Regularity theory and traces of $\mathcal {A}$-harmonic functions

Abstract:

In this paper we discuss two different topics concerning $\mathcal {A}$- harmonic functions. These are weak solutions of the partial differential equation \begin{equation*}\text {div}(\mathcal {A}(x,\nabla u))=0,\end{equation*} where $\alpha (x)|\xi |^{p-1}\le \langle \mathcal {A}(x,\xi ),\xi \rangle \le \beta (x) |\xi |^{p-1}$ for some fixed $p\in (1,\infty )$, the function $\beta$ is bounded and $\alpha (x)>0$ for a.e. $x$. First, we present a new approach to the regularity of $\mathcal {A}$-harmonic functions for $p>n-1$. Secondly, we establish results on the existence of nontangential limits for $\mathcal {A}$-harmonic functions in the Sobolev space $W^{1,q}(\mathbb {B})$, for some $q>1$, where $\mathbb {B}$ is the unit ball in $\mathbb {R}^n$. Here $q$ is allowed to be different from $p$.