论文标题
有关功能的重新排列的注释，抛物线不平等现象
A note on the rearrangement of functions in time and on the parabolic Talenti inequality
论文作者
论文摘要
Talenti不平等是对PDE受约束最佳控制以及变化的微积分的定性分析的核心特征。经典的抛物线天赋不平等指出，如果我们考虑抛物线方程$ {\ frac {\ partial u} {\ partial t}}} - Δu= f = f = f = f（t，x）$，然后替换，以任何时间$ t $，$ t $，$ f（t，\ cdot）$ fimpers $ f（t，\ cd，从以下意义上讲：让$ v $是$ {\ frac {\ partial v} {\ partial t}}}的解决方案 - ΔV= f^\＃$在球中，那么解决方案$ u $的浓度低于$ v $。可以根据某个订单关系的最大元素的存在来改写此属性。自然而然地尝试重新排列源术语，不仅在空间中而且及时地重新排列，因此当我们相对于两个变量重新排列函数时，研究了这种最大元素的存在。在本文中，我们证明这是不可能的。
Talenti inequalities are a central feature in the qualitative analysis of PDE constrained optimal control as well as in calculus of variations. The classical parabolic Talenti inequality states that if we consider the parabolic equation ${\frac{\partial u}{\partial t}}-Δu=f=f(t,x)$ then, replacing, for any time $t$, $f(t,\cdot)$ with its Schwarz rearrangement $f^\#(t,\cdot)$ increases the concentration of the solution in the following sense: letting $v$ be the solution of ${\frac{\partial v}{\partial t}}-Δv=f^\#$ in the ball, then the solution $u$ is less concentrated than $v$. This property can be rephrased in terms of the existence of a maximal element for a certain order relationship. It is natural to try and rearrange the source term not only in space but also in time, and thus to investigate the existence of such a maximal element when we rearrange the function with respect to the two variables. In the present paper we prove that this is not possible.
