Solvability of the L2 Dirichlet problem for parabolic operators with spatially periodic coefficients
Abstract
We prove global reverse Hölder estimates in L2 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 A0=A0(x,t) has a quantitative uniform modulus of continuity in time and its convergence to A0 in the transverse variable is controlled by a square-Dini modulus, then the Poisson kernel satisfies a uniform RH2 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 RH2 estimate. Consequently, the L2 Dirichlet problem is uniquely solvable among weak solutions u satisfying N∗ u∈L2. We also establish, for fully spatially periodic coefficients, a nonsymmetric A∞ 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 Dt1/2A0∈L∞x(BMOt) provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.
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BibTeX
@misc{nystrom2026solvability,
title = {{Solvability of the \(\mathrm L^2\) Dirichlet problem for parabolic operators with spatially periodic coefficients}},
author = {Kaj Nyström},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11626},
eprint = {2609.11626},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11626},
}
RIS
TY - UNPB
TI - Solvability of the L^2 Dirichlet problem for parabolic operators with spatially periodic coefficients
AU - Kaj Nyström
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11626
UR - https://arxiv.org/abs/2609.11626
AB - We prove global reverse Hölder estimates in 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 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 RH_2 estimate. Consequently, the L^2 Dirichlet problem is uniquely solvable among weak solutions u satisfying N_∗ u∈L^2. We also establish, for fully spatially periodic coefficients, a nonsymmetric A_∞ 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^1/2A^0∈L^∞_x(BMO_t) provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.
ER -