Convergence of dynamic programming principles for the p-Laplacian

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.

A mean value formula for the variational p-Laplacian

“…In the last years, several other mean value formulas for the normalized p-Laplacian have been found, and the corresponding program (equivalence of solutions in the viscosity and classical sense and study of the associated dynamic programming principle) has been developed. See for instance [2,7,9,14,[16][17][18], and [22]. We also want to mention [8] and [10], where two other nonlinear mean value formulas are studied, with some similarities with ours.…”

“…The only thing left to show is the convergence stated in Theorem 2.5. Once we have proved monotonicity and consistency as stated in Lemma 9.7, the proof follows as explained in Section 4.3 of[7].Proof of Theorem 2.5 (ii). Defineu(x) = lim sup r→0, y→x U r (y), u(x) = lim inf r→0, y→x U r (y)…”

“…The proof of the convergence is based on the numerical analysis technique introduced by Barles and Souganidis in [5]. We partially follow the outline of [7], where this technique was adapted to homogeneous problems involving the p-Laplacian. 9.2.1.…”

“…where d(y) := dist(y, ∂Ω). As it is shown in the proof of Theorem 3.4 in [7], this is a suitable test function at some point y ε ∈ Ω as in Definition 9.1. Moreover, 27 Page 24 of 33 F. del Teso and E. Lindgren NoDEA u(x 0 ) ≤ u(y ε ) for all ε > 0 small enough and…”

“…It is standard to check, as done in step three of the proof of Theorem 3.4 in [7], that ∇ϕ ε (y ε ) = 0 for all ε > 0. which allows us to use the standard condition (5.2) rather than (5.1).…”