Define $$Q = \bigcup\_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required.


Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ and $$H^1(Q) := \{ u \in L^2(0,T;H^1(\Gamma)) : u\_t \in L^2(0,T;L^2(\Gamma)) \}.$$


Given a function $u \in H^1(Q)$ with $\Delta\_\Gamma u \in L^2(Q)$, is it possible to find smooth functions (at least $C^2$ in space and $C^1$ in time) such that
$$u\_n \to u \quad \text{ in $H^1(Q)$}$$
$$\Delta\_\Gamma u\_n \to \Delta\_\Gamma u \quad \text{ in $L^2(Q)$}$$


Here $\nabla\_\Gamma$ is the usual surface or tangential gradient and $\Delta\_\Gamma$ is the Laplace-Beltrami operator.Note that the first requested convergence means 
$u\_n \to u$, $\nabla\_\Gamma u\_n \to \nabla\_\Gamma u,$ and $(u\_n)\_t \to u\_t$ all in $L^2(0,T;L^2(\Gamma)).$


I don't know how to even start this problem. Thanks for any help.



Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$?
In my first comment I wrote: "I think the question is off-topic here as not research level." Since there have been no other comments in that direction, and no (visible) downvotes, maybe the question then is on-topic anyway.


I have read the OP's questions and comments also in MSE, and there seems to be a similar, but not quite the same problem. Here (in MO) the problem is formulated as possible density of the set of (equivalence classes containing) smooth functions in a space $W(Q)$ for $Q=\Gamma\times J\_T$ with $J\_T={]}0,T{[}$ such that the elements of $W(Q)$ are (equivalence classes of) functions in $L^2(Q)$ having also first order weak partial derivatives and the weak "space" Laplacian in $L^2(Q)$ . In MSE, in place of the compact (smooth) hypersurface $\Gamma$ there is (if I understand correctly) an open set $\Omega$ in a Euclidean space. Moreover, in the MSE formulation, the space is defined by requiring all second order partials "in the space direction" to be in $L^2(Q)$ , not just the Laplacian. 


The possible validity of the equation
$$
\int\_Q \varphi^2 (\Delta v)v\_t = \int\_Q |\nabla v|^2 \varphi \varphi\_t - 2\int\_Q (\nabla v \cdot \nabla \varphi) (\varphi v\_t)\kern30mm (\*)
$$
seems to be the source of the density problem. It holds if $v$ is known to be smooth, and by continuity, it also holds for $v$ in $W(Q)$ if density the set of smooth functions in the space $W(Q)$ can be established. By elliptic regularity, the two slightly different formulations are actually equivalent since only integration over a compact set occurs in $(\*)$ as $\varphi$ has compact support. Namely, by Theorem 20.1 on page 308 in Wloka's *Partial Differential Equations*, CUP 1992, the elements in $W(Q)$ actually have also all second order weak space partials in $L^2(Q)$ .


So all that has to be done, if I have understood everything correctly, is to prove density of the set of (equivalence classes containing) smooth functions in the space $W(Q)$ that is naturally linearly homeomorphic to the space one would usually write e.g. as $H^1(J\_T,L^2(\Gamma))\cap L^2(J\_T,H^2(\Gamma))$ .


A decent proof would require too much effort. So I give only a sequence of remarks as hints hoping that a mathematically matured reader could then write a proof himself if he wished.


Arbitrarily fix an element $u$ in $W(Q)$ . Since in ($\*$) only integration over a compact subset of $Q$ is involved, by multiplying with a suitable "cut-off" function "in the time direction", we may assume $u$ defined on $\Gamma\times\mathbb R$ with support contained in some $\Gamma\times[\varepsilon,T-\varepsilon]$ . Cover $\Gamma$ with finitely many coordinate patches, and take a partition of unity subordinate to this covering. Then $u$ is expressed as a finite sum of functions $u\_k$ with the same regularity properties as $u$ . Moreover, each $u\_k$ has its support in some $U\_k\times[\varepsilon,T-\varepsilon]$ where $U\_k$ is the domain of a chart $\phi\_k:U\_k\to\mathbb R^{\,n}$ of $\Gamma$ . Letting $v\_k$ be the zero extension of $(x,t)\mapsto u\_k(\phi\_k^{-1}(x),t)$ to $\mathbb R^{\,n}\times\mathbb R$ , then $v\_k$ has the same regularity as $u$ . If each $v\_k$ can be approximated by a smooth function $\varphi\_k$ in the sense that the $L^2(\mathbb R^{\,n}\times\mathbb R)$ norms of $\varphi\_k-v\_k$ and all its first order derivatives and second order "space" derivatives become small, and also the support of $\varphi\_k$ is sufficiently close to that of $v\_k$ , then taking $\varphi=\sum\_k\varphi\_k^0$ with $\varphi\_k^0$ the zero extension to $\Gamma\times J\_T$ of $U\_k\times J\_T\owns(z,t)\mapsto\varphi\_k(\phi\_k(z),t)$ , we have $\varphi$ smooth with the $W(Q)$ norm of $\varphi-u$ small. The required approximation is possible e.g. by Wloka's Theorem 1.3 on page 9 loc. cit.


