Eugenia Malinnikova, Quantitative Unique Continuation for Solutions of Second Order Elliptic..., 4.1
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 08/07/2018

Quantitative Unique Continuation for Solutions of Second Order Elliptic Equations

Eugenia Malinnikova, Norwegian University of Science & Technology

Frequency function of solutions to second order elliptic equations and zero sets of Laplace-Beltrami eigenfunctions

Link to lecture notes at:
https://pcmi.ias.edu/sites/pcmi.ias.e...

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. We will give an introduction to Almgren’s frequency function, monotonicity formula and quantitative unique continuation results, including the three-ball inequality and Cauchy uniqueness inequality. These ideas 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 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, we will survey some of his results.
Prerequisites: Real analysis and basic theory of elliptic PDE, some familiarity with Hausdorff measure.

Presented at the Presented at the 28th Annual PCMI Summer Session will be held July 1–21, 2018, at the Prospector Conference Center in Park City, Utah. The residential, three-week Summer Session is the flagship activity of the IAS/Park City Mathematics Institute (PCMI), an outreach program of the Institute for Advanced Study. The 2018 research theme is Harmonic Analysis.

The PCMI Summer Session is an intensive program that includes several parallel sets of activities aimed at different groups of participants across the entire mathematics community. These activities include a program for mathematics researchers, eight mini-courses for graduate students on topics related to harmonic analysis, two lecture series for undergraduate students, a program for faculty from predominantly undergraduate institutions, a workshop on equity and mathematics education for undergraduate faculty, and a professional development program for K–12 school teachers. All of PCMI’s groups meet simultaneously, pursuing individual courses of study and a meaningful amount of interaction both scientifically and socially.