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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.

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  • $\begingroup$ Is your $F$ (capital $F$) actually an $f$? $\endgroup$
    – Piero D'Ancona
    Commented Apr 18, 2012 at 16:08
  • $\begingroup$ Sorry for the oversight. It's $F = f\circ X$. I'll edit my post. $\endgroup$
    – blackcat
    Commented Apr 18, 2012 at 16:10
  • $\begingroup$ Why did you define $g$ and $G$? They are never used in the question. My guess is they are used in the surface gradient and Laplacian expressions. You should be able to do integration by parts directly working on $A$ with the metric $g$. For existence/uniqueness one way would be to study the surface Laplacian and use semigroup method. $\endgroup$
    – timur
    Commented Aug 9, 2012 at 0:16

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There's a lot of literature on this. Search for Prof Charles Elliott at Warwick university and on his homepage there are many papers related to this.

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  • $\begingroup$ And PDEs on surfaces in general $\endgroup$
    – Mike
    Commented Jul 25, 2012 at 21:50

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