SMS scnews item created by Dominic Dimech at Fri 15 Dec 2023 1644
Type: Seminar
Distribution: World
Expiry: 18 Dec 2023
Calendar1: 18 Dec 2023 1500-1600
CalLoc1: Zoom webinar
CalTitle1: An extension problem for the logarithmic Laplacian
Auth: (ddim8352) in SMS-SAML

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

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

ball Calendar (ICS file) download, for import into your favourite calendar application
ball UNCLUTTER for printing
ball AUTHENTICATE to mark the scnews item as read
School members may try to .