A priori estimates for infinitely degenerate quasilinear equations

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.