Example of a PDE model with nonlinear Dirichlet boundary conditions
Is there any application for PDEs with nonlinear Dirichlet boundary conditions? That is, I am looking for an example of a partial differential equation for a state $u$ posed on a domain $\Omega$ with $g(u|_{\Gamma})=0$, where $\Gamma$ is the boundary of $\Omega$ and $g\in L^2(\Gamma)\to L^2(\Gamma)$ is a nonlinear function.
Note that I am looking for pure Dirichlet conditions. I am aware of the Stefan-Boltzmann boundary conditions for heat conduction, but they are Robin-type boundary conditions.
EDIT:
As <PERSON> has pointed out in his answer, there is no sense in assigning boundary values via a nonlinear $g\in L^2(\Gamma)\to L^2(\Gamma)$.
What if the boundary values are fixed via a function $G \in \bigl ( L^2(\Omega) \to L^2(\Gamma) \bigr)$? Something like $u|_\Gamma(x)=g(x)\|u\|$. Has anyone come across such an example.
I am insisting so much on this point, because I was investigating the incorporation of boundary conditions of a PDE by means of multipliers. When, instead of resolving the BCs in the ansatz space, looking at a PDE of the Form $ F(u)=0 \text{ in } (H^1(\Omega))',$ completed by $G(u)=0$, where $G:H^1(\Omega) \to H^{1/2}(\Gamma)$ assigns the Dirichlet conditions, I found it interesting to think of what may happen with a nonlinear $G$.
EDIT 2:
When considering unsteady processes, there can be a unique solution, even if the boundary data $g(u)=0$ allows for multiple values. An example is the <PERSON> problem formulated for the enthalpy $u$ as considered e.g. 935
by <PERSON> et al..
There, the boundary condition is given in terms of $\theta(u)=0$ on $\Gamma$, where $\theta$ is a monotone Lipshitz-continuous scalar function, with - in particular - $\theta(s)=0$ for $s\in(0,1)$.
Thus, there is no way to reduce the nonlinear Dirichlet to a linear assignment, as $\theta$ is not injective in the region of interest.
EDIT 3:
Another source of nonlinear boundary conditions are free surface problems, where the boundary itself depends on the solution. See, for example, this ArXiv preprint of a paper by <PERSON> & <PERSON>.
Again, this is in line with <PERSON> answer, as the BCs for free surface problems involve spatial derivatives and, thus, are nonlocal. In his book Nonlinear Partial Differential Equations with Applications (p. 155), the author <PERSON> provides an example for a wetting problem of flow in porous media. Apart from the formulation as a partial differential variational inequality it includes the boundary condition $$ u\frac{\partial u}{\partial n} = 0, $$ on nonpermeable interfaces. 474