Continuity of the $L^2$-norm with respect to a parameter

Question

Let $\psi:\mathbb R^d\times (0,\infty)\to\mathbb (0,\infty)$ be a Holder-continuous function such that $$ \psi(\cdot,t)\in W^{1,2}(\mathbb R^d)$$ for almost every $t>0$. Using the continuity of $\psi$, can I conclude that $\psi(\cdot,t)\in W^{1,2}(\mathbb R^d)$ for every $t>0$ and the map $$t\mapsto\int_{\mathbb R^d}|\nabla_x\psi(x,t)|^2\,d x\,$$ is continuous on $(0,\infty)\,$?

Answer 1

The answer to the question, as is, is no: Consider $\psi(x,t):=\eta(x,t)\sqrt{|(x,t)|}:\mathbb{R}^2\to \mathbb{R}$, where $\eta$ is a smooth cut-off function, identically 1 near the origin. Clearly $\eta\in W^{1,2}(\mathbb{R}^2)$ (since $\sqrt{|(x,t)|}\in W^{1,2}_{loc}(\mathbb{R}^2)$). It's also a globally Hölder continuous function in $\mathbb{R}^2$ (of order $1/2$). However, $\psi(\cdot, 0)\notin W^{1,2}(\mathbb{R})$ (of course you can translate this so that the issue is at $t>0$ instead).