Second-Order Equations with Nonnegative Characteristic Form

ISBN-13: 9781468489675 / Angielski / Miękka / 2012 / 259 str.

O. Oleinik

Opis

Second order equations with nonnegative characteristic form constitute a new branch of the theory of partial differential equations, having arisen within the last 20 years, and having undergone a particularly intensive development in recent years. An equation of the form (1) is termed an equation of second order with nonnegative characteristic form on a set G, kj if at each point x belonging to G we have a (xHk j 0 for any vector = ( l' ... ' m)' In equation (1) it is assumed that repeated indices are summed from 1 to m, and x = (x l' ---, x ). Such equations are sometimes also called degenerating m elliptic equations or elliptic-parabolic equations. This class of equations includes those of elliptic and parabolic types, first order equations, ultraparabolic equations, the equations of Brownian motion, and others. The foundation of a general theory of second order equations with nonnegative characteristic form has now been established, and the purpose of this book is to pre- sent this foundation. Special classes of equations of the form (1), not coinciding with the well-studied equations of elliptic or parabolic type, were investigated long ago, particularly in the paper of Picone 105], published some 60 years ago.

Spis treści

I. The First Boundary Value Problem.- 1. Notation. Auxiliary results. Formulation of the first boundary value problem.- 2. A priori estimates in the spaces Lp (?).- 3. Existence of a solution of the first boundary value problem in the spaces Lp (?).- 4. Existence of a weak solution of the first boundary value problem in Hilbert space.- 5. Solution of the first boundary value problem by the method of elliptic regularization.- 6. Uniqueness theorems for weak solutions of the first boundary value problem.- 7. A lemma on nonnegative quadratic forms.- 8. On smoothness of weak solutions of the first boundary value problem. Conditions for existence of solutions with bounded derivatives.- 9. On conditions for the existence of a solution of the first boundary value problem in the spaces of S. L. Sobolev.- II. On the Local Smoothness of Weak Solutions and Hypoellipticity of Second Order Differential Equations.- 1. The spaces Hs.- 2. Some properties of pseudodifferential operators.- 3. A necessary condition for hypoellipticity.- 4. Sufficient conditions for local smoothness of weak solutions and hypoellipticity of differential operators.- 5. A priori estimates and hypoellipticity theorems for the operators of Hörmander.- 6. A priori estimates and hypoellipticity theorems for general second order differential equations.- 7. On the solution of the first boundary value problem in nonsmooth domains. The method of M. V. Keldyš.- 8. On hypoellipticity of second order differential operators with analytic coefficients.- III. Additional Topics.- 1. Qualitative properties of solutions of second order equations with non- negative characteristic form.- 2. The Cauchy problem for degenerating second order hyperbolic equations.- 3. Necessary conditions for correctness of the Cauchy problem for second order equations.