Let $U$ be a bounded domain in the Euclidean space with sufficiently smooth boundary. Let $\{f_i\}$ be a orthonormal basis of $H^1_0(U)$ satisfying $-\Delta f_i = \lambda_i f_i$ where $\lambda_i \leq \lambda_{i+1}$.

Think of a fixed natural number $N$ and let $V_N$ be a subspace of $H^1_0(U)$ spanned by $f_1, ..., f_N$. Let $u$ be an element of $H^1_0(U) \cap H^2(U)$ and $\pi_N(u)$ be its orthogonal projection onto $V_N$.

I want to show that $\int_U \mid \nabla u - \nabla \pi_N(u) \mid ^2 \leq C \int_U \mid \nabla \nabla u \mid ^2$ where $C$ is specifically chosen to be $\frac{1}{\lambda_N}$.

How is this possible? I can show that there exists such a $C$ via argument by contradiction. But, I cannot find a way to 'construct' a specific $C$. Could anyone help me?

Let $U$ be a bounded domain in the Euclidean space with sufficiently smooth boundary. Let $\{f_i\}$ be a orthonormal basis of $H^1_0(U)$ satisfying $-\Delta f_i = \lambda_i f_i$ where $\lambda_i \leq \lambda_{i+1}$.

For fixed $N\in\mathbb N$ let $V_N$ be the subspace of $H^1_0(U)$ spanned by $\{f_1, \dots , f_N\}$. Let $u$ be an element of $H^1_0(U) \cap H^2(U)$ and $\pi_N(u)$ be its $L^2$ orthogonal projection onto $V_N$.

I want to show that $$ \int_U \mid \nabla u - \nabla \pi_N(u) \mid ^2 \leq C \int_U \mid \nabla \nabla u \mid ^2 $$ where $C$ is specifically chosen to be $\frac{1}{\lambda_N}$ and $\nabla\nabla u$ is the Hessian of $u$.

How is this possible? I can show that there exists such a $C$ via argument by contradiction. But, I cannot find a way to 'construct' a specific $C$. Could anyone help me?