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Suppose that we have a sequence of functions $u_n \in W^{2,2}(\mathbb{R}^N)$, $N \geq 3$, such that

$$

\|\nabla u_n\|_{L^2(\mathbb{R}^N)} = 1

\quad \text{and} \quad

\|\Delta u_n\|_{L^2(\mathbb{R}^N)} = 1,

\quad \forall n \in \mathbb{N}.

$$

Are these assumptions enough to conclude that $\|u_n\|_{W^{2,2}(\mathbb{R}^N)} \leq C < +\infty$ for some $C > 0$ and all $n \in \mathbb{N}$?

Apart $\Delta u_n$, the norm $\|u_n\|_{W^{2,2}(\mathbb{R}^N)}$ contains also the second *mixed* derivatives, which probably could make $\|u_n\|_{W^{2,2}(\mathbb{R}^N)}$ unbounded.

However, maybe the additional assumption $\|\nabla u_n\|_{L^2(\mathbb{R}^N)} = 1$ prevents this scenario?

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Take any $u \in C_c^\infty(\mathbb{R}^N)$ and integrate by parts twice:

$$ \int |\Delta u|^2 = \int \sum_{i,j} u_{ii} u_{jj} = – \int \sum_{i,j} u_{iij} u_j = \int \sum_{i,j} u_{ij} u_{ij} = \int |Du|^2. $$

That is, $\|\Delta u\|_{L^2(\mathbb{R}^N)} = \|D^2 u\|_{L^2(\mathbb{R}^N)}$. By density of $C_c^\infty(\mathbb{R}^N)$ in $W^{2,2}(\mathbb{R}^N)$, the same equality holds for any $u \in W^{2,2}(\mathbb{R}^N)$.

We can apply Sobolev-Poincare inequality $\|u\|_{L^p(\mathbb{R}^N)} \le C_N \| Du \|_{L^2(\mathbb{R}^N)}$ with $p = \frac{2N}{N-2}$ to bound the $L^p$-norm, but the $L^2$-norm seems out of reach.

**Edit.** Previous version of this answer referred to Poincare’s inequality in $\mathbb{R}^n$ with $p=2$, which is false. It can be seen by considering the family of functions $u_\lambda(x) := u(\lambda x)$ with different values of $\lambda \in \mathbb{R}$ that the only possible value for $p$ is $p = \frac{2N}{N-2}$.

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