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.
Ulmer,L p Boundary Value Problems for Elliptic and Parabolic Operators, University of Edinburgh, thesis (2024)
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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.