THE OBSTACLE PROBLEM FOR THE p-laplacian VIA OPTIMAL STOPPING OF TUG-OF-WAR GAMES

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1 THE OBSTACLE PROBLEM FOR THE p-laplacian VIA OPTIMAL STOPPING OF TUG-OF-WAR GAMES MARTA LEWICKA AND JUAN J. MANFREDI Abstract. 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 step size go to zero, not unlike the case when Brownian motion is approximated by random walks. Rather than stopping the process when the boundary is reached, the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain. 1. Introduction Let L be the second order differential operator: L = 1 2 traceσσ xd 2 xv, whose matrix coefficient function σ is Lipschitz continuous. Consider the obstacle problem in R N : 1.1 min Lv, v g = 0. In order to solve 1.1 probabilistically [18], one first solves the stochastic differential equation: 1.2 dx t = σx t dw t, starting from x at time t = 0. Denoting its solution by {X x t, t 0}, the value function is defined by taking the supremum over the set T of all stopping times valued in [0, ]: 1.3 vx = sup E [gx x τ ], τ T This value function, under appropriate regularity hypothesis on g, turns out then to be the unique solution to 1.1; for details see Chapter 5 in [18]. The purpose of the present paper is to consider the analog problem when the second order linear differential operator L is replaced by the p-laplacian: 1.4 p u = div u p 2 u, 2 p <. Since the operator p is non-linear, we do not have a suitable variant of the linear stochastic differential equation 1.2 that could be used to write a formula similar to 1.3. Instead, we will show that one can use tug-of-war games with noise as the basic stochastic process. More precisely, we will prove that the solutions to the obstacle problem for the p-laplacian for p [2,, can be interpreted as limits of values of a specific discrete tug-of-war game with noise, when the step-size ɛ, determining the allowed length of move of a token at each step of the game, converges to 0. To explain our approach, let us first recall a concept of supersolutions suggested by the tug-ofwar characterization in [13]. This notion is based on the mean value properties, and it is equivalent to the notion of viscosity supersolution in the class of continuous functions. Namely, let Ω R N Date: November 16, Key words and phrases. tug-of-war games, viscosity solutions, p-laplacian, obstacle problem. 1

2 2 MARTA LEWICKA AND JUAN J. MANFREDI be an open, bounded set with Lipschitz boundary and let F : Ω R be a Lipschitz continuous boundary data. Choose the parameters α and β as follows: α = p 2 N + p, where α 0 since p 2, β > 0, and α + β = 1. β = 2 + N N + p, Supersolutions in the sense of means: A continuous function v : Ω R { } is a supersolution in the sense of means if whenever φ C0 Ω is such that φx vx for all x Ω, with equality at one point φx 0 = vx 0 φ touches v from below at x 0, then we have: φx 0 + α 2 sup φ + α B ɛx 0 2 inf φ + β φ + oɛ 2 as ɛ 0 +. B ɛx 0 B ɛx 0 Above, by 0 hɛ + oɛ 2 as ɛ 0 + we mean that: [hɛ] lim ɛ 0 + ɛ 2 = 0. Fixing a scale ɛ, we now consider functions for which 1.5 holds with equality, i.e. without the error term ɛ 2. Let 0 < ɛ 0 1 be a small constant and define the fattened outer boundary set, together with the fattened domain: Γ = {x R N \ Ω; distx, Ω < ɛ 0 }, X = Ω Γ. ɛ-p-harmonious functions: Let 0 < ɛ ɛ 0. ɛ-p-harmonious with boundary values F : Γ R if: α 1.6 u ɛ x = 2 sup u ɛ + α B ɛx 2 inf u ɛ + β B ɛx A bounded, Borel function u : X R is u ɛ B ɛx for x Ω F x for x Γ. Then, it has been established in [14] that u = lim ɛ 0 u ɛ is a solution to the Dirichlet problem: { p u = 0 in Ω, 1.7 u = F on Ω. Let now Ψ : R n R be a bounded, Lipschitz function, which we assume to be compatible with the boundary data: F x Ψx for x Ω. The function Ψ is interpreted as the obstacle and we consider the problem: 1.8 p u 0 in Ω, u Ψ in Ω, p u = 0 in {x Ω; ux > Ψx}, u = F on Ω. That is, we want to find a p-superharmonic function u which takes boundary values F, which is above the obstacle Ψ, and which is actually p-harmonic in the complement of the contact set {x Ω: ux = Ψx}. The problem 1.8 has been extensively studied from the variational point of view; see the seminal paper by Lindqvist [10] and the book [6]. In particular, regularity requirements for the domain Ω, the boundary data F and the obstacle Ψ can be vastly generalized. We have, however, focused on the Lipschitz category for technical reasons in our proofs.

3 THE OBSTACLE PROBLEM FOR THE p-laplacian 3 Our first result shows how to solve the obstacle problem using ɛ-p-superharmonious functions. The dynamic programing principle 1.9 below is similar to the Wald-Bellman equations of optimal stopping see Chapter 1 of [21]. Theorem 1.1. Let α [0, 1 and β = 1 α. Let F : Γ R and Ψ : R N R be two bounded, Borel functions such that Ψ F in Γ. Then there exists a unique bounded Borel function u : X R which satisfies: { } max Ψx, α 1.9 u ɛ x = 2 sup u ɛ + α B ɛx 2 inf u ɛ + β u ɛ for x Ω B ɛx B ɛx F x for x Γ. After proving Theorem 1.1 in Section 2, we proceed to establishing that u = lim ɛ 0 + u ɛ is the solution to the obstacle problem 1.8. The key step is to show that {u ɛ } is equicontinuous up to scale ɛ, i.e. the functions u ɛ may be discontinuous but the discontinuities are of size roughly ɛ. Then, an easy extension of the Arzelá-Ascoli theorem from [14] shows that there are subsequences of {u ɛ } that converge uniformly to a function u. The standard stability of viscosity solutions yields then that any such limit is a viscosity solution of 1.8. By uniqueness see Lemma 4.2 and its proof in the Appendix they all must agree, and we obtain: Theorem 1.2. Let p [2, and let u ɛ : Ω Γ R be the unique ɛ-p-superharmonious function solving 1.9 with α = p 2 2+N p+n and β = p+n. Then u ɛ converge as ɛ 0, uniformly in Ω, to a continuous function u which is the unique viscosity solution to the obstacle problem 1.8. Towards the proof, the key estimate in Lemma 4.4 bounds the oscillation of u ɛ in a uniform way. Namely, given η > 0 there are small r 0, ɛ 0 > 0 so that whenever ɛ < ɛ 0, then for all x 0, y 0 Ω: 1.10 x 0 y 0 < r 0 = u ɛ x 0 u ɛ y 0 < η. To deduce 1.10 we use probabilistic techniques, interpreting u ɛ as value functions of certain tugof-war games and finding an appropriate extension of 1.3. In Section 3 we present the details of this construction, involving stochastic processes tug-of-war games with noise needed to write down the representation formulas for u ɛ. For the case of linear equations that correspond to p = 2 with variable coefficients, a similar version of the representation formula 1.11 below is due to Pham [17] and Øksendal-Reikvam [16]. Note that since ɛ > 0 is fixed, we omit it in the statement that follows. Theorem 1.3. Let α, β 0 satisfy α + β = 1. Let F : Γ R and Ψ : R N R be two bounded, Borel functions such that Ψ F in Γ. Define: G : X R G = χ Γ F + χ Ω Ψ, where χ A stands for the characteristic function of a set A X. Define the two value functions: 1.11 u I x 0 = sup τ,σ I inf σ II [G x τ ], u II x 0 = inf σ II sup τ,σ I [G x τ ], where sup and inf are taken over all strategies σ I, σ II and stopping times τ τ 0 that do not superseed the exit time τ 0 from the set Ω. Then: u I = u = u II in Ω, where u is a bounded, Borel function satisfying 1.9.

4 4 MARTA LEWICKA AND JUAN J. MANFREDI The core of this paper can be found in Section 5, where we use the representation formulas from Theorem 1.3 to establish the oscillation estimate We provide full details of the proof for the equicontinuity estimates in Section 4, the proof of Theorem 1.2 in Section 5, and the fact that our games end almost surely in Section 6. We finish this introduction by discussing other notions of solutions for 1.4 in addition to 1.5: i Weak or Sobolev supersolutions: These are functions v W 1,p loc Ω such that: v p 2 v, φ dx 0 Ω for all test functions φ C 0 Ω, R + that are non-negative in Ω. ii Potential theoretic supersolutions or p-superharmonic functions: A lower-semicontinuous function v : Ω R { } is p-superharmonic if it is not identically on any connected component of Ω and it satisfies the comparison principle with respect to p-harmonic functions, that is: if D Ω, and w C D is p-harmonic in D satisfying w v on D, then we must have: w v on D. iii Viscosity supersolutions: A lower-semicontinuous function v : Ω R { } is a viscosity p-supersolution if it is not identically on any connected component of Ω, and if whenever φ C0 Ω is such that φx vx for all x Ω with equality at one point φx 0 = vx 0 φ touches v from below at x 0, and φx 0 0, then we have: p φx 0 0. The fact that weak supersolutions are potential theoretic and viscosity supersolutions follows from the comparison principle and a regularity argument implying the lower-semicontinuity; see for example Chapter 3 in [6]. The fact that bounded p-superharmonic functions are weak supersolutions was established by Lindqvist in [10]. 1 Note that an arbitrary, not necessarily bounded p-superharmonic function v is always the pointwise increasing limit of bounded p-superharmonic functions v n = min{v, n}. The equivalence between viscosity supersolutions and p-superharmonic functions was established in [7]. Therefore, the three notions of supersolution agree on the class of bounded functions; see also the Appendix where for completeness we present the folklore argument stating that viscosity solutions are weak solutions and thus they are unique. Finally, we recall a classical useful observation. When u C 2 and ux 0, then one can express the p-laplacian as a combination of the ordinary Laplacian and the -Laplacian: 1.12 p ux = u p 2 ux + p 2 ux, where: 1.13 ux = 2 ux ux ux, ux. ux The tug-of-war interpretation of the -Laplacian has been developed in the fundamental paper [19]. The obstacle problem for 1.13 has been studied in [15]. A similar treatment as in the present paper, for the double obstacle problem, has been developed in [4]. Acknowledgments. The first author was partially supported by NSF awards DMS and DMS The second author was partially supported by NSF award DMS In fact, this is also the first reference that we have been able to locate for the classical case p = 2.

5 THE OBSTACLE PROBLEM FOR THE p-laplacian 5 2. ɛ-p-superharmonious functions: a proof of Theorem 1.1 The proof uses the monotonicity arguments of the Perron method as extended by [11], modified to accommodate the obstacle constraint. 1. The solution to 1.9 will be obtained as the uniform limit of iterations u n+1 = T u n, where, for any bounded Borel function v : X R, we define: { } max Ψx, α 2.1 T vx = 2 sup v + α B ɛx 2 inf v + β v for x Ω B ɛx B ɛx vx for x Γ and where we put: 2.2 u 0 = χ Γ F + χ Ω inf Ψ. X We easily note that u 1 = T u 0 u 0 in X. Consequently: u 2 = T u 1 T u 0 = u 1 in X and, by induction, the sequence of Borel functions {u n } is nondecreasing in X. Also, {u n } n=1 { } satisfies: Ψ u n max sup F, sup Ψ in Ω, Γ X and clearly u n = F on Γ. Therefore, the sequence converges pointwise to a bounded Borel function u : X R, satisfying: u Γ = F. 2. We now show that the convergence of {u n } to u is uniform in X. This will automatically imply that u = lim T u n = T lim u n = T u, and hence yield the desired existence result. We argue by contradiction and assume that: M = lim Fix a small δ > 0 and take n > 1 so that: sup X sup n x X u u n < M + δ and x Ω β u u n x > 0. u u n B ɛx β u u n < δ. B ɛ x X The second condition above is justified by the monotone convergence theorem. Take now x 0 Ω satisfying: ux 0 u n+1 x 0 > M δ. Note that if ux 0 = Ψx 0 then since u j x 0 increases to ux 0 and all u j x 0 Ψx 0, there would be u n x 0 = Ψx 0. Therefore ux 0 > Ψx 0 and consequently: We now compute: 2.3 m > n u m+1 x 0 u n+1 x 0 > M 2δ and u m+1 x 0 > Ψx 0. M 2δ < u m+1 x 0 u n+1 x 0 = α sup u m sup u n + α inf 2 B ɛx 0 B ɛx 0 2 u m B ɛx 0 α sup u m u n + β B ɛx 0 < αm + δ + δ = αm + α + 1δ. inf B ɛx 0 u n + β u m u n α sup u u n + β B ɛx 0 B ɛx 0 u m u n B ɛx 0 u u n B ɛx 0 This implies that M < αm + α + 3δ, which clearly is a contradiction for δ sufficiently small, since α < 1.

6 6 MARTA LEWICKA AND JUAN J. MANFREDI 3. We now prove uniqueness of solutions to 1.9. Assume, by contradiction, that u and ū are two distinct solutions and denote: M = supu ū > 0. Ω Let {x n } n 1 be a sequence of points in Ω such that lim n u ūx n = M. Observe that for large n there must be: ux n > Ψx n, since M > 0. Therefore, as in 2.3, we get: u ūx n = α 2 sup u sup ū B ɛx n B ɛx n + α 2 inf u B ɛx n inf ū B ɛx n + β u ū B ɛx n α sup u ū + β B ɛx n u ū αm + β B ɛx n Passing to the limit with n, where lim x n = x 0, we obtain: u ū. B ɛx n M αm + β u ū, B ɛx 0 and hence: M ffl B ɛx 0 u ū, since β > 0. Consequently: u ū = M almost everywhere in B ɛ x 0, and hence in particular the following set is nonempty: G = {x X; u ūx = M}. By the same argument as above, we see that in fact for all x G, the set B ɛ x \ G has measure 0. We conclude that: u ū = M a.e. in X which contradicts the fact that G Γ =, and proves the result. We further easily derive the following weak comparison principle: Corollary 2.1. Let u and ū be the unique solutions to 1.9 with the respective boundary data F and F and obstacle constraints Ψ and Ψ, satisfying the assumptions of Theorem 1.1. If F F and Ψ Ψ then u ū in Ω. Proof. Let {u n } and {ū n } be the approximating sequences for u and ū, as in the proof of Theorem 1.1. By 2.2: u 0 ū 0, which results in u n ū n for every n, in view of 1.9. Consequently, the limits u and ū satisfy the same pointwise inequality. 3. Game-theoretical interpretation of the ɛ-p-superharmonious functions We now link the ɛ-p-superharmonious function u solving 1.9 to the probabilistic setting. We define this setting in detail, as this paper is dedicated to analysts rather than probabilists. All the basic concepts can be found in the classical textbook [23] The measure spaces X,x 0, F x 0 n and X,x 0, F x 0. Fix any x 0 X and consider the space of infinite sequences ω recording positions of token during the game, starting at x 0 : X,x 0 = {ω = x 0, x 1, x 2...; x n X for all n 1}. For each n 1, let F x 0 n be the σ-algebra of subsets of X,x 0, containing sets of the form: A 1... A n := {ω X,x 0 ; x i A i for i : 1... n}, for all n-tuples of Borel sets A 1,..., A n X. Although the expression in the left hand side above is, formally, a Borel subset of R Nn, we will, with a slight abuse of notation, identify it with the set of infinite histories ω with completely undetermined positions beyond n. Let F x 0

7 THE OBSTACLE PROBLEM FOR THE p-laplacian 7 be now defined as the smallest σ-algebra of subsets of X,x 0, containing n=1 F x 0 n. Clearly, the increasing sequence {F x 0 n } n 1 is a filtration of F x 0, and the coordinate projections x n ω = x n are F x 0 measurable random variables on X,x The stopping times τ 0 and τ. Define the exit time from the set Ω: τ 0 ω = min{n 0; x n Γ} where we adopt the convention that the minimum over the empty set equals +. This way: τ 0 : X,x 0 N {+ } is F x 0 measurable and, in fact, it is a stopping time with respect to the filtration {F x 0 n }, that is: 3.1 n 0 {ω X,x 0 ; τ 0 ω n} F x 0 n. Let now τ : X,x 0 N {+ } be any stopping time i.e. a random variable satisfying 3.1, where τ 0 is replaced by τ such that: 3.2 τ τ 0. For n 1 we define the Borel sets: A τ n = {x 0, x 1,..., x n ; ω = x 0, x 1,..., x n, x n+1,... X,x 0, By 3.2, it follows that x 0,..., x n A τ n whenever x n Γ. τω n} The strategies σ I, σ II. For every n 1, let σ n I, σn II : Xn+1 X be Borel measurable functions with the property that: σ n I x 0, x 1,..., x n, σ n IIx 0, x 1,..., x n B ɛ x n X. We call σ I = {σ n I } n 1 and σ II = {σ n II } n 1 the strategies of Players I and II, respectively The probability measure P x 0. Fix two parameters α, β 0, such that: α + β = 1. Given τ, σ I, σ II as above, we define now a family of probabilistic Borel measures on X, parametrised by the finite histories x 0,..., x n : 3.3 n 1 x 1,..., x n X γ n [x 0, x 1,..., x n ] = α = 2 δ σi nx 0,x 1,...,x n + α 2 δ σii n x 0,x 1,...,x n + β L N B ɛ x n B ɛ x n δ xn when x 0,..., x n A τ n otherwise where δ y denotes the Dirac delta at a given y X, while the measure multiplied by β above stands for the N-dimensional Lebesgue measure restricted to the ball B ɛ x n and normalized by the volume of this ball. Note that since τ τ 0, then γ n [x 0, x 1,..., x n ] = δ xn whenever x n Γ. For every n 1 we now define the probability measure P n,x 0 on X,x 0, F x 0 n by setting: 3.4 P n,x 0 A 1... A n =... 1 dγ n 1 [x 0, x 1,..., x n 1 ]... dγ 0 [x 0 ] A 1 A n for every n-tuple of Borel sets A 1,..., A n X. Here, A 1 is interpreted as the set of possible successors x 1 of the initial position x 0, which we integrate dγ 0 [x 0 ], while x n A n is a possible successor of x n 1 which we integrate dγ n 1 [x 0, x 1,..., x n 1 ], etc. The following observation justifies the definition 3.4: Lemma 3.1. The family {γ n [x 0, x 1,..., x n ]} in 3.3 has the following measurability property. For every n 1 and every Borel set A X, the function: is Borel measurable. X n+1 x 0, x 1,..., x n γ n [x 0, x 1,..., x n ]A R

8 8 MARTA LEWICKA AND JUAN J. MANFREDI It is clear that the family {P n,x 0 } n 1 is consistent see [23], with the transition probabilities γ n [x 0, x 1,..., x n ]. Consequently, in virtue of the Kolmogoroff s Consistency Theorem, it generates uniquely the probability measure P x 0 = lim n P n,x 0 on X,x 0, F n so that: n 1 A 1... A n F x 0 n P x 0 A 1... A n = P n,x 0 A 1... A n. One can easily prove the following useful observation, which follows by directly checking the definition of conditional expectation: Lemma 3.2. Let v : X R be a bounded Borel function. For any n 1, the conditional expectation {v x n F x 0 n 1 } of the random variable v x n is a F x 0 n 1 measurable function on X,x 0 and hence it depends only on the initial n positions in the history ω = x 0, x 1,... X,x 0, given by: {v x n F x 0 n 1 }x 0,..., x n 1 = v dγ n 1 [x 0,..., x n 1 ]. We also have: Lemma 3.3. In the above setting, assume that β > 0. Then each game stops almost surely, i.e.: 3.5 P x 0 {τ < } = 1. For convenience of the reader, we give a self-contained proof of this observation in the Appendix ɛ-p-superharmonious functions and game values. Before proving Theorem 1.3, we need a lemma on almost optimal selections. This lemma was put forward in [11] and now we present its possible elementary proof. Lemma 3.4. Let u : X R be a bounded, Borel function. Fix δ, ɛ > 0. There exist Borel functions σ sup, σ inf : Ω X such that: 3.6 x Ω σ sup x, σ inf x B ɛ x and: 3.7 x Ω uσ sup x sup u δ, B ɛx X uσ inf x inf B ɛx u + δ. Proof. We will prove existence of σ sup, while existence of σ inf follows in a similar manner. 1. Let u = χ A for some Borel set A X. Without loss of generality δ < 1 3. We write A + B δ 0 = i=1 B δx i as the union of countably many open balls, and define: { x if x A + Bδ 0 x Ω σ sup x = x i if x B δ x i \ i 1 j=1 B δx j Clearly, σ sup above is Borel as a pointwise limit of Borel functions. 2. Let u = n k=1 α kχ Ak be a simple function, given by disjoint Borel sets A k X and α α 1 < α 2 <... < α n. Without loss of generality δ < min k+1 α k k=1...n 1 3. We now write, as before: A k + B δ 0 = i=1 B δx k i, and we subsequently define: { x if x n k=1 x Ω σ sup x = A k + B δ 0 x k i if x B δ x k i \ i 1 j=1 B δx k j j>k A j + B δ 0 3. In the general case when u is an arbitrary bounded Borel function, consider a simple function v such that u v L X δ 3. By the previous construction, there exists σ sup : Ω X which is

9 THE OBSTACLE PROBLEM FOR THE p-laplacian 9 a sup-selection for v, with the error δ 3. Then we have: x Ω uσ sup x vσ sup x δ 3 sup v 2δ B ɛx 3 sup u δ, B ɛx and therefore σ sup is also the required sup-selection for the function u. Remark 3.5. If we replace the open balls B ɛ x in the requirement 3.6 by the closed ones, then the Borel selection satisfying 3.7 may not exist. Take ɛ = 1, δ = 1 3 and let u = χ A where A R 3 is a bounded Borel set with the property that A + B 1 0 is not a Borel set. The existence of A is nontrivial see [11] and relies on the existence of a 2d Borel set whose projection on the x 1 axis is not Borel. This result extends the famous construction of Erdos and Stone [5] of a compact Cantor set A and a G δ set B such that A + B is not Borel. Proof of Theorem We first show that: 3.8 u II u in Ω. Fix η > 0 and fix any strategy σ I and a stopping time τ τ 0. By Lemma 3.4, there exists a Markovian strategy σ 0,II for Player II, such that σ n 0,II x 0,... x n = σ n 0,II x n and that: 3.9 n 1 x n X uσ0,iix n n inf u + η B ɛx n 2 n+1 Using Lemma 3.2, definition 3.3, suboptimality in 3.9 and the equation 1.9, we compute: x 0,..., x n 1 A τ n 1 {u x n + η 2 n F x 0 n 1 }x 0,..., x n 1 = u dγ n 1 [x 0,..., x n 1 ] + η 2 n X = α 2 uσn 1 I x 0,..., x n 1 + α 2 uσn 1 0,II x 0,..., x n 1 + β u + η B ɛx n 1 2 n α sup u + α inf 2 B ɛx n 1 2 u + β u + η B ɛx n 1 B ɛx n 1 2 n 1 + α 2 ux n 1 + η 2 n 1 = u x n 1 + η 2 n 1 x 0,..., x n 1. On the other hand, when x 0,..., x n 1 A τ n 1, then Ex 0 {u x n + η 2 F x n 0 n 1 }x 0,..., x n 1 = ux n 1 + η 2 directly from Lemma 3.2 and by 3.3. We therefore obtain that the sequence of n random variables {u x n + η 2 } n n 0 is a supermartingale with respect to the filtration {F x 0 n }. It follows that: u II x 0 sup τ,σ I [G x τ + η 2 τ ] sup τ,σ I [u x τ + η 2 τ ] sup τ,σ I [u x 0 + η 2 0 ] = ux 0 + η, where we used the definition of u II, the fact that G u, and the Doob s optional stopping theorem in view of the supermartingale property and the uniform boundedness of the random variables {u x τ n + η 2 τ n } n 0. Since η > 0 was arbitrary, 3.8 follows. 2. We now prove that: 3.10 u u I in Ω.

10 10 MARTA LEWICKA AND JUAN J. MANFREDI Together with 3.8 and in view of the direct observation from 1.11 that u I u II, 3.10 will imply Theorem 1.3. Fix η > 0 and fix any strategy σ II. By Lemma 3.4, there exists a strategy σ 0,I for Player I, such that σ n 0,I x 0,... x n = σ n 0,I x n and that: 3.11 n 1 x n X uσ0,ix n n sup u η B ɛx n 2 n+1. Define the stopping time τω = inf{n 0; ux n = Ψx n }, with the convention that inf over an empty set is +. Clearly: 3.12 x 0,... x n A τ n iff k = 0... n ux k > Ψx k. As before, by Lemma 3.2, the definition 3.3, and the suboptimality in 3.11, we obtain: x 0,..., x n 1 A τ n 1 τ, σ 0,I,σ II {u x n η 2 n F x 0 n 1 }x 0,..., x n 1 = u dγ n 1 [x 0,..., x n 1 ] η 2 n X = α 2 uσn 1 0,I x 0,..., x n 1 + α 2 uσn 1 II x 0,..., x n 1 + β u η B ɛx n 1 2 n α sup u + α inf 2 B ɛx n 1 2 u + β u η B ɛx n 1 B ɛx n 1 2 n 1 + α 2 = ux n 1 η 2 n 1 + α 2 u x n 1 η 2 n 1 x 0,..., x n 1, where the last equality above follows from 1.9 because of For x 0,..., x n 1 A τ n 1, we also get, as before: τ,σ 0,I,σ II {u x n η 2 F x n 0 n 1 }x 0,..., x n 1 = ux n 1 η 2. We now conclude n that {u x n η 2 } n n 0 is a submartingale with respect to the filtration {F x 0 n }, and therefore: u I x 0 inf σ II τ,σ 0,I,σ II [G x τ inf σ II τ,σ 0,I,σ II [u x 0 η 2 0 ] = ux 0 η, η 2 τ ] = inf σ II τ,σ 0,I,σ II [u x τ η 2 τ ] where we used the definition of u I, the fact that Gx τ = ux τ derived from the definition of τ, and the Doob s optional stopping theorem used to the two stopping times: τ and 0. Since η > 0 was arbitrary, we conclude The main convergence theorem: a proof of Theorem 1.2 First, we recall the definition of viscosity solutions. Definition 4.1. We say that a continuous function u : Ω R is a viscosity solution of the obstacle problem 1.8 if and only if: u = F on Ω together with u Ψ in Ω, and: i for every x 0 Ω and every φ C 2 Ω such that: 4.1 φx 0 = ux 0, φ < u in Ω \ {x 0 }, φx 0 0, there holds: p φx 0 0. ii for every x 0 Ω such that ux 0 > Ψx 0 and every φ C 2 Ω such that: 4.2 φx 0 = ux 0, φ > u in Ω \ {x 0 }, φx 0 0, there holds: p φx 0 0.

11 THE OBSTACLE PROBLEM FOR THE p-laplacian 11 The fact that variational solutions are viscosity solutions in the sense of Definition 4.1 is due to the equivalence of the local notions of viscosity and weak solutions [7] and the continuity up to the boundary of variational solutions under the regularity hypothesis on F, Ψ, and Ω see for example [3]. The fact that viscosity solutions are variational solutions is actually equivalent to the following folklore uniqueness result that, for the sake of completeness, we prove in the Appendix: Lemma 4.2. Let u and ū be two viscosity solutions to 1.8 as in Definition 4.1. Then u = ū. It is also classical that the unique solution to 1.8 is the pointwise infimum of all p-superharmonic functions that are above the obstacle see Chapters 5 and 7 in [6]. Our main approximation result is given in Theorem 4.3 below. Theorem 4.3. Let p [2,. Let F : Ω R, Ψ : Ω R be two Lipschitz continuous functions, satisfying: 4.3 Ψ F on Ω. Without loss of generality, we may assume that F, Ψ above are defined on Γ and X, respectively, and that 4.3 still holds on Γ. Let u ɛ : Ω Γ R be the unique ɛ-p-superharmonious function solving 1.9 with α = p 2 2+N p+n and β = p+n. Then u ɛ converge as ɛ 0, uniformly in Ω, to a continuous function u which is the unique viscosity solution to the obstacle problem 1.8. Proof. 1. We first prove the uniform convergence of u ɛ, as ɛ 0, in Ω. This is achieved by verifying the assumptions of the following version of the Ascoli-Arzelá theorem, valid for equibounded possibly discontinuous functions with uniformly vanishing oscillation : Lemma 4.4. [14] Let u ɛ : Ω R be a set of functions such that: i C > 0 ɛ > 0 u ɛ L Ω C, ii η > 0 r 0, ɛ 0 > 0 ɛ < ɛ 0 x 0, y 0 Ω x 0 y 0 < r 0 = u ɛ x 0 u ɛ y 0 < η Then, a subsequence of u ɛ converges uniformly in Ω, to a continuous function u. Clearly, solutions u ɛ to 1.9 as in the statement of Theorem 4.3 are uniformly bounded, by the boundedness of F. The crucial step in the proof of condition ii above is achieved by estimating the oscillation of u ɛ close to the boundary. Lemma 4.5. Under the assumptions of Theorem 4.3, let u ɛ : X R be the ɛ-p-superharmonious solution to 1.9. Then, for every η > 0 there exist r 0, ɛ 0 > 0 such that we have: 4.4 ɛ < ɛ 0 y 0 Ω x 0 Ω x 0 y 0 < r 0 = u ɛ x 0 u ɛ y 0 < η. We postpone the proof of Lemma 4.5 to Section 5. We now have: Corollary 4.6. Under the assumptions of Theorem 4.3, let {u ɛ } be the sequence of ɛ-p-superharmonious solutions to 1.9. Then {u ɛ } satisfies condition ii in Lemma 4.4. Proof. Fix η > 0 and let r 0, ɛ 0 be as in Lemma 4.5 so that 4.4 holds with η/3 instead of η. Since Ψ is Lipschitz, we may without loss of generality also assume that: 4.5 x, y X x y < r 0 = Ψx Ψy < η. Note that, consequently, we have: 4.6 ɛ < ɛ 0 x 0, y 0 Γ r0 /3 x 0 y 0 < r 0 /3 = u ɛ x 0 u ɛ y 0 η, where for any δ > 0 we denote: Γ δ = {x Ω; distx, Ω δ}.

12 12 MARTA LEWICKA AND JUAN J. MANFREDI In particular, the same implication as in 4.6 holds for x 0 Γ r0 /6 and y 0 Ω when x 0 y 0 < r 0 /6. Let now x 0, y 0 Ω \ Γ r0 /6 and assume that x 0 y 0 < r 0 /6. Define the bounded Borel function F : Γ r0 /6 R and the Lipschitz obstacle Ψ : R N R by: F z = u ɛ z x 0 y 0 + η, Ψz = Ψz x0 y 0 + η. Clearly: F Ψ in Γr0 /6, hence by Theorem 1.1 there exists a solution ũ ɛ : Ω R to 1.9 subject to the boundary data F on Γ r0 /6, and to the obstacle constraint Ψ. Note that by the uniqueness of such solution, there must be: ũ ɛ z = u ɛ z x 0 y 0 + η On the other hand: F uɛ in Γ r0 /6 by 4.6, and also: Ψ Ψ in Ω by 4.5. Corollary 2.1 now implies that ũ ɛ u ɛ in Ω and we get: u ɛ x 0 u ɛ y 0 ũ ɛ x 0 u ɛ y 0 = u ɛ y 0 + η u ɛ y 0 = η. Exchanging x 0 with y 0, the same argument yields u ɛ x 0 u ɛ y 0 < η. 2. We now prove that the uniform limit of u ɛ is a viscosity solution to the obstacle problem 1.8. Clearly, u = F on Ω and u Ψ in Ω because each ɛ-p-superharmonious function u ɛ has the same properties. We show that i in Definition 4.1 holds. Let φ be a test function as in 4.1. Since x 0 is the minimum of the continuous function u φ, one can find a sequence of points x ɛ converging to x 0 as ɛ 0, and such that: 4.7 u ɛ x ɛ φx ɛ inf u ɛ φ + ɛ 3. Ω To prove this statement, for every j 1 let a j = min u φ > 0 and let ɛ j > 0 be such that: Ω\B 1/j x 0 ɛ ɛ j u ɛ u L Ω 1 2 a j. Without loss of generality {ɛ j } is decreasing to 0. Now, for ɛ ɛ j+1, ɛ j ] let x ɛ B 1/j x 0 satisfy: u ɛ x ɛ φx ɛ inf u ɛ φ + ɛ 3. B 1/j x 0 We finally conclude 4.7 by noting that also for every x Ω \ B 1/j x 0 there holds: 4.8 u ɛ x φx ux φx u ɛ u L Ω a j 1 2 a j u ɛ u L Ω u ɛ x 0 ux 0 = u ɛ x 0 φx 0 u ɛ x ɛ φx ɛ ɛ 3. By 4.7 it follows that for all x Ω we have: u ɛ x u ɛ x ɛ φx ɛ + φx ɛ 3 and hence: u ɛ x ɛ α 2 sup u ɛ + α B ɛx ɛ 2 sup u ɛ + β B ɛx ɛ u ɛ B ɛx ɛ u ɛ x ɛ φx ɛ ɛ 3 α + 2 sup φ + α B ɛx ɛ 2 sup φ + β B ɛx ɛ which further implies, for x ɛ argmin Bɛx ɛ φ: 4.9 α ɛ 3 2 sup φ + α B ɛx ɛ 2 sup φ + β B ɛx ɛ βɛ 2 2N + 2 p 2 2 φx ɛ x ɛ x ɛ ɛ φ φx ɛ B ɛx ɛ φ, B ɛx ɛ, x ɛ x ɛ + φx ɛ + oɛ 2. ɛ

13 THE OBSTACLE PROBLEM FOR THE p-laplacian 13 For completeness, we recall now [13] the proof of the second inequality in 4.9. Taylor expand the regular function φ at x ɛ, to get: min φ = φ x ɛ = φx ɛ + φx ɛ, x ɛ x ɛ φx ɛ x ɛ x ɛ, x ɛ x ɛ + oɛ 2. B ɛx ɛ 2 On the other hand, in a similar manner: max φ φx ɛ + x ɛ x ɛ = φx ɛ φx ɛ, x ɛ x ɛ φx ɛ x ɛ x ɛ, x ɛ x ɛ + oɛ 2, B ɛx ɛ 2 and again: Consequently, we obtain: α 2 max φ + α B ɛx ɛ 2 min φ + β B ɛx ɛ ɛ 2 φ = φ x ɛ + B ɛx ɛ 2N + 2 φx ɛ + oɛ 2. α 2 φ φx ɛ B ɛx ɛ 2 φx ɛ x ɛ x ɛ, x ɛ x ɛ which yields 4.9, because α 2 / βɛ 2 2N+2 = p 2 ɛ 2. + βɛ 2 2N + 2 φx ɛ + oɛ 2, 3. After dividing by ɛ 2, 4.9 becomes the following asymptotic inequality, valid for ɛ 0: lim sup p 2 2 φx ɛ x ɛ x ɛ, x ɛ x ɛ φx ɛ 0. ɛ 0 ɛ ɛ Note now that: x ɛ x ɛ 4.11 lim = φx 0 ɛ 0 ɛ φx 0. This follows by a simple blow-up argument. Indeed, the maps φ ɛ z = 1 ɛ φxɛ + ɛz φz ɛ converge uniformly on B 1 0 to the linear map φx 0, z. Hence the limit of any converging 1 subsequence of their minimizers: ɛ x ɛ x ɛ argmin B1 0 φ ɛ must be a minimizer of the limiting function φx 0, z. This minimizer is unique and equals: φx 0 / φx 0, proving In conclusion, 4.11 and 4.10 imply that: 1 φ p 2 pφx 0 = p 2 2 φx 0 φx 0 φx 0, φx 0 φx 0 which yields the validity of condition i in Definition 4.1, in view of φx 0 0, 4. To prove condition ii in Definition 4.1, let φ be as in 4.2. One can follow the argument as in steps 2. and 3. above, taking x ɛ to be the approximate maximizers of u ɛ φ. The first inequality in 4.8 is then replaced by equality because ux 0 > Ψx 0, and we consequently obtain: u ɛ x ɛ u ɛ x ɛ φx ɛ + ɛ 3 α + 2 sup φ + α B ɛx ɛ 2 sup φ + β B ɛx ɛ while the counterpart of 4.9, written for x ɛ argmax Bɛx ɛ φ is: ɛ 3 βɛ 2 p 2 2N φx ɛ x ɛ x ɛ ɛ φ B ɛx ɛ,, x ɛ x ɛ + φx ɛ + oɛ 2. ɛ Similarly to step 3. above, we conclude: p φx 0 0. The proof of Theorem 4.3 is complete.

14 14 MARTA LEWICKA AND JUAN J. MANFREDI 5. Estimates close to the boundary: a proof of Lemma 4.5 In this Section, by C we denote constants that depend only on the general setup of the problem, i.e. on N, Ω, p, α and β, but not on u, x 0, F or Ψ. By C F, C Ψ or C F,Ψ we denote constants depending additionally on F, Ψ, or on both F and Ψ. Let x 0 Ω and y 0 Ω. Assume that we have fixed a particular strategy σ 0,II of Player II. Then, by 1.11: u ɛ x 0 u ɛ y 0 sup [G x τ F y 0 ]. τ,σ I Note that for every x X: thus: Gx F y 0 χ Γ x F x F y 0 + χ Ω x Ψx Ψy 0 C F,Ψ x y 0, 5.1 u ɛ x 0 u ɛ y 0 C F,Ψ sup τ,σ I [ x τ y 0 ]. On the other hand, for a fixed strategy σ 0,I, again in view of 1.11 it follows that: 5.2 u ɛ x 0 u ɛ y 0 inf σ II τ 0,σ 0,I,σ II [G x τ0 F y 0 ] = inf σ II τ 0,σ 0,I,σ II [F x τ0 F y 0 ] C F sup σ II τ 0,σ 0,I,σ II [ x τ y 0 ]. We will now prove that, with σ 0,I and σ 0,II chosen appropriately, one has: 0 < δ 1 ɛ < min β, δ 5.3 2C δ 3 sup [ x τ y 0 ] + sup τ,σ 0,I,σ II [ x τ y 0 ] Cδ + C δ x 0 y 0 + ɛ, τ,σ I τ,σ II where the supremum is taken over all admissible stopping times τ τ 0. The constant C δ depends only on the associated parameter δ in 5.3. Clearly, 5.3 with 5.1 and 5.2 will imply 4.4. Remark 5.1. Denote by u 0 ɛ the ɛ-p-superharmonious function subject to the same boundary condition F as u ɛ on Γ, but in the absence of any obstacle. It satisfies: u 0 ɛx 0 = sup inf σ I σ τ 0,σ I,σ II [F x τ0 ] = inf sup II σ τ 0,σ I,σ II [F x τ0 ]. II σ I Equivalently, u 0 ɛ solves 1.9 with Ψ = const < min Γ F. It is clear that: 5.4 u ɛ u 0 ɛ in Ω. The following estimate has been proven in [14]: 5.5 y 0 Ω, x 0 Ω 0 < δ 1 u 0 ɛx 0 u 0 ɛy 0 C F δ + C δ x 0 y 0 + ɛ. Note that the lower bound follows directly by 5.4 and 5.5: u ɛ x 0 u ɛ y 0 u 0 ɛx 0 u 0 ɛy 0 C F δ + C δ x 0 y 0 + ɛ. Also, the upper bound is straightforward in case when x 0 belongs to the contact set, i.e. when: u ɛ x 0 = Ψx 0, because then in view of 4.3: u ɛ x 0 u ɛ y 0 = Ψx 0 F y 0 Ψx 0 Ψy 0 C Ψ x 0 y 0. It remains hence to prove a similar bound for the case x 0 Ω \ A ɛ. We will in fact reprove the inequality 5.5, in a slightly more general setting of the obstacle ɛ-p-superharmonious function u ɛ. The scheme of proof of 5.3 below follows [14] but we fill in all the details.

15 THE OBSTACLE PROBLEM FOR THE p-laplacian 15 Proof of Lemma Let δ > 0 and z 0 R N \ Ω satisfy: B δ z 0 Ω = {y 0 }. Define strategy σ 0,II for Player II: { 5.6 σ0,iix n 0,... x n = σ0,iix n xn + ɛ ɛ 3 z 0 x n n = z 0 x n if x n Ω x n if x n Γ. Let σ I be any strategy for Player I and let τ τ 0 be any admissible stopping time. Firstly, notice that for all ɛ < δ/3 we have: 5.7 x Ω w z 0 dw x z 0 + C δ ɛ 2. B ɛx This is because the function fw = w z 0 is smooth in the domain Ω + B δ/2 0 and hence by taking Taylor s expansion and averaging, we get: ffl B f = ux + ɛ2 ɛx 2N+2 fx + oɛ2. Take C = C δ + 1. By Lemma 3.2, the definition 5.6, and 5.7 it follows that: x 0,... x n 1 A τ n 1 { xn z 0 Cɛ 2 n F x 0 n 1} x0,... x n 1 α 2 σn 1 I x 0,... x n 1 z 0 + α 2 σn 1 0,II x n 1 z 0 + β w z 0 dw Cɛ 2 n B ɛx n 1 α 2 x n 1 z 0 + ɛ + α 2 x n 1 z 0 ɛ ɛ 3 + β x n 1 z 0 + C δ ɛ 2 Cɛ 2 n x n 1 z 0 Cɛ 2 n 1, while for x 0,... x n 1 A τ n 1 one has: { Ex 0 τ,σ I,σ xn 0,II z 0 Cɛ 2 n F x 0 n 1} x0,... x n 1 = x n 1 z 0 Cɛ 2 n x n 1 z 0 Cɛ 2 n 1. In any case, we see that { x n z 0 Cɛ 2 n} n 0 is a supermartingale with respect to the filtration {F x 0 n }. Applying Doob s theorem to the bounded stopping times τ n and 0, we obtain: [ x τ n z 0 ] Cɛ 2 [τ n] x 0 z 0. Consequently, and further using the dominated and the monotone convergence theorems while passing with n and recalling 3.5, we obtain: 5.8 [ x τ y 0 ] [ x τ z 0 ] + δ x 0 y 0 + 2δ + C δ ɛ 2 [τ]. 2. Towards estimating the expectation [τ] in 5.8, we first observe the following simple general result: Lemma 5.2. Let Y R N be a bounded, open set. Let x 0 Y and let {P n,x 0 } n 1, { P n,x 0 } n 1 be two consistent sequences of probability measures on Y,x 0 defined as in 3.4, where the filtration {F x 0 n } n 1 is as in subsection 3.1, and the transition probabilities are denoted, respectively: {γ n [x 0, x 1,... x n ]} and { γ n [x 0, x 1,... x n ]}. Let A F n,x 0 have the property: 5.9 ω = x 0, x 1,... A 0 k < n γ k [x 0, x 1,... x k ] = γ k [x 0, x 1,... x k ]. Then P n,x 0 A = P n,x 0 A. Proof. We prove the lemma by induction on n. For n = 1, the result is trivially true because P 1,x 0 = P 1,x 0 = γ 0 [x 0 ]. Let now A {x 0 } Y n be a Borel set. Fix η > 0 and find the covering A i=1 Ai 1 Ai 2 where each A i 1 {x 0} Y n 1 and A i 2 Y is a Borel set, such that the rectangles {Ai 1 Ai 2 } are pairwise disjoint, and such that: P n,x 0 A i 1 A i 2 P n,x 0 A + i=1 i=1 P n,x 0 A i 1 A i 2 P n,x 0 A η.

16 16 MARTA LEWICKA AND JUAN J. MANFREDI For each i 1 consider the Borel set A i = A A i 1 Ai 2. Its projection: πa i = {x 0, x 1,... x n 1 : x n Y x 0,... x n A i } does not have to be Borel compare Remark 3.5, but it is an analytic set [22] and hence it is measurable with respect to completions of Borel measures. Hence, there exists Borel sets B1 i, Ci 1 {x 0} Y n 1 such that: B i 1 πa i C i 1 and P n 1,x 0 C i 1 \ B i 1 = P n 1,x 0 C i 1 \ B i 1 = 0. We have then: P n,x 0 B1 i A i 2 = 5.11 = B i 1 B i 1 γ n 1 [x 0,... x n 1 ]A i 2 dp n 1,x 0 γ n 1 [x 0,... x n 1 ]A i 2 d P n 1,x 0 = P n,x 0 B i 1 A i 2, where we used the induction assumption to conclude that P n 1,x 0 B1 i = P n 1,x 0 B1 i. Further, in view of 5.10 and the fact that: P n,x 0 C1 i \ B1 i A i 2 = γ n 1 [x 0,... x n 1 ]A i 2 dp n 1,x 0 = 0, there holds: P n,x 0 A C i 1 \Bi 1 P n,x 0 B1 i A i 2 = i=1 n P n,x 0 A i P n,x 0 C1 i A i 2 i=1 i=1 i=1 n P n,x 0 A i P n,x 0 A i 1 A i 2 η. In a similar manner P n,x 0 A i=1 P n,x 0 B i 1 Ai 2 η. In view of 5.11, this implies: P n,x 0 A P n,x 0 A 2η. Since η > 0 was arbitrary, the lemma follows. Consider now a new game-board Y = B R z 0 X with the same initial token position x 0. Let σ I be an extension of the strategy σ I, given by: { 5.12 x 0,... x n Y n+1 σ I n σ n x 0,... x n = I x 0,... x n if x 0,... x n X n+1 otherwise, while: { σ 0,IIx n 0,... x n = σ 0,IIx n xn + ɛ ɛ 3 z 0 x n n = z 0 x n if x n Y \ B δ z 0 otherwise. Let τ 0 : Y,x 0 N {+ } be the exit time into the ball Bδ z 0, i.e.: x n x n i=1 τ 0 ω = min{n 0; x n z 0 δ} and let τ : Y,x 0 N {+ } be a stopping time extending τ, so that τ τ 0 and τ X,x 0 = τ. Define the transition probabilities on Y by: n 1 x 1,..., x n Y γ n [x 0, x 1,..., x n ] = α 2 δ σ I nx 0,,...,x n + α 2 δ σ 0,II n xn + βmx n for x n Y \ B δ z 0 = αδ xn + βmx n for x n B δ z 0 \ B δ ɛ z 0 δ xn for x n B δ ɛ z 0,

17 THE OBSTACLE PROBLEM FOR THE p-laplacian 17 where the probability mx is uniform in the set B ɛ x Y and is given by: 5.13 mx = L N B ɛ x Y. B ɛ x Y Let now P x 0 and P n,x 0 be the Borel probability measures on Y,x 0 defined as in subsection 3.4. As in the proof of Lemma 3.3, observe that: 5.14 Px 0 { τ0 < + } = 1, so that, in addition to 3.5 there also holds: P x 0 { τ < + } = 1. In view of Lemma 5.2 we hence observe: 5.15 [τ] = = n=1 n=1 n=1 np n,x 0 ω X x 0, ; τω = n n P n,x 0 ω X x 0, ; τω = n n P n,x 0 ω Y x 0, ; τω = n = Ēx 0 [ τ] Ēx 0 [ τ 0 ]. Indeed, if ω = x 0, x 1,... X,x 0 satisfies τω = n, then for all k < n we have: x k Ω and x 0, x 1,... x k A τ k, and hence there must be: γ k[x 0,... x k ] = γ k [x 0,... x k ]. Consequently: P n,x 0 ω X x 0, ; τω = n = P n,x 0 ω X x 0, ; τω = n. 3. We now estimate the expectation Ēx 0 [ τ 0 ]. Let v 0 : 0, + R be a smooth, increasing and concave function of the form: { as v 0 s = 2 bs 2 N + c for N > 2 as 2 b log s + c for N = 2, where the positive constants a, b, c are such that the function vx = v 0 x z 0 solves the following problem: v = 2N + 2 in B R z 0 \ B δ z 0 v = 0 on B δ z 0 v n = 0 on B Rz 0. As in 5.7, we obtain: 5.16 x R N \ B δ ɛ z 0 On the other hand, for every x B R z 0 \ B δ ɛ z 0, we have: v B ɛx B R z 0 1 = v B ɛx B ɛ x B R z 0 1 B ɛ x 1 B ɛ x B R z 0 1 B ɛ x v B ɛx B R z 0 v 0 R B ɛ x B R z 0 vw dw = vx ɛ 2. B ɛx 1 v B ɛ x B ɛx\b R z 0 1 B ɛ x v 0R B ɛ x \ B R z 0 = 0.

18 18 MARTA LEWICKA AND JUAN J. MANFREDI Consequently, recalling 5.13 and in view of 5.16: 5.17 x Y \ B δ ɛ z 0 v dmx vx ɛ 2. Consider the following bounded Borel functions on Y : vx + β 2 nɛ2 if x z 0 > δ ɛ Q n x = vx if δ 2ɛ < x z 0 δ ɛ v 0 δ 2ɛ if x z 0 δ 2ɛ, and compute the conditional expectation of the random variables Q n x n, which by Lemma 3.2 equals: Ē y { 0 Qn x n F x 0 n 1} x0,... x n 1 = Q n d γ n 1 [x 0,... x n 1 ]. Y We distinguish three cases. Case 1: x n 1 Y \ B δ z 0. Using the fact that v 0 is increasing, denoting its Lipschitz constant on [δ/3, R + δ] by C δ, recalling 5.17 and observing that σ II x n 1 z 0 > δ ɛ, we get: Q n d γ n 1 [x 0,... x n 1 ] = α 2 Q n 1 n σ I x 0,... x n 1 + α 2 Q n 1 n σ 0,II x n 1 + β Q n dmx n 1 Y α 2 v 0 xn 1 z 0 + ɛ + α 2 v 0 xn 1 z 0 ɛ + ɛ 3 + β v 0 x n 1 z 0 ɛ 2 + δ 2 nɛ2 αv 0 x n 1 z 0 + βv 0 x n 1 z 0 + C δ ɛ 3 βɛ 2 + β 2 nɛ2 v 0 x n 1 z 0 + β 2 n 1ɛ2 = Q n 1 x n 1, where the last two inequalities follow from concavity of v 0, and the fact that if only: 5.18 ɛ < min β, δ, 2C δ 3 then: C δ ɛ 3 βɛ 2 + β 2 nɛ2 β 2 ɛ2 n 1. Case 2: x n 1 B δ z 0 \ B δ ɛ z 0. Then: Q n d γ n 1 [x 0,... x n 1 ] = αq n x n 1 + β Y Q n dmx n 1 αv 0 x n 1 z 0 + β v 0 x n 1 z 0 ɛ 2 + δ 2 nɛ2 = v 0 x n 1 z 0 + β 2 n 1ɛ2 = Q n 1 x n 1. Case 3: x n 1 B δ ɛ z 0. In this case, we directly have: Q n d γ n 1 [x 0,... x n 1 ] = Q n 1 x n 1. Y Consequently, it follows that {Q n x n } n 0 is a supermartingale with respect to the filtration {F x 0 n }, under the assumption Applying Doob s theorem to pairs of bounded stopping times: τ 0 n and 0, we obtain: v 0 x 0 z 0 Ēx 0 [Q τ0 n] = Ēx 0 [v 0 x τ0 n z 0 ] + β 2 ɛ2 Ē x 0 [ τ 0 n].

19 THE OBSTACLE PROBLEM FOR THE p-laplacian 19 After passing with n, in view of the monotone convergence and dominated convergence theorems, and applying 5.14 we get: β ɛ2 Ē x 0 [ τ 0 ] v 0 x 0 z 0 + Ēx 0 [ v 0 x τ0 z 0 ]. Since v 0 δ = 0, it follows that v 0 x 0 z 0 C δ x0 z 0 δ = C δ x 0 y 0. Further, whenever τ 0 ω < +, we have: v 0 x τ0 z 0 C δ ɛ. Now, 5.19 together with 5.15 and 5.8 imply: for all ɛ sufficiently small as in [ x τ y 0 ] Cδ + C δ x 0 x 0 + ɛ 4. Clearly, exchanging the roles of σ I and σ II, and defining σ 0,I by means of 5.6 while setting σ II to be fixed, the same proof as above yields: τ,σ I,0,σ II [ x τ x 0 ] Cδ + C δ x 0 y 0 + ɛ for all ɛ sufficiently small. This gives 5.3 and ends the proof of Lemma Appendix: a proof of Lemma 3.3: games end almost-surely 1. Consider a new game-board Y = R N with the same initial token position x 0 Ω. By the same symbols σ I and σ II we denote the extensions on {Y n } n=0 of the given strategies σ I and σ II, defined as in the formula 5.12, where in order to simplify notation we suppress the overline in σ. Define also the new transition probabilities: γ n [x 0,..., x n ] = α 2 δ σi nx 0,...,x n + α 2 δ σii n x 0,...,x n + β B ɛ L N B ɛ x n, and let P n,x 0 σ I,σ II and P x 0 σ I,σ II be the resulting probability measures on Y,x 0 as in subsection 3.4. By Lemma 5.2 and since τ τ 0, it follows that: 6.1 P x 0 {τ < } = = n=0 n=0 P n,x 0 {ω X x 0, ; τω = n } P n,x 0 σ I, σ II {ω X x 0, ; τω = n} = = P x 0 σ I, σ II {τ < } P x 0 σ I, σ II {τ 0 < }. n=0 P n,x 0 σ I, σ II {ω Y x 0, ; τω = n} Let now A 0 be the sector in B ɛ = B ɛ 0: { } A 0 = x R N ; x ɛ/2, ɛ and x, e 1 π/8, π/8. For M N sufficiently large to ensure that M consecutive shifts of the token by vectors chosen from A 0 will get the token, originally located at any point in Ω, out of Ω, define: } S x0 = {ω = x 0, x 1,... Y x0, ; i 0 i = i 0, i 0 + 1,..., i 0 + M x i+1 x i A 0. t is clear that: 6.2 P x 0 σ I,σ II {τ 0 < } P x 0 σ I,σ II S x0. 2. We now show that the probability in the right hand side of 6.2 equals 1. Recall that for a bounded F x 0 -measurable function f : Y x0, R, its conditional expectation σ I,σ II {f F x 0 1 } is the function: x 0, x 1 E x 1 [f ], where σ σ I,σ I, σ II are strategies on Y x0, given by: II σ I n x 1,..., x n+1 = σ n+1 I x 0, x 1,..., x n+1, σ II n x 1,..., x n+1 = σ n+1 II x 0, x 1,..., x n+1,

20 20 MARTA LEWICKA AND JUAN J. MANFREDI while the Borel random variable f : Y x1, R is similarly set to be: f x 1, x 2,... = fx 0, x 1, x 2,.... Consequently: P x 0 σ I,σ II S x0 = E x [ 1 R N σ I χsx0 ] dγ 0 [x 0 ],σ II 6.3 = α 2 Eσ0 I x 0 σ I,σ II = α 2 Pσ0 I x 0 S x σ I 0,σ II [ χsx0 ] + α 2 Eσ0 II x 0 [ σ I χsx0 ] + β,σ II σ I,σ II σ 0 I x 0 + α 2 Pσ0 II x 0 S x 0 σ 0 II x 0 + β B ɛ E x [ 1 σ χsx0 ] dx 1 I B ɛx 0,σ II B ɛx 0 P x 1 S x σ I 0,σ x 1 dx 1, II where each set S x 0 x 1 = {x 1, x 2,... Y x1, ; x 0, x 1, x 2,... S x0 } clearly contains S x1. Let now: 6.4 qx = inf P x σ σ I, σ I, σ II S x. II By an easy translation invariance argument, qx = q is actually independent of x R N. Hence, in view of 6.3 we obtain: 6.5 P x 0 σ I,σ II S x0 α 2 Pσ0 I x 0 S σ I,σ σ 0 II I x 0 + α 2 Pσ0 II x 0 S σ I,σ σ 0 II II x 0 + β P x 1 S B ɛ σ I B ɛx 0 \x 0 +A 0,σ x1 dx 1 + β II B ɛ α 2 + α 2 + β B ɛ B ɛ \ A 0 q + β P x 1 B ɛ σ I x 0 +A,σ II 0 = θq + β S x 0 x B ɛ 1 dx 1, where we defined: x 0 +A 0 P x 1 σ I,σ II θ = α + β 1 A 0 B ɛ 3. Similarly as in the previous step, for every x 1 x 0 + A 0 there holds: where the set S x 0,x 1 x we see that: and hence: P x 1 S σ I,σ x x 0 = α 1 II 2 Pσ1 I x0,x 1 σ I,σ II S x 0,x 1. σi 1x 0,x 1 + α 2 Pσ1 II x 0,x 1 σ I,σ II + β B ɛ B ɛx 0 x 0 +A 0 P x 1 σ I,σ II S x 0 x 1 dx 1 S x 0,x 1 σii 1 x 0,x 1 P x 2 S x 0,x 1 σ I,σ x 2 dx 2, II S x 0 x 1 dx 1 = {x 2, x 3,... Y x 2, ; x 2, x 2, x 2, x 3... S x0 } contains the set S x2. By P x 1 S x σ I 0,σ x 1 θq + II inf σ I, σ II P x 2 σ I, σ II S x 0,x 1 x 2 q, β B ɛ x 1 +A 0 P x 2 σ I,σ II Since 1 θ = β A 0 / B ɛ, the estimate in 6.5 becomes: P x 0 β 2 σ I,σ II S x0 = θq + 1 θθq + B ɛ 6.6 x 0 +A 0 Iterating the same argument as above M times, we arrive at: P x 0 S x 0,x 1 x 2 dx 2. x 1 +A 0 P x 2 σ I,σ II S x 0,x 1 x 2 dx 2 dx 1. σ I,σ II S x0 θq + 1 θθq + 1 θ 2 θq θ M 1 θq β M +... P x M S x 0,...,x M 1 B ɛ σ M x 1 +A 0 x M 1 +A 0 I,σ M x M dx M... dx 1. II x 0 +A 0

21 THE OBSTACLE PROBLEM FOR THE p-laplacian 21 But each probability under the iterated integrals equals to 1, because: S x 0,...,x M 1 x M x 1 x 0 + A 0, x 2 x 1 + A 0,..., x M x M 1 + A 0. Consequently, by 6.6 we get: M 1 P x 0 σ I,σ II S x0 1 θ n θq + 1 θ M = 1 1 θ M q + 1 θ M. n=0 = Y x M, for Infimizing over all strategies σ I, σ II, it follows that q 1, since θ < 1 because of β > 0. Further: This achieves 3.5 in view of 6.1 and 6.2. P x 0 σ I,σ II S x0 q = Appendix: a proof of Lemma 4.2: uniqueness of viscosity solutions 1. Firstly, note that the continuous function u is a viscosity p-supersolution to 1.7. Thus, by the classical result in [7], u is p-superharmonic in Ω, and consequently see [10] we have u W 1,p loc Ω. In the same manner, it follows from Definition 4.1 that u is a viscosity p-subsolution on the open set V = {x Ω; ux > Ψx}, hence u is p-subharmonic in V. Therefore, using the variational definitions of p-super- and p-subharmonic functions, we have that for any open, Lipschitz domain U Ω there holds: 7.1 u p u + φ p φ C0 U, R +, 7.2 U V U u p U U V u + φ p φ C 0 U V, R. Let now φ C 0 U, R be such that Ψ u + φ. We write: φ = φ+ + φ as the sum of the positive and negative parts of φ. Denote: D + = {x U; φx > 0} and D = {x U; φx < 0} V. Then we have, in view of 7.1 and 7.2: u + φ p = u + φ p + u + φ p + u p U D + D {φ=0} 7.3 = u + φ + p + u + φ p u p D + U V U V\D u p + u p u p = u p. D + U V U V\D U The above means precisely that u is a variational solution of the obstacle problem on U, with the lower obstacle Ψ and boundary data f = u U ; we denote this problem by K Ψ,f U. Existence and uniqueness of such variational solution is an easy direct consequence of the strict convexity of the functional U u p. It is also quite classical that such solutions obey a comparison principle [10]. 2. Let now u and ū be as in the statement of the Lemma. Fix ɛ > 0. By the uniform continuity of u, ū on Ω and the fact that they coincide on Ω, there exists δ > 0 such that: 7.4 ux ūx ɛ x O δ Ω := Ω + B0, δ Ω. Consider an open, Lipschitz set U satisfying: Ω \ O δ Ω U Ω. By the argument in Step 1, u is the variational solution of the problem set K Ψ,u U U, and it is also easy to observe that ū + ɛ is the variational solution of the problem K Ψ,ū U +ɛu. Since u < ū + ɛ on U in view of 7.4, the comparison principle implies that u ū + ɛ in Ū.

22 22 MARTA LEWICKA AND JUAN J. MANFREDI Reversing the same argument and taking into account 7.4, we arrive at: ux ūx ɛ x Ω. We conclude that u = ū in Ω by passing to the limit ɛ 0 in the above bound. References [1] T. Antunovic, Y. Peres, S. Sheffield and S. Somersille, Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition, Comm PDEs, 37:10, [2] S. Armstrong and C. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans AMS, 364, [3] A. Björn and J. Björn, Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces. J. Math. Soc. Japan 584, [4] L. Codenotti, M. Lewicka and J.J. Manfredi, The discrete approximations to the double-obstacle problem, and optimal stopping of the tug-of-war games, to appear. [5] P. Erdos and A. H. Stone, On the sum of two Borel sets, Proc. Amer. Math. Soc. 25, [6] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Unabridged republication of the 1993 Oxford Mathematical Monograph original, Dover Publications, Inc., Mineola, NY, [7] P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal. 33, [8] R. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math. 593, [9] R. Kohn and S. Serfaty, A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations, Comm. Pure Appl. Math. 63, [10] P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. 365, [11] H. Luiro, M. Parviainen, and E. Saksman, On the existence and uniqueness of p-harmonious functions, Differential and Integral Equations [12] H. Luiro, M. Parviainen, and E. Saksman, Harnack s inequality for p-harmonic functions via stochastic games, Comm. Partial Differential Equations 3812, [13] J. Manfredi, M. Parviainen, and J. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc , [14] J. Manfredi, M. Parviainen and J. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. 112, , [15] J. Manfredi, J. Rossi and S. Somersille, An obstacle problem for Tug-of-War games, Communications on Pure and Applied Analysis. 14, , [16] B. Øksendal and K. Reikvam, Viscosity solutions of optimal stopping problems, Stoc. and Stoc. Reports, 62, , 1998 [17] H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity solution approach, J. Math. Sys. Est. Cont. 8, 1 27, 1998 electronic. [18] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer Stochastic Modelling and Applied Probability 61, [19] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22, [20] Y. Peres, S. Sheffield, Tug-of-war with noise: a game theoretic view of the p-laplacian, Duke Math. J. 1451, , [21] G. Peskir, A. Shiryaev, Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics, ETH Zürich, Birkhäuser, [22] A. C. M. Van Rooij and W. H. Schikhof, A second course on real functions, Cambridge University Press, [23] S. Varadhan, Probability theory, Courant Lecture Notes in Mathematics, 7. Marta Lewicka and Juan J. Manfredi, University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, PA address: lewicka@pitt.edu, manfredi@pitt.edu