Differential and Integral Equations

A priori estimates for infinitely degenerate quasilinear equations

Cristian Rios, Eric T. Sawyer, and Richard L. Wheeden

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Abstract

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.

Article information

Source
Differential Integral Equations, Volume 21, Number 1-2 (2008), 131-200.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039062

Mathematical Reviews number (MathSciNet)
MR2479665

Zentralblatt MATH identifier
1224.35149

Subjects
Primary: 35J70: Degenerate elliptic equations
Secondary: 35B45: A priori estimates 35B65: Smoothness and regularity of solutions 35J62: Quasilinear elliptic equations

Citation

Rios, Cristian; Sawyer, Eric T.; Wheeden, Richard L. A priori estimates for infinitely degenerate quasilinear equations. Differential Integral Equations 21 (2008), no. 1-2, 131--200. https://projecteuclid.org/euclid.die/1356039062


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