Author	Oleฤญnik, O. A. author
Title	Second Order Equations With Nonnegative Characteristic Form [electronic resource] / by O. A. Oleฤญnik, E. V. Radkeviฤ{141}
Imprint	Boston, MA : Springer US, 1973
Connect to	http://dx.doi.org/10.1007/978-1-4684-8965-1
Descript	VII, 259 p. online resource

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 (xHk̃j ̃ 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' โ{128}ขโ{128}ขโ{128}ข , 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 preยญ 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

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 Hรถrmander -- 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. Keldyลก -- 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