Radiation field of whispering gallery waves over a concave-convex boundary

In this work the wave field arising over a concave-convex reflecting boundary is studied in the Kirchhoff approximation. The field arises as a result of the incidence of whispering gallery waves on an inflection point of the boundary from the concave side. The shortwave asymptotics of the Kirchhoff integral are obtained which is expressed in terms of special functions in a neighborhood of the inflection point of the boundary and in a neighborhood of the tangent to the boundary at the inflection point. Diagrams are constructed that illustrate the behavior of the scattered field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 49–65, 1981.     

Asymptotic expansion of the solution of the first boundary value problem for the Schrödinger system near conical points of the boundary

We study the first initial-boundary value problem for the Schrödinger system in a cylindrical domain. It is assumed that the boundary contains a conical point. We obtain an asymptotic expansion of the solution in a neighborhood of such a point.     

Ray dynamics in a neighborhood of flat points of the boundary

The behavior of rays is studied in a neighborhood of boundary points where the curvature has a zero of multiplicity one (an inflection point) and multiplicity two (a flat point of a boundary which is concave from the side of the wave field). The rays considered away from the flat point of the boundary are connected with a whispering gallery wave incident on this point, and they are constructed on the basis of the known asymptotics of this wave. The results are represented in figures obtained with the help of the computer.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 146–155, 1981.     

On the Structure of asymptotics of the solution of a second-order elliptic equation in a neighborhood of an angular point

The behavior of solutions of elliptic equations in neighborhoods of angular and conical boundary points has been well studied; the asymptotics of these solutions has been constructed. In the present paper, we propose a new approach to constructing asymptotic decompositions in a neighborhood of an angular boundary point, which allows us to describe the structure of these asymptotics in a relatively simple and illustrative way.     

Boundary value problems of electroelasticity with concentrated singularities

We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighborhood of infinity or in the neighborhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.     

The method of <Emphasis Type="Italic">A</Emphasis>-harmonic approximation and boundary regularity for nonlinear elliptic systems under the natural growth condition

We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighborhood. This result is new for the situation under the natural growth conditions.     

Best possible estimates of solutions to the Robin boundary value problem for linear elliptic nondivergence second-order equations in a neighborhood of the conical point

We investigate the behavior of strong solutions to the Robin boundary value problem for linear elliptic nondivergence second-order equations in a neighborhood of the boundary conical point. We establish precise exponent of the solution decreasing rate.     

Boundary Properties of Solutions to Second-Order Parabolic Equations in Domains with Special Symmetry

We consider the first boundary-value problem for second-order nondivergent parabolic equations with, in general, discontinuous coefficients. We study the regularity of a boundary point assuming that in a neighborhood of this point the boundary of the domain is a surface of revolution. We prove a necessary and sufficient regularity condition in terms of parabolic capacities; for the heat equation this condition coincides with Wiener's criterion.     

Wave field in the caustic shadow in a neighborhood of an inflection point of the boundary

The problem of the propagation of whispering gallery waves in a neighborhood of an inflection point of the boundary is considered. It is shown that a caustic shadow zone occurs away from the boundary along the normal. The asymptotics of the wave field in the caustic shadow are obtained, and their geometric interpretation in terms of complex rays is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 246–260, 1979.     

Second order parabolic equations in nonvariational form: Boundary Harnack principle and comparison theorems for nonnegative solutions

Summary We study non-divergence parabolic equations with continuous coefficients in a cylinder with Lipschitzian cross-section. A Harnack type inequality at the boundary is established for nonnegative solutions vanishing in a surface neighborhood of a boundary point. As an application of this result we show that (in a C ² cylinder) all nonnegative solutions which vanish in a surface neighborhood of a point on the lateral side must go to zero at an equivalent rate. We present some consequences of these results which find application to the study of non-tangential behavior of nonnegative solutions.The author was partially supported by the Italian C.N.R., G.N.A.F.A.     

Smoothness of solutions of the Dirichlet problem for the biharmonic equation in nonsmooth 2D domains

We study the smoothness of a generalized solution of the Dirichlet problem for the biharmonic equation in a two-dimensional domain. We introduce a weighted test function and derive an estimate for the absolute value of the solution in a neighborhood of an irregular boundary point.     

Concentration of the solutions near a limit ray in the neighborhood of an inflection point of the boundary

One considers the question regarding the energy concentration of a wave field in the neighborhood of a limit ray in the problem of the behavior of whispering gallery waves near an inflection point of the boundary. An estimate related to this question is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 155–158, 1988.     

Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.     

On the regularity of minimal surfaces with free boundaries in Riemannian manifolds

We show c^1,-regularity of minimal surfaces in Riemannian manifolds with a free boundary on C²-hypersurfaces with bounded second fundamental form and a uniform neighborhood on which the nearest point projection is uniquely defined and differentiable. The decisive step is the proof of continuity at the free boundary.partially supported by SFB 72 (Deutsche Forschungsgemeinschaft)     

Computation of the Second Variation of the Energy Functional of a Two-Phase Medium

A formula for the second variation of the energy functional ofa two-phase elastic medium is derived at a critical point of the functional. An estimate for the remainder is given. The critical field of displacement and the critical boundary of the interface of the phases are assumed to be sufficiently smooth. Computations are made inside a domain occupied by the elastic medium as well as in a neighborhood of the intersection of the boundary of this domai and the boundary of the interface of the phases. Bibliography: 6 titles.     

A critical case for the solvability of stefan-like problems

We consider a one-phase one-dimensional Stefan problem with general data with the aim to investigate some open questions on existence of classical solutions. We show how existence and nonexistence are discriminated by the behavior of the initial datum in the neighborhood of the starting point of the free boundary.     

On completeness of root functions of elliptic boundary problems in a domain with conical points on the boundarySur la complétude des fonctions propres et associées d'un problème au bord elliptique dans un domaine avec points coniques sur le bord

We prove the completeness of the system of eigen and associated functions (i.e., root functions) of an elliptic boundary value problem in a domain, whose boundary is a smooth surface everywhere, except at a finite number of points, such that each point possesses a neighborhood, where the boundary is a conical surface. To cite this article: Y.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 649–654.     

Local boundary controllability for the semilinear plate equation

weijiu_liu@uow.edu.

The problem of local controllability for the semilinear plate equation with Dirichlet boundary conditions is studied. By making use of Schauder's fixed point theorem and the inverse function theorem, we prove that this system is locally controllable under a super-linear assumption on the nonlinearity, that is, the initial states in a small neighborhood of 0 in a certain function space can be driven to rest by Dirichlet boundary controls. Our super-linear assumption includes the critical exponent.     

Linear second-order differential equations with degenerating coefficient of the second derivative

For an ordinary linear second-order differential equation with degenerating coefficient of the higher derivative, we obtain exact asymptotic representations of solutions near the degeneration point, which are convergent uniformly and absolutely in some neighborhood of that point. The results permit one to obtain well-posed statements of initial and boundary value problems for degenerating equations and establish oscillation properties of solutions.