EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM*

By Guy Barles, J. Busca. 2001

168 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/687c55a447bbb427ae4fd480713f599dd37fe98b

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

A mixed problem for the infinity Laplacian via Tug-of-War games (2007, background): “When ΓD ≡ ∂Ω, it is known that problem (1) has a unique viscosity solution, (as proved in [12]; see also [5], [8], and in a more general framework, [19]).”
A Visit with the ∞-Laplace Equation (2008, background): “A self-contained presentation of the proof of [9], which does not require familiarity with viscosity solution theory, is given in [8].”
Uniqueness of ∞-Harmonic Functions and the Eikonal Equation (2007, methodology): “As mentioned, this was previously established in [11] and [2], with two different proofs.”
A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations (2010, unknown): “Sufficient conditions for the PDE to have a comparison principle can be found for example in Section 5 of [15], and (for more recent results) in [3, 4].”
Inhomogeneous infinity Laplace equation (2008, unknown): “If f ∈ C(Ω), then ‖f ‖L∞(Ω) := supx∈Ω |f (x)| denotes the L∞-norm of f on Ω .”
Tug-of-war and the infinity Laplacian (2006, methodology): “) Three uniqueness proofs in this setting have been published, by Jensen [13], by Barles and Busca [6], and by Aronsson, Crandall, and Juutinen [4].”
A TOUR OF THE THEORY OF ABSOLUTELY MINIMIZING FUNCTIONS (2004, methodology): “In this regard, the first uniqueness proof was given by Jensen [47], who used “viscosity solution” theory and the theory of Sobolev spaces, and the second by Barles and Busca [12], who used clever new arguments which circumvented the need to employ results on Sobolev spaces.”
The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞ (2004, result): “This follows from a result of Barles and Busca [8], which generalizes Jensen’s uniqueness theorem.”
A finite difference approach to the infinity Laplace equation and tug-of-war games (2009, methodology): “The uniqueness of infinity harmonic functions with given boundary data is due to Jensen [12], and new proofs and extensions have appeared in the papers of Barles and Busca [3], Aronsson, Crandall, and Juutinen [2], Crandall, Gunnarsson and Wang [9], Peres, Schramm, Sheffield, and Wilson [18], and Lu”
An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions (2009, methodology): “In contrast, previous proofs employ either intricate viscosity solution methods as well as a deep result of Aleksandrov [2,3,6,7], or follow a less direct path using probabilistic arguments [10].”