Issue |
ESAIM: COCV
Volume 26, 2020
|
|
---|---|---|
Article Number | 89 | |
Number of page(s) | 34 | |
DOI | https://doi.org/10.1051/cocv/2020010 | |
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,
38610
Gières, France.
* Corresponding author: baptiste.trey@univ-grenoble-alpes.fr
Received:
4
October
2019
Accepted:
9
March
2020
This paper is dedicated to the spectral optimization problem
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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