The Calderón-Zygmund theory for elliptic problems with measure data

Giuseppe Mingione

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 2, page 195-261
  • ISSN: 0391-173X

Abstract

top
We consider non-linear elliptic equations having a measure in the right-hand side, of the type div a ( x , D u ) = 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.

How to cite

top

Mingione, Giuseppe. "The Calderón-Zygmund theory for elliptic problems with measure data." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.2 (2007): 195-261. <http://eudml.org/doc/272257>.

@article{Mingione2007,
abstract = {We consider non-linear elliptic equations having a measure in the right-hand side, of the type $ \operatorname\{div\}\ a(x,Du)=\mu , $ 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.},
author = {Mingione, Giuseppe},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {non-linear elliptic equations; Radon measure; Dirichlet problem; Marcinkiewicz spaces; Morrey spaces; VMO},
language = {eng},
number = {2},
pages = {195-261},
publisher = {Scuola Normale Superiore, Pisa},
title = {The Calderón-Zygmund theory for elliptic problems with measure data},
url = {http://eudml.org/doc/272257},
volume = {6},
year = {2007},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 2
SP - 195
EP - 261
AB - We consider non-linear elliptic equations having a measure in the right-hand side, of the type $ \operatorname{div}\ a(x,Du)=\mu , $ 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.
LA - eng
KW - non-linear elliptic equations; Radon measure; Dirichlet problem; Marcinkiewicz spaces; Morrey spaces; VMO
UR - http://eudml.org/doc/272257
ER -

References

top
  1. [1] D. R. Adams, Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25 (1971), 203–217. Zbl0219.46027MR287301
  2. [2] D. R. Adams, A note on Riesz potentials, Duke Math. J.42 (1975), 765–778. Zbl0336.46038MR458158
  3. [3] D. R. Adams and L. I. Hedberg, “Function Spaces and Potential Theory”, Grundlehren der Mathematischen Wissenschaften, Vol. 314, Springer-Verlag, Berlin, 1996. Zbl0834.46021MR1411441
  4. [4] R.A. Adams, “Sobolev Spaces”, Academic Press, New York, 1975. Zbl1098.46001MR450957
  5. [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. Zbl0957.49001MR1857292
  6. [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. Zbl0866.35037MR1354907
  7. [7] L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure, Boll. Un. Mat. Ital. A (7) 11 (1997), 439–461. Zbl0893.35131
  8. [8] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal.87 (1989), 149–169. Zbl0707.35060MR1025884
  9. [9] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations17 (1992), 641–655. Zbl0812.35043MR1163440
  10. [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 Linéaire13 (1996), 539–551. Zbl0857.35126MR1409661
  11. [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. Zbl0548.30016MR731786
  12. [12] L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano58 (1988), 253–284. Zbl0726.35036MR1069735
  13. [13] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. 126 (2) 130 (1989), 189–213. Zbl0692.35017MR1005611
  14. [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. Zbl0906.35030MR1486629
  15. [15] S. Campanato, Proprietà di inclusione per spazi di Morrey, Ricerche Mat.12 (1963), 67–86. Zbl0192.22703MR156228
  16. [16] S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 137–160. Zbl0133.06801MR167862
  17. [17] S. Campanato, Equazioni elittiche non variazionali a coefficienti continui, Ann. Mat. Pura Appl. (4) 86 (1970), 125–154. Zbl0204.11701MR277881
  18. [18] S. Campanato, Hölder continuity of the solutions of some nonlinear elliptic systems Adv. Math.48 (1983), 16–43. Zbl0519.35027MR697613
  19. [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. Zbl1232.35042MR2230582
  20. [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. Zbl0869.35050MR1441620
  21. [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. Zbl0958.35045MR1760541
  22. [22] T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal.4 (1995), 185–203. Zbl0815.35023MR1323826
  23. [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. Zbl0822.35036MR1213138
  24. [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. Zbl0942.35059MR1707751
  25. [25] M. Di Giampaolo and F. Leonetti, Boundedness of weak solutions to some linear elliptic equations with measure data, Differential Integral Equations18 (2005), 1371–1382. Zbl1212.35179MR2174977
  26. [26] G. Dolzmann, N. Hungerbühler and S. Müller, The p -harmonic system with measure-valued right-hand side, Ann. Inst. H. Poincarè Anal. Non Linèaire14 (1997), 353–364. Zbl0879.35052MR1450953
  27. [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. Zbl0937.35065MR1748270
  28. [28] L. D’Onofrio and T. Iwaniec, Notes on p -harmonic analysis, Contemp. Math.370 (2005), 25–49. Zbl1134.35332MR2126700
  29. [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. Zbl0999.49022MR1880913
  30. [30] L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with ( p , q ) growth, J. Differential Equations204 (2004), 5–55. Zbl1072.49024MR2076158
  31. [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. Zbl0883.35047MR1467088
  32. [32] M. Fuchs and J. Reuling, Non-linear elliptic systems involving measure data, Rend. Mat. Appl. (7) 15 (1995), 311–319. Zbl0838.35133MR1339247
  33. [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. Zbl0361.35003MR473443
  34. [34] E. Giusti, “Direct Methods in the Calculus of Variations”, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. Zbl1028.49001MR1962933
  35. [35] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p -harmonic operator, Manuscripta Math.92 (1997), 249–258. Zbl0869.35037MR1428651
  36. [36] C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. (Crelles J.) 431 (1992), 7–64. Zbl0776.35006MR1179331
  37. [37] J. Heinonen, T. Kilpeläinen and O. Martio, “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Oxford Mathematical Monographs., New York, 1993. Zbl0780.31001MR1207810
  38. [38] T. Iwaniec, The Gehring lemma, In: “Quasiconformal mappings and analysis” (Ann Arbor, MI, 1995), 181–204, Springer, New York, 1998. Zbl0888.30017MR1488451
  39. [39] T. Iwaniec and C. Sbordone, Quasiharmonic fields, Ann. Inst. H. Poincaré Anal. Non Linèaire18 (2001), 519–572. Zbl1068.30011MR1849688
  40. [40] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math.14 (1961), 415–426. Zbl0102.04302MR131498
  41. [41] T. Kilpeläinen, Hölder continuity of solutions to quasilinear elliptic equations involving measures, Potential Anal.3 (1994), 265–272. Zbl0813.35016
  42. [42] T. Kilpeläinen and G. Li, Estimates for p -Poisson equations, Differential Integral Equations13 (2000), 791–800. Zbl0970.35035
  43. [43] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math.172 (1994), 137–161. Zbl0820.35063
  44. [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. Zbl1158.35355MR2379685
  45. [45] T. Kilpeläinen and Xiangsheng Xu, On the uniqueness problem for quasilinear elliptic equations involving measures Rev. Mat. Iberoamericana12 (1996), 461–475. Zbl0858.35037MR1402674
  46. [46] T. Kilpeläinen and X. Zhong, Removable sets for continuous solutions of quasilinear elliptic equations, Proc. Amer. Math. Soc.130 (2002), 1681–1688. Zbl1027.35032MR1887015
  47. [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 Equations19 (1994), 959–1014. Zbl0803.35068MR1274547
  48. [48] J. Kristensen and G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal.180 (2006), 331–398. Zbl1116.49010MR2214961
  49. [49] G. M. Lieberman, Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures, Comm. Partial Differential Equations18 (1993), 1191–1212. Zbl0802.35041MR1233190
  50. [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. Zbl1107.35030MR1986696
  51. [51] P. Lindqvist, On the definition and properties of p -superharmonic functions, J. Reine Angew. Math. (Crelles J.) 365 (1986), 67–79. Zbl0572.31004MR826152
  52. [52] P. Lindqvist, “Notes on p -Laplace Equation”, University of Jyväskylä - Lectures notes, 2006. Zbl1256.35017MR2242021
  53. [53] J. L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites non Linèaires”, Dunod, Gauthier-Villars, Paris, 1969. Zbl0189.40603MR259693
  54. [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. Zbl0116.30302MR161019
  55. [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. Zbl0882.35001MR1461542
  56. [56] J. J. Manfredi, Regularity for minima of functionals with p -growth, J. Differential Equations76 (1988), 203–212. Zbl0674.35008MR969420
  57. [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. [58] A. L. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc.355 (2003), 1297–1364. Zbl1022.35039MR1946395
  59. [59] G. Mingione, The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal.166 (2003), 287–301. Zbl1142.35391MR1961442
  60. [60] G. Mingione, Calderón-Zygmund estimates for measure data problems, C. R. Acad. Sci. Paris Sèr. I Math.344 (2007), 437–442. Zbl1190.35088MR2320247
  61. [61] G. Mingione, Sub-quadratic measure data problems, in preparation. 
  62. [62] T. Miyakawa, On Morrey spaces of measures: basic properties and potential estimates, Hiroshima Math. J.20 (1990), 213–222. Zbl0728.31007MR1050438
  63. [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. Zbl0805.35035MR1291535
  64. [64] J. Ross, A Morrey-Nikolski inequality, Proc. Amer. Math. Soc.78 (1980), 97–102. Zbl0453.46026MR548092
  65. [65] T. Runst and W. Sickel, “Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations”, Walter de Gruyter & Co., Berlin, 1996. Zbl0873.35001
  66. [66] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc.207 (1975), 391–405. Zbl0319.42006MR377518
  67. [67] J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 385–387. Zbl0142.37601MR170094
  68. [68] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier15 (1965), 189–258. Zbl0151.15401MR192177
  69. [69] G. Stampacchia, The spaces ( p , λ ) , N ( p , λ ) and interpolation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 443–462. Zbl0149.09202MR199697
  70. [70] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4) 120 (1979), 160–184. Zbl0419.35041MR551065
  71. [71] M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations17 (1992), 1407–1456. Zbl0771.35047MR1187618
  72. [72] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math.138 (1977), 219–240. Zbl0372.35030MR474389
  73. [73] X. Zhong, On nonhomogeneous quasilinear elliptic equations, Dissertation, University of Jyväskylä, 1998, Ann. Acad. Sci. Fenn. Math. Diss. 117 (1998), 46 pages. Zbl0911.35048MR1648847

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.