La teoria di Calderón-Zygmund dal caso lineare a quello non lineare

Mingione, Giuseppe

Geodesic

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La teoria di Calderón-Zygmund dal caso lineare a quello non lineare

Mingione, Giuseppe

Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 269-297.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Résumé

La teoria di Calderón-Zygmund per equazioni ellittiche e paraboliche lineari ammette un analogo non lineare che si è andato man mano delineando sempre più chiaramente negli ultimi anni. Di seguito si discutono alcuni risultati validi in questo ambito.

MR Zbl

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     author = {Mingione, Giuseppe},
     title = {La teoria di {Calder\'on-Zygmund} dal caso lineare a quello non lineare},
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Mingione, Giuseppe. La teoria di Calderón-Zygmund dal caso lineare a quello non lineare. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 269-297. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a1/

Bibliographie
Cité par

[1] E. Acerbi - G. Mingione, Gradient estimates for the $p(x)$-Laplacean system. J. Reine Ang. Math. (Crelles J.), 584 (2005), 117-148. | DOI | MR

[2] E. Acerbi - G. Mingione, Gradient estimates for a class of parabolic systems. Duke Math. J. 136 (2007), 285-320. | DOI | MR | Zbl

[3] D. R. Adams, A note on Riesz potentials. Duke Math. J. 42 (1975), 765-778. | MR | Zbl

[4] D. R. Adams - L. I. Hedberg, Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften 314. Springer-Verlag, Berlin, 1996. | DOI | MR

[5] D. R. Adams - N. G. Meyers, Thinnes and Wiener criteria for nonlinear potentials. Indiana Univ. Math. J. 22 (1972/73), 169-197. | DOI | MR | Zbl

[6] R. A. Adams - J. J. F. Fournier, Sobolev Spaces. Second edition. Pure and Appl. Math. 140, Elsevier/Academic Press, Amsterdam, 2003. | MR

[7] P. Baroni - J. Habermann, Calderón-Zygmund estimates for parabolic measure data problems. J. Diff. Equ. 252 (2012), 412-447. | DOI | MR | Zbl

[8] P. Baroni - J. Habermann, New gradient estimates for parabolic equations. Houston J. Math., in stampa. | MR | Zbl

[9] P. Bénilan - L. Boccardo - T. Gallouët - R. Gariepy - M. Pierre - J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (IV), 22 (1995), 241-273. | fulltext EuDML | MR | Zbl

[10] L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure. Boll. UMI A (VII), 11 (1997), 439-461.

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

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

[13] V. Bögelein - F. Duzaar - G. Mingione, Degenerate problems with irregular obstacles. J. Reine Angew. Math. (Crelles J.), 650 (2011), 107-160. | DOI | MR

[14] J. Bourgain - H. Brezis, On the equation $\operatorname{div} Y = f$ application to control of phases. J. Amer. Math. Soc. 16 (2003), 393-426. | DOI | MR | Zbl

[15] S. S. Byun - L. Wang, Gradient estimates for elliptic systems in non-smooth domains. Math. Ann. 341 (2008), 629-650. | DOI | MR | Zbl

[16] S. S. Byun - L. Wang - S. Zhou, Nonlinear elliptic equations with BMO coefficients in Reifenberg domains. J. Funct. Anal. 250 (2007), 167-196. | DOI | MR | Zbl

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

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

[19] L. Caffarelli - I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51 (1998), 1-21. | DOI | MR

[20] A. P. Calderón - A. Zygmund, On the existence of certain singular integrals. Acta Math. 88 (1952), 85-139. | DOI | MR

[21] A. P. Calderón - A. Zygmund, On singular integrals. Amer. J. Math. 78 (1956), 289-309. | DOI | MR

[22] S. Campanato - G. Stampacchia, Sulle maggiorazioni in $L^p$ nella teoria delle equazioni ellittiche. Boll. UMI (III), 20 (1965), 393-399. | fulltext bdim | fulltext EuDML | MR | Zbl

[23] A. Cianchi, Nonlinear potentials, local solutions to elliptic equations and rearrangements. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (V), 10 (2011), 335-361. | MR | Zbl

[24] G. Dal Maso - F. Murat - L. Orsina - A. Prignet, Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (IV), 28 (1999), 741-808. | fulltext EuDML | MR | Zbl

[25] E. Dibenedetto, Degenerate parabolic equations. Universitext. Springer-Verlag, New York, 1993. | DOI | MR | Zbl

[26] E. Dibenedetto - J. J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Amer. J. Math. 115 (1993), 1107-1134. | DOI | MR | Zbl

[27] A. Di Castro, Anisotropic elliptic problems with natural growth terms. Manuscripta Math. 135 (2011), 521-543. | DOI | MR | Zbl

[28] A. Di Castro - G. Palatucci, Measure data problems, lower order terms and interpolation effects. Ann. Mat. Pura Appl. (IV) DOI: 10.1007/s10231-012-0277-7. | DOI | MR | Zbl

[29] A. Di Castro - G. Palatucci, Nonlinear parabolic problems with lower order terms and related integral estimates. Nonlinear Anal. 75 (2012), 4177-4197. | DOI | MR | Zbl

[30] E. Di Nezza - G. Palatucci - E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. math., in stampa. | DOI | MR | Zbl

[31] G. Dolzmann - N. Hungerbühler - 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. | DOI | MR

[32] F. Duzaar - G. Mingione, Gradient estimates via non-linear potentials. Amer. J. Math. 133 (2011), 1093-1149. | DOI | MR | Zbl

[33] F. Duzaar - G. Mingione, Gradient estimates via linear and nonlinear potentials. J. Funct. Anal. 259 (2010), 2961-2998. | DOI | MR | Zbl

[34] F. Duzaar - G. Mingione - K. Steffen, Parabolic systems with polynomial growth and regularity. Mem. Amer. Math. Soc. 214, no. 1005 (2011), 128. | DOI | MR

[35] L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations. Indiana Univ. Math. J. 42 (1993), 413-423. | DOI | MR | Zbl

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

[37] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983. | MR | Zbl

[38] E. Giusti, Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. | DOI | MR | Zbl

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

[40] M. Havin - V. G. Maz'Ya, A nonlinear potential theory. Russ. Math. Surveys, 27 (1972), 71-148. | MR

[41] L. I. Hedberg - T. Wolff, Thin sets in nonlinear potential theory. Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. | fulltext EuDML | MR | Zbl

[42] T. Iwaniec, Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators. Studia Math. 75 (1983), 293-312. | fulltext EuDML | DOI | MR | Zbl

[43] T. Iwaniec, p-harmonic tensors and quasiregular mappings. Ann. Math. (II), 136 (1992), 589-624. | DOI | MR | Zbl

[44] T. Iwaniec, Nonlinear Cauchy-Riemann operators in $\mathbb{R}^n$. Trans. Amer. Math. Soc. 354 (2002), 1961-1995. | DOI | MR | Zbl

[45] T. Iwaniec - C. Sbordone, On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119 (1992), 129-143. | DOI | MR | Zbl

[46] T. Iwaniec - C. Sbordone, Weak minima of variational integrals. J. Reine Angew. Math. (Crelle J.), 454 (1994), 143-161. | fulltext EuDML | DOI | MR | Zbl

[47] T. Iwaniec - C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients. J. Anal. Math. 74 (1998), 183-212. | DOI | MR | Zbl

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

[49] S. Kichenassamy - L. Verón, Singular solutions to the p-Laplace equation. Math. Ann. 275 (1986), 599-615. | fulltext EuDML | DOI | MR

[50] T. Kilpeläinen - T. Kuusi - A. Tuhola-Kujanpäa, Superharmonic functions are locally renormalized solutions. Ann. Inst. H. Poincarè, Anal. Non Lin., 28 (2011), 775-795. | DOI | MR | Zbl

[51] T. Kilpeläinen - J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (IV), 19 (1992), 591-613. | fulltext EuDML | MR

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

[53] J. Kinnunen - J. L. Lewis, Very weak solutions of parabolic systems of p-Laplacian type. Ark. Mat. 40 (2002), 105-132. | DOI | MR | Zbl

[54] J. Kinnunen - S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients. Comm. Partial Differential Equations, 24 (1999), 2043-2068. | DOI | MR | Zbl

[55] R. Korte - T. Kuusi, A note on the Wolff potential estimate for solutions to elliptic equations involving measures. Adv. Calc. Var. 3 (2010), 99-113. | DOI | MR | Zbl

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

[57] J. Kristensen - G. Mingione, Boundary regularity in variational problems. Arch. Ration. Mech. Anal. 198 (2010), 369-455. | DOI | MR | Zbl

[58] T. Kuusi - G. Mingione, Universal potential estimates. J. Funct. Anal. 262 (2012), 4205-4269. | DOI | MR | Zbl

[59] T. Kuusi - G. Mingione, Endpoint and intermediate potential estimates for nonlinear equations. Boll. UMI (IX), 4 (2011), 149-157. | fulltext bdim | fulltext EuDML | MR | Zbl

[60] T. Kuusi - G. Mingione, Nonlinear potential estimates in parabolic problems. Rendiconti Lincei, Matematica e Applicazioni, 22 (2011), 161-174. | DOI | MR | Zbl

[61] T. Kuusi - G. Mingione, The Wolff gradient bound for degenerate parabolic equations. J. Europ. Math. Soc., in stampa. | fulltext EuDML | DOI | MR | Zbl

[62] T. Kuusi - G. Mingione, Gradient regularity for nonlinear parabolic equations. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (V), in stampa. | MR | Zbl

[63] T. Kuusi - G. Mingione, Linear potentials in nonlinear potential theory. Arch. Rat. Mech. Anal. 207 (2013), 215-246. | DOI | MR | Zbl

[64] T. Kuusi - G. Mingione, A surprising linear type estimate for nonlinear elliptic equations. C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 889-892. | DOI | MR | Zbl

[65] O. A. Ladyzhenskaya - N. N. Ural'Tseva, Linear and quasilinear elliptic equations. Academic Press, New York-London 1968. | MR

[66] J. Leray - J.-L. Lions, Quelques résulatats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France, 93 (1965), 97-107. | fulltext EuDML | MR | Zbl

[67] J. L. Lewis, On very weak solutions of certain elliptic systems. Comm. Partial Differential Equations, 18 (1993), 1515-1537. | DOI | MR | Zbl

[68] P. Lindqvist, On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. (Crelles J.), 365 (1986), 67-79. | fulltext EuDML | DOI | MR | Zbl

[69] V. Maz'Ya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations. (Russian), Vestnik Leningrad. Univ. 25 (1970), 42-55. | MR

[70] G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (V), 6 (2007), 195-261. | fulltext EuDML | MR | Zbl

[71] G. Mingione, Gradient estimates below the duality exponent. Math. Ann. 346 (2010), 571-627. | DOI | MR | Zbl

[72] G. Mingione, Gradient potential estimates. J. Europ. Math. Soc. 13 (2011), 459-486. | fulltext EuDML | DOI | MR | Zbl

[73] G. Mingione, Regularity of minima: an invitation to the Dark Side of the Calculus of Variations. Applications of Mathematics, 51 (2006), 355-425. | fulltext EuDML | DOI | MR | Zbl

[74] G. Mingione, Nonlinear measure data problems. Milan J. Math. 79 (2011), 429-496. | DOI | MR | Zbl

[75] N. C. Phuc - I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type. Ann. of Math. (II), 168 (2008), 859-914. | DOI | MR | Zbl

[76] N. C. Phuc - I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities. J. Funct. Anal. 256 (2009), 1875-1906. | DOI | MR | Zbl

[77] T. Rivière, The role of conservation laws in the analysis of conformally invariant problems. In Topics in modern regularity theory (G. Mingione ed.). Sc. Normale Superiore 2012. | DOI | MR

[78] C. Scheven, Non-linear Calderón-Zygmund theory for parabolic systems with subquadratic growth. J. Evol. Equ. 10 (2010), 597-622. | DOI | MR | Zbl

[79] J. Serrin, Pathological solutions of elliptic differential equations. Ann. Sc. Norm. Sup. Pisa (III), 18 (1964), 385-387. | fulltext EuDML | MR | Zbl

[80] J. Serrin, Local behavior of solutions of quasi-linear equations. Acta Math. 111 (1964), 247-302. | DOI | MR | Zbl

[81] G. Stampacchia, The spaces $\mathcal{L}^{(p,\lambda)}$, $N^{(p,\lambda)}$ and interpolation. Ann. Sc. Norm. Sup. Pisa (III), 19 (1965), 443-462. | fulltext EuDML | MR | Zbl

[82] G. Stampacchia, $\mathcal{L}^{(p,\lambda)}$-spaces and interpolation. Comm. Pure Appl. Math. 17 (1964), 293-306. | DOI | MR | Zbl

[83] E. M. Stein - G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Math. Ser., 32. Princeton Univ. Press, Princeton, N.J. 1971. | MR | Zbl

[84] V. Šverák - X. Yan, Non-Lipschitz minimizers of smooth uniformly convex functionals. Proc. Natl. Acad. Sci. USA, 99/24 (2002), 15269-15276. | DOI | MR | Zbl

[85] G. Talenti, Elliptic equations and rearrangements. Ann Sc. Norm. Sup. Pisa Cl. Sci. (IV), 3 (1976), 697-717. | fulltext EuDML | MR | Zbl

[86] N. S. Trudinger - X. J. Wang, On the weak continuity of elliptic operators and applications to potential theory. Amer. J. Math. 124 (2002), 369-410. | MR | Zbl

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

[88] F. Yao, A new approach to Lp estimates for Calderón-Zygmund type singular integrals. Arch. Math. (Basel), 92 (2009), 137-146. | DOI | MR