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

Kaj Nyström

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arXiv · 2609.11626

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

Kaj Nyström

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Abstract

We prove global reverse Hölder 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 $X=(λ,x)$, 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 $Φ^*=λ+O(1)$. Boundary comparison with $Φ^*$ 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.

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BibTeX RIS

Kaj Nyström. 2026-09-10. Solvability of the $\mathrm L^2$ Dirichlet problem for parabolic operators with spatially periodic coefficients. https://arxiv.org/abs/2609.11626

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