Polar sets and removable singularities of partial differential equations
The question of removable singularities for partial differential equations is essentially the following: If u is a solution of such an equation in a domain V c R = with a closet set S (of measure zero) removed; and if u is assumed to belong to a class limiting its size near S (for example u CLp), what conditions can be put on the size of S to insure tha t u (after being redefined in S) is a solution in all of V ? For example, every bounded harmonic function in a punctured disk has a removable singularity at the "puncture". Here the class limiting the size of u may be taken to be the class of bounded functions; and S may be taken to be a single point. L. Carleson [4] has shown that if u is harmonic in V S (V a bounded n-dimensional domain, S a compact set ~ V) and S has finite n 2p' dimensional t tausdorff measure ( l ip+l ip ' =1) then the singularities of u on S are removable provided u ELv. Serrin [10] has extended this result to second order linear elliptic equations with tt61der continuous coefficients, and has given a different sufficient condition for second order (linear or quasilinear) elliptic equations [11]. Our aim is to treat linear equations of arbi trary order. (Some results of this nature are contained in [3]). We begin by observing (in section 1) that the question of removable singularities of solutions in Lp is closely tied to the notion of " m p polar" sets in R ~ (A compact set S is m p polar if every element in H_m.v,(R n) with support in S vanishes), a notion apparent ly first introduced by H5rmander and Lions [6]. The relationships between the two concepts is expressed in theorems 1 and 2. These theorems are proved in sections 1 and 2 where they are applied to second order equations. In section 3 generalizations of the Hm, v spaces called "A spaces" are introduced, and, using these, sharper results are obtained for equations which are assumed to be of a more special form. Section 4 deals with geometric sufficient conditions for m p polarity of sets, which seems to be of interest independently of the question of removable singularities (see for example [5]); while in section 5 similar results are obtained for the A-spaces introduced earlier. Finally, as an illustration, the latter results are applied to give geometric conditions for removabili ty of singularities of the heat equation. The author would like to take this opportunity to thank Professors Hans Weinberger and Richard Juberg for a number of helpful discussions.