In this thesis we study the $L^p$ Dirichlet problem for second order divergence-form operators having a BMO antisymmetric part. To be precise, for the divergence-form operator $L=-\divg A\nabla$, we assume that the symmetric part of the matrix $A$ is $L^\infty$ and elliptic, and that the anti-symmetric part of $A$ belongs to the space BMO. These operators are relevant to the study of incompressible flows. We show that the weak solution to $L$ is H\"older continuous near the boundary. This enables us to prove the existence of the elliptic measures associated to $L$ in non-tangentially accessible (NTA) domains; these general domains were introduced by Jerison and Kenig and include the class of Lipschitz domains. We also derive pointwise estimates for the elliptic measure, as well as its relation with the Green's function. Moreover, we are able to prove for these operators that the $L^p$ Dirichlet problem is solvable for $p$ sufficiently large in the upper half space, under the additional assumption that the coefficients are independent of the vertical variable. This result is equivalent to saying that the elliptic measure associated to $L$ belongs to the $A_\infty$ class with respect to the Lebesgue measure $dx$. We prove the $A_\infty$ condition by showing a Carelson measure estimate for the weak solution. Our method relies on kernel estimates and off-diagonal estimates for the semigourp $e^{-tL}$, solution to the Kato problem, and various estimates for the Hardy norms of certain commutators. This result extends the work of Hofmann, Kenig, Mayboroda and Pipher in 2015, which holds for elliptic operators in divergence form with non-symmetric, $L^\infty$ coefficients.
Li, Linhan,
"$L^p$ Dirichlet problem for second order elliptic operators having a BMO anti-symmetric part"
(2019).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/2jm0-xm59
