Related papers: Double-Layer Potentials for a Generalized Bi-Axial…

We study the existence of solutions $u:\R^{3}\to\R^{2}$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:\R^{2}\to\R$ is a double well symmetric potential. We…

We consider a possibly multiply connected bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},\alpha}$ for some $m\in {\mathbb{N}}$, $\alpha\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for…

This paper considers to the problems of diffraction of electromagnetic waves on a half-plane, which has a finite inclusion in the form of a Lipschitz curve. The diffraction problem formulated as boundary value problem for Helmholtz…

The purpose of this article is to provide a solution to the $m$-fold Laplace equation in the half space $R_+^d$ under certain Dirichlet conditions. The solutions we present are a series of $m$ boundary layer potentials. We give explicit…

In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function…

We present a simple and effective method for evaluating double-and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are…

In the present article for the generalized bi-axially symmetric multivariable Helmholtz equation four fundamental solutions are constructed in explicit form. Furthermore, some properties of these solutions are shown, which will be used for…

We consider an elliptic pseudo differential equation in a multi-dimensional cone and starting wave factorization concept we add some boundary conditions. For the simplest cases explicit formulas for solution are given like layer potentials…

Potential-based formulation with generalized Lorenz gauge can be used in the quantization of electromagnetic fields in inhomogeneous media. However, one often faces the redundancy of modes when finding eigenmodes from potential-based…

We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain…

For the structure of the thin electrical double layer~(EDL) and the property related to the EDL capacitance, we analyze boundary layer solutions (corresponding to the electrostatic potential) of a non-local elliptic equation which is a…

For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with…

In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates…

In models of phase coexistence, the precise form of the double-well potential is of central importance, yet it cannot be derived from first principles. In this paper, we investigate an inverse problem: starting from a prescribed transition…

Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…

In this paper, a Quadrature by Two Expansions (QB2X) numerical integration technique is developed for the single and double layer potentials of the Helmholtz equation in two dimensions. The QB2X method uses both local complex Taylor…

We consider a semilinear elliptic equation with a nonsmooth, locally \hbox{Lipschitz} potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our…

Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the…

We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For…