Nonhomogeneous Neumann Problem Research Articles

Infinitely Many Solutions For Neuman Problems Associated To Non-Homogeneous Differential Operator Through Orlicz-Sobolev Spaces

The goal of this paper is to establish the existence of an unbounded sequence of weak solutions for the following non-homogeneous Neumann problem

Existence of Solutions for a Non-homogeneous Neumann Problem

Existence of Solutions for a Non-homogeneous Neumann Problem

A high-order compact difference method on fitted meshes for Neumann problems of time-fractional reaction–diffusion equations with variable coefficients

This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L2-norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

Nonlinear nonhomogeneous Neumann problem on the Heisenberg group

Here, the existence of at least one positive radial solution of the Nonlinear nonhomogeneous Neumann problem is proved on the Heisenberg group , via variational principle, where Ω is the Korányi ball, a and b are nonnegative radial functions and holds in some suitable conditions.

Existence results for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces

Abstract Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed.

Infinitely many weak solutions for a non-homogeneous Neumann problem in Orlicz--Sobolev spaces

Under a suitable oscillatory behavior either at infinity or at zero of the nonlinear term, the existence of infinitely many weak solutions for a non-homogeneous Neumann problem, in an appropriate Orlicz--Sobolev setting, is proved. The technical approach is based on variational methods.

Existence and multiplicity results for non-homogeneous Neumann problems in Orlicz-Sobolev spaces

In this paper, existence results for positive solutions to an eigenvalue non-homogeneous Neumann problem are established. Multiplicity results are also pointed out. The proofs are based on variational methods and topological arguments in Orlicz-Sobolev spaces.

A boundary value problem associated to non-homogeneous operators in Orlicz-Sobolev spaces

ABSTRACTWe apply a three critical points theorem of B. Ricceri to establish the existence of at least three weak solutions for a class of non-homogeneous Neumann problems. Furthermore, by using another theorem of him, we prove that an appropriate oscillating behaviour of the nonlinear term ensures the existence of infinitely many weak solutions. Our analysis is based on recent variational methods for smooth functionals defined on Orlicz-Sobolev spaces.

The Trace Theorem for Anisotropic Sobolev — Slobodetskii Spaces with Applications to Nonhomogeneous Elliptic BVPs

In this paper, anisotropic Sobolev — Slobodetskii spaces in poly-cylindrical domains of any dimension N are considered. In the first part of the paper we revisit the well-known Lions — Magenes Trace Theorem (1961) and, naturally, extend regularity results for the trace and lift operators onto the anisotropic case. As a byproduct, we build a generalization of the Kruzhkov — Korolev Trace Theorem for the first-order Sobolev Spaces (1985). In the second part of the paper we observe the nonhomogeneous Dirichlet, Neumann, and Robin problems for p-elliptic equations. The well-posedness theory for these problems can be successfully constructed using isotropic theory, and the corresponding results are outlined in the paper. Clearly, in such a unilateral approach, the anisotropic features are ignored and the results are far beyond the critical regularity. In the paper, the refinement of the trace theorem is done by the constructed extension.DOI 10.14258/izvasu(2018)4-19

Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces

Abstract By using variational methods and critical point theory in an appropriate Orlicz-Sobolev setting, the authors establish the existence of infinitely many non-negative weak solutions to a non-homogeneous Neumann problem. They also provide some particular cases and an example to illustrate the main results in this paper.

English

The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error estimate has the convergence rate as 0.5.

Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous Neumann problems

We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly $$(p-1)$$ -sublinear reaction term. We prove a bifurcation-type result establishing the existence of a critical parameter value $$\lambda _*>0$$ such that for all $$\lambda >\lambda _*$$ the problem has at least two positive solutions, for $$\lambda =\lambda _*$$ it has at least one positive solution and for $$\lambda \in (0,\lambda _*)$$ there are no positive solutions. Also, for $$\lambda \ge \lambda _*$$ we show that the problem has a smallest positive solution $$\bar{u}_{\lambda }$$ and we investigate the continuity and monotonicity properties of the map $$\lambda \rightarrow \bar{u}_{\lambda }$$ .

Polyharmonic Neumann and mixed boundary value problems in the Heisenberg group

We study the polyharmonic Neumann and mixed boundary value problems on the Korányi ball in the Heisenberg group . Necessary and sufficient solvability conditions are obtained for the nonhomogeneous polyharmonic Neumann problem and Neumann–Dirichlet boundary value problems.

Perturbed Kirchhoff-type Neumann problems in Orlicz–Sobolev spaces

The aim of this paper is to establish the existence of infinitely many solutions for perturbed Kirchhoff-type non-homogeneous Neumann problems involving two parameters. To be precise, we prove that an appropriate oscillating behaviour of the nonlinear term, even under small perturbations, ensures the existence of infinitely many solutions. Our approach is based on recent variational methods for smooth functionals defined on Orlicz–Sobolev spaces.

Homogenization of fluid–porous interface coupling in a biconnected fractured media

The modelization of mass transfer through biconnected fractured porous media is studied by homogenizing the coupling between the Darcyan percolation and the viscous Stokes flow on their interface governed by the Beavers–Joseph law. The case of high transmission coefficients is considered. The asymptotic behaviour is completely described with the help of the solutions of some specific local problems and of a nonhomogeneous Neumann problem defined by the effective permeability tensor.

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.

Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems

We consider nonlinear Neumann and Dirichlet problems driven by a nonhomogeneous differential operator and a Caratheodory reaction. Our framework incorporates $p$-Laplacian equations and equations with the $(p,q)$-differential operator and with the generalized $p$-mean curvature operator. Using variational methods, together with truncation and comparison techniques and Morse theory, we prove multiplicity theorems, producing three, five or six nontrivial smooth solutions, all with sign information.

Multiplicity of Solutions for Perturbed Nonhomogeneous Neumann Problem through Orlicz‐Sobolev Spaces

We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz‐Sobolev space.

Existence of three solutions for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces

The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz–Sobolev space.

Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz–Sobolev spaces

Under an appropriate oscillating behavior of the nonlinear term, the existence of a determined open interval of positive parameters for which an eigenvalue non-homogeneous Neumann problem admits infinitely many weak solutions that strongly converges to zero, in an appropriate Orlicz–Sobolev space, is proved. Our approach is based on variational methods. The abstract result of this paper is illustrated by a concrete case.