Related papers: Wolf-Keller theorem for Neumann eigenvalues

We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge…

A Faber-Krahn type argument gives a sharp lower estimate for the first Dirichlet eigenvalue for subdomains of wedge domains in spheres, generalizing the inequality in the plane, found by Payne and Weinberger. An application is an…

We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result…

We prove weighted isoperimetric inequalities for smooth, bounded, and simply connected domains. More precisely, we show that the moment of inertia of inner parallel curves for domains with fixed perimeter attains its maximum for a disk.…

We describe the region $\mathcal{V}(z_0)$ of values of $f(z_0)$ for all normalized bounded univalent functions $f$ in the unit disk $\mathbb{D}$ at a fixed point $z_0 \in \mathbb{D}$. The proof is based on identifying $\mathcal{V}(z_0)$ as…

For $N\geq 2$, a bounded smooth domain $\Omega$ in $\mathbb{R}^N$, and $g_0, V_0 \in L^1_{loc}(\Omega)$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: \begin{align*} -\Delta_p \phi + V…

We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a combination of Pleijel's original approach and…

We study a planar magnetic Schr\"odinger operator with an Aharonov-Bohm vector potential, under Neumann boundary conditions. Through a gauge transformation, the corresponding eigenvalue problem can be formulated in terms of the Laplacian on…

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity three. We also analyze…

We consider the eigenvalues of the magnetic Laplacian on a bounded domain $\Omega$ of $\mathbb R^2$ with uniform magnetic field $\beta>0$ and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy…

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…

Let $K$ be a compact convex domain in the Euclidean plane. The mixed area $A(K,-K)$ of $K$ and $-K$ can be bounded from above by $1/(6\sqrt{3})L(K)^2$, where $L(K)$ is the perimeter of $K$. This was proved by Ulrich Betke and Wolfgang Weil…

In this paper, we define a new capacity which allows us to control the behaviour of the Dirichlet spectrum of a compact Riemannian manifold with boundary, with "small" subsets (which may intersect the boundary) removed. This result…

We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential $V$ lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the $L^1-norm$ of $V$ is…

In this article, we study Steklov eigenvalues and mixed Steklov Neumann eigenvalues on a smooth bounded domain in $\mathbb{R}^{n}$, $n \geq 2$, having a spherical hole. We focus on two main results related to Steklov eigenvalues. First, we…

Inequalities are derived for sums and quotients of eigenvalues of magnetic Schroedinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit.

We consider the mixed Dirichlet-conormal problem on irregular domains in $\mathbb{R}^d$. Two types of regularity results will be discussed: the $W^{1,p}$ regularity and a non-tangential maximal function estimate. The domain is assumed to be…

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…

The magnetic Laplacian on a planar domain under a strong constant magnetic field has eigenvalues close to the Landau levels. We study the case when the domain is a disc and the spectrum consists of branches of eigenvalues of one dimensional…

In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result…