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

On the existence of periodic solutions of some nonlinear problems of thermal control

From the text (translated from the Russian): "We consider the heat equation with a boundary condition containing a control function. The control function is a solution of an ordinary differential equation whose right-hand side contains a nonlinear functional that models the hysteresis effect. The dependence of the functional on the mean temperature over the domain causes nonlocal effects. Such problems arise in the modeling of thermal control processes in chemical reactors and climate control systems. We study the solvability and periodicity of the solutions of the problem.

On the unique solvability of the first mixed problem for the Vlasov-Poisson system of equations in an infinite cylinder

The authors establish new sufficient conditions for the unique solvability of the first initial-boundary value problem for the Vlasov-Poisson system in an infinite cylinder with an external magnetic field

On the Fredholm and unique solvability of nonlocal elliptic problems in multidimensional domains

The authors study nonlocal elliptic boundary value problems of the form aligned Au&=f_0quadtext{for }xin Q, B_{ij}u&=f_{ij}quadtext{for }xinGamma_i, i=1,dots,N, j=1,dots,m. endaligned Here Au=sum_{|alpha|leq 2m}a_alpha(x)D^alpha u, B_{ij}u=sum_{s=0}^{S_i}(sum_{|alpha|leq m_j}b_{ijalpha s}(x)D^alpha u)(omega_{is}(x))|_{Gamma_i} are boundary operators, Qsubset {bf R}^n is a bounded domain with boundary partial Q=bigcup_{i=1}^NGamma_i, omega_{is} is a diffeomorphism from a neighborhood of Gamma_i such that omega_{is}(Gamma_i) subset Q. The boundary conditions are nonlocal due to the mappings omega_{is}. The operator A is properly elliptic and the Lopatinskiĭ condition is imposed. The authors study solvability of such boundary value problems. A priori estimates of solutions are obtained and the Fredholm property is proved. Unique solvability of the problem with a parameter is shown