TY - JOUR

T1 - On the dirichlet problem with several volume constraints on the level sets

AU - Stepanov, Eugene

AU - Tilli, Paolo

PY - 2002/12/1

Y1 - 2002/12/1

N2 - 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 fΩ |∇u|2 among all those functions u ∈ H1 that simultaneously satisfy n-dimensional measure constraints on the level sets of the kind |{u = li}| = α1, i = 1,..., k, and a generalized boundary condition u ∈. Here, Κ is a closed convex subset of H1 such that Κ + H01 = Κ; the invariance of Κ under H01 provides that the condition u ∈ Κ actually depends only on the trace of u along δΩ. 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 ∈ Κ. In the particular case where Κ = H1 (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio et al.

AB - 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 fΩ |∇u|2 among all those functions u ∈ H1 that simultaneously satisfy n-dimensional measure constraints on the level sets of the kind |{u = li}| = α1, i = 1,..., k, and a generalized boundary condition u ∈. Here, Κ is a closed convex subset of H1 such that Κ + H01 = Κ; the invariance of Κ under H01 provides that the condition u ∈ Κ actually depends only on the trace of u along δΩ. 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 ∈ Κ. In the particular case where Κ = H1 (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio et al.

UR - http://www.scopus.com/inward/record.url?scp=0036974329&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036974329

VL - 132

SP - 437

EP - 461

JO - Royal Society of Edinburgh - Proceedings A

JF - Royal Society of Edinburgh - Proceedings A

SN - 0308-2105

IS - 2

ER -