Seminar Analysis: Tobias Weth (University of Frankfurt)

The Dirichlet Problem for the Logarithmic Laplacian

I will report on some recent results - obtained in joint work with Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Operator, which arises as formal derivative of fractional Laplacians at order s= 0. I will discuss the functional analytic framework for these problems and show how it allows to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore, I will discuss necessary and sufficient conditions on domains giving rise to weak and strong maximum principles for the logarithmic Laplacian. If time permits, I will also discuss regularity estimates for solutions to corresponding Poisson problems.

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