具有空间周期系数的抛物算子的 \\(\mathrm L^2\\) Dirichlet 问题的可解性
Solvability of the \(\mathrm L^2\) Dirichlet problem for parabolic operators with spatially periodic coefficients
- Uppsala University（乌普萨拉大学）
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AI总结：
本文证明具有空间周期系数抛物算子泊松核的全局逆Hölder估计，进而建立L² Dirichlet问题的唯一可解性，并推广至非对称情形与Lipschitz图域。
AI中文摘要：
我们证明了上半空间中标量散度型抛物算子泊松核在 \\(\mathrm L^2\\) 中的全局逆 Hölder 估计。系数矩阵是实的、对称的，并且在每个空间变量 \\(X=(\lambda,x)\\) 上具有周期性，但不假设时间上的周期性。如果其边界迹 \\(A^0=A^0(x,t)\\) 在时间上具有定量的均匀连续性模，并且其在横向变量上收敛到 \\(A^0\\) 的过程受平方-Dini 模控制，则泊松核在低于周期的尺度上满足一致的 \\(\mathrm{RH}_2\\) 估计。然后利用完全的空间周期性将该估计推广到所有尺度。大尺度论证中的主要新成分是一个正的伴随高度坐标 \\(\Phi^*=\lambda+O(1)\\)。与 \\(\Phi^*\\) 的边界比较产生了所需的 Green 函数的大尺度衰减，从而得到全局 \\(\mathrm{RH}_2\\) 估计。因此，在满足 \\(N_* u\in\mathrm L^2\\) 的弱解 \\(u\\) 中，\\(\mathrm L^2\\) Dirichlet 问题是唯一可解的。我们还针对完全空间周期系数，在平方-Dini 收敛到与横向无关的边界迹的条件下，建立了非对称的 \\(A_\infty\\) 定理，其中时间正则性仅需可测性。我们说明了在所述的系数和周期性相容性假设下，结果如何转移到与时间无关的 Lipschitz 图域，并为所考虑的周期均匀化族推导了尺度一致的边界估计。条件 \\(D_t^{1/2}A^0\in\mathrm L^\infty_x(\mathrm{BMO}_t)\\) 为对称定理中所需的时间连续性提供了一个自然的充分判据。
英文摘要：
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.
