Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift
Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift
复制标题
带漂移的分数拉普拉斯障碍问题解的最优正则性
DOI：
10.1016/j.jfa.2014.10.009
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
C. Pop
中科院分区：
文献类型：
--
作者：
A. Petrosyan;C. Pop
我们证明了在亚临界状态下，由带漂移的分数阶拉普拉斯算子定义的定常障碍问题解的存在性、唯一性和最优正则性。与[4]一样，我们通过考虑[2]中引入的合适的扩张算子来定位我们的问题。与文献[4]中所构造的延拓方程的结构不同，障碍函数的正则性较弱，并具有一些奇异性。考虑到问题的新特点，我们证明了一个新的单调性公式，我们然后使用它来建立最佳的正则性的解决方案。
We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. As in [4], we localize our problem by considering a suitable extension operator introduced in [2]. The structure of the extension equation is different from the one constructed in [4], in that the obstacle function has less regularity, and exhibits some singularities. To take into account the new features of the problem, we prove a new monotonicity formula, which we then use to establish the optimal regularity of solutions.