Consider the function $\varphi$ defined on the domain $\mathcal{D}\subset \mathbb{R}^d$ and its trace $\gamma(\varphi)$ on $\partial \mathcal{D}$ (I assume a smooth boundary). We have that $\varphi \in L^2(\mathcal{D})$ and $\gamma(\varphi)\in L^2(\partial \mathcal{D})$. Now we introduce the mollifier $\rho_\varepsilon \in \overline{\mathcal{D}}$ and we note the regularisation $\varphi_\varepsilon = \varphi * \rho_\varepsilon$ defined on $\mathcal{D}$.
Question: Is it possible to show that the trace of $\varphi_\varepsilon$ on $\partial \mathcal{D}$ converges towards $\gamma(\varphi)$ on $L^2(\partial \mathcal{D})$ with just the given hypothesis or do I need more regularity (such as continuity of $\varphi$ on $\mathcal{D}$ but also up to $\partial \mathcal{D}$)?