| Title:
|
Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data (English) |
| Author:
|
Dall'Aglio, Andrea |
| Author:
|
Segura de León, Sergio |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
379-390 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove boundedness and continuity for solutions to the Dirichlet problem for the equation \[ -{\rm div}(a(x,\nabla u))=h(x,u)+\mu ,\quad \text {in} \ \Omega \subset \mathbb R^N, \] where the left-hand side is a Leray-Lions operator from $W_0^{1,p} (\Omega )$ into $W^{-1,p'}(\Omega )$ with $1<p<N$, $h(x,s)$ is a Carathéodory function which grows like $|s|^{p-1}$ and $\mu $ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu $. (English) |
| Keyword:
|
bounded solution |
| Keyword:
|
$p$-Laplacian |
| Keyword:
|
renormalized solution |
| Keyword:
|
measure data |
| MSC:
|
35B45 |
| MSC:
|
35B65 |
| MSC:
|
35J15 |
| MSC:
|
35J25 |
| MSC:
|
35J60 |
| MSC:
|
35J92 |
| idZBL:
|
Zbl 07088791 |
| idMR:
|
MR3959951 |
| DOI:
|
10.21136/CMJ.2018.0322-17 |
| . |
| Date available:
|
2019-05-24T08:56:37Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147731 |
| . |
| Reference:
|
[1] Hamid, H. Abdel, Bidaut-Veron, M. F.: On the connection between two quasilinear elliptic problems with source terms of order $0$ or $1$.Commun. Contemp. Math. 12 (2010), 727-788. Zbl 1205.35135, MR 2733197, 10.1142/S0219199710003993 |
| Reference:
|
[2] Abdellaoui, B., Dall'Aglio, A., Peral, I.: Some remarks on elliptic problems with critical growth in the gradient.J. Differ. Equations 222 (2006), 21-62 corrigendum ibid. 246 2988-2990 2009. Zbl 1357.35089, MR 2200746, 10.1016/j.jde.2005.02.009 |
| Reference:
|
[3] Abdellaoui, B., Dall'Aglio, A., León, S. Segura de: Multiplicity of solutions to elliptic problems involving the 1-Laplacian with a critical gradient term.Adv. Nonlinear Stud. 17 (2017), 333-353. Zbl 1370.35115, MR 3641646, 10.1515/ans-2017-0011 |
| Reference:
|
[4] Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J. L.: An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations.Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22 (1995), 241-273. Zbl 0866.35037, MR 1354907 |
| Reference:
|
[5] Boccardo, L., Gallouët, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13 (1996), 539-551. Zbl 0857.35126, MR 1409661, 10.1016/S0294-1449(16)30113-5 |
| Reference:
|
[6] Boccardo, L., Leonori, T.: Local properties of solutions of elliptic equations depending on local properties of the data.Methods Appl. Anal. 15 (2008), 53-63. Zbl 1173.35488, MR 2482209, 10.4310/MAA.2008.v15.n1.a6 |
| Reference:
|
[7] Brézis, H., Kato, T.: Remarks on the Schrödinger operator with singular complex potentials.J. Math. Pures Appl., IX. Sér. 58 (1979), 137-151. Zbl 0408.35025, MR 0539217 |
| Reference:
|
[8] Maso, G. Dal, Murat, F., Orsina, L., Prignet, A.: Renormalization solutions of elliptic equations with general measure data.Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28 (1999), 741-808. Zbl 0958.35045, MR 1760541 |
| Reference:
|
[9] Giusti, E.: Direct Methods in the Calculus of Variations.World Scientific, Singapore (2003). Zbl 1028.49001, MR 1962933, 10.1142/9789812795557 |
| Reference:
|
[10] Grenon, N.: Existence results for semilinear elliptic equations with small measure data.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19 (2002), 1-11. Zbl 1011.35054, MR 1902548, 10.1016/S0294-1449(01)00079-8 |
| Reference:
|
[11] Jaye, B. J., Verbitsky, E.: Local and global behaviour of solutions to nonlinear equations with natural growth terms.Arch. Ration. Mech. Anal. 204 (2012), 627-681. Zbl 1255.35137, MR 2909911, 10.1007/s00205-011-0491-2 |
| Reference:
|
[12] Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients.Ann. Inst. Fourier 15 (1965), 189-257 French. Zbl 0151.15401, MR 0192177, 10.5802/aif.204 |
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