Geometric PDE and Applied Analysis Seminar (January 25, 2024)

Title: Bernstein functional calculus and a generalized Helmholtz problem

Speaker: Prof. Daniel Hauer (University of Sydney)

Abstract: 
In this talk, I aim to characterize all distributional solutions of the generalized Helmholtz equation 
$f(-\Delta)u=f(k^2)u$
on the Euclidean space $\mathbb{R}^d$ for every real $k\neq 0$ and a non-constant Berstein function $f$. Note, that $f(-\Delta)$ is a non-local operator and the prototype would be the fractional operator $(-\Delta)^s$ for $0 < s <1$. To attack this problem, we first need to introduce a notion of distributional solutions of the generalized Helmholtz equation. This involves showing that the negative Laplacian is non-negative on a Lizorkin space.
The results presented in this talk are obtained in joint work with Robert Denk (University of Konstanz, Germany) and David Lee (Laboratoire Jacques-Louis Lions, Paris, France).