R^n的一般子集上映射的可微性与Sard定理及其应用  

Differentiability and Sard Theorem for Mappings Defined on General Subsets of R^n and Their Applications

作　　者：李德生[1] 

机构地区：[1]烟台大学数学与信息科学系,山东烟台264005

出　　处：《烟台大学学报（自然科学与工程版）》2003年第3期163-170,共8页Journal of Yantai University(Natural Science and Engineering Edition)

基　　金：国家自然科学基金资助项目(10251002);山东省自然科学基金资助项目(Y2002A10).

摘　　要：将Rn的开子集上非线性映射的导算子,一致可微性等概念推广到定义在Rn的一般子集上的映射,然后建立相应的Sard定理,并将所得结果用于一类含参数的椭圆问题:∫Ωudx=α下解的通有有限性,-Δu+f(u)=(λ), u υ=0在约束条件:m(u):=1｜Ω｜其中f严格单调递增,∈C1([0,1];L2(Ω)).我们证明存在零测集E R1使得对所有α∈R1\E,该问题只有有限个解(u,λ).The uniform differentiability as well as the classical Sard theorem are extended to nonlinear mappings which are defined on general subsets of Rn.The results are applied to establish a generic finiteness result for the elliptic problem:-Δu+f(u)=(λ),uυ=0 with a constraint:m(u):=1|Ω|∫Ωudx=α,where f is strictly increasing,and ∈C1(;L2(Ω)).It is proved that there exists a null set ER1 such that for all α∈R1\E,the problem has at most finite number of solutions (u,λ).

关 键 词：开子集 非线性映射 导算子 可微性 Sard定理 椭圆问题 通有有限性 

分 类 号：O177.91[理学—数学]