# 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.

## 16 Citations

### Convergence of the natural p-means for the p-Laplacian

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2021

We prove uniform convergence in Lipschitz domains of approximations to p-harmonic functions obtained using the natural p-means introduced by Ishiwata, Magnanini, and Wadade [Calc. Var. Partial…

### Convergence of Natural $p$-Means for the $p$-Laplacian in the Heisenberg Group

- Mathematics, Philosophy
- 2021

In this paper we prove uniform convergence of approximations to p-harmonic functions by using natural p-mean operators on bounded domains of the Heisenberg group H which satisfy an intrinsic exterior…

### Evolution Driven by the Infinity Fractional Laplacian

- Mathematics
- 2022

We consider the evolution problem associated to the inﬁnity fractional Laplacian introduced by Bjorland, Caﬀarelli and Figalli (2012) as the inﬁnitesimal generator of a non-Brownian tug-of-war game.…

### Asymptotic mean-value formulas for solutions of general second-order elliptic equations

- MathematicsAdvanced Nonlinear Studies
- 2022

Abstract We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as…

### Boundary Aware Tug-of-War with Noise: Case p ∈ (2, ∞)

- Mathematics
- 2020

This chapter improves on the construction of the Tug-of-War game in Chap. 3 and proves convergence of its values to p-harmonic functions, in the simplified nondegenerate case. This first convergence…

### An asymptotic expansion for the fractional p-Laplacian and for gradient-dependent nonlocal operators

- Mathematics
- 2020

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic…

### A Finite Difference Method for the Variational p-Laplacian

- MathematicsJ. Sci. Comput.
- 2022

To the best of the knowledge, this is the first monotone finite difference discretization of the variational p -Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

### p-harmonic functions by way of intrinsic mean value properties

- MathematicsAdvances in Calculus of Variations
- 2021

Abstract Let Ω be a bounded domain in ℝn{\mathbb{R}^{n}}. Under appropriate conditions on Ω, we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to…

### Uniform limit of discrete convex functions

- Mathematics
- 2019

It is proved that the uniform limit on compact subsets of discrete convex mesh functions which are uniformly bounded is a continuous convex function.

### Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian

- Mathematics
- 2022

In this paper we prove the first quantitative convergence rates for the graph infinity Laplace equation for length scales at the connectivity threshold. In the graph-based semi-supervised learning…

## References

SHOWING 1-10 OF 32 REFERENCES

### General existence of solutions to dynamic programming equations

- Mathematics
- 2014

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…

### A dynamic programming principle with continuous solutions related to the $p$-Laplacian, $1 < p < \infty$

- Mathematics
- 2015

We study a Dynamic Programming Principle related to the $p$-Laplacian for $1 < p < \infty$. The main results are existence, uniqueness and continuity of solutions.

### Regularity for nonlinear stochastic games

- MathematicsAnnales de l'Institut Henri Poincaré C, Analyse non linéaire
- 2018

### The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems

- Mathematics
- 1995

The aim of this article is to study the Dirichlet problem for second-order semilinear degenerate elliptic PDEs and the connections of these problems with stochastic exit time control problems.

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

- Mathematics
- 2014

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

- Computer ScienceJ. Comput. Appl. Math.
- 2013

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

- Mathematics
- 2012

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…

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

- MathematicsMath. Comput.
- 2005

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

- Mathematics, Philosophy
- 2012

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