{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/expansion-of-the-fundamental-solution-of-a","title":"Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients","arxiv_id":"2110.15104","date":"2021-10-28","proceeding":null,"authors":["Federico Franceschini","Federico Glaudo"],"abstract":"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 $Lu=\\delta_0$. 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 $\\mathbb Z^2$.","url_abs":"https://arxiv.org/abs/2110.15104v2","url_pdf":"https://arxiv.org/pdf/2110.15104v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"expansion-of-the-fundamental-solution-of-a","repo_url":"https://github.com/dario2994/fundamental_solution_expansion","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}