Submitted Paper

Inserted: 30 jan 2006

Year: 2006

Abstract:

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

\{u=0\}
=\alpha,\quad
\{u=1\}
=\beta, \end{eqnarray} on $\Omega\subset*R*^n$, 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 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.

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

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