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Elliptic obstacle problems with measure data: Potentials and low order regularity

Christoph Scheven

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We consider obstacle problems with measure data related to elliptic equations of $p$-Laplace type, and investigate the connections between low order regularity properties of the solutions and non-linear potentials of the data. In particular, we give pointwise estimates for the solutions in terms of Wolff potentials and address the questions of boundedness and continuity of the solution.

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Publ. Mat., Volume 56, Number 2 (2012), 327-374.

First available in Project Euclid: 19 June 2012

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Zentralblatt MATH identifier

Primary: 35J87: Nonlinear elliptic unilateral problems and nonlinear elliptic variational inequalities [See also 35R35, 49J40] 35B65: Smoothness and regularity of solutions 31B35: Connections with differential equations
Secondary: 35R05: Partial differential equations with discontinuous coefficients or data 35R06: Partial differential equations with measure

Elliptic obstacle problems $p$-Laplacean measure data problems regularity of solutions non-linear potential theory


Scheven, Christoph. Elliptic obstacle problems with measure data: Potentials and low order regularity. Publ. Mat. 56 (2012), no. 2, 327--374.

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  • D. R. Adams and N. G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J. 22 (1972/73), 169\Ndash197.
  • P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L. Vázquez, An $L^{1}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(2) (1995), 241\Ndash273.
  • A. Bensoussan and J.-L. Lions, “Applications of variational inequalities in stochastic control”, Translated from the French, Studies in Mathematics and its Applications 12, North-Holland Publishing Co., Amsterdam-New York, 1982.
  • L. Boccardo and G. R. Cirmi, Existence and uniqueness of solution of unilateral problems with $L^{1}$ data, J. Convex Anal. 6(1) (1999), 195\Ndash206.
  • L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87(1) (1989), 149\Ndash169. \small\tt DOI: 10.1016/0022-1236(89)90005-0.
  • L. Boccardo and T. Gallouët, Problèmes unilatéraux avec données dans $L^{1}$, C. R. Acad. Sci. Paris Sér. I Math. 311(10) (1990), 617\Ndash619.
  • L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17(3–4) (1992), 641\Ndash655. \small\tt DOI: 10.1080/03605309208820857.
  • L. Boccardo, T. Gallouët, and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 13(5) (1996), 539\Ndash551.
  • L. Boccardo and F. Murat, Almost everywhere convergenceof the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19(6) (1992), 581\Ndash597. \small\tt DOI: 10.1016/0362- \small\tt 546X(92)90023-8.
  • G. Dal Maso, F. Murat, L. Orsina, and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(4) (1999), 741\Ndash808.
  • F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133(4) (2011), 1093\Ndash1149. \small\tt DOI: 10.1353/ajm.2011.0023.
  • F. Duzaar and G. Mingione, Gradient continuity estimates, Calc. Var. Partial Differential Equations 39(3–4) (2010), 379\Ndash418. \small\tt DOI: 10.1007/s00526-010-0314-6.
  • F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(6) (2010), 1361\Ndash1396. \small\tt DOI: 10.1016/j.anihpc.2010. \small\tt 07.002.
  • F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259(11) (2010), 2961\Ndash2998. \small\tt DOI: 10.1016/j.jfa.2010.08.006.
  • E. Giusti, “Direct methods in the calculus of variations”, WorldScientific Publishing Co., Inc., River Edge, NJ, 2003. \small\tt DOI: 10.1142/ \small\tt 9789812795557.
  • J. Heinonen, T. Kilpeläinen, and O. Martio, “Nonlinear potential theory of degenerate elliptic equations”, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.
  • T. Kilpeläinen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal. 27(3) (1996), 661\Ndash683. \small\tt DOI: 10.1137/0527036.
  • T. Kilpeläinen and J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19(4) (1992), 591\Ndash613.
  • T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172(1) (1994), 137\Ndash161. \small\tt DOI: 10.1007/BF02392793.
  • T. Kilpeläinen and X. Zhong, Growth of entire $\mathcal{A}$-subharmonic functions, Ann. Acad. Sci. Fenn. Math. 28(1) (2003), 181\Ndash192.
  • D. Kinderlehrer and G. Stampacchia, “An introduction to variational inequalities and their applications”, Pure and Applied Mathematics 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.
  • J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann.Mat. Pura Appl. (4) 185(3) (2006), 411\Ndash435. \small\tt DOI: 10.1007/s10231- \small\tt 005-0160-x.
  • T. Kuusi and G. Mingione, A surprising linear type estimate for nonlinear elliptic equations, C. R. Math. Acad. Sci. Paris 349(15\Ndash16) (2011), 889\Ndash892. \small\tt DOI: 10.1016/j.crma.2011.07.025.
  • T. Kuusi and G. Mingione, Linear potentials in nonlinear potential theory, Preprint.
  • C. Leone, Existence and uniqueness of solutions for nonlinear obstacle problems with measure data, Nonlinear Anal., Ser. A: Theory Methods 43(2) (2001), 199\Ndash215. \small\tt DOI: 10.1016/S0362-546X(99) \small\tt 00190-X.
  • P. Lindqvist, On the definition and properties of $p$-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67\Ndash79. \small\tt DOI: 10.1515/ \small\tt crll.1986.365.67.
  • J.-L. Lions, “Quelques méthodes de résolution des problèmes aux limites non linéaires”, Dunod; Gauthier-Villars, Paris, 1969.
  • J. Malý and W. P. Ziemer, “Fine regularity of solutions of elliptic partial differential equations”, Mathematical Surveys and Monographs 51, American Mathematical Society, Providence, RI, 1997.
  • V. G. Maz'ja and V. P. Havin, A nonlinear potential theory, Uspehi Mat. Nauk 27(6) (1972), 67\Ndash138; English translation in: Russian Math. Surveys 27(6) (1972), 71\Ndash148 (1974).
  • G. Mingione, Gradient potential estimates, J. Eur. Math. Soc. (JEMS) 13(2) (2011), 459\Ndash486. \small\tt DOI: 10.4171/JEMS/258.
  • P. Oppezzi and A. M. Rossi, Unilateral problems with measure data, Nonlinear Anal., Ser. A: Theory Methods 43(8) (2001), 1057\Ndash1088. \small\tt DOI: 10.1016/S0362-546X(99)00244-8.
  • C. Scheven, Existence and gradient estimates in nonlinear problems with irregular obstacles, Habilitation thesis (2011).
  • C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, Preprint (2011).
  • C. Scheven, Potential estimates for solutions to parabolic obstacle problems, Ann. Acad. Sci. Fenn. Math. (to appear). \small\tt DOI: 10.5186/aasfm.2012.3730.
  • C. Scheven, Gradient potential estimates in nonlinear elliptic obstacle problems with measure data, J. Funct. Anal. 262(6) (2012), 2777\Ndash-2832. \small\tt DOI: 10.1016/j.jfa.2012.01.003.
  • N. S. Trudinger and X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124(2) (2002), 369\Ndash410. \small\tt DOI: 10.1353/ajm.2002.0012.
  • W. P. Ziemer, “Weakly differentiable functions”, Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120, Springer-Verlag, New York, 1989.