Calculus of Variations and Geometric Measure Theory

E. Stepanov - P. Tilli

On the Dirichlet problem with several volume constraints on the level sets

created on 10 Mar 2000
modified on 01 Oct 2002


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


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 $
=\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 Hölder 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.