On smoothness of solutions of some elliptic functional-differential equations with degeneration

We consider second-order differential-difference equations in bounded domains in the case where several degenerate difference operators enter the equation. The degeneration leads to the fact that the multiplicity of the zero eigenvalue for the corresponding differential-difference operator becomes infinite. Regularity of generalized solutions for such equations is known to fail in the interior of the domain. However, we prove that the projections of solutions onto the orthogonal complement to the kernel of the "leading" difference operator remain regular in certain subdomains which form a decomposition of the original domain. © 2013 Pleiades Publishing, Ltd.