Optimal Lipschitz extensions and the infinity laplacian

By M. Crandall, Lawrence C. Evans, Ronald F. Gariepy. 2001

284 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c2c06d83bb0cbd31c2b4c87ea59a5081702639f5

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions (2009, background): “In addition to [4], elementary proofs of this result can be found in [2, 6].”
A remark on infinity-harmonic functions. (2006, unknown): “The results in Crandall-Evans-Gariepy [5] imply that the above definition is in fact equivalent to say that for any open set V ⊂ Ω and v ∈ W 1,∞(V ), u|∂V = v|∂V ⇒ ||Du||L∞(V ) ≤ ||Dv||L∞(V ).”
Uniqueness of ∞-Harmonic Functions and the Eikonal Equation (2007, result): “It is shown in Aronsson et al. (2004, Section 4), that u ≥ 0 in U is equivalent x ∈ Br y U ⇒ u x − u y ≤ max z−y =r u z − u y r x − y (1.5) This result was first proved by Crandall et al. (2001).”
Principles of comparison with distance functions for absolute minimizers (2007, methodology): “We extend the principle of comparison with cones introduced by Crandall, Evans and Gariepy in [12] for the Minimizing Lipschitz Extension Problem to a wide class of supremal functionals.”
C1,α regularity for infinity harmonic functions in two dimensions (2008, background): “A result in this direction was obtained by Crandall and Evans [4] (see also Crandall-Evans-Gariepy [3]) which showed that at small scales u is close to a plane.”
On the evolution governed by the infinity Laplacian (2006, background): “1), a large number of estimates for the sub- and supersolutions can be derived from the fact that these sets of functions are characterized via a comparison property that involves a special class of solutions, cone functions, see [8], [3].”
Various Properties of Solutions of the Infinity-Laplacian Equation (2005, background): “More recently, Crandall et al. (2001) have focused attention upon the “comparison with cones” property of solutions.”
A finite difference approach to the infinity Laplace equation and tug-of-war games (2009, unknown): “The notion of comparisons with cones was introduced in the work of Crandall, Evans, and Gariepy [7] for infinity harmonic functions.”
Everywhere differentiability of infinity harmonic functions (2011, background): “From standard comparison with cones arguments (see [5]) we see that max B(0,1) |Du| ≤ |a| + Cλ.”
Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian (2010, background): “Using the exact solution (η+)s to the one dimensional s-fractional Laplacian on the positive half-line [3, Propositions 5.4 and 5.5] (that is, for all η > 0, ∫∞ −∞ ((η+τ)+)s−(η+)s |τ |1+2s dτ = 0 in the principal value sense) we have ∆s∞gǫ(x) ≥ ∫ ǫ− 1 s−t t′ 1− ǫ(t+ η)s η1+2s dη − ∫ ∞ ǫ− 1 s−t ǫ(t+ ”
Inhomogeneous infinity Laplace equation (2008, background): “A function f :S → R is said to be Lipschitz continuous on S if there is a constant L such that ∣∣f (x) − f (y)∣∣ L|x − y|, for any x and y in S.”
A TOUR OF THE THEORY OF ABSOLUTELY MINIMIZING FUNCTIONS (2004, background): “Various adaptations of the concept have since been used in [18], [47], [64], [57], [49] and [29] to investigate the extension problem in different settings.”
A Visit with the ∞-Laplace Equation (2008, methodology): “In [29] it was proved directly - without reference to existence or uniqueness - that comparison with cones, and hence ∆∞u = 0, implies that u is absolutely minimizing for F∞ (Proposition 6.”