| Title:
|
On some $L^{p}$-estimates for solutions of elliptic equations in unbounded domains (English) |
| Author:
|
Monsurrò, Sara |
| Author:
|
Transirico, Maria |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
140 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
507-515 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this review article we present an overview on some a priori estimates in $L^p$, $p>1$, recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^p$-bounds, $p>2$, for the solution of the associated Dirichlet problem, obtained in correspondence with two different sign assumptions. Then, putting together the above mentioned bounds and using a duality argument, we extend the estimate also to the case $1<p<2$, for each sign assumption, and for a data in $L^p\cap L^2$. (English) |
| Keyword:
|
elliptic equation |
| Keyword:
|
discontinuous coefficient |
| Keyword:
|
a priori bound |
| MSC:
|
35B45 |
| MSC:
|
35J25 |
| MSC:
|
35R05 |
| idZBL:
|
Zbl 06537680 |
| idMR:
|
MR3432549 |
| DOI:
|
10.21136/MB.2015.144466 |
| . |
| Date available:
|
2015-11-17T20:56:58Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144466 |
| . |
| Reference:
|
[1] Adams, R. A.: Sobolev Spaces.Pure and Applied Mathematics 65. A Series of Monographs and Textbooks Academic Press, New York (1975). Zbl 0314.46030, MR 0450957 |
| Reference:
|
[2] Boccia, S., Monsurrò, S., Transirico, M.: Elliptic equations in weighted Sobolev spaces on unbounded domains.Int. J. Math. Math. Sci. 2008 (2008), Article No. 582435, 12 pages. Zbl 1158.35346, MR 2448275 |
| Reference:
|
[3] Boccia, S., Monsurrò, S., Transirico, M.: Solvability of the Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains.Bound. Value Probl. (electronic only) 2008 (2008), Article No. 901503, 13 pages. Zbl 1158.35346, MR 2448193 |
| Reference:
|
[4] Bottaro, G., Marina, M.: Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati.Boll. Unione Mat. Ital., IV. Ser. 8 Italian (1973), 46-56. Zbl 0291.35021, MR 0326148 |
| Reference:
|
[5] Caso, L., Cavaliere, P., Transirico, M.: An existence result for elliptic equations with {VMO}-coefficients.J. Math. Anal. Appl. 325 (2007), 1095-1102. Zbl 1152.35025, MR 2270071, 10.1016/j.jmaa.2006.02.048 |
| Reference:
|
[6] Caso, L., Cavaliere, P., Transirico, M.: Solvability of the Dirichlet problem in {$W^{2,p}$} for elliptic equations with discontinuous coefficients in unbounded domains.Matematiche 57 (2002), 287-302. MR 2167127 |
| Reference:
|
[7] Caso, L., D'Ambrosio, R., Monsurrò, S.: Some remarks on spaces of Morrey type.Abstr. Appl. Anal. 2010 (2010), Article No. 242079, 22 pages. Zbl 1208.42006, MR 2735004 |
| Reference:
|
[8] Cavaliere, P., Longobardi, M., Vitolo, A.: Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains.Matematiche 51 (1996), 87-104. Zbl 0905.35017, MR 1456365 |
| Reference:
|
[9] Donato, P., Giachetti, D.: Quasilinear elliptic equations with quadratic growth in unbounded domains.Nonlinear Anal. 10 (1986), 791-804. Zbl 0602.35036, MR 0851147, 10.1016/0362-546X(86)90038-6 |
| Reference:
|
[10] Lions, P. L.: Remarks on linear second order elliptic equations in divergence form on unbounded domains.Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 78 French (1985), 205-212. Zbl 0656.35030, MR 0919011 |
| Reference:
|
[11] Lions, P. L.: Remarks on linear second order elliptic equations in divergence form on unbounded domains II.Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 79 French (1985), 178-183. MR 0944371 |
| Reference:
|
[12] Miranda, C.: Alcune osservazioni sulla maggiorazione in {$L^{\nu}$} delle soluzioni deboli delle equazioni ellittiche del secondo ordine.Ann. Mat. Pura Appl. (4) 61 Italian (1963), 151-169. MR 0177187 |
| Reference:
|
[13] Miranda, C.: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui.Ann. Mat. Pura Appl. (4) 63 Italian (1963), 353-386. Zbl 0156.34001, MR 0170090 |
| Reference:
|
[14] Monsurrò, S., Transirico, M.: A priori bounds in {$L^p$} for solutions of elliptic equations in divergence form.Bull. Sci. Math. 137 (2013), 851-866. Zbl 1286.35092, MR 3116216, 10.1016/j.bulsci.2013.02.002 |
| Reference:
|
[15] Monsurrò, S., Transirico, M.: A $W^{2,p}$-estimate for a class of elliptic operators.Int. J. Pure Appl. Math. 83 (2013), 489-499. MR 3071216, 10.12732/ijpam.v83i3.9 |
| Reference:
|
[16] Monsurrò, S., Transirico, M.: A weighted $W^{2,p}$-a priori bound for a class of elliptic operators.J. Inequal. Appl. (electronic only) 2013 (2013), Article No. 263, 11 pages. Zbl 1284.35157, MR 3071216 |
| Reference:
|
[17] Monsurrò, S., Transirico, M.: An {$L^p$}-estimate for weak solutions of elliptic equations.Abstr. Appl. Anal. 2012 (2012), Article No. 376179, 15 pages. Zbl 1250.35074, MR 2947671 |
| Reference:
|
[18] Monsurrò, S., Transirico, M.: Dirichlet problem for divergence form elliptic equations with discontinuous coefficients.Bound. Value Probl. (electronic only) 2012 (2012), Article No. 67, 12 pages. Zbl 1278.35065, MR 3085725 |
| Reference:
|
[19] Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus.Ann. Inst. Fourier 15 French (1965), 189-257 Colloques Int. Centre nat. Rech. Sci. 146 (1965), 189-258. Zbl 0151.15401, MR 0192177, 10.5802/aif.204 |
| Reference:
|
[20] Transirico, M., Troisi, M.: Second-order elliptic equations with discontinuous coefficients and of variational type in unbounded open subsets.Boll. Unione Mat. Ital., VII. Ser., B 2 Italian. English summary (1988), 385-398. MR 0946077 |
| Reference:
|
[21] Transirico, M., Troisi, M., Vitolo, A.: Spaces of Morrey type and elliptic equations in divergence form on unbounded domains.Boll. Unione Mat. Ital., VII. Ser., B 9 (1995), 153-174. Zbl 0881.35031, MR 1328515 |
| . |
