Higher dimensional problems with volume constraints—Existence and $\Gamma$-convergence

Abstract

We study variational problems with volume constraints (also called level set constraints) of the form

on $\Omega \subset {\mathbb{R}}^n$, where $u\in H^1(\Omega )$ and $\alpha +\beta <|\Omega |$. The volume constraints force a phase transition between the areas on which $u=0$ and $u=1$.

We give some sharp existence results for the decoupled homogenous and isotropic case $f(u,\nabla u)=\psi (|\nabla u|)+\theta (u)$ under the assumption of $p$-polynomial growth and strict convexity of $\psi$. We observe an interesting interaction between $p$ and the regularity of the lower order term which is necessary to obtain existence and find a connection to the theory of dead cores. Moreover we obtain some existence results for the vector-valued analogue with constraints on $|u|$.

In the second part of this article we derive the $\Gamma$-limit of the functional $E$ for a general class of functions $f$ in the case of vanishing transition layers, i.e. when $\alpha +\beta \to |\Omega |$. As limit functional we obtain a nonlocal free boundary problem.