Abstract
These notes are about the celebrated \(\infty \)-Laplace Equation, which so far as I know first was derived by G. Aronsson in 1967, see [A1]. This fascinating equation has later been rediscovered in connexion with Image Processing and in Game Theory.
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Notes
- 1.
Formally it would be classified as parabolic, since its discriminant is \(u_x^2u_y^2 -(u_xu_y)^2 = 0,\) but the main features are “elliptic”. The characteristic curves are \(u_xdy-u_ydx = 0.\) In several variables, it is degenerate elliptic.
- 2.
The stream lines are orthogonal to the level surfaces \(u = \) constant.
- 3.
A function \(u \in C^2(\Omega )\) obtained by the so-called Casas-Torres image interpolation algorithm satisfies
$$\begin{aligned} u(x) = \dfrac{u\left( x+h\frac{\nabla u(x)}{|\nabla u(x)|}\right) + u\left( x-h\frac{\nabla u(x)}{|\nabla u(x)|}\right) }{2} + o(h^2). \end{aligned}$$See J.R. Casas and L. Torres, Strong edge features for image coding, in ‘Mathematical Morphology and its Application to Image and Signal Processing’ edited by R. Schaefer, P. Maragos, and M. Butt, Kluwer Academic Press, Atlanta 1996, pp. 443–450.
- 4.
A serious omission is the differentiability proof of the gradient. I have not succeded in simplifying the proofs in [S] and [ES], and so it is best to directly refer to the original publications.
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Lindqvist, P. (2016). Introduction. In: Notes on the Infinity Laplace Equation. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-31532-4_1
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DOI: https://doi.org/10.1007/978-3-319-31532-4_1
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