Spectral optimization problems with internal constraint
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 477-495.

We consider spectral optimization problems with internal inclusion constraints, of the form

min {λ k (Ω):DΩ d ,|Ω|=m},
where the set D is fixed, possibly unbounded, and λ k is the k-th eigenvalue of the Dirichlet Laplacian on Ω. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

DOI : 10.1016/j.anihpc.2012.10.002
Classification : 49J45, 49R05, 35P15, 47A75, 35J25
Mots clés : Shape optimization, Capacity, Eigenvalues, Sobolev spaces, Concentration-compactness
@article{AIHPC_2013__30_3_477_0,
     author = {Bucur, Dorin and Buttazzo, Giuseppe and Velichkov, Bozhidar},
     title = {Spectral optimization problems with internal constraint},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {477--495},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.10.002},
     zbl = {1287.49049},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.002/}
}
TY  - JOUR
AU  - Bucur, Dorin
AU  - Buttazzo, Giuseppe
AU  - Velichkov, Bozhidar
TI  - Spectral optimization problems with internal constraint
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 477
EP  - 495
VL  - 30
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.002/
DO  - 10.1016/j.anihpc.2012.10.002
LA  - en
ID  - AIHPC_2013__30_3_477_0
ER  - 
%0 Journal Article
%A Bucur, Dorin
%A Buttazzo, Giuseppe
%A Velichkov, Bozhidar
%T Spectral optimization problems with internal constraint
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 477-495
%V 30
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.002/
%R 10.1016/j.anihpc.2012.10.002
%G en
%F AIHPC_2013__30_3_477_0
Bucur, Dorin; Buttazzo, Giuseppe; Velichkov, Bozhidar. Spectral optimization problems with internal constraint. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 477-495. doi : 10.1016/j.anihpc.2012.10.002. http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.002/

[1] H.W. Alt, L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144 | EuDML | Zbl

[2] M.S. Ashbaugh, Open problems on eigenvalues of the Laplacian, Analytic and Geometric Inequalities and Applications, Math. Appl. vol. 478, Kluwer Acad. Publ., Dordrecht (1999), 13-28 | Zbl

[3] J. Baxter, G. Dal Maso, Stopping times and Γ-convergence, Trans. Amer. Math. Soc. 303 no. 1 (1987), 1-38 | Zbl

[4] T. Briançon, J. Lamboley, Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 4 (2009), 1149-1163 | EuDML | Zbl

[5] D. Bucur, Uniform concentration-compactness for Sobolev spaces on variable domains, J. Differential Equations 162 (2000), 427-450 | Zbl

[6] D. Bucur, Minimization of the k-th eigenvalue of the Dirichlet Laplacian, Arch. Ration. Mech. Anal. 206 no. 3 (2012), 1073-1083 | Zbl

[7] D. Bucur, G. Buttazzo, Variational Methods in Shape Optimization Problems, Progr. Nonlinear Differential Equations Appl. vol. 65, Birkhäuser Verlag, Basel (2005) | Zbl

[8] D. Bucur, G. Buttazzo, I. Figueiredo, On the attainable eigenvalues of the Laplace operator, SIAM J. Math. Anal. 30 no. 3 (1999), 527-536 | Zbl

[9] G. Buttazzo, Spectral optimization problems, Rev. Mat. Complut. 24 no. 2 (2011), 277-322 | Zbl

[10] G. Buttazzo, G. Dal Maso, Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim. 23 (1991), 17-49 | Zbl

[11] G. Buttazzo, G. Dal Maso, An existence result for a class of shape optimization problems, Arch. Ration. Mech. Anal. 122 (1993), 183-195 | Zbl

[12] G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston (1993)

[13] G. Dal Maso, U. Mosco, Wienerʼs criterion and Γ-convergence, Appl. Math. Optim. 15 (1987), 15-63

[14] G. Dal Maso, F. Murat, Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. 24 (1997), 239-290 | EuDML | Numdam | Zbl

[15] E. Davies, Heat Kernels and Spectral Theory, Cambridge University Press (1989) | Zbl

[16] B. Fuglede, Finely Harmonic Functions, Lecture Notes in Math. vol. 289, Springer-Verlag, Berlin–New York (1972) | Zbl

[17] A. Henrot, Minimization problems for eigenvalues of the Laplacian, J. Evol. Equ. 3 no. 3 (2003), 443-461 | Zbl

[18] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Front. Math., Birkhäuser Verlag, Basel (2006) | Zbl

[19] A. Henrot, M. Pierre, Variation et Optimisation de Formes, Anal. Geom., Math. Appl. vol. 48, Springer-Verlag, Berlin (2005)

[20] V. Latvala, A theorem on fine connectedness, Potential Anal. 12 no. 3 (2000), 221-232 | Zbl

[21] D. Mazzoleni, A. Pratelli, Existence of minimizers for spectral problems, http://cvgmt.sns.it (2011)

[22] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 no. 2 (1984), 109-145 | EuDML | Numdam | Zbl

[23] W.P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin (1989) | Zbl

Cité par Sources :