Is this a norm in the sobolev space $W_{1,p}$

Question

Consider a function

$h(f)=[\int_{\Omega} (|f|^p+ |\triangledown f|^p)dx]^{1/p}$

Where $f \in W_{1,p}$

I need to prove this is a norm, and I only struggle with the triangle inequality. So, could anyone give me ideas for that.

Any help would be appreciated!

Answer 1

Let $g(u)=(f(u), \triangledown f(u))$. Let $\|(a,b)\|'=(|a|^{p}+|b|^{p})^{1/p}$. Then your norm is $(\int \|g(u)\|'^{p})^{1/p}$. Triangle inequality should be clear from this.

Answer 2

First of all note that, thanks to the linearity of the gradient, the proof is exactly the same as for $L^p$. So we are lead to prove that the $L^p$-norm is a norm in $L^p$. In fact, withouth loss of generality let's suppose that $\Vert f+g\Vert_{L^p}\neq 0$, otherwise the inequality trivially holds. Thus, $$ \Vert f+g\Vert_{L^p}^p=\int \vert f+g\vert^p= \int \vert f+g\vert\,\vert f+g\vert^{p-1}\leq \int (\vert f\vert+\vert g\vert)\vert f+g\vert^{p-1}. $$ Expanding the right-hand side we have $$ \Vert f+g\Vert_{L^p}^p=\int \vert f\vert \,\vert f+g\vert^{p-1}+\int \vert g\vert\,\vert f+g\vert^{p-1}. $$ Now, applying Hölder's inequality with parameters $p$ and $\tfrac{p}{p-1}$ we obtain $$ \Vert f+g\Vert_{L^p}^p \leq \left(\left(\int \vert f\vert^p\right)^{1/p}+\left(\int \vert g\vert^p\right)^{1/p}\right)\left(\int\vert f+g\vert^{(p-1)(p/(p-1))}\right)^{1-1/p}. $$ Finally, note that the right-hand side is exactly $$ \qquad \qquad \qquad \qquad \qquad(\Vert f\Vert_{L^p}+\Vert g\Vert_{L^p})\dfrac{\Vert f+g\Vert_{L^p}^p}{\Vert f+g\Vert_{L^p}}. \qquad \qquad \qquad \qquad \qquad (1) $$ Thus, multiplying both sides of the inequality by the second term in $(1)$ we obtain the desired result.

Now, for the gradient terms, just note that following exactly the same procedure you can prove that $$ \Vert \nabla (f+g)\Vert_{L^p}\leq \Vert \nabla f\Vert_{L^p}+\Vert \nabla g\Vert_{L^p}. $$ I let you how to conclude from this point (which should be easy). Please let me know if you still have problems concluding.