论文标题
在Sobolev空间和鳍歧管上的密度定理上
On Sobolev spaces and density theorems on Finsler manifolds
论文作者
论文摘要
在这里，为芬斯勒结构$ f $定义了Sobolev空间的自然扩展，并且显示出所有真正的$ c^{\ infty} $的集合在正向上完整的Finsler verold $（m，f）$上具有紧凑的支持，在Extended Sobolev sobolev space $ h_1^p（m）中是密集的。结果，dirichlet方程的弱解决方案$ u $ u $ΔU= f $可以由$ c^\ infty $函数近似，并在$ m $上进行紧凑的支持。此外，让$ w \ subset m $是$ c^r $ bouncy $ \ partial w $的常规域，然后在$ c^r（w）\ cap c^0（\ overline w）$ in $ h_k^p（w）$中，其中$ c^r（w）\ cap c^0（\ cap c^0）中的所有真实功能集。最后，说明了几个示例，并显示了不平等$ k \ leq r $的清晰度。
Here, a natural extension of Sobolev spaces is defined for a Finsler structure $F$ and it is shown that the set of all real $C^{\infty}$ functions with compact support on a forward geodesically complete Finsler manifold $(M, F)$, is dense in the extended Sobolev space $H_1^p (M)$. As a consequence, the weak solutions $u$ of the Dirichlet equation $Δu=f$ can be approximated by $C^\infty$ functions with compact support on $M$. Moreover, let $W \subset M$ be a regular domain with the $C^r$ boundary $\partial W$, then the set of all real functions in $C^r (W) \cap C^0 (\overline W)$ is dense in $H_k^p (W)$, where $k\leq r$. Finally, several examples are illustrated and sharpness of the inequality $k\leq r$ is shown.
