ref.bib

@Inbook{Simon1997,
author="Simon, Leon",
title="The Minimal Surface Equation",
bookTitle="Geometry V: Minimal Surfaces",
year="1997",
publisher="Springer Berlin Heidelberg",
address="Berlin, Heidelberg",
pages="239--266",
abstract="The minimal surface equation (MSE) for functions u: $\Omega$ {\textrightarrow} ℝ, $\Omega$ a domain of ℝ2, can be written {\$}{\$}{\backslash}left( {\{}1 + u{\_}{\{}{\}}^2{\}} {\backslash}right){\{}u{\_}{\{}xx{\}}{\}} - 2{\{}u{\_}x{\}}{\{}u{\_}y{\}}{\{}u{\_}{\{}xy{\}}{\}} + {\backslash}left( {\{}1 + u{\_}x^2{\}} {\backslash}right){\{}u{\_}{\{}yy{\}}{\}} = 0{\$}{\$}or equivalently {\$}{\$} {\{}u{\_}{\{}xx{\}}{\}} + {\{}u{\_}{\{}yy{\}}{\}} - {\{}{\backslash}left( {\{}1 + |Du{\{}|^2{\}}{\}} {\backslash}right)^{\{} - 1{\}}{\}}{\backslash}left( {\{}u{\_}x^2{\{}u{\_}{\{}xx{\}}{\}} + 2{\{}u{\_}x{\}}{\{}u{\_}y{\}}{\{}u{\_}{\{}xy{\}}{\}} + u{\_}y^2{\{}u{\_}{\{}yy{\}}{\}}{\}} {\backslash}right) = 0{\$}{\$}where {\$}{\$}{\{}u{\_}x{\}} = {\backslash}frac{\{}{\{}{\backslash}partial u{\backslash}left( {\{}x,y{\}} {\backslash}right){\}}{\}}{\{}{\{}{\backslash}partial x{\}}{\}},{\{}u{\_}y{\}} = {\backslash}frac{\{}{\{}{\backslash}partial u{\backslash}left( {\{}x,y{\}} {\backslash}right){\}}{\}}{\{}{\{}{\backslash}partial y{\}}{\}}{\$}{\$}. Generally, for domains $\Omega$ ⊂ ℝnand functions $\Omega$ {\textrightarrow} ℝ depending on the n variables (x1, {\ldots}, xn) ∈ $\Omega$, n ≥ 2, the MSE can be written {\$}{\$}{\backslash}sum{\backslash}limits{\_}{\{}i,j = 1{\}}^n {\{}{\backslash}left( {\{}{\{}{\backslash}delta {\_}{\{}ij{\}}{\}} - {\backslash}frac{\{}{\{}{\{}u{\_}i{\}}{\{}u{\_}j{\}}{\}}{\}}{\{}{\{}{\backslash}left( {\{}1 + |Du{\{}|^2{\}}{\}} {\backslash}right){\}}{\}}{\}} {\backslash}right){\{}u{\_}{\{}ij{\}}{\}} = 0{\}} {\$}{\$}where {\$}{\$}{\{}u{\_}i{\}} = {\{}D{\_}i{\}}u {\backslash}equiv {\backslash}frac{\{}{\{}{\backslash}partial u{\}}{\}}{\{}{\{}{\backslash}partial {\{}x^i{\}}{\}}{\}}{\$}{\$}and uij= DiDju. Notice that this is a quasilinear elliptic equation: that is, it is linear in the second derivatives, and the coefficient matrix {\$}{\$}{\backslash}left( {\{}{\{}{\backslash}delta {\_}{\{}ij{\}}{\}} - {\backslash}frac{\{}{\{}{\{}u{\_}i{\}}{\{}u{\_}j{\}}{\}}{\}}{\{}{\{}{\backslash}left( {\{}1 + |Du{\{}|^2{\}}{\}} {\backslash}right){\}}{\}}{\}} {\backslash}right){\$}{\$}is positive definite1 depending only on the derivatives up to first order. The equation can alternatively be written in ``divergence form'' (1){\$}{\$}{\backslash}sum{\backslash}limits{\_}{\{}i = 1{\}}^n {\{}{\{}D{\_}i{\}}{\}} {\backslash}left( {\{}{\backslash}frac{\{}{\{}{\{}D{\_}i{\}}u{\}}{\}}{\{}{\{}{\backslash}sqrt {\{}1 + |Du{\{}|^2{\}}{\}} {\}}{\}}{\}} {\backslash}right) = 0{\$}{\$}which is readily checked using the chain rule and the fact that {\$}{\$}{\backslash}frac{\{}{\backslash}partial {\}}{\{}{\{}{\backslash}partial {\{}p{\_}j{\}}{\}}{\}}{\backslash}left( {\{}{\backslash}frac{\{}p{\}}{\{}{\{}{\backslash}sqrt {\{}1 + |p{\{}|^2{\}}{\}} {\}}{\}}{\}} {\backslash}right) = {\{}{\backslash}left( {\{}1 + |p{\{}|^2{\}}{\}} {\backslash}right)^{\{} - 1/2{\}}{\}}{\backslash}left( {\{}{\{}{\backslash}delta {\_}{\{}ij{\}}{\}} - {\backslash}frac{\{}{\{}{\{}p{\_}i{\}}{\{}p{\_}j{\}}{\}}{\}}{\{}{\{}1 + |p{\{}|^2{\}}{\}}{\}}{\}} {\backslash}right){\$}{\$}.",
isbn="978-3-662-03484-2",
doi="10.1007/978-3-662-03484-2_5",
url="https://doi.org/10.1007/978-3-662-03484-2_5"
}