Caccioppoli estimates through an anisotropic Picone’s identity
HTML articles powered by AMS MathViewer
- by Jaroslav Jaroš PDF
- Proc. Amer. Math. Soc. 143 (2015), 1137-1144 Request permission
Abstract:
Caccioppoli-type estimates for a class of nonlinear differential operators which include the $p$-Laplacian and the pseudo $p$-Laplacian as special cases are obtained by means of the differential identity involving an arbitrary norm in $\mathbb {R}^n$ which generalizes the well-known multidimensional Picone’s formula.References
- Boumediene Abdellaoui and Ireneo Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian with a critical potential, Ann. Mat. Pura Appl. (4) 182 (2003), no. 3, 247–270. MR 2000444, DOI 10.1007/s10231-002-0064-y
- Walter Allegretto and Yin Xi Huang, A Picone’s identity for the $p$-Laplacian and applications, Nonlinear Anal. 32 (1998), no. 7, 819–830. MR 1618334, DOI 10.1016/S0362-546X(97)00530-0
- Angelo Alvino, Vincenzo Ferone, Guido Trombetti, and Pierre-Louis Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire 14 (1997), no. 2, 275–293 (English, with English and French summaries). MR 1441395, DOI 10.1016/S0294-1449(97)80147-3
- M. Belloni, V. Ferone, and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. 54 (2003), no. 5, 771–783. Special issue dedicated to Lawrence E. Payne. MR 2019179, DOI 10.1007/s00033-003-3209-y
- G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J. 25 (1996), no. 3, 537–566. MR 1416006, DOI 10.14492/hokmj/1351516749
- Andrea Cianchi and Paolo Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009), no. 4, 859–881. MR 2545870, DOI 10.1007/s00208-009-0386-9
- Francesco Della Pietra and Nunzia Gavitone, Symmetrization for Neumann anisotropic problems and related questions, Adv. Nonlinear Stud. 12 (2012), no. 2, 219–235. MR 2951716, DOI 10.1515/ans-2012-0203
- Francesco Della Pietra and Nunzia Gavitone, Anisotropic elliptic problems involving Hardy-type potentials, J. Math. Anal. Appl. 397 (2013), no. 2, 800–813. MR 2979615, DOI 10.1016/j.jmaa.2012.08.008
- D. R. Dunninger, A Sturm comparison theorem for some degenerate quasilinear elliptic operators, Boll. Un. Mat. Ital. A (7) 9 (1995), no. 1, 117–121 (English, with Italian summary). MR 1324611
- Vincenzo Ferone and Bernd Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc. 137 (2009), no. 1, 247–253. MR 2439447, DOI 10.1090/S0002-9939-08-09554-3
- Tadeusz Iwaniec and Carlo Sbordone, Caccioppoli estimates and very weak solutions of elliptic equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), no. 3, 189–205 (2004). Renato Caccioppoli and modern analysis. MR 2064266
- John Lewis, Peter Lindqvist, Juan J. Manfredi, and Sandro Salsa, Regularity estimates for nonlinear elliptic and parabolic problems, Lecture Notes in Mathematics, vol. 2045, Springer, Heidelberg; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2012. Notes of the CIME course held in Cetraro, June 22–27, 2009; Edited by Ugo Gianazza and Lewis; Fondazione CIME/CIME Foundation Subseries. MR 2920528, DOI 10.1007/978-3-642-27145-8
- Vitali Liskevich, Sofya Lyakhova, and Vitaly Moroz, Positive solutions to nonlinear $p$-Laplace equations with Hardy potential in exterior domains, J. Differential Equations 232 (2007), no. 1, 212–252. MR 2281194, DOI 10.1016/j.jde.2006.09.001
- Constantin P. Niculescu and Lars-Erik Persson, Convex functions and their applications, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 23, Springer, New York, 2006. A contemporary approach. MR 2178902, DOI 10.1007/0-387-31077-0
- M. Picone, Un teorema sulle soluzioni delle equazioni lineari ellittiche autoaggiunte alle derivate parziali del secondo-ordine, Atti Accad. Naz. Lincei Rend., 20 (1911), 213-219.
- Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Vanishing and finiteness results in geometric analysis, Progress in Mathematics, vol. 266, Birkhäuser Verlag, Basel, 2008. A generalization of the Bochner technique. MR 2401291, DOI 10.1007/978-3-7643-8642-9
- Charles A. Swanson, Picone’s identity, Rend. Mat. (6) 8 (1975), no. 2, 373–397 (English, with Italian summary). MR 402188
- Guofang Wang and Chao Xia, A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 99–115. MR 2754338, DOI 10.1007/s00205-010-0323-9
- Guofang Wang and Chao Xia, An optimal anisotropic Poincaré inequality for convex domains, Pacific J. Math. 258 (2012), no. 2, 305–325. MR 2981956, DOI 10.2140/pjm.2012.258.305
- Guofang Wang and Chao Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations 252 (2012), no. 2, 1668–1700. MR 2853556, DOI 10.1016/j.jde.2011.08.001
Additional Information
- Jaroslav Jaroš
- Affiliation: Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia
- Email: jaros@fmph.uniba.sk
- Received by editor(s): March 15, 2013
- Received by editor(s) in revised form: May 31, 2013
- Published electronically: October 17, 2014
- Communicated by: Lei Ni
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 1137-1144
- MSC (2010): Primary 35J60
- DOI: https://doi.org/10.1090/S0002-9939-2014-12294-5
- MathSciNet review: 3293729