Asia-Pacific Analysis and PDE Seminar: Hauer -- An extension problem for the logarithmic Laplacian

On Monday, 18 December 2023 at 

- 12 PM for Beijing, Hong Kong and Perth 
- 1 PM for Seoul and Tokyo 
- 3 PM for Canberra, Melbourne and Sydney 
- 5 PM for Auckland 

Daniel Hauer (@ Sydney University, Australia) is speaking at the Asia-Pacific Analysis 

Title: An extension problem for the logarithmic Laplacian 

Abstract: Motivated by the fact that for positive s tending to zero the fractional
Laplacian converges to the identity and for s tending to 1 to the local Laplacian, Chen
and Weth [Comm.  PDE 44 (11), 2019] introduced the logarithmic Laplacian as the first
variation of the fractional Laplacian at s=0.  In particular, they showed that the
logarithmic Laplacian admits an integral representation and can, alternatively, be
defined via the Fourier-transform with a logarithmic symbol.  The logarithmic Laplacian
turned out to be an important tool in various mathematical problems; for instance, to
determine the asymptotic behavior as the order s tends to zero of the eigenvalues of the
fractional Laplacian equipped with Dirichlet boundary conditions (see, e.g., [Feulefack,
Jarohs, Weth, J.  Fourier Anal.  Appl.  28(2), no.  18, 2022]), in the study of the
logarithmic Sobolev inequality on the unit sphere [Frank, K\”onig, Tang, Adv.  Math.
375, 2020], or in the geometric context of the 0-fractional perimeter, see [De Luca,
Novaga, Ponsiglione, ANN SCUOLA NORM-SCI 22(4), 2021].  Caffarelli and Silvestre [Comm.
Part.  Diff.  Eq.  32(7-9), (2007)] showed that for every sufficiently regular $u$, the
values of the fractional Laplacian at $u$ can be obtained by the co-normal derivative of
an s-harmonic function $w_u$ on the half-space (by adding one more space dimension) with
Dirichlet boundary data $u$.  This extensionproblem represents the important link
between an integro-differential operator (the nonlocal fractional Laplacian) and a local
2nd-order differential operator.  This property has been used frequently in the past in
many problems governed by the fractional Laplacian.  

In this talk, I will present an extension problem for the logarithmic Laplacian, which
shows that this nonlocal integro-differential operator can be linked with a local
Poisson problem on the (upper) half-space, or alternatively (after reflection) in a
space of one more dimension.  As an application of this extension property, I show that
the logarithmic Laplacian admits a unique continuous property.  

The results presented here were obtained in joint work with Huyuan Chen (Jiangxi Normal
University, China \& The University of Sydney, Australia) and Tobias Weth
(Goethe-Universit\”at Frankfurt, Germany) 

To join this Zoom Webinar, you can copy and paste the following link into your internet
browser: https://uni-sydney.zoom.us/j/81441242510 

More information and how to attend this talk can be found at the seminar webpage