Abstract
This paper is dedicated to the spectral optimization problemmin{λ1(Ω)+⋯+λk(Ω)+Λ|Ω| : Ω⊂Dquasi-open}whereD⊂ ℝdis a bounded open set and 0 <λ1(Ω) ≤⋯ ≤λk(Ω) are the firstkeigenvalues onΩof an operator in divergence form with Dirichlet boundary condition and Hölder continuous coefficients. We prove that the firstkeigenfunctions on an optimal set for this problem are locally Lipschtiz continuous inDand, as a consequence, that the optimal sets are open sets. We also prove the Lipschitz continuity of vector-valued functions that are almost-minimizers of a two-phase functional with variable coefficients.
Highlights
Introduction and main resultsLet D be a bounded open subset of Rd and Λ be a positive constant
We prove a Lipschitz regularity result for vectorvalued functions which are almost-minimizers of a two-phase functional with variable coefficients (Thm. 1.2)
The present paper extends this result to the case of an operator with variable coefficients, but with a completely different proof
Summary
Let D be a bounded open subset of Rd and Λ be a positive constant. We consider the spectral optimization problem min λ1(Ω) + · · · + λk(Ω) + Λ|Ω| : Ω ⊂ D quasi-open (1.1). It is natural to expect that the same holds for an operator with variable coefficients, but we will not address this question in the present paper since we are mainly interested in the Lipschitz continuity of the eigenfunctions on optimal shapes for the problem (1.1) for which the equation (1.3) is already known. Soon before the present paper was published online, a new preprint of the same authors, in collaboration with Engelstein and Smit Vega Garcia (see [7]), appeared on Arxiv They prove a regularity result for functions satisfying a suitable quasi-minimality condition for operators with variable coefficients. We always extend functions of the spaces H01(Ω) and H01(Ω, Rk) by zero outside Ω so that we have the inclusions H01(Ω) ⊂ H1(Rd) and H01(Ω, Rk) ⊂ H1(Rd, Rk)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
