论文标题
Möbius-Invariant willmore的功能分析属性和$ \ Mathbb {r}^n $的规律性
Functional analytic properties and regularity of the Möbius-invariant Willmore flow in $\mathbb{R}^n$
论文作者
论文摘要
在本文中，我们继续作者对Möbius-Invariant willmore Flow Mover Movermatizations in $ \ Mathbb {r}^n $以及$ n $ -sphere $ \ mathbb {s s}^n $中的调查。在本文的主要定理中，我们证明了Möbius-Invariant willmore的“ Deturck修改”的进化操作员的基本属性，并通过作者对此主题的结果结合了对“有限的$ \ nscal {h} ^ h} _ _ {$ callty for line for for to and call and in for for for for for for line”。 Denk，Duong，Hieber，Prüss和Simonett，以及Amann's and Lunardi在Semigroups和插值理论方面的工作。确切地说，我们证明了进化运算符$ [f \ mapsto \ Mathcal {p}^*（\，\ cdot \，0，f）] $的局部真实分析性$ W^{4- \ frac {4} {p}，p}（σ，\ Mathbb {r}^n）$，对于任何$ p \ in（3，\ infty）$，大约任何固定的平滑parametrization $ f_0 $ f_0：σ\ t \ longrightArrow \ longrightArrow \ m m iansbb {r} $ \ mathbb {r}^n $。此外，我们证明了整个最大流量线$ \ MATHCAL {p}^*（\，\ cdot \，0，f_0）$，开始以平滑而无脐带的初始沉浸式$ f_0 $，是积极的分析，因此，这是Fréchet的衍生产品，因此 $D_{F}\mathcal{P}^*(\,\cdot\,,0,F_0)$ of the evolution operator in $F_0$ can be uniquely extended to a family of continuous linear operators $G^{F_0}(t_2,t_1)$ in $L^p(Σ,\mathbb{R}^n)$, whose ranges are在$ l^{p}（σ，\ mathbb {r}^n）$中，对于每对固定的time $ t_2 \ geq t_1 $，在最大存在的间隔内$（0，t_ {max}（max}（f_0））$。
In this article we continue the author's investigation of the Möbius-invariant Willmore flow moving parametrizations of umbilic-free tori in $\mathbb{R}^n$ and in the $n$-sphere $\mathbb{S}^n$. In the main theorems of this article we prove basic properties of the evolution operator of the "DeTurck modification" of the Möbius-invariant Willmore flow and of its Fréchet derivative by means of a combination of the author's results about this topic with the theory of "bounded $\mathcal{H}_{\infty}$-calculus" for linear elliptic operators due to Amann, Denk, Duong, Hieber, Prüss and Simonett, and with Amann's and Lunardi's work on semigroups and interpolation theory. Precisely, we prove local real analyticity of the evolution operator $[F\mapsto \mathcal{P}^*(\,\cdot\,,0,F)]$ of the "DeTurck modification" of the Möbius-invariant Willmore flow in a small open ball in $W^{4-\frac{4}{p},p}(Σ,\mathbb{R}^n)$, for any $p\in (3,\infty)$, about any fixed smooth parametrization $F_0:Σ\longrightarrow \mathbb{R}^n$ of a compact and umbilic-free torus in $\mathbb{R}^n$. We prove moreover that the entire maximal flow line $\mathcal{P}^*(\,\cdot\,,0,F_0)$, starting to move in a smooth and umbilic-free initial immersion $F_0$, is real analytic for positive times, and that therefore the Fréchet derivative $D_{F}\mathcal{P}^*(\,\cdot\,,0,F_0)$ of the evolution operator in $F_0$ can be uniquely extended to a family of continuous linear operators $G^{F_0}(t_2,t_1)$ in $L^p(Σ,\mathbb{R}^n)$, whose ranges are dense in $L^{p}(Σ,\mathbb{R}^n)$, for every fixed pair of times $t_2\geq t_1$ within the interval of maximal existence $(0,T_{max}(F_0))$.
