Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

TL;DR

The paper develops a Fefferman–Kenig–Pipher–type perturbation theory for the $L^p$ Dirichlet problem of divergence-form elliptic operators with potentially unbounded antisymmetric parts in $BMO$, on bounded chord-arc domains. By introducing a generalized Carleson measure $d ilde ext{m}'(Z)=rac{eta_r(Z)^2}{\delta(Z)}dZ$ with $eta_r(Z)=(int_{B(Z,rac{\delta(Z)}{2})}|A_1-A_0|^r)^{1/r}$, the authors show that $\\\omega_0\,\in A_ ext{infty}(d\\sigma)$ implies \\omega_1\in A_ ext{infty}(d\\sigma)$, and that solvability of the $L^p$ Dirichlet problem for $L_0$ transfers to some $L^q$ Dirichlet problem for $L_1$ (with $q\\ge p$; equality when the Carleson norm is small). The approach extends the classical symmetric-case theory to non-symmetric, BMO-antisymmetric coefficients, and provides a concrete route to obtain $L^p$ solvability on Lipschitz domains under Carleson-type conditions on the coefficients. The results are built upon a careful decomposition of the boundary, a detailed analysis of the difference of solutions $F=u_1-u_0$ via the Green's function, and rigorous control of near and far contributions, culminating in a small-norm perturbation conclusion and an application to Lipschitz domains with Carleson-coefficient control.

Abstract

In the present paper we study perturbation theory for the $L^p$ Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $L_0 = \mbox{div}(A_0\nabla)$ and $L_1 = \mbox{div}(A_1\nabla)$ such that the $L^p$ Dirichlet problem for $L_0$ is solvable for some $p>1$; we show that if $A_0 - A_1$ satisfies certain Carleson condition, then the $ L^q$ Dirichlet problem for $L_1$ is solvable for some $q \geq p$. Moreover if the Carleson norm is small then we may take $q=p$. We use the approach first introduced in Fefferman-Kenig-Pipher '91 on the unit ball, and build on Milakis-Pipher-Toro '11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $L^p$ Dirichlet problem on a bounded Lipschitz domain for an operator $L = \mbox{div}(A\nabla)$, where $A$ satisfies a Carleson condition similar to the one assumed in Kenig-Pipher '01 and Dindoš-Petermichl-Pipher '07 but with unbounded antisymmetric part.

Theorems & Definitions (55)

• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.9
• Definition 2.1
• Definition 2.2
• Definition 2.3: NTA
• Definition 2.4: CAD
• Definition 2.5
• Proposition 2.6