Suppose $X:A \subset \mathbb{R}^n \to \mathbb{R}^{n+1}$ is a parametrisation of a surface $\Gamma$ such that $X(\theta_1, \theta_2) = (\theta_1, \theta_2, h(\theta_1, \theta_2))$ (i.e., $\Gamma$ is a graph).

Define the $g_{ij} = X_{\theta_i} \cdot X_{\theta_j}$ where the $\theta_i$ are the coordinates in $A$. Define $g^{ij}$ to be the elements of the inverse matrix of the matrix G = ($g_{ij})_{i,j}$.

Let $f:\Gamma \to \mathbb{R}$, and define $F:A \to \mathbb{R}$ by $F = f\circ X$. I can show that $\nabla_\Gamma f = p(F_{\theta_1}, F_{\theta_2}, h_{\theta_1}, h_{\theta_2})$ (some function $p$) and similarly that $\Delta_\Gamma f = \frac{\partial}{\partial \theta_1}q_1(F_{\theta_1}, F_{\theta_2}, h_{\theta_1}, h_{\theta_2}) + \frac{\partial}{\partial \theta_2}q_2(F_{\theta_1}, F_{\theta_2}, h_{\theta_1}, h_{\theta_2})$ (some functions $q_1$ and $q_2$). (I know $p, q_1$ and $q_2$ explicitly).

I want to find a weak form of $$ f_t - \Delta_\Gamma f = 0$$ with zero Dirichlet boundary condition and initial condition $f_0$.

The obvious thing to do would be to multiply by a $v \in H^1_0(\Gamma)$ and integrate, which turns the second term in the PDE to to a dot product of gradient terms, but this doesn't really use the formula I worked out for the gradient term (or Laplace term) above (except if I just plug it in). How can I deal with this? I feel like I should be able to integrate out those $\frac{\partial}{\partial \theta_i}$ terms in the Laplacian of $f$..

Also, how does one start to show uniqueness/existence of a solution?

Any hints/references would be appreciated.