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?
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4$\begingroup$ As written, $\mathrm{dist}(x(t),\partial U)$ is only a function of $t$. Hence, it could only satisfy an ODE. Also, the condition to be the shortest distance to $\partial U$ is very non-local, in terms of the geometry around the trajectory $x(t)$, so a local equation for the rate of change of this quantity seems unlikely. Does the question really ask what you are after? $\endgroup$– Igor KhavkineCommented Dec 6, 2014 at 17:34
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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 $$\dot d(t)=-d(t)^{-1}\langle (P-I)x(t),\dot{x}(t)\rangle.$$
