*(English)*Zbl 1042.35002

From the preface: This revision of the 1983 second edition (see Zbl 0562.35001) corresponds to the Russian edition, published in 1989 [Nauka, Moscow; Zbl 0691.35001], in which we essentially updated the previous version to 1984. The additional text relates to the boundary Hölder derivative estimates of Nikolai Krylov, which provided a fundamental component of the further development of the classical theory of elliptic (and parabolic), fully nonlinear equations in higher dimensions. In our presentation we adapted a simplification of Krylov’s approach due to Luis Caffarelli.

The theory of nonlinear elliptic second order equations has continued to flourish during the last fifteen years and, in a brief epilogue to this volume, we signal some of the major advances. Although a proper treatment would necessitate at least another monograph, it is our hope that this book, most of whose text is now more than twenty years old, can continue to serve as background for these and future developments.

Since our first edition (see the review in Zbl 0361.35003) we have become indebted to numerous colleagues, all over the globe. It was particularly pleasant in recent years to make and renew friendships with our Russian colleagues, Olga Ladyzhenskaya, Nina Ural’tseva, Nina Ivochkina, Nikolai Krylov and Mikhail Safonov, who have contributed so much to this area. Sadly, we mourn the passing away in 1996 of Ennio De Giorgi, whose brilliant discovery forty years ago opened the door to higher-dimensional nonlinear theory.

##### MSC:

35-02 | Research monographs (partial differential equations) |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35B45 | A priori estimates for solutions of PDE |

35J25 | Second order elliptic equations, boundary value problems |

35B50 | Maximum principles (PDE) |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. (PDE) |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |