So it is natural and meaningful for us to study nonconforming finite element approximations of steklov eigenvalue problems. An optimization problem for nonlinear steklov eigenvalues with a boundary potential. Steklov approximations of harmonic boundary value problems on. This project investigates the application of the steklov eigenvalue expansion method sem to two types of boundary value problems. Introduction the homogenization of spectral problems in domains with microstructure is an important and, in many cases, di. The sum of the entries on the main diagonal is called the trace of a. On the asymptotic behaviour of eigenvalues of a boundary.
On the other hand, betterknown is the laplacian eigenvalue problem. On the eigenvalues of a biharmonic steklov problem. Optimization of steklovneumann eigenvalues sciencedirect. The steklov problem yields also exceptional values, socalled steklov eigenvalues, and these eigenvalues are discussed intensively in the literature. In this paper, we nd upper bounds for the eigenvalues of the laplacian in the conformal class of a compact riemannian manifold m. Steklov problem, homogenization of spectral problems. Optimization of steklov neumann eigenvalues habib ammari. We recommend the excellent expository article, which surveys results on eigenvalue asymptotics, questions of isospectrality and rigidity, and geometric bounds on.
Conformal upper bounds for the eigenvalues of the laplacian and steklov problem asma hassannezhad department of mathematical sciences, sharif university of technology, tehran, iran received 1 february 2011. Pdf simulation of a nonlinear steklov eigenvalue problem. A multilevel correction method for steklov eigenvalue problem. Eigenvalue inequalities for mixed steklov problems. The spectral element method for the steklov eigenvalue.
Shape optimization for the eigenvalues of a biharmonic. Pdf eigenvalue inequalities for mixed steklov problems. Free boundary minimal surfaces and the steklov eigenvalue problem 1. Using spectral approximation theory, it is shown theoretically that the. Rearrangements and minimization of the principal eigenvalue of a nonlinear steklov problem. Introduction of crucial importance in the study of boundary value problems for di. For second order selfadjoint eigenvalue problems banerjee and osborn 3 prove that nite element approximations. Pdf on a fourth order steklov eigenvalue problem researchgate. Steklov eigenproblems and the representation of solutions. Free boundary minimal surfaces and the steklov eigenvalue. In this paper, a multilevel correction scheme is proposed to solve the steklov eigenvalue problem by nonconforming finite element methods.
Note that most steklov eigenvalue problems considered in the literature are related to partial differential equations of second order. With the scheme, the solution of the steklov eigenvalue problem on a fine grid is reduced to the solution of the steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Riemannian surfaces with boundary, as well as for more general eigenvalue problems. Over the past years, there has been a growing interest in the steklov problem from.
Conformal upper bounds for the eigenvalues of the laplacian and steklov problem asma hassannezhad abstract. There are relatively few papers treating the e ect of numerical integration on eigenvalue approximation. Finally, numerical experiments on the square and the lshaped domain are carried out to get very accurate approximations by the spectral element method. Simulation of a nonlinear steklov eigenvalue problem using finiteelement approximation. The lagrange finite element is used for discretization and the convergence is proved using the spectral perturbation theory for compact operators. Simulation of a nonlinear steklov eigen value problem using finite element approximation. The above mixed steklovneumann eigen value problem is also called the sloshing problem. Steklov eigenproblems and the representation of solutions of. So this correction method can improve the overall ef. Nilima nigam abstract this paper examines the laplace equation with mixed boundary conditions, the n. In order to state our main conclusions below clearly, we would like to make some agreements.
In this paper, we present a multilevel correction scheme to solve the steklov eigen value problem by nonconforming. In this paper we analyse possible extensions of the classical steklov eigenvalue problem to the fractional setting. Pdf we study the spectrum of a biharmonic steklov eigenvalue problem in a bounded domain of r n. We consider an eigenvalue problem for the biharmonic operator with steklov type boundary conditions. Research article a multilevel correction scheme for the. The method of fundamental solutions applied to boundary. We prove reillytype upper bounds for di erent types of eigen value problems on submanifolds of euclidean spaces with density. The steklov eigenvalue problem let be a bounded domain of dimension n with smooth boundary m of dimension n 1.
The neumann problem describes the vibration of a homogeneous free membrane. The first type of problem is a mixed dirichletneumann boundary value problem mixed dn bvp involving a secondorder uniformly elliptic equation subjected. Convergence and quasioptimality of adaptive fem for steklov eigenvalue problems. Guaranteed eigenvalue bounds for the steklov eigenvalue. With the proposed method, the solution of the steklov eigenvalue problem will not be much more dif. A twogrid discretization scheme for the steklov eigenvalue. Spectral method with the tensorproduct nodal basis for the. Twoparameter eigenvalues steklov problem involving the plaplacian 153 it remains to prove that the sequence k is unbounded. Also, it was brought to our attention that in 1994, giovanni alessandrini and rolando magnanini. The values of the parameter such that the equation has nontrivial solutions. A comparison theorem for the first nonzero steklov eigenvalue.
We develop methods based on fundamental solutions to compute the steklov, wentzell and. Steklov problems arise in a number of important applications, notably, in hydrodynamics through the steklov type sloshing eigenvalue problem describing small oscillations of fluid in an open vessel, and in medical and geophysical imaging via the link between the steklov problem and the celebrated dirichlettoneumann map. We extend some classical inequalities between the dirichlet and neumann eigenvalues of the laplacian to the context of mixed steklovdirichlet and steklovneumann eigenvalue problems. Let be an immersed submanifold of dimension kin rn. In this article, we give a sharp lower bound for the first nonzero eigenvalue of the steklov eigenvalue problem in \\omega. Shape optimization for neumann and steklov eigenvalues. Note that all the other eigenvalues k of this sequence has also a variational characterization but we dont know if all the spectrum is contained in this sequence. Pdf a virtual element method for the steklov eigenvalue.
Approximation of eigenvalues there are two classes of numerical methods. The method of eigenfunctions is closely related to the fourier method, or the method of. Steklov eigenproblems and the representation of solutions of elliptic boundary value problems giles auchmuty department of mathematics, university of houston, houston, texas, usa abstract this paper describes some properties and applications of steklov eigenproblems for prototypical secondorder elliptic operators on bounded regions in rn. A new type of iteration method is proposed in this paper to solve the steklov eigenvalue problem by the finite element method. A multilevel correction method for steklov eigenvalue.
On a class of fourth order steklov eigen value problems. Comparatively, nonlinear eigenvalue problems for the pxlaplacian have been investigated. A reilly inequality for the rst steklov eigenvalue. However, to the best of our knowledge, there have been no reports on spectral method for steklov eigenvalue problems.
In this scheme, solving the steklov eigenvalue problem is transformed into a series of solutions of boundary value problems on multilevel meshes by the multigrid method and solutions of the steklov eigenvalue problem on the coarsest mesh. It says that the first eigenvalue of a geodesic ball of radius r and center p, b r. Since steklov eigenvalue problems have important physical background and many applications, for instance, they appear in the analysis of stability of mechanical oscillators immersed in a viscous fluid see and the references therein, in the analysis of the antiplane shearing on a system of collinear faults under slipdependent friction law. Pdf on the eigenvalues of a biharmonic steklov problem. The spectral element method for the steklov eigenvalue problem scientific. There clearly are many further questions about the efficacy of such approximations but the primary observation is that low order steklov approximations do provide good interior approximations to solutions of harmonic boundary value problems. We study bounds on the riesz means of the mixed steklov neumann and steklov. Over the past years, there has been a growing interest in the steklov problem from the viewpoint of spectral geometry. The steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. May 24, 2015 furthermore, xing wang and jiuyi zhu proved a polynomial lower bound of the nodal set under the assumption that 0 is a regular value for the steklov eigenfunction a lower bound for the nodal sets of steklov eigenfunctions, arxiv. Bounds to these are discussed in , 16, 18, 19, 31, asymptotic formulas to domain perturbations can be found in 21, 28 and for boundary perturbations in 28, 30.
Twoparameter eigenvalues steklov problem involving the p. The steklov problem on compact riemannian manifolds with boundary was introduced by v. The nonselfadjointness of the problem leads to nonhermitian matrix eigenvalue problem. We refer to the recent survey and the references therein for an account of this topic. Nonconforming finite element approximations of the steklov. They are proved in problems 1617 and again in the next section. Spectral geometry of the steklov problem on orbifolds. Reillytype inequalities for paneitz and steklov eigenvalues julien roth abstract.
A multilevel correction scheme for the steklov eigenvalue problem. Its spectrum coincides with that of the dirichlettoneumann operator. The latter one is also known as the sloshing problem, and has been actively studied. The differential equation is said to be in sturmliouville form or selfadjoint form. Shape optimization for low neumann and steklov eigenvalues. Steklov was the first to demonstrate strictly for a very broad class of surfaces the existence of an infinite sequence of proper eigen values and corresponding eigen functions defining them in a way different from poincares. The steklovlike eigenvalue problem associated with the equation. Fractional eigenvalue problems that approximate steklov. Asymptotics of sloshing eigenvalues michael levitin leonid parnovski iosif polterovich david a. The corresponding eigenfunctions form an orthonormal basis. Research article a multilevel correction scheme for the steklov eigenvalue problem qichaozhao,yiduyang,andhaibi school of mathematics and computer science, guizhou normal university, guiyang, china. In the new scheme, the cost of solving eigenvalue problems is almost the same as solving the associated boundary value problems. A virtual element method for the steklov eigenvalue problem article pdf available in mathematical models and methods in applied sciences 2508.
The vector x is the right eigenvector of a associated with the eigenvalue. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an steklov eigenvalue problem on the coarsest finite element space. We characterize it in general and give its explicit. In this paper, we consider the following model problem. We obtain it as a limiting neumann problem for the biharmonic operator in a process of mass concentration at the boundary. Combining the correction technique proposed by lin and xie and the shifted inverse iteration, a multilevel correction scheme for the steklov eigenvalue problem is proposed in this paper. Iterative techniques for solving eigenvalue problems. Pdf spectral indicator method for a nonselfadjoint. There is an extensive literature concerning the steklov eigenvalue problem. Hence, the sobolev inequalities and their optimal constants is a subject of interest in the analysis of pdes and related topics. We consider the steklov eigenvalue problem for the laplace operator. It is known that it has a discrete set of eigenv alues see for example 2, chapter ii i 0. In the paper, a twogrid discretization scheme is discussed for the steklov eigenvalue problem.
Eigenvalue comparisons in steklov eigenvalue problem and some. Spectral indicator method for a nonselfadjoint steklov. A complete description of the set of eigenvalues is provided in this nonhomogeneous case p q. Computational methods for extremal steklov problems. On a class of fourth order steklov eigen value problems alberto ferrero dipartimento di scienze e innovazione tecnologica, universit a degli studi del piemonte orientale \amedeo avogadro, italy, alberto. Consider the neumann and steklov eigenvalue problems on.
Steklov in 1902 see also for a historical discussion and has recently seen a surge of interest from the spectral geometry community. Spectral indicator method for a nonselfadjoint steklov eigenvalue. The legacy of vladimir andreevich steklov nikolay kuznetsov, tadeusz kulczycki, mateusz kwa. It is known that upper eigenvalue bounds can easily be obtained using the rayleighritz method with trial functions, e. This inequality makes an important role in the study of existence and regularity of solutions of some boundary value problems. Optimization of steklovneumann eigenvalues habib ammari. The steklov eigenvalue problem belongs to the class of eigenvalue problems involving selfadjoint di erential operators, such as the laplacian eigenvalue problem.
All secondorder linear ordinary differential equations can be recast in the form on the lefthand side of by multiplying both sides of the equation by an appropriate integrating factor although the same is not true of secondorder partial differential equations, or if y is a vector. The above mixed steklov neumann eigen value problem is also called the sloshing problem. Research article spectral method with the tensorproduct. We say uis a steklov eigenfunction if it satis es the equation 1. Shape optimization for the eigenvalues of a biharmonic steklov problem luigi provenzano joint work with davide buoso pepworkshop 2014, aveiro november 06, 2014. These upper bounds depend only on the dimension and a conformal invariant that we call \minconformal. This includes the eigenvalues of panetizlike operators as well as three types of generalized steklov problems. The spectral element method for the steklov eigenvalue problem. In ch, cheng established a comparison principle for the first eigen value for the dirichlet problem. The present article highlights some of nikolay kuznetsov heads the laboratory for mathematical modelling of wave phenomena at the institute for problems.
Eigenvalue comparisons in steklov eigenvalue problem and. These simple examples were chosen primarily to illustrate the phenomenology observed in computing steklov approximations. Shape optimization problems for steklov eigenvalues with mixed boundary conditions have also been studied 10. Eigenvalue bounds of mixed steklov problems asma hassannezhad and ari laptev with an appendix by f.
International journal of computer mathematics 2015, 119. The aim of the present note is to obtain upper bounds for the rst nonzero eigenvalue 1 of the p steklov problem. How large can the nth eigenvalue of the steklov problem be on a bounded simplyconnected planar domain of a given mass. The theoretical analysis and numerical experiments indicate that the scheme proposed in this paper is efficient for both simple and multiple eigenvalues of the steklov eigenvalue problem. The present paper is motivated by the following question 1. Eigenvalue problems eigenvalue problems often arise when solving problems of mathematical physics. We consider the steklov eigen value problem 2 where the domain. We consider an eigenvalue problem for the biharmonic operator with steklovtype boundary conditions. We study the dependence of the spectrum upon the domain. We propose an efficient numerical method for a nonselfadjoint steklov eigenvalue problem.
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