Convergence of dynamic programming principles for the p-Laplacian

@article{delTeso2018ConvergenceOD,
  title={Convergence of dynamic programming principles for the p-Laplacian},
  author={F{\'e}lix del Teso and Juan J. Manfredi and Mikko Parviainen},
  journal={Advances in Calculus of Variations},
  year={2018},
  volume={15},
  pages={191 - 212}
}
Abstract We provide a unified strategy to show that solutions of dynamic programming principles associated to the p-Laplacian converge to the solution of the corresponding Dirichlet problem. Our approach includes all previously known cases for continuous and discrete dynamic programming principles, provides new results, and gives a convergence proof free of probability arguments. 
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References

SHOWING 1-10 OF 32 REFERENCES

General existence of solutions to dynamic programming equations

We provide an alternative approach to the existence of solutions to dynamic programming equations arising in the discrete game-theoretic interpretations for various nonlinear partial differential…

Solutions of nonlinear PDEs in the sense of averages

Convergence of approximation schemes for fully nonlinear second order equations

The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is…

The obstacle problem for the p-laplacian via optimal stopping of tug-of-war games

We present a probabilistic approach to the obstacle problem for the p-Laplace operator. The solutions are approximated by running processes determined by tug-of-war games plus noise, and letting the…

Finite difference methods for the Infinity Laplace and pp-Laplace equations

A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions

This article considers the problem of building absolutely minimizing Lipschitz extensions to a given function. These extensions can be characterized as being the solution of a degenerate elliptic…

On the definition and properties of p-harmonious functions

We consider functions that satisfy the identity ue(x) = α 2 { sup Be(x) ue + inf Be(x) ue } + β ∫

On the existence and uniqueness of $p$-harmonious functions

We give a self-contained and short proof for the existence, uniqueness and measurability of so called $p$-harmonious functions. The proofs only use elementary analytic tools. As a consequence, we…

Tug-of-war and the infinity Laplacian

We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a…

An asymptotic mean value characterization for p-harmonic functions

We characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to Δ p u = div(|∇u| p-2 ∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and…