(The same system with slightly different questions has been asked in MSE.)

Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \mathbb{R}^3$ such that

a) The lengths of all axes-aligned segments remain unchanged,

b) The boundary remains planar, i.e, $\mathbf{r}|_{\partial \Omega}\subset \mathbb{R}^2$,

c) The volume between the plane and the surface is maximal.

Writing $\mathbf{r}=\left(u(x,y),v(x,y),w(x,y) \right)$, being $(x,y)$ rectangular coordinates on $\Omega$, the 1st condition is expressed as

\begin{align} (u_x)^2+(v_x)^2+(w_x)^2&=1\\ (u_y)^2+(v_y)^2+(w_y)^2&=1, \tag{*} \end{align}


\begin{align} w\left(\partial \Omega\right)=0 \tag{**} \end{align}

for the 2nd condition, and the functional

\begin{align} V\left[u(x,y),v(x,y),w(x,y)\right]=\int_\Omega dxdy\:\left(u_x v_y-u_y v_x\right)w, \tag{***} \end{align}

is to be maximized to satisfy the 3rd condition. Introducing the multipliers $\lambda_1=\lambda_1(x,y)$ and $\lambda_2=\lambda_2(x,y)$ to enforce the constraints, we find the Euler-Lagrange equations

\begin{align} 0&=u_x v_y-u_y v_x+\left(\lambda_1 w_x \right)_x+\left(\lambda_2 w_y \right)_y\\ 0&=-\left(wv_y-\lambda_1 u_x\right)_x+\left(wv_x+\lambda_2 u_y\right)_y\\ 0&=\left(wu_y+\lambda_1 v_x\right)_x-\left(w u_x-\lambda_2 v_y\right)_y. \end{align}

The boundary conditions of $u$ and $v$ are free, meaning that among all the possible conditions we seek for the ones that maximize $V$ (for $\lambda_1$ and $\lambda_2$ as well).

Is the problem well-posed? What are the conditions for the existence of a (unique?) maximizer? Is it possible to come up with the maximizing boundary conditions of $u,\:v,\: \lambda_1,\:\lambda_2$? Is there a general protocol to eliminate the multipliers in this kind of problems?

I suspect the existence of a maximizer can be safely established. The picture below shows a numerical simulation for $\Omega$ a unit square.

enter image description here

  • 5
    $\begingroup$ I see. A pride of lions, a murder of crows, an exaltation of larks, a model of pillows. $\endgroup$
    – Will Jagy
    Jan 16 at 23:33
  • 8
    $\begingroup$ @WillJagy, an overflow of mathematicians? $\endgroup$
    – LSpice
    Jan 16 at 23:52
  • 4
    $\begingroup$ Similar questions were studied in Paulsen, What is the shape of a mylar balloon?, Amer. Math. Monthly 101 (1994) and Pak, Schlenker, Profiles of inflated surfaces, Journal of Nonlinear Mathematical Physics, Volume 17, 2010 - Issue 2 $\endgroup$ Jan 17 at 5:09
  • 2
    $\begingroup$ @LSpice here's a real one, based on observed behavior: a wedge of swans $\endgroup$
    – Will Jagy
    Jan 17 at 23:08
  • 1
    $\begingroup$ I don't know how related this is, but when I saw the pillow I was reminded of Igor Pak's article "Inflating Polyhedral Surfaces". $\endgroup$
    – M. Winter
    Jan 22 at 11:41


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