论文标题
非对称固有的HOPF-LAX SEMIGROUP与内在的Lagrangian
Non-symmetric intrinsic Hopf-Lax semigroup vs. intrinsic Lagrangian
论文作者
论文摘要
在本文中，我们分析了作者在度量空间中本质上Lipschitz部分中提出的内在HOPF-LAX半群的“对称”。确实，在通常的情况下，我们有$ d（x，y）= d（y，x）$的任何点$ x $，$ y $属于公制空间$ x $;另一方面，在我们的内在上下文中，我们有$ d（f（x），π^{ - 1}（y））\ ne d（f（y），π^{ - 1}（x）），每$ x，y \ in x $ in x $。因此，我们获得“经典”内在的HopF-Lax半群，即“对称的” Hopf-lax semigroup是汉密尔顿 - 雅各比型方程的订阅，这并非微不足道。在这里，一个重要的观察结果是，$ f $只是商地图$π$的连续部分，它不能固有的lipschitz。 但是，在埃文斯之后，本说明的主要结果是表明“新”内在的HOPF-LAX半群可以满足合适的变异问题，其中功能包含固有的Lagrangian。因此，我们还定义并证明并证明了这种内在的Lagrangian的内在Fenchel-Legendre变换的某些基本属性，该转换取决于$π$的连续部分。
In this paper, we analyze the 'symmetrized' of the intrinsic Hopf-Lax semigroup introduced by the author in the context of the intrinsically Lipschitz sections in the setting of metric spaces. Indeed, in the usual case, we have that $d(x,y) =d(y,x)$ for any point $x$ and $y$ belong to the metric space $X$; on the other hand, in our intrinsic context, we have that $d(f(x),π^{-1} (y)) \ne d(f(y),π^{-1} (x)),$ for every $x,y \in X$. Therefore, it is not trivial that we get the same result obtained for the "classical" intrinsic Hopf-Lax semigroup, i.e., the 'symmetrized' Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equation. Here, an important observation is that $f$ is just a continuous section of a quotient map $π$ and it can not intrinsic Lipschitz. However, following Evans, the main result of this note is to show that the "new" intrinsic Hopf-Lax semigroup satisfies a suitable variational problem where the functional contained an intrinsic Lagrangian. Hence, we also define and prove some basic properties of the intrinsic Fenchel-Legendre transform of this intrinsic Lagrangian that depends on a continuous section of $π$.
