Related papers: On principal frequencies and inradius in convex se…

In this paper, we deal with the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\Omega_r = \Omega_0 \setminus \overline{B}_r$, where $\Omega_0 \subset \mathbb{R}^n$, $n \geq 2$, is an…

The Faber-Krahn deficit $\delta\lambda$ of an open bounded set $\Omega$ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $\Omega$ and on the ball having same measure as $\Omega$. For any…

In this paper, by mainly using the rearrangement technique and suitably constructing trial functions, under the constraint of fixed weighted volume, we can successfully obtain several isoperimetric inequalities for the first and the second…

We establish an explicit lower bound of the first eigenvalue of the Laplacian on K\"ahler manifolds based off the comparison results of Li and Wang. The lower bound will depend on the diameter, dimension, holomorphic sectional curvature and…

This paper is about a shape optimization problem related to the Dirichlet Laplacian eingevalues in the Euclidean plane. More precisely we study the shape of the minimizer in the class of open sets of constant width. We prove that the disk…

We establish a lower bound for the principal $p-$frequency $\lambda_{1,p}(\Omega)$ on a bounded domain $\Omega$ in a non-compact Riemannian manifold of dimension $n.$ Under the assumption that the Ricci curvature satisfies…

We discuss several properties of eigenvalues and eigenfunctions of the $p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which…

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality,…

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean…

Our main result is an abstract good-$\lambda$ inequality that allows us to consider three self-improving properties related to oscillation estimates in a very general context. The novelty of our approach is that there is one principle…

We give a new lower bound for the first gap $\lambda_2 - \lambda_1$ of the Dirichlet eigenvalues of the Schr{\"o}dinger operator on a bounded convex domain $\Omega$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new…

In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature…

The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to…

We use the Lusternik-Schnirelman theory to prove the existence of a nondecreasing sequence of variational eigenvalues for the subelliptic $p$-Laplacian subject to the Dirichlet boundary condition.

We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy-Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary $p$-Laplacian, but extending it at a wide…

We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean…

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schr\"odinger operator with convex potential and Dirichlet boundary data on a convex domain is…

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$,…

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--\'{E}mery…

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately…