Related papers: Eigenfunction localization and nodal geometry on d…
Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…
In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general…
We study the bottom of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_x^2 + \xi$ on $[0,L]$ driven by a white noise $\xi$ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as…
The article [HPS] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering…
We show that the location of the first singularity of the Upsilon function of an algebraic knot is determined by the first term of its Puiseux characteristic sequence. In many cases this gives better bounds than the tau invariant on the…
It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method,…
We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions,…
In this paper, we construct a dumbbell domain for which the associated principle $\infty$-eigenvalue is not simple. This gives a negative answer to the outstanding problem posed by Juutinen-Lindquivst-Manfredi ("The $\infty$-eigenvalue…
We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary--the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_\lambda$ be its nodal set. Assume that zero is a regular value of…
In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane.
Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture. We…
For a bounded Lipschitz domain $\Sigma$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that $$\mu_{3-\beta_1} \leq \lambda_{1},$$ where $\mu_k$ ($\lambda_k$ resp.) is the $k$-th Neumann (Dirichlet resp.)…
We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain. The technique is based on the `mirror coupling' of reflected Brownian motions.
We show that if $u$ is a solution to a linear elliptic differential equation of order $2m\geq 2$ in the half-space with $t$-independent coefficients, and if $u$ satisfies certain area integral estimates, then the Dirichlet and Neumann…
We consider the eigenvalues of the Laplacian on an open, bounded, connected set in $\mathbb{R}^n$ with $C^2$ boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a…
We consider two-dimensional waveguide with a rectangular obstacle symmetric about the axis of the waveguie. We study the behaviour of the Neumann eigenvalues located below the first threshold when the sides of the obstacle approach the…
In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM…
We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions or combinations of either for different parts…
In this paper, we investigate the monotonicity of the first Steklov--Dirichlet eigenvalue on eccentric annuli with respect to the distance, namely $t$, between the centers of the inner and outer boundaries of an annulus. We first show the…
The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields and Neumann realization in polyhedral domains is characterized by a hierarchy of model problems. We investigate properties of the…