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

2019/09/27 by Baptiste Trey, Trey, Baptiste · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1909.12597

openalex publication_date 2019/09/27 · openalex created_date 2022/11/17 · openalex updated_date 2026/07/28

Abstract

This paper is dedicated to the spectral optimization problem\n\
min
big

lambda₁(
Omega)+
cdots+
lambda_k(
Omega) +\n
Lambda|
Omega|
:

Omega
subset D
text quasi-open
big
\n where D\⊂\ℝ^d is a bounded open set and\n0<\λ₁(\Ω)\≤\⋯\≤\λ_k(\Ω) are the first k\neigenvalues on \Ω of an operator in divergence form with Dirichlet\nboundary condition and H "older continuous coefficients. We prove that the\nfirst k eigenfunctions on an optimal set for this problem are locally\nLipschtiz continuous in D and, as a consequence, that the optimal sets are\nopen sets. We also prove the Lipschitz continuity of vector-valued functions\nthat are almost-minimizers of a two-phase functional with variable\ncoefficients.\n

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• Vectorial Bernoulli Problems and Free Boundary Systems

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