From linear to nonlinear Calderón-Zygmund theory (Q2869423)

This is a survey article of the plenary lecture delivered by Giuseppe Mingione at the XIX-th Congress of the Italian Mathematical Union held on September 2011, and it provides a brilliant presentation of the nonlinear Calderón--Zygmund theory elaborated in the last decade mainly by the author and collaborators.NEWLINENEWLINENEWLINEStarting from the classical Calderón--Zygmund program for the Poisson equation, Mingione guides friendly the reader through its relaxation in the case of linear divergence form equations due to Campanato and Stampacchia, as well as through the sharp maximal operators approach of T.~Iwaniec to the \(p\)-Laplacian, in order to arrive at the modern issues related to regularity problems for quasilinear divergence form operators and based on the harmonic analysis free technique developed by \textit{E. Acerbi} and the author [Duke Math. J. 136, No. 2, 285--320 (2007; Zbl 1113.35105)]. The reader will find a fine analysis of the rôle played by the Wolff potentials in deriving the recent striking pointwise estimates for the weak solutions to degenerate quasilinear equations with measure data.NEWLINENEWLINEIt should be noted that Mingione succeeds, in an excellent manner, to present the philosophy of the involved techniques employed in the modern regularity theory of PDEs. That is why, the reading of the article is a real pleasure and it is highly recommended for both Ph.D.-students and professional researchers working on the subject. Complementary matter and details are available in [the author, Jahresber. Dtsch. Math.-Ver. 112, No. 3, 159--191 (2010; Zbl 1218.35104)] and [\textit{T. Kuusi} and the author, Bull. Math. Sci. 4, No. 1, 1--82 (2014; \url{doi:10.1007/s13373-013-0048-9})].