Asymptotics of the eigenvalues of the Sturm-Liouville problem with discrete self-similar weight

We study the asymptotics of the spectrum of the boundary-value problem $$ - y'' - \lambda \rho y = 0,y(0) = y(1) = 0 $$ for the case in which the weight ρ ∈ W? ₂ ^?1 [0, 1] is the generalized (in the sense of distributions) derivative of a self-similar function P ∈ L ₂[0, 1] of zero spectral order.     

On the generalized Sturm-Liouville problem for an equation with discontinuous solutions

Discontinuous solutions of the Sturm-Liouville problem whose differential operator is a generalized differentiation in a certain sense are studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 29, Voronezh Conference-1, 2005.     

Indefinite Sturm-Liouville problem for some classes of self-similar singular weights

We continue studying the asymptotics of the spectrum for the boundary value problem ?y″-λρy = 0, y(0) = y(1) = 0, where ρ is a function in the space with a self-similar primitive. The cases of nonarithmetic and degenerate arithmetic self-similarity of such a primitive are considered.     

On a generalization of the Neumann problem for the Laplace equation

We investigate solvability of a fractional analogue of the Neumann problem for the Laplace equation. As a boundary operator we consider operators of fractional differentiation in the Hadamard sense. The problem is solved by reduction to an integral Fredholm equation. A theorem on existence and uniqueness of the problem solution is proved.     

Neumann problem for the Korteweg-de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg-de Vries equation on a half-line

(0.1)     

Inverse nodal problem for the Sturm-Liouville operator with a weight

In this work,we consider the inverse nodal problem for the Sturm-Liouville problem with a weight and the jump condition at the middle point.It is shown that the dense nodes of the eigenfunctions can uniquely determine the potential on the whole interval and some parameters.     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

The geometric Neumann problem for the Liouville equation

Let Ω denote the upper half-plane ${\mathbb{R}_+^2}$ or the upper half-disk ${D_{\varepsilon}^+\subset \mathbb{R}_+^2}$ of center 0 and radius ${\varepsilon}$ . In this paper we classify the solutions ${v\in\;C^2(\overline{\Omega}\setminus\{0\})}$ to the Neumann problem $$\left\{\begin{array}{lll}{\Delta v+2 Ke^v=0\quad {\rm in}\,\Omega\subseteq \mathbb{R}^2_+=\{(s, t)\in \mathbb{R}^2: t >0 \},}\\ {\frac{\partial v}{\partial t}=c_1e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s >0 \},}\\ {\frac{\partial v}{\partial t}=c_2e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s <0 \},}\end{array}\right.$$ where ${K, c_1, c_2 \in \mathbb{R}}$ , with the finite energy condition ${\int_{\Omega} e^v < \infty}$ As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.     

Solution of the Neumann problem for the Laplace equation

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.     

Inverse spectral problem for a generalized Sturm-Liouville equation with complex-valued coefficients

The present paper is the first to prove that one of the columns of the monodromy matrix and two of the three coefficients (piecewise analytic on the interval [0, 1]) of the equation (f(x)y′)′+(r(x)−λ ² q(x))y = 0 uniquely determine the third coefficient on this interval provided that the values of the functions f(x) and q(x) lie in the lower (or upper) open complex halfplane and on the positive part of the real axis. This unknown coefficient can be reconstructed by finding the unique zero minimum of a specially constructed functional depending on the solutions of the corresponding Cauchy problem and the given elements of the monodromy matrix.     

Neumann problem for generalized n-Poisson equation

Using a hierarchy of integral operators having higher-order Neumann functions and their derivatives as kernels, the Neumann problem for a 2nth order linear partial complex differential equation is discussed. The solvability of the problem is obtained.     

Nonlocal Neumann problem for a degenerate parabolic equation

In the spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of the nonlocal Neumann problem for nonuniformly parabolic equations without restrictions on the power order of coefficient degeneration. We find an estimate of the solution of this problem in the spaces considered. Chernovtsy State University, Chernovtsy. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1232–1243, September, 1999.     

On solvability conditions for the Neumann problem for a polyharmonic equation in the unit ball

Some necessary and sufficient solvability conditions are obtained for the nonhomogeneous Neumann problem for a polyharmonic equation in the unit ball.     

On a Sturm-Liouville problem

In this paper we give a lower bound for the first eigenvalue of an ordinary differential operator which represents the radial part of the Laplace operator restricted to a spherical cap of a sphere, possibly of fractional dimension. The results are obtained by purely analytical methods.Research supported by CONICIT.     

Inverse scattering problem for the Sturm-Liouville equation with spectral parameter in the discontinuity condition

We give a complete solution of the inverse scattering problem for the Sturm-Liouville equation with spectral parameter in the discontinuity condition in the absence of discrete spectrum.