.
Today a lot of people are using this kind of marriage to prevent from being deported from Britain. These bogus marriages have become a problem not just for the Home Department authorities, but also for the Family Courts. A marriage abuse team is given the authority of finding out facts behind such marriages, especially to find out if any money transactions have taken place between the partner and his sponsor. They make late calls to suspected houses to find out if the couple is living together. They ask questions separately to find out any discrepancy thereof. Other than Immigration authorities, very few people knew about it earlier; but today, it has become a popular mode of finding an entry into UK. If marriages are proved false, partners of such marriages could be immediately deported. There are improbable reports that men disguise themselves as women and get married to gain the citizenship. Authorities also have come across many stories where money changes hand for marrying an individual who is in need of a citizenship. There are also possibilities of professional brides, who get married under different names for money. Tightening up the laws has drawn many criticisms too. "The fact was that under this provision an increasing number of men were refused even when their marriage was known to the Home Office to be genuine. This was never intended by Parliament or even necessarily by the Home Secretary," says Juss, S (1993, p. 91).
After suspecting this nexus, Government has given more power to the marriage registrars. Home office considers this an awful problem that cannot be fully stopped. Sometimes, even though they are aware of the problems, they are unable to take any action depending only on suspicion. Government is trying hard to stop this deception. "At the end of the immigration process, a traveler may eventually be granted leave to remain indefinitely and in due course apply to the Home Office for naturalization as a British Citizen. Alternatively, he or she may be removed as an irregular migrant or face deportation on the grounds of conducive to the public good," (Blake and Hussain, 2003, 1.12, p. 22).
Cases: On 10.10.2000, the case of Pin-Wah Jeff Chang vs Home Department of UK came up in front of Immigration Appeal Tribunal. The Appellant, a citizen of Malaysia, questioned the refusal of right to remain in the United Kingdom as the spouse of an EEA national and denying him the right to appeal. There was a question if it was a valid marriage or a marriage of convenience. There was also a doubt if the relationship was a lawful one or not. They had to consider if the Appellant had been denied any of the Community Rights. It was pointed out that it was a marriage of convenience to ignore the entry rules of UK, and the complaint was dismissed.
In case of R (Kimani) v Lambeth LBC (2004), Kimani, Kenyan national who arrived at UK for asylum with her son, married an Irish national, but permission was not given for her to stay in UK was rejected for contracting a marriage of convenience.
Under Primary Purpose-Rule (in effect from 1980 to 1997) British citizens, married to non-EU citizens had to prove that marriage was not for avoiding deportation. Onus is on the parties to clarify the genuineness of it and this deters many. Labor Government modified these laws.
70% of Bangladeshi and Pakistani marriages in UK are arranged. Entry Clearance rules are tightened up to the maximum now. The minimum age of the bride to be allowed to UK was raised from 16 to 18 years. At this age, they cannot act as sponsors either and after entry a period of 12 month is compulsory and now it has been extended up to two years. British National Criminal Intelligence Service, An-Nisa Society, Anti-Slavery International, Anti-Trafficing Programme, Apna Ghar, Asylum Aid, Belgrave Baheno, Enfield Saheli, Institute of Race Relations are some Government and Non Government organizations who are fighting these evils.
Blake and Hussain, (2003), Immigration, Asylum & Human Rights, Oxford University Press.
Cohen, Steve, (2003), No One is Illegal, Trentham Books, Stoke on Trent.
Clayton, Gina, (2004), Textbook on Immigration and Asylum Law, Oxford University Press.
Cotran et al, (2001), Butterworths Handbook on Immigration Law, Butterworths, London.
Evans, J.M. (1976) Immigration Law, Sweet & Maxwell, London.
Encouraging Citizenship, Report of the Commission on Citizenship, (2001), London, HMSO.
Guild, Elspeth, (2001), Immigration Law in the European Community, Kluwer Law International, London.
Juss, Satvinder, (1993), Immigration, Nationality and Citizenship, Mansell Publishing Ltd., London.
"Contemporary marriage issues: Forced and migration." LawTeacher.net. 11 2013. All Answers Ltd. 04 2019 <https://www.lawteacher.net/free-law-essays/family-law/contemporary-issues-in-marriage-law.php?vref=1>.
"Contemporary marriage issues: Forced and migration." LawTeacher. LawTeacher.net, November 2013. Web. 22 April 2019. <https://www.lawteacher.net/free-law-essays/family-law/contemporary-issues-in-marriage-law.php?vref=1>.
LawTeacher. November 2013. Contemporary marriage issues: Forced and migration. [online]. Available from: https://www.lawteacher.net/free-law-essays/family-law/contemporary-issues-in-marriage-law.php?vref=1 [Accessed 22 April 2019].
LawTeacher. Contemporary marriage issues: Forced and migration [Internet]. November 2013. [Accessed 22 April 2019]; Available from: https://www.lawteacher.net/free-law-essays/family-law/contemporary-issues-in-marriage-law.php?vref=1.


text:
U of L student, rugby player offering rainbow of hope; Karlee Durfey's support for ill niece has grown to help others
By Herald on December 28, 2020.
Three-year-old Bo Smith is in remission after a cancer diagnosis in 2019 and her aunt Karlee Durfey, a University of Lethbridge Pronghorn rugby player, is helping with an initiative to provide support for families affected by childhood cancer. Photos submitted by the University of Lethbridge
Trevor Kenney
University of Lethbridge kinesiology student Karlee Durfey and her family understand fully how your world can be turned upside down in a manner of moments and how that can lead to the creation of something totally unexpected. The day-to-day priorities of school, work and relationships fade to the background upon the sudden illness of a child.
Shortly after Durfey's two-year-old niece, Bo Smith, complained of tummy pain in December 2019, a routine trip to the doctor resulted in a shocking cancer diagnosis. For the next few months, their lives revolved around Bo's health as she fought her cancer battle at the Alberta Children's Hospital. A GoFundMe page was created, and support rolled in from friends and family in their home community of southern Alberta, helping offset the loss of work, the trips to Calgary and the hotel stays.
A year later, Bo, minus a kidney, is a healthy three-year-old and in remission. Her checkups will be many and the spectre of her brush with cancer will always be a part of their lives. Now, on the other side of the crisis, her family is looking to support other families whose worlds are similarly being upended.
"We spent quite a few days at the hospital and honestly, it gets boring, so we were looking for things to do," says Durfey, who is in her fourth year and also plays for the Pronghorns women's rugby team.
"So, my mom took us to a sewing store, and we started embroidering. The first thing I ever embroidered was a rainbow — Bo loves rainbows — they kind of signify the light after the rain, after a hard thing."
When the clouds lifted on their own lives, Durfey said her sister (Bo's mom) had an idea.
"I was still embroidering and she said we should embroider clothes and sell them, not necessarily to make a profit but to give to families that needed help when their kids are sick — just like the support our family got," says Durfey.
"She spent a lot of time in the hospital and saw a lot of families and a lot of kids going through the same thing who could really use some help."
Not long after, Bo Smith & Co was created. An online shop, it offers T-shirts and sweatshirts designed by Durfey and her sister that feature the rainbow she first embroidered in a Calgary hospital room. The model, of course, is Bo.
"We donate any money we raise straight to families we know who need help," says Durfey. "We've also collaborated with another mom who had a little girl sadly pass away recently. We've given gas cards to families needing to travel up to Calgary, money for hotels and food, things like that. Those are huge costs you don't think of and where my sister needed help."
For Durfey, the entire experience has been eye-opening and heartwarming.
"Childhood cancer, before this, was never really a topic for me. I never really understood it," she says. "I feel like most people dealing with cancer they never truly understand it until maybe someone very close to them is going through it."
She's thankful for the support she received from her Pronghorn teammates and coaches, faculty members at the U of L and the southern Alberta community that continues to patronize Bo Smith & Co. The demand for product has outstripped their capabilities, even with her other two sisters chipping in to help.
They now order shirts from a wholesale company, and rather than hand embroider each individual piece, have two machines to help with the process.
"I feel like most people love the opportunity to give money to kids with cancer and this has opened up a lot of people's minds and hearts to recognizing there are many kids going through this," she says.
In November, Bo Smith & Co teamed up with former Pronghorns track star Aaron Hernandez to create mustache shirts to raise awareness for prostate cancer and Durfey is enthused about other initiatives she says are in the works. In addition to a possible career as an occupational therapist, she wants to see how much more they can do with Bo Smith & Co.
"We plan on collaborating with other people to spread awareness, and not only for childhood cancers. We really want to help anyone we can," she says. "And we understand not everybody has the means to donate or the money to buy a shirt, so simply sharing our story, sharing our website and our Instagram and following us — even that helps."
Check out bosmithandco.ca or bo.smith.co on Instagram.

redpajama_set_name: RedPajamaCommonCrawl


If I lived in AZ, CQ, FL, NY, TX, or WA, I'd have this on my iPhone in two seconds. Integrity Logic – Geology AZ .
Wheel won't turn on car left 3 months: Rust or Brake Issue?
2009-09-24 2009-09-24 9 Comments on Wheel won't turn on car left 3 months: Rust or Brake Issue?
Collection Agencies for Deadbeat Clients?
2009-09-23 2019-03-27 10 Comments on Collection Agencies for Deadbeat Clients?


text:
Convicted ex-immigration minister Støjberg vows to return from parliamentary expulsion
Inger StøjbergPolitics
A majority in the Danish parliament on December 21st voted to fire convicted ex-minister Inger Støjberg from the chamber. Photo: Mads Claus Rasmussen/Ritzau Scanpix
Denmark's parliament on Tuesday voted to expel former migration minister Inger Støjberg, who was convicted last week of violating migrants' rights by separating asylum-seeking couples.
Following a lengthy debate, 98 members voted for her immediate expulsion and 18 against, making her the first parliamentarian to be kicked out in 30 years.
Støjberg, who was handed a 60-day jail term by a special court last week though is unlikely to serve any time in prison, had to leave the chamber immediately, waving goodbye as she stepped away.
"I would rather be voted out by my colleagues here in parliament for trying to protect some girls than voted out by the Danish people for turning a blind eye," she told reporters after exiting the chamber.
However, she said she was open to returning to politics. She is free to return to parliament should she be given a mandate by voters at the next election, scheduled to take place in 2023.
"I don't think you should expect this to be the last word from me," she told reporters.
Her order to separate asylum-seeking couples when the woman was under 18 with no individual examination of the cases was found to have violated the European Convention on Human Rights.
READ ALSO: Danish ex-minister gets prison sentence in impeachment trial
In 2016, the government separated 23 couples on Støjberg's orders without examining their cases following instructions from the minister.
The policy was found to be unlawful because the action was taken without allowing for exceptions or consideration of individual circumstances.
The couples, most of whom had only a small age difference, were then placed in different centres while their cases were reviewed.
In seven of the cases, staff at the centres reported that the separated asylum seekers experienced suicidal thoughts or attempted to kill themselves.
Most political parties were in favour of removing the 48-year-old self-styled champion of "Danish values", a hugely popular politician who served as minister from 2015 to 2019.
"It is not compatible with being a member of the parliament to receive a prison sentence," said Karsten Lauritzen, parliamentary chairman of the Liberals (Venstre), the party Støjberg left in February.
Since 1953, only four members of parliament have been excluded.
Støjberg has insisted the policy was designed to fight against forced marriages and said after her trial that she was "being punished for trying to protect the girls", while not commenting on the similar ages of most of the couples.
As minister, Støjberg was at the forefront as Denmark' centre-right government, propped up by the anti-immigration Danish People's Party (DF), tightened restrictive migration policies from 2015-2019.
She passed a law allowing for migrants' assets to be confiscated to finance their care in Denmark and boasted of having passed more than 110 amendments restricting the rights of foreigners.
She also published a picture of herself with a cake to celebrate the passing of a 50th law curbing immigration; called for the public to report pizzerias where staff did not speak Danish; and told a false story about a daycare banning pork from children's lunches.
Conversely, she was the architect of an apprenticeship system which was praised by companies for helping them bring refugees onto Denmark's labour market.
Since leaving government and the Liberal party, she has echoed former US president Donald Trump by using the phrase "drain the swamp" when addressing a demonstration against the current government.
Despite the election of the Social Democrats to power two years ago, the Scandinavian country still has one of the most restrictive migration policies in Europe.
READ ALSO: OPINION: Why reaction to Støjberg verdict is important for democracy in Denmark
Denmark aims for all domestic flights to be green by 2030
Danish politician to face retrial in EU fraud case
Convicted Danish ex-minister faces expulsion from parliament
Danish parliament set to expel prison-bound ex-immigration minister
OPINION: Why reaction to Støjberg verdict is important for democracy in Denmark
Danish ex-minister gets prison sentence in impeachment trial
How damaging is local election result for Danish PM Frederiksen?
Danish anti-immigration party leader heads for exit after election defeat
The video series shining a light on personal European stories


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redpajama_set_name: RedPajamaGithub


Transcendence and Self-Transcendence, Workshop organized in the frame of the research project "The Return of the Religious in Postmodern Thought" (FWF I2785), October 20-21, 2016, IWM/Vienna.
Pro Patria Mori. Solidarity and Sacrifice in the First World War, April 14th/15, 2016, IWM/ Vienna (in cooperation with the ERC Project GRAPH – The Great War and Modern Philosophy, KU Leuven).
Philosophie, Geschichte, Politik. Europa und die Sorge für die Seele, March 31/April 1 2016, Center for Theoretical Study (CTS), Prague.


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Soft, quick-drying jersey knit or single layer mesh footbed cover for maximum comfort and breathability.


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