Vol 63, No 2 (2017): Proceedings of the Crimean Autumn Mathematical School-Symposium

The diffusion equation is considered in an infinite 1-periodic medium. For its fundamental solution we find approximations at large values of time t. Precision of approximations has pointwise and integral estimates of orders O(t(-d+j+1)/2) and O(t(-j+1)/2), j=0,1,…, respectively. Approximations are constructed based on the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used for construction of approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in Lp-spaces, 1≤p≤∞. For the analogous equation in an ε-periodic medium (here ε is a small parameter) we obtain approximations of the operator exponential in Lp-operator norms for a fixed time with precision of order O(εn), n=1,2,….

For mappings with unbounded characteristics we prove theorems on removal of isolated singularities on Riemannian manifolds. We prove that if a mapping satisfies certain inequality of absolute values and its quasiconformity characteristic has a majorant of finite average oscillation at a fixed singular point, then it has a limit at this point.

Based on the approach to abstract matching boundary-value problems introduced in [18], we consider matching spectral problems for one and two domains. We study in detail the arising operator pencil with self-adjoint operator coefficients. This pencil acts in a Hilbert space and depends on two parameters. Both possible cases are considered, where one parameter is spectral and the other is fixed, and properties of solutions are obtained depending on this. Also we study initial-boundary value problems of mathematical physics generating matching problems. We prove theorems on unique solvability of a strong solution ranging in the corresponding Hilbert space.

Higher-order differential operators on a finite interval with jump conditions inside the interval are studied. Properties of spectral characteristics are obtained, and completeness and expansion theorems are proved for this class of operators.