*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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