Remez inequalities for solutions of elliptic PDEs

School of Mathematics Colloquium
Thursday, March 28, 2019 - 11:00am for 1 hour (actually 50 minutes)
Skiles 006
Eugenia Malinnikova – Norwegian University of Science and Technology –
Mayya Zhilova
The Remez inequality for polynomials quantifies the way the maximum of a polynomial over an interval is controlled by its maximum over a subset of positive measure. The coefficient in the inequality depends on the degree of the polynomial; the result also holds in higher dimensions. We give a version of the Remez inequality for solutions of second order linear elliptic PDEs and their gradients. In this context, the degree of a polynomial is replaced by the Almgren frequency of a solution. We discuss other results on quantitative unique continuation for solutions of elliptic PDEs and their gradients and give some applications for the estimates of eigenfunctions for the Laplace-Beltrami operator. The talk is based on a joint work with A. Logunov.