Some Research Of A -harmonic Equation On Riemannian Manifold

We say that an abstract topological space is a manifold if each point in the space has a neighbourhood which is homeomorphic to an open subset of Euclid space. But from the perspective of globability, the structure of manifold may be very complicated. However, we still can simply understand and express these complicated structure analogous to Euclid spaces. Riemann manifold, as a special manifold which is equiped with Riemann metric, plays an important role in mathematical and physical fields. Therefore, the theory of Riemann manifold attracts more and more people's attentions.In this paper, using the homeomorphic mappings from manifold to Euclid space, we obtain some resules in manifold which are similar with the ones in Euclid space. By this way, on Riemann manifold we give the definition of obstacle problem for the A -harmonic equation, and prove the existence of the solution to the obstacle problem of A -harmonic equation. Then, on Riemann manifold we can get the existence of solution to A -harmonic equation. Finally, on Riemann manifold we give the Poincaréinequality and Weak Reverse H?lder inequality of A -harmonic tensor about local global and weighted integral inequality.