Calculus of Variations and Geometric Measure Theory

M. O. Rieger

Higher dimensional problems with volume constraints - Existence and Gamma-convergence

created by rieger on 30 Jan 2006


Submitted Paper

Inserted: 30 jan 2006

Year: 2006


We study variational problems with volume constraints (also called level set constraints) of the form \begin{eqnarray} \mbox{Minimize }E(u):=\int\Omega f(u,\nabla u)\,dx,\nonumber

=\beta, \end{eqnarray
} on $\Omega\subset*R*^n$, where $u\in H^1(\Omega)$ and $\alpha+\beta<
$. 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 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 $

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
$. As limit functional we obtain a nonlocal free boundary problem.

Keywords: volume constrained problems, level set constraint, Gamma convergence