This paper is dedicated to the spectral optimization problem
where D ⊂ ℝ$$ is a bounded open set and 0 < λ1(Ω) ≤⋯ ≤ λ$$(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.
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DOI : 10.1051/cocv/2020010
Mots-clés : Spectral optimization problem, almost-minimizer, free boundary problem, the two-phase problem
@article{COCV_2020__26_1_A89_0,
author = {Trey, Baptiste},
title = {Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2020010},
mrnumber = {4173855},
zbl = {1459.35027},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2020010/}
}
TY - JOUR AU - Trey, Baptiste TI - Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2020010/ DO - 10.1051/cocv/2020010 LA - en ID - COCV_2020__26_1_A89_0 ER -
%0 Journal Article %A Trey, Baptiste %T Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2020010/ %R 10.1051/cocv/2020010 %G en %F COCV_2020__26_1_A89_0
Trey, Baptiste. Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 89. doi : 10.1051/cocv/2020010. http://www.numdam.org/articles/10.1051/cocv/2020010/
[1] and , Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981) 105–144. | MR | Zbl
[2] , and , Variational problems with two phases and their free boundary. Trans. Am. Math. Soc. 282 (1984) 431–461. | DOI | MR | Zbl
[3] , and , Lipschitz continuity of state functions in some optimal shaping. Calc. Var. Partial Differ. Equ. 23 (2005) 13–32. | DOI | MR | Zbl
[4] , , and , Lipschitz regularity of the eigenfunctions on optimal domains. Arch. Ration. Mech. Anal. 216 (2015) 117–151. | DOI | MR | Zbl
[5] , and , Global energy minimizers for free boundary problems and full regularity in three dimensions. In Noncompact problems at the intersection of geometry, analysis, and topology. Proceedings of the conference on noncompact variational problems and general relativity held in honor of Haim Brezis and Felix Browder at Rutgers University, New Brunswick, NJ, USA, October 14–18, 2001. American Mathematical Society (AMS), Providence, RI (2004) 83–97. | MR | Zbl
[6] and , Regularity of almost minimizers with free boundary. Calc. Variat. Partial Differ. Equ. 54 (2015) 455–524. | DOI | MR | Zbl
[7] , , and , Regularity for almost-minimizers of variable coefficient Bernoulli-type functional. Preprint (2019). | arXiv | MR
[8] , and , Free boundary regularity for almost-minimizers. Adv. Math. 350 (2019) 1109–1192. | DOI | MR | Zbl
[9] and , Elliptic partial differential equations of second order. Reprint ofthe 1998. Springer (Berlin), (2001). | DOI | MR | Zbl
[10] and , Variation et optimisation de formes. Une analyse géométrique, vol. 48. Springer, Berlin (2005). | MR | Zbl
[11] and , Existence and regularity of Faber Krahn minimizers in a Riemannian manifold. Preprint (2019). | arXiv | MR
[12] , Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory, vol. 135. Cambridge University Press, Cambridge (2012). | MR | Zbl
[13] and , Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients. Commun. Pure Appl. Math. 64 (2011) 271–311. | DOI | MR | Zbl
[14] , and , Regularity of the optimal sets for some spectral functionals. Geom. Funct. Anal. 27 (2017) 373–426. | DOI | MR | Zbl
[15] , and , Existence and regularity of optimal shapes for elliptic operators with drift. Calc. Variat. Part. Differ. Equ. 58 (2019) 199. | DOI | MR | Zbl
[16] , and , Free boundary regularity for a multiphase shape optimization problem. Commun. Partial Differ. Equ. (2019) 1–32. | MR | Zbl
[17] , Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form. Preprint (2020). | arXiv | MR
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