# A model of pillows

(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}

while

\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.

• I see. A pride of lions, a murder of crows, an exaltation of larks, a model of pillows. Jan 16 at 23:33
• @WillJagy, an overflow of mathematicians? Jan 16 at 23:52
• 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 Jan 17 at 5:09
• @LSpice here's a real one, based on observed behavior: a wedge of swans Jan 17 at 23:08
• 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". Jan 22 at 11:41