Fractional derivatives are commonly used to model a variety of phenomena, but… what does it mean to have a logarithmic derivative? And what would it be used for?

In this talk we focus on the logarithmic Laplacian, a pseudodifferential operator that appears as a first order expansion of the fractional Laplacian of order 2s as s goes to zero. This operator can also be represented as an integrodifferential operator with a zero order kernel.

We will discuss how this operator can be used to study the behavior of linear and nonlinear fractional problems in the small order limit. This analysis will also reveal a deep and interesting mathematical structure behind the set of solutions of Dirichlet logarithmic problems.