相关论文: Growth rates of Laplace eigenfunctions on the unit…

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq…

Let $X$ be a compact Riemann surface and let $L$ be a positive line bundle on $X$. We obtain the growth speed of unit ball volume in $H^0(X,L^n)$ towards the energy at equilibrium. As an application, we also obtain the speed of Fekete…

The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size {\epsilon} are derived. An explicit formula to compute the first term from…

We establish asymptotic bounds on the L^p norms of spectrally localized functions in the case of two-dimensional Dirichlet forms with coefficients of Lipschitz regularity. These bounds are new for the range p>6. A key step in the proof is…

The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the…

In this work we study the homogenization for eigenvalues of the fractional $p-$Laplace in a bounded domain both with Dirichlet and Neumann conditions. We obtain the convergence of eigenvalues and the explicit order of the convergence rates.

Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet $L$-functions to a large prime modulus $q$. As applications, we give alternative proofs of…

In this paper, we investigate properties of classes of functions related to certain elliptic operators. Firstly, we prove that a main result of Dyakonov (Acta Math. 178(1997), 143--167) on analytic functions can be extended to this more…

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…

In this paper, we investigate some properties to solutions $f$ to the Yukawa PDE: $\Delta f=\lambda f$ in the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$, where $\lambda$ is a nonnegative constant. First, we prove that the answer to an open…

A class of subharmonic functions are proved to have the growth estimates $u(x)= o(x_n^{1-\frac{\alpha}{p}}|x|^{\frac{\gamma}{p}+\frac{n-1}{q}-n+\frac{\alpha}{p}})$ at infinity in the upper half space of ${\bf R}^{n}$, which generalizes the…

For any Riemannian metric $ds^2$ on a compact surface of genus $g$, Yang and Yau proved that the normalized first eigenvalue of the Laplacian $\lambda_1(ds^2)Area(ds^2)$ is bounded in terms of the genus. In particular, if $\Lambda_1(g)$ is…

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.…

Denote by $C^{\alpha}(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are H\"older continuous with the exponent $\alpha$, and denote by $C^{1, \alpha}(\mathbb{D})$ the space which consists of differentiable…

We consider the spectrum of the Laplace operator on 3D rod structures, with a small cross section depending on a small parameter $\varepsilon$. The boundary conditions are of Dirichlet type on the basis of this structure and Neumann on the…

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\_1(D)$ and $c\_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c\_1(D)\sqrt{\lambda}\|\phi\|\_\infty \le…

We study the growth of Laplacian eigenfunctions $ -\Delta \phi_k = \lambda_k \phi_k$ on compact manifolds $(M,g)$. H\"ormander proved sharp polynomial bounds on $\| \phi_k\|_{L^{\infty}}$ which are attained on the sphere. On a `generic'…

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…

We obtain the sharp version of the uncertainty principle recently introduced in [47], and improved by [13], relating the size of the zero set of a continuous function having zero mean and the optimal transport cost between the mass of the…

Given a bounded domain $\Omega$ in $\mathbb{R}^N$, $N\geq 1$ we study the asymptotic behavior as $\varepsilon \to 0$ of the eigencurves of $$ -\Delta_p u_\varepsilon=\alpha_\varepsilon m(\tfrac{x}{\varepsilon})(u_\varepsilon^+ )^{p-1} -…