Free boundary regularity for a multiphase shape optimization problem

Abstract

In this paper we prove a C 1,α regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As a consequence, we deduce the complete regularity of solutions of a multiphase shape optimization problem for the first eigenvalue of the Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a new application of the epiperimetric-inequality of [18] up to the boundary. While the framework that leads to this application is valid in every dimension, the epiperimetric inequality is known only in dimension two, thus the restriction on the dimension. 1 Remark 1.2. The Hölder continuity of the (exterior) normal vector n Ω is the best regularity result that one can expect. Indeed, recently Chang-Lara and Savin [10] showed that even for minimizers the regularity of the constrained free boundaries cannot exceed C 1, 1 /2 . Moreover, we notice that the result analogous to Theorem 1.1 was proved in any dimension in [10], by a viscosity approach, but only for minimizers of the functional J op .Analogously, we say that the nonnegative function u : B 2 → R is a almost-minimizer of the one-phase functional J op in B 2 , if u ∈ H 1 (B 2 ) and there are constants r 1 > 0, C 1 > 0 and δ 1 > 0 such that, for every x 0 ∈ B 1 ∩ ∂Ω u and r ∈ (0, r 1 ), we haveThe regularity of the unconstrained one-phase free boundary ∂Ω u follows directly by Theorem 1.1. For the sake of completeness, we give the precise statement in Corollary 1.3 below.Corollary 1.3 (Regularity of the unconstrained one-phase free boundaries). Let B 2 ⊂ R 2 and u : B 2 → R be a non-negative and Lipschitz continuous function. If u is a almost-minimizer of the functional J op in B 2 , then the free boundary B 1 ∩ ∂Ω u is locally the graph of a C 1,α function.

Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients

“…As already pointed out, the proof of Theorem 1.1 goes through the study of the Lipschitz regularity of vector-valued almost-minimisers for a two-phase functional with variable coefficients. Our approach is to reduce from the non-constant coefficients case to the constant coefficientsone by a change of variables and is inspired by [16], where the authors prove free boundary regularity of almost-minimizers of the one-phase and two-phase functionals in dimension 2 using an epiperimetric inequality. The second contribution which was a strong inspiration for our work is of David and Toro in [8].…”

Existence and regularity of optimal shapes for elliptic operators with drift

“…] and [?]. An alternative approach in dimension two, based on the epiperimetric inequality from [38], was recently introduced in [37], where Theorem 1.2 (6) is proved in the case τ = 0 and d = 2. We notice that the method from [37] can be applied to give an alternative proof of Theorem 1.2 (6) in the case τ > 0, but the restriction on the dimension is required by the epiperimetric inequality and for now cannot be removed.…”

“…Thus, the blow-up limits in this case are still one-homogeneous. We refer for instance to [37,Proposition 4.3] and Lemma 5.37 below. where ν ∈ ∂B 1 is some unit vector and q is a constant such that q ≥ Λ u e Φ(x 0 ) .…”

“…Thus, the blow-up v 0 of v is one-homogeneous solution of the constrained problem for Λ = Λ u e Φ(x 0 ) . This implies (in any dimension) that v 0 (x) = qx + d for some q ≥ √ Λ (see [37,Proposition 4.3]). Finally, (6) follows by the definition of the regular part and claim (5).…”

Regularity of almost-minimizers of Hölder-coefficient surface energies

“…We are interested in studying the boundary regularity of those sets of locally finite perimeter which are almost-minimizers of the F A -surface energy in an open set when compared to their local compactly contained variations. Recent work addressing regularity of almost-minimizers for other variational problems can be found in [41,16,8,26] and the notions of almost-minimizers we consider are similar. Fix universal constants n ≥ 2, 0 < λ ≤ Λ < +∞, κ ≥ 0, α ∈ (0, 1) and r 0 ∈ (0, +∞), and let A = (a ij (x)) n i,j=1 be a symmetric, uniformly elliptic, and Hölder continuous with respect to λ, Λ, and α and fix an open set U in R n .…”