Given g an α-H¨older continuous function defined on the boundary of a bounded domain Ω and given ψ a continuous obstacle defined in Ω, in this article, we find u an α-H¨older extension of g in Ω with u ≥ ψ. This function u minimizes the α-H¨older semi-norm of all possible extensions with these properties and it is a viscosity solution of the associated obstacle problem for the infinity fractional Laplace operator.Fil: Moreno Mérida, Lourdes. Universidad de Granada; EspañaFil: Vidal, Raúl Emilio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

Moreno Mérida, Lourdes

Vidal, Raúl Emilio

Rendiconti del Circolo Matematico di Palermo (1952 -)

English

Lourdes Moreno Mérida

Raúl Emilio Vidal

Crossref

The obstacle problem for the infinity fractional laplacian

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THE OBSTACLE PROBLEM FOR THE INFINITYFRACTIONAL LAPLACIANLOURDES MORENO MÉRIDA AND RAÚL EMILIO VIDALAbstract. Given g an α-Hölder continuous function defined on theboundary of a bounded domain Ω and given ψ a continuous obstacledefined in Ω, in this article, we find u an α-Hölder extension of g in Ωwith u ≥ ψ. This function u minimizes the α-Hölder semi-norm of allpossible extensions with these properties and it is a viscosity solutionof the associated obstacle problem for the infinity fractional Laplaceoperator.1. IntroductionLet Ω be an open, bounded domain of IRN and α ∈ (0, 1). In this paperwe will consider the infinite fractional Laplace operator given byLu(x) = supy∈Ω, y 6=xu(y)− u(x)|y − x|α+ infy∈Ω, y 6=xu(y)− u(x)|y − x|α, for x ∈ Ω .Motivated by the results of Chambolle, Lindgren and Monneau (see [6]), wewill be interested in solutions of the associated Dirichlet obstacle problem.Concretely, given an α-Hölder function g defined on ∂Ω and a continuousobstacle ψ defined on Ω, we aim to prove the existence and uniqueness of atleast a super infinity fractional harmonic function constrained to lie abovethe obstacle and to take the datum on ∂Ω. More precisely, we consider thefollowing obstacle problem(1)−Lu(x) = 0, in {x ∈ Ω : u(x) > ψ(x)},−Lu(x) ≥ 0, in {x ∈ Ω : u(x) = ψ(x)},u(x) ≥ ψ(x), if x ∈ Ω,u(x) = g(x), if x ∈ ∂Ω,and we will study the existence and uniqueness of a viscosity solution thatseems to be natural in this framework.By a viscosity subsolution (resp. supersolution) of (1) we mean an up-per semicontinuous (resp. lower semicontinuous) function u from Ω to IRsatisfying that u ≤ g (resp. u ≥ g) on ∂Ω and the following property:∀ϕ ∈ C1(Ω) ∩ C(Ω) such thatu ≤ ϕ , in Ω (resp. u ≥ ϕ) ,Key words and phrases. Infinity fractional Laplace operator, viscosity solutions, obsta-cle problem.2010 Mathematics Subject Classification: 35D40, 35J60, 35J65.12 L. MORENO MÉRIDA AND R. E. VIDALu(x0) = ϕ(x0) , for some x0 ∈ Ω ,thenmin{−Lϕ(x0), ϕ(x0)− ψ(x0)} ≤ 0 ( resp. ≥ 0) .A viscosity solution is a function which is both a subsolution and a su-persolution.We observe that, if u is a continuos function defined on Ω satisfying u = gon ∂Ω, then we easily deduce, from the above definition, the following char-acterization of viscosity sub and super solution. Concretely, u is a viscositysubsolution of (1) if for any ϕ ∈ C1(Ω) ∩ C(Ω) such thatu ≤ ϕ , in Ω ,ψ(x0) < u(x0) = ϕ(x0) , for some x0 ∈ Ω ,then−Lϕ(x0) ≤ 0 .On the other hand, u is a viscosity supersolution of (1) if for any ϕ ∈C1(Ω) ∩ C(Ω) such thatu ≥ ϕ , in Ω ,u(x0) = ϕ(x0) , for some x0 ∈ Ω ,then−Lϕ(x0) ≥ 0 and u(x0) ≥ ψ(x0).It is interesting to note that the obstacle function ψ is always a viscositysubsolution of (1). See [11] for more information about viscosity solutions.We also emphasize that in order to have a solution of our obstacle problem(1), it is necessary (due to the boundary conditions) that(2) ψ(x) ≤ g(x) , ∀x ∈ ∂Ω ,holds true.Problem (1) involves a boundary problem in the fractional setting. Thiskink of problems have been studied extensively, see for instance [7] and [8].Specifically, Problem (1) involves the infinity fractional Laplace operatorThe infinity Laplace operator was considered widely in the literature in thelocal case, see [1], [2], [5], [12] [14], and [15]; as well as in the nonlocal case(especially fractional), see for instance [4], [6] and [10]. Moreover, in [4], [12]and [15] it is studied the existence of a solution for some obstacle problems.On one hand, in [12], the authors consider the (local) infinity Laplace oper-ator. In particular, they propose a game which involves an obstacle functionand they prove that certain limit of some specific values functions is a viscos-ity solution of the obstacle problem for the infinity Laplacian. On the otherside, in [4], the authors consider a nonlocal tug of war game. Motivated by[14] the authors consider here a nonlocal version of the game.Recently, in [6], given an α-Hölder continuous function g defined on ∂Ω,it is obtained a viscosity solution for the Dirichlet problem{ −Lu(x) = 0, if x ∈ Ω,u(x) = g(x), if x ∈ ∂Ω,where L is the infinity fractional Laplace operator defined before. Our aimis to extend these results considering the study of the obstacle problem (1).THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 3Specifically, if we will denote the α-Hölder semi-norm of a function u definedon Ω by 1[u]α = supx,y∈Ω, x 6=y|u(y)− u(x)||y − x|α,our main results are the following theorems.Theorem 1. Let Ω be an open, bounded domain of IRN , g ∈ C(∂Ω) andψ ∈ C(Ω) such that (2) holds true. Then there exists at most a viscositysolution u of the obstacle problem (1).Theorem 2. Let Ω be an open, bounded and Lipschitz domain of IRN ,α ∈ (0, 1). If g ∈ C0,α(∂Ω) and ψ ∈ C(Ω) satisfy (2), then there existsa unique viscosity solution u of the obstacle problem (1) which belongs toC0,α(Ω). Moreover, the solution u is the best α-Hölder continuous extensionof the datum g which lies above the obstacle ψ, in the sense that[u]α ≤ [z]α ,for any arbitrary α-Hölder extension z of the datum g which satisfies z ≥ ψ.Following the arguments of [6], a possible approach to study our problem(1) is to approximate our infinity Laplace operator L with a sequence ofapproximate operators (see Section 2 below). In this sense, in what follows,given p > N , Np < α < 1 and s := α−Np , we consider the fractional Sobolevspace W s,p(Ω) defined byW s,p(Ω) ={u ∈ Lp(Ω) :∫Ω∫Ω(|u(y)− u(x)||y − x|α)pdydx < +∞},and we recall that (W s,p(Ω), ||.||s,p) is a Banach space, where||u||s,p =(∫Ω|u|p +∫Ω∫Ω(|u(y)− u(x)||y − x|α)pdydx)1/p, u ∈W s,p(Ω) .We define the functional Ep : Ws,p(Ω) −→ IR by(3) Ep(u) =∫Ω∫Ω(|u(y)− u(x)||y − x|α)pdydx ,and we study the minimization problem in a specific set. Observe that theoperator of the Euler Lagrange equation associated to this functional is(4) Lpu(x) =∫Ω(|u(y)− u(x)||y − x|α)p−1 sig(u(y)− u(x))|y − x|αdy ,where sig(x) = x|x| for x 6= 0. At least formally, we emphasize that theoperator (Lp(u))1/(p−1) should tend to our infinity fractional Laplace oper-ator L when p goes to ∞. We remark that this formal limit procedure onlyworks when the right hand side is zero (when it is not zero one may expecta different limit equation). This will be the key point in our approach. Wewant to prove that the unique minimum (belonging to a suitable set) up ofEp is a viscosity solution of the obstacle problem associated to the operatorLp. Afterwards, we want to pass to the limit when p tends to infinity. We1Recall that (C0,α(Ω), ||.||∞ + [.]α) is a Banach Space4 L. MORENO MÉRIDA AND R. E. VIDALwill prove that the limit of the sequence up of approximate solutions is aviscosity solution of (1).The article is organized as follows: in Section 2, we study the properties ofthe approximate obstacle problems (associated to the approximate operatorsLp) and in Section 3, we prove our main results.2. Approximate problemsWe consider the approximate operators Lp given by (4) and we study inthis section the approximate obstacle problems(5)−Lpu(x) = 0, in {x ∈ Ω : u(x) > ψ(x)},−Lpu(x) ≥ 0, in {x ∈ Ω : u(x) = ψ(x)},u(x) ≥ ψ(x), if x ∈ Ω,u(x) = g(x), if x ∈ ∂Ω.In the next lemmas, we prove that the functional Ep given by (3) hasa unique minimum (in a specific set) which is a viscosity solution of theapproximate obstacle problem (5).Lemma 1. Let Ω be an open, bounded and Lipschitz domain of IRN , α ∈(0, 1), p > 2Nα . If g ∈ C0,α(∂Ω), ψ ∈ C(Ω) and (2) holds, then the functionalEp given by (3) takes a unique minimum up in the setXg,ψ = {v ∈W s,p(Ω) : v ≥ ψ en Ω , v = g en ∂Ω} .Moreover, up belongs to C(Ω).Proof. Firstly we observe that, any α-Hölder extension of g which lies abovethe obstacle ψ belongs to the set Xg,ψ. Thus, this set is not empty (see [15,Proposition 3.3] for the existence of this extension). In addition, if we takeu ∈ Xg,ψ and we fix k ≥ ||ψ||∞ + ||g||∞, then Tk(u) ∈ Xg,ψ andEp(u) ≥ Ep(Tk(u)) ,where the function Tk : IR −→ IR is defined byTk(s) = −k , si s < −k ,s , si |s| ≤ k ,k , si s > k .As a consequence, we have thatminXg,ψEp(u) = minXkg,ψEp(u) ,whereXkg,ψ = {u ∈ Xg,ψ : ||u||∞ ≤ k}.Since the set Xkg,ψ is weakly closed (with the weak topology of Ws,p(Ω)), toprove the existence of a minimum in this set, we will study the coercivityand the weak lower semicontinuity of the functional Ep.On the one hand, we take a sequence {un} ⊂ Xkg,ψ such that ||un||s,p →+∞. Since ||un||∞ ≤ k, this necessarily means that Ep(un) → +∞. Thatis, our functional is coercive.THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 5On the other hand, we take a sequence {un} ⊂ Xkg,ψ such that un weaklyconverges to a function u in W s,p(Ω). Since W s,p(Ω) is compactly embeddedin Lp(Ω) (see [9, Corollary 1.2]) and the norm ||.||s,p is a w.l.s.c. function,thenlim inf Ep(un) ≥ Ep(u)as we desired.Consequently, the functional Ep has a minimum up in the set Xg,ψ andagain by [9, Theorem 8.2], up ∈ C(Ω). Moreover, since the functional isconvex, this minimum is unique. Lemma 2. Let Ω be an open, bounded and Lipschitz domain of IRN , g ∈C0,α(∂Ω) and ψ ∈ C(Ω) satisfying (2). If p > 2N/α, then the minimum up(given by Lemma 1) is a viscosity solution of (5).Proof. Firstly, recall that the minimum up (given by Lemma 1) belongs toC(Ω) and satisfies up = g on ∂Ω and up ≥ ψ in Ω. We will prove that up isa viscosity sub and super solution. On the one hand, we claim that up is aviscosity subsolution of (5). Indeed, we take ϕ ∈ C1(Ω) ∩ C(Ω) such thatup ≤ ϕ , in Ω ,ψ(x0) < up(x0) = ϕ(x0) , for some x0 ∈ Ω ,and we prove −Lϕ(x0) ≤ 0. Without loss of generality, we suppose thatϕ touches up only at the point x0; otherwise it is sufficient to replace ϕ byϕ(x) + δ|x− x0|2 with δ small enough. We define the functionsϕε = max(up, ϕ− ε) ,andϕε = min(up, ϕ− ε).Since, we suppose that ϕ(x0) = up(x0) > ψ(x0), for ε small enough, ϕε ≥ ψin Ω and moreover, ϕε = up on ∂Ω. Hence, ϕε belongs to Xg,ψ and usingthat up is a minimum of Ep in this set, we haveEp(ϕε) ≥ Ep(up) .From this inequality and using the following convexity inequality (see [6,Lemma 6.2])|max(a, c)−max(b, d)|p + |min(a, c)−min(b, d)|p ≤ |a− b|p + |c− d|p ,for all p ≥ 1 , we deduce thatEp(ϕε) + Ep(ϕε) ≤ Ep(up) + Ep(ϕ) ≤ Ep(ϕε) + Ep(ϕ) ,that isEp(ϕε) ≤ Ep(ϕ) .The convexity of Ep impliesEp((1− t)ϕ+ tϕε) ≤ (1− t)Ep(ϕ) + tEp(ϕε) ≤ Ep(ϕ) ,and then we haveEp((1− t)ϕ+ tϕε)− Ep(ϕ)t≤ 0 .Let callf(t) = Ep((1− t)ϕ+ tϕε).6 L. MORENO MÉRIDA AND R. E. VIDALFrom the above inequality and using the convexity of the function f , wehavef ′(0) ≤ f(t)− f(0)t≤ 0 ,and thenp∫Ω∫ΩH(x, y) dydx ≤ 0 ,whereH(x, y)=∣∣∣∣ϕ(y)−ϕ(x)|y − x|α∣∣∣∣p−1sgn(ϕ(y)−ϕ(x))|y − x|α (ϕε(y)−ϕ(y)+ε−ϕε(x)+ϕ(x)−ε) .Therefore, a change of variable implies∫Ω(ϕε − ϕ+ ε)(x)(−Lpϕ(x)) dx ≤ 0 .Now we argue by contradiction. Suppose that −Lpϕ(x0) > 0. By continuity,which holds under our assumptions, there is a small ball Br(x0) such that−Lpϕ > 0 in Br(x0). Since ϕε = max(up, ϕ − ε), for ε small enough, wehave supp(ϕε − ϕ + ε) ⊂ Br(x0). We also observe that ϕε − ϕ + ε ≥ 0.Consequently, we deduce0 <∫Br(x0)(ϕε − ϕ+ ε)(x)(−Lpϕ(x)) dx ≤ 0 ,which is a contradiction.In the same way, one can prove that u is a viscosity supersolution. 3. Proofs of the main resultsProof of Theorem 1. Suppose that u and v are two viscosity solutions of theobstacle problem (1) and define the setW = {x ∈ Ω : u(x) > v(x)} .We claim that W is an empty set. Indeed, arguing by contradiction, wesuppose that W is not empty. Since v ≥ ψ in Ω, we have v ≥ ψ and u > ψin W . Consequently, the functions u and v satisfy{−Lu = 0 in W ,u = u on ∂W ,{−Lv ≥ 0 in W ,v = u on ∂W ,which implies, using the comparison principle [6, Proposition 11.2], thatv ≥ u in W . This is a contradiction and the claim is proved. Reversing therole of u and v gives that u = v which conclude the proof. To prove Theorem 2 we need the following technical result.Lemma 3. [6, Lemma 6.5] For ϕ ∈ C1(Ω), p ≥ 1 and α ∈ (0, 1), we definefp(y) =ϕ(y)− ϕ(xp)|y − xp|αand f(y) =ϕ(y)− ϕ(x0)|y − x0|α,where xp −→ x0 ∈ Ω as p −→∞. Then,limp−→∞∣∣∣∣∣∣∣∣∣∣ f+p (y)|y − xp|α/p∣∣∣∣∣∣∣∣∣∣Lp(Ω)=∣∣∣∣f+∣∣∣∣L∞(Ω),THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 7with f±(x) = max(±f(x), 0). The same also holds for f−p .Proof of Theorem 2. Let {up} be a sequence of viscosity solutions up of (5)given by Lemma 1. Our aim is to pass to the limit when p goes to infinite.Firstly, given α ∈ (0, 1), we prove that the sequence {up} is bounded inW s,q(Ω) for any q > 2N/α. Indeed, by construction, there is a positiveconstant k such that||up||∞ ≤ k , ∀p .Now, we take any p > 2N/α and we fix a number q such that 2N/α < q < p.Let z be a Hölder extension of g such that z ≥ ψ (see [15, Proposition 3.3]).Since the functional Ep takes a unique minimum up in the set Xg,ψ, thenEp(up) ≤ Ep(z) ≤ |Ω|2[z]pα,and by Hölder inequality(6) Eq(up) ≤ Ep(up)qp |Ω|2(p−q)p ≤ |Ω|2[z]qα ,which implies that the sequence {up} is bounded in W s,q(Ω). By the Sobolevembedding (see [9, Theorem 8.2]) we deduce that, up to a subsequence, upstrongly converges to a function u in C(Ω). Moreover, since up = g on ∂Ω,and up ≥ ψ in Ω, then we also have that the function u satisfiesu = g , on ∂Ω , u ≥ ψ , in Ω .Now, we will prove that u is a viscosity sub and super solution of problem(1). On the one hand, we claim that u is a viscosity subsolution. Indeed,we take ϕ ∈ C1(Ω) ∩ C(Ω) such thatu ≤ ϕ , in Ω ,ψ(x0) < u(x0) = ϕ(x0) , for some x0 ∈ Ω ,and, without loss of generality, we suppose that ϕ touches u only at thepoint x0 (x0 is a strict maximum of u− ϕ). Hence,Mp := supΩ(up − ϕ) = (up − ϕ)(xp),wherexp 7→ x0 , Mp 7→ 0.Moreover, since ϕ(x0) > ψ(x0), we can suppose that ϕ(xp) > ψ(xp) for plarge enough. This shows that{up ≤ ϕp := ϕ+Mp ,ψ(xp) < up(xp) = ϕ(xp).The fact that up is a viscosity solution implies0 ≥ −Lpϕp(xp) = −Lpϕ(xp),that is,0 ≥ −∫Ω∣∣∣∣ϕ(y)− ϕ(xp)|y − xp|α∣∣∣∣p−1 sgn(ϕ(y)− ϕ(xp))|y − xp|α dy ,or equivalently,∣∣∣∣∣∣∣∣∣∣(ϕ(y)− ϕ(xp)|y − xp|α+αp−1)+∣∣∣∣∣∣∣∣∣∣Lp−1(Ω)≥∣∣∣∣∣∣∣∣∣∣(ϕ(y)− ϕ(xp)|y − xp|α+αp−1)−∣∣∣∣∣∣∣∣∣∣Lp−1(Ω).8 L. MORENO MÉRIDA AND R. E. VIDALThanks to Lemma 3, we can pass to the limit in this inequality to obtainsupy∈Ω(max(ϕ(y)− ϕ(x0)|y − x0|α, 0))+ infy∈Ω(min(ϕ(y)− ϕ(x0)|y − x0|α, 0))≥ 0.Since ϕ is C1 at x0, it is clear that L+ϕ(x0) ≥ 0 and L−ϕ(x0) ≤ 0, whereL+ϕ(x0) = supy∈Ω, y 6=x0ϕ(y)− ϕ(x0)|y − x0|α, L−ϕ(x0) = infy∈Ω, y 6=x0ϕ(y)− ϕ(x0)|y − x0|α.Summing up, we deduce−Lϕ(x0) ≤ 0 ,as we desired.Finally, in the same way, one can proved that u is a viscosity super solu-tion. Therefore, u is a viscosity solution of the obstacle problem (1).To conclude, we characterize the function limit u. In order to do it, let zbe any Hölder extension of g such that z ≥ ψ. By (6), we haveEq(up) ≤ |Ω|2[z]qα,which implies, passing to the limit as p goes to ∞,(Eq(u))1/q ≤ |Ω|2/q[z]α.As a consequence, when q tends to ∞, we obtain∣∣∣∣∣∣∣∣u(y)− u(x)|y − x|α∣∣∣∣∣∣∣∣L∞(Ω×Ω)≤ [z]α,i.e., we have proved that u ∈ C0,α(Ω) and moreover[u]α ≤ [z]α ,for any arbitrary Hölder extension z of the datum g which satisfies z ≥ ψ,as we desired.Acknowledgements. The first author was supported by MINECO-FEDERgrant MTM2015-68210-P, Junta de Andalućıa FQM-116 and Ministerio deEducación, Cultura y Deporte (Spain) FPU grant FPU12/02395. The sec-ond author was partially supported by IEMath-Granada, CONICET andSecyt-UNC.References[1] Aronsson, G. On certain singular solutions of the partial differential equation u2xuxx +2uxuyuxy + u2yuyy = 0. Manuscripta mathematica, vol. 47, no 1-3, (1984), p. 133-151.[2] Aronsson, G., Crandall, M., and Juutinen, P. A tour of the theory of absolutely mini-mizing functions. Bulletin of the American mathematical society, vol. 41, no 4, (2004),p. 439-505.[3] Barles, G.; Chasseigne, E. and Imbert, C. On the Dirichlet problem for second-orderelliptic integro-differential equations. Indiana University Mathematics Journal, vol. 57,no 1, (2008), p. 213-146.[4] Bjorland C., Caffarelli L. and Figalli A. Nonlocal TugofWar and the Infinity FractionalLaplacian. Communications on Pure and Applied Mathematics, vol. 65, no 3, (2012),p. 337-380.[5] Bhattacharya, T.; Dibenedetto, E. and Manfredi, J. Limits as p → ∞ of ∆pup = fand related extremal problems. Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo SpecialeNonlinear PDEs, (1989), p. 15-68.THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 9[6] Chambolle, A., Lindgren, E. and Monneau, R. A Hölder infinity Laplacian. Optimisa-tion and Calculus of Variations, vol. 18, no 03, (2012), p. 799-835.[7] Caffarelli, L., Salsa, S., and Silvestre, L. Regularity estimates for the solution and thefree boundary of the obstacle problem for the fractional Laplacian. Inventiones mathe-maticae, vol 171, no 2, (2008), p. 425-461.[8] Dipierro, S., Savin, O., and Valdinoci, E. A nonlocal free boundary problem. SIAMJournal on Mathematical Analysis, vol. 47, no 6, (2015), p. 4559-4605.[9] Di Nezza, E.; Palatucci, G. and Valdinoci, E. Hitchhikers guide to the fractional Sobolevspaces. Bulletin des Sciences Mathmatiques. vol. 136, no 5, (2012), p. 521-573.[10] Ferreira, R. and Pérez-Llanos, M. Limit problems for a fractional p-Laplacian asp→∞. Nonlinear Differ. Equ. Appl. no 2, Art. 14, 28pp., (2016). doi:10.1007/s00030-016-0368-z[11] Lions Crandall, M.; Ishit, H. and Lions, P. Users guide to viscosity solutions of secondorder partial differential equations. Bulletin of the American Mathematical Society, vol.27, no 1, (1992), p. 1-67.[12] Manfredi, J.; Rossi, J. and Somersille, S. An obstacle problem for Tug-of-War games.Communications on Pure and Applied Analysis. Vol. 14(1), (2015), p. 217-228.[13] Mcshne, E. Extension of range of functions. Bulletin of the American MathematicalSociety, vol. 40, no 12, (1934), p. 837-842.[14] Peres, Y., Schramm, O., Sheffield, S. and Wilson, D. Tug-of-war and the infinityLaplacian. Journal of the American Mathematical Society, vol. 22, no 1, (2009), p.167-210.[15] Rossi, J.; Teixeira, E. and Urbano J. Optimal regularity at the free boundary forthe infinity obstacle problem. Interfaces and Free Boundaries. Vol. 17(3), (2015), p.381-398.(L. Moreno-Mérida)Universidad de Granada, Granada, España.email: lumore@ugr.es(R. E. Vidal)FaMAF, Universidad Nacional de Cordoba, (5000), Cordoba, Argentina.email: vidal@mate.uncor.edu

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THE OBSTACLE PROBLEM FOR THE INFINITYFRACTIONAL LAPLACIANLOURDES MORENO ME´RIDA AND RAU´L EMILIO VIDALAbstract. Given g an α-Ho¨lder continuous function defined on theboundary of a bounded domain Ω and given ψ a continuous obstacledefined in Ω, in this article, we find u an α-Ho¨lder extension of g in Ωwith u ≥ ψ. This function u minimizes the α-Ho¨lder semi-norm of allpossible extensions with these properties and it is a viscosity solutionof the associated obstacle problem for the infinity fractional Laplaceoperator.1. IntroductionLet Ω be an open, bounded domain of IRN and α ∈ (0, 1). In this paperwe will consider the infinite fractional Laplace operator given byLu(x) = supy∈Ω, y 6=xu(y)− u(x)|y − x|α + infy∈Ω, y 6=xu(y)− u(x)|y − x|α , for x ∈ Ω .Motivated by the results of Chambolle, Lindgren and Monneau (see [6]), wewill be interested in solutions of the associated Dirichlet obstacle problem.Concretely, given an α-Ho¨lder function g defined on ∂Ω and a continuousobstacle ψ defined on Ω, we aim to prove the existence and uniqueness of atleast a super infinity fractional harmonic function constrained to lie abovethe obstacle and to take the datum on ∂Ω. More precisely, we consider thefollowing obstacle problem(1)−Lu(x) = 0, in {x ∈ Ω : u(x) > ψ(x)},−Lu(x) ≥ 0, in {x ∈ Ω : u(x) = ψ(x)},u(x) ≥ ψ(x), if x ∈ Ω,u(x) = g(x), if x ∈ ∂Ω,and we will study the existence and uniqueness of a viscosity solution thatseems to be natural in this framework.By a viscosity subsolution (resp. supersolution) of (1) we mean an up-per semicontinuous (resp. lower semicontinuous) function u from Ω to IRsatisfying that u ≤ g (resp. u ≥ g) on ∂Ω and the following property:∀ϕ ∈ C1(Ω) ∩ C(Ω) such thatu ≤ ϕ , in Ω (resp. u ≥ ϕ) ,Key words and phrases. Infinity fractional Laplace operator, viscosity solutions, obsta-cle problem.2010 Mathematics Subject Classification: 35D40, 35J60, 35J65.12 L. MORENO ME´RIDA AND R. E. VIDALu(x0) = ϕ(x0) , for some x0 ∈ Ω ,thenmin{−Lϕ(x0), ϕ(x0)− ψ(x0)} ≤ 0 ( resp. ≥ 0) .A viscosity solution is a function which is both a subsolution and a su-persolution.We observe that, if u is a continuos function defined on Ω satisfying u = gon ∂Ω, then we easily deduce, from the above definition, the following char-acterization of viscosity sub and super solution. Concretely, u is a viscositysubsolution of (1) if for any ϕ ∈ C1(Ω) ∩ C(Ω) such thatu ≤ ϕ , in Ω ,ψ(x0) < u(x0) = ϕ(x0) , for some x0 ∈ Ω ,then−Lϕ(x0) ≤ 0 .On the other hand, u is a viscosity supersolution of (1) if for any ϕ ∈C1(Ω) ∩ C(Ω) such thatu ≥ ϕ , in Ω ,u(x0) = ϕ(x0) , for some x0 ∈ Ω ,then−Lϕ(x0) ≥ 0 and u(x0) ≥ ψ(x0).It is interesting to note that the obstacle function ψ is always a viscositysubsolution of (1). See [11] for more information about viscosity solutions.We also emphasize that in order to have a solution of our obstacle problem(1), it is necessary (due to the boundary conditions) that(2) ψ(x) ≤ g(x) , ∀x ∈ ∂Ω ,holds true.Problem (1) involves a boundary problem in the fractional setting. Thiskink of problems have been studied extensively, see for instance [7] and [8].Specifically, Problem (1) involves the infinity fractional Laplace operatorThe infinity Laplace operator was considered widely in the literature in thelocal case, see [1], [2], [5], [12] [14], and [15]; as well as in the nonlocal case(especially fractional), see for instance [4], [6] and [10]. Moreover, in [4], [12]and [15] it is studied the existence of a solution for some obstacle problems.On one hand, in [12], the authors consider the (local) infinity Laplace oper-ator. In particular, they propose a game which involves an obstacle functionand they prove that certain limit of some specific values functions is a viscos-ity solution of the obstacle problem for the infinity Laplacian. On the otherside, in [4], the authors consider a nonlocal tug of war game. Motivated by[14] the authors consider here a nonlocal version of the game.Recently, in [6], given an α-Ho¨lder continuous function g defined on ∂Ω,it is obtained a viscosity solution for the Dirichlet problem{ −Lu(x) = 0, if x ∈ Ω,u(x) = g(x), if x ∈ ∂Ω,where L is the infinity fractional Laplace operator defined before. Our aimis to extend these results considering the study of the obstacle problem (1).THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 3Specifically, if we will denote the α-Ho¨lder semi-norm of a function u definedon Ω by 1[u]α = supx,y∈Ω, x 6=y|u(y)− u(x)||y − x|α ,our main results are the following theorems.Theorem 1. Let Ω be an open, bounded domain of IRN , g ∈ C(∂Ω) andψ ∈ C(Ω) such that (2) holds true. Then there exists at most a viscositysolution u of the obstacle problem (1).Theorem 2. Let Ω be an open, bounded and Lipschitz domain of IRN ,α ∈ (0, 1). If g ∈ C0,α(∂Ω) and ψ ∈ C(Ω) satisfy (2), then there existsa unique viscosity solution u of the obstacle problem (1) which belongs toC0,α(Ω). Moreover, the solution u is the best α-Ho¨lder continuous extensionof the datum g which lies above the obstacle ψ, in the sense that[u]α ≤ [z]α ,for any arbitrary α-Ho¨lder extension z of the datum g which satisfies z ≥ ψ.Following the arguments of [6], a possible approach to study our problem(1) is to approximate our infinity Laplace operator L with a sequence ofapproximate operators (see Section 2 below). In this sense, in what follows,given p > N , Np < α < 1 and s := α− Np , we consider the fractional Sobolevspace W s,p(Ω) defined byW s,p(Ω) ={u ∈ Lp(Ω) :∫Ω∫Ω( |u(y)− u(x)||y − x|α)pdydx < +∞},and we recall that (W s,p(Ω), ||.||s,p) is a Banach space, where||u||s,p =(∫Ω|u|p +∫Ω∫Ω( |u(y)− u(x)||y − x|α)pdydx)1/p, u ∈W s,p(Ω) .We define the functional Ep : Ws,p(Ω) −→ IR by(3) Ep(u) =∫Ω∫Ω( |u(y)− u(x)||y − x|α)pdydx ,and we study the minimization problem in a specific set. Observe that theoperator of the Euler Lagrange equation associated to this functional is(4) Lpu(x) =∫Ω( |u(y)− u(x)||y − x|α)p−1 sig(u(y)− u(x))|y − x|α dy ,where sig(x) = x|x| for x 6= 0. At least formally, we emphasize that theoperator (Lp(u))1/(p−1) should tend to our infinity fractional Laplace oper-ator L when p goes to ∞. We remark that this formal limit procedure onlyworks when the right hand side is zero (when it is not zero one may expecta different limit equation). This will be the key point in our approach. Wewant to prove that the unique minimum (belonging to a suitable set) up ofEp is a viscosity solution of the obstacle problem associated to the operatorLp. Afterwards, we want to pass to the limit when p tends to infinity. We1Recall that (C0,α(Ω), ||.||∞ + [.]α) is a Banach Space4 L. MORENO ME´RIDA AND R. E. VIDALwill prove that the limit of the sequence up of approximate solutions is aviscosity solution of (1).The article is organized as follows: in Section 2, we study the properties ofthe approximate obstacle problems (associated to the approximate operatorsLp) and in Section 3, we prove our main results.2. Approximate problemsWe consider the approximate operators Lp given by (4) and we study inthis section the approximate obstacle problems(5)−Lpu(x) = 0, in {x ∈ Ω : u(x) > ψ(x)},−Lpu(x) ≥ 0, in {x ∈ Ω : u(x) = ψ(x)},u(x) ≥ ψ(x), if x ∈ Ω,u(x) = g(x), if x ∈ ∂Ω.In the next lemmas, we prove that the functional Ep given by (3) hasa unique minimum (in a specific set) which is a viscosity solution of theapproximate obstacle problem (5).Lemma 1. Let Ω be an open, bounded and Lipschitz domain of IRN , α ∈(0, 1), p > 2Nα . If g ∈ C0,α(∂Ω), ψ ∈ C(Ω) and (2) holds, then the functionalEp given by (3) takes a unique minimum up in the setXg,ψ = {v ∈W s,p(Ω) : v ≥ ψ en Ω , v = g en ∂Ω} .Moreover, up belongs to C(Ω).Proof. Firstly we observe that, any α-Ho¨lder extension of g which lies abovethe obstacle ψ belongs to the set Xg,ψ. Thus, this set is not empty (see [15,Proposition 3.3] for the existence of this extension). In addition, if we takeu ∈ Xg,ψ and we fix k ≥ ||ψ||∞ + ||g||∞, then Tk(u) ∈ Xg,ψ andEp(u) ≥ Ep(Tk(u)) ,where the function Tk : IR −→ IR is defined byTk(s) = −k , si s < −k ,s , si |s| ≤ k ,k , si s > k .As a consequence, we have thatminXg,ψEp(u) = minXkg,ψEp(u) ,whereXkg,ψ = {u ∈ Xg,ψ : ||u||∞ ≤ k}.Since the set Xkg,ψ is weakly closed (with the weak topology of Ws,p(Ω)), toprove the existence of a minimum in this set, we will study the coercivityand the weak lower semicontinuity of the functional Ep.On the one hand, we take a sequence {un} ⊂ Xkg,ψ such that ||un||s,p →+∞. Since ||un||∞ ≤ k, this necessarily means that Ep(un) → +∞. Thatis, our functional is coercive.THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 5On the other hand, we take a sequence {un} ⊂ Xkg,ψ such that un weaklyconverges to a function u in W s,p(Ω). Since W s,p(Ω) is compactly embeddedin Lp(Ω) (see [9, Corollary 1.2]) and the norm ||.||s,p is a w.l.s.c. function,thenlim inf Ep(un) ≥ Ep(u)as we desired.Consequently, the functional Ep has a minimum up in the set Xg,ψ andagain by [9, Theorem 8.2], up ∈ C(Ω). Moreover, since the functional isconvex, this minimum is unique. Lemma 2. Let Ω be an open, bounded and Lipschitz domain of IRN , g ∈C0,α(∂Ω) and ψ ∈ C(Ω) satisfying (2). If p > 2N/α, then the minimum up(given by Lemma 1) is a viscosity solution of (5).Proof. Firstly, recall that the minimum up (given by Lemma 1) belongs toC(Ω) and satisfies up = g on ∂Ω and up ≥ ψ in Ω. We will prove that up isa viscosity sub and super solution. On the one hand, we claim that up is aviscosity subsolution of (5). Indeed, we take ϕ ∈ C1(Ω) ∩ C(Ω) such thatup ≤ ϕ , in Ω ,ψ(x0) < up(x0) = ϕ(x0) , for some x0 ∈ Ω ,and we prove −Lϕ(x0) ≤ 0. Without loss of generality, we suppose thatϕ touches up only at the point x0; otherwise it is sufficient to replace ϕ byϕ(x) + δ|x− x0|2 with δ small enough. We define the functionsϕ = max(up, ϕ− ) ,andϕ = min(up, ϕ− ).Since, we suppose that ϕ(x0) = up(x0) > ψ(x0), for  small enough, ϕ ≥ ψin Ω and moreover, ϕ = up on ∂Ω. Hence, ϕ belongs to Xg,ψ and usingthat up is a minimum of Ep in this set, we haveEp(ϕ) ≥ Ep(up) .From this inequality and using the following convexity inequality (see [6,Lemma 6.2])|max(a, c)−max(b, d)|p + |min(a, c)−min(b, d)|p ≤ |a− b|p + |c− d|p ,for all p ≥ 1 , we deduce thatEp(ϕ) + Ep(ϕ) ≤ Ep(up) + Ep(ϕ) ≤ Ep(ϕ) + Ep(ϕ) ,that isEp(ϕ) ≤ Ep(ϕ) .The convexity of Ep impliesEp((1− t)ϕ+ tϕ) ≤ (1− t)Ep(ϕ) + tEp(ϕ) ≤ Ep(ϕ) ,and then we haveEp((1− t)ϕ+ tϕ)− Ep(ϕ)t≤ 0 .Let callf(t) = Ep((1− t)ϕ+ tϕ).6 L. MORENO ME´RIDA AND R. E. VIDALFrom the above inequality and using the convexity of the function f , wehavef ′(0) ≤ f(t)− f(0)t≤ 0 ,and thenp∫Ω∫ΩH(x, y) dydx ≤ 0 ,whereH(x, y)=∣∣∣∣ϕ(y)−ϕ(x)|y − x|α∣∣∣∣p−1sgn(ϕ(y)−ϕ(x))|y − x|α (ϕ(y)−ϕ(y)+−ϕ(x)+ϕ(x)−) .Therefore, a change of variable implies∫Ω(ϕ − ϕ+ )(x)(−Lpϕ(x)) dx ≤ 0 .Now we argue by contradiction. Suppose that −Lpϕ(x0) > 0. By continuity,which holds under our assumptions, there is a small ball Br(x0) such that−Lpϕ > 0 in Br(x0). Since ϕ = max(up, ϕ − ), for  small enough, wehave supp(ϕ − ϕ + ) ⊂ Br(x0). We also observe that ϕ − ϕ +  ≥ 0.Consequently, we deduce0 <∫Br(x0)(ϕ − ϕ+ )(x)(−Lpϕ(x)) dx ≤ 0 ,which is a contradiction.In the same way, one can prove that u is a viscosity supersolution. 3. Proofs of the main resultsProof of Theorem 1. Suppose that u and v are two viscosity solutions of theobstacle problem (1) and define the setW = {x ∈ Ω : u(x) > v(x)} .We claim that W is an empty set. Indeed, arguing by contradiction, wesuppose that W is not empty. Since v ≥ ψ in Ω, we have v ≥ ψ and u > ψin W . Consequently, the functions u and v satisfy{ −Lu = 0 in W ,u = u on ∂W ,{ −Lv ≥ 0 in W ,v = u on ∂W ,which implies, using the comparison principle [6, Proposition 11.2], thatv ≥ u in W . This is a contradiction and the claim is proved. Reversing therole of u and v gives that u = v which conclude the proof. To prove Theorem 2 we need the following technical result.Lemma 3. [6, Lemma 6.5] For ϕ ∈ C1(Ω), p ≥ 1 and α ∈ (0, 1), we definefp(y) =ϕ(y)− ϕ(xp)|y − xp|α and f(y) =ϕ(y)− ϕ(x0)|y − x0|α ,where xp −→ x0 ∈ Ω as p −→∞. Then,limp−→∞∣∣∣∣∣∣∣∣∣∣ f+p (y)|y − xp|α/p∣∣∣∣∣∣∣∣∣∣Lp(Ω)=∣∣∣∣f+∣∣∣∣L∞(Ω) ,THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 7with f±(x) = max(±f(x), 0). The same also holds for f−p .Proof of Theorem 2. Let {up} be a sequence of viscosity solutions up of (5)given by Lemma 1. Our aim is to pass to the limit when p goes to infinite.Firstly, given α ∈ (0, 1), we prove that the sequence {up} is bounded inW s,q(Ω) for any q > 2N/α. Indeed, by construction, there is a positiveconstant k such that||up||∞ ≤ k , ∀p .Now, we take any p > 2N/α and we fix a number q such that 2N/α < q < p.Let z be a Ho¨lder extension of g such that z ≥ ψ (see [15, Proposition 3.3]).Since the functional Ep takes a unique minimum up in the set Xg,ψ, thenEp(up) ≤ Ep(z) ≤ |Ω|2[z]pα,and by Ho¨lder inequality(6) Eq(up) ≤ Ep(up)qp |Ω|2(p−q)p ≤ |Ω|2[z]qα ,which implies that the sequence {up} is bounded in W s,q(Ω). By the Sobolevembedding (see [9, Theorem 8.2]) we deduce that, up to a subsequence, upstrongly converges to a function u in C(Ω). Moreover, since up = g on ∂Ω,and up ≥ ψ in Ω, then we also have that the function u satisfiesu = g , on ∂Ω , u ≥ ψ , in Ω .Now, we will prove that u is a viscosity sub and super solution of problem(1). On the one hand, we claim that u is a viscosity subsolution. Indeed,we take ϕ ∈ C1(Ω) ∩ C(Ω) such thatu ≤ ϕ , in Ω ,ψ(x0) < u(x0) = ϕ(x0) , for some x0 ∈ Ω ,and, without loss of generality, we suppose that ϕ touches u only at thepoint x0 (x0 is a strict maximum of u− ϕ). Hence,Mp := supΩ(up − ϕ) = (up − ϕ)(xp),wherexp 7→ x0 , Mp 7→ 0.Moreover, since ϕ(x0) > ψ(x0), we can suppose that ϕ(xp) > ψ(xp) for plarge enough. This shows that{up ≤ ϕp := ϕ+Mp ,ψ(xp) < up(xp) = ϕ(xp).The fact that up is a viscosity solution implies0 ≥ −Lpϕp(xp) = −Lpϕ(xp),that is,0 ≥ −∫Ω∣∣∣∣ϕ(y)− ϕ(xp)|y − xp|α∣∣∣∣p−1 sgn(ϕ(y)− ϕ(xp))|y − xp|α dy ,or equivalently,∣∣∣∣∣∣∣∣∣∣(ϕ(y)− ϕ(xp)|y − xp|α+αp−1)+∣∣∣∣∣∣∣∣∣∣Lp−1(Ω)≥∣∣∣∣∣∣∣∣∣∣(ϕ(y)− ϕ(xp)|y − xp|α+αp−1)−∣∣∣∣∣∣∣∣∣∣Lp−1(Ω).8 L. MORENO ME´RIDA AND R. E. VIDALThanks to Lemma 3, we can pass to the limit in this inequality to obtainsupy∈Ω(max(ϕ(y)− ϕ(x0)|y − x0|α , 0))+ infy∈Ω(min(ϕ(y)− ϕ(x0)|y − x0|α , 0))≥ 0.Since ϕ is C1 at x0, it is clear that L+ϕ(x0) ≥ 0 and L−ϕ(x0) ≤ 0, whereL+ϕ(x0) = supy∈Ω, y 6=x0ϕ(y)− ϕ(x0)|y − x0|α , L−ϕ(x0) = infy∈Ω, y 6=x0ϕ(y)− ϕ(x0)|y − x0|α .Summing up, we deduce−Lϕ(x0) ≤ 0 ,as we desired.Finally, in the same way, one can proved that u is a viscosity super solu-tion. Therefore, u is a viscosity solution of the obstacle problem (1).To conclude, we characterize the function limit u. In order to do it, let zbe any Ho¨lder extension of g such that z ≥ ψ. By (6), we haveEq(up) ≤ |Ω|2[z]qα,which implies, passing to the limit as p goes to ∞,(Eq(u))1/q ≤ |Ω|2/q[z]α.As a consequence, when q tends to ∞, we obtain∣∣∣∣∣∣∣∣u(y)− u(x)|y − x|α∣∣∣∣∣∣∣∣L∞(Ω×Ω)≤ [z]α,i.e., we have proved that u ∈ C0,α(Ω) and moreover[u]α ≤ [z]α ,for any arbitrary Ho¨lder extension z of the datum g which satisfies z ≥ ψ,as we desired.Acknowledgements. The first author was supported by MINECO-FEDERgrant MTM2015-68210-P, Junta de Andaluc´ıa FQM-116 and Ministerio deEducacio´n, Cultura y Deporte (Spain) FPU grant FPU12/02395. The sec-ond author was partially supported by IEMath-Granada, CONICET andSecyt-UNC.References[1] Aronsson, G. On certain singular solutions of the partial differential equation u2xuxx +2uxuyuxy + u2yuyy = 0. Manuscripta mathematica, vol. 47, no 1-3, (1984), p. 133-151.[2] Aronsson, G., Crandall, M., and Juutinen, P. A tour of the theory of absolutely mini-mizing functions. Bulletin of the American mathematical society, vol. 41, no 4, (2004),p. 439-505.[3] Barles, G.; Chasseigne, E. and Imbert, C. On the Dirichlet problem for second-orderelliptic integro-differential equations. Indiana University Mathematics Journal, vol. 57,no 1, (2008), p. 213-146.[4] Bjorland C., Caffarelli L. and Figalli A. Nonlocal TugofWar and the Infinity FractionalLaplacian. Communications on Pure and Applied Mathematics, vol. 65, no 3, (2012),p. 337-380.[5] Bhattacharya, T.; Dibenedetto, E. and Manfredi, J. Limits as p → ∞ of ∆pup = fand related extremal problems. Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo SpecialeNonlinear PDEs, (1989), p. 15-68.THE OBSTACLE PROBLEM FOR THE INFINITY FRACTIONAL LAPLACIAN 9[6] Chambolle, A., Lindgren, E. and Monneau, R. A Ho¨lder infinity Laplacian. Optimisa-tion and Calculus of Variations, vol. 18, no 03, (2012), p. 799-835.[7] Caffarelli, L., Salsa, S., and Silvestre, L. Regularity estimates for the solution and thefree boundary of the obstacle problem for the fractional Laplacian. Inventiones mathe-maticae, vol 171, no 2, (2008), p. 425-461.[8] Dipierro, S., Savin, O., and Valdinoci, E. A nonlocal free boundary problem. SIAMJournal on Mathematical Analysis, vol. 47, no 6, (2015), p. 4559-4605.[9] Di Nezza, E.; Palatucci, G. and Valdinoci, E. Hitchhikers guide to the fractional Sobolevspaces. Bulletin des Sciences Mathmatiques. vol. 136, no 5, (2012), p. 521-573.[10] Ferreira, R. and Pe´rez-Llanos, M. Limit problems for a fractional p-Laplacian asp→∞. Nonlinear Differ. Equ. Appl. no 2, Art. 14, 28pp., (2016). doi:10.1007/s00030-016-0368-z[11] Lions Crandall, M.; Ishit, H. and Lions, P. Users guide to viscosity solutions of secondorder partial differential equations. Bulletin of the American Mathematical Society, vol.27, no 1, (1992), p. 1-67.[12] Manfredi, J.; Rossi, J. and Somersille, S. An obstacle problem for Tug-of-War games.Communications on Pure and Applied Analysis. Vol. 14(1), (2015), p. 217-228.[13] Mcshne, E. Extension of range of functions. Bulletin of the American MathematicalSociety, vol. 40, no 12, (1934), p. 837-842.[14] Peres, Y., Schramm, O., Sheffield, S. and Wilson, D. Tug-of-war and the infinityLaplacian. Journal of the American Mathematical Society, vol. 22, no 1, (2009), p.167-210.[15] Rossi, J.; Teixeira, E. and Urbano J. Optimal regularity at the free boundary forthe infinity obstacle problem. Interfaces and Free Boundaries. Vol. 17(3), (2015), p.381-398.(L. Moreno-Me´rida)Universidad de Granada, Granada, Espan˜a.email: lumore@ugr.es(R. E. Vidal)FaMAF, Universidad Nacional de Cordoba, (5000), Cordoba, Argentina.email: vidal@mate.uncor.edu

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The obstacle problem for the infinity fractional laplacian

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