The L^p Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

2011/01/27 by Martin Dindoš, Martin Dindos, Dindos, Martin +2
Computer Science · Mathematics · #35J25 #42B37 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35J25 #msc:42B37

paper · pdf · doi:10.48550/arxiv.1101.5389

19 pages

arxiv created 2011/01/27 · openalex publication_date 2011/01/27 · arxiv updated 2011/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish Dahlberg's perturbation theorem for non-divergence form operators L = A∇². If L₀ and L₁ are two operators on a Lipschitz domain such that the L^p Dirichlet problem for the operator L₀ is solvable for some p in (1,∞) and the coefficients of the two operators are sufficiently close in the sense of Carleson measure, then the L^p Dirichlet problem for the operator L₁ is solvable for the same p. This is an improvement of the A_∞ version of this result proved by Rios in "The L^p Diriclet problem and nondivergence harmonic measure" (Trans. AMS 355, 2 (2003)). As a consequence we also improve a result from Dindos, Petermichl and Pipher, "The L^p Dirichlet problem for second order elliptic operators and a p-adapted square function" (J. Fun. Anal. 249 (2007)) for the L^p solvability of non-divergence form operators by substantially weakening the condition required on the coefficients of the operator. The improved condition is exactly the same one as is required for divergence form operators L = div A∇.

Related

• Perturbation theory for solutions to second order elliptic operators with complex coefficients and the L^p Dirichlet problem
• The L^p Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lamé system
• Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part
• The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients: The two-dimensional case
• Extrapolation for the L^p Dirichlet Problem in Lipschitz domains