Best constant in Poincaré inequalities with traces: a free discontinuity approach. (English) Zbl 1428.49043
Summary: For \(\Omega \subset \mathbb{R}^N\) open bounded and with a Lipschitz boundary, and \(1 \leq p < + \infty\), we consider the Poincaré inequality with trace term \[C_p(\Omega) \| u \|_{L^p(\Omega)} \leq \| \nabla u \|_{L^p(\Omega; \mathbb{R}^N)} + \| u \|_{L^p(\partial \Omega)}\] on the Sobolev space \(W^{1, p}(\Omega)\). We show that among all domains \(\Omega\) with prescribed volume, the constant is minimal on balls. The proof is based on the analysis of a free discontinuity problem.
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 26A45 | Functions of bounded variation, generalizations |
| 35R35 | Free boundary problems for PDEs |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 49K20 | Optimality conditions for problems involving partial differential equations |
Keywords:
free discontinuity problems; functions of bounded variation; sets of finite perimeter; Robin boundary conditionsReferences:
| [1] | Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs (2000), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0957.49001 |
| [2] | Bossel, M.-H., Membranes élastiquement liées inhomogènes ou sur une surface: une nouvelle extension du théorème isopérimétrique de Rayleigh-Faber-Krahn, Z. Angew. Math. Phys., 39, 5, 733-742 (1988) · Zbl 0672.73015 |
| [3] | Bucur, D.; Buttazzo, G.; Nitsch, C., Symmetry breaking for a problem in optimal insulation, J. Math. Pures Appl. (9), 107, 4, 451-463 (2017) |
| [4] | Bucur, D.; Daners, D., An alternative approach to the Faber-Krahn inequality for Robin problems, Calc. Var. Partial Differ. Equ., 37, 1-2, 75-86 (2010) · Zbl 1186.35118 |
| [5] | Bucur, D.; Giacomini, A., A variational approach to the isoperimetric inequality for the Robin eigenvalue problem, Arch. Ration. Mech. Anal., 198, 3, 927-961 (2010) · Zbl 1228.49049 |
| [6] | Bucur, D.; Giacomini, A., Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity approach, Arch. Ration. Mech. Anal., 218, 2, 757-824 (2015) · Zbl 1458.35286 |
| [7] | Bucur, D.; Giacomini, A., Shape optimization problems with Robin conditions on the free boundary, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 6, 1539-1568 (2016) · Zbl 1352.49045 |
| [8] | Burago, Yu. D.; Zalgaller, V. A., Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285 (1988), Springer-Verlag: Springer-Verlag Berlin, Translated from the Russian by A.B. Sosinskiĭ, Springer Series in Soviet Mathematics · Zbl 0633.53002 |
| [9] | Caffarelli, L. A.; Kriventsov, D., A free boundary problem related to thermal insulation, Commun. Partial Differ. Equ., 41, 7, 1149-1182 (2016) · Zbl 1351.35268 |
| [10] | Carlier, Guillaume; Comte, Myriam, On a weighted total variation minimization problem, J. Funct. Anal., 250, 1, 214-226 (2007) · Zbl 1120.49011 |
| [11] | Cianchi, A.; Fusco, N., Functions of bounded variation and rearrangements, Arch. Ration. Mech. Anal., 165, 1, 1-40 (2002) · Zbl 1028.49035 |
| [12] | Dai, Q.; Fu, Y., Faber-Krahn inequality for Robin problems involving p-Laplacian, Acta Math. Appl. Sin. Engl. Ser., 27, 1, 13-28 (2011) · Zbl 1209.35093 |
| [13] | Daners, D., A Faber-Krahn inequality for Robin problems in any space dimension, Math. Ann., 335, 4, 767-785 (2006) · Zbl 1220.35103 |
| [14] | De Giorgi, E.; Carriero, M.; Leaci, A., Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal., 108, 3, 195-218 (1989) · Zbl 0682.49002 |
| [15] | DiBenedetto, E., \(C^{1 + \alpha}\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7, 8, 827-850 (1983) · Zbl 0539.35027 |
| [16] | Maggi, F.; Villani, C., Balls have the worst best Sobolev inequalities. II. Variants and extensions, Calc. Var. Partial Differ. Equ., 31, 1, 47-74 (2008) · Zbl 1135.46016 |
| [17] | Maggi, Francesco; Villani, Cédric, Balls have the worst best Sobolev inequalities, J. Geom. Anal., 15, 1, 83-121 (2005) · Zbl 1086.46021 |
| [18] | Morgan, F.; Howe, S.; Harman, N., Steiner and Schwarz symmetrization in warped products and fiber bundles with density, Rev. Mat. Iberoam., 27, 3, 909-918 (2011) · Zbl 1229.49046 |
| [19] | Morgan, F.; Pratelli, A., Existence of isoperimetric regions in \(R^n\) with density, Ann. Glob. Anal. Geom., 43, 4, 331-365 (2013) · Zbl 1263.49049 |
| [20] | Siaudeau, A., Ahlfors-régularité des quasi-minima de Mumford-Shah, J. Math. Pures Appl. (9), 82, 12, 1697-1731 (2003) · Zbl 1034.49045 |
| [21] | Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110, 353-372 (1976) · Zbl 0353.46018 |
| [22] | Walter, W., Sturm-Liouville theory for the radial \(\Delta_p\)-operator, Math. Z., 227, 1, 175-185 (1998) · Zbl 0915.34022 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
