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Mat. Zametki, 2005, Volume 77, Issue 4, Pages 584–591 (Mi mz2519)  

This article is cited in 1 scientific paper (total in 1 paper)

Removable singularities of weak solutions to linear partial differential equations

A. V. Pokrovskii

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: Suppose that $P(x,D)$ is a linear differential operator of order $m>0$ with smooth coefficients whose derivatives up to order $m$ are continuous functions in the domain $G\subset\mathbb R^n$ $(n\geqslant1)$, $1<p<\infty$, $s>0$ and $q=p/(p-1)$. In this paper, we show that if $n,m,p$ and $s$ satisfy the two-sided bound $0\leqslant n-q(m-s)<n$, then for a weak solution of the equation $P(x,D)u=0$ from the Sharpley–DeVore class $C_p^s(G)_{loc}$, any closed set in $G$ is removable if its Hausdorff measure of order $n-q(m-s)$ is finite. This result strengthens the well-known result of Harvey and Polking on removable singularities of weak solutions to the equation $P(x,D)u=0$ from the Sobolev classes and extends it to the case of noninteger orders of smoothness.

DOI: https://doi.org/10.4213/mzm2519

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English version:
Mathematical Notes, 2005, 77:4, 539–545

Bibliographic databases:

UDC: 517.956
Received: 20.06.2003
Revised: 13.09.2004

Citation: A. V. Pokrovskii, “Removable singularities of weak solutions to linear partial differential equations”, Mat. Zametki, 77:4 (2005), 584–591; Math. Notes, 77:4 (2005), 539–545

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Ruzhansky M., Sugimoto M., “Criteria for Bochner's extension problem”, Asymptot. Anal., 66:3–4 (2010), 125–138  mathscinet  zmath  isi
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