Related papers: Exact Poincar\'e Constants in three-dimensional An…
We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by…
Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these…
In the present paper, the simplest scalar ODE case is studied for polynomials $$ \dot{w}=f(w)=(w-e_0)\cdot\ldots\cdot(w-e_{d-1}) $$ of degree $d$ with $d$ simple complex zeros. The explicit solution by separation of variables and explicit…
We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface $\mM $ with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$…
We analyze the gravity-induced effects associated with a massless scalar field in a higher-dimensional spacetime being the tensor product of $(d-n)$-dimensional Minkowski space and $n$-dimensional spherically/cylindrically-symmetric space…
The eigenvalue problem for the p-Laplace operator with p>1 on planar domains with the zero Dirichlet boundary condition is considered. The Constrained Descent Method and the Constrained Mountain Pass Algorithm are used in the Sobolev space…
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 consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $\Omega$. The considered problems are well studied for the case when $\Omega$ is a unit disc, but barely studied for…
In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $\lambda_{F}(p,\Omega)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal…
Physical properties of gravitational instantons which are derivable from minimal surfaces in 3-dimensional Euclidean space are examined using the Newman-Penrose formalism for Euclidean signature. The gravitational instanton that corresponds…
We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear…
Proposed is a new approach to finding exact solutions of nonlinear $p$-brane equations in $D$-dimensional Minkowski space based on the use of various initial value constraints. It is shown that the constraints $\Delta^{(p)}\vec{x}=0$ and…
We obtain optimal estimates of the Poincar\'e constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincar\'e constant is determined by the second largest end. The proof is based on the argument by…
In a recent paper (Phys. Rev. D 94, 125016 (2016)), the authors argued that the singularities of the two-point functions on the Poincar\'e domain of the $n$-dimensional anti-de Sitter spacetime ($\text{PAdS}_n$) have the Hadamard form,…
In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter $D$ of sphere $\mathbb S^n$ is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$. We…
We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an…
We consider the Neumann-Poincar\'e operator on a three-dimensional axially symmetric domain which is generated by rotating a planar domain around an axis which does not intersect the planar domain. We investigate its spectral structure when…
We discuss the local behaviour of vector fields in the plane $\R^2$ around a regular singular point, using recently introduced reduced normal forms, i.e. Poincar\'e and Lie renormalized forms [{\it Lett. Math. Phys.} {\bf 42} (1997),…
This paper concerns the eigenvalues of the Neumann-Poincar\'e operator, a boundary integral operator associated with the harmonic double-layer potential. Specifically, we examine how the eigenvalues depend on the support of integration and…
This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's…