CONVERGENCE OF DYNAMIC PROGRAMMING PRINCIPLES FOR THE p-LAPLACIAN

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.


Introduction
Consider a bounded Lipschitz domain Ω ⊂ R d and a Lipschitz function g : ∂Ω → R. We are interested in the Dirichlet problem where ∆ p u = div (|∇u| p−2 ∇u) is the p-Laplace operator. Given a small ε > 0 consider a thin strip around the boundary Γ ε = {x ∈ R d \ Ω : d(x, ∂Ω) ≤ ε} and extend g to Γ ε . For a representative example choose α and β non-negative such that α + β = 1 and a function v : Ω ∪ Γ ε → R, consider the dynamic programming principle When p ≥ 2, α = p−2 p+d , and β = 2+d p+d , it turns out that there is always a unique solution to (1.2) that we label u ε and call (ε, p)-harmonious. Manfredi-Parviainen-Rossi proved in [MPR12] that u ε converges uniformly to u the solution of (1.1) when ε → 0. This theorem has been extended to the case 1 < p < 2, to the case of variable p(x) and to other dynamic programming principles with continuous solutions ( [AHP17]), to the single and double obstacle problems ( [LM17,CLM17]), and to the Heisenberg group ( [LMR18]). All of these proofs use probability tools based on the fact that function u ε can be interpreted as the value function of a tug-of-war game with noise [PSSW09,PS08]. One of the main contributions of this paper is to present a unified and purely analytical proof of the convergence results for general dynamic programming principles associated to the Dirichlet problem (1.1). We do it in two steps. First, we consider the case of smooth domains (Ω is of class C 2 ) where we show that for the p-Laplacian with p ∈ (1, ∞], the concept of generalized viscosity solution of Barles and Ishii ([BP87,BP88,Ish89]) that takes into account the boundary condition is equivalent to the classical notion of viscosity solution. This equivalence allows us to follow the scheme of Barles-Souganidis [BS91] to show convergence. In the language of [BS91] we show the that p-Laplacian satisfies the "strong uniqueness property". For Lipschitz domains, we adapt a boundary regularity argument from [MPR12] showing the existence of good barriers near boundary points. These barriers are built on certain ring domains using the fundamental solution of the p-Laplacian with pole at the center of the ring, which are C 2 -domains. As mentioned above our method is quite general requiring only certain properties of the average operator that appears in the dynamic programming principle (see section §2). As a consequence of this generality we cover also discrete dynamic programming principles that when restricted to a lattice can be seen as convergent numerical schemes for the Dirichlet problem (1.1).

Organization of the paper:
In Section §2 we present the general framework with precise assumptions and definition and state the main results. Section §3 is devoted to the study of generalized viscosity solutions and the strong uniqueness property for the p-Laplacian. In Section §4 we give an interpretation of dynamic programming principles as schemes and use this framework to prove the desired convergence. In Section §5, we extend our results to a class of dynamic programming principles with continuous solutions. Finally, in Section §6 we present a series of examples included in our general theory.

General framework and main results
2.1. Dirichlet problem for the p-Laplacian. In this section we revisit the concept of viscosity solution for the Dirichlet problem associated to the p-Laplacian. First, consider the p-Laplacian operator defined for p ∈ (1, ∞) and a smooth function φ as ∆ p φ(x) := div |∇φ(x)| p−2 ∇φ(x) . It is also interesting to consider the so-called normalized p-Laplacian defined whenever ∇φ(x) = 0 as The limit cases p = 1 and p = ∞ are given by Remark 2.1. We recall the following relations between the above defined operators valid for any p ∈ (1, ∞): and Let p ∈ (1, ∞] and Ω ⊂ R d be a Lipschitz bounded domain. Consider the Dirichlet problem given by for some g ∈ C(∂Ω). Recall the classical concept of viscosity solutions for (Dp) (see eg. [JLM01]).
Observe that we only use test functions with non-vanishing gradient. Existence of weak solutions of (Dp) when p ∈ (1, ∞), Ω is a bounded Lipschitz domain, and g ∈ C(∂Ω) is classical, see [HKM06]. The fact that weak solutions are viscosity solutions, as well as many other properties, is established in [JLM01]. Uniqueness is a folklore result that can be found in appendix 2 of [LM17].
Extended domain: Ω E := Ω ∪ O (note that Ω E is a closed set). Two basic elements in the construction of a dynamic programming principle are the concepts of averages and mean value properties.
Definition 2.6. We say that family of averages {A ε } ε>0 is an (asymptotic) mean value property for the p-Laplacian if for every . for some constant c p,d > 0 and where the constant in o(ε 2 ) can be taken uniformly for all x ∈ Ω.
Remark 2.7. An alternative to (2.7) is given by Given g ∈ C(∂Ω) let G to be a continuous extension of g to Ω E . Associated to a mean value property A ε we have a dynamic programming principle given by Our first general result (Theorem 2.10) concerning C 2 -domains, will require the following assumption on (DPP) For all ε > 0 there exists u ε ∈ B(Ω E ), a solution of (DPP) with a bound on u ε L ∞ (Ω E ) uniform in ε. (A s ) Remark 2.8. Note that the uniform bound in (A s ) follows trivially if (DPP) satisfies the standard maximum principle given by for all ε > 0.
Our second general result (Theorem 2.11) for Lipschitz domains requires slightly more restrictive (but still very natural) assumptions on (DPP): For all ε > 0 there exists u ε ∈ B(Ω E ), a solution of (DPP), and Let u 1 ε and u 2 ε be a subsolution and a supersolution of (DPP) with boundary data G 1 and G 2 respectively. If Remark 2.9. (a) (A cp ) is the comparison principle for (DPP).
Let {A ε } ε>0 be a mean value property for the p-Laplacian. Let also {u ε } ε>0 be a sequence of solutions of the corresponding (DPP) satisfying (A s ). Then we have that where v is the unique viscosity solution of (Dp).
Once convergence for smooth domains is established, we will be able to generalize it to general Lipschitz domains using barrier arguments by requiring a little bit more of our (DPP). We have the following result.
Let {A ε } ε>0 be a mean value property for the p-Laplacian. Let also {u ε } ε>0 be a sequence of solutions of the corresponding (DPP) satisfying (A s ) and (A cp ). Then we have that where v is the unique viscosity solution of (Dp).
The proofs of Theorem 2.10 and Theorem 2.11 will require several steps that we will present in sections §3 and §4. We emphasize that the results of Theorem 2.10 and Theorem 2.11 apply to more general dynamic programming principles than (DPP). The motivation of studying such generalisations is that, unlike (DPP), they are known to produce continuous solutions (cf. [Har16]). For the sake of clarity, we delay the statements of these results to section §5 (see Theorem 5.2 and Theorem 5.3).

Strong uniqueness property for the Dirichlet problem in C 2 -domains
Our main strategy to prove convergence of solutions of (DPP) to the solution of (Dp) is suggested by an argument introduced in [BS91], where the so-called strong uniqueness property plays a crucial role. We introduce a generalized version of viscosity solutions that takes into account the boundary condition in a weaker sense as done in [BP87,BP88] and [Ish89]. As we will see later, these solutions naturally appear as limits of solutions of approximation problems akin to dynamic programming principles.
Definition 3.1. Under the same hypothesis as in Definition 2.2 (resp. Definition 2.3), we say that u is a generalized viscosity p-subsolution (resp. p-supersolution) of (Dp) if the interior condition (2.3) (resp. (2.5)) holds and the condition at the boundary (2.4) (resp. (2.6)) is replaced by The strong uniqueness property states the following: , Ω ⊂ R d be a C 2domain, and g ∈ C(∂Ω). Let u and v be a generalized viscosity p-subsolution and a p-supersolution respectively. Then u ≤ v in Ω.
To show Proposition 3.2 we will prove that, in fact, the notions of viscosity p-subsolution and generalized viscosity p-subsolution are equivalent for C 2 -domains and then use the standard comparison principle for viscosity solutions.
The proof of Theorem 3.3 will be given later in this section. As mentioned before, Proposition 3.2 follows trivially from Theorem 3.3 and the following standard comparison principle for viscosity solutions (cf. [JLM01], [LM17]): Theorem 3.4. (Comparison principle for viscosity solutions) Let p ∈ (1, ∞], Ω ⊂ R d be a Lipschitz domain, and g ∈ C(∂Ω). Let u and v be viscosity p-subsolution and psupersolution respectively of (Dp). Then we have u ≤ v in Ω.
Proof of Theorem 3.3. We will prove the result for subsolutions since for supersolutions follows exactly in the same way. It is trivial to show that (a) =⇒ (b) since To prove (b) =⇒ (a) we need some more work. The proof is inspired in [BB95], where the case p = 2 is presented. The idea of the proof is to try to find a suitable test function φ at x 0 ∈ ∂Ω such that −∆ p φ(x 0 ) > 0. This will immediately imply that and will conclude the proof.
1. Regularity of the domain and the distance function.
We want to emphasize that the point of making the condition local in U is to avoid the ridge points of the domain, where the distance function is not smooth. 2. Properties of a suitable test function (I), after [BB95]. We include the details of this step for the convenience of the reader. Fix a point x 0 ∈ ∂Ω. For small ε > 0 consider the function In particular, if we consider the set then we also have that u(y) − φ ε (y) ∈ USC(K ε ). Therefore u − φ ε attains its maximum in K ε at some point y ε ∈ K ε . Observe that Since φ ε (x 0 ) = 0, we have that By the upper semicontinuity of u we get Since both terms in the left hand side are non-negative, we conclude that they both go to zero. Since we also have In particular we cannot have d(y ε ) = ε for small ε. Since y ε → x 0 then y ε ∈ ∂U . We conclude that y ε is a point of local maximum for u − φ ε in K ε , thus we can extend φ ε so that y ε is point of local maximum for u − φ ε in Ω.

Properties of a suitable test function (II).
In order to prove that φ ε is a suitable test function at y ε for Definition 3.1 we show that ∇φ(y ε ) = 0. Differentiating we get We recall that, from steps 1 and 2, we have that |y ε − x 0 | ≤ cε, ∆d is bounded, |∇d| = 1, and d(y ε ) = o(ε 2 ). We conclude that for some constantc > 0 and ε small enough. In particular, we always have ∇φ ε (y ε ) = 0. 4. Positivity of −∆ p φ ε (y ε ). We will estimate −∆φ ε (y ε ) and −∆ N ∞ φ ε (y ε ) separately and the result for −∆ p φ ε (y ε ) will follow from the interpolation between these operators given in Remark 2.1. Differentiating (3.6) and using one more time that |y ε − x 0 | ≤ cε, ∆d is bounded and |∇d| = 1 we get The previous estimate, together with d(y ε ) = o(ε 2 ) shows that for some C > 0 we have that We will find an estimate for ∆ N ∞ φ ε (y ε ) using so that the expression D 2 φ ε (y) · ∇φ ε (y), ∇φ ε (y) has 45 terms. One can compute them explicitly, however it is interesting first to notice the order of magnitude of each term when y = y ε . More precisely, using once again that |y ε − x 0 | ≤ cε, D 2 d is bounded, |∇d| = 1 and d(y ε ) = o(ε 2 ) we have that where by ∼ we mean that the absolute size is at the most of the denoted magnitude. It is clear that the positive dominating term in ∆ ∞ φ ε (y ε ) is bounded by C/ε 6 for ε small enough. One the other hand, the negative dominating term is precisely given by We conclude then that ∆ ∞ φ ε (y) ≤ Cε −6 − ε −7 , and Combining (3.7) and (3.8) we get that Since by assumption p > 1 and |∇φ ε (y ε )| = 0 we get that for ε small enough we have The same conclusion follows for p = ∞ directly from (3.8).

Conclusion.
From the definition of generalized viscosity solution we get From (3.9), it follows that u(y ε ) ≤ g(y ε ). By taking limits, using (3.5), and the continuity of g the result follows.

Proof of convergence: A numerical analysis approach
In order to prove convergence of the solution of (DPP) to the solution of (Dp) we need to ensure three properties of (DPP): Stability, Monotonicity and Consistency. These names are taken from the numerical analysis framework. Stability is precisely given by assumption (A s ). Monotonicity of (DPP) is a standard property that follows directly from the properties of an average operator. On the other hand, the consistency needed below in this context is slightly different than the usual one, but it will be a consequence of the concept of mean value property. Let us define the following operator for a smooth functions φ that depends explicitly on ε > 0, x ∈ Ω E , φ(x) and φ (through the average A ε ): Then, (DPP) can be formulated as the following scheme:  Let Ω ⊂ R d be a bounded domain and A ε be an average. Then S defined by (4.1) is monotone, that is, for all ε > 0, x ∈ Ω E , t ∈ R and u, v ∈ B(Ω E ) such that u ≤ v we have that S(ε, x, t, v) ≤ S(ε, x, t, u)

4.2.
Consistency. The following consistency result incorporates both the interior equation and the boundary condition of (DPP). Note that it is related to the concept of generalized viscosity solution.
Lemma 4.3. Let {A ε } ε>0 be a mean value property for the p-Laplacian and let S be given by (4.1). Then, for all x ∈ Ω and φ ∈ C ∞ b (Ω E ) such that ∇φ(x) = 0, we have that Proof. We present the proof for the lim sup, since the other one is similar. First note that, Fix first x ∈ Ω and let B denote a ball centred at x and radius smaller than d(x, ∂Ω)/2 (so that B ⊂ Ω) and such that ∇φ(y) = 0 for all y ∈ B (we can assume this by regularity of φ and the fact that ∇φ(x) = 0). Then, by the fact that y ∈ Ω (limit as ξ → 0), the uniformity in the mean value property in (2.7) (limit as ε → 0) and by regularity of −∆ N p φ (limit as y → x), we have lim sup Next, let x ∈ ∂Ω. We can approach x both from points y ∈ Ω and y ∈ O. Again, we can assume that ∇φ(y) = 0 for all y close enough to x. We then have lim sup ε→0, y→x, ξ→0 S(ε, y, φ(y) + ξ, φ + ξ)

4.3.
Proof of convergence for C 2 -domains. We are now ready to prove our most general convergence result regarding C 2 -domains.
Proof of Theorem 2.10. This argument is from [BS91] in the context of numerical schemes. Define By (A s ) both v and v are bounded functions in Q T . Also, by definition, v is USC and v is LSC and v ≤ v. Assume for a moment (we will prove it later) that v is a generalized viscosity p-subsolution and v is a generalized viscosity p-supersolution of (Dp). Then, the strong uniqueness property given by Proposition 3.2, ensures also that v ≥ v. In fact, this proves that v := v = v is a generalized viscosity p-solution of (Dp) and also that u ε → v as ε → 0 uniformly in Ω. The equivalence of notations of viscosity solutions given by Theorem 3.3 concludes the proof. Next, we prove that v is a generalized viscosity p-subsolution of (Dp). Let φ ∈ C ∞ b (Ω E ) and x 0 ∈ Ω such that v(x 0 ) = φ(x 0 ), v < φ if x = x 0 (v − φ reaches a global maximum on Ω) and ∇φ(x 0 ) = 0. Then for all x ∈ Ω, we have that Thus, we can find a sequence {ε n } n≥0 and {y n } n≥0 ⊂ Ω E , such that ε n → 0 + , y n → x 0 , u εn (y n ) → v(x 0 ) and ∇φ(y n ) = 0 with y n being a global max of u εn − φ. Let now ξ n := u εn (y n ) − φ(y n ). We have that ξ n → 0 and u εn (x) − φ(x) ≤ ξ n , that is, Since A ε is an average, the monotonicity given by Lemma 4.2 ensures that 0 = S(ε n , y n , u εn (y n ), u εn ) = S(ε n , y n , φ(y n ) + ξ n , u εn ) ≥ S(ε n , y n , φ(y n ) + ξ n , φ + ξ n ).
As usual, we can assume that ∇φ(y) = 0 for all y close enough to x 0 . Consistency given by Lemma 4.3 then shows Moreover, since ∇φ(x 0 ) = 0 we have that −∆ N p φ(x 0 ) ≤ 0 if and only if −∆ p φ(x 0 ) ≤ 0 and thus v is a generalized viscosity p-subsolution of (Dp), and the proof is completed. 4.4. Proof of convergence for Lipschitz domains. Next we prove Theorem 2.11, i.e. we present the proof of convergence for Lipschitz domains following the proof sketched in [MPR12] in the case of p-harmonious functions. See Figure 1 below for a graphical explanation of the proof.
Since Ω is Lipschitz, it is clear that Ω satisfies the following regularity condition which is the one that we actually use in the proof: There existsδ > 0 and µ ∈ (0, 1) such that for every δ ∈ (0,δ) and y ∈ ∂Ω there exists a ball B µδ (z) strictly contained in B δ (y) \ Ω.
Set Ω ε = Ω ∪ Γ ε . Let u ε be as in Theorem 2.11. Fix δ ∈ (0,δ). For y ∈ ∂Ω consider: Assume momentarily that p = d. Fix a number θ ∈ (0, 1) depending only on µ, d and p, to be determined later. For k ≥ 0 define δ k = δ/4 k−1 and (4.4) M ε k (y) := m ε (y) + θ k (M ε − m ε (y)). By the regularity assumption on Ω, there exist balls B µδ k+1 (z k ) contained in B δ k+1 (y) \ Ω for all k ∈ N. Note that µ is independent of k and δ. Next, we present the basic iteration lemma which follows. We construct smooth barriers based on the fundamental solution on appropriate shrinking rings. We use the convergence result on smooth domains to get estimates in the Lipschitz case. The proof, see (4.11), will show that the right choice of for θ is where ξ = d−p p−1 . Lemma 4.4. Fix η > 0 and let y ∈ ∂Ω and ε k > 0. Under the above notations, suppose that for all ε < ε k we have: Proof. The idea of the proof is to iterate over annular domains and use a comparison principle against a p-harmonic function with suitable boundary values. In this way, we get an upper bound for u ε inside the annular domain that is smaller than the upper bound at the outer boundary of the annular domain we started with. On the next round, we take this improved upper bound as a boundary value at the outer boundary of the next smaller annular domain and repeat the argument to gain again the same multiplying factor θ, and continuing in this way prove the result. The key estimates showing that we gain the uniform θ at each round are (4.9)-(4.11) below. 1. For notational convenience, denote m = m ε (y), M = M ε and M k = M ε k (y). Consider the barrier: Note that when ξ = 0 we have that U k is increasing in |x−z k |, has non vanishing gradient, and solves the problem In the case p = d, we use U k (x) = a k log(|x − z k |) + b k with suitable coefficients a k and b k . We give details only in the case p = d. We will establish several upper bounds for ε k+1 , and take ε k+1 to be the minimum of such bounds. First, let ε k+1 = µδ k+1 2 . For ε ≤ ε k+1 , extend the barrier U k to the ring Let U ε k be the solution of the (DPP) in the ring R k = B δ k (z k ) \ B µδ k+1 (z k ) with boundary value U k on R k,ε \ R k , the outer ε-neighbourhood of R k . Since R k is a smooth domain, by Theorem 2.10 we have that U ε k converges to U k uniformly in R k, as ε → 0. Hence, given for ε ≤ ε k+1 and for every x ∈ R k,ε .
where [ · ] denotes the integer part function and θ is as in (4.5).
It follows now that v and v defined in (4.2) satisfy for all y ∈ ∂Ω. By the same argument as in the proof of Theorem 2.10 we conclude that v is a viscosity p-subsolution and v is a viscosity p-supersolution, since (4.12) ensures that the boundary condition is taken in the usual sense. Again, by construction v ≤ v, and by the comparison principle for viscosity solutions given by Theorem 3.4 we have that v ≤ v. Thus, we get v = v = lim ε→0 u ε . Another family of dynamic programming principles can be formulated as The most interesting property of (DPP2) is that in some cases they are known to produce continuous solutions (cf. [LPS14,Har16]).
Remark 5.1. Note that we can recover (DPP) from (DPP2) by choosing For clarity of the presentation we re-estate here assumptions (A s ), (A s ) and (A cp ) in the context of (DPP2) as well as the corresponding main results.
For all ε > 0 there exists u ε ∈ B(Ω E ) solution of (DPP2) with a bound on u ε L ∞ (Ω E ) uniform in ε.
For all ε > 0 there exists u ε ∈ B(Ω E ) solution of (DPP2), and Let u 1 ε and u 2 ε be a subsolution and a supersolution of (DPP2) with boundary data G 1 and G 2 respectively. If Theorem 5.2. Assume p ∈ (1, ∞], Ω ⊂ R d be a bounded C 2 -domain and g ∈ C(∂Ω). Let {A ε } ε>0 be a mean value property for the p-Laplacian. Let also {u ε } ε>0 be a sequence of solutions of the corresponding (DPP2) satisfying (H s ). Then we have that u ε → v uniformly in Ω as ε → 0, where v is the unique viscosity solution of (Dp).
Let {A ε } ε>0 be a mean value property for the p-Laplacian. Let also {u ε } ε>0 be a sequence of solutions of the corresponding (DPP2) satisfying (H s ) and (H cp ). Then we have that where v is the unique viscosity solution of (Dp).
At this point, it is clear that the outlines of the proof of Theorem 5.2 and Theorem 5.3 are exactly as in Theorem 2.10 and Theorem 2.11 and we skip the unnecessary details. The only nontrivial adaptation is to show that the scheme associated to (DPP2) is monotone and stable. More precisely, (DPP2) can be written as which leads to the following scheme defined for smooth functions as Then, (DPP2) can be formulated as We have the following monotonicity result: Lemma 5.4. Let Ω ⊂ R d be a bounded domain and A ε be an average. Then S defined by (5.3) is monotone, that is, for all ε > 0, x ∈ Ω E , t ∈ R and u, v ∈ B(Ω E ) such that u ≤ v we have that S(ε, x, t, v) ≤ S(ε, x, t, u).
Proof. If x ∈ Ω \ I ε or x ∈ O, everything is as in the proof proof Lemma 4.2 for the scheme (4.1). On the other hand, since 0 ≤ δ ε ≤ 1, if x ∈ I ε , then we have For consistency, we have to work a little bit more, but the result remains the same: Lemma 5.5. Let {A ε } ε>0 be a mean value property for the p-Laplacian and let S be given by (5.3). Then, for all x ∈ Ω and φ ∈ C ∞ b (Ω E ) such that ∇φ(x) = 0, we have that As before in the proof of Lemma 4.3, we can assume ∇φ = 0 in a neighbourhood of x by regularity of φ. If x ∈ Ω, take ε < ρ := d(x, ∂Ω)/2. Then B := B ρ (x) ⊂ Ω \ I ε , and thus, using the same idea of the proof of Lemma 4.3, lim sup ε→0, y→x, ξ→0 Next, let x ∈ ∂Ω. We can approach x from points y ∈ Ω \ I ε , y ∈ I ε and y ∈ O. We have to treat the cases of lim sup and lim inf separately since the argument is slightly different. In both cases, two of the three limits follow as in the proof of Lemma 4.3, so we skip the details for them: Note that, On the other hand, since φ and G are uniformly continuous continuous, there exist a modulus of continuity Λ G,φ such that Note also that, lim sup ε→0,Iε y→x (1 − δ ε (y)) = 1 since it is trivially smaller than one, and the choice y = x has 1 as limit. Then lim sup ε→0,Iε y→x and we conclude that lim sup ε→0, y→x, ξ→0 In the case of the lim inf we aggregate the terms in a different way, arriving to lim inf ε→0, y→x, ξ→0 But we now have that lim inf ε→0,Iε y→x δ ε (y) = 0.

Some representative examples included in our theory
In this section we present some examples of dynamic programming principles for the p-Laplacian that naturally fall into our general framework. Some of them have been extensively studied in the literature, as we will comment below. We recall that Ω ⊂ R d is a bounded Lipschitz domain, {A ε } ε>0 a mean value property (in the sense of Definition 2.6) and G is a continuous extension of g ∈ C(∂Ω) to Ω. We have considered two families of dynamic programming principles: We will present a series of mean value properties for the p-Laplacian with p ∈ (1, ∞] and comment the literature on them for dynamic programming principles of the form of (DPP) and (DPP2).
6.1. Dynamic programming principles in R d . We will always assume that ε > 0, : Ω → R. We will denote by B ε (x) to the ball of radius ε centred at some x ∈ R d , and write just B ε if it is centred at x = 0.
6.1.1. Case p = 2: The Laplacian. The most iconic case is the so-called mean value property for ∆, given by Clearly A 2 ε is an average for fixed ε, and the family {A 2 ε } ε>0 is a mean value property for the Laplacian: Indeed, using the Taylor theorem for φ up to order four, integrating by |B ε | −1 Bε(x) and using the symmetry of the operator and domain allow to conclude that Remark 6.1. One can replace B ε by other symmetric domains like e.g. squares centred with sides of length 2ε or spheres ∂B ε (x).
6.1.2. Case p = ∞: The ∞-Laplacian. The following mean value property was introduced by Le Gruyer and Archer in [LGA98]: Trivially A ∞ ε is an average for fixed ε. It is now standard to show that family {A ∞ ε } ε>0 is a mean value property for the ∞-Laplacian. Recall that we have the expansion Properties (A s ) and (A cp ) for (DPP) with A ∞ ε given by (6.1) are shown in e.g [LGA98] by analytical methods and in [PSSW09] using tug-of-war games. 6.1.3. Case p ∈ (2, ∞): The p-Laplacian. The following mean value property was introduced by Manfredi, Parviainen and Rossi in [MPR10]: φ(y)dy with α p = p−2 p+d and β p = 2+d p+d (α p + β p = 1). Note that ε is a convex combination of averages, and thus is itself an average for a fixed ε. Property (6.3) together with the interpolation formula (2.2) also ensures that {A ∞ ε } ε>0 is a mean value property for the p-Laplacian: Properties (A s ) and (A cp ) for (DPP) with A p ε given by (6.2) are shown in [MPR10] by probabilistic methods and in [LPS14] with analytic tools. A p ε also been studied for the regularized version (DPP2) in [AHP17]. Bε(x) φ + ε 3 1 |B ε | Bε(x) φ(y)dy.
Note that for fixed ε, A p→∞ ε coincides with A p ε given by (6.2) for a certain choice of p depending on ε. As a consequence, properties (A s ) and (A cp ) are also true for (DPP). Moreover, thus, A p→∞ ε is a mean value property of the ∞-Laplacian. More precisely, let u ε be the solution of (DPP) with the mean value property (6.4) and u be the viscosity solution of the Dirichlet problem for the ∞-Laplacian given by (Dp). Then u ε → u uniformly as ε → 0. This strategy provides an alternative dynamic programming principle for the ∞-Laplacian.
for the p-Laplacian is also standard and we omit the details. We refer to [CLM17] for precise details and applications of A p ε,h to the obstacle problem. Note that the second term in (6.6) is precisely the composed midpoint quadrature rule for Bε which is known to produce an error of order O(h). To produce more precise numerical methods in the form of a dynamic programming principle, one can consider higher order monotone quadrature rules that will have the form where ω(y) > 0. Precise form of the weights ω can be obtained for example from the Newton-Cotes formulas. They are known to produce positive weights (and thus monotone schemes) for orders up to O(h 7 ).
6.2.2. Case p = 2: The Laplacian. The best known discrete mean value property for the Laplacian is given by the discrete Laplacian: From the Taylor theorem we see that ∆ h φ(x) := ∆φ(x) + O(h 2 ). We naturally define It is clear that {A 2 h } h>0 is a mean value property for the Laplacian since .
Properties (A s ) and (A cp ) for (DPP) (and consequently for (NM)) with A p h given by (6.7) is standard in numerical analysis framework (see for example [Obe13]). Remark 6.2. As shown in [Obe05], it is also possible to construct monotone schemes (and consequently discrete dynamic programming principles) for the ∞ − Laplacian by means of the so-called absolutely minimizing Lipschitz extensions. As expected, one can combine them with A 2 h given by (6.7) to produce also discrete dynamic programming principles for the p-Laplacian for p ∈ [2, ∞] (cf. [Obe13]).