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).