| Title:
|
On weighted estimates of solutions of nonlinear elliptic problems (English) |
| Author:
|
Skrypnik, Igor V. |
| Author:
|
Larin, Dmitry V. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
124 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
173-184 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper is devoted to the estimate
\vert u(x,k)\vert\leq K\vert k\vert\left\{\mathop cap\nolimits_{p,w}(F)\frac{\rho^p}{w(B(x,\rho))}\right\} ^{\frac1{p-1}},
$2\<p<n$ for a solution of a degenerate nonlinear elliptic equation in a domain ${B(x_0,1)\setminus F}$, $F\subset B(x_0,d)=\{x\in\Bbb R^n |x_0-x|<d\}$, $d<\frac12$, under the boundary-value conditions $u(x,k)=k$ for $x\in\partial F$, $ u(x,k)=0$ for $x\in\partial B(x_0,1)$ and where $0<\rho\leq\mathop dist(x,F)$, $w(x)$ is a weighted function from some Muckenhoupt class, and $\mathop cap_{p,w}(F)$, $w(B(x,\rho))$ are weighted capacity and measure of the corresponding sets. (English) |
| Keyword:
|
degeneracy |
| Keyword:
|
Muckenhoupt class |
| Keyword:
|
pointwise estimate |
| Keyword:
|
nonlinear elliptic equation |
| Keyword:
|
capacity |
| Keyword:
|
a-priori estimate |
| MSC:
|
35B45 |
| MSC:
|
35J70 |
| idZBL:
|
Zbl 0937.35074 |
| idMR:
|
MR1780690 |
| DOI:
|
10.21136/MB.1999.126242 |
| . |
| Date available:
|
2009-09-24T21:36:49Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126242 |
| . |
| Reference:
|
[1] Skrypnik I. V.: Nonlinear elliptic boundary value problems.B. G. Teubner Verlag, Leipzig, 1986. Zbl 0617.35001, MR 0915342 |
| Reference:
|
[2] Skrypnik I. V.: New conditions of homogenization of nonlinear Dirichlet problems in perforated domains.Ukrainian Math. J. 48 (1996), no. 5, 675-694. MR 1417035, 10.1007/BF02384225 |
| Reference:
|
[3] Heinonen J., Kilpelainen T., Martio O.: Nonlinear potential theory of degenerate elliptic equations.Clarendon Press, Oxford, 1993. MR 1207810 |
| Reference:
|
[4] Chanillo S., Wheeden R. L.: Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions.Amer. J. Math. 107 (1985), 1191-1226. Zbl 0575.42026, MR 0805809, 10.2307/2374351 |
| Reference:
|
[5] Kufner A.: Weighted Sobolev spaces.B.G.Teubner Verlag, Leipzig, 1980. Zbl 0455.46034, MR 0664599 |
| Reference:
|
[6] Gutiérrez C. E., Nelson G. S.: Bounds for the fundamental solution of degenerate parabolic equations.Commun. Partial Differential Equations 13 (1988), no. 5, 635-649. MR 0919445, 10.1080/03605308808820555 |
| Reference:
|
[7] Leonardi S., Skrypnik I. I.: Necessary condition for regularity of a boundary point for a degenerate quasilinear parabolic equations.Catania Univ., Catania, 1995, preprint. |
| . |
