PDE for the area-preserving non-parametric curve shortening flow

Question

In dimension $1$, it is well-known that the motion of a (unparametrized) plane curve $\{(x,f(x,t);\,x \in I \subset \mathbb R)\}$ under the curve shortening flow can be encoded by the PDE $$\partial_t f = \frac{\partial_{xx} f}{1+|\partial_x f|^2} = \partial_x\left(\arctan(\partial_x f)\right),\quad x\in I.$$ We also suppose that $f(x,0) \geq 0$ for all $x \in I$ so that $f(x,t) \geq 0$ for all $t \geq 0$ and $x\in I$ (thanks to the parabolic maximum principle). It is clear that if $I = \mathbb R$, then the area under the plane curve is conserved since $$\int_{\mathbb R} \partial_t f \,dx = \arctan(\partial_x f)\Big|_{x=-\infty}^{x=\infty} = 0.$$ But if we insist that $I = [a,b]$ being a finite subset of $\mathbb R$, along with the Dirichlet boundary conditions $f(a,t) = f(b,t) \equiv 0$ for all $t\geq 0$. Then the PDE above does not preserve the area anymore, as $\int_a^b \partial_t f \,dx \neq 0$. Is there a way to modify the aforementioned PDE so that the corresponding curve shortening flow with Dirichlet boundary datum is area-preserving? I googled hard for the relevant literature but unfortunately I didn't find anything useful for my question. Thanks for any help!