# Plateau problem for fluxes of curves

Consider $$\mathbb{R}_t \times \mathbb{R}_x ^n$$ , let $$b_1(t,x)$$ and $$b_2 (t,x)$$ be two velocity fields with all the regularity you want and consider the flow of the point $$(0,x_0)$$ for a time $$T$$. Now, I can view the trajectories of $$(0,x_0)$$ as curves in $$\mathbb{R}^{n+1}$$ like $$\gamma_1 =(t, \Phi_1 (t,x_0))$$ and $$\gamma _2= (t, \Phi_2 (t,x_0))$$ with $$\Phi_i$$ the flux relative to $$b_i$$. The question is: if I consider $$\tilde{\gamma} =\gamma_1 \cup \gamma_2 \cup I_T$$ where $$I_T$$ is a segment in $$\mathbb{R}^{n+1}$$ joining $$(T,\Phi_1 (T, x_0) )$$ and $$(T, \Phi_2 (T, x_0))$$ , I can consider the Plateau problem associated to the rectifiable set $$\tilde{\gamma}$$, and by the general theory I'll have a minimal surface whose boundary is $$\tilde{\gamma}$$. The question is: given how my $$\gamma$$ is constructed, is there a way to estimate the area of such surface in terms of $$b_1$$ and $$b_2$$?

EDIT: Actually, what I want is to integrate such an estimate, starting from a certain mass distribution at time 0, so I would like something whixh takes that into account.

Almgren (https://mathscinet.ams.org/mathscinet-getitem?mr=855173) proves that the Plateau solution will satisfy the 2-dimensional Euclidean isoperimetric inequality. So you can bound $$\textrm{Area(Plateau Solution)} \leq (4\pi)^{-1}(L(\gamma_1)+L(\gamma_2)+L(I_T))^2.$$ If you just want a coarse estimate you should be able to bound each term by the $$L^\infty$$ norm of $$b_1,b_2$$.

Of course, this inequality is unlikely to be close to optimal. (For example, if $$b_1=b_2$$ then the area of the Plateau solution is $$=0$$ while the RHS above is nonzero). If you want an estimate that works better in this case you might construct the explicit competitor $$\Gamma : = \bigcup_{t=0}^T I_{\Phi_1(t,x_0),\Phi_2(t,x_1)} \times \{t\}$$ namely you union up all the lines between the points in the flow. You should be able to compute the area of this in terms of $$b_1,b_2$$ using the co-area formula.

EDIT: As requested here is a formula for the area of $$\Gamma$$. I did it via a parametrization instead of co-area but I think co-area would also work.

I parametrize $$\Gamma$$ by $$F: [0,1]\times [0,T]\to\mathbb{R}^{n+1}$$ $$F(s,t) : = (s\Phi_1(t)+(1-s)\Phi_2(t),t)$$ (dropping the $$x_0$$ notation). Then we have $$\partial_s F = \Phi_1-\Phi_2, \qquad \partial_t F = s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2 + \mathbf{e}_{n+1}.$$ In particular, the induced metric is \begin{align*} g_{ss} & = |\Phi_1-\Phi_2|^2 \\ g_{st} & = \langle \Phi_1-\Phi_2,s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2\rangle\\ g_{tt} & = 1 + |s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2|^2. \end{align*} Hence $$\det g = (1 + |s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2|^2)|\Phi_1-\Phi_2|^2 - \langle \Phi_1-\Phi_2,s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2\rangle^2.$$ This gives the explicit formula $$\textrm{area}(\Gamma) = \int_{[0,1]\times [0,T]} \sqrt{(1 + |s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2|^2)|\Phi_1-\Phi_2|^2 - \langle \Phi_1-\Phi_2,s\partial_t\Phi_1 +(1-s)\partial_t \Phi_2\rangle^2}.$$ If you want to write this in terms of $$b_1$$ and $$b_2$$ you can do the following (perhaps one can be more precise here to get a better estimate). First of all $$|\partial_t \Phi_i| = |b_i| \leq \Vert b_i\Vert_{C^0}$$ Secondly, by the argument here we can bound $$|\Phi_1-\Phi_2| \leq L^{-1}(e^{Lt}-1)|b_1-b_2|_{L^\infty}.$$ for $$L$$ the Lipschitz constant of one of the vector fields.

Using this above we should get $$\textrm{area}(\Gamma) \leq (1 + (|b_1|_{C^0}+|b_2|_{C^0}) |b_1-b_2|_{C^0}^2 L^{-2}(e^{LT}-LT-1).$$

Note that I discarded the $$g_{st}$$ term, which could improve this estimate if you had some way of knowing it was big. But it does have the nice feature that it vanishes if $$b_1=b_2$$ and it's $$O(T^2)$$ for $$T$$ small.

I'm sure you can improve the estimate used above in the case of large $$T$$ (e.g. if $$b_i$$ are bounded, then the flows should diverge at most linearly).

• Yeah, I kept it simple in the question but I'll edit it. In fact I don't have a single point, but rather a mass distribution at time 0 which flows, so I should, in a sense, integrate my estimate over all points $x_0$. Could you please give an idea on how to use the coarea formula on your example? I thought about that competitor you are mentioning, but how do you explicitly compute its area? Jun 8 at 16:56
• Marvelous! Thanks a lot! Jun 9 at 17:04