Consider the standard heat equation in with Dirichlet boundary data. $$\begin{cases} \partial_tu - \Delta u = 0 & \text{in } \Omega_T := (0,T) \times \Omega, \\ \hfill u = g & \text{in } S_T := (0,T) \times \partial \Omega, \\ \hfill u = 0 & \text{on } \{0\} \times \Omega. \end{cases}\tag{H}$$ Here we assume $\Omega \subset \mathbb R^n$ is a bounded and sufficiently regular domain.
Question: What are the minimal conditions to impose on $g$, as a function on $S_T$, to infer the existence of a solution $u$ with $\nabla u \in L^2(\Omega_T)$?
Existence is understood in the sense of any reasonable weak formulation. This is standard if $g \equiv 0$ and we instead add a forcing term, or prescribe initial conditions at $t=0$, by means of Galerkin methods. Moreover if $g$ admits a suitable regular extension to $\tilde g$ on $\Omega_T$, then we can reduce to this case solving for $w = u-g$ instead - however it is not clear how much regularity one needs to assume on $g$.
The corresponding problem for the Laplacian (where we drop the time component) is known, and it is necessary and sufficient to prescribe $g \in W^{\frac12,2}(\partial\Omega)$, which is the trace space of $W^{1,2}(\Omega)$. In this case the elliptic problem $$\begin{cases} -\Delta u = 0 &\text{in } \Omega \\ \hfill u = g & \text{on } \partial\Omega \end{cases}$$ admits a unique solution $u$ and we have the equivalence $$ \lVert u \rVert_{W^{1,2}(\Omega)} \simeq \lVert g \rVert_{W^{\frac12,2}(\partial\Omega)},$$ where the implicit constants depend on $n$ and $\Omega$ (which is assumed to be e.g. Lipschitz).
My question concerns finding an equivalent sharp space for the corresponding parabolic problem. If a solution $u$ exists for instance, by considering the functions $u(t,\cdot)$ for almost all $t$, we infer the necessary condition that $$ g = u \rvert_{S_T} \in L^2(0,T;W^{\frac12,2}(\partial\Omega)). \tag{$\dagger$}$$ Refined question: Is $(\dagger)$ also sufficient? If not, what should one additionally impose?
I have also used fractional and time-based Sobolev spaces in the above, which can be found separately in standard texts like Evans Chapter 6 and Adams & Fournier, and also together in the monograph of Ladyženskaja, Solonnikov & Uralʹceva (the latter book I have consulted, but I do not believe it addresses this problem).
Some thoughts
If a solution $u$ exists, then due to the structure of the equation we see that $$ \partial_t u = \Delta u \in L^2(0,T;W^{-1,2}(\Omega)), $$ so $u$ also has some time regularity. By means of energy inequalities, one may also expect that $u \in L^{\infty}(0,T;L^2(\Omega))$. However these spaces do not give us further information about $g$, due to the lack of a trace operator from $L^2(\Omega)$ nor $W^{-1,2}(\Omega)$.
This is in contrast if we wish to infer the existence of a solution with $\nabla^2 u \in L^2(\Omega_T)$, where one finds that the necessary and sufficient condition is $$ g \in L^2(0,T;W^{\frac32,2}(\partial\Omega)) \cap W^{\frac34,2}(0,T;L^2(\Omega)).$$ Here we see it is necessary to additionally assume some time regularity for $g$, however it is not clear whether this remains necessary for gradient regularity.
Also, similarly to the elliptic case, using the linearity one can reduce this to the problem of traces. If $\tilde g$ is a suitable extension of $g$ to $\Omega_T$, then we can solve for $w = u -g$, which satisfies $$ \partial_t w - \Delta w = - (\partial_t \tilde g - \Delta \tilde g). $$ If the right-hand side lies in $L^2(0,T;W^{-1,2}(\Omega))$ one can infer existence using the Galerkin method, but this requires that $$ \tilde g \in W^{1,2}(0,T;W^{-1,2}(\Omega)) \cap L^2(0,T;W^{1,2}(\Omega)), $$ and it is not clear if such an extension exists if we merely assume $(\dagger)$.
I expect this problem to be well-studied, but I have not found any results in this direction. Any references, be it textbooks or papers, would be much appreciated.
Another attempt (Added later)
Instead of a Galerkin approximation, which uses a finite dimensional approximation in space, one can also use a minimising movements approach by means of a time-discretisation. Here I assume the boundary data $g$ satisfies $(\dagger)$.
For this let $h > 0$, and we consider $t_k = hk$ for $0 \leq k \leq \lfloor T/h \rfloor$. We can then seek approximation $u_h$ given by $u_h(t,x) = u_{h,k}(x)$ whenever $t_k \leq t < t_{k+1}$ and we choose each $u_{h,k}$ to solve $$ \begin{cases} \frac{u_{h,k} - u_{h,k-1}}h - \Delta u_{h,k} = 0 & \text{in } \Omega, \\ u_{h,k} = g_{h,k} & \text{on } \partial\Omega, \end{cases}$$ for $k \geq 1$, setting $u_{h,0} \equiv 0$. We would like to take $g_{h,k}(x)$ to be $g(t_k,x)$, however $g$ is not assumed to be regular enough for this to be well-defined. Instead we can use a Steklov average $$ g_{h,k}(x) = \frac1{h} \int_{t_k}^{t_{k+1}} g(s,x) \,\mathrm{d}s \in W^{\frac12,2}(\partial\Omega).$$ Note that there is room to play with the choice of $g_{h,k}$, and this just seemed like the most natural approximation. Now it is not difficult to inductively show that $u_{h,k}$ exists for each $k$ to build $u_h$. One way is to take a variational viewpoint, where $u_{h,k}$ will be a minimiser of the functional $$ u \mapsto \int_{\Omega} \frac12 \lvert \nabla u \rvert^2 + \frac1{2h} \lvert u - u_{h,k-1} \rvert^2 \,\mathrm{d}x \tag{V} $$ over functions $u \in W^{1,2}(\Omega)$ for which $u\rvert_{\partial\Omega} = g_{h,k}$. The strategy is now to obtain suitable a-priori estimates, so we can appeal to compactness and obtain a limit solving our original problem.
Unfortunately, it does not appear obvious how one can obtain nice a-priori estimates for these approximate solutions $u_h$. Typically one would use the variational formulation (V) and invoke an energy comparison argument with $u_{h,k-1}$, however this is not possible as $u_{h,k-1}$ is not admissible (it generally does not equal $g_{h,k}$ on the boundary).
This suggests building suitable extensions $G_{h,k}$ of $g_{h,k}$ to perform energy comparison arguments. However this appears to require control on the quantity $$ \frac1{2h} \int_{\Omega} \lvert G_{h,k} - G_{h,k-1} \rvert^2 \,\mathrm{d}x,$$ and for it to vanish as $h \to 0$, uniformly in $k$. This is a discrete version of finding an extension $G$ of $g$ satisfying $(\dagger)$ to $\Omega_T$ which lies in $C([0,T];L^2(\Omega))$, and unfortunately does not appear to be any easier. Note that a simple harmonic extension does not suffice here, as $g$ itself does not possess any regularity in time.
However this does suggest it would be enough to build an extension of $g$ lying in $C([0,T];L^2(\Omega))$, as opposed to the $W^{1,2}(0,T;W^{-1,2}(\Omega))$ which seems more complicated to me.