Estimates of Solutions of Elliptic Differential-Difference Equations with Degeneration

Abstract

We consider a second-order differential-difference equation in a bounded domain Q ⊂ ℝn. We assume that the differential-difference operator contains difference operators with degenerations corresponding to the differential operators. Also, it is assumed that the considered differential-difference operator cannot be expressed by a composition of a difference operator and a strongly elliptic differential operator. The presence of degenerate difference operators does not allow us to obtain the Gårding inequality. We prove a priori estimates implying that the considered differential-difference operator is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space $$ {W}_2^1(Q) $$ . However, using the obtained estimates, we prove smoothness of solutions at least in subdomains Qr generated by translations of the boundary, where $$ \underset{r}{\cup}\overline{Q_r}=\overline{Q} $$ .