Let $$\Omega $$ be a bounded domain in $${\mathbb {R}}^n$$ with $$C^{1}$$ boundary and let $$u_\lambda Dirichlet Laplace eigenfunction eigenvalue $$\lambda . We show that the $$(n-1)$$ -dimensional Hausdorff measure of zero set does not exceed $$C(\Omega )\sqrt{\lambda }$$ This result is new even for case domains $$C^\infty -smooth boundary.

This paper is devoted to the spectral analysis of the Laplacian with constant magnetic field on a cone of aperture α and Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit α → 0 and establish a full asymptotic...

We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of fundamental solution to heat equation with a compactly supported potential. It is assumed that corresponding stationary operator has at least one positive eigenvalue. Two regions different types behavior are distinguished: inside certain conical surface in $(t,x)$ space, asymptotics determined by principal eigenvalue and eigenfuncti...

The use of numerical tools to solve challenging problems in mathematics has exploded in the past several decades. The purpose of this paper is to compare the results of two different types of numerical methods in finding solutions to the eigenvalue problem for a second-order elliptic differential equation subject to boundary and transmission conditions. The transmission conditions result from j...

This paper is devoted to the spectral analysis of the Laplacian with constant magnetic field on a cone of aperture α and Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit α → 0 and establish a full asymptotic...

We consider a reaction-diffusion-advection equation with a nonlinearity of the KPP type in a cellular flow. We show that the minimal pulsating traveling front speed c∗(A) in a flow of amplitude A satisfies the upper and lower bounds C1A ≤ c∗(A) ≤ C2A for A 1. We also analyze a related eigenvalue problem and establish an “averaging along the streamlines” principle for the positive eigenfunction ...

The simulation of drift instabilities in the plasma edge leads to cubic polynomial PDE eigenvalue problems with parameter dependent coefficients. The aim is to determine the wave number which leads to the maximum growth rate of the amplitude of the wave. This requires the solution of a large number of PDE eigenvalue problems. Since we are only interested in a smooth eigenfunction corresponding ...

Let m 2 Lloc(R N ); 0 6= m+ in Kato’s class. We investigate the spectral function 7! s( + m)where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m is sufficiently large, we show that there exists a unique 1 > 0 such that s( + 1m) = 0. Under suitable conditions on m it follows that 0 is an eigenvalue of + 1m wi...

Let $D \subset \mathbb{R}^d$ be a bounded, connected domain with smooth boundary, and let $-\Delta u = \mu\_1 u$ the first nontrivial eigenfunction of Laplace operator Neumann boundary conditions. We prove that $$ \max\_{x \in D} \~u(x) \leq 60 \cdot \partial we emphasize this constant is uniform among all domains in dimensions. In particular, hot spots conjecture cannot fail by an arbitrarily ...