The multi-layer free boundary problem for the p-Laplacian in convex domains

Abstract

The main result of this paper concerns existence of classical solutions to the multi-layer Bernoulli free boundary problem with nonlinear joining conditions and the p-Laplacian as governing operator. The present treatment of the two-layer case involves technical refinements of the one-layer case, studied earlier by two of the authors. The existence treatment of the multi-layer case is largely based on a reduction to the two-layer case, in which uniform separation of the free boundaries plays a key role.

Shape optimization of systems governed by Bernoulli free boundary problems

“…Indeed, in [1] the following results have been proven: (2). Then for every ω ∈ O problem (P(ω)) has a unique solution (Ω(ω), u(ω)).…”

“…= ω such that the outer component Γ e (Ξ) of the double connected domain Ξ = Ω \ ω is the free boundary in (1).…”

A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition

“…Usefulness. As mentioned in the introduction, the result in this paper could probably be extended to results similar to those in [8] and [2] for our case. This together with results similar to those in [7] might yield a proof of the existence of solutions to the following problem    ∆ p u = f in Ω , u = 1 on K , u = 0 on ∂Ω \ K , for certain class of functions f .…”

“…See for example [1], [3] and [5]. We also see a possibility to extend the results in [2] and [8] to be valid in our case.…”

“…In [2], the authors prove this lemma in a more simple case, in the sense that the convex domain K doesn't belong to the boundary of Ω. Lemma 3 prove that the gradient of u Ω is bounded even if K ∩ Ω = ∅.…”