No abstract

This paper addresses some topological and analytical issues concerning Sobolev mappings between compact Riemannian manifolds. Among the results we obtained are unified proofs of various generalizations of results obtained in a recent work of Brezis and Li. In particular we solved two conjectures in [BL]. We also give a topological obstruction for the weak sequential density of smooth maps in a given Sobolev mapping space. Finally we show a necessary and sufficient topological condition under which the smooth maps are strongly dense in the Sobolev spaces. The earlier result, Theorem 1 of [B2], was shown to be not correct.

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension two. The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities.

Abstract. For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q curvature and dimension at least 5, we prove the existence of a conformal metric with constant Q curvature. Our approach is based on the study of extremal problem for a new functional involving the Paneitz operator.

Motivated by Carleman's proof of isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper half-space. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville-type theorems.