论文标题
Schauder对具有单一Lévy措施的非本地方程的估计
Schauder's estimates for nonlocal equations with singular Lévy measures
论文作者
论文摘要
在本文中，我们在\ mr^d中建立了Schauder对以下非本地方程的估算： 其中$α\ in（1/2,2）$和$ b：\ Mathbb r _+\ times \ Mathbb r^d \ to \ Mathbb r $是一个无限的本地$β$β$ -BORDERhölder在$ x $中均在$ t $中，在$ t $中均在$ t $中，$ x $ $α$ - 稳定的类似运算符，带有形式：\ begin {align*} {\ Mathscr l}^{（α）} _ {κ，σ} u（t，x）：= \ int _ {\ Mathbb r^d} \ big（u（t，x+σ（t，x+σ（t，x，x）z）-U（t，x）-u（t，x） - x） - σ（t，x）z^{（t，x）z^{α{α{α{α）}} u（t，x）\ big）κ（t，x，z）ν^{（α）}（\ mathord {\ rm d} z）， \ end {align*}其中$ z^{（α）} = z \ mathbf {1} _ {α\ in（1,2）}+z \ z \ mathbf {1} _ {| z | | \ leq 1} R_+\times\mathbb R^{2d}\to\mathbb R_+ $ is bounded from above and below, $ σ:\mathbb R_+\times\mathbb R^{d}\to \mathbb R^d\otimes \mathbb R^d$ is a $ γ$-order Hölder continuous function in $ x $ uniformly in $ t $和$ν^{（α）} $是一种单数非脱位$α$稳定的lévy措施。
In this paper, we establish Schauder's estimates for the following non-local equations in \mR^d : $$ \partial_tu=\mathscr L^{(α)}_{κ,σ} u+b\cdot\nabla u+f,\ u(0)=0, $$ where $α\in(1/2,2)$ and $ b:\mathbb R_+\times\mathbb R^d\to\mathbb R$ is an unbounded local $β$-order Hölder function in $ x $ uniformly in $ t $, and $\mathscr L^{(α)}_{κ,σ}$ is a non-local $α$-stable-like operator with form: \begin{align*} {\mathscr L}^{(α)}_{κ,σ}u(t,x):=\int_{\mathbb R^d}\Big(u(t,x+σ(t,x)z)-u(t,x)-σ(t,x)z^{(α)}\cdot\nabla u(t,x)\Big)κ(t,x,z)ν^{(α)}(\mathord{\rm d} z), \end{align*} where $z^{(α)}=z\mathbf{1}_{α\in(1,2)}+z\mathbf{1}_{|z|\leq 1}\mathbf{1}_{α=1}$, $ κ:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R_+ $ is bounded from above and below, $ σ:\mathbb R_+\times\mathbb R^{d}\to \mathbb R^d\otimes \mathbb R^d$ is a $ γ$-order Hölder continuous function in $ x $ uniformly in $ t $, and $ ν^{(α)} $ is a singular non-degenerate $ α$-stable Lévy measure.
