Green function for $T_\alpha$-Laplacian in higher dimensions

TL;DR

This paper derives an explicit Green function for the $T_\alpha$-Laplacian operator in higher dimensions and uses it to solve a non-homogeneous Dirichlet boundary value problem.

Contribution

It provides the first explicit formula for the Green function associated with the $T_\alpha$-Laplacian in higher dimensions and applies it to solve boundary value problems.

Findings

Explicit Green function formula derived for the $T_\alpha$-Laplacian.

Representation theorem for solutions to the boundary value problem.

Application to non-homogeneous boundary conditions.

Abstract

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 $x=x(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$ we use $|x|=\sqrt{x_1^2+x_2^2+\ldots+x_n^2}, \nabla =(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\ldots,\frac{\partial}{\partial x_n}),\Delta=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\ldots+\frac{\partial^2}{\partial x_n^2}.$ The purpose of this paper is to investigate a Dirichlet problem, corresponding to above mentioned PDE. We will specificaly consider non-homogenous boundary value problem. In that purpose the explicit formula for Green function assosiated to the operator (\ref{alfaLap}) will be calculated, and also, we will present the corresponding representation theorem.