Removable singularities of weak solutions to linear partial differential equations

A. V. Pokrovskii

Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 584-591

Résumé

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 $G\subset\mathbb R^n$ $(n\geqslant1)$, $1$, $s>0$ and $q=p/(p-1)$. In this paper, we show that if $n,m,p$ and $s$ satisfy the two-sided bound $0\leqslant n-q(m-s)$, then for a weak solution of the equation $P(x,D)u=0$ from the Sharpley–DeVore class $C_p^s(G)_{\text{loc}}$, 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.