The first boundary value problem for strongly elliptic differential- difference equations (Q1079750)

The paper deals with the first boundary value problem for strongly elliptic differential-difference equation of order 2m in a bounded domain \(D\subset {\mathbb{R}}^ n\) with either smooth or rectangular boundary. Linear difference operators with smooth coefficients are considered. The corresponding differential-difference operator L is called strongly elliptic, if \(Re(Lu,u)_ 0\geq c_ 1 \| u\|^ 2_ m-c_ 2 \| u\|^ 2_ 0\) holds for all \(u\in \overset\circ C^{\infty}(D)\) with \(c_ 1,c_ 2>0\). Necessary and sufficient conditions of strong ellipticity are obtained in terms of algebraic properties of the coefficients of the difference operators. Further it is shown that strongly elliptic operators are Fredholm operators in \(L_ 2\). The smoothness of generalized solutions is preserved in special subdomains; these subdomains are defined by the differences used; conditions for smoothness on subdomain boundaries are also given. A set K of singularities is investigated.