Boundary value problems for elliptic functional-differential equations and their applications

The paper under review discusses the state of the art in boundary value problems for strongly elliptic functional-differential equations in bounded domains, extending the author's earlier work [{it Elliptic functional-differential equations and applications}, Oper. Theory Adv. Appl., 91, Birkhäuser, Basel, 1997; [msn] MR1437607 [/msn]]. The paper is divided into two parts, called chapters. Theoretical aspects are discussed in Chapter I. The first two sections are devoted to preliminary results. Sections 3--5 concern differential-difference equations. In Section 3, necessary and sufficient conditions for strong ellipticity are obtained in algebraic form. The spectrum is discussed in Section 4, and the smoothness of generalized solutions is established in Section 5. Section 6 concerns a special kind of strongly elliptic functional-differential equations. par Chapter II is devoted to applications. Non-local elliptic differential-difference equations are discussed in Section 7. In Section 8, it is shown that strongly elliptic differential-difference operators with Dirichlet conditions satisfy the Kato square root conjecture. Applications to a special elasticity problem are considered is Section 9. Section 10 concerns nonlinear laser systems.