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The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local)…
In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that…
In this paper we use the method of layer potentials to study $L^2$ boundary value problems in a bounded Lipschitz domain $\Omega$ for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the…
Let ${\frak M}^\alpha$ be the spherical maximal operators of complex order $\alpha$ on ${\mathbb R^n}$. In this article we show that when $n\geq 2$, suppose \begin{eqnarray*} \|{\frak M}^{\alpha} f \|_{L^p({\mathbb R^n})} \leq C\|f…
It is a classical fact in Euclidean geometry that the regular polygon maximizes area amongst polygons of the same perimeter and number of sides, and the analogue of this in non-Euclidean geometries has long been a folklore result. In this…
We prove Lieb-Thirring-type bounds for fractional Schr\"odinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of…
We prove that the geodesic disks are the unique maximisers of the first non-trivial Neumann eigenvalue among all simply connected domains of the sphere $\mathbb S^2$ with fixed area.
In this paper, we begin by showing a new generalization of the celebrated Cauchy-Schwarz inequality for the inner product. Then, this generalization is used to present some bounds for the Euclidean operator radius and the Euclidean operator…
We prove the existence of an open set $\Omega\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large…
Consider the class of optimal partition problems with long range interactions \[ \inf \left\{ \sum_{i=1}^k \lambda_1(\omega_i):\ (\omega_1,\ldots, \omega_k) \in \mathcal{P}_r(\Omega) \right\}, \] where $\lambda_1(\cdot)$ denotes the first…
We describe the generic behavior of the resonance counting function for a Schr\"odinger operator with a bounded, compactly-supported real or complex valued potential in $d \geq 1$ dimensions. This note contains a sketch of the proof of our…
The discrete Laplace operator on a triangulated polyhedral surface is related to geometric properties of the surface. This paper studies extremum problems for eigenvalues of the discrete Laplace operators. Among all triangles, an…
In this paper, we prove the uniform estimates for the resolvent $(\Delta - \alpha)^{-1}$ as a map from $L^q$ to $L^{q'}$ on real hyperbolic space $\mathbb{H}^n$ where $\alpha \in \mathbb{C}\setminus [(n - 1)^2/4, \infty)$ and $2n/(n + 2)…
We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the…
We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack…
We establish sharp Trudinger-Moser inequalities with logarithmic weights for the $k$-Hessian equation and investigate the existence of maximizers. Our analysis extends the classical results of Tian and Wang to $k$-admissible function spaces…
This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this…
We consider a class of Hilbert-Schmidt integral operators with an isotropic, stationary kernel acting on square integrable functions defined on flat tori. For any fixed kernel which is positive and decreasing, we show that among all…
In this article, we establish the existence of a norm-one projection from the space of all \emph{two-Lipschitz} operators onto the space of all bounded bilinear operators under certain conditions on the corresponding codomain spaces, using…
The goal of this note is to establish non-tangential convergence results for Schr\"{o}dinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the…