Type: Seminar

Distribution: World

Expiry: 18 Dec 2023

CalTitle1: An extension problem for the logarithmic Laplacian

Auth: dominic@al463fm6pfv.staff.wireless.sydney.edu.au (ddim8352) in SMS-SAML

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\â€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\â€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