Memoirs of the American Mathematical Society 1994; 78 pp; softcover Volume: 112 ISBN10: 0821826042 ISBN13: 9780821826041 List Price: US$37 Individual Members: US$22.20 Institutional Members: US$29.60 Order Code: MEMO/112/540
 This work studies equivariant linear second order elliptic operators \(P\) on a connected noncompact manifold \(X\) with a given action of a group \(G\). The action is assumed to be cocompact, meaning that \(GV=X\) for some compact subset \(V\) of \(X\). The aim is to study the structure of the convex cone of all positive solutions of \(Pu=0\). It turns out that the set of all normalized positive solutions which are also eigenfunctions of the given \(G\)action can be realized as a real analytic submanifold \(\Gamma _0\) of an appropriate topological vector space \(\mathcal H\). When \(G\) is finitely generated, \(\mathcal H\) has finite dimension, and in nontrivial cases \(\Gamma _0\) is the boundary of a strictly convex body in \(\mathcal H\). When \(G\) is nilpotent, any positive solution \(u\) can be represented as an integral with respect to some uniquely defined positive Borel measure over \(\Gamma _0\). Lin and Pinchover also discuss related results for parabolic equations on \(X\) and for elliptic operators on noncompact manifolds with boundary. Readership Analysts, specialists in partial differential equations and mathematical physics, and graduate students in analysis. Table of Contents  Introduction
 Some notions connected with group actions
 Some notions and results connected with elliptic operators
 Elliptic operators and group actions
 Positive multiplicative solutions
 Nilpotent groups: extreme points and multiplicative solutions
 Nonnegative solutions of parabolic equations
 Invariant operators on a manifold with boundary
 Examples and open problems
 Appendix: analyticity of \(\Lambda (\xi , \scr L)\)
 References
