Let $\Omega\subset\mathbb{R}^2$ be an open subset such that $\partial\Omega$ is a closed, simple curve.
I'm trying to find an example of an $u:\overline{\Omega}\to\mathbb{R}$ such that $\Sigma:=\text{graph}(u)$ is a minimal surface and, yet, there exists another minimal surface $\Sigma'$ with $\partial\Sigma'=\partial\Sigma$ and $\text{Area}(\Sigma')<\text{Area}(\Sigma)$.
Does such an example exist?
This is an excellent question. The answer is "no." The way to see this is via a calibration argument.
Def: Let $(M^n, g)$ be a Riemannian manifold. A calibration on $M$ is a $p$-form $\varphi \in \Omega^p(M)$ satisfying:
- $d\varphi = 0$, and
- $|\varphi(v_1, \ldots, v_p)| \leq 1$ for every orthonormal set $\{v_1, \ldots, v_p\}$ in $T_xM$.
An oriented $p$-dim subspace $V \subset T_xM$ is calibrated by $\varphi$ iff $\varphi(v_1, \ldots, v_p) = 1$ for some oriented orthonormal basis $\{v_1, \ldots, v_p\}$ of $V$.
An oriented $p$-dim submanifold $N^p \subset M^n$ is calibrated by $\varphi$ iff each tangent space $T_xN \subset T_xM$ is calibrated by $\varphi$.
The following theorem is due to F. Reese Harvey and H. Blaine Lawson (1982):
Fundamental Theorem on Calibrations: Let $(M^n, g)$ be a Riemannian manifold and $\varphi$ a calibration. Let $N, N' \subset M$ be two compact, oriented, $p$-dim submanifolds with $\partial N = \partial N'$ and $N$ homologous to $N'$. If $N$ is calibrated by $\varphi$, then $\text{Area}(N) \leq \text{Area}(N')$.
Proof: Using that $N$ is calibrated first, then Stokes' Theorem second, then the definition of calibration third, we have $$\text{Area}(N) = \int_{N} \varphi = \int_{N'} \varphi \leq \text{Area}(N'). \ \ \ \lozenge $$
Let $u \colon \overline{\Omega} \to \mathbb{R}$ be such that $\Sigma := \text{Graph}(u)$ is a minimal surface in $\mathbb{R}^3$. Regarding $\Sigma$ as the level set $\{v(x,y) = 0\}$, where $v(x,y) := z - u(x,y)$, we see that a unit normal vector field to $\Sigma$ is $$N_u = \frac{\nabla v}{\Vert \nabla v \Vert} = \frac{(-u_x, -u_y, 1)}{\sqrt{1 + (u_x)^2 + (u_y)^2}}.$$ Define the $2$-form $\varphi_u \in \Omega^2(\mathbb{R}^3)$ via $$\varphi_u(X,Y) = \det(X,Y, N_u) = (X \times Y) \cdot N_u.$$ The following exercise, together with the Fundamental Theorem above, gives the result:
Exercise: (a) The $2$-form $\varphi_u$ is a calibration on $\mathbb{R}^3$. That is:
- $d\varphi_u = 0$, and
- $|\varphi_u(X,Y)| \leq 1$ for every orthonormal set $\{X,Y\}$ in $T_x\mathbb{R}^3$.
(b) The surface $\Sigma = \text{Graph}(u)$ is calibrated by $\varphi_u$. That is: If $\{X,Y\}$ is an oriented orthonormal set having $X,Y \in T_x\Sigma$, then $\varphi_u(X,Y) = 1$.
The calibration argument works perfectly if $\bar\Omega$ is convex; if not, then there are counterexamples.
If $\bar\Omega$ is not convex, then the minimizer can escape from $\bar\Omega\times \mathbb R$ (where the form $\varphi_u$ is defined). In this case we have no control on its area.
For example, a minimal graph $z=u(x,y)$ might not minimize area if $u$ is defined on domain $\bar\Omega\subset \mathbb R^2$ as on the picture:
Indeed, imagine that boundary values are zero except the horizontal lines where they depend only on the $x$-coordinate. In this case (for suitable $u$) it is cheaper to take a disc in the $(x,y)$-plane with two thin belts approaching the center from both sides. (Projection of an area minimizer to $(x,y)$ plane will enter in the gaps between half-discs.)
