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Caccioppoli estimates through an anisotropic Picone's identity

Author: Jaroslav Jaroš
Journal: Proc. Amer. Math. Soc. 143 (2015), 1137-1144
MSC (2010): Primary 35J60
Published electronically: October 17, 2014
MathSciNet review: 3293729
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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.

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  • [1] 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 (2005d:35055),
  • [2] 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 (99c:35051),
  • [3] Angelo Alvino, Vincenzo Ferone, Guido Trombetti, and Pierre-Louis Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 2, 275-293 (English, with English and French summaries). MR 1441395 (98d:35068),
  • [4] 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 (2005e:35177),
  • [5] 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 (97i:53079)
  • [6] Andrea Cianchi and Paolo Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009), no. 4, 859-881. MR 2545870 (2010i:35264),
  • [7] 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
  • [8] 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,
  • [9] 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 (96b:35011)
  • [10] Vincenzo Ferone and Bernd Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc. 137 (2009), no. 1, 247-253. MR 2439447 (2009m:35135),
  • [11] 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 (2005d:35077)
  • [12] 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, 2012. Notes of the CIME course held in Cetraro, June 22-27, 2009; Edited by Ugo Gianazza and Lewis; Fondazione C.I.M.E./CIME Foundation Subseries. MR 2920528
  • [13] 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 (2008f:35125),
  • [14] Constantin P. Niculescu and Lars-Erik Persson, Convex functions and their applications, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, New York, 2006. A contemporary approach. MR 2178902 (2006m:26001)
  • [15] 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.
  • [16] 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 (2009m:58001)
  • [17] Charles A. Swanson, Picone's identity, Rend. Mat. (6) 8 (1975), no. 2, 373-397 (English, with Italian summary). Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday, II. MR 0402188 (53 #6009)
  • [18] 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 (2012c:35303),
  • [19] Guofang Wang and Chao Xia, An optimal anisotropic Poincaré inequality for convex domains, Pacific J. Math. 258 (2012), no. 2, 305-325. MR 2981956,
  • [20] 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 (2012j:35106),

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Additional Information

Jaroslav Jaroš
Affiliation: Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia

Keywords: Picone identity, Caccioppoli inequality
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
Article copyright: © Copyright 2014 American Mathematical Society

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