Regularity theory and traces of 𝒜-harmonic functions

Abstract

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