Lipschitz Regularity of the Eigenfunctions on Optimal Domains
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- 作者：Dorin Bucur (1)
Dario Mazzoleni (2)
Aldo Pratelli (3)
Bozhidar Velichkov (4)
1. Laboratoire de Math茅matiques (LAMA) ; Universit茅 de Savoie ; Campus Scientifique ; 73376 ; Le-Bourget-Du-Lac ; France
2. Dipartimento di Matematica ; Universit脿 degli Studi di Pavia ; via Ferrata ; 1 ; 27100 ; Pavia ; Italy
3. Department Mathematik ; Friederich-Alexander Universit盲t Erlangen-N眉rnberg ; Cauerstrasse ; 11 ; 91058 ; Erlangen ; Germany
4. Scuola Normale Superiore di Pisa ; Piazza dei Cavalieri 7 ; 56126 ; Pisa ; Italy - 刊名：Archive for Rational Mechanics and Analysis
- 出版年：2015
- 出版时间：April 2015
- 年：2015
- 卷：216
- 期：1
- 页码：117-151
- 全文大小：395 KB
- 参考文献：1. Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. / J. Reine Angew. Math. 325, 105鈥?44 (1981)
2. Alt, H.W., Caffarelli, L.A., Friedman, A.: Variational problems with two phases and their free boundaries. / Trans. Amer. Math. Soc. 282(2), 431鈥?61 (1984) CrossRef
3. Antunes, P., Freitas, P.: Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. / J. Optim. Theory Appl. 154(1), 235鈥?57 (2012) CrossRef
4. Ashbaugh, M.S.: Open problems on eigenvalues of the Laplacian. In: / Analytic and Geometric Inequalities and Applications. Mathematical Applications, vol. 478, pp. 13鈥?8. Kluwer Acad. Publ., Dordrecht, 1999
5. Brian莽on, T., Lamboley, J.: Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints. / Ann. I. H. Poincar茅 26(4), 1149鈥?163 (2009) CrossRef
6. Brian莽on, T., Hayouni, M., Pierre, M.: Lipschitz continuity of state functions in some optimal shaping. / Calc. Var. Partial Differ. Equ. 23(1), 13鈥?2 (2005) CrossRef
7. Bucur, D.: Minimization of the / k-th eigenvalue of the Dirichlet Laplacian. / Arch. Rat. Mech. Anal. 206(3), 1073鈥?083 (2012) CrossRef
8. Bucur, D., Buttazzo, G.: / Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations, vol. 65. Birkh盲user Verlag, Basel, 2005
9. Bucur, D., Buttazzo, G., Velichkov, B.: Spectral optimization problems with internal constraint. / Ann. I. H. Poincar茅 30(3), 477鈥?95 (2013) CrossRef
10. Bucur, D., Velichkov, B.: Multiphase shape optimization problems. / SIAM J. Contr. Optim. (2014, accepted)
11. Buttazzo, G.: Spectral optimization problems. / Rev. Mat. Complut. 24(2), 277鈥?22 (2011) CrossRef
12. Buttazzo, G., Dal Maso, G.: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. / Appl. Math. Optim. 23, 17鈥?9 (1991) CrossRef
13. Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. / Arch. Rational Mech. Anal. 122, 183鈥?95 (1993) CrossRef
14. Caffarelli, L., Jerison, D., Kenig, C.: Some new monotonicity theorems with applications to free boundary problems. / Ann. Math. 155(2), 369鈥?04 (2002) CrossRef
15. Dal Maso, G., Mosco, U.: Wiener criteria and energy decay for relaxed Dirichlet problems. / Arch. Rat. Mech. Anal. 95, 345鈥?87 (1986) CrossRef
16. Dal Maso, G., Mosco, U.: Wiener鈥檚 criterion and \({\Gamma}\) -convergence. / Appl. Math. Optim. 15, 15鈥?3 (1987) CrossRef
17. Davies, E.: / Heat Kernels and Spectral Theory. Cambridge University Press, 1989
18. De Philippis, G., Velichkov, B.: Existence and regularity of minimizers for some spectral optimization problems with perimeter constraint. / Appl. Math. Optim. 69, 199鈥?31 (2014) CrossRef
19. Gilbarg, D., Trudinger, N.S.: / Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Classics in Mathematics. Springer, Berlin (2001)
20. Henrot, A., Pierre, M.: / Variation et Optimisation de Formes. Une Analyse G茅om茅trique, Math茅matiques and Applications, vol. 48. Springer, Berlin (2005)
21. Mazzoleni, D.: Boundedness of minimizers for spectral problems in \({\mathbb{R}^N}\) , preprint. http://cvgmt.sns.it/person/977/ (2013)
22. Mazzoleni, D.: Existence and regularity results for solutions of spectral problems. Ph.D. Thesis, Universit脿 脿 di Pavia and Friedrich-Alexander Universit盲t Erlangen-N眉rnberg (in preparation) (2014)
23. Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. / J. Math. Pures Appl. 100(3), 433鈥?53 (2013) CrossRef
24. Oudet, E.: Numerical minimization of eigenmodes of a membrane with respect to the domain. / ESAIM:COCV 10(3), 315鈥?30 (2004) CrossRef
25. Talenti, G.: Elliptic equations and rearrangements. / Ann. Scuola Normale Superiore di Pisa 3(4), 697鈥?18.
26. Rayleigh, J.W.S.: / The Theory of Sound, 1st edn. Macmillan, London (1877)
27. Wolf, S.A., Keller, J.B.: Range of the first two eigenvalues of the Laplacian. / Proc. R. Soc. Lond., 447, 397鈥?12 (1994) CrossRef - 刊物类别：Physics and Astronomy
- 刊物主题：Physics
Mechanics
Electromagnetism, Optics and Lasers
Mathematical and Computational Physics
Complexity
Fluids - 出版者：Springer Berlin / Heidelberg
- ISSN：1432-0673
文摘
We study the optimal sets \({\Omega^\ast\subseteq\mathbb{R}^d}\) for spectral functionals of the form \({F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}\) , which are bi-Lipschitz with respect to each of the eigenvalues \({\lambda_1(\Omega), \lambda_2(\Omega)}, \ldots, {\lambda_p(\Omega)}\) of the Dirichlet Laplacian on \({\Omega}\) , a prototype being the problem $$\min{\big\{\lambda_1(\Omega)+\cdots+\lambda_p(\Omega)\;:\;\Omega\subseteq\mathbb{R}^d,\ |\Omega|=1\big\}}.$$ We prove the Lipschitz regularity of the eigenfunctions \({u_1,\ldots,u_p}\) of the Dirichlet Laplacian on the optimal set \({\Omega^\ast}\) and, as a corollary, we deduce that \({\Omega^\ast}\) is open. For functionals depending only on a generic subset of the spectrum, as for example \({\lambda_k(\Omega)}\) , our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.