Volume 26, 2020
|Number of page(s)||34|
|Published online||13 November 2020|
Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients
Université Grenoble Alpes, CNRS UMR 5582, Institut Fourier,
* Corresponding author: firstname.lastname@example.org
Accepted: 9 March 2020
where D ⊂ ℝd is a bounded open set and 0 < λ1(Ω) ≤⋯ ≤ λk(Ω) are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and Hölder continuous coefficients. We prove that the first k eigenfunctions on an optimal set for this problem are locally Lipschtiz continuous in D and, 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.
Mathematics Subject Classification: 35R35 / 49Q10 / 47A75
Key words: Spectral optimization problem / almost-minimizer / free boundary problem / the two-phase problem
© The authors. Published by EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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