Shape derivative of the Cheeger constant. (English) Zbl 1315.49018

Summary: This paper deals with the existence of the shape derivative of the Cheeger constant \({h}_{1}({\Omega})\) of a bounded domain {\(\Omega\)}. We prove that if {\(\Omega\)} admits a unique Cheeger set, then the shape derivative of \({h}_{1}({{\Omega}})\) exists, and we provide an explicit formula. A counter-example shows that the shape derivative may not exist without the uniqueness assumption.


49Q10 Optimization of shapes other than minimal surfaces
49Q20 Variational problems in a geometric measure-theoretic setting
Full Text: DOI Link


[1] F. Alter and V. Caselles, Uniqueness of the Cheeger set of a convex body. Nonlin. Anal.70 (2009) 32-44. · Zbl 1167.52005
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press (2000). · Zbl 0957.49001
[3] G. Carlier and M. Comte, On a weighted total variation minimization problem. J. Funct. Anal.250 (2007) 214-226. · Zbl 1120.49011
[4] 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. · Zbl 1198.49042
[5] 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.
[6] F. Demengel, Functions locally almost 1-harmonic. Appl. Anal.83 (2004) 865-896. · Zbl 1135.35333
[7] 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. · Zbl 1114.35072
[8] J. Garcia Melian and J. Sabina De Lis, On the perturbation of eigenvalues for the p-Laplacian. C.R. Acad. Sci. Paris332 (2001) 893-898. · Zbl 0989.35103
[9] E. Giusti, Minimal surfaces and functions of bounded variation. In vol. 80 of Monogr. Math. Birkhäuser Verlag, Basel (1984). · Zbl 0545.49018
[10] 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. · Zbl 1099.58012
[11] A. Henrot and M. Pierre, Variation et optimisation de formes. Springer (2005). · Zbl 1098.49001
[12] 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. · Zbl 1105.35029
[13] B. Kawohl and T. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane. Pacific J. Math.225 (2006) 103-118. · Zbl 1133.52002
[14] B. Kawohl and F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem. Commun. Contemp. Math.9 (2007) 1-29. · Zbl 1146.35023
[15] D. Krejčiřík and A. Pratelli, The Cheeger constant of curved strips. Pacific J. Math.254 (2011) 309-333. · Zbl 1247.28003
[16] 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. · Zbl 1054.35046
[17] 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. · Zbl 1304.35469
[18] E. Parini, An introduction to the Cheeger problem. Surveys Math. Appl.6 (2011) 9-22. · Zbl 1399.49023
[19] J.D. Rossi and N. Saintier, On the 1st eigenvalue of the \infty-Laplacian with Neumann boundary conditions, To appear in Houston J. · Zbl 1352.35084
[20] N. Saintier, Estimates of the best Sobolev constant of the embedding of BV({\(\Omega\)}) into L^{1}(\partial{\(\Omega\)}) and related shape optimization problems. Nonlinear Anal.69 (2008) 2479-2491. · Zbl 1151.49036
[21] 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. · Zbl 0940.49025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.