Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2021
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| _version_ | 1866916230190858240 |
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| author | Franceschini, Federico Glaudo, Federico |
| author_facet | Franceschini, Federico Glaudo, Federico |
| contents | Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$ such that $Lu=δ_0$.
As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of $|x|$, plus homogeneous polynomials multiplied by $\log(|x|)$ if the dimension $n$ is even) which improves the classical result of F. John (1950). The control we have on the "complexity" of each homogeneous term is optimal and in particular, when $L$ is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to K. Kodaira (1949).
The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by $\mathbb Z^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_15104 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients Franceschini, Federico Glaudo, Federico Analysis of PDEs Classical Analysis and ODEs 35A08 (Primary) 33C55, 35C20, 05C22 (Secondary) Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$ such that $Lu=δ_0$. As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of $|x|$, plus homogeneous polynomials multiplied by $\log(|x|)$ if the dimension $n$ is even) which improves the classical result of F. John (1950). The control we have on the "complexity" of each homogeneous term is optimal and in particular, when $L$ is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to K. Kodaira (1949). The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by $\mathbb Z^2$. |
| title | Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients |
| topic | Analysis of PDEs Classical Analysis and ODEs 35A08 (Primary) 33C55, 35C20, 05C22 (Secondary) |
| url | https://arxiv.org/abs/2110.15104 |