Let $U \subset \mathbb{R}^2$ be a domain im $\mathbb{R}^3$ with smooth boundary. Let a point move inside $\mathbb{R}^3$ along the smooth curve $x(t)$. We denote by $\mbox{dist}(x(t), \partial U)$ the distance of $x(t)$ to the boundary $\partial U$.
Do you know any partial differential equation which $\mbox{dist}(x(t), \partial U)$ has to satisfy? The equation should hopefully be of parabolic type.
I would like to use this equation in connection with a maximum principle and apply it to a surface moving inside the domain by some curvature flow. Can you help me?



Point moving inside smooth domain?
Let $P$ be the shortest point projection onto $\partial U$. Then
$$d(t):=dist(x(t),\partial U)=\|P(x(t))-x(t)\|\_2$$
and therefore
$$\dot d(t)=\frac{d}{dt}(\|P(x(t))-x(t)\|^2\_2)^{\frac{1}{2}}=d(t)^{-1}\langle (P-I)x(t),(DP-I)\dot{x}(t)\rangle.$$
As $DP\dot{x}(t) \perp (P-I)x(t)$ we have
$$\dot d(t)=-d(t)^{-1}\langle (P-I)x(t),\dot{x}(t)\rangle.$$


