Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators

Hofmann, Steve¹ ; Kenig, Carlos² ; Mayboroda, Svitlana³ ; Pipher, Jill⁴

¹ Department of Mathematics, University of Missouri, Columbia, Missouri 65211
² Department of Mathematics, University of Chicago, Chicago, Illinois, 60637
³ School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
⁴ Department of Mathematics, Brown University, Providence, Rhode Island 02912

Journal of the American Mathematical Society, Tome 28 (2015) no. 2, pp. 483-529

Résumé

We consider divergence form elliptic operators $L= {-}\mathrm {div} A(x) \nabla$, defined in the half space $\mathbb {R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A(x)$ is bounded, measurable, uniformly elliptic, $t$-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation $Lu=0$, and we then combine these estimates with the method of “$\epsilon$-approximability” to show that $L$-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class $A_\infty$ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in $L^p$, for some $p\infty$). Previously, these results had been known only in the case $n=1$.

Détail

DOI : 10.1090/S0894-0347-2014-00805-5

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