2025/09/22 by Tianyu Guan, Guan, Tianyu, Lihe Wang +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2509.17690
openalex publication_date 2025/09/22 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
In this paper, we study the boundary pointwise regularity for the Poisson problem on domains with rough boundaries, specifically uniform domains. In general, it is not straightforward to define weak solutions for non-zero boundary data on such domains. To address this, we introduce a novel definition of weak solutions tailored to the setting of uniform domains. Remarkably, this definition allows for the analysis of the regularity of weak solutions. In particular, by establishing an energy inequality, we prove the boundary pointwise Cα regularity by using compactness methods under the admissible condition. Furthermore, by exploiting the the linear structure of solutions with respective to the harmonic functions, we establish boundary pointwise C1,α and C2,α regularities when the boundary data and the domain boundary are pointwise C1,α and C2,α, respectively.