Removable singularities of weak solutions to linear partial differential equations

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 ⁿ (n 1), 1 < p > , 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 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.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 584–591.Original Russian Text Copyright © 2005 by A. V. Pokrovskii.This revised version was published online in April 2005 with a corrected issue number.