Researches On Applications Of Differential Forms In A-harmonic Equations On Riemannian Manifolds

The theory of differential forms on Riemannian manifolds plays an important role in global analysis of Riemannian manifolds, partial differential equations, physics and mechanics etc. The A-harmonic equation is a class of quasilinear partial elliptic equations of second order, it is widely used in the potential theory, fluid mechanics, elasticity theory, the theory of relativity, the electromagnetic field, plasticity, differential geometry and geometric function theory etc. It is therefore very important and meaningful to study A-harmonic equations with the theory of differential forms. In this dissertation we focus on the A-harmonic equation of differential forms on Riemannian manifolds, and its generalized Dirichlet boundary value problems, obstacle problems and variational problems on Riemannian manifolds with boundary. The research includes the following four aspects:Firstly, due to the divergence structure of A-harmonic equations, operator theory can be applied to study the properties of solutions. By Morrey’s lemma and isoperimetric inequality on Riemannian manifolds, we prove the H ¨older continuity for A-harmonic tensors on compact subsets of manifolds.Secondly, we consider the generalized Dirichlet problems of A-harmonic equations on Riemannian manifolds with boundary. Based on the solvability, we prove an integral estimate and a weak reverse H ¨older inequality. We also define the weak convergence for sequences of differential forms and prove the stability of solutions about the varying nonhomogeneous terms.Thirdly, by the definition of A-harmonic tensors and properties of the operator A, we study the weakly A-harmonic tensors with integrability below the natural exponent p. We prove the Poincar′e-Sobolev inequality on Riemannian manifolds, and together with the Hodge decomposition of Lpdifferential forms inRn, a weak reverse H ¨older inequality on an open subset ? ofRnis obtained. Furthermore, based on the close connection between A-harmonic equation and obstacle problem, we consider the ψ-obstacle problem of its generalized Dirichlet problems on a bounded regular domain ? inRn. We prove the existence and uniqueness of its very weak solution, and prove that it quasiminimizes the rDirichlet integral on ??r,θ?,ψp?, Λl′1q. We also establish the stability of very weak solutions about varying obstacle differential forms.Finally, note that the variational structure of A-harmonic equations, we can apply the calculus of variations to prove the existence of solutions. We generalize the variational methods to spaces of differential forms and prove the existence of minimizers of the corresponding extreme value problem. This is also an important application of the theory of differential forms.