Show a function is Sobolev

Question

Let $T_{h}$ be a subdivision of a domain $\Omega \subset \mathbb{R}^d$ into elements $K$ with boundary $\delta K$so that the Gauss divergence theorem holds.

If for a function $f$ it holds that $f \in C(\bar \Omega)$ such that $f \in C^\infty(\bar K)$ for all $K \in T_h$, then $f \in W^{1,2}( \Omega)$.

I am fairly new to this topic. Can anyone please help/explain/give me an idea how to show this?

Thank you!

Answer 1

In order to show that $f \in W^{1,2}(\Omega)$, we have to show that it possesses weak derivatives. Recall that $v_i$ is $\partial_i f$ (in a weak sense), if $$\int_\Omega f \, \partial_i \varphi \, \mathrm{d}x = - \int_\Omega v_i \, \varphi \, \mathrm{d}x$$ for all $\varphi \in C_0^\infty(\Omega)$. Now, let $\varphi \in C_0^\infty(\Omega)$ be given. We have $$\int_\Omega f \, \partial_i \varphi \, \mathrm{d}x = \sum_K \int_K f \, \partial_i \varphi \, \mathrm{d}x.$$ Now, you should

• use Gauß divergence theorem on each $K$
• utilize that $K$ is a subdivision to get rid of the boundary integrals over $\delta K$
• write everything as an integral over $\Omega$.

If everything works, you arrive at $$\int_\Omega f \, \partial_i \varphi \, \mathrm{d}x = -\int_\Omega \partial_i f \, \varphi \, \mathrm{d} x,$$ where $\partial f_i$ is defined piecewise on each element $K$.