Shape derivative of the Cheeger constant

Parini, Enea; Saintier, Nicolas

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 348-358.

This paper deals with the existence of the shape derivative of the Cheeger constant $h_1\left(\Omega \right)$ of a bounded domain $\Omega$. We prove that if $\Omega$ admits a unique Cheeger set, then the shape derivative of $h_1\left(\Omega \right)$ exists, and we provide an explicit formula. A counter-example shows that the shape derivative may not exist without the uniqueness assumption.

Classification : 49Q10, 49Q20
Mots clés : Shape derivative, CHEEGER constant, 1-Laplacian

F. Alter and V. Caselles, Uniqueness of the Cheeger set of a convex body. Nonlin. Anal. 70 (2009) 32–44. | DOI | MR | Zbl

L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press (2000). | MR | Zbl

G. Carlier and M. Comte, On a weighted total variation minimization problem. J. Funct. Anal. 250 (2007) 214–226. | DOI | MR | Zbl

V. Caselles, A. Chambolle and M. Novaga, Some remarks on uniqueness and regularity of Cheeger sets, Rendiconti del Seminario Matematico della Università di Padova 123 (2010) 191–202. | Numdam | MR | Zbl

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

F. Demengel, Functions locally almost 1-harmonic. Appl. Anal. 83 (2004) 865–896. | DOI | MR | Zbl

J. Garcia-Azorero, J. Manfredi, I. Peral and J.D. Rossi, Steklov eigenvalues for the $\infty$-Laplacian. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 17 (2006) 199–210. | DOI | MR | Zbl

J. Garcia Melian and J. Sabina De Lis, On the perturbation of eigenvalues for the p-Laplacian. C.R. Acad. Sci. Paris 332 (2001) 893–898. | MR | Zbl

E. Giusti, Minimal surfaces and functions of bounded variation. In vol. 80 of Monogr. Math. Birkhäuser Verlag, Basel (1984). | MR | Zbl

E. Hebey and N. Saintier, Stability and perturbations of the domain for the first eigenvalue of the 1-Laplacian. Arch. Math. 86 (2006) 340–351. | DOI | MR | Zbl

A. Henrot and M. Pierre, Variation et optimisation de formes. Springer (2005). | MR

B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carolin. 44 (2003) 659–667. | MR | Zbl

B. Kawohl and T. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane. Pacific J. Math. 225 (2006) 103–118. | DOI | MR | Zbl

B. Kawohl and F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem. Commun. Contemp. Math. 9 (2007) 1–29. | DOI | MR | Zbl

D. Krejčiřík and A. Pratelli, The Cheeger constant of curved strips. Pacific J. Math. 254 (2011) 309–333. | DOI | MR | Zbl

P.D. Lamberti, A differentiability result for the first eigenvalue of the $p$-Laplacian upon domain perturbation, Nonlinear analysis and Applications: to V. Lakshmikantham on his 80th birthday. Vol. 1, 2. Kluwer Acad. Publ., Dordrecht (2003) 741–754. | MR | Zbl

J.C. Navarro, J.D. Rossi, N. Saintier and A. San Antolin, The dependence of the first eigenvalue of the $\infty$-Laplacian with respect to the domain, Glasg. Math. J. 56 (2014) 241–249. | DOI | MR | Zbl

E. Parini, An introduction to the Cheeger problem. Surveys Math. Appl. 6 (2011) 9–22. | MR | Zbl

J.D. Rossi and N. Saintier, On the 1st eigenvalue of the $\infty$-Laplacian with Neumann boundary conditions, To appear in Houston J. | MR

N. Saintier, Estimates of the best Sobolev constant of the embedding of $BV\left(\Omega \right)$ into $L^1\left(\partial \Omega \right)$ and related shape optimization problems. Nonlinear Anal. 69 (2008) 2479–2491. | DOI | MR | Zbl

E. Stredulinsky and W.P. Ziemer, Area minimizing sets subject to a volume constraint in a convex set. J. Geom. Anal. 7 (1997) 653–677. | DOI | MR | Zbl