On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains
Konstantin Pankrashkin
备注：
12 pages, contribution to the proceedings of the conference "Mathematical Challenge of Quantum Transport in Nanosystems" (Pierre Duclos Workshop), Saint-Petersburg, Russia, March 13-15, 2013
Let $Ω\subset \RR^2$ be a domain having a compact boundary $Σ$ which is Lipschitz and piecewise $C^4$ smooth, and let $ν$ denote the inward unit normal vector on $Σ$. We study the principal eigenvalue $E(β)$ of the Laplacian in $Ω$ with the Robin boundary conditions $\partial f/\partialν+βf=0$ on $Σ$, where $β$ is a positive number. Assuming that $Σ$ has no convex corners we show the estimate $E(β)=-β^2- γ_\mxβ+ O\big(β^\{2}{3}\big)$ as $β\to+\infty$, where $γ_\mx$ is the maximal curvature of the boundary.
This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard--Helffer, Helffer--Persson--Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter $h$, we focus on the case when $h$ is large. We hope to come back to the analysis when $h$ is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of …
This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter $h$). For the square, our first paper focused on the case where $h$ is large and extended results that were obtained by Pleijel, Bérard--Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of $h$, where $h$ is positive and not close to $0$. In this…
In the present paper we introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth open set. The considered perturbation is of lower order and corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if, and only if, the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of…
We consider the problem of the recovery of a Robin coefficient on a part $γ\subset \partial Ω$ of the boundary of a bounded domain $Ω$ from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on $\partial Ω\setminus γ$. We prove uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method.
The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $α/L(Ω)$, and $α$ lies between $-2π$ and $2π$. Corollaries include Szegő's sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock's inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter. The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization o…
Let $Ω\subset \mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $\partial Ω$. For $σ>0$, we consider the Robin boundary value problem \[ -Δf =λf, \qquad \frac{\partial f}{\partial n} + σf = 0 \mbox{ on } \partial Ω\] where $ \frac{\partial f}{\partial n} $ is the derivative in the direction of the outward pointing normal to $\partial Ω$. Let $0<λ^σ_0\leq λ^σ_1\leq \dots $ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps \[ d_n(σ):=λ_n^σ-λ_n^0 . \] For a wide class of planar domains we show that there is a limiting mean value, …
This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a strong magnetic field and Robin boundary condition on a smooth planar domain and with a negative boundary parameter. We study the singular limit when the Robin parameter tends to infinity which is equivalent to a semi-classical limit involving a small positive semi-classical parameter. The main result is a comparison between the spectrum of the Robin Laplacian with an effective operator defined on the boundary of the domain via the Born-Oppenheimer approximation. More precisely, the low-lying eig…
In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $Ω$ corresponding to a negative Robin parameter does not exceed the analogous quantity …
Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Institute of Mathematics, Ufa Federal Research Centre, RAS（俄罗斯科学院乌法联邦研究中心数学研究所）；Ufa University of Science and Technology（乌法科学技术大学）；Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia（西波希米亚大学应用科学学院数学与NTIS系）
We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary c…
