*Published Paper*

**Inserted:** 10 mar 2000

**Last Updated:** 1 oct 2002

**Journal:** Proc. Royal Soc. Edinburgh - Ser. A. (Mathematics)

**Volume:** 132

**Number:** 2

**Pages:** 437-461

**Year:** 2002

**Abstract:**

We consider minimization problems involving the Dirichlet integral under
an arbitrary number of volume constraints on the level sets and a generalized
boundary condition. More precisely, given a bounded open domain
$\Omega\subset R^n$ with smooth boundary, we study the problem of minimizing
$\int_\Omega

\nabla u

^2$ among all those functions $u\in H^1$ which
simultaneously satisfy $n$-dimensional measure constraints on the level sets of the
kind $

\{u=l_i\}

=\alpha_i$, $i=1,\ldots,k$, and a generalized
boundary condition $u\in K$. Here
$K$ is a closed
convex subset of $H^1$ such that $K+H^1_0=K$: the invariance of $K$
under $H^1_0$ provides that the condition $u\in K$
actually depends only on the trace of $u$ along $\partial\Omega$.

By a penalization approach, we prove the existence of minimizers and their HÃ¶lder continuity, generalizing previous results which are not applicable when a boundary condition is prescribed.

Finally, in the case of just two volume constraints, we investigate the $\Gamma$-convergence of the above (rescaled) functionals when the total measure of the two prescribed level sets tends to saturate the whole domain $\Omega$. It turns out that the resulting $\Gamma$-limit functional can be split into two distinct parts: the perimeter of the interface due to the Dirichlet energy which concentrates along the jump, and a boundary integral term due to the constraint $u K$. In the particular case where $K=H^1$ (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio, Fonseca, Marcellini and Tartar.

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