论文标题
Sobolev函数无紧凑的近似值
Sobolev functions without compactly supported approximations
论文作者
论文摘要
Sobolev空间理论中的基底属性和有用的工具是空间中平滑紧凑型功能的密度，$ w^{k，p}（\ r^n）$（即，在$ l^p $中具有较弱订单$ 0 $ $ 0 $ $ k $的功能$ 0 $ 0 $ $ k $）。在Riemannian歧管上，众所周知，在合适的几何假设下，同一特性仍然有效。但是，正如我们在本文中所证明的那样，在完整的非压缩歧管上，它一般而言。这解决了例如E. Hebey [\ textit {非线性分析：sobolev空间和不等式}，《数学中的courant entuct》，第1卷，第1卷。 5，1999，第48-49页]。
A basilar property and a useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}(\R^n)$ (i.e. the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On Riemannian manifolds, it is well known that the same property remains valid under suitable geometric assumptions. However, on a complete non-compact manifold it can fail to be true in general, as we prove in this paper. This settles an open problem raised for instance by E. Hebey [\textit{Nonlinear analysis on manifolds: Sobolev spaces and inequalities}, Courant Lecture Notes in Mathematics, vol. 5, 1999, pp. 48-49].
