论文标题
与Hölder连续内核非本地PDE的Calderon-Zygmund类型估计
Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel
论文作者
论文摘要
我们研究内部$ l^p $ regularity理论，也称为Calderon-Zygmund理论， \ int _ {\ Mathbb {r}^n} \ int _ {\ Mathbb {r}^n}^n} \ frac {k（x，y）\（x，x，x）-u（x）-u（y（y）） c_c^\ infty（\ mathbb {r}^n）中的φ\ rangle \quadφ\。 \]对于$ s \ in（0,1）$，$ t \ in [s，2s] $，$ p \ in [2，\ infty）$，$ k $ a椭圆形，对称，hölder连续内核，如果$ f \ in \ in \ in \ weft（h^{ $ h^{2s-t，p} _ {loc}（ω）$长达$ 2S-t <1 $。 可不同性的增加与$ K $的Hölder系数无关。例如，我们的结果表明，如果$ f \ in l^{p} _ {loc} $，则$ u \ in H^{2S-δ，p} _ {loc} _ {loc} $对于任何$δ\ in（0，s] $ in（0，s] $）in（0，s] $ in（s] $）$ 2S-Δ<1 $。 \ nabla u）= f $（即$ s = 1 $），其中$ c^γ$-Hölder连续系数$ \ bar {k} $允许估计$ h^{1+γ} $的订单$ h^{1+γ} $，实际上，这是许多不合时宜的差异。改进，但一直到$ \ min \ {2s-t，1 \} $。 证明与（更简单）方程的比较来争论\ [ \ int _ {\ MathBb {r}^n} k（z，z）（-Δ）^{\ frac {t} {2}} {2}} u（z）\，（ - Δ） c_c^\ infty（\ mathbb {r}^n）。 \]并表明，只要$ k $是hölder连续的，$ s，t，2s-t \ in（0,1）$，则是“ commutator” \ [ \ int _ {\ Mathbb {r}^n} k（z，z）（-Δ）^{\ frac {t} {2}} {2}} u（z）\，（ - δ）^{\ frac {\ frac {2s-t} {2s-t} {2}} {2}} {2} {2} {Z）\，dz-（z）\，dz - c \ c \ n \ int _ {\ MathBb {r}^n} \ frac {k（x，y）\（u（x）-u（x）-u（y（y））\，（φ（x）-φ（y））} {| x-y |^|^|^{n+2s}}} \，dx \，dx \，dx \，dy \，dy \ \，dy \，dy \]行为。
We study interior $L^p$-regularity theory, also known as Calderon-Zygmund theory, of the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y)\ (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy = \langle f, φ\rangle \quad φ\in C_c^\infty(\mathbb{R}^n). \] For $s \in (0,1)$, $t \in [s,2s]$, $p \in [2,\infty)$, $K$ an elliptic, symmetric, Hölder continuous kernel, if $f \in \left (H^{t,p'}_{00}(Ω)\right )^\ast$, then the solution $u$ belongs to $H^{2s-t,p}_{loc}(Ω)$ as long as $2s-t < 1$. The increase in differentiability is independent of the Hölder coefficient of $K$. For example, our result shows that if $f\in L^{p}_{loc}$ then $u\in H^{2s-δ,p}_{loc}$ for any $δ\in (0, s]$ as long as $2s-δ< 1$. This is different than the classical analogue of divergence-form equations ${\rm div}(\bar{K} \nabla u) = f$ (i.e. $s=1$) where a $C^γ$-Hölder continuous coefficient $\bar{K}$ only allows for estimates of order $H^{1+γ}$. In fact, it is another appearance of the differential stability effect observed in many forms by many authors for this kind of nonlocal equations -- only that in our case we do not get a "small" differentiability improvement, but all the way up to $\min\{2s-t,1\}$. The proof argues by comparison with the (much simpler) equation \[ \int_{\mathbb{R}^n} K(z,z) (-Δ)^{\frac{t}{2}} u(z) \, (-Δ)^{\frac{2s-t}{2}} φ(z)\, dz = \langle g,φ\rangle \quad φ\in C_c^\infty(\mathbb{R}^n). \] and showing that as long as $K$ is Hölder continuous and $s,t, 2s-t \in (0,1)$ then the "commutator" \[ \int_{\mathbb{R}^n} K(z,z) (-Δ)^{\frac{t}{2}} u(z) \, (-Δ)^{\frac{2s-t}{2}} φ(z)\, dz - c\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y)\ (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy \] behaves like a lower order operator.
