| Title:
|
Existence of entropy solutions for degenerate quasilinear elliptic equations in $L^1$ (English) |
| Author:
|
Cavalheiro, Albo Carlos |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
22 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
57-69 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article, we prove the existence of entropy solutions for the Dirichlet problem $$ (P)\begin {cases} -\mathrm{div} [{\omega }(x){\cal A} (x,u,{\nabla }u)]=f(x)-\mathrm{div} (G),&\text {in }\Omega \\ u(x) = 0,&\text {on }{\partial \Omega } \end {cases} $$ where $\Omega $ is a bounded open set of $\real ^N$, $N\geq 2$, $f \in L^1(\Omega )$ and $G/{\omega } \in [L^{p'}(\Omega , \omega )]^N$. (English) |
| Keyword:
|
degenerate elliptic equations |
| Keyword:
|
entropy solutions |
| Keyword:
|
weighted Sobolev spaces |
| MSC:
|
35A01 |
| MSC:
|
35J25 |
| MSC:
|
35J60 |
| MSC:
|
35J62 |
| MSC:
|
35J70 |
| idZBL:
|
Zbl 1302.35180 |
| idMR:
|
MR3233727 |
| . |
| Date available:
|
2014-08-27T09:00:24Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143906 |
| . |
| Reference:
|
[1] Bélinan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vasquez, J.L.: An $L^1$ theory of existence and uniqueness of solutions of nonlinear elliptic equations.Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22, 2, 1995, 241-273, MR 1354907 |
| Reference:
|
[2] Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solutions for nonlinear elliptic unilateral problems.Ann. Mat. Pura Appl., 152, 1988, 183-196, MR 0980979 |
| Reference:
|
[3] Cavalheiro, A.C.: The solvability of Dirichlet problem for a class of degenerate elliptic equations with $L^1$-data.Applicable Analysis, 85, 8, 2006, 941-961, Zbl 1142.35042, MR 2250779 |
| Reference:
|
[4] Cavalheiro, A.C.: Existence of solutions for some degenerate quasilinear elliptic equations.Le Matematiche, LXIII, II, 2008, 101-112, Zbl 1189.35124, MR 2531653 |
| Reference:
|
[5] Piat, V. Chiadò, Cassano, F. Serra: Relaxation of degenerate variational integrals.Nonlinear Anal., 22, 1994, 409-429, MR 1266369, 10.1016/0362-546X(94)90165-1 |
| Reference:
|
[6] Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations.Comm. PDEs, 7, 1982, 77-116, Zbl 0498.35042, MR 0643158, 10.1080/03605308208820218 |
| Reference:
|
[7] Folland, G.B.: Real Analysis.1984, Wiley-Interscience, New York, Zbl 0549.28001, MR 0767633 |
| Reference:
|
[8] Garcia-Cuerva, J., Francia, J.L. Rubio de: Weighted Norm Inequalities and Related Topics.116, 1985, North-Holland Mathematics Studies, MR 0807149 |
| Reference:
|
[9] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations.1993, Oxford Math. Monographs, Clarendon Press, Zbl 0780.31001, MR 1207810 |
| Reference:
|
[10] Kufner, A., John, O., Fučík, S.: Function Spaces.1977, Noordhoof International Publishing, Leyden, MR 0482102 |
| Reference:
|
[11] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function.Trans. Am. Math. Soc., 165, 1972, 207-226, Zbl 0236.26016, MR 0293384, 10.1090/S0002-9947-1972-0293384-6 |
| Reference:
|
[12] Stein, E.: Harmonic Analysis.1993, Princeton University Press, Zbl 0821.42001, MR 1232192 |
| Reference:
|
[13] Torchinsky, A.: Real-Variable Methods in Harmonic Analysis.1986, Academic Press, San Diego, Zbl 0621.42001, MR 0869816 |
| Reference:
|
[14] Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics.1736, 2000, Springer-Verlag, MR 1774162 |
| . |
