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2008 A priori estimates for infinitely degenerate quasilinear equations
Cristian Rios, Eric T. Sawyer, Richard L. Wheeden
Differential Integral Equations 21(1-2): 131-200 (2008). DOI: 10.57262/die/1356039062


We prove a priori bounds for derivatives of solutions $w$ of a class of quasilinear equations of the form \begin{equation*} \mathrm {div} \mathcal{A} ( x,w ) \nabla w+\vec{\gamma} ( x,w ) \cdot \nabla w+f ( x,w ) =0, \end{equation*} where $x \! = \! ( x_{1},\dots ,x_{n} ) $, and where $f$, $\vec{\gamma} = ( \gamma^{i} ) _{1\leq i\leq n}$ and $\mathcal{A}= ( a_{ij} ) _{1\leq i,j\leq n}$ are $\mathcal{C}^{\infty }$. The rank of the square symmetric matrix $\mathcal{A}$ is allowed to degenerate, as all but one eigenvalue of $\mathcal{A}$ are permitted to vanish to infinite order. We estimate derivatives of $w$ of arbitrarily high order in terms of just $w$ and its first derivatives. These estimates will be applied in a subsequent work to establish existence, uniqueness and regularity of weak solutions of the Dirchlet problem.


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Cristian Rios. Eric T. Sawyer. Richard L. Wheeden. "A priori estimates for infinitely degenerate quasilinear equations." Differential Integral Equations 21 (1-2) 131 - 200, 2008.


Published: 2008
First available in Project Euclid: 20 December 2012

zbMATH: 1224.35149
MathSciNet: MR2479665
Digital Object Identifier: 10.57262/die/1356039062

Primary: 35J70
Secondary: 35B45 , 35B65 , 35J62

Rights: Copyright © 2008 Khayyam Publishing, Inc.

Vol.21 • No. 1-2 • 2008
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