2023/11/06 by Mingming Cao, Cao, Mingming, Pablo Hidalgo-Palencia +7 · 1 citation
Computer Science · Mathematics · #26A16 #31B05 #31B25 #35B65 #35J25 (Primary) #42B35 (Secondary) #42B37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2311.03270
openalex publication_date 2023/11/06 · openalex created_date 2023/11/08 · openalex updated_date 2026/08/01
In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets Ω⊂ ℝn+1, n ≥ 2, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if Ω is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when Ω is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition. As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems. All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.