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.

We consider non-linear elliptic equations having a measure in the right-hand side, of the type diva(x,Du)=μ, 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.

Classification: 35J60, 35J70
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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] D. R. Adams, 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] D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778. | MR | Zbl

[3] D. R. Adams and L. I. Hedberg, “Function Spaces and Potential Theory”, Grundlehren der Mathematischen Wissenschaften, Vol. 314, Springer-Verlag, Berlin, 1996. | MR | Zbl

[4] R.A. Adams, “Sobolev Spaces”, Academic Press, New York, 1975. | MR | Zbl

[5] L. Ambrosio, N. Fusco and D. Pallara,“Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. | MR | Zbl

[6] P. Benilan, 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 (1995), 241-273. | EuDML | Numdam | MR | Zbl

[7] L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure, Boll. Un. Mat. Ital. A (7) 11 (1997), 439-461. | Zbl

[8] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169. | MR | Zbl

[9] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (1992), 641-655. | MR | Zbl

[10] L. Boccardo, T. Gallouët and L. Orsina, 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

[11] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in n Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257-324. | MR | Zbl

[12] L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58 (1988), 253-284. | MR | Zbl

[13] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. 126 (2) 130 (1989), 189-213. | MR | Zbl

[14] L. Caffarelli and I. Peral, On W 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1-21. | MR | Zbl

[15] S. Campanato, Proprietà di inclusione per spazi di Morrey, Ricerche Mat. 12 (1963), 67-86. | MR | Zbl

[16] S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 137-160. | Numdam | MR | Zbl

[17] S. Campanato, Equazioni elittiche non variazionali a coefficienti continui, Ann. Mat. Pura Appl. (4) 86 (1970), 125-154. | MR | Zbl

[18] S. Campanato, Hölder continuity of the solutions of some nonlinear elliptic systems Adv. Math. 48 (1983), 16-43. | MR | Zbl

[19] G. R. Cirmi and S. Leonardi, Regularity results for the gradient of solutions to linear elliptic equations with L 1,λ data, Ann. Mat. Pura e Appl. (4) 185 (2006), 537-553. | MR | Zbl

[20] A. Dall'Aglio, Approximated solutions of equations with L 1 -data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl. (4) 170 (1996), 207-240. | MR | Zbl

[21] 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 (1999), 741-808. | Numdam | MR | Zbl

[22] T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995), 185-203. | MR | Zbl

[23] G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 241-256. | MR | Zbl

[24] G. Di Fazio, M. A. Ragusa and D. K. Palagachev, 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

[25] M. Di Giampaolo and F. Leonetti, Boundedness of weak solutions to some linear elliptic equations with measure data, Differential Integral Equations 18 (2005), 1371-1382. | MR | Zbl

[26] G. Dolzmann, N. Hungerbühler and S. Müller, The p-harmonic system with measure-valued right-hand side, Ann. Inst. H. Poincarè Anal. Non Linèaire 14 (1997), 353-364. | Numdam | MR | Zbl

[27] G. Dolzmann, N. Hungerbühler and S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right-hand side, J. Reine Angew. Math. (Crelles J.) 520 (2000), 1-35. | MR | Zbl

[28] L. D'Onofrio and T. Iwaniec, Notes on p-harmonic analysis, Contemp. Math. 370 (2005), 25-49. | MR | Zbl

[29] L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with (p,q) growth, Forum Math. 14 (2002), 245-272. | MR | Zbl

[30] L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with (p,q) growth, J. Differential Equations 204 (2004), 5-55. | MR | Zbl

[31] V. Ferone and N. Fusco, VMO solutions of the N-Laplacian with measure data, C. R. Acad. Sci. Paris Sèr. I Math. 325 (1997), 365-370. | MR | Zbl

[32] M. Fuchs and J. Reuling, Non-linear elliptic systems involving measure data, Rend. Mat. Appl. (7) 15 (1995), 311-319. | MR | Zbl

[33] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977; second edition: 1998. | MR | Zbl

[34] E. Giusti, “Direct Methods in the Calculus of Variations”, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. | MR | Zbl

[35] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math. 92 (1997), 249-258. | MR | Zbl

[36] C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. (Crelles J.) 431 (1992), 7-64. | MR | Zbl

[37] J. Heinonen, T. Kilpeläinen and O. Martio, “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Oxford Mathematical Monographs., New York, 1993. | MR | Zbl

[38] T. Iwaniec, The Gehring lemma, In: “Quasiconformal mappings and analysis” (Ann Arbor, MI, 1995), 181-204, Springer, New York, 1998. | MR | Zbl

[39] T. Iwaniec and C. Sbordone, Quasiharmonic fields, Ann. Inst. H. Poincaré Anal. Non Linèaire 18 (2001), 519-572. | Numdam | MR | Zbl

[40] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. | MR | Zbl

[41] T. Kilpeläinen, Hölder continuity of solutions to quasilinear elliptic equations involving measures, Potential Anal. 3 (1994), 265-272. | Zbl

[42] T. Kilpeläinen and G. Li, Estimates for p-Poisson equations, Differential Integral Equations 13 (2000), 791-800. | Zbl

[43] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161. | Zbl

[44] T. Kilpeläinen, N. Shanmugalingam and X. Zhong, Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures, Preprint 2006. | MR | Zbl

[45] T. Kilpeläinen and Xiangsheng Xu, On the uniqueness problem for quasilinear elliptic equations involving measures Rev. Mat. Iberoamericana 12 (1996), 461-475. | MR | Zbl

[46] T. Kilpeläinen and X. Zhong, Removable sets for continuous solutions of quasilinear elliptic equations, Proc. Amer. Math. Soc. 130 (2002), 1681-1688. | MR | Zbl

[47] H. Kozono and M. Yamazaki, 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

[48] J. Kristensen and G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006), 331-398. | MR | Zbl

[49] G. M. Lieberman, 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] G. M. Lieberman, 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] P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. (Crelles J.) 365 (1986), 67-79. | MR | Zbl

[52] P. Lindqvist, “Notes on p-Laplace Equation”, University of Jyväskylä - Lectures notes, 2006. | MR | Zbl

[53] J. L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites non Linèaires”, Dunod, Gauthier-Villars, Paris, 1969. | MR | Zbl

[54] W. Littman, G. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scu. Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43-77. | Numdam | MR | Zbl

[55] J. Malý and W.P. Ziemer, “Fine regularity of solutions of elliptic partial differential equations”, Mathematical Surveys and Monographs, Vol. 51. American Mathematical Society, Providence, RI, 1997. | MR | Zbl

[56] J. J. Manfredi, Regularity for minima of functionals with p-growth, J. Differential Equations 76 (1988), 203-212. | MR | Zbl

[57] J. J. Manfredi, “Regularity of the Gradient for a Class of Nonlinear Possibly Degenerate Elliptic Equations”, Ph.D. Thesis, University of Washington, St. Louis.

[58] A. L. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc. 355 (2003), 1297-1364. | MR | Zbl

[59] G. Mingione, The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal. 166 (2003), 287-301. | MR | Zbl

[60] G. Mingione, Calderón-Zygmund estimates for measure data problems, C. R. Acad. Sci. Paris Sèr. I Math. 344 (2007), 437-442. | MR | Zbl

[61] G. Mingione, Sub-quadratic measure data problems, in preparation.

[62] T. Miyakawa, On Morrey spaces of measures: basic properties and potential estimates, Hiroshima Math. J. 20 (1990), 213-222. | MR | Zbl

[63] J. M. Rakotoson, Uniqueness of renormalized solutions in a T-set for the L 1 -data problem and the link between various formulations, Indiana Univ. Math. J. 43 (1994), 685-702. | MR | Zbl

[64] J. Ross, A Morrey-Nikolski inequality, Proc. Amer. Math. Soc. 78 (1980), 97-102. | MR | Zbl

[65] T. Runst and W. Sickel, “Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations”, Walter de Gruyter & Co., Berlin, 1996. | Zbl

[66] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. | MR | Zbl

[67] J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 385-387. | Numdam | MR | Zbl

[68] G. Stampacchia, 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] G. Stampacchia, The spaces (p,λ) , N (p,λ) and interpolation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 443-462. | Numdam | MR | Zbl

[70] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4) 120 (1979), 160-184. | MR | Zbl

[71] M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407-1456. | MR | Zbl

[72] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. | MR | Zbl

[73] X. Zhong, On nonhomogeneous quasilinear elliptic equations, Dissertation, University of Jyväskylä, 1998, Ann. Acad. Sci. Fenn. Math. Diss. 117 (1998), 46 pages. | MR | Zbl