Density results with linear combinations of translates of fundamental solutions

Abstract

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 may possess holes and they are required to satisfy a rather mild boundary regularity requirement, namely the segment condition. We study approximations with respect to the norms of the spaces C k (Ω ) and the spaces of uniformly Hölder continuous functions lip k,σ (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.

A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs

“…We take a closer look at Smyrlis' papers now in order to show similar results for the QEP (4.1). The results in these three articles are based on the following lemma [262][263][264]. Based on this lemma, Smyrlis proved a theorem for the MFS for elliptic differential equations similar to the one by Browder [41] which we mentioned in the previous section [263].…”

“…The proof follows the ones given in [262][263][264]. We split it up into several steps and start by proving an estimate for the partial derivatives of the fundamental solutions of (4.1).…”

“…This formula also explains how we understand the convolution of a matrix-valued and a vector-valued distribution, in the sense of the definition given by Hörmander [132,Definition 4.2.2], which is valid if at least one of the distributions has compact support. The introduction of the above notations and definitions is motivated by three recent papers by Smyrlis [262][263][264]. In these papers, density results for elliptic equations on domains which have the segment property (see Definition 2.14) but may possess holes are proven.…”

“…The results in these three articles are based on the following lemma [262][263][264]. Based on this lemma, Smyrlis proved a theorem for the MFS for elliptic differential equations similar to the one by Browder [41] which we mentioned in the previous section [263]. Unfortunately, this result is not exactly what we are looking for, because in this theorem a set of source points which is dense in an arbitrary, but fixed, open set outside the closure of an open domain is needed.…”

“…Before doing this, we give a (necessarily incomplete) overview of some of the literature on the method of fundamental solutions, particularly with regard to density results and (rather scarce) convergence results. Based mainly on work by Smyrlis [262][263][264], we prove a density result for the method of fundamental solutions in poroelasticity developed from regularizing the corresponding boundary integral equations. To the best of our knowledge, density results for implicit evolution equations, systems of differential equations with a dependence on time, which are, thus, non-elliptic, or systems which cannot be easily decoupled into several separate equations have not been proven before.…”