Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition

We consider the Cauchy-Neumann problem of the heat equation in the exterior domain of a ball in R^N, and study the movement of hot spots of the solution as t→∞.     

A mixed problem for the Laplace operator in a domain with moderately close holes

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ε and we define a perforated domain Ω_ε obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ε → 0. For ε small, we denote by u_ε the solution of a mixed problem for the Laplace equation in Ω_ε. We describe what happens to u_ε as ε → 0 in terms of real analytic maps and we compute an asymptotic expansion.     

Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge-Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method.     

An expansion theorem for an eigenvalue problem of an arbitrary multiply connected domain with an eigenparameter in a general type of boundary conditions

The object of this paper is to establish an expansion theorem for a regular right-definite eigenvalue problem with an eigenvalue parameter which is contained in the Schrödinger partial differential equation and in a general type of boundary conditions on the boundary of an arbitrary multiply connected bounded domain inR ⁿ (n2). We associate with this problem an essentially self-adjoint operator in a suitably defined Hilbert space and then we develop an associated eigenfunction expansion theorem.     

Hölder continuity of quasiconformal harmonic mappings from the unit ball to a spatial domain with C1 boundary

We prove that every quasiconformal mapping from the harmonic $\beta$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $\alpha$-Hölder continuous for $\alpha <1?\beta$, with the Hölder coefficient that does not depend neither on the mapping nor on $\beta$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $\alpha <1$. This is an approach towards the extension of some results from the complex plane obtained by Warschawski (1951) for conformal mappings and Kalaj (2022) for quasiconformal harmonic mappings.     

The Approximability Behaviour of Some Combinatorial Problems with Respect to the Approximability of a Class of Maximum Independent Set Problems

We prove that the existence of a polynomial time-approximation algorithm (where < 1 is a fixed constant)for a class of independent set problems, leads to a polynomial timeapproximation algorithm with approximation ratio strictly smallerthan 2 for vertex covering, while the non-existence of such analgorithm induces a lower bound on the ratio of every polynomialtime approximation algorithm for vertex covering. We also prove asimilar result for a (maximization) convex programming problemincluding quadratic programming as subproblem.     

Small ball probabilities for a Wiener process under weighted sup-norms,with an application to the supremum of bessel local times

LetR be the radial part of ad-dimensional Wiener process, starting from 0. In this paper, small ball probabilities are evaluated for sup_0<11(t ^–p R(t)) and sup _t 0(e ^–1 R(t)), withp[0, 1/2]. Chung's law of the iterated logarithm is established for the supremum of the local times of a two-dimensional Bessel process.     

A new approach to solve complex valued systems arising from the solution of Maxwell equations in the frequency domain through real‐equivalent formulations

The purpose of this article is to propose algebraic techniques to solve symmetric complex‐valued systems. In particular, we focus on the systems which arise from the modeling of electromagnetic fields in the frequency domain. These systems usually present a very ill‐conditioned number, so specific preconditioning techniques are supposed to be used. Finally, both analytical and numerical reference tests are proposed on the simple case of the magnetic field produced by a solenoid containing conductive or non‐conducting materials.