Second order equations of elliptic and parabolic type. Transl. from the Russian by T. Rozhkovskaya.

*(English)*Zbl 0895.35001
Translations of Mathematical Monographs. 171. Providence, RI: American Mathematical Society (AMS). xii, 203 p. (1998).

This book is based on a series of lectures given by the author in 1967-1968 at Moscow State University. It presents a self-contained and unified treatment of several aspects of the theory of elliptic and parabolic differential equations of the second order.

The first chapter is concerned with elliptic equations in nondivergence form whose solutions are studied using subfundamental and superfundamental solutions which are constructed by means of the Riesz potential. The qualitative behavior of solutions near boundary points (Wiener type theorems) and at infinity (theorems of Phragmen-Lindelöf type and Liouville’s theorem) is derived from a basic growth lemma. The same technique is used to obtain a priori estimates for the Hölder norms of solutions and to prove the Harnack inequality under the so called Cordes condition \(n\leq e<n+2\), where \(e\) is the ellipticity constant, and \(n\) is the dimension of the space. These estimates are used to establish the existence of a solution to a boundary value problem for quasilinear equations.

The second chapter is devoted to elliptic equations in divergence form. Weak solutions of the Dirichlet problem are defined and their existence established by variational methods. Hölder continuity of weak solutions (established first by De Giorgi) is poved by means of the methods developed in the first chapter. The third chapter is devoted to parabolic problems. The material presented in this chapter is a natural extension of the material presented in the two preceeding chapters. It includes weak and strong maximum principles and uniqueness results. An appendix provides some additional results, such as the proof of the fixed point theorem of Schauder, the isoperimetric inequality, and the Schauder estimates. This book is an original and accessible account on the topics of elliptic and parabolic partial differential equations. Each section contains a number of bibliographic remarks.

The first chapter is concerned with elliptic equations in nondivergence form whose solutions are studied using subfundamental and superfundamental solutions which are constructed by means of the Riesz potential. The qualitative behavior of solutions near boundary points (Wiener type theorems) and at infinity (theorems of Phragmen-Lindelöf type and Liouville’s theorem) is derived from a basic growth lemma. The same technique is used to obtain a priori estimates for the Hölder norms of solutions and to prove the Harnack inequality under the so called Cordes condition \(n\leq e<n+2\), where \(e\) is the ellipticity constant, and \(n\) is the dimension of the space. These estimates are used to establish the existence of a solution to a boundary value problem for quasilinear equations.

The second chapter is devoted to elliptic equations in divergence form. Weak solutions of the Dirichlet problem are defined and their existence established by variational methods. Hölder continuity of weak solutions (established first by De Giorgi) is poved by means of the methods developed in the first chapter. The third chapter is devoted to parabolic problems. The material presented in this chapter is a natural extension of the material presented in the two preceeding chapters. It includes weak and strong maximum principles and uniqueness results. An appendix provides some additional results, such as the proof of the fixed point theorem of Schauder, the isoperimetric inequality, and the Schauder estimates. This book is an original and accessible account on the topics of elliptic and parabolic partial differential equations. Each section contains a number of bibliographic remarks.

Reviewer: G.Philippin (Quebec)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35K10 | Second-order parabolic equations |

35J15 | Second-order elliptic equations |

35Bxx | Qualitative properties of solutions to partial differential equations |