Nonlocal Neumann Boundary Conditions Research Articles

Non-local diffusion problems with Neumann type boundary values

In this paper, we study the non-local diffusion problemJ⁎u−u−ut=f(u) with non-local Neumann boundary condition. We first establish some integration by parts formulas similar to the Laplacian. Then the existence and uniqueness of the solution and the maximum principle are proved. We also investigate the properties of the Neumann condition for one equation and for a family of equations.

The fractional p(.,.)-Neumann boundary conditions for the nonlocal p(.,.)-Laplacian operator

In this paper, we deal with the fractional -Laplacian problem with nonlocal Neumann boundary conditions here are parameters and are two nonnegative weighted functions. The domain is smooth and bounded, are continuous bounded functions. Applying the Ekeland principle and the variational method, under appropriate assumptions, we show that the above problem has nontrivial weak solutions.

Numerical Methods for a Diffusive Class of Nonlocal Operators

In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, \(\nu \), of diffusive type. In particular, we assume \(\nu \) is symmetric and exponentially decaying at infinity. We consider problems posed in bounded domains and in \({\mathbb {R}}\). In the case of bounded domains with nonlocal Dirichlet boundary conditions, we show the convergence of the scheme for kernels that have positive tails, but that can take on negative values. When the equations are posed on all of \({\mathbb {R}}\), we show that our scheme converges for nonnegative kernels. Since nonlocal Neumann boundary conditions lead to an equivalent formulation as in the unbounded case, we show that these last results also apply to the Neumann problem.

A priori estimates for the Fractional <i>p</i>-Laplacian with nonlocal Neumann boundary conditions and applications

<p style='text-indent:20px;'>We first prove that solutions of fractional <i>p</i>-Laplacian problems with nonlocal Neumann boundary conditions are bounded and then we apply such a result to study some resonant problems by means of variational tools and Morse theory.</p>

Linking over cones for the Neumann fractional p-Laplacian

We consider nonlinear problems governed by the fractional p-Laplacian in presence of nonlocal Neumann boundary conditions and we show three different existence results: the first two theorems deal with a p-superlinear term, the last one with a source having p-linear growth. For the p-superlinear case we face two main difficulties. First: the p-superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.

Fractional Diffusion Limit of a Kinetic Equation with Diffusive Boundary Conditions in the Upper-Half Space

We investigate the fractional diffusion approximation of a kinetic equation in the upper-half plane with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we derive a fractional diffusion equation with a nonlocal Neumann boundary condition for the density of particles. Interestingly, this asymptotic equation is different from the one derived by L. Cesbron in [7] in the case of specular reflection conditions at the boundary and does not seem to have receive a lot of attention previously.

A topological degree approach to a nonlocal Neumann problem for a system at resonance

In this paper, we investigate the existence of solutions for a system of second-order differential equations with two nonlocal Neumann boundary conditions at resonance. The existence results are obtained by applying the Mawhin continuation theorem, where the required a priori estimates are derived using some conditions of the Nirenberg type.

Neumann conditions for the higher order [formula omitted]-fractional Laplacian [formula omitted] with [formula omitted

In this paper we study a variational Neumann problem for the higher order fractional Laplacian (−Δ)s, s>1. In the process we introduce some nonlocal Neumann boundary conditions that appear in a natural way from a Gauss-like integration formula.

Properties of solutions for a reaction\u2013diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition

In this paper, the authors consider the reaction–diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition. They prove that the solution either exists globally or blows up in finite time depending on the initial data, the weighting function on the border, and nonlinear indexes in the equation by using the comparison principle.

On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions

We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.

Initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal boundary condition*

We consider an initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal Neumann boundary condition. We prove a comparison principle and the existence of a local solution and study the problem of uniqueness and nonuniqueness.

Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition

In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence result. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.

On an inverse problem of pressure recovery arising from soil venting facilities

The technique of soil venting is commonly used for remediation of the unsaturated zone of the soil, which is contaminated by gaseous organic pollutants in the pores. We study a steady-state model with a finite number of extraction wells, where the air flow field in the subsurface is described by a linear elliptic boundary value problem, with a nonlocal Neumann boundary condition, accompanied of a Dirichlet side condition, containing unknown parameters. We develop a constructive method for the parameter identification, i.e., for the determination of the unknown pressure values at the active wells from their measured total discharges.

On a numerical method for 2D magnetic field computations in a lamination with enforced total flux

The paper deals with a numerical method for the evaluation of the electromagnetic loss in a lamination of an electric machine, based upon the Maxwell equations. The underlying problem consists in the computation of the unidirectional magnetic field in a cross-section of the laminate orthogonal to the enforced flux. The method starts from a suitable variational formulation of the governing parabolic problem with a nonlocal Neumann boundary condition accompanied by a Dirichlet side condition, involving nonlinear and hysteresis effects through the differential permeability coefficient. The variational problem is solved numerically by a finite element method, combined with a finite difference technique, which is deviced so as to take into account the nonlinear and hysteresis behaviour of the material. The numerical method is found to be effective and reliable.

Limiting behaviour of a spectral problem in fluid-solid structures

Conca, c., J. Planchard and M. Vanninathan, Limiting behaviour of a spectral problem in fluid-solid structures, Asymptotic Analysis 6 (1993) 365-389 The aim of this paper is the asymptotic analysis of a spectral problem which involves Helmholtz' equation coupled with a nonlocal Neumann boundary condition on the boundary of a periodic perforated domain of [J;l2. This eigenvalue problem represents the vibrations Ceigenfrequencies and eigenmotions) of a tube-bundle immersed in a perfect compressible fluid. Our analysis of convergence shows that the vibrations of this fluid-solid structure with a large number of tubes are close to the spectrum of an unbounded operator in a Hilbert space. Using the method of Bloch and exploiting the periodic structure of the problem, we derive the spectral family of this limit operator and we prove that its spectrum can be completely determined by only computing local eigenvalue problems in the basic cell representing the periodic structure in the problem.