arXiv subjects

Phi Le

Publications and source records attributed to Phi Le.

3 recordsLinked to original sources

arXiv2018

Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

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 $n\geq2$ and the boundary data is in $L^p_μ(\mathbb{R}^n)$ 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 $μ$. 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 $A_\infty$ with respect to the $μ$-weighted Lebesgue measure on $\mathbb{R}^n$. 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.

math.AP↗

arXiv2016

BMO solvability and absolute continuity of harmonic measure

We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $Ω$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\infty$ property) of elliptic-harmonic measure with respect to surface measure on $\partial Ω$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $Ω$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, under the additional assumption that $Ω$ satisfies an interior Corkscrew condition, in the special case that $L$ is the Laplacian.

math.AP↗

arXiv2015

The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.

math.CA↗