Symmetry breaking in a constrained Cheeger type isoperimetric inequality

Brandolini, Barbara; Della Pietra, Francesco; Nitsch, Carlo; Trombetti, Cristina

doi:10.1051/cocv/2014016

Brandolini, Barbara ¹ ; Della Pietra, Francesco ¹ ; Nitsch, Carlo ¹ ; Trombetti, Cristina ¹

¹ Universitàdegli Studi di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Complesso Monte S. Angelo - Via Cintia, 80126 Napoli, Italia.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 359-371.

Résumé

The study of the optimal constant 𝒦_q(Ω) in the Sobolev inequality ||u||_L^q(Ω)≤1/𝒦_q(Ω)||Du||(ℝⁿ), 1≤q<1*, for BV functions which are zero outside $Ω$ and with zero mean value inside $Ω$ , leads to the definition of a Cheeger type constant. We are interested in finding the best possible embedding constant in terms of the measure of $Ω$ alone. We set up an optimal shape problem and we completely characterize, on varying the exponent $q$ , the behaviour of optimal domains. Among other things we establish the existence of a threshold value 1≤q̅<1* above which the symmetry of optimal domains is broken. Several differences between the cases $n = 2$ and $n \geq 3$ are emphasized.

Reçu le : 2013-11-07

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv/2014016

Classification : 49Q20, 39B05
Mots-clés : Cheeger inequality, optimal shape, symmetry and asymmetry

Affiliations des auteurs :

Brandolini, Barbara ¹ ; Della Pietra, Francesco ¹ ; Nitsch, Carlo ¹ ; Trombetti, Cristina ¹

¹ Universitàdegli Studi di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Complesso Monte S. Angelo - Via Cintia, 80126 Napoli, Italia.

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     title = {Symmetry breaking in a constrained {Cheeger} type isoperimetric inequality},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {359--371},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {2},
     year = {2015},
     doi = {10.1051/cocv/2014016},
     mrnumber = {3348402},
     zbl = {1319.49066},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2014016/}
}

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Brandolini, Barbara; Della Pietra, Francesco; Nitsch, Carlo; Trombetti, Cristina. Symmetry breaking in a constrained Cheeger type isoperimetric inequality. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 359-371. doi : 10.1051/cocv/2014016. https://www.numdam.org/articles/10.1051/cocv/2014016/

Bibliographie
Cité par

L. Barbosa and P. Bérard, Eigenvalue and “Twisted” eigenvalue problems, applications to CMC surfaces. J. Math. Pures Appl. 79 (2000) 427–450. | DOI | MR | Zbl

B. Brandolini, P. Freitas, C. Nitsch and C. Trombetti, Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv. Math. 228 (2011) 2352–2365. | DOI | MR | Zbl

F. Brock, F. Chiacchio and A. Mercaldo, Weighted isoperimetric inequalities in cones and applications. Nonlinear Anal. 75 (2012) 5737–5755. | DOI | MR | Zbl

A.P. Buslaev, V.A. Kondrat’Ev and A.I. Nazarov, On a family of extremal problems and related properties of an integral. Mat. Zametki 64 (1998) 830–838. English transl. Math. Notes 64 (1998) 719–725. | MR | Zbl

J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian.Problems in analysis: A symposium in honor of Salomon Bochner. Princeton University Press (1970) 195–199. | MR | Zbl

A. Cianchi, A sharp trace inequality for functions of bounded variation in the ball.Proc. of Royal Soc. Edinburgh, in vol. 142. Cambridge University Press (2012) 1179–1191. | MR | Zbl

G. Croce and B. Dacorogna, On a generalized Wirtinger inequality. Discr. Contin. Dyn. Syst. 9 (2003) 1329–1341. | DOI | MR | Zbl

G. Croce, A. Henrot and G. Pisante, An isoperimetric inequality for a nonlinear eigenvalue problem. Ann. Inst. Henri Poincaré Anal. non Linéaire 29 (2012) 21–34. | DOI | Numdam | MR | Zbl

B. Dacorogna, W. Gangbo and N. Subía, Sur une généralisation de l’inégalité de Wirtinger. Ann. Inst. Henri Poincaré Anal. Non Linéaire 9 (1992) 29–50. | DOI | Numdam | MR | Zbl

F. Della Pietra and N. Gavitone, Symmetrization for Neumann anisotropic problems and related questions. Adv. Nonlinear Stud. 12 (2012) 219–235. | DOI | MR | Zbl

F. Della Pietra and N. Gavitone, Relative isoperimetric inequality in the plane: the anisotropic case. J. Convex. Anal. 20 (2013) 157–180. | MR | Zbl

L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré-Sobolev inequalities. Arch. Rational Mech. Anal. 206 (2012) 821–851. | DOI | MR | Zbl

P. Freitas and A. Henrot, On the First Twisted Dirichlet Eigenvalue. Commun. Anal. Geom. 12 (2004) 1083–1103. | DOI | MR | Zbl

I.V. Gerasimov and A.I. Nazarov, Best constant in a three-parameter Poincaré inequality. Probl. Mat. Anal. 61 (2011) 69–86, (Russian). English transl.: J. Math. Sci. 179 (2011) 80–99. | MR | Zbl

G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge University Press, Cambridge (1988). | MR | Zbl

V.G. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations. Springer, Heidelberg (2011) | MR | Zbl

A.I. Nazarov, On exact constant in the generalized Poincaré inequality. Probl. Mat. Anal. 24 (2002) 155–180, (Russian). English transl.: J. Math. Sci. 112 (2002) 4029–4047. | MR | Zbl

A.I. Nazarov, On symmetry and asymmetry in a problem of shape optimization. (2012) 1–5. Available at . | arXiv

E. Parini, An introduction to the Cheeger problem. Surv. Math. Appl. 6 (2011) 9–21. | MR | Zbl

E. Parini, The second eigenvalue of the $p$ -Laplacian as $p$ goes to $1$ . Inter. J. Differ. Equ. (2010) DOI:. | DOI | MR | Zbl

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