Density results with linear combinations of translates of fundamental solutions

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by linear combinations of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie outside of Ω. The domains under consideration satisfy a rather mild boundary regularity requirement, namely the Segment Condition. We study approximations with respect to the norms of the spaces Ck(Ω), Ck,σ(Ω) and Wk,p(Ω), and we establish density and non–density results for elliptic operators with constant coefficients. We also provide applications of our density results related to the method of fundamental solutions and to the theory of universal series.