On boundary value problem solvability theory for a class of high-order operator-differential equations

In this note, we establish sufficient conditions for the correct and unique solvability of various boundary value problems for a class of even-order operator-differential equations on the half-axis. These conditions are unimprovable in terms of operator coefficients of the equation. We note that the principal part of the equation under study suffers a discontinuity.     

On integral equations of axisymmetric potential theory

It is shown that two integral equations of the first kind, muchused in, respectively, axisymmetric electrostatics and hydrodynamics,are wrong in the sense that they do not in general possess solutions.A theorem is established giving the precise conditions necessaryfor solutions to exist, but perhaps more important practicallyis the fact, brought out by examples, that the necessary conditionsare far from sufficient. An alternative integral equation inthe electrostatic case is proposed and justified, one havinga similar form and the same computational advantages, but freefrom existence difficulties. The apparent paradox that ‘solutions’are found to problems when the governing equations may not possesssolutions is explained by the fact that these purported solutionsare obtained by numerical or asymptotic analysis, when an approximatingequation possesses a solution, but one which cannot be saidto approximate to the solution of the problem if the equationto which, formally, it approximates cannot, through being meaningless,represent the problem. The arguments are given mainly in theelectrostatic context, but it is shown how they are modifiedto carry over to the hydrodynamical one.     

Criteria of solvability for multidimensional Riccati equations

We study the solvability problem for the multidimensional Riccati equation ??u=|?u|^q+ω, whereq>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation ?Δu?ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions onR ⁿ in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type?Lu=f(x, u, ?u)+ω where , andL is a uniformly elliptic operator.     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

On solvability of boundary value problems for systems of differential equations

We consider systems of differential equations (1)x=g(t, x) together with boundary conditions of the form (2)x(0)=x ₀,x(T)=x ₁; (3)x(0)=Qx(T), x(0)=Qx(T); and (4)B ₁ x(0)–B ₂ x(0)=0=C ₁ x(T)+C ₂ x(T). Herex=(x ₁,...,x _n )^T andg=g(t, x) are realn-vectors andQ, B _i ,C _i ,i=1, 2, denoten×n matrices withQ nonsingular andB ₂,C ₂ positive definite. We examine the existence of solutions of (1) satisfying (3) or (4) and which also stay in a certain regionin(t, x) space. Conditions in terms of the Jacobian matrixG (t, x)=g _x (t, x) and an auxiliary positive definite symmetric matrixP=P(t) C ² [0,T] are given which yield the existence of the desired solution of (1), (3) or (1), (4).Dedicated to Professor Hans W. Knobloch on the occassion of his sixtieth birthdayResearch supported by NSERC Canada Grant A7673 and NSF Grant DMS-8501311.     

The solvability of integral lightguide equations

Integral equations of a three-dimensional periodic lightguide are studied for arbitrary values of the frequency parameter. The solution to the homogeneous integral equation vanishes for those values of the spectral parameter that are different from some isolated points. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 89–100.     

On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation

This work is devoted to the investigation of the nonclassical problem for a multidimensional elliptic equation with two integral boundary conditions. By introducing special multipliers we prove the uniqueness of the solution and obtain new a priori estimates, which permit one to establish the existence of a solution in the corresponding Sobolev spaces.     

Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations

In this paper we present a battery of results related to how Galerkin semidiscretization in space affects some formulations of wave scattering and propagation problems when retarded boundary integral equations are used.     

On boundary integral equations in electroelasticity

A class of plane electroelasticity problems of the steady vibrations of bodies with a smooth boundary is studied. A system of boundary integral equations for the components of the displacement vector and the potential is formulated on the basis of the fundamental solution constructed and its analysis.     

Integral equations for the Dirichlet and Neumann boundary value problems in a plane domain with a cusp on the boundary

The boundary equations of the logarithmic potential theory corresponding to the interior Dirichlet problem and the exterior Neumann problem for a plane domain with a cusp on the boundary are studied. Solvability theorems are proved for these integral equations in the spacesL ^p. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 881–892, June, 1996.     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

Conditions for the solvability of a system of integral equations by quadratures

By reduction to Goursat problems, we obtain various versions of conditions ensuring that the solution of the system in question can be constructed in closed form.