Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs – such as the Krylov–Safonov and Evans–Krylov results, Caffarelli's regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ – and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.

‘The regularity theory of elliptic partial differential equations is one of the bedrocks of modern mathematics since it elegantly and creatively uses virtually all possible mathematical tools to construct a solid set of concepts with ubiquitous applications. This book tells a story about this regularity theory, especially from the point of view of viscosity solutions for fully nonlinear equations and in the light of perturbative methods. As all good stories, the important part is not the happy ending in itself, but the whole plot through the series of adventures and vicissitudes (namely, the beautiful theorems) in which the reader will be captured page after page.'

Enrico Valdinoci - University of Western Australia