Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form

[2505.03137] Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2505.03137 (math) [Submitted on 6 May 2025 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form Authors: Hongjie Dong , Dong-ha Kim , Seick Kim View PDF HTML (experimental) Abstract: We study the Dirichlet problem for second-order elliptic operators in double divergence form, which arise as formal adjoints of non-divergence form operators and include the stationary Fokker-Planck-Kolmogorov equation. Assuming that the leading coefficients have Dini mean oscillation and that the lower-order coefficients satisfy natural integrability conditions, we construct the Perron solution in arbitrary bounded domains. We prove that a boundary point is regular with respect to the operator if and only if it satisfies the classical Wiener criterion for the Laplacian. In particular, the Dirichlet problem is uniquely solvable in every bounded domain that is regular for the Laplacian. Comments: 29 pages Subjects: Analysis of PDEs (math.AP) MSC classes: 35J25, 35J67 Cite as: arXiv:2505.03137 [math.AP] (or arXiv:2505.03137v3 [math.AP] for this version) https://doi.org/10.48550/arXiv.2505.03137 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Seick Kim [ view email ] [v1] Tue, 6 May 2025 03:24:32 UTC (23 KB) [v2] Wed, 1 Jul 2026 10:48:09 UTC (24 KB) [v3] Thu, 27 Aug 2026 05:26:00 UTC (24 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2025-05 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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