# Higher dimensional problems with volume constraints—Existence and $Γ$-convergence

### Marc Oliver Rieger

University of Zürich, Switzerland

## Abstract

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

on $Ω⊂R_{n}$, where $u∈H_{1}(Ω)$ and $α+β<∣Ω∣$. 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 $f(u,∇u)=ψ(∣∇u∣)+θ(u)$ under the assumption of $p$-polynomial growth and strict convexity of $ψ$. 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 $Γ$-limit of the functional $E$ for a general class of functions $f$ in the case of vanishing transition layers, i.e. when $α+β→∣Ω∣$. As limit functional we obtain a nonlocal free boundary problem.

## Cite this article

Marc Oliver Rieger, Higher dimensional problems with volume constraints—Existence and $Γ$-convergence. Interfaces Free Bound. 10 (2008), no. 2, pp. 155–172

DOI 10.4171/IFB/184