论文标题
非散发形式的分数椭圆方程：定义，应用和Harnack不等式
Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality
论文作者
论文摘要
我们定义了分数幂$ l^s =（ - a^{ij}（x）\ partial_ {ij}）^s $，$ 0 <s <1 $，nondivergence fore e椭圆形操作员$ l = -a^{ij}（ij}（ij}（ij}（x）\ partial_ {ij {ij {ij {ij {系数上的最小规律性假设$ a^{ij}（x）$和边界$ \partialΩ$。我们表明，这些分数运算符出现在几种应用中，例如分数Monge - Ampère方程，弹性和金融。解决非局部泊松问题的解决方案$ u $ $$ \ begin {case}（-a^{ij}（x）（x）\ partial_ {ij}）^su = f＆\ hbox {in} 〜Ω \\ U = 0＆\ u = 0＆\ hbox {on}〜\ f in}〜\\ poartial}〜\ \ poartialmopial extersive y degnied y degenation y degenation case y reandized restime;我们开发了在Monge中滑动抛物面的方法 - Ampère几何形状，并证明内部harnack不平等和Hölder估计到系数$ a^{ij}（x）$有界的延伸问题解决方案。反过来，这意味着内部harnack不平等和Hölder估计的解决方案$ u $对于分数问题。
We define the fractional powers $L^s=(-a^{ij}(x)\partial_{ij})^s$, $0 < s < 1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\partial_{ij}$ in bounded domains $Ω\subset\mathbb{R}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\partialΩ$. We show that these fractional operators appear in several applications such as fractional Monge--Ampère equations, elasticity, and finance. The solution $u$ to the nonlocal Poisson problem $$\begin{cases} (-a^{ij}(x) \partial_{ij})^su = f&\hbox{in}~Ω\\ u=0&\hbox{on}~\partialΩ\end{cases}$$ is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge--Ampère geometry and prove the interior Harnack inequality and Hölder estimates for solutions to the extension problem when the coefficients $a^{ij}(x)$ are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions $u$ to the fractional problem.
