Abstract. Let Γ be an H-group. In 1974 Marvin Knopp conjectured that the Eichler cohomology group, with base space taken to be the set of all functions holomorphic in the upper half-plane, of polynomial growth at the real line (including ∞), and with a weight k,multiplier system v linear fractional action of Γ, is isomorphic to the space of cusp forms on Γ of weight 2 − k and multiplier system v, in the range 0 < k < 2. In this article the authors prove the conjecture by making essential use of Hans Petersson's "principal parts condition" for automorphic forms (1955).
We study PDE of the form max{F (D 2 u, x) − f (x), H(Du)} = 0 where F is uniformly elliptic and convex in its first argument, H is convex, f is a given function and u is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients. We also generalize previous results known for the case where M → F (M, x) is the maximum of finitely many linear functions.
We consider the problem of approximating Nash equilibria of N functions f 1 , . . . , f N of N variables. In particular, we deduce conditions under which systems of the form uj (t) = −∇ x j f j (u(t)) (j = 1, . . . , N ) are well posed and in which the large time limits of their solutions u(t) = (u 1 (t), . . . , u N (t)) are Nash equilibria for f 1 , . . . , f N . To this end, we will invoke the theory of maximal monotone operators. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces.
Let be a domain in R n and suppose, locally, there exists a function u which, in the sense of distributions, solves the problemwhere B 1 (0) is the unit ball in R n and L is a fourth order elliptic operator, i.e. |α|=4 a α ξ α = 0 for ξ ∈ R n , ξ = 0. With no sign assumption on u and by using blow-up techniques we will prove that u is in C 3,1 loc .
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