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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 1311911
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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

Juan J. Manfredi
Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

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

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