An extension problem for the logarithmic Laplacian
A talk in the Bielefeld Stochastic Afternoon series by
Daniel Hauer
| Abstract: | Motivated by the fact that for positive $s$ tending to zero the
fractional Laplacian $(-\Delta)^s$ converges to the identity and for $s$
tending to 1 to the (negative) local Laplacian $-\Delta$, 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, König,
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 extension problem 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) and Tobias Weth (Goethe-Universität Frankfurt, Germany) Within the CRC this talk is associated to the project(s): A5 |