We consider non-linear elliptic equations having a measure in the right-hand side, of the type and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.
@article{ASNSP_2007_5_6_2_195_0, author = {Mingione, Giuseppe}, title = {The {Calder\'on-Zygmund} theory for elliptic problems with measure data}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {195--261}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {2}, year = {2007}, mrnumber = {2352517}, zbl = {1178.35168}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2007_5_6_2_195_0/} }
TY - JOUR AU - Mingione, Giuseppe TI - The Calderón-Zygmund theory for elliptic problems with measure data JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 SP - 195 EP - 261 VL - 6 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2007_5_6_2_195_0/ LA - en ID - ASNSP_2007_5_6_2_195_0 ER -
%0 Journal Article %A Mingione, Giuseppe %T The Calderón-Zygmund theory for elliptic problems with measure data %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 195-261 %V 6 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2007_5_6_2_195_0/ %G en %F ASNSP_2007_5_6_2_195_0
Mingione, Giuseppe. The Calderón-Zygmund theory for elliptic problems with measure data. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 2, pp. 195-261. http://archive.numdam.org/item/ASNSP_2007_5_6_2_195_0/
[1] Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25 (1971), 203-217. | EuDML | Numdam | MR | Zbl
,[2] A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778. | MR | Zbl
,[3] “Function Spaces and Potential Theory”, Grundlehren der Mathematischen Wissenschaften, Vol. 314, Springer-Verlag, Berlin, 1996. | MR | Zbl
and ,[4] “Sobolev Spaces”, Academic Press, New York, 1975. | MR | Zbl
,[5] “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. | MR | Zbl
, and ,[6] An -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 241-273. | EuDML | Numdam | MR | Zbl
, , , , and ,[7] Problemi differenziali ellittici e parabolici con dati misure, Boll. Un. Mat. Ital. A (7) 11 (1997), 439-461. | Zbl
,[8] Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169. | MR | Zbl
and ,[9] Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (1992), 641-655. | MR | Zbl
and ,[10] Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincarè Anal. Non Linéaire 13 (1996), 539-551. | Numdam | MR | Zbl
, and ,[11] Analytical foundations of the theory of quasiconformal mappings in Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257-324. | MR | Zbl
and ,[12] Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58 (1988), 253-284. | MR | Zbl
,[13] Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. 126 (2) 130 (1989), 189-213. | MR | Zbl
,[14] On estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1-21. | MR | Zbl
and ,[15] Proprietà di inclusione per spazi di Morrey, Ricerche Mat. 12 (1963), 67-86. | MR | Zbl
,[16] Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 137-160. | Numdam | MR | Zbl
,[17] Equazioni elittiche non variazionali a coefficienti continui, Ann. Mat. Pura Appl. (4) 86 (1970), 125-154. | MR | Zbl
,[18] Hölder continuity of the solutions of some nonlinear elliptic systems Adv. Math. 48 (1983), 16-43. | MR | Zbl
,[19] Regularity results for the gradient of solutions to linear elliptic equations with data, Ann. Mat. Pura e Appl. (4) 185 (2006), 537-553. | MR | Zbl
and ,[20] Approximated solutions of equations with -data. Application to the -convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl. (4) 170 (1996), 207-240. | MR | Zbl
,[21] Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 741-808. | Numdam | MR | Zbl
, , and ,[22] Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995), 185-203. | MR | Zbl
,[23] Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 241-256. | MR | Zbl
and ,[24] Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal. 166 (1999), 179-196. | MR | Zbl
, and ,[25] Boundedness of weak solutions to some linear elliptic equations with measure data, Differential Integral Equations 18 (2005), 1371-1382. | MR | Zbl
and ,[26] The -harmonic system with measure-valued right-hand side, Ann. Inst. H. Poincarè Anal. Non Linèaire 14 (1997), 353-364. | Numdam | MR | Zbl
, and ,[27] Uniqueness and maximal regularity for nonlinear elliptic systems of -Laplace type with measure valued right-hand side, J. Reine Angew. Math. (Crelles J.) 520 (2000), 1-35. | MR | Zbl
, and ,[28] Notes on -harmonic analysis, Contemp. Math. 370 (2005), 25-49. | MR | Zbl
and ,[29] Regularity results for minimizers of irregular integrals with growth, Forum Math. 14 (2002), 245-272. | MR | Zbl
, and ,[30] Sharp regularity for functionals with growth, J. Differential Equations 204 (2004), 5-55. | MR | Zbl
, and ,[31] solutions of the -Laplacian with measure data, C. R. Acad. Sci. Paris Sèr. I Math. 325 (1997), 365-370. | MR | Zbl
and ,[32] Non-linear elliptic systems involving measure data, Rend. Mat. Appl. (7) 15 (1995), 311-319. | MR | Zbl
and ,[33] “Elliptic Partial Differential Equations of Second Order”, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977; second edition: 1998. | MR | Zbl
and ,[34] “Direct Methods in the Calculus of Variations”, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. | MR | Zbl
,[35] Inverting the -harmonic operator, Manuscripta Math. 92 (1997), 249-258. | MR | Zbl
, and ,[36] Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. (Crelles J.) 431 (1992), 7-64. | MR | Zbl
,[37] “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Oxford Mathematical Monographs., New York, 1993. | MR | Zbl
, and ,[38] The Gehring lemma, In: “Quasiconformal mappings and analysis” (Ann Arbor, MI, 1995), 181-204, Springer, New York, 1998. | MR | Zbl
,[39] Quasiharmonic fields, Ann. Inst. H. Poincaré Anal. Non Linèaire 18 (2001), 519-572. | Numdam | MR | Zbl
and ,[40] On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. | MR | Zbl
and ,[41] Hölder continuity of solutions to quasilinear elliptic equations involving measures, Potential Anal. 3 (1994), 265-272. | Zbl
,[42] Estimates for -Poisson equations, Differential Integral Equations 13 (2000), 791-800. | Zbl
and ,[43] The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161. | Zbl
and ,[44] Maximal regularity via reverse Hölder inequalities for elliptic systems of -Laplace type involving measures, Preprint 2006. | MR | Zbl
, and ,[45] On the uniqueness problem for quasilinear elliptic equations involving measures Rev. Mat. Iberoamericana 12 (1996), 461-475. | MR | Zbl
and ,[46] Removable sets for continuous solutions of quasilinear elliptic equations, Proc. Amer. Math. Soc. 130 (2002), 1681-1688. | MR | Zbl
and ,[47] Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959-1014. | MR | Zbl
and ,[48] The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006), 331-398. | MR | Zbl
and ,[49] Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures, Comm. Partial Differential Equations 18 (1993), 1191-1212. | MR | Zbl
,[50] A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal. 201 (2003), 457-479. | MR | Zbl
,[51] On the definition and properties of -superharmonic functions, J. Reine Angew. Math. (Crelles J.) 365 (1986), 67-79. | MR | Zbl
,[52] “Notes on -Laplace Equation”, University of Jyväskylä - Lectures notes, 2006. | MR | Zbl
,[53] “Quelques Méthodes de Résolution des Problèmes aux Limites non Linèaires”, Dunod, Gauthier-Villars, Paris, 1969. | MR | Zbl
,[54] Regular points for elliptic equations with discontinuous coefficients, Ann. Scu. Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43-77. | Numdam | MR | Zbl
, and ,[55] “Fine regularity of solutions of elliptic partial differential equations”, Mathematical Surveys and Monographs, Vol. 51. American Mathematical Society, Providence, RI, 1997. | MR | Zbl
and ,[56] Regularity for minima of functionals with -growth, J. Differential Equations 76 (1988), 203-212. | MR | Zbl
,[57] “Regularity of the Gradient for a Class of Nonlinear Possibly Degenerate Elliptic Equations”, Ph.D. Thesis, University of Washington, St. Louis.
,[58] Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc. 355 (2003), 1297-1364. | MR | Zbl
,[59] The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal. 166 (2003), 287-301. | MR | Zbl
,[60] Calderón-Zygmund estimates for measure data problems, C. R. Acad. Sci. Paris Sèr. I Math. 344 (2007), 437-442. | MR | Zbl
,[61] Sub-quadratic measure data problems, in preparation.
,[62] On Morrey spaces of measures: basic properties and potential estimates, Hiroshima Math. J. 20 (1990), 213-222. | MR | Zbl
,[63] Uniqueness of renormalized solutions in a -set for the -data problem and the link between various formulations, Indiana Univ. Math. J. 43 (1994), 685-702. | MR | Zbl
,[64] A Morrey-Nikolski inequality, Proc. Amer. Math. Soc. 78 (1980), 97-102. | MR | Zbl
,[65] “Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations”, Walter de Gruyter & Co., Berlin, 1996. | Zbl
and ,[66] Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. | MR | Zbl
,[67] Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 385-387. | Numdam | MR | Zbl
,[68] Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier 15 (1965), 189-258. | Numdam | MR | Zbl
,[69] The spaces , and interpolation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 443-462. | Numdam | MR | Zbl
,[70] Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4) 120 (1979), 160-184. | MR | Zbl
,[71] Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407-1456. | MR | Zbl
,[72] Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. | MR | Zbl
,[73] On nonhomogeneous quasilinear elliptic equations, Dissertation, University of Jyväskylä, 1998, Ann. Acad. Sci. Fenn. Math. Diss. 117 (1998), 46 pages. | MR | Zbl
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