Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

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Colleges, School and Institutes


We prove that the Dirichlet problem for degenerate elliptic equations div(A∇u)=0 in the upper half-space (x,t)∈Rn+1+ is solvable when n≥2 and the boundary data is in Lpμ(Rn) for some p<∞. The coefficient matrix A is only assumed to be measurable, real-valued and t-independent with a degenerate bound and ellipticity controlled by an A2-weight μ. 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 A∞ with respect to the μ-weighted Lebesgue measure on Rn. The Carleson measure estimate allows us to avoid applying the method of ϵ-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.


Original languageEnglish
Pages (from-to)2095-2146
Number of pages52
JournalAnalysis and PDE
Issue number8
Publication statusPublished - 28 Oct 2019


  • square functions, nontangential maximal functions, harmonic measure, Radon–Nikodym derivative, Carleson measure, divergence form elliptic equations, Dirichlet problem, A2 Muckenhoupt weights, reverse Hölder inequality, Harmonic measure, Radon-Nikodym derivative, Square functions, Divergence form elliptic equations, Nontangential maximal functions, Reverse Hölder inequality, A Muckenhoupt weights