We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on \(\textsf {CAT} (\kappa )\)-spaces and prove that they can be characterized by the same differential inclusion \(y_t'\in -\partial ^-\textsf {E} (y_t)\) one uses in the smooth setting and more precisely that \(y_t'\) selects the element of minimal norm in \(-\partial ^-\textsf {E} (y_t)\). This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of \(L^2\) and CAT(0) valued maps: we define the Laplacian of such \(L^2\) map as the element of minimal norm in \(-\partial ^-\textsf {E} (u)\), provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is \(L^2\)-dense. Basic properties of this Laplacian are then studied.

我们回顾了\(\textsf {CAT} (\kappa )\)空间上的凸函数和下半连续泛函框架中的梯度流理论，并证明它们可以用相同的微分包含\(y_t'\in -\partial ^-\textsf {E} (y_t)\)用于平滑设置，更准确地说，\(y_t'\)选择\(-\partial ^-\textsf {E} 中的最小范数元素(y_t)\)。这概括了之前在这个方向上的结果，其中能量也被假定为 Lipschitz。然后我们将这样的结果应用于\(L^2\)和CAT (0) 值映射空间上的 Korevaar-Schoen 能量泛函：我们定义了这样的\(L^2\)的拉普拉斯算子映射为\(-\partial ^-\textsf {E} (u)\)中最小范数的元素，前提是它不为空。梯度流理论确保允许拉普拉斯算子的映射集是\(L^2\)密集的。然后研究这个拉普拉斯算子的基本性质。