Sobolev空间W^(m,p)(Ω)中残差泛函的极值理论

An Extremum Theory of the Residual Functional in Sobolev Spaces W^(m,p)(Ω)

本文在Sobolev空间中讨论残差泛函J(u)的概念及性质,论证了残差泛函J(u)的弱紧性、强制性和下半连续性及凸性条件.根据临界点理论在Sobolev空间中建立起该残差泛函的极值原理,给出J(u)=0极小值存在定理.此外还证明了等价定理和J(R_n(c))=0的五种等价形式.

In the present paper the concept and properties of the residual functional in Sobolev space are investigated. The -weak compactness, force condition, lower semi-continuity and convex of the residual functional are proved. In Sobolev space, the minimum principle of the residual functional is proposed. The minimum existence theoreom for J(u) = 0 is given by the modern critical point theory. And the equivalence theorem or five equivalence forms for the residual functional equation are also proved.