Geodesic
On weighted estimates of solutions of nonlinear elliptic problems
Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 173-184

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MR Zbl
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\
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\$ for a solution of a degenerate nonlinear elliptic equation in a domain ${B(x_0,1)\setminus F}$, $F\subset B(x_0,d)=\{x\in\Bbb R^n |x_0-x|$, $d\frac12$, 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.
DOI : 10.21136/MB.1999.126242
Classification : 35B45, 35J70
Keywords: degeneracy; Muckenhoupt class; pointwise estimate; nonlinear elliptic equation; capacity; a-priori estimate 
Skrypnik, Igor V.; Larin, Dmitry V. On weighted estimates of solutions of nonlinear elliptic problems. Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 173-184. doi: 10.21136/MB.1999.126242
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[1] Skrypnik I. V.: Nonlinear elliptic boundary value problems. B. G. Teubner Verlag, Leipzig, 1986. | MR | Zbl

[2] Skrypnik I. V.: New conditions of homogenization of nonlinear Dirichlet problems in perforated domains. Ukrainian Math. J. 48 (1996), no. 5, 675-694. | DOI | MR

[3] Heinonen J., Kilpelainen T., Martio O.: Nonlinear potential theory of degenerate elliptic equations. Clarendon Press, Oxford, 1993. | MR

[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. | DOI | MR | Zbl

[5] Kufner A.: Weighted Sobolev spaces. B.G.Teubner Verlag, Leipzig, 1980. | MR | Zbl

[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. | DOI | MR

[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.

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