$\u211d^n$上分数阶拉普拉斯算子的$L^p$同构及其应用
$L^p$ isomorphisms for the fractional Laplacian on $\mathbb{R}^n$ and their application
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中文总结 AI 辅助
该研究受经典拉普拉斯方程加权Sobolev空间同构理论启发，建立了分数阶拉普拉斯算子的$L^p$同构理论，证明了解的存在唯一性，并将其应用于分数阶Stokes系统的求解。
中文摘要 AI 辅助
Amrouche、Girault和Giroire于1994年提出的经典理论通过加权Sobolev空间之间的同构求解了$\u211d^n$上的拉普拉斯方程。受其框架启发，我们建立了分数阶拉普拉斯算子对应的$L^p$理论：我们引入加权分数阶Sobolev空间$\u039b^{s,p}(\u211d^n)$，其最高阶范数由$\\|(-\u25b3)^{\frac{s}{2}}u\\|_{L^p}$度量，并证明对所有$s\in(0,2)$和$p\in(1,\u221e)$，算子$(-\u25b3)^{\frac{s}{2}}:\u039b^{s,p}(\u211d^n)/\u2119_{[s-n/p]}\to L^p(\u211d^n)$是同构映射；因此对每个$f\in L^p(\u211d^n)$，方程$(-\u25b3)^{\frac{s}{2}}u=f$可解，且解在一个显式有限维多项式空间的意义下唯一。更一般地，通过对偶性将空间尺度扩展到负阶后，适当阶数的分数阶拉普拉斯算子可将其对应空间彼此同构映射且复合运算精确成立。该理论的关键要素是一族适配于$\\|(-\u25b3)^{\frac{s}{2}}u\\|_{L^p}$的加权Hardy不等式和Poincaré不等式，本文通过将问题归约为梯度情形，在$1\le s<2$的条件下证明了这些不等式。由于这些不等式仅先验地在稠密函数类上成立，而$C_{\mathrm{c}}^\infty(\u211d^n)$在$\u039b^{s,p}(\u211d^n)$中的稠密性本身并非平凡结论，我们首先在Schwartz类的闭包上证明了同构性——由于两个解的差是整$s$-调和函数因而是多项式，该闭包与$\u039b^{s,p}(\u211d^n)$重合，稠密性也作为副产品得到。作为应用，我们得到了$\u211d^n$上分数阶Stokes系统解的存在性与唯一性，给出了显式核函数与相容性条件。
英文摘要
A classical theory of Amrouche, Girault and Giroire (1994) resolves the Laplace equation on $\mathbb{R}^n$ through an isomorphism between weighted Sobolev spaces. Inspired by their framework, we develop the corresponding $L^p$ theory for the fractional Laplacian: we introduce weighted fractional Sobolev spaces $Λ^{s,p}(\mathbb{R}^n)$, gauged at top order by $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, and prove that $(-\triangle)^{\frac{s}{2}}:Λ^{s,p}(\mathbb{R}^n)/\mathcal P_{[s-n/p]}\to L^p(\mathbb{R}^n)$ is an isomorphism for all $s\in(0,2)$ and $p\in(1,\infty)$; thus $(-\triangle)^{\frac{s}{2}}u=f$ is solvable for every $f\in L^p(\mathbb{R}^n)$, uniquely modulo an explicit finite-dimensional space of polynomials. More generally, extending the scale by duality to negative orders, the fractional Laplacians of the appropriate orders map its spaces isomorphically onto one another and compose exactly. A key ingredient is a family of weighted Hardy and Poincaré inequalities adapted to $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, established here for $1\le s<2$ by a reduction to the gradient. Since these inequalities hold a priori only on a dense class, while the density of $C_{\mathrm{c}}^\infty(\mathbb{R}^n)$ in $Λ^{s,p}(\mathbb{R}^n)$ is itself nontrivial, we first prove the isomorphism on the closure of the Schwartz class, which coincides with $Λ^{s,p}(\mathbb{R}^n)$ since the difference of two solutions is an entire $s$-harmonic function, hence a polynomial; the density follows as a by-product. As an application, we obtain existence and uniqueness, with explicit kernels and compatibility conditions, for the fractional Stokes system in $\mathbb{R}^n$.
发表机构
- School of Mathematical Sciences Shanghai Jiao Tong University（上海交通大学数学科学学院）
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