Periodic homogenization and harmonic measures

2025/04/24 by Guy David, David, Guy, Antoine Gloria +5 · 1 citation
Computer Science · #35B27 #35J25 #42B37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2504.17396

openalex publication_date 2025/04/24 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/30

Abstract

Since the seminal work of Kenig and Pipher, the Dahlberg-Kenig-Pipher (DKP) condition on oscillations of the coefficient matrix became a standard threshold in the study of absolute continuity of the harmonic measure with respect to the Hausdorff measure on the boundary. It has been proved sufficient for absolute continuity in the domains with increasingly complex geometry, and known counterexamples show that in a certain sense it is necessary as well. In the present note, we introduce into the subject ideas from homogenization theory to exhibit a new class of operators for which the elliptic measure is well-behaved, featuring the coefficients violating the DKP condition, and on the contrary, oscillating so quickly, that the homogenization takes place.

Citations

• Failure to slide: a brief note on the interplay between the Kenig-Pipher condition and the absolute continuity of elliptic measures
• A regularity theory for random elliptic operators
• Layer Potential Methods for Elliptic Homogenization Problems
• Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains
• The Lp Dirichlet problem for second order elliptic operators and a p-adapted square function
• An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
• A New Approach to Absolute Continuity of Elliptic Measure, with Applications to Non-symmetric Equations
• Homogenization of Differential Operators and Integral Functionals

Cited by

• The L^p regularity problem for parabolic operators with transversally independent coefficients

Related

• Lipschitz regularity of homogenization with continuous coefficients: Dirichlet problem
• Homogenization of the Oscillating Dirichlet Boundary Condition in General Domains
• Lectures on Periodic Homogenization of Elliptic Systems
• Small A_∞ results for Dahlberg-Kenig-Pipher operators in sets with uniformly rectifiable boundaries
• Failure to slide: a brief note on the interplay between the Kenig-Pipher condition and the absolute continuity of elliptic measures