arXiv 2303.027092023-03-05math.FAmath.CAmath.DG

圆上的一个Lorentz不变的Sobolev不等式

A Lorentz invariant sharp Sobolev inequality on the circle

Pengyu Le

备注：: 38 pages

摘要：

证明了圆$$\int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} θ\geq - \frac{4π^2}{\int_{\mathbb{S}^1} v^{-2} \mathrm{d} θ},$$上的Sobolev不等式，当$v^{-2} (θ) = \frac{k\sqrt{1-α^2}}{1+ α\cos(θ- θ_o)}$满足等式时，不等式成立，其中$k>0$，$α\in (-1,1)$，$θ_0 \in \mathbb{R}$满足等式.如果PDMATHAMR在圆上的某个地方为零，则$$4 \int_{\mathbb{S}^1} (v')^2 \mathrm{d} θ\geq\int_{\mathbb{S}^1} v^2 \mathrm{d} θ.$$证明该不等式的基本工具是圆上的重排不等式和变分法。研究了泛函$\mathcal{F}[v] = \int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} θ$在约束$\int_{\mathbb{S}^1} v^{-2} \mathrm{d} θ= 2π$下的变分问题.泛函$\mathcal{F}$的一个重要的几何见解是它在洛伦兹群下是不变的，因为$\mathcal{F}[v]$是$3$ -昏暗闵可夫斯基时空中光锥中由函数$v^{-2}$参数化的类空曲线的两个零展开的乘积的积分。该约束下的$\mathcal{F}$的全局极小值由光锥的类空平面截面给出。我们介绍了一种结合对称递减重排和洛伦兹变换的方法。这种方法不仅限于本文的讨论范围，而且适用于$\mathbb{S}^{n}, n \geq 1$上的其它Lorentz不变变分问题。作为一个例子，我们用这种方法给出了$\mathbb{S}^n, n\geq 3$上的一个尖锐Sobolev不等式的证明.

英文摘要：

We prove the following sharp Sobolev inequality on the circle $$\int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} θ\geq - \frac{4π^2}{\int_{\mathbb{S}^1} v^{-2} \mathrm{d} θ},$$ with the equality being achieved when $v^{-2} (θ) = \frac{k\sqrt{1-α^2}}{1+ α\cos(θ- θ_o)}$where$k>0$, $α\in (-1,1)$, $θ_0 \in \mathbb{R}$. If $v$ vanishes somewhere on the circle, then $$4 \int_{\mathbb{S}^1} (v')^2 \mathrm{d} θ\geq\int_{\mathbb{S}^1} v^2 \mathrm{d} θ.$$ The basic tools to prove the inequality are the rearrangement inequality on the circle and the variational method. We investigate the variational problem of the functional $\mathcal{F}[v] = \int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} θ$ under the constraint $\int_{\mathbb{S}^1} v^{-2} \mathrm{d} θ= 2π$. An important geometric insight of the functional $\mathcal{F}$ is that it is invariant under the Lorentz group, since $\mathcal{F}[v]$ is the integral of the product of two null expansions of a spacelike curve parameterised by the function $v^{-2}$ in a lightcone in $3$-dim Minkowski spacetime. The global minimiser of $\mathcal{F}$ under the constraint is simply given by the spacelike plane section of the lightcone. We introduce a method which combines the symmetric decreasing rearrangement and the Lorentz transformation. This method isnot confined to the scope of this paper, but is applicable to other Lorentz invariant variational problems on $\mathbb{S}^{n}, n \geq 1$. As an example, we sketch a proof of the sharp Sobolev inequality on $\mathbb{S}^n, n\geq 3$ by this method.

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