We prove the first positive results concerning boundary value problems in the upper half-space of second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so, we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In the way, we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. The major new challenge, compared to the earlier results by one of us under time and transversally independence of the coefficients, is to handle non-local half-order derivatives in time which are unavoidable in our situation.

中文翻译：

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L$^2$ 具有可测系数的抛物线系统边值问题的适定性

我们证明了关于二阶抛物线系统上半空间边值问题的第一个积极结果，仅假设椭圆部分的系数具有可测性和一些横向规律。为此，我们通过边界处的抛物线狄拉克算子引入和开发一阶策略，以获得特别是涉及平方函数和非切向极大函数的自然类中解的格林表示，适定结果为$L^2$-Sobolev 空间中的数据以及层电位的可逆性和扰动结果。以这种方式，我们解决了抛物线算子的加藤平方根问题，其中椭圆部分的系数可测量地取决于所有变量。新的重大挑战，