Unique continuation principles and equidistribution properties of eigenfunctions

本课题主要研究二阶椭圆型偏微分方程解的振荡性和集中性。它们可以是自伴随算子的本征函数，也可以是不带任何特殊边界条件的非齐次方程的解。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。我们将特别关注多尺度结构的问题。我们可以把提到的局部L^2平均值解释为解的振幅分布的样本。我们想研究在区域内均匀放置的样本是否能很好地控制整个区域上解的L^2范数。特别地，我们的目标是推导出可观测常数的显式界。

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.