Related papers: A counterexample to Payne's nodal line conjecture …

We present examples of bounded planar domains with one single hole for which the nodal line of a second Dirichlet eigenfunction is closed and does not touch the boundary. This shows that Payne's nodal line conjecture can at most hold for…

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 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…

We construct a counterexample to the ``hot spots'' conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and…

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} =…

Let $\Omega$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is left invariant under the action of the dihedral group $D_n$ of isometries of $\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction must…

In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $\Omega$ in $\mathbb{R}^2$ must intersect the boundary $\partial\Omega$ at exactly two points.

In 1990, P\"utter shown that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain $\Omega$ intersects the boundary $\partial\Omega$ provided $\Omega$ has the circular symmetry. By…

We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the $k$-th eigenvalue of the Dirichlet Laplacian…

The paper addresses the the number of nodal domains for eigenfunctions of Schr\"{o}dinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem…

The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem…

According to Courant's theorem, an eigenfunction as\-sociated with the $n$-th eigenvalue $\lambda\_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear…

We present a proof of the Harbourne-Hirschowitz conjecture for linear systems with base points of multiplicity seven or less. This proof uses a well-known degeneration of the projective plane, as well as a combinatorial technique that…

\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…

Let $P$ be a finite set of points in the plane in general position, that is, no three points of $P$ are on a common line. We say that a set $H$ of five points from $P$ is a $5$-hole in $P$ if $H$ is the vertex set of a convex $5$-gon…

This paper considers overdetermined boundary problems. Firstly, we give a proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two dimensional case and under an additional condition for the case of…

In 1960, Payne and Weinberger proved that among all domains that lie within a wedge (an angle whose measure is less than or equal to $\pi$), and have a given value of a certain integral the circular sector has the lowest fundamental…

It has been conjectured by P\'{o}lya and Szeg\"{o} seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with $n$ sides and fixed area is the regular polygon. Despite its…

Poncelet's theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of this theorem that we are aware of,…

In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta_p u = f(u)$ in bounded Steiner symmetric domains $\Omega \subset \mathbb{R}^N$ under the…