Student Harmonic Analysis and PDE Seminar (HADES): Zeroes of harmonic functions and propagation of smallness
Seminar | March 6 | 3:30-5 p.m. | 740 Evans Hall
Aleksandr Logunov, Institute for Advanced Study
The classical Liouville theorem claims that any positive harmonic function in $R^n$ is a constant function. Nadirashvili conjectured that any non-constant harmonic function in $R^3$ has a zero set of infinite area. The conjecture is true and the following principle holds for harmonic functions: "the faster the function grows the bigger the area of its zero set is" and vice versa. Propagation of smallness techniques are useful to study zero sets of elliptic PDE. We will discuss two useful tools: three spheres theorem and quantitative Cauchy uniqueness.