相关论文: Pleijel's nodal domain theorem for free membranes

In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This…

We prove the Pleijel theorem in non-collapsed RCD spaces, providing an asymptotic upper bound on the number of nodal domains of Laplacian eigenfunctions. As a consequence, we obtain that the Courant nodal domain theorem holds except at most…

We show the existence of planar domains with one hole for which the first non-trivial Neumann eigenfunction has a closed nodal line fully contained inside the domain. This is optimal, as it is known since Pleijel's 1956 result that the…

In this paper, we revisit Courant's nodal domain theorem for the Dirichlet eigenfunctions of a square membrane, and the analyses of A. Stern and {\AA}. Pleijel.

We are concerned in this paper with the real eigenfunctions of Schr\"odinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant's theorem is…

In 1956 $\AA$. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an…

We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular we remove the restriction, imposed in previous work, that the Robin parameter be non-negative. We also improve the upper bound in the…

We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel's theorem, which states that, as soon as the dimension is strictly larger…

A slight improvement of Pleijel's estimate on the number of nodal domains is obtained, exploiting a refinement of the Faber-Krahn inequality and packing density of discs.

We consider the zeros on the boundary $\partial \Omega$ of a Neumann eigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. We prove that the number of its boundary zeros is $O (\lambda)$ where $-\Delta \phi_{\lambda} =…

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…

We consider the Courant-sharp eigenvalues of the Robin Laplacian for bounded, connected, open sets in $\mathbb{R}^n$, $n \geq 2$, with Lipschitz boundary. We prove Pleijel's theorem which implies that there are only finitely many…

\AA. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is $\geq…

Consider partitions of an open, bounded domain in $\mathbb{R}^n$. Then an average element of the partition has either its Fraenkel asymmetry or its deviation from the smallest element in the partition bounded away from 0 by a universal…

{\AA} Pleijel has proved that in the case of the Laplacian on the square with Neumann condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues. We identify five…

We prove sharp bilinear estimates for Dirichlet or Neumann eigenfunctions in domains in the plane. These are the natural analog of earlier estimates for the boundaryless case of Burq, G\'erard, and Tzvetkov.

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,…

We consider Dirichlet-to-Neumann operators associated to $\Delta+q$ on a Lipschitz domain in a smooth manifold, where $q$ is an $L^{\infty}$ potential. We prove a Courant-type bound for the nodal count of the extensions $u_k$ of the $k$th…

Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of…

The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free…