For elliptic matrices with bounded measurable coefficients independent of the transversal variable, the parabolic Regularity problem is solvable in some L^p range, dual to the known Dirichlet range for the adjoint operator.
Periodic homogenization and harmonic measures
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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.
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The $L^p$ regularity problem for parabolic operators with transversally independent coefficients
For elliptic matrices with bounded measurable coefficients independent of the transversal variable, the parabolic Regularity problem is solvable in some L^p range, dual to the known Dirichlet range for the adjoint operator.