几类带测度的非局部椭圆问题的研究

This program is devoted to study some nonlocal elliptic equations involving measures. The first one is to consider the fractional elliptic equations involving measure as forcing term, the second one is to investigate the fractional elliptic equations involving measure as outside source, the last one is to study regional fractional elliptic equations involving measure. We will consider the existence, uniqueness and regularity of weak solutions to the nonlocal elliptic problems above. Moreover, the asymptotic behavior of weak solutions are charactered when the measure is Dirac mass. It is well-known that the fractional laplacian is a nonlocal linear operator, while the laplacian is local. Although the fractional laplacian keeps some properties of laplacian, but the nonlocal property of the fractional laplacian makes big difference between the fractional elliptic problem and the second order elliptic problem. The regional fractional laplacian is a nonlocal operator between the fractional laplacian and the laplacian. By studying our nonlocal elliptic problems, we want to make a good understanding about the commonness and difference between the nonlocal elliptic problems and the second order elliptic problems.

本项目拟讨论三类带测度的非局部椭圆问题，第一类是以测度作为方程非齐次项的分数阶椭圆问题，第二类是以测度作为边值型条件的分数阶椭圆问题，第三类是带测度的区域型分数阶椭圆问题。关于以上三类问题，我们拟研究其弱解的存在性、唯一性和正则性，以及测度为Dirac测度时弱解的渐近性。众所周知，分数阶拉普拉斯算子是一个非局部的线性算子，而拉普拉斯是一个局部的线性算子。虽然分数阶拉普拉斯算子保持了拉普拉斯算子的部分性质，然而它的非局部性质使得对具有分数阶拉普拉斯算子的椭圆问题的研究与二阶椭圆问题之间存在着很大的差异。区域型分数阶拉普拉斯算子是介于分数阶拉普拉斯和拉普拉斯之间的一种非局部算子，因此，对这类椭圆问题的研究有利于我们进一步揭示非局部椭圆问题与二阶椭圆问题之间的共性与区别。

本项目研究了一些带有分数阶拉普拉斯算子的椭圆问题在引入测度时，弱解的存在性、唯一性以及渐近性态。首先，我们研究了以Dirac测度作为边值型条件的分数阶椭圆问题，证明了其弱解的存在性和唯一性；其次，我们考虑了以测度作为方程非齐次项的分数阶椭圆问题，用带测度的椭圆问题的弱解刻划了椭圆问题的边界爆破解；最后，我们研究了带Hausdorff测度的分数阶椭圆方程、带测度的分数阶热方程以及Choquard方程等等。