# Eugenia Malinnikova

(NTNU)

## Frequency function and unique continuation

### Video

** Lundi 12 mars 2018, 9h30 / Monday, March 12, 2018, 9:30 am **

In 1966, Shmuel Agmon introduced the method of logarithmic convexity for weighted norms of solutions of second order equations. These ideas were developed by Almgren, and later by Garofalo and Lin, in particular, to prove unique continuation results for a wide class of second order elliptic equations. In the first lecture, we give an introduction to Almgren’s frequency function, monotonicity formula and quantitative unique continuation results, including the three-ball inequality and Cauchy uniqueness inequality.

## Application of the frequency function to the study of nodal sets

### Video

Mercredi 14 mars 2018, 9h30 / * Wednesday, March 14, 2018, 9:30 am *

The properties of the frequency function explained in the first lecture can be applied to estimates of the zero sets of eigenfunctions of the Laplace-Beltrami operator on compact manifolds. In dimension two, the first result was obtained by Donnelly and Fefferman in the 1980s. Their estimate for the size of the zero set from above was slightly improved in our recent work with A. Logunov, the new ingredients of the proof being the combinatorial properties of the frequency function. More detailed analysis of the frequency function led Logunov to new estimates for the zero set of eigenfunctions in higher dimensions, and we will survey some of his results as well.

## Remez inequality, unique continuation and propagation of smallness for second order elliptic PDEs

### Video

Vendredi 16 mars 2018, 16h00 / * Friday, March 16, 2018, 4pm, *

*Cette conférence s'adresse à un large auditoire.*

*This lecture is aimed at a general mathematical audience.*

The celebrated Remez inequality for polynomials states that the maximum of the polynomial over an interval is controlled by its maximum over a subset of positive measure of the interval. The coefficient in the inequality depends on the degree of the polynomial, and the equality is attained by Chebyshev polynomials. In my recent joint work with A. Logunov, we obtained a generalization of the Remez inequality to the solutions of general second order elliptic PDEs and their gradients. In this context, the degree of a polynomial is replaced by the Almgren frequency of the solution which we investigate using geometric methods. In the talk, I will present this result and its connections to other important features of the solutions of elliptic PDEs, such as quantitative unique continuation and propagation of smallness. In particular, our approach yields an answer to an old question of Landis and improves upon earlier results of Nadirashvili and Vesella.

## LIEU / VENUE

Salle / Room 6254

Centre de recherches mathématiques

Pavillon André-Aisenstadt

Université de Montréal

2920, chemin de la Tour

Le café sera serviavant chaque conférence.

*Coffee will be served before each talk*