Universal approximation by translates of fundamental solutions of elliptic equations

摘要

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by universal series of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie on a prescribed surface outside of Ω, known as the pseudo-boundary. 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 C~e(Ω) and we establish the existence of universal series. Analogous results are obtainable with respect to the norms of Holder spaces C~e'v(Ω). The sequence a = {a_n}_(n∈N) of coefficients of the universal series may be chosen in ∩_(p>1)~(lp)(N) but it can not be chosen in l~1(N).