A TOUR OF THE THEORY OF ABSOLUTELY MINIMIZING FUNCTIONS

By G. Aronsson, M. Crandall, Petri Juutinen. 2004

385 citations recorded by the source corpus.

A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/51554f3256135d125576c88b123193bc1dc5614c

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space (2009, methodology): “Using the considerations developed in the case dom(T) = {a, b} and the “glueing” property of Lipschitz constants (see [ ArCJ , p. 60]) we obtain the following theorem which is just a reformulation of Glaeser’s result in terms of Lipschitz minimality.”
A finite difference approach to the infinity Laplace equation and tug-of-war games (2009, background): “V which are equal to u on @V . We refer to Aronsson, Crandall, and Juutinen [ 2 ] for more background and details.”
A mixed problem for the infinity Laplacian via Tug-of-War games (2007, background): “AMLE extensions were introduced by Aronsson in [2], see the survey [3] for more references and applications of this subject.”
A Seeded Image Segmentation Framework Unifying Graph Cuts And Random Walker Which Yields A New Algorithm (2007, background): “This form can be recognized as a combinatorial formulation of the minimal Lipschitz extension [2].”
Principles of comparison with distance functions for absolute minimizers (2007, methodology): “2 in [1]) one should consider cones x 7→ a|x−x0|+ b with a non-negative coe cient a (see Remark 6.”
C1,α regularity for infinity harmonic functions in two dimensions (2008, background): “withinanopenregion U ⊆ R n .WereferthereadertothesurveypaperbyAronsson,Crandall and Juutinen [ 1 ], which explains the interest in this highly degenerate and highly nonlinear elliptic PDE, and just note here that (1.1) arises as a sort of Euler-Lagrange equation for a model problem in the “calculus ”
A Hölder infinity Laplacian (2012, neutral): “Keywords: Lipschitz extensions, Hölder extensions, infinity Laplacian, non-local and non-linear equations, viscosity solutions.”
Uniqueness of ∞-Harmonic Functions and the Eikonal Equation (2007, methodology): “The exact definition of the set on which u is modified involves instead the “local Lipschitz constant,” introduced in Aronsson et al. (2004), as explained in Section 1.”
Inhomogeneous infinity Laplace equation (2008, background): “A systematic treatment of the theory of absolute minimizers can be found in the manuscript [ 5 ] by Aronsson, Crandall and Juutinen, and the references therein.”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, background): “2) clearly serves to ensure that Dp(x) keeps far from either the total variation functional [287], and the so called ∞-Laplacean [20].”
On the evolution governed by the infinity Laplacian (2006, background): “Indeed, the geometric interpretation of the viscosity solutions of the equation −∆∞u = 0 as absolutely minimizing Lipschitz extensions, see [1], [3], has attracted considerable interest in image processing, the main usage being in the reconstruction of damaged digital images.”
DIRICHLET PROBLEMS FOR THE 1-LAPLACE OPERATOR, INCLUDING THE EIGENVALUE PROBLEM (2007, neutral): “[9], [5] and references therein.”