AI 中文总结
本文在矩形区域构造满足正交性、边界条件的曲线网，将其视为控制问题，为设计特定椭圆散度型算子提供基础，具有理论与潜在应用价值。
AI 中文摘要
我们在矩形区域$R$中构造具有特殊性质的曲线网。该网由“水平”和“垂直”两组曲线构成，每组曲线均构成$R$的叶状结构，且每条水平曲线与每条垂直曲线正交相交。此外，每条从$R$左边缘出发的水平曲线到达右边缘时高度相同，每条从$R$下边缘出发的垂直曲线到达上边缘时的经度可预先指定。该经度作为从下边缘到上边缘的映射，可预先选取为恒等映射的任意小扰动。我们的研究动机是设计一些椭圆散度型算子，例如定义在平面内由Reifenberg平坦曲线界定的某个区域上的椭圆散度型算子，使其标量系数及对应的椭圆测度关于边界上的自然测度绝对连续（该内容将在另一篇论文中详述）。我们在矩形区域中构造的曲线网也可被视为一个控制问题，其本身具有研究价值，且可能存在其他应用。
英文摘要
We construct nets of curves in a rectangle $R$ with special properties. The net is formed by two families of curves, "horizontal" and "vertical", each family foliates $R$, and every horizontal curve meets every vertical curve orthogonally. Moreover, every horizontal curve started at the left edge of $R$ arrives to the right edge of $R$ at the same height, and every vertical curve started at the bottom edge of $R$ arrives to the top edge of $R$ at a prescribed in advance longitude. The latter longitude function as a map from bottom to top can be chosen in advance as any small perturbation of the identity. Our motivation for this was to design some elliptic divergence form operators, for instance on one of the domains bounded by a Reifenberg flat curve in the plane, with the scalar coefficient and the corresponding elliptic measure being absolutely continuous with respect to the natural measure on the boundary (an object of a separate paper). The construction of our net of curves in a rectangle can also be interpreted as control problem, has an interest on its own, and, likely, has other applications.
Comments55 pages, 9 figures
