A Variable Coefficient Free Boundary Problem for Lᵖ-solvability of Parabolic Dirichlet Problems in Graph Domains

We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nystr\"om BHMN25. In particular, we show that if is the region above the graph of a Lip (1, 1/2) (parabolic Lipschitz) function and L is a parabolic operator in divergence form = ₜ - div A \ with A satisfying an L¹ Carleson condition on its spatial and time derivatives, then the Lᵖ-solvability of the Dirichlet problem for L and L^* implies that the graph function has a half-order time derivative in BMO. Equivalently, the graph is parabolic uniformly rectifiable. In the case of A symmetric, we only require that the Dirichlet problem for L is solvable, which requires us to adapt a clever integration by parts argument by Lewis and Nystr\"om. A feature of the present work is that we must overcome the lack of translation invariance in our equation, which is a fundamental tool in similar works, including BHMN25.