On Wiener's type regularity of a boundary point for higher order elliptic equations
Boundary integral equations of elasticity in domains with piecewise smooth boundaries
Estimates for differential operators of vector analysis involving -norm
Приложения некоторых интегральных неравенств к теории квазилинейных эллиптических уравнений
Об одном операторе вложения и функциях множества типа -ёмкости
Solvability and asymptotic behavior of solutions of ordinary differential equations with variable operator coefficients
Boundary singularities of solutions to quasilinear elliptic equations
Asymptotic formulae for solutions to boundary value problems for linear and quasilinear elliptic equations and systems near a boundary point are discussed. The boundary is not necessarily smooth. The main ingredient of the proof is a spectral splitting and reduction of the original problem to a finite-dimensional dynamical system. The linear version of the corresponding abstract asymptotic theory is presented in our new book “Differential equations with operator coefficients”, Springer, 1999.
Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary
We derive an asymptotic formula of a new type for variational solutions of the Dirichlet problem for elliptic equations of arbitrary order. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.
The form boundedness criterion for the relativistic Schrödinger operator
We establish necessary and sufficient conditions on the real- or complex-valued potential defined on for the relativistic Schrödinger operator to be bounded as an operator from the Sobolev space to its dual .
Asymptotic analysis of surface waves due to high-frequency disturbances
The present paper is devoted to the asymptotic analysis of the linear unsteady surface waves. We study two problems concerned with high-frequency surface and submerged disturbances. The two-scale asymptotic series are obtained for the velocity potential. The principal terms in the asymptotics of some hydrodynamical characteristics of the wave motion (the free surface elevation, the energy, etc.) are described.
On «power-logarithmic» solutions of the Dirichlet problem for elliptic systems in , where is a d-dimensional cone
A description of all «power-logarithmic» solutions to the homogeneous Dirichlet problem for strongly elliptic systems in a -dimensional cone is given, where is an arbitrary open cone in and .
Gradient regularity via rearrangements for -Laplacian type elliptic boundary value problems
A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.
Продолжение функций из классов С. Л. Соболева во внешность области с вершиной пика на границе II
Продолжение функций из классов С. Л. Соболева во внешность области с вершиной пика на границе, I.
On pointwise interpolation inequalities for derivatives
Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where is the gradient of order , is the Hardy-Littlewood maximal operator, and is the Riesz potential of order , are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space is described.
Water-wave problem for a vertical shell
The uniqueness theorem is proved for the linearized problem describing radiation and scattering of time-harmonic water waves by a vertical shell having an arbitrary horizontal cross-section. The uniqueness holds for all frequencies, and various locations of the shell are possible: surface-piercing, totally immersed and bottom-standing. A version of integral equation technique is outlined for finding a solution.
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