Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

Abstract

We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{div}(A\nabla u)=0$ in the upper half-space $(x,t)\in {\mathbb{R}}_{+}^{n+1}$ is solvable when $n\ge 2$ and the boundary data is in $L_{\mu }^p({\mathbb{R}}^n)$ for some $p<\infty$. The coefficient matrix $A$ is only assumed to be measurable, real-valued and $t$-independent with a degenerate bound and ellipticity controlled by an $A_2$-weight $\mu$. 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 $A_{\infty }$ with respect to the $\mu$-weighted Lebesgue measure on ${\mathbb{R}}^n$. The Carleson measure estimate allows us to avoid applying the method of $\epsilon$-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.

arXiv comment48 pages

References in corpus (4)

• The Green function estimates for strongly elliptic systems of second order
• The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^{\infty }$ coefficients
• A priori estimates for boundary value elliptic problems via first order systems
• Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators

Cited by in corpus (4)

• On the Kato problem and extensions for degenerate elliptic operators
• The Dirichlet problem for elliptic operators having a BMO anti-symmetric part
• The Green function with pole at infinity applied to the study of the elliptic measure
• Absolute continuity of degenerate elliptic measure