Regularity properties of solutions of elliptic equations in $ \mathbb{R}^{2} $ in limit	cases

Alberico, Angela; Ferone, Vincenzo

Regularity properties of solutions of elliptic equations in

R^{2}

in limit cases

Alberico, Angela ; Ferone, Vincenzo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 6 (1995), p. 237-250 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

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.

In questa Nota si studia il problema di Dirichlet per un'equazione lineare ellittica in un insieme aperto, limitato di $R^{2}$ . Sono provate proprietà di regolarità per le soluzioni, quando i dati sono funzioni di $L^{1}$ oppure misure di Radon. In particolare sono date ipotesi ottimali che garantiscono la continuità delle soluzioni.

Publié le : 1995-12-01

MR 1382708 | Zbl 0860.35015

@article{RLIN_1995_9_6_4_237_0,
     author = {Angela Alberico and Vincenzo Ferone},
     title = {Regularity properties of solutions of elliptic equations in \( \mathbb{R}^{2} \) in limit	cases},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {6},
     year = {1995},
     pages = {237-250},
     zbl = {0860.35015},
     mrnumber = {1382708},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1995_9_6_4_237_0}
}

Alberico, Angela; 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, Tome 6 (1995) pp. 237-250. http://gdmltest.u-ga.fr/item/RLIN_1995_9_6_4_237_0/

Bibliographie

[1] Adams, D. R., A sharp inequality of J. Moser for higher order derivatives. Annals of Math., 128, 1988, 385-398. | MR 960950 | Zbl 0672.31008

[2] Alvino, A., Un caso limite della disuguaglianza di Sobolev in spazi di Lorentz. Rend. Acc. Sci. Fis. Mat. Napoli, 44, 1977, 105-112. | Zbl 0412.46024

[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. | Zbl 0371.35009

[4] Alvino, A. - Ferone, V. - Trombetti, G., Moser-type inequalities in Lorentz spaces. Potential Anal., to appear. | MR 1389498 | Zbl 0856.46020

[5] Bennett, C. - Rudnick, K., On Lorentz-Zygmund spaces. Dissert. Math., 175, 1980, 1-67. | MR 576995 | Zbl 0456.46028

[6] Boccardo, L. - Gallouët, T., Non-linear elliptic and parabolic equations involving measures data. Journal of Functional Analysis, 87, 1989, 149-169. | Zbl 0707.35060

[7] Boccardo, L. - Murat, F., A property of nonlinear elliptic equations when the right-hand side is a measure. Potential Anal., 3, 1994, 257-263. | MR 1290666 | Zbl 0807.35034

[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. | MR 1132783 | Zbl 0746.35006

[9] Brezis, H. - Strauss, W. A., Semi-linear second-order elliptic equations in $L^{1}$ . J. Math. Soc. Japan, 25, 1973, 565-590. | MR 336050 | Zbl 0278.35041

[10] Chanillo, S. - Li, Y. Y., Continuity of solutions of uniformly elliptic equations in $R^{2}$ . Manuscripta Math., 77, 1992, 415-433. | MR 1190215 | Zbl 0797.35031

[11] Chiti, G., Rearrangements of functions and convergence in Orlicz spaces. Appl. Anal., 9, 1979, 23-27. | MR 536688 | Zbl 0424.46023

[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. | MR 372140 | Zbl 0275.46024

[13] Ferone, V., Estimates and regularity for solutions of elliptic equations in a limit case. Boll. U.M.I., (7) 8B, 1994, 257-270. | MR 1278335 | Zbl 0799.35056

[14] Ferone, V. - Fusco, N., Continuity properties of minimizers of integral functionals. J. Math. Anal. Appl., to appear. | MR 1402586 | Zbl 0865.49008

[15] Liskevich, V. A., Some limit cases in estimates for solutions of second order elliptic equations. Houston J. Math., 19, 1993, 661-673. | MR 1251616 | Zbl 0797.35019

[16] Serrin, J., Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa, (3) 18, 1964, 385-387. | MR 170094 | Zbl 0142.37601

[17] Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, Grenoble, 15, 1965, 189-258. | MR 192177 | Zbl 0151.15401

[18] Talenti, G., Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa, (4) 3, 1976, 697-718. | MR 601601 | Zbl 0341.35031