Green function for $T_α$-Laplacian in higher dimensions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929523955597312 |
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| author | Mateljević, M. Mutavdžić, N. Purtić, B. |
| author_facet | Mateljević, M. Mutavdžić, N. Purtić, B. |
| contents | Through this article we will use a notation
\begin{equation}\label{alfaLap} T_αu(x)=(1-|x|^2)Δu(x)+2 α\langle x,\nabla u(x)\rangle + (n-2-α) αu(x). \end{equation}
Here, $|x|<1$ and $α>-1$. Also, for $x=x(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$ we use $|x|=\sqrt{x_1^2+x_2^2+\ldots+x_n^2}, \nabla =(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\ldots,\frac{\partial}{\partial x_n}),Δ=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\ldots+\frac{\partial^2}{\partial x_n^2}.$
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01271 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Green function for $T_α$-Laplacian in higher dimensions Mateljević, M. Mutavdžić, N. Purtić, B. Complex Variables Analysis of PDEs Through this article we will use a notation \begin{equation}\label{alfaLap} T_αu(x)=(1-|x|^2)Δu(x)+2 α\langle x,\nabla u(x)\rangle + (n-2-α) αu(x). \end{equation} Here, $|x|<1$ and $α>-1$. Also, for $x=x(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$ we use $|x|=\sqrt{x_1^2+x_2^2+\ldots+x_n^2}, \nabla =(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\ldots,\frac{\partial}{\partial x_n}),Δ=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\ldots+\frac{\partial^2}{\partial x_n^2}.$ 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. |
| title | Green function for $T_α$-Laplacian in higher dimensions |
| topic | Complex Variables Analysis of PDEs |
| url | https://arxiv.org/abs/2410.01271 |