论文标题
第四阶椭圆运算符的障碍型问题的调查
A survey on obstacle-type problems for fourth order elliptic operators
论文作者
论文摘要
在本文中，我们简要概述了与一类四阶椭圆运营商相关的障碍型问题理论中的一些已知结果，我们强调了我们最近与合作者在这个方向上的工作。由第四阶操作员控制的障碍型问题自然出现在板弯曲现象的线性化基尔乔夫 - 爱理论中。此外，正如杨在\ cite {y13}中首次观察到的那样，与加权双层式操作员相关的边界障碍物问题可以看作是延伸问题，本着caffarelli-silvestre引入的延伸问题，用于caffarelli-silvestre，用于fractional laplacian laplacian $（ - δ）$（ - δ）^s $。在我们最近的工作中，我们研究了这种类型的一些问题，其中我们关注问题的适当性，解决方案的规律性以及自由边界的结构。在我们的方法中，我们将潜在理论中的经典技术和变化的计算结合在一起，以及更现代的方法，例如操作员的定位和单调性公式。
In this article we give a brief overview of some known results in the theory of obstacle-type problems associated with a class of fourth-order elliptic operators, and we highlight our recent work with collaborators in this direction. Obstacle-type problems governed by operators of fourth order naturally arise in the linearized Kirchhoff-Love theory for plate bending phenomena. Moreover, as first observed by Yang in \cite{Y13}, boundary obstacle-type problems associated with the weighted bi-Laplace operator can be seen as extension problems, in the spirit of the one introduced by Caffarelli-Silvestre, for the fractional Laplacian $(-Δ)^s$ in the case $1<s<2$. In our recent work, we investigate some problems of this type, where we are concerned with the well-posedness of the problem, the regularity of solutions, and the structure of the free boundary. In our approach, we combine classical techniques from potential theory and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas.
