Related papers: Fundamental gaps of spherical triangles

We generalize the Fenchel theorem to strong spacelike (which means that the tangent vector and the curvature vector span a spacelike 2-plane at each point) closed curves with index 1 in the 3-dimensional Lorentz space, showing that the…

We study the counting function of rational approximations with given bounds on the denominator and satisfying the critical Dirichlet exponent on the sphere $S^d$, $d\geq 3$. We give an effective estimate for this counting function, with an…

The structure of the spectrum of the three-dimensional Dirichlet Laplacian in the 3D polyhedral layer of fixed width is studied. It appears that the essential spectrum is defined by the smallest dihedral angle that forms the boundary of the…

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…

In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius $R$ and outer…

This paper is concerned with the Dirichlet eigenvalue problem associated to the $\infty$-Laplacian in metric spaces. We establish a direct PDE approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without…

Let alpha be an arbitrary angle in a random spherical triangle Delta and a be the side opposite alpha. (The sphere has radius 1; vertices of Delta are independent and uniform.) If some other side is constrained to be pi/2, then…

We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any…

We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {\it Dirichlet…

We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function nondecreasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. This includes, in…

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect…

We consider second order elliptic systems of partial differential equations subject to Dirichlet and Neumann boundary conditions. We prove analyticity of the elementary symmetric functions of the eigenvalues, and compute Hadamard-type…

The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a…

We discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $\Omega$}, & U|_{\partial\Omega}&=\phi. \end{eqnarray*} Here $\Omega$…

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than…

We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation…

We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to 1/2. The absence of…

We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical…

It is commonly believed that the normalized gaps between consecutive ordinates $t_n$ of the zeros of the Riemann zeta function on the critical line can be arbitrarily large. In particular, drawing on analogies with random matrix theory, it…