Significant differences exist between the Laplacian and the bi-Laplacian, particularly regarding the behavior of solutions under Dirichlet boundary conditions. Notably, the bi-Laplacian lacks the maximum principle. This presentation will explore the transition from the Laplacian to the bi-Laplacian through fractional powers of the Laplacian. We will discuss several unique characteristics of this class of operators and their associated function spaces, including the maximum principle, polarizations of functions, and the Faber–Krahn inequality in one dimension.

This presentation is based on a series of works with Alberto Saldana (University of Mexico) and Nicola Abatangelo (University of Bologna).