The study of the optimal constant 𝒦q(Ω) in the Sobolev inequality ||u||Lq(Ω)≤1/𝒦q(Ω)||Du||(ℝn), 1≤q<1*, for BV functions which are zero outside and with zero mean value inside , 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 alone. We set up an optimal shape problem and we completely characterize, on varying the exponent , 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 and are emphasized.
DOI : 10.1051/cocv/2014016
Mots clés : Cheeger inequality, optimal shape, symmetry and asymmetry
@article{COCV_2015__21_2_359_0,
author = {Brandolini, Barbara and Della Pietra, Francesco and Nitsch, Carlo and Trombetti, Cristina},
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 = {http://www.numdam.org/articles/10.1051/cocv/2014016/}
}
TY - JOUR AU - Brandolini, Barbara AU - Della Pietra, Francesco AU - Nitsch, Carlo AU - Trombetti, Cristina TI - Symmetry breaking in a constrained Cheeger type isoperimetric inequality JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 359 EP - 371 VL - 21 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014016/ DO - 10.1051/cocv/2014016 LA - en ID - COCV_2015__21_2_359_0 ER -
%0 Journal Article %A Brandolini, Barbara %A Della Pietra, Francesco %A Nitsch, Carlo %A Trombetti, Cristina %T Symmetry breaking in a constrained Cheeger type isoperimetric inequality %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 359-371 %V 21 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014016/ %R 10.1051/cocv/2014016 %G en %F COCV_2015__21_2_359_0
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. http://www.numdam.org/articles/10.1051/cocv/2014016/
and , Eigenvalue and “Twisted” eigenvalue problems, applications to CMC surfaces. J. Math. Pures Appl. 79 (2000) 427–450. | DOI | MR | Zbl
, , and , Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv. Math. 228 (2011) 2352–2365. | DOI | MR | Zbl
, and , Weighted isoperimetric inequalities in cones and applications. Nonlinear Anal. 75 (2012) 5737–5755. | DOI | MR | Zbl
, and , 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
and , On a generalized Wirtinger inequality. Discr. Contin. Dyn. Syst. 9 (2003) 1329–1341. | DOI | MR | Zbl
, and , An isoperimetric inequality for a nonlinear eigenvalue problem. Ann. Inst. Henri Poincaré Anal. non Linéaire 29 (2012) 21–34. | DOI | Numdam | MR | Zbl
, and , Sur une généralisation de l’inégalité de Wirtinger. Ann. Inst. Henri Poincaré Anal. Non Linéaire 9 (1992) 29–50. | DOI | Numdam | MR | Zbl
and , Symmetrization for Neumann anisotropic problems and related questions. Adv. Nonlinear Stud. 12 (2012) 219–235. | DOI | MR | Zbl
and , Relative isoperimetric inequality in the plane: the anisotropic case. J. Convex. Anal. 20 (2013) 157–180. | MR | Zbl
, , , and , The longest shortest fence and sharp Poincaré-Sobolev inequalities. Arch. Rational Mech. Anal. 206 (2012) 821–851. | DOI | MR | Zbl
and , On the First Twisted Dirichlet Eigenvalue. Commun. Anal. Geom. 12 (2004) 1083–1103. | DOI | MR | Zbl
and , 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
, 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
, An introduction to the Cheeger problem. Surv. Math. Appl. 6 (2011) 9–21. | MR | Zbl
E. Parini, The second eigenvalue of the -Laplacian as goes to . Inter. J. Differ. Equ. (2010) DOI:. | DOI | MR | Zbl
Cité par Sources :
