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Scale-free quantitative unique continuation principle

  • Ivan Veselic (TU Chemnitz)
A3 01 (Sophus-Lie room)

Abstract

This is joint work with Constanza Rojas-Molina.

For many elliptic partial differential equations it is known that a solution cannot vanish of arbitrary order unless it is identically zero. This can be proven using a Carleman estimate. The latter is also useful to prove a quantitative version of the unique continuation principle.

We are interested in a particular quantitative and scale free version of this result: Consider an eigenfunction of a Schroedinger equation on cube of size L with periodic boundary conditions. Assume that the $L^2$-norm of the eigenfunction is one.

The cube of size L can be decomposed into unit cubes. Place in each unit cube arbitrarily a ball with fixed but small radius. We wnat to derive a lower bound on the $L^2$-norm of the eigenfunction when integrated over the union of the small balls, which is independent of the size $L$, and depends in an explicit way on the other parameters of enterig the problem.

Upcoming Events of this Seminar

  • Monday, 14.07.25 tba with Alexandra Holzinger
  • Tuesday, 15.07.25 tba with Anna Shalova
  • Tuesday, 12.08.25 tba with Sarah-Jean Meyer
  • Friday, 15.08.25 tba with Thomas Suchanek
  • Friday, 22.08.25 tba with Nikolay Barashkov
  • Friday, 29.08.25 tba with Andreas Koller