Dense Subsets of [formula omitted]1-Solutions to Linear Elliptic Partial Differential Equations

Abstract

Let Ω⊂RN (N⩾2) be an unbounded domain, and Lm be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing L1-solutions to Lm (in Ω) approximate all L1-solutions to Lm (in Ω), provided there exist real numbers Rj→∞, ε⩾0, and a sequence {yj} such that B(yj, ε)∩Ω=∅ and|Λ(yj, Rj, RN\\Ω)|RNj>ε ∀ j,where |·| means the volume andΛ(z, R, D)≔∪x∈B(z, R)∩Dz+t(x−z)|x−z|;t⩽1,for z∈RN, R>0 and D⊂RN. For m=2, we can replace the volume density by the capacity-density. It appears that the problem is related to the characterization of largest sets on which a nonzero polynomial solution to Lm may vanish, along with its (m−1)-derivatives. We also study a similar approximation problem for polyanalytic functions in C.