On a nonlinear eigenmode problem in semiconductor theory

A model problem in semiconductor physics concerning the natural electrical oscillations in a semiconductor with allowance for strong dissipation and negative differential conductivity is considered. This is a nonlinear boundary value problem, and its solvability is proved by applying S.I. Pohozaev’s method of spherical fibering.     

Solving a free boundary problem with nonconstant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with nonconstant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function, which is driven by the objective's shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.     

On a problem in the theory of nonlinear Fredholm operators

We prove that for any Fredholm operator A(x) of class C¹ and zero index in a Hilbert space, in a neighborhood of any star compactum T lying in the domain of A we can define a completely continuous and continuously differentiable operator C, so that the linear operator A(x)+ C(x) has a bounded inverse for all xT.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 541–548, November, 1971.The author wishes to thank M. A. Krasnosel'skii for posing the problem, and P. P. Zabreiko and S. G. Krein for their interest in this work.     

On the dispersion function in particle transport theory

Elementary considerations are used to show in regard to particle transport theory that the dispersion function relevant to each of the Fourier-component problems resulting from a finite Legendre expansion of the scattering law cannot have a zero forv (–1, 1), and a condition for the endpoints ±1 to be zeros is reported. It is also shown that the dispersion function and the characteristic function cannot, for a given Fourier-component problem, vanish simultaneously for any value ofz [–1, 1].

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Zusammenfassung Mit elementaren Betrachtungen der Teilchentransport-Theorie wird gezeigt, daß die Dispersionsfunktion für jedes Fourier-Komponentenproblem, welches durch eine endliche Legendre-Entwicklung des Streuungsproblems entsteht, keine Nullstellen fürv (–1, 1) hat; eine Bedingung für Nullstellen an den Endpunkten ±1 wird angegeben. Es wird gezeigt, daß die Dispersionsfunktion und die charakteristische Funktion für ein gegebenes Fourier-Komponenten-Problem nicht gleichzeitig verschwinden können für Wertez [–1, 1].

     

Remarks on a nonlinear transport problem

A nonlinear transport problem of hyperbolic–elliptic type is studied. Estimates of potentials over varying domains and the method of characteristics enable one to treat the initial value problem for Hölder continuous data as an abstract evolution equation via Picard–Lindelöf theorem. In addition, existence for all times is proved. Similar techniques yield the existence of shock front solutions with smooth interfaces at least for a small time interval. By a priori estimates of approximating solutions, the results extend to the case of only bounded initial values. A modification of the system applies to the construction of a diffeomorphism with prescribed Jacobian determinant.     

On the invariance of a boundary-value problem for a nonlinear evolution equation

We obtain an invariance group for one boundary-value problem in the physics of the sea. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1281–1283, September, 1998.     

On a nonlinear eigenvalue problem

In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R ¹. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.     

On one mathematical problem in the theory of nonlinear oscillations

We consider one mathematical problem that was discussed by the author and A. M. Samoilenko at the Third International Conference on the Theory of Nonlinear Oscillations (Transcarpathia, 1967). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 56–62, January, 2008.     

On a general nonlinear problem with distributed delays

The paper considers a general system of ordinary differential equations appearing in the neural network theory. The activation functions are assumed to be continuous and bounded by power type functions of the states and distributed delay terms. These activation functions are not necessarily Lipschitz continuous as it is commonly assumed in the literature. We obtain sufficient conditions for exponential decay of solutions.     

A particle transport problem with non-homogeneous reflection boundary conditions

We consider particle transport in a three-dimensional convex region V, bounded by the regular surface ∂V. We assume that particles are specularly reflected by ∂V and that a source q is assigned on ∂V; more general non-homogeneous boundary conditions are also discussed. The problem is non-linear because the boundary condition is not homogeneous. We prove existence of a unique strict solution and by using the theory of semigroups we derive the explicit expression of such a solution in terms of the boundary source q. In the appendix, we indicate how some properties of affine operators can be used to derive the solution. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load

The paper is concerned with the existence of periodic solutions for the Lazer-McKenna suspension bridge equation with damping and nonconstant load. By using the Lyapunov-Schmidt reduction methods, the author discuss the relationship between the sign-changing solutions and the source terms. The result answers partly the open problem in Lazer and McKenna (SIAM Rev. 32 (1990) 537-578).