On some new boundary-value problems in nonsmooth domains

The wave factorization of an elliptic pseudodifferential operator and the solvability of the equation with such an operator are described.     

Determination of boundary-value conditions in some nonlinear problems

Problems of a certain type are reduced to classical form by finding conditions on discontinuities of initial and boundary values.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 62–68, 1990.     

Elliptic boundary-value problems in nonsmooth domains

In a bounded domain, we study elliptic boundary-value problems for equations and systems of the Douglis-Nirenberg structure in complete scales of Banach spaces. The boundary of the domain contains conic points, edges, etc. A theorem on local increase in the smoothness of generalized solutions and a theorem on complete collection of isomorphisms are proved. Applications are considered. It is shown that the results obtained are also valid for transmission problems, nonlocal elliptic problems, elliptic problems with a parameter, and parabolic problems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 701–709, May, 1995.This research was partially supported by a grant of the American Mathematical Society and by the Ukrainian State Committee on Science and Technology.     

Degenerate nonlinear boundary-value problems

We establish necessary and sufficient conditions for the existence of solutions of weakly nonlinear degenerate boundary-value problems for systems of ordinary differential equations with a Noetherian operator in the linear part. We propose a convergent iterative procedure for finding solutions and establish the relationship between necessary and sufficient conditions.     

On some boundary-value problems for functional-differential inclusions in Banach spaces

The general boundary-value problem for a semilinear functional-differential inclusion in a separable Banach space is considered. A many-valued integral operator whose fixed points are integral solutions to the problem is constructed. Conditions ensuring this many-valued operator to be condensing with respect to the vector measure of noncompactness are investigated. Application of topological degree theory allows one to establish some existence theorems for the boundary-value problem. The Cauchy problem and the periodic problem are considered as special cases. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 15, Differential and Functional Differential Equations. Part 1, 2006.     

On the method of complementary functions for nonlinear boundary-value problems

We show that, like the method of adjoints, the method of complementary functions can be effectively used to solve nonlinear boundary-value problems.This work was supported by the Alexander von Humboldt Foundation. The author is thankful to Prof. G. Hämmerlin for providing the facilities and to Miss J. Gumberger for performing numerical tests. The author is also indebted to Dr. S. M. Roberts for his suggestions on the first draft of this paper.     

Superconvergence phenomena in nonlinear two-point boundary-value problems

Superconvergence estimates are derived for a class of strongly nonlinear two-point boundary-value problems. The analysis considers solution and gradient superconvergence points as well as flux postprocessing formulas. Finally, the extension to parabolic problems is considered. © 1993 John Wiley & Sons, Inc.     

Identification of parameters in nonlinear boundary-value problems

This paper presents a method for identification of parameters in nonlinear boundary-value problems. The successive approximations technique proposed uses the theory of Lagrange multipliers and the Newton-Raphson method. This method does not require storage of functions and is quadratically convergent. Numerical results are presented.This research was sponsored by the National Institutes of Health, Grant No. GM-16197-01. Computing assistance was obtained from the Health Sciences Computing Facility, University of California at Los Angeles, NIH Grant No. FR-3.     

Elliptic boundary-value problems in domains of the exterior-of-a-cusp type

One investigates general elliptic boundary-value problems in a domain DRⁿ with isolated singularities of the boundary. A common property of the considered singular points is the fact that in their neighborhoods the complement to the domain is thin. For the singular pointO this means that the central projection of the set onto the sphere shrinks to a point as One proves theorems on the normal solvability in Sobolev spaces with weighted norms, one obtains coercive estimates of the solutions, one investigates the properties of the solutions in dependence on the properties of the right-hand sides.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 105–148, 1984.     

On two-point boundary-value problems

Summary Boundary-value problems are considered for systems of m ordinary linear differential equations of order n. For such a problem L, a canonically associated first-order problem M, with mn equations, is introduced in such a may that associates of adjoint problems are adjoint first-order problems. The solution of a boundary-value problem L is given in terms of a generalized Green's function. This work has been supported by Sandia Corporation, a prime contractor to the U. S. Atomic Energy Commission.     

Positive solutions of nonlinear three-point boundary-value problems

Let a∈C[0,1], b∈C([0,1],(−∞,0]). Let φ₁(t) be the unique solution of the linear boundary value problem

u″(t)+a(t)u′(t)+b(t)u(t)=0,t∈(0,1),u(0)=0,u(1)=1.     

Stationary boundary-value problems for Boltzmann's nonlinear equation

For Boltzmann's nonlinear equation one investigates the solvability of the stationary flow problem. One describes the asymptotic behavior of its solution for ¦x¦.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 100–104, 1981.     

Integrable boundary-value problems and nonlinear Fourier harmonics

For the nonlinear Schrödinger equation, the integrable boundary-value problem on a segment is considered. The concept of nonlinear ?-harmonics similar to the ordinary Fourier harmonics in the linear case is suggested. A solution of the initial boundary-value problem on the semiaxis is constructed by means of reduction to the Cauchy problem on the whole axis. Bibliography: 11 titles.     

On some conditions on mappings of spaces onto cylindrical domains

Criteria are given for a local homeomorphismf: Rⁿ Rⁿ to be a homeomorphism of Rⁿ onto a strip. In the planar case, this gives a condition forf to be a transformation of R² onto a band.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 459–462, April, 1972.     

On the solution of some boundary-value problems arising in elastico-viscous fluid mechanics

Résumé En mécanique des fluides visquex et élastiques, on rencontre souvent des équations différentielles du troisième ordre avec seulement deux conditions aux limites. On voit que la méthode ordinairement employée est approximative, n'est pas uniformément valable et donne des résultats faux. On donne également une autre méthode de résolution.