| Title:
|
Integrability for very weak solutions to boundary value problems of $p$-harmonic equation (English) |
| Author:
|
Gao, Hongya |
| Author:
|
Liang, Shuang |
| Author:
|
Cui, Yi |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
101-110 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with very weak solutions $u\in \theta + W_0^{1,r}(\Omega )$, $\max \{1,p-1\}<r<p<n$, to boundary value problems of the \mbox {$p$-harmonic} equation $$ \begin {cases} -\mbox {div}(|\nabla u(x)|^{p-2} \nabla u(x))=0, & x\in \Omega , \\ u(x)=\theta (x), & x\in \partial \Omega . \end {cases} \eqno (*) $$ We show that, under the assumption $\theta \in W^{1,q}(\Omega )$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$ u\in \begin {cases} \theta +L_{\rm weak}^{q^*}(\Omega ) & \mbox {for } q<n, \\ \theta +L_{\rm weak}^\tau (\Omega ) & \mbox {for } q=n \mbox { and any } \tau <\infty , \\ \theta +L^\infty (\Omega ) & \mbox {for } q>n, \end {cases} $$ provided that $r$ is sufficiently close to $p$. (English) |
| Keyword:
|
integrability |
| Keyword:
|
very weak solution |
| Keyword:
|
boundary value problem |
| Keyword:
|
$p$-harmonic equation |
| MSC:
|
35D30 |
| MSC:
|
35J25 |
| idZBL:
|
Zbl 06587876 |
| idMR:
|
MR3483225 |
| DOI:
|
10.1007/s10587-016-0242-5 |
| . |
| Date available:
|
2016-04-07T14:58:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144870 |
| . |
| Reference:
|
[1] Bensoussan, A., Frehse, J.: Regularity Results for Nonlinear Elliptic Systems and Applications.Applied Mathematical Sciences 151 Springer, Berlin (2002). Zbl 1055.35002, MR 1917320, 10.1007/978-3-662-12905-0_9 |
| Reference:
|
[2] Campanato, S.: Sistemi ellittici in forma divergenza. Regolarità all'interno.Pubblicazioni della Classe di Scienze: Quaderni Italian Scuola Normale Superiore Pisa, Pisa (1980). MR 0668196 |
| Reference:
|
[3] Gao, H.: Regularity for solutions to anisotropic obstacle problems.Sci. China, Math. 57 (2014), 111-122. Zbl 1304.35298, MR 3146520, 10.1007/s11425-013-4601-5 |
| Reference:
|
[4] Gao, H., Chu, Y. M.: Quasiregular Mappings and $A$-harmonic Equations.Science Press Beijing (2013), Chinese. |
| Reference:
|
[5] Gao, H., Huang, Q.: Local regularity for solutions of anisotropic obstacle problems.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 4761-4765. Zbl 1250.49038, MR 2927542, 10.1016/j.na.2012.03.026 |
| Reference:
|
[6] Gao, H., Liu, C., Tian, H.: Remarks on a paper by Leonetti and Siepe.J. Math. Anal. Appl. 401 (2013), 881-887. Zbl 1266.35065, MR 3018035, 10.1016/j.jmaa.2012.12.037 |
| Reference:
|
[7] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems.Annals of Mathematics Studies 105 Princeton University Press, Princeton (1983). Zbl 0516.49003, MR 0717034 |
| Reference:
|
[8] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order.Grundlehren der mathematischen Wissenschaften Springer, Berlin (1977). MR 0473443 |
| Reference:
|
[9] Greco, L., Iwaniec, T., Sbordone, C.: Inverting the {$p$}-harmonic operator.Manuscr. Math. 92 (1997), 249-258. MR 1428651, 10.1007/BF02678192 |
| Reference:
|
[10] Giusti, E.: Metodi diretti nel calcolo delle variazioni.Unione Matematica Italiana Bologna Italian (1994). MR 1707291 |
| Reference:
|
[11] Iwaniec, T.: {$p$}-harmonic tensors and quasiregular mappings.Ann. Math. (2) 136 (1992), 589-624. MR 1189867 |
| Reference:
|
[12] Iwaniec, T., Martin, G.: Geometric Function Theory and Nonlinear Analysis.Oxford Mathematical Monographs Oxford University Press, Oxford (2001). Zbl 1045.30011, MR 1859913 |
| Reference:
|
[13] Iwaniec, T., Migliaccio, L., Nania, L., Sbordone, C.: Integrability and removability results for quasiregular mappings in high dimensions.Math. Scand. 75 (1994), 263-279. MR 1319735, 10.7146/math.scand.a-12519 |
| Reference:
|
[14] Iwaniec, T., Sbordone, C.: Weak minima of variational integrals.J. Reine Angew. Math. 454 (1994), 143-161. MR 1288682 |
| Reference:
|
[15] Iwaniec, T., Scott, C., Stroffolini, B.: Nonlinear Hodge theory on manifolds with boundary.Ann. Mat. Pura Appl. (4) 177 (1999), 37-115. MR 1747627 |
| Reference:
|
[16] Kufner, A., John, O., Fučík, S.: Function Spaces.Monographs and Textsbooks on Mechanics of Solids and Fluids; Mechanics: Analysis Noordhoff International Publishing, Leyden; Academia, Praha (1977). MR 0482102 |
| Reference:
|
[17] Ladyzhenskaya, O. A., Ural'tseva, N. N.: Linear and Quasilinear Elliptic Equations.Mathematics in Science and Engineering Academic Press, New York (1968). Zbl 0177.37404, MR 0244627 |
| Reference:
|
[18] Leonetti, F., Siepe, F.: Global integrability for minimizers of anisotropic functionals.Manuscr. Math. 144 (2014), 91-98. Zbl 1287.49041, MR 3193770, 10.1007/s00229-013-0641-y |
| Reference:
|
[19] Leonetti, F., Siepe, F.: Integrability for solutions to some anisotropic elliptic equations.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 2867-2873. Zbl 1244.35053, MR 2878481, 10.1016/j.na.2011.11.028 |
| Reference:
|
[20] Lindqvist, P.: Notes on the \mbox{$p$-Laplace} Equation.Report. University of Jyväskylä Department of Mathematics and Statistics 102 University of Jyväskylä, Jyväskylä (2006). MR 2242021 |
| Reference:
|
[21] Meyers, N. G., Elcrat, A.: Some results on regularity for solutions of nonlinear elliptic systems and quasi-regular functions.Duke Math. J. 42 (1975), 121-136. MR 0417568, 10.1215/S0012-7094-75-04211-8 |
| Reference:
|
[22] C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations.Die Grundlehren der mathematischen Wissenschaften 130 Springer, New York (1966). Zbl 0142.38701, MR 0202511, 10.1007/978-3-540-69952-1 |
| Reference:
|
[23] Stampacchia, G.: Èquations elliptiques du second ordre à coefficients discontinus.Séminaire de mathématiques supérieures 16 (été 1965) Les Presses de l'Université de Montréal, Montreal, Que. French (1966). MR 0251373 |
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