Related papers: On Courant-like bound for Neumann domain count

This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $\Omega$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the…

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the…

Let $\mathbb{M}$ denote a complete, simply connected Riemannian manifold with sectional curvature $K_{\mathbb{M}} \leq k$ and Ricci curvature $\text{Ric}_{\mathbb{M}} \geq (n-1)K$, where $k,K \in \mathbb{R}$. Then for a bounded domain…

We are concerned with the analysis of a mean field type equation and its linearization, which is a nonlocal operator, for which we estimate the number of nodal domains for the radial eigenfunctions and the related uniqueness properties.

We consider the sequence of nodal counts for eigenfunctions of the Laplace-Beltrami operator in two dimensional domains. It was conjectured recently that this sequence stores some information pertaining to the geometry of the domain, and we…

For an arbitrary nonempty, open set $\Omega \subset \mathbb{R}^n$, $n \in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(- \Delta)^m\big|_{C_0^{\infty}(\Omega)}$, $m \in \mathbb{N}$,…

We present a new and direct proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below.

We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space $\R^N$, $N\geq4$. These domains arise as perturbations of complements of straight cylinders in $\R^N$, and by definition they support a positive…

{\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 the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where…

It is an open problem in general to prove that there exists a sequence of $\Delta_g$-eigenfunctions $\phi_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(\phi_{j_k}) $ of nodal domains tends to infinity with the eigenvalue.…

We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In $\R^{2}$ the domains we consider are the isosceles right triangle and the rectangle with edge ratio $\sqrt{2}$ (also known as the A4 paper). In…

We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $\Omega\subset\R^{n}$ whose boundary has an $(n-2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n-2}$, we prove that,…

This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel,…

The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are the interior zeros of $f'$. The Neumann domains of $f$ are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of…

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.

We obtain the simplicity of the first Neumann eigenvalue of convex thin domain with boundary in $R^n$ and compact thin manifolds with non-negative Ricci curvature. For convex thin domain in $R^2$, we get the simplicity of the first k…

In this paper, we establish a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on bounded domains of a complete, non-compact Riemannian manifold with non-negative Ricci curvature.

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet…

The question of unique continuation of harmonic functions in a domain $\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset $\omega$ of the…