Related papers: Variation of Green Functions: Normal Derivatives
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…
We study higher derivative extension of the functional renormalization group (FRG). We consider FRG equations for a scalar field that consist of terms with higher functional derivatives of the effective action and arbitrary cutoff…
Through this article we will use a notation \begin{equation}\label{alfaLap} T_{\alpha}u(x)=(1-|x|^2)\Delta u(x)+2 \alpha \langle x,\nabla u(x)\rangle + (n-2-\alpha) \alpha u(x). \end{equation} Here, $|x|<1$ and $\alpha>-1$. Also, for…
Consider a five-point discretization of a two-dimensional finite-gap for a fixed energy Schr\"{o}dinger operator. We construct the Green's function of the operator. In appears as the explicit formula in terms of the integral by the specific…
In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…
Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the…
We study scalar perturbations to a Robertson-Walker cosmological metric in terms of a pseudo-Newtonian potential, which emerges naturally from the solution of the field equations. This potential is given in terms of a Green function for…
We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the…
We investigate boundary estimates for elliptic operators with stationary random coefficients exhibiting integrable correlations, arising from stochastic homogenization theory. As practical applications, we establish decay estimates for…
A subordinate Brownian motion $X$ is a L\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\phi$ of the corresponding subordinator…
This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace…
In this paper, we present a new derivative via the Laplace transform. The Laplace transform leads to a natural form of the fractional derivative which is equivalent to a Riemann-Liouville derivative with fixed terminal point. We first…
We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a…
Let $L$ be a second-order, homogeneous, constant (complex) coefficient elliptic system in ${\mathbb{R}}^n$. The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the…
We explore the connections between Green's functions for certain differential equations, covariance functions for Gaussian processes, and the smoothing splines problem. Conventionally, the smoothing spline problem is considered in a setting…
The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we…
In the present paper, we consider an elliptic divergence form operator in the half-space and prove that its Green function is almost affine, or more precisely, that the normalized difference between the Green function and a suitable affine…
The species cutoff is a moduli-dependent quantity signaling the onset of quantum gravitational phenomena, whose form can be oftentimes determined from higher-derivative and higher-curvature corrections within low-energy gravitational EFTs.…
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the…
Energy-dependent Green's functions for the two and three dimensional $\delta$-function plus harmonic oscillator potential systems are derived by incorporating the renormalization and the self-adjoint extension into the Green's function…