相关论文: Closed nodal lines and interior hot spots of the s…

In this paper following the same methods in [M. Kadakal, O. Sh. Mukhtarov, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl., 54 (2007) 1367-1379] we investigate discontinuous two-point boundary value problems…

A simple sufficient condition on curved end of a straight cylinder is found that provides a localization of the principal eigenfunction of the mixed boundary value for the Laplace operator with the Dirichlet conditions on the lateral side.…

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…

We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain.…

We consider restrictions along closed geodesics and geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain a non-trivial bound on the L^2-norm of such restrictions as the…

In this paper, we study the restrictions of both the harmonic functions and the eigenfunctions of the symmetric Laplacian to edges of pre-gaskets contained in the Sierpinski gasket. For a harmonic function, its restriction to any edge is…

In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general…

Let $\Omega$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d\geq2$. Consider the Laplacian operator in $\Omega$ with Dirichlet and Neumann boundary conditions on opposite sides of $\Omega$. We prove some…

Let $ \Omega \subset R^2$ be a bounded piecewise smooth domain and $\phi_\lambda$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $\lambda^2$ and nodal set ${ N}_{\phi_{\lambda}} = {x \in \Omega; \phi_{\lambda}(x) = 0}.$ Let $H…

For the magnetic Laplacian on a bounded planar domain, imposing Neumann boundary conditions produces eigenvalues below the lowest Landau level. If the domain has two boundary components and one imposes a Neumann condition on one component…

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean…

We show there are no extremal metrics for the eigenvalues of the Neumann Laplacian on any compact manifold. Nonetheless, we construct examples of conformally extremal metrics for the eigenvalues of this operator in any annulus and…

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…

In this paper, we focus primarily on the symmetry properties of the second Neumann eigenfunction $u$ with respect to the symmetry axis or symmetry center of the relevant domain $Q$, such as isosceles trapezoids, parallelograms, kite…

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 study the first eigenvalue of the magnetic Laplacian with Neumann boundary conditions in the unit disk $\mathbb D$ in $\mathbb R^2$. There is a rather complete asymptotic analysis when the constant magnetic field tends to…

In this paper, we investigate on a bounded open set of $\mathbb{R}^N$ with smooth boundary, an eigenvalue problem involving the sum of nonlocal operators $(-\Delta)_p^{s_1}+ (-\Delta)_q^{s_2}$ with $s_1,s_2\in (0,1)$, $p,q\in (1,\infty)$…

Let $M$ denote a compact, connected Riemannian manifold of dimension $n\in{\mathbb N}$. We assume that $ M$ has a smooth and connected boundary. Denote by $g$ and ${\rm d}v_g$ respectively, the Riemannian metric on $M$ and the associated…

We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex…