On elliptic problems with indefinite superlinear boundary conditions

In this paper we prove the existence of positive solutions to some nondivergent elliptic equations with indefinite nonlinear boundary conditions. The proof is based on a new Liouville-type theorem about the nonnegative solutions to some canonical indefinite elliptic equations, which is also proved in this paper by the method of moving planes.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.     

On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions

General elliptic boundary value problems with the spectral parameter appearing linearly both in the elliptic equation and in boundary conditions are considered. It is proved that the corresponding matrix operator from the Boutet de Monvel algebra is similar to an almost diagonal operator. This result is applied to prove the completeness and the summability (in the sense of Abel) of the root vectors of this operator.The support of the Rashi Foundation is gratefully acknowledged.The support of the Israel Ministry of Science and Technology is gratefully acknowledged.     

On the solution of boundary value problems on an inhomogeneous plane with a crack and a barrier connected in series

We consider boundary value problems for an equation in divergence form on a plane divided into two inhomogeneous half-planes by a film inclusion in the form of a strongly permeable crack and a weakly permeable barrier connected in series; this models a contact of heterogeneous media under inhomogeneous external conditions. The desired potentials have prescribed singular points (sources, drains, etc.). The coefficients of the equation are nonconstant and may increase or decrease when moving away from the film inclusion along a family of parabolas. We obtain representations of solutions of the considered problems via harmonic functions with the corresponding singular points on the plane.     

On radiative-transfer problems with reflective boundary conditions and internal emission

An integral-transform technique and theF _N method are used to solve basic radiative-transfer problems. Boundary conditions that account for specular and diffuse reflection as well as external illumination are used, and internal emission is allowed. Emphasis is given to the azimuthally symmetric component of the complete radiation field.

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Zusammenfassung Eine Integral-Transformation und dieF _N -Methode werden zur Lösung von grundlegenden Problemen der Strahlungsübertragung herangezogen. Es werden Randbedingungen mit spekulärer und diffuser Reflexion wie auch mit äußerer Bestrahlung benützt, und innere Emission ist zugelassen. Besonders betrachtet werden die Komponenten des gesamten Strahlungsfeldes, welches azimutale Symmetrie besitzt.

     

On free boundary problems with arbitrary initial and flux conditions

The paper is concerned with the free boundary problem of a semi-infinite body with an arbitrarily prescribed initial condition and an arbitrarily prescribed flux at its face. An analytically exact solution to the problem is found and established, which is expressed in terms of some functions and polynomials of the error integral family and timet. Convergence of the series solution is considered and proved. For illustration, the solution is applied to a special problem, which is discussed in detail. Some remarks about the solution are also included.

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Résumé Cet article traite du problème à limites variables pour un corps semi-infini, les conditions initiales et un flux arbitraire sur la face étant donnés. On obtient une solution analytique exacte de ce problème. Elle est formulée à l'aide de fonctions et de polynomes de la famille des intégrales d'erreur et en fonction du tempst. La convergence des séries apparaissant dans la solution est étudiée et vérifiée. Comme illustration, les résultats sont appliqués à un problème particulier qui est étudié en détail. Quelques remarques complètent le travail.

     

On averaging of nonlinear elliptic problems with inhomogeneous boundary conditions

A sequence of solutions of nonlinear elliptic problems is considered in the case where the Dirichlet conditions are given on the one part of the boundary and the Neumann conditions are given on the other part. The boundary-value problem is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 269–276, February, 1995.     

On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem on ; on , where is a bounded region in , is an indefinite weight function and may be positive, negative or zero.

     

On the existence for viscoelastodynamic problems with unilateral boundary conditions

This note deals with a damped wave equation and the evolution of a Kelvin–Voigt viscoelastic material, both problems being subject to unilateral boundary conditions. The functional properties of all the traces are precisely identified through Fourier analysis, which implies the existence of a solution satisfying almost everywhere the unilateral boundary conditions. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed boundary value problems of diffraction by a half‐plane with an obstacle perpendicular to the boundary

The paper is devoted to the analysis of wave diffraction problems modeled by classes of mixed boundary conditions and the Helmholtz equation, within a half‐plane with a crack. Potential theory together with Fredholm theory, and explicit operator relations, are conveniently implemented to perform the analysis of the problems. In particular, an interplay between Wiener–Hopf plus/minus Hankel operators and Wiener–Hopf operators assumes a relevant preponderance in the final results. As main conclusions, this study reveals conditions for the well‐posedness of the corresponding boundary value problems in certain Sobolev spaces and equivalent reduction to systems of Wiener–Hopf equations. Copyright © 2013 John Wiley & Sons, Ltd.     

On an evolution equation with acoustic boundary conditions

In this paper, we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with weak internal damping and quadratic term, coupled with mixed boundary conditions of Dirichlet type and acoustic type. Our goal is to extend some of the results of Frota‐Goldstein work in the sense of considering a weaker internal damping and one more quadratic nonlinearity in the elastic string equation. Copyright © 2011 John Wiley & Sons, Ltd.     

On a boundary value problem with integral boundary conditions

We study the existence of positive solutions of second-order ordinary differential equations with integral boundary conditions. The result generalizes the conditions obtained in [1] for the existence of positive solutions.     

Plastic yielding of notched strips under conditions of plane stress

Zusammenfassung Das Problem dünner gekerbter Flachstäbe, die unter Zug stehen, ist bereits vonHill [2] für den Fall, wo die Kerben an jeder Seite des Flachstabes symmetrisch angeordnet sind, untersucht worden. Die Verformung kommt dem ebenen Spannungszustand gleich, undHill vermag mit Hilfe der Charakteristikentheorie eine obere Grenzlösung für die zwei Fälle von V-förmig und kreisförmig gekerbten Flachstäben zu geben. Später konnteBischop [1] zeigen, dassHills obere Grenze die wahre Lösung für den V-förmig gekerbten Fall darstellt, bleibt aber für den kreisförmig gekerbten Fall den Beweis dafür schuldig.In der vorliegenden Arbeit wird für beide Fälle ein statisch zulässiges mögliches Spannungsfeld (untere Grenze) gefunden; dennoch liegt die Hauptaufgabe in der Behandlung einfach gekerbter Flachstäbe bei Biegung in der Flachebene. Obere Grenzen können bei Beigung sowohl für einfach V-förmig wie für kreisförmig gekerbte Flachstäbe angegeben werden, doch ist es möglich, auch eine allgemeine untere Grenzlösung für beide Fälle zur Darstellung zu bringen.     

On eigenvalue problems with Robin type boundary conditions having indefinite coefficients

A Robin type boundary condition with a sign-changing coefficient is treated. First, the associated linear elliptic eigenvalue problem is studied, where the existence of a principal eigenvalue is discussed by the use of a variational approach. Second, the associated semilinear elliptic boundary value problem of logistic type is studied and the one parameter-dependent structure of positive solutions is investigated, where results obtained are due to the construction of suitable super- and subsolutions by using the principal positive eigenfunctions of the linear eigenvalue problem.     

On self-adjoint operators generated by nonhomogeneous elliptic problems with discontinuous boundary conditions and conjugation conditions

We construct and study self-adjoint operators corresponding to problems mentioned in the title. We describe a correlation between domains of definition of fractional powers of these operators and Sobolev spaces. We state new results on the solvability of several problems for nonlinear parabolic equations which have not yet been studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 374–381, March, 1991.