Related papers: Courant-sharp eigenvalues of the three-dimensional…

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed guaranteed computation methods for both eigenvalues and…

In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally…

In this paper, we study $n$-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an $n$-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with $f_3$ constant is isometric to the…

In the last decades, many mathematicians have studied the curl operator in compact three-manifolds , mainly the structure of its spectrum and some isoperimetric problems associated with it. In this paper, we will see that all the compact…

One isoperimetric inequality for the fundamental sloshing eigenvalue is derived under the assumption that containers have vertical side walls and either finite or infinite depth. It asserts that among all such containers, whose free…

This paper is a continuation and an extension of our recent work [3] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In [3], the analytic behaviour of the Laplacian…

It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which…

We prove that every metric graph which is a tree has an orthonormal sequence of Laplace-eigenfunctions of full support. This implies that the number of nodal domains $\nu_n$ of the $n$-th eigenfunction of the Laplacian with standard…

We study an eigenvalue problem for the infinity-Laplacian on bounded domains. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The work also contains existence results when the parameter, in the…

We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…

We introduce an analogue of Payne's nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of…

The lower-order cr-invariant variational problem for Legendrian curves in the 3-sphere is studied and its Euler-Lagrange equations are deduced. Closed critical curves are investigated. Closed critical curves with non-constant cr-curvature…

We investigate the overdetermined torsion problem $\begin{cases} -\Delta u = 1 & \text{in}\ \Omega\\ u=0 & \text{on}\ \partial \Omega\\ \frac{\partial u}{\partial \nu}=\text{const.} & \text{on}\ \partial \Omega, \end{cases}$ where $\Omega$…

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with…

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.)…

In this study, we address the eigenvalue problem given by: \begin{equation*} \begin{cases} -\Div (w\nabla u_i)=\la_iu_i &\text{in } \Om\subset \mathbb{R}^n,\\ u_i=0 &\text{on } \pt \Om, \end{cases} \end{equation*} where $w > 0$ within $\Om$…

We prove quantitative equidistribution properties for orthonormal bases of eigenfunctions of the Laplacian on the rational $d$-torus. We show that the rate of equidistribution of such eigenfunctions is of polynomial decay. We also prove…

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means…

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-\'Emery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian…

Using the multilinear theory of oscillatory integral operators, we establish a square function inequality with implications to moment inequalities for discrete exponential sums which frequencies lie on curved hyper-surfaces. In particular a…