Perturbations of elliptic operators in chord arc domains (Q2875851)

Let \(\Omega \subset \mathbb{R}^n\) be a bounded domain and \(L\) an elliptic operator of the form \(Lu = \text{div}(A \nabla u)\) for some bounded and strictly positive symmetric matrix \(A\) with measurable coefficients. Suppose further that \(\Omega\) is regular for \(L\), that is for every continuous boundary data \(g \in C(\partial \Omega)\) the generalized solution \(u\) of the problem \(Lu = 0\) with Dirichlet boundary condition \(u = g\) on \(\partial \Omega\) satisfies \(u \in C(\overline{\Omega})\). It then follows from the Riesz Representation Theorem that for every \(x \in \Omega\) there exists a regular Borel probability measure \(\omega_L^x\) such that the solution of the Dirichlet problem for \(L\) with boundary data \(g\) is given by NEWLINE\[NEWLINEu(x) = \int_{\partial \Omega} g(q) \, d\omega^x_L(q).NEWLINE\]NEWLINE One calls \(\omega^x_L\) the \(L\)-elliptic measure of \(\Omega\) with pole \(x\). Let \(\sigma\) be the normalized surface measure with respect to \(\Omega\) and let \(k^x\) denote the Radon--Nikodým derivative of \(\omega^x_L\) with respect to \(\sigma\) (if it exists).NEWLINENEWLINEEscauriaza showed in [\textit{L. Escauriaza}, Isr. J. Math. 94, 353--366 (1996; Zbl 0853.35047)] that if \(\Omega \subset \mathbb{R}^n\) is a Lipschitz domain and if for two elliptic operators \(L_0\) and \(L_1\) with associated derivatives \(k_0^x\) and \(k_1^x\) as above \(L_1\) is a small perturbation of \(L_0\) (more precisely, the perturbation has a vanishing Carleson constant), then \(\log k_0^x \in VMO(\sigma)\)---the space of functions with vanishing mean oscillation---if and only if \(\log k_1^x \in VMO(\sigma)\).NEWLINENEWLINEIn the main result of this article the authors generalize this result to more general class of chord arc domains.NEWLINENEWLINEFor the entire collection see [Zbl 1285.00036].