The obstacle problem for the fractional Laplacian with drift
The obstacle problem for the fractional Laplacian with drift
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带漂移的分数拉普拉斯障碍问题
DOI：
10.1016/j.aim.2018.07.021
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发表时间：
2015
影响因子：
1.7
通讯作者：
Mariana Smit Vega Garcia
中科院分区：
文献类型：
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作者：
Mariana Smit Vega Garcia
研究了rn + 1(0.1)（∂t−Δ） s u= V (x, t) u, s∈(0,1)中的非局部方程在时间上的强唯一延拓性。我们的主要结果定理1.2可以被认为是在s= 1的情况下[30]中得到的结果的非局部对应物。为了证明定理1.2，我们发展了方程（0.1）可拓问题的正则性理论。有了这样的理论，我们建立了：(i)一个调整后的频率函数的基本单调性结果，它在本文中起着中心作用，见下面的定理1.3；（ii）对与可拓问题相关的所谓Almgren重标进行了广泛的放大分析。我们认为我们的工作也将对自由边界问题中的某些基本开放问题以及非局部分离问题的研究产生兴趣。
We study the strong unique continuation property backwards in time for the nonlocal equation in R n+ 1 (0.1)(∂ t− Δ) s u= V (x, t) u, s∈(0, 1). Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained in [30] for the case when s= 1. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation (0.1). With such theory in hands we establish:(i) a basic monotonicity result for an adjusted frequency function which plays a central role in this paper, see Theorem 1.3 below;(ii) an extensive blowup analysis of the so-called Almgren rescalings associated with the extension problem. We feel that our work will also be of interest eg in the study of certain basic open questions in free boundary problems, as well as in nonlocal segregation problems.