I am interested in finding a direct yes or no answer for the following:
In general, on a bounded domain $\Omega$ in $R^n$ with say $C^1$ boundary, can we that say any function $f$ $\in$ $C_c^{\infty}$($\Omega$), where $C_c^{\infty}$($\Omega$) is the space of all compactly supported smooth functions defined on $\Omega$, must also belong to a sobolev space $W^{k,p}(\Omega)$, for $1 \le p < \infty$?
I appreciate anyone's help, and enlightenment. If not can someone please give an example?
Thanks,
Sandy
Best Answer
Yes, this is the case. You can show that for any $\Omega \subset \mathbb R^n$ open and any $k \in \mathbb N$, $1 \leq p \leq \infty$ the inclusion $C_c^\infty \subset W^{k,p}$ holds true. First acknowledge that the case $k=0$ is trivial. Then convince yourself that the case $k=1$ contains all the difficulty. Now to see this for $k=1$ let $u,\varphi \in C_c^\infty$, pick $i = 1,...,n$, you want to find $v \in L^p$ such that $$ \int_\Omega u \frac{\partial \varphi}{\partial x_i} = - \int_\Omega v \varphi. $$ If life was good then $v = \frac{\partial u}{\partial x_i}$ the classical partial derivative would be ok. And indeed that's the case, you can first observe $$ \int_\Omega u \frac{\partial \varphi}{\partial x_i} + \int_\Omega \frac{\partial u}{\partial x_i} \varphi = \int_\Omega \frac{\partial (u\varphi)}{\partial x_i}. $$ At this point either you know integration by part (Gauss Green formula, divergence theorem or Stokes formula) or you can show by basic real analysis that $$ \forall \psi \in C^1_c(\Omega),\quad \int_\Omega \frac{\partial \psi}{\partial x_i} = 0. $$