Simulation of a nonlinear Steklov eigenvalue problem using finite-element approximation

Elliptic problems with parameters in the boundary conditions are called Steklov problems. With the tool of computational approximation (finite-element method), we estimate the solution of a nonlinear Steklov eigenvalue problem for a second-order, self-adjoint, elliptic differential problem. We discussed the behavior of the nonlinear problem with the help of computational results using Matlab.     

Eigenvalues of the Wentzell–Laplace operator and of the fourth order Steklov problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell–Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.     

Eigenproblem for p-Laplacian and nonlinear elliptic equation with nonlinear boundary conditions

In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalue problem as simplicity, isolation and its strict monotonicity. Afterward, we character also the second eigenvalue and its strictly partial monotony. On the other hand, in some sense, we establish the non-resonance below the first and furthermore between the first and second eigenvalues of nonlinear Steklov–Robin.     

New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.     

Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model

We consider a relatively simple model for pool-boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling medium via a nonlinear boundary condition imposed on the fluid-heater interface. This results in a standard heat-transfer problem with a nonlinear Neumann boundary condition on part of the boundary. In a recent paper [Speetjens M, Reusken A, Marquardt W. Steady-state solutions in a nonlinear pool-boiling model. IGPM report 256, RWTH Aachen. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2006.11.002] we analysed this nonlinear heat-transfer problem for the case of two space dimensions and in particular studied the qualitative structure of steady-state solutions. The study revealed that, depending on system parameters, the model allows both multiple homogeneous and multiple heterogeneous temperature distributions on the fluid-heater interface. In the present paper we show that the analysis from Speetjens et al. (doi:10.1016/j.cnsns.2006.11.002) can be generalised to the physically more realistic case of three space dimensions. A fundamental shift-invariance property is derived that implies multiplicity of heterogeneous solutions. We present a numerical bifurcation analysis that demonstrates the multiple solution structure in this mathematical model by way of a representative case study.     

A nonlinear boundary eigenvalue problem for TM-polarized electromagnetic waves in a nonlinear layer

We consider the propagation of TM-polarized electromagnetic waves in a nonlinear dielectric layer located between two linear media. The nonlinearity in the layer is described by the Kerr law. We reduce the problem to a nonlinear boundary eigenvalue problem for a system of ordinary differential equations. We obtain a dispersion relation and a first approximation for eigenvalues of the problem. We compare the results with those obtained for the case of a linear medium in the layer.     

Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.     

An optimization problem for nonlinear Steklov eigenvalues with a boundary potential

In this paper, we analyze an optimization problem for the first (nonlinear) Steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed L¹$L^1$-norm.     

Bifurcation from trivial solution for elliptic systems with nonlinear boundary conditions

In this paper, we investigate semilinear elliptic systems having a parameter with nonlinear Neumann boundary conditions over a smooth bounded domain. The objective of our study is to analyse bifurcation component of positive solutions from trivial solution and their stability. The results are obtained via classical bifurcation theorem from a simple eigenvalue, by studying the eigenvalue problem of elliptic systems.     

Rearrangements and minimization of the principal eigenvalue of a nonlinear Steklov problem

This paper, motivated by Del Pezzo et al. (2006) [1], discusses the minimization of the principal eigenvalue of a nonlinear boundary value problem. In the literature, this type of problem is called Steklov eigenvalue problem. The minimization is implemented with respect to a weight function. The admissible set is a class of rearrangements generated by a bounded function. We merely assume the generator is non-negative in contrast to [1], where the authors consider weights which are positively away from zero, in addition to being two-valued. Under this generality, more physical situations can be modeled. Finally, using rearrangement theory developed by Geoffrey Burton, we are able to prove uniqueness of the optimal solution when the domain of interest is a ball.     

A Comparison of Preconditioners for the Steklov–Poincaré Formulation of the Fluid-Structure Coupling in Hemodynamics

A Fluid–Structure Interaction (FSI) problem can be reinterpreted as a heterogeneous problem with two subdomains. It is possible to describe the coupled problem at the interface between the fluid and the structure, yielding a nonlinear Steklov–Poincaré problem. The linear system can be linearized by Newton iterations on the interface and the resulting linear problem can be solved by the preconditioned GMRES method. In this work we investigate the behavior of preconditioners of Neumann–Neumann and Dirichlet–Neumann type. We find that, in the context of hemodynamics, the Dirichlet– Neumann, i.e., using Dirichlet boundary conditions on the fluid side and Neumann on the structure side, outperforms the Neumann–Neumann method, except when a weighting is used such that it basically reduces to the Dirichlet–Neumann method. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this case.     

Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight

We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henry?s approach to the problem.Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided.The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the \(p\) -Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator \(\Delta _p\) when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.     

On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Dirichlet-to-Neumann (or Steklov) problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.     

Steady-state solutions in a nonlinear pool boiling model

We consider a relatively simple model for pool boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling fluid via a nonlinear boundary condition imposed on the fluid–heater interface. This results in a standard heat equation with a nonlinear Neumann boundary condition on part of the boundary. In this paper, we analyse the qualitative structure of steady-state solutions of this heat equation. It turns out that the model allows both multiple homogeneous and multiple heterogeneous solutions in certain regimes of the parameter space. The latter solutions originate from bifurcations on a certain branch of homogeneous solutions. We present a bifurcation analysis that reveals the multiple-solution structure in this mathematical model. In the numerical analysis a continuation algorithm is combined with the method of separation-of-variables and a Fourier collocation technique. For both the continuous and discrete problem a fundamental symmetry property is derived that implies multiplicity of heterogeneous solutions. Numerical simulations of this model problem predict phenomena that are consistent with laboratory observations for pool boiling processes.     

Steklov特征值问题的边界元近似

本文将Ｌａｐｌａｃｅ算子的Ｓｔｅｋｌｏｖ特征值问题归化为一个边界变分问题，从而使原问题的空间维数降低了一维，基于此变分问题给出了Ｓｔｅｋｌｏｖ特征值问题的边界元近似解，计算实例表明此方法是十分有效的。     

On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains

We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.     

An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.