Related papers: Construction and properties for the Green's functi…
Green's functions for Neumann boundary conditions have been considered in Math Physics and Electromagnetism textbooks, but special constraints and other properties required for Neumann boundary conditions have generally not been noticed or…
On a bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 2$, we consider existence, uniqueness and "regularity" issues for the Green function $G_\lambda$ of the quasi-linear operator $u \to -\Delta_p u-\lambda |u|^{p-2}u$ with $1<p \leq…
The paper treats boundary value problems for the fractional Laplacian $(-\Delta )^a$, $a>0$, and more generally for classical pseudodifferential operators ($\psi $do's) $P$ of order $2a$ with even symbol, applied to functions on a smooth…
We show existence, uniqueness and positivity for the Green's function of the operator $(\Delta_g + \alpha)^k$ in a closed Riemannian manifold $(M,g)$, of dimension $n>2k$, $k\in \mathbb{N}$, $k\geq 1$, with Laplace-Beltrami operator…
A new perspective of the Green's function in a boundary value problem as the only eigenstate in an auxiliary formulation is introduced. In this treatment, the Green's function can be perceived as a defect state in the presence of a…
In this paper we will deduce several properties of the Green's functions related to the Hill's equation coupled to various boundary value conditions. In particular, the idea is to study the Green's functions of the second order differential…
In this paper we will show several properties of the Green's functions related to various boundary value problems of arbitrary even order. In particular, we will write the expression of the Green's functions related to the general…
We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=\Omega\times (-\infty,\infty)$, where $\Omega\subset \mathbb{R}^n$…
The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables…
We construct the Neumann Green function and establish scale invariant regularity estimates for solutions to the Neumann problem for the elliptic operator $Lu=-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c} \cdot \nabla u+du$…
The subject of the present paper is the phenomenon of vanishing of the Green function of the operator $-\Delta + V$ on $\mathbb R^3$ at the points where a potential $V$ has positive critical singularities. More precisely, imposing minimal…
We prove a regularity result for Green's functions that are associated to elliptic second order divergence-type linear PDO's with coefficients in C^{1,\alpha}(\bar{\Omega}). Here \alpha\in (0,1) and \Omega\subset \R^n is a bounded…
We construct an expression for the Green function of a differential operator satisfying nonlocal, homogeneous boundary conditions starting from the fundamental solution of the differential operator. This also provides the solution to the…
In this paper we obtain the explicit expression of the Green's function related to a general $n$ order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, a $n$ dimensional parameter…
Using Gegenbauer polynomials and the zonal harmonic functions we build an explicit representation formula for the Green function with Neumann boundary conditions in the annulus.
We construct Green's functions for second order parabolic operators of the form $Pu=\partial_t u-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c} \cdot \nabla u+du$ in $(-\infty, \infty) \times \Omega$, where $\Omega$ is an open…
The aim of this paper is to show certain properties of the Green's functions related to the Hill's equation coupled with different two point boundary value conditions. We will obtain the expression of the Green's function of Neumann,…
We construct Green's functions for elliptic operators of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$ in domains $\Omega\subseteq\mathbb R^n$, under the assumption $d\geq\text{div}b$, or $d\geq\text{div}c$. We show that,…
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional…
We prove that for an open domain $D \subset \mathbb{R}^d $ with $d \geq 2 $ , for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \in D$ , there exists a unique Green's function centred in $ y $…