# 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 @Wolfgang Bangerth 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 Stefan problem formulated for the enthalpy $u$ as considered e.g. by Nochetto 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 Sprittles & Shikhmurzaev.

Again, this is in line with @Wolfgang Bangerth's 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 Roubicek 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.

• Are you interested when g couples mutliple points on the boundary together, or when g simply applies a non-linear constraint at each point individually? – Godric Seer May 8 '13 at 13:48
• It should be a pointwise constraint (I have edited the question accordingly). But a coupling of the boundarie points, e.g. via an integral relation, might be also ok. – Jan May 8 '13 at 14:05

This makes no sense. Let us assume that $g(z)=0$ has exactly one solution $z=z^\ast$, then your boundary condition $g(u|_\Gamma)=0$ is equivalent to $u|_\Gamma=z^\ast$, i.e., a linear Dirichlet condition.
On the other hand, if there are multiple solutions of $g(z)=0$, then you are saying that the value $u|_\Gamma$ could have multiple values, but this does not lead to a unique solution of the equation and so it makes no sense.
In a similar vein, it makes no sense to have nonlinear Neumann boundary conditions. The only thing that does make sense is to have nonlinear Robin-type conditions, $(\partial u/\partial n+g(u))|_\Gamma = 0$. A typical case is the Planck radiation condition $(D\partial u/\partial n+\gamma u^4)|_\Gamma = 0$.
• I can think of some sense. Namely, the law that is imposed at the boundary is nonlinear. I agree, that well-posedness requires the existence of an implicit function, such that locally one has $g(u)=0 \Leftrightarrow u=z^*$ . In practise, however, this $z^*$ may be accessible only pointwise via nonlinear solves, so that the actual implementation bases on nonlinear boundary conditions. This is hypothetic, but that is my question. – Jan May 9 '13 at 11:02
• Well, but that doesn't make any sense. If you have a boundary condition $g(u,x)=0$, then for every $x$ you have a Dirichlet condition of the form $u(x)|_\Gamma=z^\ast(x)$ -- i.e., a linear Dirichlet boundary condition. Maybe it's been computed by a nonlinear solver, but the equation is linear: it can be solved in one shot, without having to iterate out the equation. – Wolfgang Bangerth May 9 '13 at 16:48
• I see your point. If $u|_\Gamma$ is defined by external data, it is in fact a linear assignment. Conversely, a nonlinear relation might appear if $u|_\Gamma$ depends on internal states. I.e. when the side condition of the PDE is given via $g(u)=0$ where $g$ maps from the state space onto* the boundary. *onto has to be understood wrt. to the setting. In a weak formulation this will be into a dual of the space of traces. – Jan May 10 '13 at 8:57