Carleson perturbations of locally Lipschitz elliptic operators

In 1-sided chord-arc domains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we demonstrate that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -absolute continuity of the elliptic measure with respect to the surface measure remains stable under <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Carleson perturbations. This stability holds provided that either the elliptic operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 0 equals minus d i v upper A 0 nabla"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi mathvariant="normal"> ∇ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L_0=-divA_0\nabla</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is being perturbed, or the perturbed operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1 equals minus d i v upper A 1 nabla"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi mathvariant="normal"> ∇ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L_1=-divA_1\nabla</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies the condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript upper X element-of normal upper Omega Endscripts d i s t left-parenthesis upper X comma partial-differential normal upper Omega right-parenthesis StartAbsoluteValue nabla upper A Subscript i Baseline left-parenthesis upper X right-parenthesis EndAbsoluteValue greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> </mml:mrow> </mml:munder> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sup _{X\in \Omega }dist(X,\partial \Omega )|\nabla A_i(X)| &gt;\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on its coefficients. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Carleson perturbations are slightly more general than those previously discussed in the literature. The proof hinges on the availability of a comprehensive elliptic theory and a domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that allows uniform non-tangential access to any point on its boundary. Consequently, while the current theory of <inline-formula content-t