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1.
We introduce an analogue of Payne’s nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain \(\Omega \) with rotational symmetry of order two (i.e. \(x\in \Omega \) iff \(-x\in \Omega \)) “should normally” be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this assertion to the nodal line conjecture, while demonstrating that the extra structure of the problem makes it easier to obtain stronger statements: it is true for all simply connected planar domains, while there is a counterexample domain homeomorphic to a disk with two holes.  相似文献   

2.
The well-known Schiffer conjecture saying that for a smooth bounded domain ΩRn, if there exists a positive Neumann eigenvalue such that the corresponding Neumann eigenfunction u is constant on the boundary of Ω, then Ω is a ball. In this paper, we shall prove that the Schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding Neumann eigenfunction is constant on the boundary. We also prove a similar result to the Berenstein conjecture for the overdetermined Dirichlet eigenvalue problem.  相似文献   

3.
Cranston and Salisbury have obtained an integral test for the existence of a maximal number of minimal Martin boundary points at 0 in certain domains (cf. [8]). This paper will extend the result as follows: let D R d be an open Greenian set (with respect to the Laplacian) consisting of n disjoint open connected cones with Lipschitz boundary and a subset of the boundary of these cones. Let be some local Kato measure supported by the boundary of the cones and consider the Schrödinger operator 1/2-µ. We will assume a boundary Harnack principle and give a sufficient integral criterion for the existence of exactly n minimal Martin boundary points at 0. In certain cases there is a necessary criterion, too. When the sufficient integral criterion holds we will give a necessary and a sufficient condition for the existence of a certain process related to the Schrödinger operator that connects two different admissible boundary points. In the paper of Cranston and Salisbury the case = 0, d = 2 is treated, but many of the arguments work as well in the general situation.  相似文献   

4.
The object of this paper is to establish an expansion theorem for a regular right-definite eigenvalue problem with an eigenvalue parameter which is contained in the Schrödinger partial differential equation and in a general type of boundary conditions on the boundary of an arbitrary multiply connected bounded domain inR n (n2). We associate with this problem an essentially self-adjoint operator in a suitably defined Hilbert space and then we develop an associated eigenfunction expansion theorem.  相似文献   

5.
We prove the existence of nontrivial compact extremal domains for the first eigenvalue of the Laplacian in manifolds ${\mathbb{R}^{n}\times \mathbb{R}{/}T\, \mathbb{Z}}$ with flat metric, for some T > 0. These domains are close to the cylinder-type domain ${B_1 \times \mathbb{R}{/}T\, \mathbb{Z}}$ , where B 1 is the unit ball in ${\mathbb{R}^{n}}$ , they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations. Such domains can be extended by periodicity to nontrivial and noncompact domains in Euclidean spaces whose first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boundary.  相似文献   

6.
We prove two stability-type estimates involving the Schwarz rearrangement of the normalized first eigenfunction u 1?>?0 of certain linear elliptic operators whose first eigenvalue λ1 is close to the lowest possible one (i.e., ${\lambda_1^\star}$ , the first eigenvalue of the Dirichlet Laplacian in a suitable ball). In particular, we prove that if ${\lambda_1\approx \lambda_1^\star}$ then the L -distance between the rearrangement ${u_1^\star}$ and the normalized first eigenfunction of the Dirichlet Laplacian corresponding to ${\lambda_1^\star}$ is less than a suitable power of the difference ${\lambda_1-\lambda_1^\star}$ times a universal constant. We also show that the L -distance between the first eigenfunction of the Dirichlet Laplacian in a ball whose first eigenvalue equals λ1 and the rearrangement ${u_1^\star}$ can be controlled with a power of the value assumed by ${u_1^\star}$ on the boundary of that ball.  相似文献   

7.
In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n-dimensional Euclidean space Rn, the conjecture of Pólya is well known: the kth eigenvalue λk of the Dirichlet eigenvalue problem of Laplacian satisfies
  相似文献   

8.
The sum of the first n ≥ 1 energy levels of the planar Laplacian with constant magnetic field of given total flux is shown to be maximal among triangles for the equilateral triangle, under normalization of the ratio (moment of inertia)/(area)3 on the domain. The result holds for both Dirichlet and Neumann boundary conditions, with an analogue for Robin (or de Gennes) boundary conditions too. The square similarly maximizes the eigenvalue sum among parallelograms, and the disk maximizes among ellipses. More generally, a domain with rotational symmetry will maximize the magnetic eigenvalue sum among all linear images of that domain. These results are new even for the ground state energy (n = 1).  相似文献   

9.
The asymptotic expansions of the trace of the heat kernel for small positive t, where λν are the eigenvalues of the negative Laplacian in Rn (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ?Ω1 and a smooth outer boundary ?Ω2 where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ j (j=1,…,m) of ?Ω1 and on the components of ?Ω2 are considered such that and and where the coefficients in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.  相似文献   

10.
It is shown that solutions to the Dirichlet problem for quasilinear elliptic systems in a domain ofR n n3 with smooth boundary datum can be singular at the boundary.  相似文献   

11.
Given an admissible measure μ on where is an open set, we define a realization of the Laplacian in with general Robin boundary conditions and we show that generates a holomorphic C 0 -semigroup on which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form . We also show that if contains smooth functions, then μ is of the form (where σ is the (n-1)-dimensional Hausdorff measure and β a positive measurable bounded function on ); i.e. we have the classical Robin boundary conditions. RID="h1" ID="h1"Dédié à Philippe Bénilan RID="*" ID="h1"This work is part of the DGF-Project: "Regularit?t und Asymptotik für elliptische und parabolische Probleme".  相似文献   

12.
It is shown that the Dirichlet problem for where Ω??n is critical in that it has first eigenvalue one, is globally solvable for any continuous positive initial datum vanishing at ?Ω. Moreover, for p<3 all solutions are bounded and tend to some nonnegative eigenfunction of the Laplacian as t→∞, while if p?3 then there are both bounded and unbounded solutions. Finally, it is shown that unlike the case p∈[0,1), all steady states are unstable if p?1. Copyright © 2004 John Wiley & Sons, Ltd.  相似文献   

13.
Summary The manifold metric between two points in a planar domain is the minimum of the lengths of piecewiseC 1 curves in the domain connecting these two points. We define a bounded simply connected planar region to be a pseudo Jordan domain if its boundary under the manifold metric is topologically homeomorphic to the unit circle. It is shown that reflecting Brownian motionX on a pseudo Jordan domain can be constructed starting at all points except those in a boundary subset of capacity zero.X has the expected Skorokhod decomposition under a condition which is satisfied when G has finite 1-dimensional lower Minkowski content.  相似文献   

14.
This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of RNRN (N≥3N3) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.  相似文献   

15.
Under mild conditions a certain subspace M, consisting of functions which are analytic in a simply connected domain Ω and continuous on the boundary Gamma;, is shown to have real parts which are dense, in the sup norm, in the set of all solutions to the Dirichlet problem for continuous boundary data. Similar results hold for Lp boundary data. Numerical solutions of sample Dirichlet problems are computed. © 1994 John Wiley & Sons, Inc.  相似文献   

16.
Consider the problem of heat flow in a convex domain in ℝn with Dirichlet boundary condition and constant initial temperature. We show that the solution has a fixed hot spot if the domain is invariant under the action of an essential symmetry group. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.  相似文献   

17.
A well-known conjecture about the second eigenfunction of a bounded domain in ℝ2 states that the nodal line has to intersect the boundary in exactly two points. We give sufficient conditions on the domain for this assertion to hold. For special doubly symmetric domains we also prove that λ2 is simple and that the nodal line of the second eigenfunction lies on one of the axes.  相似文献   

18.
In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space R n . If λ k+1 is the (k + 1)th eigenvalue of Dirichlet Laplacian on Ω, then, we prove that, for n ≥ 41 and and, for any n and with , where j p,k denotes the k-th positive zero of the standard Bessel function J p (x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Pólya, we know that our estimates are optimal in the sense of order of k.Q.-M. Cheng was partially Supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of ScienceH. Yang was partially Supported by Chinese NSF, SF of CAS and NSF of USA  相似文献   

19.
20.
 Let M be an n-dimensional simply connected Hadamard manifold with Ricci curvature satisfying and be a bounded domain having smooth boundary. In this paper, we prove that the first n nonzero Neumann eigenvalues of the Laplacian on Ω satisfy , where is a computable constant depending only on and , Ω being the volume of Ω. This result generalizes the corresponding estimate for bounded domains in a Euclidean space obtained recently by M. S. Ashbaugh and R. D. Benguria. (Received 19 May 1998; in revised form 21 September 1998)  相似文献   

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