Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the Lp Dirichlet Problem

• Published in: Acta mathematica Sinica. English series Vol. 35; no. 6; pp. 749 - 770
• Main Authors: Dindoš, Martin, Pipher, Jill
• Format: Journal Article
• Language: English
• Published: Beijing Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.06.2019
• Subjects: Mathematics; Mathematics and Statistics; Complex coefficients elliptic PDEs; perturbation theory; 35J25; 35J57; boundary value problems
• DOI: 10.1007/s10114-019-8214-y
• ISSN: 1439-8516

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L 0 = div A 0 ( x )∇ + B 0 ( x ). ∇ is a p -elliptic operator satisfying the assumptions of Theorem 1.1 then the L p Dirichlet problem for the operator L 0 is solvable in the upper half-space ℝ + n . In this paper we prove that the L p solvability is stable under small perturbations of L 0 . That is if L 1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L 0 and L 1 are sufficiently close in the sense of Carleson measures, then the L p Dirichlet problem for the operator L 1 is solvable for the same value of p. As a corollary we obtain a new result on L p solvability of the Dirichlet problem for operators of the form L = div A ( x )∇ + B ( x ) · ∇ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiate and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoš, Petermichl and Pipher.