Unique continuation principles and equidistribution properties of eigenfunctions

本研究项目致力于研究二阶椭圆型偏微分方程解的振动性和集中性。它们可以是自伴算子的本征函数，也可以是没有任何特殊边界条件的非齐次方程的解。其目的是估计局部L^2平均值的变化，即在有界区域内小球或立方体上解的平方(绝对值)的平均值。将特别注意多尺度结构的问题。可以将上述L^2局部平均值解释为解的振幅分布的样本。我们要研究在区域内均匀放置样本是否能很好地控制整个区域上解的L^2范数。特别地，我们的目标是得到可观测性常数的一个显式界。

The research project is devoted to the study of oscillation and concentration properties of solutions of second order elliptic partial differential equations. They could be eigenfunctions of a self-adjoint operator, as well as solutions of an inhomogeneous equation without any special kind of boundary conditions. The aim is to estimate the variation of local L^2 averages, i.e. averages of the (absolute value of the) square of the solution over a small ball or cube, within a bounded region. Particular attention will be paid to problems with a multiscale structure. One can interpret the mentioned local L^2 averages as samples for the distribution of the amplitude of the solution. We want to study whether uniformly placed samples within the region give a good control of the L^2 norm of the solution on the whole region. In particular, we aim to derive an explicit bound on the observability constant.