We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Rad\'o-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte's embedding theorem for planar graphs - RKC's discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.

中文翻译：

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非凸平面谐波图

我们制定了二维域之间的一系列可逆映射的新特征。我们的工作遵循两个经典结果：Rad\'o-Kneser-Choquet (RKC) 定理，它建立了调和映射到凸平面域的可逆性；和 Tutte 平面图的嵌入定理 - RKC 的离散对应物 - 证明了满足离散调和原理的三角域的分段线性映射到凸平面多边形的可逆性。在这两个定理中，目标域的凸性对于确保可逆性至关重要。在连续和离散情况下，我们通过用限制较少的条件替换凸性来扩展这些特征。在连续情况下，Alessandrini 和 Nesi 通过添加沿边界方向保持的附加条件，将可逆调和映射表征为具有平滑边界的非凸域。我们通过定义沿边界的法向导数的条件来扩展他们的结果，我们称之为锥条件；这种条件易于处理且几何直观，编码了局部可逆性的弱概念。锥形条件使我们能够将 Alessandrini 和 Nesi 扩展到具有分段平滑边界的非凸域的调和映射的情况。在离散情况下，我们使用圆锥条件的模拟来表征三角剖分的可逆离散谐波分段线性映射。