Solvability of the \(\mathrm L^2\) Dirichlet problem for parabolic operators with spatially periodic coefficients

Abstract

We prove global reverse H\"older estimates in \(\mathrm L^2\) for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is real, symmetric, and periodic in every spatial variable \(X=(\lambda,x)\), while no periodicity in time is assumed. If its boundary trace \(A^0=A^0(x,t)\) has a quantitative uniform modulus of continuity in time and its convergence to \(A^0\) in the transverse variable is controlled by a square-Dini modulus, then the Poisson kernel satisfies a uniform \(\mathrm{RH}_2\) estimate at scales below the period. Full spatial periodicity is then used to propagate this estimate to every scale. The main new ingredient in the large-scale argument is a positive adjoint height coordinate \(\Phi^*=\lambda+O(1)\). Boundary comparison with \(\Phi^*\) yields the required large-scale decay of the Green function and hence the global \(\mathrm{RH}_2\) estimate. Consequently, the \(\mathrm L^2\) Dirichlet problem is uniquely solvable among weak solutions \(u\) satisfying \(N_\ast u\in\mathrm L^2\). We also establish, for fully spatially periodic coefficients, a nonsymmetric \(A_\infty\) theorem under square-Dini convergence to a transverse-independent boundary trace, with no temporal regularity beyond measurability. We explain how the results transfer to time-independent Lipschitz graph domains under the stated coefficient and periodicity-compatibility hypotheses, and we derive scale-uniform boundary estimates for the periodic homogenization families considered below. The condition \(D_t^{1/2}A^0\in\mathrm L^\infty_x(\mathrm{BMO}_t)\) provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.