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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Regularity theory and traces of $\mathcal {A}$-harmonic functions
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by Pekka Koskela, Juan J. Manfredi and Enrique Villamor PDF
Trans. Amer. Math. Soc. 348 (1996), 755-766 Request permission

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
  • © Copyright 1996 American Mathematical Society
  • 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