Variable Exponent Spaces Of Differential Forms And Their Applications

The theory of differential forms have been playing an important part in many fieldsof mathematics and engineering. As an extension of functions, the research on spacesof differential forms are developing rapidly. Meanwhile, many mathematicians pay theirattention to studying properties of solutions to A-harmonic equations of differential forms,which generalize A-harmonic equations of functions.Recently, a lot of nonlinear problems with variable growth are arising in naturalscience and engineering. Consequently, constant exponent spaces of differential formsappears a limitation in applications. Therefore，as generalizing constant exponent func-tion spaces to variable exponent function spaces, it is of great significance to generalizeconstant exponent spaces of differential forms to variable exponent spaces of differentialforms.In this dissertation, on one hand we expand some research on variable exponentfunction spaces to variable exponent spaces of differential forms. After discussing someproperties of variable exponent spaces of differential forms in details, we prove the ex-istence and uniqueness of solutions to some classes of nonlinear systems by using theclassical variational theory. On the other hand, the theory of variable exponent functionspaces on the Euclidean spaces is extended to that on completed Riemannian manifold-s and the existence and uniqueness of weak solutions to p（m）-harmonic equations in abounded domain of completed Riemannian manifolds are studied.The main work of this dissertation is as follows:Firstly, we introduce variable exponent Lebesgue spaces L^p（x）（, Λ^k） and exteriorSobolev spaces W^1,p（x）（, Λ^k）of differential forms. Combining the properties of Caldero′n-Zygmund operators in variable exponent function spaces, we study the properties of ho-motopic operator T in variable exponent Lebesgue spaces of differential forms, which ex-tend the results in constant exponent Lebesgue spaces of differential forms. Furthermore,the properties of spaces of differential forms κ^1,p（x）（, Λ^k）, related to exterior Sobolevspaces W^1,p（x）（, Λ^k）, are also considered. Then we show the existence of weak solutionsto a class of nonlinear system by partition of the domain.Secondly, we introduce weighted variable exponent Lebesgue spaces L^p（x）（, Λ^k, ω）and weighted exterior Sobolev spaces W^1,p（x）（, Λ^k, ω） of differential forms. We get the existence and uniqueness of solutions to a class of obstacle problems after discussingsome properties of these spaces. Furthermore, the existence and uniqueness of weaksolutions to Dirichlet problems for nonhomogeneous A-harmonic equations with variablegrowth are obtained.Finally, we introduce variable exponent Lebesgue spaces L^p（m）（Λ^kM）and exteriorSobolev spaces W^1,p（m）（Λ^kM） of differential forms on completed Riemannian manifolds.By applying Vitali convergence theorem, Lebesgue dominated convergence theorem, Luzintheorem and etc, many important properties of spaces are discussed in details, such ascompleteness, separability, reflexivity and absolute convergence. We also establish thecompact imbedding theorem from W^1,p（m）（Λ^kM） into L^q（m）（Λ^kM） under certain condition-s on variable exponent p（m） and q（m）. Then we show the existence and uniqueness ofweak solutions to Dirichlet problems for nonhomogeneous p（m）-harmonic equations ina bounded domain of completed Riemannian manifolds. In particular, we generalize thetheory of variable exponent function spaces on Euclidean spaces to the case on Rieman-nian manifolds.