Consider an symmetric 2nd order differential operator on a bounded domain with smooth boundary $$A=-\sum_{i,j=1}^n \partial_j (a^{ij}(x)\partial_i)$$ be uniformly elliptic if there exists $C_0>0$ such that $\sum_{i,j=1}^n a^{ij}(x)\xi_i\xi_j \ge C_0\sum_{i=1}^n \xi_i^2$. I aim to show the following interpolation inequality: for all $\epsilon>0$, there exists $C(\epsilon)>0$ such that $$\|\partial_i u\|_{0}\le \epsilon \|Au\|_{0}+C(\epsilon)\|u\|_{0},\quad \forall u\in H^2(\Omega)\cap H^1_0(\Omega).$$
Here $\|\cdot\|_0$ denotes the $L^2$ norm and we assume $a^{ij}\in C^1(\bar{\Omega})$.
I know the inequality holds for $A$ be Laplacian, since we know for all $\epsilon>0$, there exists $C(\epsilon)>0$ such that $$\|\partial_i u\|_{0}\le \epsilon \|\partial_{ii} u\|_{0}+\dfrac{C}{\epsilon}\|u\|_{0},\quad \forall u\in H^2(\Omega)\cap H^1_0(\Omega),$$ but I don't know how to use the uniform ellipticity to get the desired result.
Let $u \in H^2 \cap H^1_0$. According to ellipticity we know that $$ C_0 \|\nabla u \|_{L^2}^2 = \int_\Omega C_0 \partial_i u \partial_i u \le \int_\Omega a^{ij} \partial_i u \partial_j u. $$ Next we integrate by parts, using the fact that $u=0$ on $\partial \Omega$: $$ \int_\Omega a^{ij} \partial_i u \partial_j u = \int_\Omega -\partial_j(a^{ij} \partial_i u) u + \int_{\partial \Omega} a^{ij} \partial_i u u n_j = \int_\Omega -\partial_j(a^{ij} \partial_i u) u. $$ Combining these and using Cauchy's inequality we find that $$ \|\nabla u\|^2_{L^2} \le \frac{1}{C_0} \int_\Omega Au \cdot u \le \int_\Omega \varepsilon |A u|^2 + \frac{1}{4 \varepsilon^2 C_0^2}|u|^2 = \varepsilon^2 \|A u\|_{L^2}^2 + \frac{1}{4\varepsilon^2 C_0^2} \|u\|_{L^2}^2. $$ Upon taking square roots and using the trivial inequality $\sqrt{x^2 +y^2} \le |x|+ |y|$ we find that $$ \|\nabla u\|_{L^2} \le \varepsilon \|A u\|_{L^2} + \frac{1}{2\varepsilon C_0} \|u\|_{L^2}, $$ which is the desired inequality.