Hello,
We know that is $W\in H^1(\mathbb{R}^n)$ then if we take the classical mollifier $m_\eta$ (its support is included in $B(0,\eta)$), we have the estimate (see Evans Partial Differential Equations for example)
$\left\| \Delta W*m_\eta\right\| \le \frac{C}{\eta} \left\| DW \right\|$
where $C$ is a constant independent of $\eta$, $DW$ is the gradient of $W$ and the norm is the $L^2$-norm.
Can we get the same type of estimate for $W_n\in\mathcal{C}_c^{\infty}(\Omega)$ (the set of functions with compact support in $\Omega$ a open subset of $\mathbb{R}^n$), knowing that $W_n$ converges to $W$ in $H^1$ ie can we find a bound for the Laplacian of $W_n$ depending only on the $H_1$-seminorm (or norm) of $W$ and the speed of convergence of $(W_n)_n$ to $W$ (we may suppose some regularity for the boundary of $\Omega$)?
Thank you in advance
