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

Authors:
Pekka Koskela, Juan J. Manfredi and Enrique Villamor

Journal:
Trans. Amer. Math. Soc. **348** (1996), 755-766

MSC (1991):
Primary 35B65; Secondary 31B25

DOI:
https://doi.org/10.1090/S0002-9947-96-01430-4

MathSciNet review:
1311911

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Abstract | References | Similar Articles | Additional Information

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 $p$.

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Additional Information

**Pekka Koskela**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

MR Author ID:
289254

Email:
pkoskela@math.jyu.fi

**Juan J. Manfredi**

Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

MR Author ID:
205679

Email:
manfredit@pitt.edu

**Enrique Villamor**

Affiliation:
Department of Mathematics, Florida International University, Miami, Florida 33199

Email:
villamor@fiu.edu

Received by editor(s):
June 7, 1994

Received by editor(s) in revised form:
January 23, 1995

Additional Notes:
Research of the first author was partially supported by the Academy of Finland and NSF grant DMS-9305742

Research of the second author was partially supported by NSF grant DMS-9101864

Article copyright:
© Copyright 1996
American Mathematical Society