论文标题
伯恩斯坦技术方程式的技术
The Bernstein technique for integro-differential equations
论文作者
论文摘要
我们将经典的Bernstein技术扩展到整数差异操作员的设置。结果，我们为分数方程的解决方案提供了第一和单面的第二个衍生估计，包括一些小于两个小于两个的订单的完全非线性方程 - 我们证明，随着它们的顺序接近两个。我们的方法足够强大，可以应用于某些PUCCI型极端方程，并将其用于分数操作员的障碍问题，尽管即使在线性情况下，其中几个结果也是新的。我们还提出了一些有趣的开放问题，其中一个涉及“纯”线性分数拉普拉斯式，另一个是与一般核与线性算子相关的PUCCI-Type凸方程的单方面第二个衍生物估计值的有效性。
We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two -- for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some intriguing open questions, one of them concerning the "pure" linear fractional Laplacian, another one being the validity of one-sided second derivative estimates for Pucci-type convex equations associated to linear operators with general kernels.
