Inverse estimate of gradient of Sobolev function

Question

I need an estimate for $\| \nabla w\|_{L^2{(\Omega \subset \mathbb{R}^n)}}$, such that it is $< c\| w\|,\ w \in H_0^1(\Omega)\ $. Is this possible?

Answer 1

Let us take $\phi\in C^{\infty}_{0}(\mathbb{R}^n)\setminus\{0\}$ with $\operatorname*{supp}\phi \subset B(0;1)$. For each $1>\epsilon>0$, let us define $\phi_\epsilon\in C^{\infty}_{0}(B(0;1))\setminus\{0\}$, by $$ \phi_\epsilon (x):=\frac{1}{\epsilon^{\frac{n}{2}}}\varphi\left(\frac{x}{\epsilon}\right),\text{ for all }x\in B(0;1). $$ Clearly $\operatorname*{supp}\phi_\epsilon \subset B(0;\epsilon)$. for all $\epsilon>0$. Note that, $\Vert\phi_\epsilon\Vert_{L^2}=\Vert\phi\Vert_{L^2}$, for all $\epsilon>0$, and $$\Vert \nabla\phi_\epsilon\Vert_{L^2}=\frac{1}{\epsilon}\Vert \nabla \phi\Vert_{L^2},\text{ for all }\epsilon>0, $$ which shows that there can be no estimate of the form $$ \Vert \nabla u\Vert_{L^2}\leqslant C\Vert u\Vert_{L^2}, \text{ for all }u\in H^1_{0}(B(0;1)). $$