[Seminar] "Bernstein functional calculus and a generalized Helmholtz problem" by Prof. Daniel Hauer
Date
Location
Description
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).
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