Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 37, pp. 1–6.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXPLICIT SOLUTIONS OF JENSEN’S AUXILIARY EQUATIONS

VIA EXTREMAL LIPSCHITZ EXTENSIONS

FERNANDO CHARRO

Abstract. In this note we prove that McShane and Whitney’s Lipschitz ex-

tensions are viscosity solutions of Jensen’s auxiliary equations which are known
to have a key role in Jensen’s celebrated proof of uniqueness of infinity har-

monic functions, and therefore of absolutely minimizing Lipschitz extensions.
To the best of the author’s knowledge, this result does not appear to be known

in the literature in spite of the vast amount of work on the topic.

1. Introduction

Given a Lipschitz function F : ∂Ω → R with Lipschitz constant λ one can
consider the problem of finding a Lipschitz extension of the function to the interior
of Ω. This problem has received great attention for many years, we refer the
interested reader to [3] for a survey on the topic.

Note that the best Lipschitz constant one can hope for the extension is λ itself.
This Lipschitz constant is achieved by the explicit extensions

u(x) = inf
z∈∂Ω

(
F (z) + λ|x− z|

)
(1.1)

and
u(x) = sup

z∈∂Ω

(
F (z)− λ|x− z|

)
(1.2)

due to McShane [7] and Whitney [9], respectively. It is easy to see that u, u coincide
with F at ∂Ω and are Lipschitz continuous with constant λ. In fact, u = F on
∂Ω follows by noticing that for all x ∈ ∂Ω, the definition of u and the Lipschitz
continuity of F yield

u(x) ≤ F (x) ≤ F (z) + λ |x− z|, for all z ∈ ∂Ω, (1.3)

and similarly for u. On the other hand, the Lipschitz condition for u can be verified
observing that if x, y ∈ Rn, then

u(x) ≤ inf
z∈∂Ω

(
F (z) + λ(|y − z|+ |x− y|)

)
= u(y) + λ|x− y|, (1.4)

and then reversing the roles of x, y (the case of u is similar).
Furthermore, these extensions are extremal in the sense that any other Lipschitz

extension u satisfies
u ≤ u ≤ u. (1.5)

2010 Mathematics Subject Classification. 35J70, 46T20, 49K20.

Key words and phrases. Lipschitz extension; McShane-Whitney extension; infinity Laplacian.
c©2020 Texas State University.

Submitted July 2, 2019. Published April 23, 2020.

1



2 F. CHARRO EJDE-2020/37

To see this, note that by the Lipchitz continuity of u,

u(z)− λ |x− z| ≤ u(x) ≤ u(z) + λ |x− z|
for all x ∈ Rn and z ∈ ∂Ω (note that u(z) = F (z)).

Whenever McShane and Whitney’s Lipschitz extensions, u and u coincide, (1.5)
provides uniqueness and optimality of the extension. However, this rarely happens,
see [3]. Then, a natural question arises, how to find the “best” extension of F :
∂Ω → R to the interior of Ω. Or, in other words, how to find u with the least
possible Lipschitz constant in every open set whose closure is compactly contained
in Ω. This extension exists and is unique, and is called an Absolutely Minimizing
Lipschitz Extension (AMLE) following [2]. It turns out that such AMLE is infinity
harmonic (see [3, 5]), i.e., it satisfies −∆∞u = 0 in Ω in the viscosity sense, where

∆∞u(x) = 〈D2u(x)∇u(x),∇u(x)〉
is the well-known infinity Laplace operator (see [6] for a survey of its applications).

In this note we prove that McShane and Whitney’s extensions are viscosity
solutions of Jensen’s auxiliary equations, which are known to have a key role in
Jensen’s celebrated proof of uniqueness of infinity harmonic functions (and hence
of AMLE) in [5]. This question arose in connection with a modified Tug-of-War
game studied in [1] which models Jensen’s auxiliary equations in graphs. To the
best of our knowledge, this result does not seem to be known in the literature in
spite of the vast amount of work around the topic.

In the sequel, given g : K ⊂ Rn → R, Lipschitz continuous on K, we will denote
by Lg(K) the smallest constant λ ≥ 0 for which |g(x) − g(y)| ≤ λ|x − y| for all
x, y ∈ K. If λ ≥ Lg(K), then we will say that λ is “a Lipschitz constant for g”.
The main result of the paper is the following.

Theorem 1.1. Let F : ∂Ω→ R be a Lipschitz function with least Lipschitz constant
LF (∂Ω). Then, for every λ ≥ LF (∂Ω), McShane’s extension u defined in (1.1) is
the unique viscosity solution of

min{|∇u(x)| − λ,−∆∞u(x)} = 0 in Ω

u(x) = F (x) on ∂Ω.
(1.6)

Similarly, Whitney’s extension u defined in (1.2) is the unique viscosity solution of

max {λ− |∇u(x)|,−∆∞u(x)} = 0 in Ω

u(x) = F (x) on ∂Ω.
(1.7)

On the other hand, whenever λ < LF (∂Ω), the functions u, u still satisfy the equa-
tions in (1.6) and (1.7) in the interior of Ω but fail to achieve the boundary condition
u = F on ∂Ω.

As a motivation, we have the following example.

Example 1.2. Let λ > 0, Ω ⊂ Rn and consider uλ(x) = λ dist(x, ∂Ω). It can be
checked by direct computation that uλ is the unique viscosity solution to

min{|∇u| − λ,−∆∞u} = 0 in Ω,

u = 0 on ∂Ω.

This agrees with Theorem 1.1 since for every λ ≥ 0 = LF (∂Ω) we have

u(x) = λ inf
z∈∂Ω

|x− z| = λ dist(x, ∂Ω).



EJDE-2020/37 EXTREMAL LIPSCHITZ EXTENSIONS AND JENSEN’S EQUATIONS 3

The fact that an AMLE is infinity harmonic (again, see [3, 5]) makes it a sub-
solution of (1.6) and a supersolution of (1.7), respectively. Then, the comparison
principle for Jensen’s equations (1.6) and (1.7) (see [5, Theorems 2.1 and 2.15])
offers another perspective on (1.5), which follows by comparison. In the next result
we show that this is a general fact that does not depend on the infinity-harmonicity
of the AMLE, i.e., we prove that any Lipschitz extension is a subsolution of (1.6)
and a supersolution of (1.7), respectively.

Theorem 1.3. Let F : ∂Ω→ R be Lipschitz continuous, and let u be any Lipschitz
extension of F to Ω, i.e., a Lipschitz function u : Ω → R such that u = F on ∂Ω
and has Lipschitz constant Lu(Ω) = LF (∂Ω). Then, for every λ ≥ LF (∂Ω)

min {|∇u(x)| − λ,−∆∞u(x)} ≤ 0 in Ω

u(x) = F (x) on ∂Ω.
(1.8)

and
max {λ− |∇u(x)|,−∆∞u(x)} ≥ 0 in Ω

u(x) = F (x) on ∂Ω.
(1.9)

in the viscosity sense.

This can also be understood in view of Rademacher’s Theorem: A Lipschitz
function u on an open subset of the Euclidean space is differentiable almost every-
where and the number ‖∇u‖∞ is bounded from above by the Lipschitz constant
of u (if in addition the domain is convex, then the least Lipschitz constant equals
‖∇u‖∞).

Remark 1.4. Theorems 1.1 and 1.3 also hold with ∆N
∞u in place of ∆∞u, where

∆N
∞u(x) :=

〈D
2u(x) ∇u(x)

|∇u(x)| ,
∇u(x)
|∇u(x)| 〉, if ∇u(x) 6= 0

limy→x
2(u(y)−u(x))
|y−x|2 , otherwise

(1.10)

is the normalized infinity Laplacian, well known for its role in the modeling of
random Tug-of-War games, see [6] and the references therein.

We would like to finish this introduction pointing out that the Taylor expansion
arguments in the proof of Theorem 1.1 have an interesting connection with the
numerical analysis of equations (1.6) and (1.7). More precisely, equations (1.6) and
(1.7) can be respectively approximated by

min
{1

ε

(
u(x)− inf

y∈Bε(x)∩Ω
u(y)− ελ

)
,

1

ε2

(
2u(x)− sup

y∈Bε(x)∩Ω

u(y)− inf
y∈Bε(x)∩Ω

u(y)
)}

= 0
(1.11)

and

max
{1

ε

(
u(x)− sup

y∈Bε(x)∩Ω

u(y) + ελ
)
,

1

ε2

(
2u(x)− sup

y∈Bε(x)∩Ω

u(y)− inf
y∈Bε(x)∩Ω

u(y)
)}

= 0,

(1.12)

which can be regarded as discrete elliptic schemes in the sense of [8] (and, therefore,
monotone in the sense of [4]).



4 F. CHARRO EJDE-2020/37

Moreover, in a similar way to the Taylor expansion arguments in the proof of
Theorem 1.1, one can show that schemes (1.11) and (1.12) are consistent (see
[4, Section 2] for the definition). This means, roughly speaking, that the finite-
difference operator converges in the viscosity sense towards the continuous operator
of the PDE as ε → 0. Monotonicity and consistency, altogether with stability
are important requirements for convergence, as established in the seminal paper
[4]. Informally, the authors in [4] prove that any monotone, stable, and consistent
scheme converges provided that the limiting equation satisfies a type of comparison
principle known as “strong uniqueness property”, which is usually difficult to prove.

It seems an interesting question to tackle the convergence of schemes (1.11)
and (1.12) and their numerical implementation; however, we will not discuss that
problem here.

2. Proofs of Theorems 1.1 and 1.3

Proof of Theorem 1.3. Let us prove the result for (1.8) since the proof for (1.9) is
similar. Let x̂ ∈ Ω and φ ∈ C2(Ω) such that φ touches u at x̂ from above in a
neighborhood of x̂. Our goal is to prove

min
{
|∇φ(x̂)| − λ,−∆∞φ(x̂)

}
≤ 0. (2.1)

Note that we can assume ∇φ(x̂) 6= 0 since we are done otherwise. Then, the contact
condition and a Taylor expansion yield

u(x) ≤ φ(x) = u(x̂) + 〈∇φ(x̂), x− x̂〉+ o(|x− x̂|) as x→ x̂

Choose x = x̂− α∇φ(x̂), with α > 0 small enough. Then

−λα|∇φ(x̂)| ≤ u
(
x̂− α∇φ(x̂)

)
− u(x̂) ≤ −α|∇φ(x̂)|2 + o(α)

by the Lipschitz continuity of u. Dividing both sides by −α|∇φ(x̂)| and letting
α→ 0, we obtain |∇φ(x̂)| ≤ λ as desired. �

Proof of Theorem 1.1. Assume first that λ ≥ LF (∂Ω), and let us prove that u is a
viscosity solution of (1.6). First, we will show the supersolution case. Observe that
for every z ∈ ∂Ω, the cone C(x) = F (z) + λ|x− z| satisfies

min
{
|∇C(x)| − λ,−∆∞C(x)

}
= 0 in Ω,

in the classical sense, and therefore u is a viscosity supersolution in Ω because it is
an infimum of supersolutions. Moreover, u = F , as discussed in (1.3).

Alternatively, let x̂ ∈ Ω and φ ∈ C2(Ω) such that φ touches u at x̂ from below
in a neighborhood of x̂. Our goal is to prove that

min{|∇φ(x̂)| − λ,−∆∞φ(x̂)} ≥ 0. (2.2)

Note that by the Lipschitz continuity of F , the function z 7→ F (z) + λ|x − z| is
continuous for each fixed x, and we have that

φ(x̂) = u(x̂) = min
z∈∂Ω

(
F (z) + λ|x̂− z|

)
= F (ẑ) + λ|x̂− ẑ|

for some ẑ ∈ ∂Ω. On the other hand,

φ(x) ≤ u(x) ≤ F (ẑ) + λ|x− ẑ|



EJDE-2020/37 EXTREMAL LIPSCHITZ EXTENSIONS AND JENSEN’S EQUATIONS 5

and we find that φ touches the cone C(x) = F (ẑ) + λ|x − ẑ| at x̂ from below in a
neighborhood of x̂. Then ∇φ(x̂) = ∇C(x̂) and D2φ(x̂) ≤ D2C(x̂), and we deduce
that

−∆∞φ(x̂) ≥ −∆∞C(x̂) = 0, and |∇φ(x̂)| = |∇C(x̂)| = λ,

which, yield (2.2).
We proceed now to prove that u is a viscosity subsolution of (1.6). Note that we

can apply Theorem 1.3. However, we are going to show a different argument which
shows an interesting connection with the numerical analysis of equations (1.6) and
(1.7).

To this aim, let x̂ ∈ Ω and φ ∈ C2(Ω) such that φ touches u at x̂ from above in
a neighborhood of x̂. Our goal is to prove

min {|∇φ(x̂)| − λ,−∆∞φ(x̂)} ≤ 0. (2.3)

By the continuity of u (see (1.4)), for ε small enough we can write

min
x∈Bε(x̂)

u(x) = min
x∈Bε(x̂)

inf
z∈∂Ω

(
F (z) + λ|x− z|

)
≥ inf
z∈∂Ω

(
F (z) + λ|x̂− z| − ελ

)
= u(x̂)− ελ,

where we have used that |x̂− z| ≤ ε+ |x− z| for every x ∈ Bε(x̂). Therefore,

1

ε

(
φ(x̂)− min

x∈Bε(x̂)
φ(x)

)
≤ 1

ε

(
u(x̂)− min

Bε(x̂)
u
)
≤ λ.

We claim that

min
x∈Bε(x̂)

φ(x) = φ
(
x̂− ε

[ ∇φ(x̂)

|∇φ(x̂)|
+ o(1)

])
as ε→ 0. (2.4)

Then, a first-order Taylor expansion yields

1

ε

(
φ(x̂)− min

x∈Bε(x̂)
φ(x)

)
= |∇φ(x̂)|+ o(1) as ε→ 0

and we deduce |∇φ(x̂)| ≤ λ and, hence, that (2.3) holds.
We prove claim (2.4) for the sake of completeness. Note that we can assume

∇φ(x̂) 6= 0 since otherwise |∇φ(x̂)| ≤ λ holds and there is nothing to prove. Write

min
x∈Bε(x̂)

φ(x) = φ(x̂− εvε)

for some vε ∈ B1(0). Observe that |vε| = 1 for every ε small enough because,
otherwise, there would be a subsequence x̂− εkvεk of interior minimum points of φ
in Bεk(x̂) for which ∇φ(x̂− εkvεk) = 0, a contradiction as εk → 0.

It remains to show that, actually,

vε =
∇φ(x̂)

|∇φ(x̂)|
+ o(1) as ε→ 0. (2.5)

Let ω be any fixed direction with |ω| = 1. Then

φ(x̂− εvε) = min
x∈Bε(x̂)

φ(x) ≤ φ(x̂− εω),

and a Taylor expansion of φ around x̂ gives

〈∇φ(x̂), vε〉+ o(1) ≥ −φ(x̂− ε ω) + φ(x̂)

ε
= 〈∇φ(x̂), ω〉+ o(1) as ε→ 0.

Since the previous argument holds for any direction ω, we have (2.5) as desired.



6 F. CHARRO EJDE-2020/37

The proof that u is a viscosity solution of (1.7) is similar.
To conclude, let us point out that in the case λ < LF (∂Ω) we can follow the

argument above and show that the functions u, u respectively satisfy the equations
in (1.6) and (1.7) in the interior of Ω. In fact, (1.1), (1.2) are still Lipschitz contin-
uous with constant λ in the interior of Ω by (1.4). However, (1.3) does not work
and we can only say u ≤ F ≤ u on ∂Ω (which holds by definition) and u, u fail to
achieve the boundary condition. �

Acknowledgments. This research was partially supported by MINECO grants
MTM2016-80474-P and MTM2017-84214-C2-1-P, Spain.

References

[1] Marcos Antón, Fernando Charro, Pei-Yong Wang; Totalitarian random tug-of-war games in

graphs, Comm. on Stochastic Analysis, 13 (2019), no. 3.

[2] Gunnar Aronsson; Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967),
no. 6, 551–561.

[3] Gunnar Aronsson, Michael Crandall, Petri Juutinen; A tour of the theory of absolutely mini-

mizing functions, Bulletin of the American mathematical society, 41 (2004), no. 4, 439–505.
[4] Guy Barles, Panagiotis E. Souganidis; Convergence of approximation schemes for fully non-

linear second order equations, Asymptotic analysis, 4 (1991), 271–283.
[5] Robert Jensen; Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,

Archive for Rational Mechanics and Analysis, 123 (1993), 51–74.

[6] Peter Lindqvist; Notes on the infinity laplace equation, Springer, 2016.
[7] E. J. McShane; Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12,

837–842.

[8] Adam M. Oberman; Convergent difference schemes for degenerate elliptic and parabolic equa-
tions: Hamilton–Jacobi equations and free boundary problems, SIAM Journal on Numerical

Analysis, 44 (2006), 879–895.

[9] Hassler Whitney; Analytic extensions of differentiable functions defined in closed sets, Trans-
actions of the American Mathematical Society, 36 (1934), no. 1, 63–89.

Fernando Charro

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
Email address: fcharro@wayne.edu


	1. Introduction
	2. Proofs of Theorems ?? and ??
	Acknowledgments

	References