The fractional Cheeger problem

Interfaces and Free Boundaries

Volumn 16, Issue 3, 2014, Pages 419-458

  Brasco, L ^a   Lindgren, E ^b   Parini, E ^a  

^a AIX MARSEILLE UNIVERSITY   (France)

^b ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

Almost minimal surfaces; Cheeger constant; Non local eigenvalue problems

Indexed keywords

EID: 84905001290     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/325     Document Type: Article

References (33)

1. ADAMS, R. A. & FOURNIER, J. J. F., Sobolev spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. Zbl1098.46001 MR2424078
2. ALMGREN, F. J. JR. & LIEB, E. H., Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), 683-773. Zbl0688.46014 MR1002633
3. ALTER, F. & CASELLES, V., Uniqueness of the Cheeger set of a convex body, Nonlinear Anal. 70 (2009), 32-44. Zbl1167.52005 MR2468216
4. AMBROSIO, L., DE PHILIPPIS, G. & MARTINAZZI, L., Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math. 134 (2011), 377-403. Zbl1207.49051 MR2765717
5. AMBROSIO, L., FUSCO, N. & PALLARA, D., Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000). Zbl0957.49001 MR1857292
6. CAFFARELLI, L., ROQUEJOFFRE, J.-M. & SAVIN, O., Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111-1144. MR2675483
7. CAPUTO, M.-C. & GUILLEN, N., Regularity for non-local almost minimal boundaries and applications, preprint (2010) available at http://arxiv.org/abs/1003.2470
8. CARLIER, G. & COMTE, M., On a weighted total variation minimization problem, J. Func. Anal. 250 (2007), 214-226. Zbl1120.49011 MR2345913
9. CASELLES, V., CHAMBOLLE, A. & NOVAGA, M., Uniqueness of the Cheeger set of a convex body, Pacific J. Math. 232 (2007), 77-90. Zbl1221.35171 MR2358032
10. CHEEGER, J., A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, N. J., 1970, pp. 195-199. Zbl0212.44903 MR0402831
11. DÁVILA, J., On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations 15 (2002), 519-527. Zbl1047.46025 MR1942130
12. DI NEZZA, E., PALATUCCI, G. & VALDINOCI, E., Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. Zbl1252.46023 MR2944369
13. DYDA, B., A fractional order Hardy inequality, Illinois J. Math. 48 (2004), 575-588. Zbl1068.26014 MR2085428
14. EKELAND, I., Convexity methods in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1990. Zbl0707.70003 MR1051888
15. FRANK, R. L. & SEIRINGER, R., Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), 3407-3430. Zbl1189.26031 MR2469027
16. FRANZINA, G. & PALATUCCI, G., Fractional p-eigenvalues, to appear on Riv. Mat. Univ. Parma (2013), available at http://cvgmt.sns.it/paper/2168/
17. FRANZINA, G. & VALDINOCI, E., Geometric analysis of fractional phase transition interfaces, in Geometric Properties for Parabolic and Elliptic PDEs, Springer INdAM Series Volume 2, 2013, 117-130. Zbl06192744 MR3050230
18. FRIDMAN, V. & KAWOHL, B., Isoperimetric estimates for the first eigenvalue of the p -Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolinae 44 (2003), 659-667. Zbl1105.35029 MR2062882
19. FUSCO, N., MILLOT, V. & MORINI, M., A quantitative isoperimetric inequality for fractional perimeters, J. Funct. Anal. 261 (2011), 697-715. Zbl1228.46030 MR2799577
20. GIUSTI, E., Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. Zbl1028.49001 MR1962933
21. GRIESER, D., The first eigenvalue of the Laplacian, isoperimetric constants, and the max flow min cut theorem, Arch. Math. (Basel) 87 (2006), 75-85. Zbl1105.35062 MR2246409
22. KAWOHL, B. & LACHAND-ROBERT, T., Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math. 225 (2006), 103-118. Zbl1133.52002 MR2233727
23. KREJČIŘÍK, D. & PRATELLI, A., The Cheeger constant of curved strips, Pacific J. Math. 254 (2011), 309-333. Zbl1247.28003 MR2900018
24. LEONI, G. & SPECTOR, D., Characterization of Sobolev and BV spaces, J. Func. Anal. 261 (2011), 2926-2958 Zbl1236.46032 MR2832587
25. LINDGREN, E. & LINDQVIST, P., Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795-826. Zbl06259060 MR3148135
26. LITTIG, S. & SCHURICHT, F., Convergence of the eigenvalues of the p-Laplace operator as p goes to 1, Calc. Var. Partial Differential Equations 40 (2014), 707-727. Zbl1282.35267 MR3148132
27. LUDWIG, M., Anisotropic fractional perimeters, J. Differ. Geom. 96 (2014), 77-93 . Zbl06282558 MR3161386
28. MAGNANINI, R.&PAPI, G.,An inverse problem for the Helmholtz equation, Inverse Problems 1 (1985), 357-370. Zbl0608.35076 MR0824135
29. MAZ'JA, V. G., Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Zbl0692.46023 MR0817985
30. PARINI, E., An introduction to the Cheeger problem, Surv. Math. Appl. 6 (2011), 9-21. MR2832554
31. SAVIN, O. & VALDINOCI, E., Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), 33-39. Zbl1275.35065 MR3090533
32. STRANG, G., Maximal flow through a domain, Math. Programming 26 (1983), 123-143. Zbl0513. 90026 MR0700642
33. VISINTIN, A., Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 (1991), 175-201. Zbl0736.49030 MR1111612