Regularity properties of solutions of elliptic equations in R 2 in limit cases

Angela Alberico; Vincenzo Ferone

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

  • Volume: 6, Issue: 4, page 237-250
  • ISSN: 1120-6330

Abstract

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In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of R 2 is studied. Regularity properties of the solutions are proved, when the data are L 1 -functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.

How to cite

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Alberico, Angela, and Ferone, Vincenzo. "Regularity properties of solutions of elliptic equations in \( \mathbb{R}^{2} \) in limit cases." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.4 (1995): 237-250. <http://eudml.org/doc/244102>.

@article{Alberico1995,
abstract = {In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of \( \mathbb\{R\}^\{2\} \) is studied. Regularity properties of the solutions are proved, when the data are \( L^\{1\} \)-functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.},
author = {Alberico, Angela, Ferone, Vincenzo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Elliptic equations; Lorentz spaces; Continuity properties},
language = {eng},
month = {12},
number = {4},
pages = {237-250},
publisher = {Accademia Nazionale dei Lincei},
title = {Regularity properties of solutions of elliptic equations in \( \mathbb\{R\}^\{2\} \) in limit cases},
url = {http://eudml.org/doc/244102},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Alberico, Angela
AU - Ferone, Vincenzo
TI - Regularity properties of solutions of elliptic equations in \( \mathbb{R}^{2} \) in limit cases
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/12//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 4
SP - 237
EP - 250
AB - In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of \( \mathbb{R}^{2} \) is studied. Regularity properties of the solutions are proved, when the data are \( L^{1} \)-functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.
LA - eng
KW - Elliptic equations; Lorentz spaces; Continuity properties
UR - http://eudml.org/doc/244102
ER -

References

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  3. ALVINO, A., Formule di maggiorazione e regolarizzazione per soluzioni di equazioni ellittiche del secondo ordine in un caso limite. Atti Acc. Lincei Rend, fis., s. 8, vol. 52, 1977, 335-340. Zbl0371.35009
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  8. BREZIS, H. - MERLE, F., Uniform estimates and blow-up behavior for solutions of — ⁢ Δ ⁢ u = V ⁢ x ⁢ e u in two dimensions. Comm. in P.D.E., 16, 1991, 1223-1253. Zbl0746.35006MR1132783DOI10.1080/03605309108820797
  9. BREZIS, H. - STRAUSS, W. A., Semi-linear second-order elliptic equations in L 1 . J. Math. Soc. Japan, 25, 1973, 565-590. Zbl0278.35041MR336050
  10. CHANILLO, S. - LI, Y. Y., Continuity of solutions of uniformly elliptic equations in R 2 . Manuscripta Math., 77, 1992, 415-433. Zbl0797.35031MR1190215DOI10.1007/BF02567065
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  12. CHONG, K. M. - RICE, N. M., Equimeasurable rearrangements of functions. Queen's papers in pure and applied mathematics, no. 28, Queen's University, Ontario, 1971. Zbl0275.46024MR372140
  13. FERONE, V., Estimates and regularity for solutions of elliptic equations in a limit case. Boll. U.M.I., (7) 8B, 1994, 257-270. Zbl0799.35056MR1278335
  14. FERONE, V. - FUSCO, N., Continuity properties of minimizers of integral functionals. J. Math. Anal. Appl., to appear. Zbl0865.49008MR1402586DOI10.1006/jmaa.1996.0301
  15. LISKEVICH, V. A., Some limit cases in estimates for solutions of second order elliptic equations. Houston J. Math., 19, 1993, 661-673. Zbl0797.35019MR1251616
  16. SERRIN, J., Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa, (3) 18, 1964, 385-387. Zbl0142.37601MR170094
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  18. TALENTI, G., Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa, (4) 3, 1976, 697-718. Zbl0341.35031MR601601

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