论文标题
$ f $ concavity的表征由dirichlet热流保留
Characterization of $F$-concavity preserved by the Dirichlet heat flow
论文作者
论文摘要
$ f $ -Concavity是功率凹的概括，实际上，是凹的概念的最大可用概括。我们表征了$ f $ concancavity由$ {\ Mathbb r}^n $上凸域中的Dirichlet热流保留的，并完成了Dirichlet热流的保存凹陷性能的研究，该研究由Brascamp和Lieb于1976年始于1976年，并在最近的一些论文中开发。更精确： （1）我们发现了热症状，这是Dirichlet热流保留的最强$ f $ concovity； （2）我们表明，对数covene是最弱的$ f $ concancavity由dirichlet热流保留的；准民用也仅以$ n = 1 $保留； （3）我们证明，如果$ f $ conccavity与对数covavity不一致，并且并不比对数concavity和$ n \ ge 2 $强，那么存在$ f $ -Concave的初始基准，因此，与dirichlet热流相应的解决方案甚至不是quasi-concave，甚至没有任何可靠性的compressive compavence compandity compavence。 此外，我们发现$ f $ concovity的足够且必要的条件可以通过Dirichlet热流来保存。我们还研究了具有可变系数的线性抛物线方程的Cauchy-dirichlet问题的解决方案的凹度特性，并研究了具有可变系数的线性抛物线方程，以及非线性抛物线方程（例如半线性热方程，多孔介质方程）和抛物线寄生虫$ p $ laplace方程。
$F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex domains on ${\mathbb R}^n$, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely: (1) we discover hot-concavity, which is the strongest $F$-concavity preserved by the Dirichlet heat flow; (2) we show that log-concavity is the weakest $F$-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for $n=1$; (3) we prove that if $F$-concavity does not coincide with log-concavity and it is not stronger than log-concavity and $n\ge 2$, then there exists an $F$-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for $F$-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy--Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic $p$-Laplace equation.
