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

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 Ω ⊂ Rⁿ with smooth boundary, we study the problem of minimizing ∫_Ω |∇u|² among all those functions u ∈ H¹ that simultaneously satisfy n-dimensional measure constraints on the level sets of the kind |{u = l_i}| = α_i, i = 1,…, k, and a generalized boundary condition u ∈ K. Here, K is a closed convex subset of H¹ such that ; the invariance of K under provides that the condition u ∈ K actually depends only on the trace of u along ∂Ω.

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

Finally, in the case of just two volume constraints, we investigate the Γ-convergence of the above (rescaled) functionals when the total measure of the two prescribed level sets tends to saturate the whole domain Ω. It turns out that the resulting Γ-limit functional can be split into two distinct parts: the perimeter of the interface due to the Dirichlet energy that concentrates along the jump, and a boundary integral term due to the constraint u ∈ K. In the particular case where K = H¹ (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio et al.