Higher dimensional problems with volume constraints—Existence and <em>Γ</em>-convergence

Abstract

We study variational problems with volume constraints (also called level set constraints) of the form \begin{eqnarray*} \mbox{Minimize }E(u):=\int_\G f(u,\nabla u)\,dx,\nonumber\\ |\{x\in\Omega,\;u(x)=a\}|=\alpha,\quad |\{x\in\Omega,\;u(x)=b\}|=\beta, \end{eqnarray*} on $\Omega\subset\R^n$, where $u\in H^1(\G)$ and $\alpha+\beta<|\G|$. 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 $\alpha+\beta\to|\G|$. As limit functional we obtain a nonlocal free boundary problem.