The book has been completely rewritten for this new edition. While most of the material found in the earlier editions has been retained, though in changed form, there are considerable additions, in which extensive use is made of Fourier transform techniques, Hilbert space, and finite difference methods. A condensed version of the present work was presented in a series of lectures as part of the Tata Institute of Fundamental Research -Indian Insti- tute of Science Mathematics Programme in Bangalore in 1977. I am indebted to Professor K. G. Ramanathan for the opportunity to participate in this excit- ing educational venture, and to Professor K. Balagangadharan for his ever ready help and advice and many stimulating discussions. Very special thanks are due to N. Sivaramakrishnan and R. Mythili, who ably and cheerfully prepared notes of my lectures which I was able to use as the nucleus of the present edition. A word about the choice of material. The constraints imposed by a partial differential equation on its solutions (like those imposed by the environment on a living organism) have an infinite variety of con- sequences, local and global, identities and inequalities. Theories of such equations usually attempt to analyse the structure of individual solutions and of the whole manifold of solutions by testing the compatibility of the differential equation with various types of additional constraints.