We present a new, easy, and elementary proof of Jensen's Theorem on the
uniqueness of infinity harmonic functions. The idea is to pass to a finite
difference equation by taking maximums and minimums over small balls.Comment: 4 pages; comments added, proof simplifie

A.M. Oberman

Charles K. Smart

E. Le Gruyer

G. Aronsson

G. Barles

M.G. Crandall

R. Jensen

Scott N. Armstrong

Y. Peres

English

arXiv

Springer - Publisher Connector

Calc. Var. (2010) 37:381–384
DOI 10.1007/s00526-009-0267-9 Calculus of Variations
An easy proof of Jensen’s theorem on the uniqueness
of infinity harmonic functions
Scott N. Armstrong · Charles K. Smart
Received: 20 June 2009 / Accepted: 10 July 2009 / Published online: 18 August 2009
© The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract We present a new, easy, and elementary proof of Jensen’s Theorem on the
uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equa-
tion by taking maximums and minimums over small balls.
Mathematics Subject Classification (2000) Primary 35J70
In this short article, we present a new proof of the famous result of Jensen [7], which estab-
lishes the uniqueness of viscosity solutions of the infinity Laplace equation
− ∞u := −
n∑
i, j=1
uxi x j uxi ux j = 0, (1)
in a bounded domain  ⊆ Rn , subject to a given Dirichlet boundary condition. Our argument
is elementary, and other than our use of a well-known equivalence (Theorem 3, below), our
presentation is self-contained. In contrast, previous proofs employ either intricate viscosity
solution methods as well as a deep result of Aleksandrov [2,3,6,7], or follow a less direct
path using probabilistic arguments [10].
Definition 1 A viscosity subsolution (supersolution, solution) of (1) is called infinity sub-
harmonic (superharmonic, harmonic).
We refer to the survey articles [2,4] for an introduction to the infinity Laplace equation,
as well as the definition of viscosity solution. In this article we do not apply the definition of
viscosity solution directly. Instead, we use the notion of comparisons with cones.
S. N. Armstrong (B) · C. K. Smart
Department of Mathematics, University of California, Berkeley, CA 94720, USA
e-mail: sarm@math.berkeley.edu
C. K. Smart
e-mail: smart@math.berkeley.edu
123
382 S. N. Armstrong, C. K. Smart
Definition 2 A cone function with vertex x0 ∈ Rn is a function of the form ϕ(x) = a +
b|x − x0|, where a, b ∈ R. A function u ∈ C() is said to enjoy comparisons with cones
from above if it possesses the following property: for every open V ⊆ Rn for which V¯ ⊆ ,
and every cone function ϕ with vertex x0 ∈ Rn\V ,
u ≤ ϕ on ∂V implies u ≤ ϕ in V .
We say that u enjoys comparisons with cones from below if −u enjoys comparisons with
cones from above. Finally, u enjoys comparisons with cones if it enjoys comparisons with
cones from above and below.
The following result of Crandall et al. [5] establishes the equivalence between the notions
of infinity harmonic functions and functions enjoying comparisons with cones.
Theorem 3 ([5]) A function u ∈ C() is infinity subharmonic (superharmonic) if and only
if u enjoys comparisons with cones from above (below).
A less general version of Theorem 3 was also obtained by Jensen [7, Lemma 3.1]. In
addition to [5], elementary proofs of this result can be found in [2,4].
We now introduce some notation. For ε > 0 and x ∈ Rn , we write B(x, ε) := {y ∈ Rn :
|x − y| < ε}. Let ε denote the set of points x ∈  for which B¯(x, ε) ⊆ . If u ∈ C()
and x ∈ ε , then we denote
S+ε u(x) := max
y∈B¯(x,ε)
u(y) − u(x)
ε
and S−ε u(x) := max
y∈B¯(x,ε)
u(x) − u(y)
ε
.
The first step in our proof of Jensen’s theorem is the following comparison lemma for a
finite difference equation. We adapt an argument due to Le Gruyer [8], who proved a similar
lemma for a difference equation on a finite graph.
Lemma 4 Assume that u, v ∈ C() ∩ L∞() satisfy the inequalities
S−ε u(x) − S+ε u(x) ≤ 0 ≤ S−ε v(x) − S+ε v(x) for every x ∈ ε. (2)
Then
sup

(u − v) = sup
\ε
(u − v).
Proof Arguing by contradiction, we suppose that
sup

(u − v) > sup
\ε
(u − v).
The set E := {x ∈  : (u − v)(x) = sup(u − v)} is nonempty, closed and contained in
ε . Define F := {x ∈ E : u(x) = maxE u}, and select a point x0 ∈ ∂ F . Since u − v attains
its maximum at x0, we see that
S−ε v(x0) ≤ S−ε u(x0). (3)
Consider the case S+ε u(x0) = 0. From the first inequality in (2) we also have S−ε u(x0)= 0.
From (3), we deduce that S−ε v(x0)= 0. Now the second inequality of (2) implies S+ε v(x0)= 0.
In particular, u and v are constant in B¯(x0, ε), contradicting our assumption that x0 ∈ ∂ F .
It remains to examine the case S+ε u(x0) > 0. Select z ∈ B¯(x0, ε) such that εS+ε u(x0) =
u(z) − u(x0). Since u(z) > u(x0) and x0 ∈ F , we see that z ∈ E . From this we deduce that
εS+ε v(x0) ≥ v(z) − v(x0) > u(z) − u(x0) = εS+ε u(x0). (4)
123
An easy proof of the uniqueness of infinity harmonic functions 383
Combining (3) and (4), we obtain
S−ε v(x0) − S+ε v(x0) < S−ε u(x0) − S+ε u(x0).
This contradicts (2), and the lemma follows. unionsq
The next lemma allows us to modify solutions of the PDE (1) to obtain solutions of the
finite difference inequalities (2). We use the notation
uε(x) := max
B¯(x,ε)
u and uε(x) := min
B¯(x,ε)
u, x ∈ ε,
which allows us to write εS+ε u(x) = uε(x) − u(x) and εS−ε u(x) = u(x) − uε(x).
Lemma 5 If u is infinity subharmonic in , then
S−ε uε(x) − S+ε uε(x) ≤ 0 for every x ∈ 2ε, (5)
and if v is infinity superharmonic in , then
S−ε vε(x) − S+ε vε(x) ≥ 0 for every x ∈ 2ε. (6)
Proof We first prove (5). Fix a point x0 ∈ 2ε. Select y0 ∈ B¯(x0, ε) and z0 ∈ B¯(x0, 2ε)
such that u(y0) = uε(x0) and u(z0) = u2ε(x0). We have
ε(S−ε uε(x0) − S+ε uε(x0)) = 2uε(x0) − (uε)ε(x0) − (uε)ε(x0)
≤ 2uε(x0) − u2ε(x0) − u(x0)
= 2u(y0) − u(z0) − u(x0).
A simple calculation verifies that the inequality
u(w) ≤ u(x0) + u(z0) − u(x0)2ε |w − x0|
holds for all w ∈ ∂ (B(x0, 2ε)\{x0}). Since u enjoys comparisons with cones from above, the
inequality therefore holds for every w ∈ B(x0, 2ε)\{x0}, and thus for every w ∈ B¯(x0, 2ε).
Substituting w = y0 and using the fact that |y0−x0| ≤ ε, we obtain 2u(y0)−u(x0)−u(z0) ≤
0. This completes the proof of (5).
To obtain (6), apply (5) to −v and simplify, using the fact that (−v)ε = −vε . unionsq
Jensen’s Theorem ([7]) Assume that u, v ∈ C(¯) are infinity subharmonic and superhar-
monic, respectively. Then
max
¯
(u − v) = max
∂
(u − v). (7)
Proof According to Lemmas 4 and 5,
sup
ε
(uε − vε) = sup
ε\2ε
(uε − vε)
for every ε > 0. We pass to the limit ε → 0 to obtain (7). unionsq
Further remarks Crandall (pers. commun.) has pointed out that our arguments apply, nearly
verbatim, if we replace the Euclidean norm in the definition of cone function with a general
norm. In this case, the PDE (1) is replaced by the notion of an absolutely minimizing Lipschitz
extension (we refer to [2] for the definition and background). We thereby obtain a new proof
of [10, Theorem 1.4] for bounded domains, which establishes the uniqueness of absolutely
123
384 S. N. Armstrong, C. K. Smart
minimizing Lipschitz extensions in this general setting. The hypotheses of Lemma 4 can be
further relaxed to metric spaces with mild structural conditions.
Our use of the functions uε and uε appears more natural when we recall that a function
u ∈ C() is infinity subharmonic if and only if the map ε → uε(x) is convex for every
x ∈  (see [4, Lemma 4.1]). Crandall (pers. commun.) has shown that our proof of Jensen’s
theorem can be efficiently presented by using this equivalence in lieu of Theorem 3.
Generalized versions of Lemmas 4 and 5 appear in [1], wherein the idea of passing from a
PDE to a finite difference inequality by maxing over ε-balls is used to obtain new results for
the (1-homogeneous) infinity Laplace equation with a nonzero function f on the right-hand
side. The finite difference equation S+ε u = S−ε u has been previously considered on finite
graphs by Oberman [9] and Le Gruyer [8], both of whom proved existence and uniqueness
of solutions to the discretized finite difference equation (with given boundary data).
Acknowledgements The authors warmly thank Michael Crandall for his many valuable suggestions and
remarks. This short article was also improved by the helpful comments of Stephanie Somersille, Kelli Talaska,
and Yifeng Yu.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommer-
cial License which permits any noncommercial use, distribution, and reproduction in any medium, provided
the original author(s) and source are credited.
References
1. Armstrong, S.N., Smart, C.K.: A finite difference approach to the infinity Laplace equation and tug-of-war
games. preprint, arXiv:0906.2871 [math.AP]
2. Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull.
Am. Math. Soc. (N.S.) 41(4), 439–505 (2004)
3. Barles, G., Busca, J.: Existence and comparison results for fully nonlinear degenerate elliptic equations
without zeroth-order term. Commun. Partial Differ. Equ. 26 (11–12), 2323–2337 (2001)
4. Crandall, M.G.: A Visit with the ∞-Laplace Equation. Calculus of Variations and Nonlinear Partial
Differential Equations. Lecture Notes in Mathematics, vol. 1927, pp. 75–122. Springer, Berlin (2008)
5. Crandall, M.G., Evans, L.C., Gariepy, R.F.: Optimal Lipschitz extensions and the infinity Laplacian. Calc.
Var. Partial Differ. Equ. 13(2), 123–139 (2001)
6. Crandall, M.G., Gunnarsson, G., Wang, P.: Uniqueness of ∞-harmonic functions and the eikonal equation.
Commun. Partial Differ. Equ. 32(10–12), 1587–1615 (2007)
7. Jensen, R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration.
Mech. Anal. 123(1), 51–74 (1993)
8. Le Gruyer, E.: On absolutely minimizing Lipschitz extensions and PDE ∞(u) = 0. NoDEA Nonlinear
Differ. Equ. Appl. 14(1–2), 29–55 (2007)
9. Oberman, A.M.: A convergent difference scheme for the infinity Laplacian: construction of absolutely
minimizing Lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2005)
10. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. J. Am. Math.
Soc. 22(1), 167–210 (2009)
123


An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions

Crossref

We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls

Armstrong, Scott N.

Smart, Charles K.

eScholarship - University of California

UC BerkeleyUC Berkeley Previously Published WorksTitleAn easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functionsPermalinkhttps://escholarship.org/uc/item/6h00m8q0JournalCalculus of Variations and Partial Differential Equations, 37(3)ISSN1432-0835AuthorsArmstrong, Scott N.Smart, Charles K.Publication Date2010-03-01DOI10.1007/s00526-009-0267-9 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaCalc. Var. (2010) 37:381–384DOI 10.1007/s00526-009-0267-9 Calculus of VariationsAn easy proof of Jensen’s theorem on the uniquenessof infinity harmonic functionsScott N. Armstrong · Charles K. SmartReceived: 20 June 2009 / Accepted: 10 July 2009 / Published online: 18 August 2009© The Author(s) 2009. This article is published with open access at Springerlink.comAbstract We present a new, easy, and elementary proof of Jensen’s Theorem on theuniqueness of infinity harmonic functions. The idea is to pass to a finite difference equa-tion by taking maximums and minimums over small balls.Mathematics Subject Classification (2000) Primary 35J70In this short article, we present a new proof of the famous result of Jensen [7], which estab-lishes the uniqueness of viscosity solutions of the infinity Laplace equation− ∞u := −n∑i, j=1uxi x j uxi ux j = 0, (1)in a bounded domain  ⊆ Rn , subject to a given Dirichlet boundary condition. Our argumentis elementary, and other than our use of a well-known equivalence (Theorem 3, below), ourpresentation is self-contained. In contrast, previous proofs employ either intricate viscositysolution methods as well as a deep result of Aleksandrov [2,3,6,7], or follow a less directpath using probabilistic arguments [10].Definition 1 A viscosity subsolution (supersolution, solution) of (1) is called infinity sub-harmonic (superharmonic, harmonic).We refer to the survey articles [2,4] for an introduction to the infinity Laplace equation,as well as the definition of viscosity solution. In this article we do not apply the definition ofviscosity solution directly. Instead, we use the notion of comparisons with cones.S. N. Armstrong (B) · C. K. SmartDepartment of Mathematics, University of California, Berkeley, CA 94720, USAe-mail: sarm@math.berkeley.eduC. K. Smarte-mail: smart@math.berkeley.edu123382 S. N. Armstrong, C. K. SmartDefinition 2 A cone function with vertex x0 ∈ Rn is a function of the form ϕ(x) = a +b|x − x0|, where a, b ∈ R. A function u ∈ C() is said to enjoy comparisons with conesfrom above if it possesses the following property: for every open V ⊆ Rn for which V̄ ⊆ ,and every cone function ϕ with vertex x0 ∈ Rn\V ,u ≤ ϕ on ∂V implies u ≤ ϕ in V .We say that u enjoys comparisons with cones from below if −u enjoys comparisons withcones from above. Finally, u enjoys comparisons with cones if it enjoys comparisons withcones from above and below.The following result of Crandall et al. [5] establishes the equivalence between the notionsof infinity harmonic functions and functions enjoying comparisons with cones.Theorem 3 ([5]) A function u ∈ C() is infinity subharmonic (superharmonic) if and onlyif u enjoys comparisons with cones from above (below).A less general version of Theorem 3 was also obtained by Jensen [7, Lemma 3.1]. Inaddition to [5], elementary proofs of this result can be found in [2,4].We now introduce some notation. For ε > 0 and x ∈ Rn , we write B(x, ε) := {y ∈ Rn :|x − y| < ε}. Let ε denote the set of points x ∈  for which B̄(x, ε) ⊆ . If u ∈ C()and x ∈ ε , then we denoteS+ε u(x) := maxy∈B̄(x,ε)u(y) − u(x)εand S−ε u(x) := maxy∈B̄(x,ε)u(x) − u(y)ε.The first step in our proof of Jensen’s theorem is the following comparison lemma for afinite difference equation. We adapt an argument due to Le Gruyer [8], who proved a similarlemma for a difference equation on a finite graph.Lemma 4 Assume that u, v ∈ C() ∩ L∞() satisfy the inequalitiesS−ε u(x) − S+ε u(x) ≤ 0 ≤ S−ε v(x) − S+ε v(x) for every x ∈ ε. (2)Thensup(u − v) = sup\ε(u − v).Proof Arguing by contradiction, we suppose thatsup(u − v) > sup\ε(u − v).The set E := {x ∈  : (u − v)(x) = sup(u − v)} is nonempty, closed and contained inε . Define F := {x ∈ E : u(x) = maxE u}, and select a point x0 ∈ ∂ F . Since u − v attainsits maximum at x0, we see thatS−ε v(x0) ≤ S−ε u(x0). (3)Consider the case S+ε u(x0) = 0. From the first inequality in (2) we also have S−ε u(x0)= 0.From (3), we deduce that S−ε v(x0)= 0. Now the second inequality of (2) implies S+ε v(x0)= 0.In particular, u and v are constant in B̄(x0, ε), contradicting our assumption that x0 ∈ ∂ F .It remains to examine the case S+ε u(x0) > 0. Select z ∈ B̄(x0, ε) such that εS+ε u(x0) =u(z) − u(x0). Since u(z) > u(x0) and x0 ∈ F , we see that z ∈ E . From this we deduce thatεS+ε v(x0) ≥ v(z) − v(x0) > u(z) − u(x0) = εS+ε u(x0). (4)123An easy proof of the uniqueness of infinity harmonic functions 383Combining (3) and (4), we obtainS−ε v(x0) − S+ε v(x0) < S−ε u(x0) − S+ε u(x0).This contradicts (2), and the lemma follows. 	The next lemma allows us to modify solutions of the PDE (1) to obtain solutions of thefinite difference inequalities (2). We use the notationuε(x) := maxB̄(x,ε)u and uε(x) := minB̄(x,ε)u, x ∈ ε,which allows us to write εS+ε u(x) = uε(x) − u(x) and εS−ε u(x) = u(x) − uε(x).Lemma 5 If u is infinity subharmonic in , thenS−ε uε(x) − S+ε uε(x) ≤ 0 for every x ∈ 2ε, (5)and if v is infinity superharmonic in , thenS−ε vε(x) − S+ε vε(x) ≥ 0 for every x ∈ 2ε. (6)Proof We first prove (5). Fix a point x0 ∈ 2ε. Select y0 ∈ B̄(x0, ε) and z0 ∈ B̄(x0, 2ε)such that u(y0) = uε(x0) and u(z0) = u2ε(x0). We haveε(S−ε uε(x0) − S+ε uε(x0)) = 2uε(x0) − (uε)ε(x0) − (uε)ε(x0)≤ 2uε(x0) − u2ε(x0) − u(x0)= 2u(y0) − u(z0) − u(x0).A simple calculation verifies that the inequalityu(w) ≤ u(x0) + u(z0) − u(x0)2ε|w − x0|holds for all w ∈ ∂ (B(x0, 2ε)\{x0}). Since u enjoys comparisons with cones from above, theinequality therefore holds for every w ∈ B(x0, 2ε)\{x0}, and thus for every w ∈ B̄(x0, 2ε).Substituting w = y0 and using the fact that |y0−x0| ≤ ε, we obtain 2u(y0)−u(x0)−u(z0) ≤0. This completes the proof of (5).To obtain (6), apply (5) to −v and simplify, using the fact that (−v)ε = −vε . 	Jensen’s Theorem ([7]) Assume that u, v ∈ C(̄) are infinity subharmonic and superhar-monic, respectively. Thenmax̄(u − v) = max∂(u − v). (7)Proof According to Lemmas 4 and 5,supε(uε − vε) = supε\2ε(uε − vε)for every ε > 0. We pass to the limit ε → 0 to obtain (7). 	Further remarks Crandall (pers. commun.) has pointed out that our arguments apply, nearlyverbatim, if we replace the Euclidean norm in the definition of cone function with a generalnorm. In this case, the PDE (1) is replaced by the notion of an absolutely minimizing Lipschitzextension (we refer to [2] for the definition and background). We thereby obtain a new proofof [10, Theorem 1.4] for bounded domains, which establishes the uniqueness of absolutely123384 S. N. Armstrong, C. K. Smartminimizing Lipschitz extensions in this general setting. The hypotheses of Lemma 4 can befurther relaxed to metric spaces with mild structural conditions.Our use of the functions uε and uε appears more natural when we recall that a functionu ∈ C() is infinity subharmonic if and only if the map ε → uε(x) is convex for everyx ∈  (see [4, Lemma 4.1]). Crandall (pers. commun.) has shown that our proof of Jensen’stheorem can be efficiently presented by using this equivalence in lieu of Theorem 3.Generalized versions of Lemmas 4 and 5 appear in [1], wherein the idea of passing from aPDE to a finite difference inequality by maxing over ε-balls is used to obtain new results forthe (1-homogeneous) infinity Laplace equation with a nonzero function f on the right-handside. The finite difference equation S+ε u = S−ε u has been previously considered on finitegraphs by Oberman [9] and Le Gruyer [8], both of whom proved existence and uniquenessof solutions to the discretized finite difference equation (with given boundary data).Acknowledgements The authors warmly thank Michael Crandall for his many valuable suggestions andremarks. This short article was also improved by the helpful comments of Stephanie Somersille, Kelli Talaska,and Yifeng Yu.Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommer-cial License which permits any noncommercial use, distribution, and reproduction in any medium, providedthe original author(s) and source are credited.References1. Armstrong, S.N., Smart, C.K.: A finite difference approach to the infinity Laplace equation and tug-of-wargames. preprint, arXiv:0906.2871 [math.AP]2. Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull.Am. Math. Soc. (N.S.) 41(4), 439–505 (2004)3. Barles, G., Busca, J.: Existence and comparison results for fully nonlinear degenerate elliptic equationswithout zeroth-order term. Commun. Partial Differ. Equ. 26 (11–12), 2323–2337 (2001)4. Crandall, M.G.: A Visit with the ∞-Laplace Equation. Calculus of Variations and Nonlinear PartialDifferential Equations. Lecture Notes in Mathematics, vol. 1927, pp. 75–122. Springer, Berlin (2008)5. Crandall, M.G., Evans, L.C., Gariepy, R.F.: Optimal Lipschitz extensions and the infinity Laplacian. Calc.Var. Partial Differ. Equ. 13(2), 123–139 (2001)6. Crandall, M.G., Gunnarsson, G., Wang, P.: Uniqueness of ∞-harmonic functions and the eikonal equation.Commun. Partial Differ. Equ. 32(10–12), 1587–1615 (2007)7. Jensen, R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration.Mech. Anal. 123(1), 51–74 (1993)8. Le Gruyer, E.: On absolutely minimizing Lipschitz extensions and PDE ∞(u) = 0. NoDEA NonlinearDiffer. Equ. Appl. 14(1–2), 29–55 (2007)9. Oberman, A.M.: A convergent difference scheme for the infinity Laplacian: construction of absolutelyminimizing Lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2005)10. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. J. Am. Math.Soc. 22(1), 167–210 (2009)123

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An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions

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