关于函数时间重排和抛物Talenti不等式的注记
A note on the rearrangement of functions in time and on the parabolic Talenti inequality
摘要：
Talenti不等式是偏微分方程约束最优控制定性分析和变分法中的一个重要特征。经典的抛物型Talenti不等式指出，如果我们考虑抛物型方程${\frac{\partial u}{\partial t}}-Δu=f=f(t,x)$，那么，对于任何时候，用其Schwarz重排$f^\#(t,\cdot)$代替$t$，$f(t,\cdot)$在以下意义上增加了溶液的浓度：让$v$是球中${\frac{\partial v}{\partial t}}-Δv=f^\#$的溶液，那么溶液$u$的浓度小于$v$。这一性质可以用对于某个顺序关系存在一个极大元来表示。不仅在空间上而且在时间上尝试重新排列源项是很自然的，因此当我们关于两个变量重新排列函数时，研究这样一个极大元的存在性。在本文中，我们证明这是不可能的。
英文摘要：
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.
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