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

Christoph Scheven

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 19 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.pm/1340127809

Mathematical Reviews number (MathSciNet)
MR2978327

Zentralblatt MATH identifier
1295.35373

Subjects
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

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

Citation

Scheven, Christoph. Elliptic obstacle problems with measure data: Potentials and low order regularity. Publ. Mat. 56 (2012), no. 2, 327--374. https://projecteuclid.org/euclid.pm/1340127809


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References

  • 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.