Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
29.11.1977
E-mail: , , , , ,
Keywords: Boundary Value Problems for Functional Differential Equations; Nonlocal Elliptic Problems.
Subject:
1) Boundary Value Problems for Differential-Difference Equations: a) Fredholm solvability of a boundary value problem for one class of differential-difference equations is proved in the one-dimensional case; b) Smoothness of generalized solutions (which may violate inside the interval) is investigated. It is proved that smoothness of a generalized solution preserves if one impacts a finite number of orthogonality conditions on the right-hand side. 2) Elliptic problems with nonlocal conditions near a boundary of domain: a) For model problems in plane and dihedral angles (that arise when studying nonlocal problems in bounded domains), the Green formula and adjoint problems are obtained. Necessary and sufficient conditions for one-valued and Fredholm solvability of model problems in the Kondrat'ev weighted spaces are proved (earlier only sufficient conditions were obtained by A. L. Skubachevskii). b) Coefficients in asymptotic (near some special set) formulas of solutions to nonlocal problems are calculated. These coefficients depend on eigenvectors and associate vectors of adjoint problems. c) Fredholm solvability of nonlocal elliptic problems in bounded domains is proved for the case of nonlinear (near some special set) argument transformations. It is shown that index of a problem with nonlinear argument transformations is equal to index of the corresponding problem with linear argument transformation. d) Fredholm solvability of elliptic equations with nonlocal conditions near a boundary is proved in the Sobolev spaces (with no weight). Asymptotics of solutions to nonlocal problem is considered in the Sobolev spaces. e) Smoothness of solutions to 2nd order elliptic equations with nonlocal conditions is studied in the Sobolev spaces.
Main publications:
Gurevich P. L. Solvability of the boundary value problem for some differential–difference equations // Functional Differential Equations. 1998, v. 5, no. 1–2, p. 139–157.
Gurevich P. L. On fredholm solvability of boundary value problem for differential–difference equations // Abstracts of international conference on differential and functional-differential equations. Moscow, 1999, p. 42–43.
Gurevich P. L. Nonlocal problems for elliptic equations in dihedral angles and the Green formula // Mitteilungen aus dem Mathem. Seminar Giessen, Math. Inst. Univ. Giessen, Germany, Heft 247, 2001, p. 1–74.
Gurevich P. L. On the Green formula for nonlocal elliptic problems // Abstracts of International Conf. "Differential Equations and Related Topics" dedicated to the Centenary Anniversary of I. G. Petrovskii, Moscow, MSU, 2001. P. 159–160.
