Evolution Driven by the Inﬁnity Fractional Laplacian Félix del TesoJørgen EndalyEspen R. JakobsenzJuan Luis Vázquezx October 13 2022

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Evolution Driven by the Inﬁnity Fractional Laplacian

Félix del Teso?Jørgen Endal†Espen R. Jakobsen‡Juan Luis Vázquez§

October 13, 2022

Abstract

We consider the evolution problem associated to the inﬁnity fractional Laplacian introduced by

Bjorland, Caﬀarelli and Figalli (2012) as the inﬁnitesimal generator of a non-Brownian tug-of-war

game. We ﬁrst construct a class of viscosity solutions of the initial-value problem for bounded

and uniformly continuous data. An important result is the equivalence of the nonlinear operator

in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially

symmetric and monotone functions. Thanks to this and a comparison theorem between classical

and viscosity solutions, we are able to establish a global Harnack inequality that, in particular,

explains the long-time behavior of the solutions.

Keywords: Fractional partial diﬀerential equations, inﬁnity fractional Laplacian, inﬁnity Laplacian,

viscosity solutions.

2020 Mathematics Subject Classiﬁcation. 35R11, 35K55, 35A01, 35B45.

Contents

1 Introduction 2

2 Preliminaries and statement of main results 3

3 Properties of a approximation scheme 7

4 Deﬁnitions, existence and properties of viscosity solutions 12

5 Review of basic results on the fractional heat equation 16

6 Smooth solutions and the 1d fractional heat equation 17

7 Comparison and local truncation errors 20

8 Global Harnack principle 22

9 Extensions and open problems 23

?Departamento de Matemáticas, Universidad Autónoma de Madrid. felix.delteso@uam.es

†Department of Mathematical Sciences, Norwegian University of Science and Technology. jorgen.endal@ntnu.no

‡Department of Mathematical Sciences, Norwegian University of Science and Technology. espen.jakobsen@ntnu.no

§Departamento de Matemáticas, Universidad Autónoma de Madrid. juanluis.vazquez@uam.es

arXiv:2210.06414v1 [math.AP] 12 Oct 2022

1 Introduction

In this paper we study a parabolic equation associated to the (normalized) inﬁnity fractional Laplacian

operator. We recall that the local version of the game had been introduced by Peres et al. in 2009

([31]) where it is shown that the standard inﬁnity Laplace equation is solved by the value function for

a two-players random turn “tug–of–war” game. The game is as follows: a token is initially placed at a

position x0∈Ωand every turn a fair coin is tossed to choose which of the players plays. This player

moves the token to any point in the ball of radius ε > 0around the current position. If, eventually,

iterating this process, the token reaches a point xe∈∂Ω, the players are awarded (or penalized) f(xe)

(payoﬀ function). For a PDE overview of the inﬁnity Laplacian operator and its role as an absolute

minimizer for the L∞norm of the gradient, see [24,25].

In 2012 Bjorland, Caﬀarelli and Figalli ([5]) introduced equations involving the so-called inﬁnity

fractional Laplacian as a model for a nonlocal version of the “tug-of-war” game. Following their

explantation, instead of ﬂipping a coin at every step, every player chooses a direction and it is an

s-stable Lévy process that chooses both the active player and the distance to travel. The inﬁnity

fractional Laplacian, with symbol ∆s

∞, is a nonlinear integro-diﬀerential operator, the original deﬁnition

is given in Lemma 2.1 below. However, for the purpose of this paper, we also consider the alternative

equivalent deﬁnition introduced in [5] (see also [14]) given by

∆s

∞φ(x) := Cssup

|y|=1

inf

|˜y|=1 ˆ∞

(φ(x+ηy) + φ(x−η˜y)−2φ(x)) dη

η1+2swhere s∈(1/2,1). (1.1)

The constant is usually taken as Cs= (4ssΓ(1

2+s))/(π1

2Γ(1 −s)) but the value is irrelevant for our

results. In their paper [5] the authors study two stationary problems involving the inﬁnity fractional

Laplacian posed in bounded space domains, namely, a Dirichlet problem and a double-obstacle problem.

Here, we consider the evolution problem

(∂tu(x, t)=∆s

∞u(x, t), x ∈Rn, t > 0,

u(x, 0) = u0(x), x ∈Rn,

(1.2)

(1.3)

with s∈(1/2,1) and n≥2. When n= 1 the operator −∆s

∞is just the usual linear fractional Laplacian

operator (−∆)sof order s, and equation (1.2) is just the well-known fractional heat equation [6,21].

See also a detailed study of that equation using PDE techniques in [18,3,7,36]. Note that for n≥2

the operator is nonlinear so a new theory is needed. A non-normalized version is introduced in [10]

along with a well-posedness theory for the corresponding equations of the type (1.2)–(1.3). However,

the two problems are not equivalent nor closely related.

Here we develop an existence theory of suitable viscosity solutions for the parabolic problem (1.2)–

(1.3), based on approximation with monotone schemes. We show that the obtained class of solutions

enjoys a number of good properties. As in the elliptic case [5], we lack a uniqueness result in the

context of viscosity solutions. However, we are able to prove an important comparison theorem relat-

ing two types of solutions, classical and viscosity ones, see Theorem 2.6. As a counterpart, we also

obtain uniqueness and comparison of classical solutions. Moreover, we show that for smooth, radially

symmetric functions and nonincreasing along the radius in Rnwith n≥2, the operator −∆s

∞reduces

to the classical fractional Laplacian (−∆)sin dimension n= 1 (Theorem 6.1). A similar example

regarding nondecreasing one-dimensional proﬁles can be found in Lemma 6.3. In this way we may con-

struct a large class of classical solutions that make the comparison theorem relevant (Theorems 2.10

and 2.12). Note that no similar reduction applies in general, even in the radial case (see Subsection

6.2 for a counterexample).

Using the developed tools, we study the asymptotic behavior of the constructed solutions, and obtain

a global Harnack type principle, see Theorem 2.13.

1.1 Related literature

It is interesting to compare the nonlocal model (1.2) with the local version of the inﬁnity Laplacian

that has been studied by many authors, both in the stationary and evolution cases, cf. [2,19,24,1,31,

32,33,25]. Asymptotic expansions for the game theoretical p-Laplacian in the local case and related

approximation schemes in the elliptic case are studied in [27,28,17] and in the parabolic case in [26].

For the variational version of the p-Laplacian operator see [15].

There exist in the literature other nonlocal generalizations of the p-Laplacian and the inﬁnity Lapla-

cian. Let us mention (i) the normalized version [5,4] with asymptotic expansions and game theoretical

approach [8,14,23]; (ii) nonnormalized version [10] both elliptic and parabolic; (iii) Hölder inﬁnity

Laplacian [9]; and (iv) the (variational) fractional p-Laplacian [12,29,35,30,34,11,37].

2 Preliminaries and statement of main results

First let us ﬁx some notation that we will use along the paper.

For given δ > 0, standard molliﬁers are denoted by ρδ. Following [5], we say that φ∈C1,1(x)at some

x∈Rnif there exists px∈Rnand Cx, ηx>0such that

|φ(x+y)−φ(x)−px·y| ≤ Cx|y|2for all |y|< ηx. (2.1)

Note that C2

b(BR(x)) ⊂C1,1(x). Here Ck

b(U)is the space of functions on the set Uwith bounded

continuous derivatives of all orders in [0, k]. Let us also deﬁne:

B(Rn) := {φ:Rn→R|φis pointwisely bounded},

UC(Rn) := {φ:Rn→R|φis uniformly continuous},

BU C(Rn) := B(Rn)∩UC(Rn)with kφkCb(Rn):= sup

x∈Rn|φ(x)|,

and for β∈(0,1], we deﬁne |φ|C0,β (Rn)= supx,y∈Rn|φ(x)−φ(y)|/|x−y|βand

C0,β(Rn) := {φ∈Cb(Rn)|kφkC0,β <∞} where kφkC0,β =kφkCb+|φ|C0,β .

A modulus of continuity is a nondecreasing function ω:R+→R+such that limr→0+ω(r) = 0. For a

function f∈BU C(Rn), we deﬁne the corresponding modulus of continuity as follows:

ωf(r) = sup

|y|≤rkf(·+y)−fkCb(Rn).

For a Hölder continuous function f∈C0,β(Rn),ωf(r)≤ |f|C0,β rβ.

We will also need ei:= (0,0,...,0,1,0,...,0) ∈Rn, where 1is at the ith component.

2.1 Alternative characterization of the inﬁnity fractional Laplacian

We have the following alternative characterization of operator ∆s

∞that we will use throughout:

Lemma 2.1 (Alternative characterization).Assume φ∈C1,1(x)∩B(Rn). Then:

•If ∇φ(x)6= 0, then

∆s

∞φ(x) = Csˆ∞

0φ(x+ηζ) + φ(x−ηζ)−2φ(x)dη

η1+2swhere ζ:= ∇φ(x)/|∇φ(x)|.

•If ∇φ(x)=0, then

∆s

∞φ(x) = Cssup

|y|=1 ˆ∞

0φ(x+ηy)−φ(x)dη

η1+2s+Csinf

|y|=1 ˆ∞

0φ(x−ηy)−φ(x)dη

η1+2s.

The equivalence when ∇φ(x)=0follows from the fact that the integrals in this case are well-deﬁned

and can be combined to get (1.1). When ∇φ(x)6= 0, it can be shown that the supremum and inﬁmum

of (1.1) is actually taken at ζ, see Proposition 2.2 in [14]. To sketch the proof, assume for simplicity

that the supremum in (1.1) is taken at y, and let us argue that y=ζ. Indeed, by splitting the integral

and using the deﬁnitions of C1,1and the inﬁmum,

∆s

∞φ(x)≤Csˆ∞

0φ(x+ηy) + φ(x−ηζ)−2φ(x)dη

η1+2s≤Cs∇φ(x)·(y−ζ)ˆηx

ηdη

η1+2s+C.

Now, since ∆s

∞φ(x)is well-deﬁned and the integral diverges if y6=ζ, we must have y=ζ. A similar

argument holds for the inﬁmum.

2.2 Existence of solutions and basic properties

We are able to construct a suitable class of viscosity solutions of (1.2)–(1.3). The two steps are as

follows:

(i) Approximating ∆s

∞by removing the singularity, i.e., we introduce

Lε[φ](x) := Cssup

|y|=1

inf

|˜y|=1 ˆ∞

εφ(x+ηy) + φ(x−η˜y)−2φ(x)dη

η1+2s.

(ii) Discretizing in time by letting τ > 0and tj:= jτ for j∈N, i.e., tj∈τN, and then considering

the semidiscrete problem











Uj+1(x)−Uj(x)

τ=Lε[Uj](x), x ∈Rn, j ∈N,

U0(x) = u0(x), x ∈Rn.

(2.2)

(2.3)

We study the properties of (2.2)–(2.3) in Section 3. Existence of viscosity solutions follows by taking

the limit in this approximate scheme, as well as properties inherited from the approximations.

Theorem 2.2 (Existence and a priori results).If u0∈BU C(Rn), then there is at least one viscosity

solution u∈Cb(Rn×[0,∞)) of (1.2)–(1.3). Moreover:

(a) (Cb-bound) For all t > 0,ku(·, t)kCb(Rn)≤ ku0kCb(Rn).

(b) (Uniform continuity in space) For all y∈Rnand all t > 0,

ku(·+y, t)−u(·, t)kCb(Rn)≤ωu0(|y|).

t > 0,

ku(·, t)−u(·,˜

t)kCb(Rn)≤˜ω(|t−˜

t|)where ˜ω(r) := infδ>0nωu0(δ) + rsupε>0kLε[u0,δ]kCb(Rn)o

is a modulus satisfying ˜ω(r)≤ωu0(r1/3) + Cr1/3+r,C:= csku0kCb(Rn)k∇ρk2−2s

L1(Rn)kD2ρk2s−1

L1(Rn),

and ρis a standard molliﬁer.

Remark 2.3. The deﬁnition of viscosity solutions is given Section 4(Deﬁnition 4.3). We obtain

viscosity solution as limits of monotone approximations of the problem in Section 3.

Note that if u0is Hölder continuous and s∈(1/2,1), then the above modulii will be (more) explicit.

Lemma 2.4. If u0∈C0,β(Rn)for β∈(0,1], then

ωu0(δ) = |u0|C0,β δβand kLε[u0,δ]kCb(Rn)≤c(s, ρ)|u0|C0,β δβ−2s.

The above result will be proved at the end of Section 4.

It follows after a minimization in δthat ˜ω(r) = c(s, ρ)|u0|C0,β r1

2s, and the solution uwill be Hölder

continuous with the correct parabolic regularity.

Corollary 2.5 (Existence and a priori results).If u0∈C0,β (Rn)for β∈(0,1], then there is at least

one viscosity solution u∈Cb(Rn×[0,∞)) of (1.2)–(1.3). Moreover:

(a) (Cb-bound) For all t > 0ku(·, t)kCb(Rn)≤ ku0kCb(Rn).

(b) (Hölder in space) For all y∈Rnand all t > 0,

ku(·+y, t)−u(·, t)kCb(Rn)≤ |u0|C0,β |y|β.

t > 0,

ku(·, t)−u(·,˜

t)kCb(Rn)≤C|u0|C0,β |t−˜

t|β

2s.

2.3 Classical solutions, radial solutions, comparison, and uniqueness

There could be other ways of obtaining viscosity solutions, and unfortunately, we lack general com-

parison and uniqueness results. Nevertheless, we can obtain that classical solutions are unique and we

can compare our constructed viscosity solutions with classical sub- and supersolutions of (1.2)–(1.3).1

Theorem 2.6 (Comparison between classical and viscosity solutions).Assume u0∈BUC(Rn).

Let u, u ∈C2

b(Rn×[0,∞)) be respective classical sub- and supersolution of (1.2)–(1.3), and let

u∈BU C(Rn×[0,∞)be a viscosity solution of (1.2)–(1.3)as constructed in Theorem 2.2. Then

u≤u≤uin Rn×(0,∞).

The above result is proved in Section 7. We want to emphasize that it is done in a rather nonstandard

way, since we inherit the comparison from the approximation scheme when the solution is classical. In

general, this cannot be done in the context of viscosity solutions since the approximation scheme only

converges up to a subsequence.

Remark 2.7. By Theorem 2.6, we can in addition get comparison of constructed viscosity solutions

as long as the initial datas are separated by an initial data which produces a classical solution.

An immediate consequence of Theorem 2.6:

Corollary 2.8 (Comparison of classical sub- and supersolutions).Let u, v ∈C2

b(Rn×[0,∞)) be

respective classical sub- and supersolutions of (1.2)–(1.3)with initial data u0, v0. If u0≤v0, then

u≤v.

Corollary 2.9 (Uniqueness of solutions).Classical solutions of (1.2)–(1.3)in C2

b(Rn×[0,∞)) are

unique.

Theorem 2.6 might be an empty statement unless we provide a class of classical solutions of (1.2)–(1.3).

The following result, proved in Section 6, solves this issue.

1We will work with classical solutions in C2

b. Actually, we can reduce to C1

bfor the temporal variable, and to C1,1∩B

for the spatial variables.

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EvolutionDrivenbytheInnityFractionalLaplacianFélixdelTeso?JørgenEndalyEspenR.JakobsenzJuanLuisVázquezxOctober13,2022AbstractWeconsidertheevolutionproblemassociatedtotheinnityfractionalLaplacianintroducedbyBjorland,CaarelliandFigalli(2012)astheinnitesimalgeneratorofanon-Browniantug-of-wargame.We...

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Evolution Driven by the Inﬁnity Fractional Laplacian Félix del TesoJørgen EndalyEspen R. JakobsenzJuan Luis Vázquezx October 13 2022

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