Steklov Problem Research Articles

Concentration along Geodesics for a Nonlinear Steklov Problem Arising in Corrosion Modeling

We consider the problem of finding pairs $(\lambda,\mathfrak{u})$, with $\lambda>0$ and $\mathfrak{u}$ a harmonic function in a three-dimensional torus-like domain $\mathcal{D}$, satisfying the nonlinear boundary condition $\partial_{\nu}\mathfrak{u}=\lambda\,\sinh\mathfrak{u}$ on $\partial\mathcal{D}$. This type of boundary condition arises in corrosion modeling (Butler--Volmer condition). We prove the existence of solutions which concentrate along some geodesics of the boundary $\partial\mathcal{D}$ as the parameter $\lambda$ goes to zero.

On a singularly perturbed Steklov problem in a domain perforated along the boundary

We study the asymptotic behavior of solutions and eigenelements to a boundary value problem for the Laplace equation in a domain perforated along part of the boundary. On the boundary of holes, we set the homogeneous Dirichlet boundary condition and the Steklov spectral condition on the mentioned part of the outer boundary of the domain. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem. Nous étudions le comportement asymptotique des solutions et des éléments propres à un problème aux limites pour l'équation de Laplace dans un domaine perforé le long d'une partie de la frontière. Sur la frontière de trous, nous posons la condition de Dirichlet homogène et la condition spectrale de Steklov sur la part mentionnée de la frontière extérieure du domaine. En supposant que la microstructure de la frontière est périodique, nous construisons le problème aux limites et prouvons le théorème d'homogénéisation.

Asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operator

For a given bounded domain Ω with smooth boundary in a smooth Riemannian manifold (M,g), by decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferential operator, and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as t→0+. In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.

A virtual element method for the Steklov eigenvalue problem

The aim of this paper is to develop a virtual element method for the two-dimensional Steklov eigenvalue problem. We propose a discretization by means of the virtual elements presented in [L. Beirão da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199–214]. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. We also prove higher-order error estimates for the computation of the eigensolutions on the boundary, which in some Steklov problems (computing sloshing modes, for instance) provides the quantity of main interest (the free surface of the liquid). Finally, we report some numerical tests supporting the theoretical results.

Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz-Sobolev spaces

Employing variational methods and critical point theory, in an appropriate Orlicz-Sobolev setting, we establish the existence of infinitely many solutions for Steklov problems associated to non-homogeneous differential operators. We also provide some particular cases and a concrete example in order to illustrate the main results.

A few shape optimization results for a biharmonic Steklov problem

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue.

Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains

Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains

Asymmetric Steklov problems with sign-changing weights

We study two asymmetric Steklov problems with indefinite weights involving the p-Laplacian operator. We prove the existence of a first nontrivial eigenvalue for the first problem and the second one serves as an application in the description of the beginning of the Fučik spectrum with weights. We thereby extend several known results related to these problems.

Landesman-lazer type conditions and multiplicity results for nonlinear elliptic problems with neumann boundary values

We establish the existence and multiplicity of solutions for Steklov problems under nonresonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of Steklov type.

On the superlinear Steklov problem involving the p(x)-Laplacian

This paper is concerned with the existence and multiplicity of solutions for p(x)-Laplacian Steklov problem without the well-known Ambrosetti-Rabinowitz type growth conditions. By means of critical point theorems with Cerami condition, under weaker conditions, existence and multiplicity results of the solutions are proved.

Existence of infinitely many solutions for a Steklov problem involving the p(x)-Laplace operator

In this article, we study the nonlinear Steklov boundary-value problem Dp(x)u =juj p(x) 2 u in W, jruj p(x) 2 ¶u ¶n = f(x, u) onW. We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

Perturbation of the Steklov Problem on a Small Part of the Boundary

We consider the boundary value problem and spectral problem for the Laplace operator with the homogeneous Dirichlet condition on a small part of the boundary and the Fourier or Steklov condition on the remaining part of the boundary. We formulate the limit problem and prove the convergence of solutions, eigenvalues, and eigenfunctions of the original problem to solutions, eigenvalues, and eigenfunctions of the limit problem. Bibliography :2 4titles.

Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small-cavity size of e > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all e > 0 small enough, and such that . Here denotes the outward unit normal of , and δ2,2 ≡ 1 and δ2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 John Wiley & Sons, Ltd.

Steklov problem for a second-order ordinary differential equation with a continual derivative

For a second order linear ordinary differential equation with a continual derivative, we construct a fundamental solution. By using the fundamental solution, we find the solution of the Cauchy problem for the considered equation.

Spectral problems in Lipschitz domains

The paper is devoted to spectral problems for strongly elliptic second-order systems in bounded Lipschitz domains. We consider the spectral Dirichlet and Neumann problems and three problems with spectral parameter in conditions at the boundary: the Poincaré–Steklov problem and two transmission problems. In the style of a survey, we discuss the main properties of these problems, both self-adjoint and non-self-adjoint. As a preliminary, we explain several facts of the general theory of the main boundary value problems in Lipschitz domains. The original definitions are variational. The use of the boundary potentials is based on results on the unique solvability of the Dirichlet and Neumann problems. In the main part of the paper, we use the simplest Hilbert L 2-spaces H s , but we describe some generalizations to Banach spaces H s p of Bessel potentials and Besov spaces B s p at the end of the paper.

Inequalities for the Steklov eigenvalues

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimates. Let Ω be a bounded smooth domain in an n(⩾2)-dimensional Hadamard manifold an let 0=λ0<λ1⩽λ2⩽… denote the eigenvalues of the Steklov problem: Δu=0 in Ω and (∂u)/(∂ν)=λu on ∂Ω. Then ∑i=1nλi-1⩾(n2|Ω|)/(|∂Ω|) with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball. Let M be an n(⩾ 2)-dimensional compact connected Riemannian manifold with boundary and non-negative Ricci curvature. Assume that the mean curvature of ∂M is bounded below by a positive constant c and let q1 be the first eigenvalue of the Steklov problem: Δ2u=0 in M and u= (∂2u)/(∂ν2)−q(∂ u)/(∂ν)=0 on ∂M. Then q1⩾c with equality holding if and only if M is isometric to a ball of radius 1/c in Rn.

Non-local Boundary Condition Steklov Problem for A Laplace Equation in Bounded Domain

In classic mathematical physics course, under the local boundary conditions the Dirichlet (first kind), Neuman (second kind) and at last special case of Poincare (third kind) boundary value problems were considered for a Laplace equation being a canonic form of elliptic type of equations. Later on for a Laplace equation, under non-local boundary condition the Steklov problem was investigated in [3] and a sufficient condition for Fredholm property was found. Note that here boundary conditions contains non -local and global terms and the investigation method consist of obtaining necessary conditions, regularization of them and reducing the stated boundary problem to the system of second kind Fredholm integral equation with non-singular kernel.

Asymptotic behavior of the eigenvalues of the Steklov problem on a junction of domains of different limiting dimensions

The asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem on a junction of rectangles: a thin rectangle with a width of ɛ > 0 and a rectangle with unit dimensions, is studied. In addition to asymptotic formulas for the main series of eigenvalues (in the low-frequency region), other series with stable characteristics are found in the medium-frequency region and explicit formulas for the correction terms are derived. In the framework of the linear theory of surface waves, the results of this work describe the effect of wave localization in shallow water.

Numerical solution to the Steklov problem

The Steklov problem is considered in a planar domain with a smooth boundary. A numerical algorithm without saturation is constructed. The algorithm allows calculating 3000 eigenvalues with 9 decimal digits.

Gaps in the spectrum of the neumann problem on a perforated plane

Here, ∂ ν is the outward normal derivative. Problem (2) is obviously related to acoustic wave propagation in polluted media. Additionally, the separation of vari ables in the elasticity system in an isotropic space with cylindrical holes [1] or in the Steklov problem for sur face waves propagating over a liquid layer with vertical cylindrical columns [2] (Fig. 1b) also leads to a Neu mann spectral problem. The boundary value problem (2) is associated [3, Section 10.1] with a positive self adjoint operator in L2(ΠR) with the domain H(ΠR) (Lebesgue and Sobo ΠR 2 \ R α1 1 2 – α2 1 2 – , ⎝ ⎠ ⎛ ⎞ ,