Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

Abstract:

Let $\Omega \subset \mathbb {R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain; that is, a domain which satisfies interior corkscrew and Harnack chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path connectedness), and whose boundary $\partial \Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators, and let $\omega _{L_0}$, $\omega _L$ be the associated elliptic measures. We show that if $\omega _{L_0}\in A_\infty (\sigma )$, where $\sigma =H^n{\left |_{ {\partial \Omega }}\right .}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega _L\in A_\infty (\sigma )$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse Hölder classes; that is, if for some $1<p<\infty$, one has $\omega _{L_0}\in RH_p(\sigma )$, then $\omega _{L}\in RH_p(\sigma )$. Equivalently, if the Dirichlet problem with data in $L^{p’}(\sigma )$ is solvable for $L_0$, then it is for $L$ also. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg; Fefferman, Kenig, and Pipher; and Milakis, Pipher, and Toro in more benign settings. As a consequence of our methods, we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $A_\infty (\sigma )$, then necessarily $\Omega$ is in fact a nontangentially accessible domain (and hence chord-arc), and therefore its boundary is uniformly rectifiable.