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29
On the definition and properties of p-harmonious functions
, 2009
"... We consider functions that satisfy the identity uε(x) = ..."
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Cited by 24 (11 self)
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We consider functions that satisfy the identity uε(x) =
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO Tug-of-war Games
"... We characterize solutions to the homogeneous parabolic p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for t ..."
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Cited by 18 (9 self)
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We characterize solutions to the homogeneous parabolic p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for these game approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.
A new proof of the equivalence of weak and viscosity solutions for the p-Laplace equation. —Manuscript 2011
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Harnack’s inequality for p-harmonic functions via stochastic games
- Comm. Partial Differential Equations
, 1985
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Asymptotic statistical characterizations of p-harmonic functions of two variables
- Rocky Mountain J. Math
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Finite Difference Methods for the infinity Laplace and p-Laplace equations
"... Abstract. We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical p-Laplacian. The discretiza-tions simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosi ..."
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Cited by 3 (0 self)
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Abstract. We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical p-Laplacian. The discretiza-tions simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosity solution of the underlying equation. We build a semi-implicit solver, which solves the Laplace equation as each step. It is fast in the sense that the number of itera-tions is independent of the problem size. This is an improvement over previous explicit solvers, which are slow due to the CFL-condition. 1.
A Game-Tree approach to discrete infinity Laplacian with running costs
, 2013
"... We give a self-contained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tug-of-war games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduc ..."
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Cited by 2 (1 self)
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We give a self-contained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tug-of-war games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce game-trees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the unique-ness. Our arguments are entirely deterministic, and also do not rely on (semi-)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for well-posedness.
