Gradient estimates below the duality exponent

Mathematische Annalen

Volumn 346, Issue 3, 2009, Pages 571-627

  Mingione, Giuseppe ^a  

^a UNIVERSITY OF PARMA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 73249152185     PISSN: 00255831     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00208-009-0411-z     Document Type: Article

References (61)

1. Acerbi E., Mingione G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136, 285-320 (2007).
2. Adams D. R.: Traces of potentials arising from translation invariant operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25(3), 203-217 (1971).
3. Adams D. R.: A note on Riesz potentials. Duke Math. J. 42, 765-778 (1975).
4. p-potential theory. Department of Mathematics, University of Umea (1981).
5. Adams D. R., Hedberg L. I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften 314. Springer-Verlag, Berlin (1996).
6. Adams D. R., Lewis J. L.: On Morrey-Besov inequalities. Studia Math. 74, 169-182 (1982).
7. Adams R. A.: Sobolev Spaces. Academic Press, New York (1975).
8. Alberico A., Cianchi A.: Optimal summability of solutions to nonlinear elliptic problems. Nonlinear Anal. TMA 74, 1775-1790 (2007).
9. Alvino A., Ferone V., Trombetti G.: Estimates for the gradient of solutions of nonlinear elliptic equations with L1-data. Ann. Mat. Pura Appl. (IV) 178, 129-142 (2000).
10. Bennet, C., Sharpley, R.: Interpolation of operators. Pure Appl. Math. series, 129. Academic Press, New York (1988).
11. Blasco O., Ruiz A., Vega L.: Non-interpolation in Morrey-Campanato and block spaces. Ann. Scu. Norm. Sup. Pisa Cl. Sci. 28(4), 31-40 (1999).
12. Boccardo L.: Problemi differenziali ellittici e parabolici con dati misure [Elliptic and parabolic differential problems with measure data]. Boll. Un. Mat. Ital. A 11(7), 439-461 (1997).
13. Boccardo L.: Quelques problemes de Dirichlet avec donneées dans de grand espaces de Sobolev [Some Dirichlet problems with data in large Sobolev spaces]. C. R. Acad. Sci. Paris Sér. I Math. 325, 1269-1272 (1997).
14. Boccardo, L.: Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data. Ann. Mat. Pura e Appl. doi: 10. 1007/s10231-008-0090-5.
15. Boccardo, L.: A Calderón-Zygmund theory for infinite energy minima of some integral functionals. Rendiconti Lincei, Matematica e Applicazioni (to appear).
16. Boccardo L., Gallouët T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149-169 (1989).
17. Boccardo L., Gallouët T.: Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differ. Equ. 17, 641-655 (1992).
18. Bojarski B., Iwaniec T.: Analytical foundations of the theory of quasiconformal mappings in ℝ. Ann. Acad. Sci. Fenn. Ser. A I Math. 8, 257-324 (1983).
19. Caffarelli L.: Elliptic second order equations. Rend. Sem. Mat. Fis. Milano 58, 253-284 (1988).
20. Caffarelli L., Peral I.: On W1, p estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51, 1-21 (1998).
21. Campanato S.: Equazioni elittiche non variazionali a coefficienti continui. Ann. Mat. Pura Appl. 86(IV), 125-154 (1970).
22. 1,λ data. Ann. Mat. Pura e Appl. 185(4), 537-553 (2006).
23. Cirmi, G. R., Leonardi, S.: Higher differentiability for linear elliptic systems with measure data (2009, in press).
24. Del Vecchio T.: Nonlinear elliptic equations with measure data. Potential Anal. 4, 185-203 (1995).
25. DiBenedetto E., Manfredi J. J.: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 115, 1107-1134 (1993).
26. Dolzmann G., Hungerbühler N., Müller S.: Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side. J. Reine Angew. Math. 520, 1-35 (2000).
27. Duoandikoetxea, J.: Fourier analysis. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI (2001).
28. Esposito L., Leonetti F., Mingione G.: Regularity results for minimizers of irregular integrals with (p, q) growth. Forum Math. 14, 245-272 (2002).
29. Esposito L., Leonetti F., Mingione G.: Sharp regularity for functionals with (p, q) growth. J. Differ. Equ. 204, 5-55 (2004).
30. Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge (2003).
31. Grafakos L.: Classical and Modern Fourier Analysis. Pearson Edu. Inc., Upper Saddle River (2004).
32. Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, New York (1993).
33. Iwaniec T.: Projections onto gradient fields and Lp-estimates for degenerated elliptic operators. Studia Math. 75, 293-312 (1983).
34. Iwaniec, T.: The Gehring Lemma. Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), pp. 181-204. Springer, New York (1998).
35. Iwaniec T., Verde A.: On the operator L(f)= f log {pipe}f{pipe}. J. Funct. Anal. 169, 391-420 (1999).
36. Kilpeläinen T.: Hölder continuity of solutions to quasilinear elliptic equations involving measures. Potential Anal. 3, 265-272 (1994).
37. Kilpeläinen T., Li G.: Estimates for p-Poisson equations. Differ. Int. Equ. 13, 791-800 (2000).
38. Kilpeläinen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137-161 (1994).
39. Kilpeläinen T., Xu X.: On the uniqueness problem for quasilinear elliptic equations involving measures. Rev. Mat. Iberoamericana 12, 461-475 (1996).
40. Kinnunen, J.: New thoughts on the Hardy-Littlewood maximal function. Lectures delivered at the University of Florence (2007).
41. Kinnunen J., Tuominen H.: Pointwise behaviour of M1, 1 Sobolev functions. Math. Z. 257, 613-630 (2007).
42. Kristensen J., Mingione G.: The singular set of ω-minima. Arch. Ration. Mech. Anal. 177, 93-114 (2005).
43. Kristensen J., Mingione G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331-398 (2006).
44. Lieberman G. M.: Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures. Comm. PDE 18, 1191-1212 (1993).
45. Lieberman G. M.: A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients. J. Funct. Anal. 201, 457-479 (2003).
46. Lindqvist P.: On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. (Crelles J.) 365, 67-79 (1986).
47. Lindqvist, P.: Notes on p-Laplace equation. Lectures notes, University of Jyväskylä (2006).
48. Lindqvist P., Manfredi J. J.: Note on a remarkable superposition for a nonlinear equation. Proc. AMS 136, 133-140 (2008).
49. Manfredi J. J.: Regularity for minima of functionals with p-growth. J. Differ. Equ. 76, 203-212 (1988).
50. Manfredi, J. J.: Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations. Ph. D. Thesis, University of Washington, St. Louis.
51. Mazzucato A. L.: Besov-Morrey spaces: function space theory and applications to non-linear PDE. Trans. Am. Math. Soc. 355, 1297-1364 (2003).
52. Mingione G.: The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166, 287-301 (2003).
53. Mingione G.: The Calderón-Zygmund theory for elliptic problems with measure data. Ann Scu. Norm. Sup. Pisa Cl. Sci. 6(5), 195-261 (2007).
54. Mingione, G.: Towards a non-linear Calderón-Zygmund theory. Quaderni di Matematica (Alvino, Mercaldo, Murat, Peral eds.) (to appear).
55. Ross J.: A Morrey-Nikolski inequality. Proc. Am. Math. Soc. 78, 97-102 (1980).
56. (p,λ) and interpolation. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (III) 19, 443-462 (1965).
57. Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton Math. Ser. 30. Princeton University Press, Princeton (1970).
58. Stein, E. M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Math. Ser. 32. Princeton University Press, Princeton (1971).
59. Talenti G.: Elliptic equations and rearrangements. Ann. Scu. Norm. Sup. Pisa Cl. Sci. 3(4), 697-717 (1976).
60. Trudinger N. S., Wang X. J.: On the weak continuity of elliptic operators and applications to potential theory. Am. J. Math. 124, 369-410 (2002).
61. Zhong X.: On nonhomogeneous quasilinear elliptic equations. Ann. Acad. Sci. Fenn. Math. Diss. 117, 46 (1998).