I would like to know if there is a result like this:
Given $\Omega\in \mathbb{R}^d$ a bounded domain where we can apply Poincaré and Sobolev inequalities, we consider $f:\Omega\to \mathbb{R}$ such that $f_{| \partial\Omega}=0$. Prove that there exists a constant $C$ non depending on $d$ such that $$d\|f\|_{L^p(\Omega)}^2\leq C \sum_{k=1}^d\int |\partial_{x_k}f|^2dx_1\dots dx_d,$$ where $p=\frac{d-2}{2d}$.
My attempt
By sobolev embedding I know that there exists $C=C(p,d)$ such that $\|f\|_{L^p(\Omega)}\leq C\|f\|_{W^{1,2}(\Omega)}$, with $p=\frac{d-2}{2d}$. Now, if we use the Poincaré inequality we obtain a new constant $C=C(p,d)$ such that $\|f\|_{L^p(\Omega)}\leq C \|\nabla f\|_{L^2(\Omega)} $. Does anyone know how to conclude from here or a different proof of this result?