Some problems regarding the existence and regularity of minimizers of variational problems with gradient constraint

Problem's Description

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. Let $N$ be a compact convex set and assume that $\sigma: N\to \mathbb{R}$ is a bounded convex function such that $u\in C^1(N^\circ)$ and strictly convex on $N^\circ$. Consider the functional \begin{align*} E[u]=\int_{\Omega}\sigma(\nabla u(x))dx, \end{align*} defined for all Lipschitz functions $u\in C^{0,1}(\Omega)$ such that $\nabla u(x)\in N$ for a.e. $x\in \Omega$ and called this closed convex subset of Lipschitz functions $\mathscr{A}_N(\Omega)$. Furthermore, chose a $\phi \in \mathscr{A}_N(\Omega)$ and let \begin{align*} \mathscr{A}_N(\Omega,\phi):=\{u\in \mathscr{A}_N(\Omega): u\vert_{\partial \Omega}=\phi\}. \end{align*}

Define the obstacles \begin{align*} M_\phi(x)&=\sup_{u\in \mathscr{A}_N(\Omega,\phi)}u(x),\\ m_\phi(x)&=\inf_{u\in \mathscr{A}_N(\Omega,\phi)}u(x). \end{align*}

It can be shown that $m_\phi,M_\Phi\in \mathscr{A}_N(\Omega,\phi)$. Let $h$ be a minimizer of \begin{align*} \inf_{u\in \mathscr{A}_N(\Omega,\phi)}E[u]. \end{align*}

Problem 1:

Does the minimization problem have a unique minimizer when $n>2$?

Note that this is true if $n=2$ due to a result of D. De Silva and O. Savin.

Problem 2:

Is it true that $h$ is $C^1$ on the set $\Omega\setminus \{x\in \Omega: h(x)=m(x)\text{ or }h(x)=M(x)\}$ if $n>3$?

Note that this is true if $n=2$ due to a result of D. De Silva and O. Savin. \newline

Let $h$ be a minimizer and define the set $\mathcal{L}_h$ to be \begin{align*} \mathcal{L}_h=\{x\in \Omega: \text{$h$ is $C^1$ in a neighbourhood of $x$ and $\nabla h(x)\in N^\circ$}\}. \end{align*}

Problem 3:

Give a necessary and sufficient condition in terms of the boundary value $\phi$ for when $\mathcal{L}_h\neq \varnothing.$

Problem 4:

Assume that $\mathcal{L}_h\neq \varnothing.$ What can be said about the regularity of $\partial \mathcal{L}_h$? What can be said about the regularity of $h$ at $\partial \mathcal{L}_h$?

1. Article Minimizers of convex functionals arising in random surfaces

   • D. De Silva and O. Savin

   Duke Mathematical Journal 151 (3), 487-532, 2010