论文标题
由双椭圆形问题引起的索波列夫空间中的常规功能近似
Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems
论文作者
论文摘要
本文处理了$ c^m（\overlineΩ）$函数的非平凡密度结果，并在{\ mathbb n} \ cup \ cup \ {\ infty \} $中，在space $ w^{k，k，\ ell，p}（e el，p）中u_ {|γ} \ in w^{\ ell，p}（γ）\ right \}，$$在$（u，u _ {|γ}）$的规范中，$（u，u_ {|γ}）$ in $ w^{k，p}（k，p}（ω）（ω）\ times w^{\ ell，p} $ suble， r}^n $，$ n \ ge 2 $，带有$ c^m $的边界$γ$，$ k \ le \ el \ el \ le m $和$ 1 \ le p <\ infty $。 当处理涉及两个椭圆运算符的双重椭圆问题时，这一点令人感兴趣，一个$ω$，另一个在$γ$中。 此外，当$γ$的相对开放的部分施加了差异均匀的边界条件时，我们还将考虑这种情况，并且作为初步步骤，当$ω= {\ mathbb r}^n $或$ ch = {\ MathbB r}^n $或$ ch = = {\ Mathb r}^n n _+$+$ $和$ $ $ $ r r ry时，我们将证明是类似的结果。 \关键字{密度结果\和sobolev spaces \以及光滑函数\以及laplace-beltrami操作员。
The paper deals with a nontrivial density result for $C^m(\overlineΩ)$ functions, with $m\in{\mathbb N}\cup\{\infty\}$, in the space $$W^{k,\ell,p}(Ω;Γ)= \left\{u\in W^{k,p}(Ω): u_{|Γ}\in W^{\ell,p}(Γ)\right\},$$ endowed with the norm of $(u,u_{|Γ})$ in $W^{k,p}(Ω)\times W^{\ell,p}(Γ)$, where $Ω$ is a bounded open subset of ${\mathbb R}^N$, $N\ge 2$, with boundary $Γ$ of class $C^m$, $k\le \ell\le m$ and $1\le p<\infty$. Such a result is of interest when dealing with doubly elliptic problems involving two elliptic operators, one in $Ω$ and the other on $Γ$. Moreover we shall also consider the case when a Dirichlet homogeneous boundary condition is imposed on a relatively open part of $Γ$ and, as a preliminary step, we shall prove an analogous result when either $Ω={\mathbb R}^N$ or $Ω={\mathbb R}^N_+$ and $Γ=\partial{\mathbb R}^N_+$. \keywords{Density results\and Sobolev spaces \and Smooth functions \and the Laplace--Beltrami operator.
