We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive \(\mathcal {R}\)-sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.

中文翻译：

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具有负正则性的边界数据的椭圆和抛物型边值问题

我们研究了混合比例尺在半空间上具有混合光滑度的椭圆和抛物线型边值问题。目的是用负规则性的边界数据解决边界值问题，并描述边界处解的奇异性。为此，我们推导了泊松算子在具有混合平滑度的混合比例尺中的映射特性。在法线方向上的平滑度不太大的情况下，我们还导出了均质边界数据的\（\ mathcal {R} \）- securality结果。