Abstract
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 language | English |
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Pages (from-to) | 2095-2146 |
Number of pages | 52 |
Journal | Analysis and PDE |
Volume | 12 |
Issue number | 8 |
DOIs | |
Publication status | Published - 28 Oct 2019 |
Keywords
- 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
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
- Numerical Analysis