I'm reading the book Minimal Surfaces and Functions of Bounded Variation by Enrico Giusti. I'm wondering if anyone could help me understand the intuition of the theorem below.
We define $$\DeclareMathOperator{\dm}{d\!}\DeclareMathOperator{\Div}{div\!} P(E, \Omega) = \int_\Omega |D \varphi_E|=\sup\left\{\int_E \Div g\dm x: g \in C_0^1(\Omega,\Bbb R^n), |g(x)| \leq 1 \right\}$$ where $\varphi_E$ is the characteristic function of $E$. When $\Omega =\Bbb R^n$ we write $P(E)$.
A Borel set $E$ has locally finite perimeter (also called Caccioppoli set) if for any open bounded set $\Omega$ we have $P(E, \Omega) < \infty$
We have the following theorem.
(Existence of minimal surfaces.) Let $\Omega$ be a bounded open set in $\Bbb R^n$ and let $L$ be a Caccioppoli set. Then there exists a set $E$ coinciding with $L$ outside $\Omega$ such that $$P(E) \leq P(F) $$ for all sets $F = L$ outside $\Omega$. We call $\partial E$ a minimal surface
I understand the proof but what I don't understand is what this theorem is actually saying, I mean the geometric intution. What I understand is that this theory was developed to solve the Plateu's problem, in which we prescribed the boundary and find a set with the boundary prescribed and with minimal area. However, in this theorem we don't exactly prescribe the boundary, but only a part of it (that's the part $L = E$ outside $\Omega$ but the boundary of the competitors is not fixed inside $\Omega$), and we don't minimize the area of the set but the perimeter
So I don't see how this solves the Plateu's problem. How does this theorem solves Plateu's problem? (once the regularity theory is developed)? or is it another kind of minimal surface problem not equivalent to Plateu's problem?
The intuitive meaning of the approach to the existence of minimal surfaces of prescribed boundary described by Enrico Giusti [1] (and due to Ennio De Giorgi) can be inferred by reading carefully remark 1.21 at page 18 and having a look at carefully analyzing the picture at page 19, reproduced here with some edits in order to be more understandable.
Here in this picture the have
$$
\begin{split}
\Omega &=\{x\in\Bbb R^2 \mid |x|<2\},\\
L & = \{x\in\Bbb R^2 \mid x_1^2+(x_2-1)^2<4\}
\end{split}
$$
First observation. The boundary of a hypersurface in $\Bbb R^n$ is not an hypersurface but a 2-codimensional surface. The approach of De Giorgi is to define this 2-codimensional surface as the intersection of two hypersurfaces, precisely $\partial L$ and $\partial\Omega$. Thus the boundary of the sought for minimal hypersurface is the 2-codimensional set $\partial L\cap\partial\Omega$. In the example above, since we are in $\Bbb R^2$, the boundary of the sought-for minimal surface const of the two blue points shown in the picture below
Thus we understand immediately that what we are seeing for is not the whole set $E$ but the part of its boundary bounded by the 2-codimensional surface $\partial L\cap\partial\Omega$, in this case the straight line segment whose boundary are the two shown points: the remaining part of $E$ is thus simply fixed arbitrarily by the shape of $L$ away of $\Omega$ and does not defines the boundary data for the Plateau's problem.
Second observation. As stated in the first observation, we need only the part of $E$ which is bounded by $\partial L\cap\partial\Omega$, thus a proper set $E$ should be chosen inside a proper class: and this class is the one of sets which are exactly equal to $L$ on $L\setminus\Omega$ but are free to vary outside, as shown in the picture below
In the example case the minimizing set is
$$
E=\left\{(x_1,x_2)\in L\mid x_2>\frac{1}{2}\right\}.
$$
Reference
Enrico Giusti (1984), Minimal surfaces and functions of bounded variations, Monographs in Mathematics, 80, Basel–Boston–Stuttgart: Birkhäuser Verlag, pp. XII+240, ISBN 978-0-8176-3153-6, MR 0775682, Zbl 0545.49018