Unique continuation principles and equidistribution properties of eigenfunctions

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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.