## Differential and Integral Equations

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

### A priori estimates for infinitely degenerate quasilinear equations

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

#### 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