2017/11/30 by Steve Hofmann, Phi Le, Phi Long Le +1
Mathematics · #Advanced Harmonic Analysis Research #Boundary (topology) #Boundary value problem #Bounded function #Combinatorics #Degenerate energy levels #Differential Equations and Boundary Problems #Dirichlet distribution #Dirichlet problem #Elliptic curve #Lebesgue integration #Lebesgue measure #Lipschitz continuity #Mathematical analysis #Mathematics #Measure (data warehouse) #Nabla symbol #Nonlinear Partial Differential Equations #Order (exchange) #Physics #Pure mathematics #Standard probability space #Upper and lower bounds #math.AP #msc:35J25 #msc:35J70 #msc:42B20 #msc:42B25
paper · pdf · doi:10.2140/apde.2019.12.2095
published as Analysis & PDE 12 (2019) 2095-2146 · 48 pages
openalex created_date 2017/12/04 · arxiv created 2018/12/06 · openalex publication_date 2019/10/28 · arxiv updated 2019/10/30 · openalex updated_date 2026/08/05
We prove that the Dirichlet problem for degenerate elliptic equations [math] in the upper half-space [math] is solvable when [math] and the boundary data is in [math] for some [math] . The coefficient matrix [math] is only assumed to be measurable, real-valued and [math] -independent with a degenerate bound and ellipticity controlled by an [math] -weight [math] . It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is in [math] with respect to the [math] -weighted Lebesgue measure on [math] . The Carleson measure estimate allows us to avoid applying the method of [math] -approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.