论文标题
迅速的非线性估计值，用于繁殖正定张量场的衍生物
Sharp nonlinear estimates for multiplying derivatives of positive definite tensor fields
论文作者
论文摘要
概括了在对称正定定义矩阵中的值的函数中概括了提高到不同功能的标量函数衍生物的简单乘积公式。这些公式在推导各种非线性估计值的推导方面是基本的，尤其是在PDE理论中。为了绕过矩阵及其导数的非交换性，我们应用了一些众所周知的积分表示公式，然后我们观察到，矩阵幂的衍生物是相对于指数的对数方面函数。这与将整体平均值和内部产品结合在矩阵上的看似简单不平等的有效性直接相关。在许多示例中说明了我们的结果的最佳性。
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of various non-linear estimates, especially in the PDE theory. To get around the non-commutativity of the matrix and its derivative, we apply some well-known integral representation formulas and then we make an observation that the derivative of a matrix power is a logarithmically convex function with respect to the exponent. This is directly related to the validity of a seemingly simple inequality combining the integral averages and the inner product on matrices. The optimality of our results is illustrated on numerous examples.
