The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H³, and solve both of them. We construct the unique mean curvature one surface in H³ that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.     

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)     

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

The Neumann boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution to a nonlocal Cahn-Hilliard equation on a bounded domain. The equation generates a gradient flow for a free energy functional with nonlocal interaction. Also we apply a nonlinear Poincaré inequality to show the existence of an absorbing set in each constant mass affine space.     

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.     

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.     

The Neumann problem for the Helmholtz equation in a starlike planar domain

The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson.     

The Neumann problem for a mixed-type equation in a rectangular domain

We establish a uniqueness criterion for the Neumann problem for a mixed-type equation with the Lavrent’ev-Bitsadze operator in a rectangular domain. We construct a solution as the sum of a series of eigenfunctions of the corresponding Sturm-Liouville problem.     

Solvability of the Neumann problem for a Sobolev pseudoparabolic equation

The dynamic potential constructed in this paper is used to analyze the existence of a classical solution to the Neumann problem for a Sobolev equation.     

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.     

Liouville type equation with exponential Neumann boundary condition and with singular data

In this paper we will analyze the blow-up behaviors of solutions to the singular Liouville type equation with exponential Neumann boundary condition. We generalize the Brezis–Merle type concentration-compactness theorem to this Neumann problem. Then along the line of the Li–Shafrir type quantization property we show that the blow-up value \(m(0) \in 2\pi \mathbb N\cup \{ 2\pi (1+\alpha )+2\pi (\mathbb N\cup \{0\})\}\) if the singular point 0 is a blow-up point. In the end, when the boundary value of solutions has an additional condition, we can obtain the precise blow-up value \(m(0)=2\pi (1+\alpha )\).     

The Neumann problem of an elliptic equation with strong resonance

In the present paper, we consider a class of strong resonance problems for nonlinear elliptic Neumann BVPs, and establish existence and multiplicity results for positive, negative and sign-changing solutions by using a Morse theoretical approach.     

On the Neumann problem for the Sturm-Liouville equation with Cantor-type self-similar weight

The second and third boundary value problems for the Sturm-Liouville equation in which the weight function is the generalized derivative of a Cantor-type self-similar function are considered. The oscillation properties of the eigenfunctions of these problems are studied, and on the basis of this study, known asymptotics of their spectra are substantially refined. Namely, it is proved that the function s in the well-known formula $$N(\lambda ) = \lambda ^D \cdot [s(\ln \lambda ) + o(1)]$$ decomposes into the product of a decreasing exponential and a nondecreasing purely singular function (and, thereby, is not constant).     

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.     

Iterative method for solving the Neumann boundary value problem for biharmonic type equation

The solution of boundary value problems (BVP) for fourth order differential equations by their reduction to BVP for second order equations, with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the Neumann BVP for biharmonic type equation. The convergence rate of the method is proved and some numerical experiments are performed for testing it in dependence on the choice of an iterative parameter.