Perlen-Kolloquium: Giuseppe Mingione (Università di Parma)

From linear to nonlinear Calderòn-Zygmund Theory

Calderòn-Zygmund theory lies at the core of the classical analysis of linear partial differential equations of the past century. The basic question it tries to address can be put as: Given an equation with assigned data, how the qualitative properties of solutions are influenced by those of the data? In other words, if the given data have some degree of regularity, how such a regularity reflects in that of the solutions? Optimal answers have been given in case the equations considered are linear, starting with the seminal work of Calderòn and Zygmund in the fifties. The last years have witnessed a great effort towards a parallel theory that tries to get the same conclusions for nonlinear equations, obviously via different methods. I will try to give an overview of the whole development, starting by the classics, finally coming to the most recent results.

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