论文标题
分数Hadamard公式和应用
A fractional Hadamard formula and applications
论文作者
论文摘要
我们考虑亚临界分数sobolev常数中最佳常数的域依赖性， $$ λ_{s，p}（ω）：= \ inf \ weft \ {[u] _ { $$ 其中$ s \ in（0,1）$，$ω$由类$ c^{1,1} $和$ p \ in [1，\ frac {2n} {n-2s}）$ IF $ 2S <n $，$ p \ in [1，\ infty）in [1，\ infty）$ in [1，\ frac {2n} {n-2s} $ in [1，\ frac} $ p \ in [1，\ infty）$ If $ 2S <n $ p \ If $ 2S \ geq n geq n = 1 $ $。明确地，我们为映射$ω\mapstoλ_{s，p}（ω）$在域扰动下的映射$ω\mapstoλ_{s，p}（ω）$的单方面形状导数提供了公式。如果$λ_{s，p}（ω）$承认一个独特的正乘积（例如$ p = 1 $或$ p = 2 $），我们的结果意味着对于$ω$的dirichlet laplacian on dirichlet laplacian的第一个特征的经典变异公式的非本地版本。多亏了我们的单方面形状衍生物的公式，我们表征了$λ_{s，p}（ω）$的光滑本地最小化器，在体积放置变形下，如果$ p \ in \ in \ in \ in \ {1 \} \ cup cup [2，\ infty）$。最后，我们考虑了$λ_{s，p}（ω）$的最大化问题。我们证明，对于$ p \ in \ {1,2 \} $，值$λ_{s，p}（b \ setminus \ edmiminus \ overline b'）$在两个球是同心时是最大的。
We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ λ_{s,p}(Ω):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(Ω),\,\, \|u\|_{L^p(Ω)}=1 \right\}, $$ where $s\in (0,1)$, $Ω$ is bounded of class $C^{1,1}$ and $p\in [1, \frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we derive formula for the one-sided shape derivative of the mapping $Ω\mapsto λ_{s,p}(Ω)$ under domain perturbations. In the case where $ λ_{s,p}(Ω)$ admits a unique positive minimizer (e.g. $p=1$ or $p=2$), our result implies a nonlocal version of the classical variational Hadamard formula for the first eigenvalue of the Dirichlet Laplacian on $Ω$. Thanks to the formula for our one-sided shape derivative, we characterize smooth local minimizers of $λ_{s,p}(Ω)$ under volume-preserving deformations, and we find that they are balls if $p\in \{1\}\cup [2,\infty)$. Finally, we consider the maximization problem for $λ_{s,p}(Ω)$ among annular-shaped domains of fixed volume of the type $B\setminus \overline B'$, where $B$ is a fixed ball and $B'$ is ball whose position is varied within $B$. We prove that, for $p\in \{1,2\}$, the value $λ_{s,p}(B\setminus \overline B')$ is maximal when the two balls are concentric.
