Sketches of nonlinear Calderón-Zygmund theory. (Q2846679)

In the article an up-to-date review of results, largely by the author of the article, on regularity of weak solutions to partial differential equations and systems of \(p\)-Laplace type, \(p>1\), is presented. The theory is clearly explained with references to original articles where the results are proved and with lot of remarks connecting the linear and nonlinear theory.NEWLINENEWLINEThe article is split into 6 sections. After the introductory first section the questions of maximal \(L^q\) regularity of energy solutions to stationary and evolutionary problems is considered in the second section. The rest of the article deals with the stationary theory. The next two sections are devoted to the situation when the right-hand side functional prevents the solutions from satisfying energy estimates. First, the notion of a solution obtained by limits of approximations (SOLA) is introduced and then the regularity of such solutions is studied in the classes of Lebesgue, Morrey, Lorentz and Morrey-Lorenz spaces. The next section deals with differentiability of SOLA and the last section addresses the pointwise estimates of solutions by means of Wolff potentials.NEWLINENEWLINEFor the entire collection see [Zbl 1262.35002].