LIMITS FOR MONGE-KANTOROVICH MASS TRANSPORT PROBLEMS.

JESUS GARCIA-AZORERO, JUAN J. MANFREDI, IRENEO PERAL AND JULIO D. ROSSI

Abstract. In this paper we study the limit of Monge-Kantorovich mass transfer problems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, Ω. Given two absolutely continuos measures (with respect to the surface measure) supported on the boundary ∂Ω, by performing a suitable extension of the measures to a strip of width ε near the boundary of the domain Ω we consider the mass transfer problem for the extensions. Then we study the limit as ε goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. Moreover we look for the possible approximations of these problems by solutions to equations involving the p−Laplacian for large values of p.

1. Introduction.

The main goal of this article is to look at the Monge-Kantorovich mass transport problem when the masses are supported on small strips of width ε near surfaces and study the limit as ε goes to zero.

First, let us briefly present what are the main features of the problem under consideration. Assume that we have a bounded domain with smooth boundary ∂Ω and a continuous funtion g : ∂Ω ↦ ℝ with

∫_{∂Ω} g dσ = ∫_{∂Ω∩{g>0}} g_{+} dσ − ∫_{∂Ω∩{g<0}} g_{−} dσ = 0,

where dσ denotes the area measure on ∂Ω. Hence, we have two subsets Γ_{+} = ∂Ω ∩ {g > 0}, Γ_{−} = ∂Ω ∩ {g < 0} and two positives functions (densities) g_{+} and g_{−} such that

∫_{Γ_{+}} g_{+} dσ = ∫_{Γ_{−}} g_{−} dσ.

Key words and phrases. Mass transport, quasilinear elliptic equations, Neumann boundary conditions.

2000 Mathematics Subject Classification. 35J65, 35J50, 35J55 .

To solve the Monge-Kantorovich mass transfer problem, that is among all mappings T̃ : Γ_{+} → Γ_{−} that preserve the measures given by the two densities, choose one that minimizes the transport cost

C(T̃) = ∫_{Γ_{+}} |x − T̃(x)|g_{+}(x) dσ.

Applying the Kantorovich optimality principle to the mass transfer problem for the measures g^{+}H^{N−1}⌞ ∂Ω and g^{−}H^{N−1}⌞ ∂Ω that are concentrated on ∂Ω we obtain the maximization problem

(1.1) max {∫_{∂Ω} wg dσ: w ∈ W^{1,∞}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{∞} ≤ 1} .

The maximizers of (1.1) are maximal Kantorovich potentials, see [1], [14].

As first noticed in [8] (see also [6]), mass transfer optimization problems appear as limits of the p−laplacian. In our case, consider u_{p,0} the solution of the maximization problem

(1.2) max {∫_{∂Ω} wg dσ: w ∈ W^{1,p}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{L^{p}(Ω)} ≤ 1} .

In [9] the limit as p → ∞ of the family u_{p,0} is studied. It is proved there that a uniform limit of a subsequence {u_{p_{i},0}}, p_{i} → ∞, v_{∞}, is a solution to (1.1). Since we are interested in large values of p we will assume along this paper that p > N. This paper can be regarded as a natural continuation of [9].

These variational problems can be achieved as a singular limit of mass transport problems where the measures are supported in small strips near the boundary. Let us consider the subset of Ω,

ω_{δ} = {x ∈ Ω : dist(x, ∂Ω) < δ} .

Note that this set has measure |ω_{δ}| ∼ δ|∂Ω| for small values of δ. Then for sufficiently small σ > 0 we can define the parallel interior boundary Γ_{s} = {z − sν(z), z ∈ ∂Ω} where ν(z) denotes the outwards normal unit at z ∈ ∂Ω. Note that Γ_{0} = ∂Ω. Then we can also look at the set ω_{δ} as the neighborhood of Γ_{0} defined by

ω_{δ} = {y = z − sν(z), z ∈ ∂Ω, s ∈ (0, δ)} = ⋃_{0<s<δ} Γ_{s}

for sufficiently small δ, say 0 < δ < δ_{0}. We also denote Ω_{s} = {x ∈ Ω : dist(x, ∂Ω) > s} and for s small we have that ∂Ω_{s} = Γ_{s}.

Let us consider the transport problem for a suitable extension of g. To define this extension, as we have mentioned, let us denote by dσ and dσ_{s} the surface measures on the sets ∂Ω and Γ_{s} respectively. Given a function ϕ defined on Ω̄, and given y ∈ Γ_{s} (with s small) , there exists z ∈ ∂Ω such that y = z − sν(z). Hence, we can change variables:

∫_{Γ_{s}} ϕ(y)dσ_{s} = ∫_{∂Ω} ϕ(z − sν(z))G(s, z) dσ

where G(s, z) depends on Ω (more precisely, it depends on the surface measures dσ and dσ_{s}), and by the regularity of ∂Ω, G(s, z) → 1 as s → 0 uniformly for z ∈ ∂Ω.

Using these ideas, we define the following extension of g in Ω. Consider η : [0, ∞) → [0, ∞) a C_{0}^{∞} such that η(s) = 1 if 0 ≤ s ≤ \frac{1}{2}, η(s) = 0 if s > 1, 0 ≤ η(s) ≤ 1 and ∫_{0}^{∞} η(s) ds = A. Defining η_{ε}(s) = \frac{1}{Aε}η (\frac{s}{ε}), we get ∫_{0}^{∞} η_{ε}(s) ds = 1. For δ < ε consider Γ_{s} and

g_{ε}(y) = η_{ε}(s)\frac{g(z)}{G(s, z)}, y = z − sν(z).

We have g_{ε} ≡ 0 in Ω − ω_{ε} and g_{ε} ∈ C(Ω). Moreover,

∫_{Ω} g_{ε}(x) dx = ∫_{0}^{ε} ∫_{Γ_{s}} g_{ε}(y) dσ_{s} ds
= ∫_{0}^{ε} ∫_{∂Ω} g_{ε}(z − sν(z))G(s, z) dσ ds
= ∫_{0}^{ε} η(s) ∫_{∂Ω} g(z) dσ ds = 0.

Associated to this extension we could consider the following two variational problems. First, the maximization problem in W^{1,p}(Ω),

(1.3) max {∫_{ω_{ε}} wg_{ε}: w ∈ W^{1,p}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{L^{p}(Ω)} ≤ 1} ,

and the maximization problem in W^{1,∞}(Ω),

(1.4) max {∫_{ω_{ε}} wg_{ε}: w ∈ W^{1,∞}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{L^{∞}(Ω)} ≤ 1} .

We call u_{p,ε} a solution to (1.3) and u_{∞,ε} a solution to (1.4).

Our first result says that we can take the limits as ε → 0 and p → ∞ in these variational problems. With the above notations we have the following commutative diagram

(1.5)  \begin{aligned}& u_{∞,ε} & ⟶ & u_{∞,0} \\ p → ∞ & ↑ & & ↑ \\ & u_{p,ε} & ⟶ & u_{p,0} \\ & & ε → 0 \end{aligned}

This diagram can be understood in two senses, either taking into account the maximal variational properties satisfied by the functions, or considering the PDEs that the functions satisfy.

From the variational viewpoint, we can state our first result:

Theorem 1. Diagram (1.5) is commutative in the following sense:

(1) Maximizers of (1.3), u_{p,ε}, converge along subsequences uniformly in \overline{Ω} to u_{p,0} a maximizer of (1.2) as ε → 0.

(2) Maximizers of (1.3), u_{p,ε}, converge along subsequences uniformly in \overline{Ω} to u_{∞,ε} a maximizer of (1.4) as p → ∞.

(3) Maximizers of (1.4), u_{∞,ε}, converge along subsequences uniformly in \overline{Ω} to u_{∞,0} a maximizer of (1.1) as ε → 0.

(4) Maximizers of (1.2), u_{p,0}, converge along subsequences uniformly in \overline{Ω} to u_{∞,0} a maximizer of (1.1) as p → ∞.

Now, we turn our attention to the PDE verified by the limits in the viscosity sense (see Section 3 for the precise definition) or in the weak sense.

When p → ∞ we find the ∞−Laplacian, a well known nonlinear operator, given by

Δ_{∞}u = ∑_{i,j=1}^{N} \frac{∂u}{∂x_{j}} \frac{∂^{2}u}{∂x_{j}∂x_{i}} \frac{∂u}{∂x_{i}} ,

see [5], [12]. The ∞−Laplacian appears naturally when one considers absolutely minimizing Lipschitz extensions of a boundary function f, see [2], [3], and [11].

Up to a Lagrange multiplier λ_{p} the functions u_{p,0} are viscosity (and weak) solutions to the problem,

(1.6)  \{ \begin{matrix}−Δ_{p}u = 0 & in Ω, \\ |Du|^{p−2} \frac{∂u}{∂ν} = λ_{p} g & on ∂Ω. \end{matrix}

Let us to point out that it is easily seen that λ_{p} → 1 as p → ∞ (see the remark at the end of Section 2). Hence, to simplify the notation, we will drop this Lagrange multiplier in the sequel.

In [9] (see also [10]) the limit as p → ∞ of the family u_{p,0} is studied in the viscosity setting. It is proved that the problem that is satisfied by a uniform limit u_{∞,0} in the viscosity sense is as follows,

(1.7)  \{ \begin{matrix}Δ_{∞}u = 0 & in Ω, \\ B(x, u, Du) = 0, & on ∂Ω, \end{matrix}

Here

B(x, u, Du) ≡ \{ \begin{matrix}min \{|Du| − 1 , \frac{∂u}{∂ν}\} & if g > 0, \\ max\{1 − |Du| , \frac{∂u}{∂ν}\} & if g < 0, \\ H(|Du|)\frac{∂u}{∂ν} & if g = 0, \end{matrix}

and H(a) is given by

H(a) = \{ \begin{matrix}1 & if a ≥ 1, \\ 0 & if 0 ≤ a < 1. \end{matrix}

Moreover, u_{∞,0} satisfies in the sense of viscosity the estimates:

|Du| ≤ 1 and − |Du| ≥ −1.

On the other hand, when we deal with the problems in the strips, again up to a Lagrange multiplier that converges to one, the functions u_{p,ε} are weak (and hence viscosity) solutions to the problem,

(1.8)  \{ \begin{matrix}−Δ_{p}u = g_{ε} & in Ω, \\ |Du|^{p−2} \frac{∂u}{∂ν} = 0 & on ∂Ω. \end{matrix}

Passing to the limit as p → ∞ in these problems we get that the function u_{∞,ε} satisfy the following properties in the viscosity sense: in all the set Ω, it holds:

(1.9)  \{ \begin{matrix}|Du| ≤ 1 & in Ω, \\ −|Du| ≥ −1 & in Ω, \\ \frac{∂u}{∂ν} = 0 & on ∂Ω, \end{matrix}

and, in the different regions determined by g_{ε}:

(1.10) \{ \begin{matrix}−Δ_{∞}u = 0 & in Ω \ ω_{ϵ}, \\ |Du| = 1 & in \{g_{ϵ} > 0\}, \\ −|Du| = −1 & in \{g_{ϵ} < 0\}, \\ −Δ_{∞}u ≥ 0 & in Ω ∩ ∂\{g_{ϵ} > 0\} ∩ (∂\{g_{ϵ} < 0\})^{c}, \\ −Δ_{∞}u ≤ 0 & in Ω ∩ ∂\{g_{ϵ} < 0\} ∩ (∂\{g_{ϵ} > 0\})^{c}. \end{matrix}

We have the following results that we split in two theorems. First, we have,

Theorem 2.

(1) The limit u_{p,0} of a uniformly converging sequence u_{p,ε} of weak solutions to (1.8) as ε → 0 is a weak solution to (1.6) (and hence a viscosity solution).

(2) The limit u_{∞,0} of a uniformly converging sequence u_{p,0} of viscosity solutions to (1.6) as p → ∞ is a viscosity solution to (1.7).

Let us to point out that when ε → 0, g_{ε} concentrates on the boundary, and therefore the sequence {g_{ε}} is not uniformly bounded. This makes difficult to pass to the limit in the viscosity setting when ε → 0. Hence in this case we consider the variational characterization of the sequence {u_{p,ε}} (that is equivalent to the fact of being a weak solution). To the best of our knowledge, it is not known that the notions of viscosity and weak solutions coincide for solutions to (1.8), cf. [13] where such equivalence is only proved for Dirichlet boundary conditions.

Now, we deal with the rest of the commutative diagram. To pass to the limit in the sequence u_{∞,ε} we need the variational characterization, and we also need a uniqueness result for the limit problem, which has been proved in [9]. This uniqueness result says that:

If Ω is convex and {g = 0}^{o} = ∅, then there is a unique function which satisfies the extremal property (1.1).

Let us to point out that the hypothesis {g = 0}^{o} = ∅ implies also the uniqueness of the extremals to (1.4). Therefore, under this hypothesis there exists a unique u_{∞,ε} reached as a limit of the solutions u_{p,ε} as p → ∞.

Now, we can state our second theorem.

Theorem 3.

(1) The limit u_{∞,ε} of a uniformly converging sequence u_{p,ε} of viscosity solutions to (1.8) as p → ∞ is a viscosity solution to (1.9)-(1.10).

(2) Assume that Ω is convex and {g = 0}^{o} = ∅. Consider the viscosity solutions u_{∞,ε} to (1.9)-(1.10), obtained as a uniform limit as p → ∞ of the solutions u_{p,ε}. Then, the sequence {u_{∞,ε}} converges uniformly to a viscosity solution to (1.7), u_{∞,0}.

The rest of the paper is organized as follows: in Section 2 we pass to the limit in the variational sense and prove Theorem 1 and in Section 3 we deal with viscosity solutions and prove Theorems 2 and 3.

2. Proof of Theorem 1

This section is devoted to the proof of Theorem 1 that shows that diagram (1.5) commutes in the variational sense.

Proof of Theorem 1. The proof of the uniform convergence (along subsequences) of u_{p,0} to u_{∞,0} is contained in [9].

Let us prove that u_{p,ε} converges to u_{p,0} as ε → 0. We have

∥Du_{p,ε}∥_{L^{p}(Ω)} ≤ 1.

Therefore we can extract a subsequence (that we still call u_{p,ε}) such that

u_{p,ε} ⇀ v, as ε → 0,

weakly in W^{1,p}(Ω) and, since p > N,

u_{p,ε} → v, as ε → 0,

uniformly in Ω (in fact, convergence holds in C^{β}). This limit v verifies the normalization constraint

∫_{Ω} v = 0

and moreover

∥Dv∥_{L^{p}(Ω)} ≤ 1.

On the other hand, thanks to the uniform convergence and to the definition of the extension g_{ε} we obtain,

lim_{ε→0} ∫_{ω_{ε}} g_{ε}u_{p,ε} = lim_{ε→0} ∫_{0}^{ε} ∫_{Γ_{s}} g_{ε}(y)u_{p,ε}(y) dσ_{s} ds

= lim_{ε→0} ∫_{0}^{ε} ∫_{∂Ω} g_{ε}(z − sν(z))u_{p,ε}(z − sν(z))G(s, z) dσ ds

= lim_{ε→0} ∫_{0}^{ε} η_{ε}(s) ∫_{∂Ω} g(z)u_{p,ε}(z − sν(z)) dσ ds

= ∫_{∂Ω} gv dσ

and hence

(2.1) ∫_{Ω} |Dv|^{p} − ∫_{∂Ω} gv dσ ≤ lim inf_{ε→0} (∫_{Ω} |Du_{p,ε}|^{p} − ∫_{ω_{ε}} g_{ε}u_{p,ε}) .

On the other hand for every w ∈ C^{1}(\overline{Ω}) we have

∫_{Ω} |Dw|^{p} − ∫_{∂Ω} gw dσ = lim_{ε→0} ∫_{Ω} |Dw|^{p} − ∫_{ω_{ε}} g_{ε}w.

Hence, the extremal characterization of u_{p,ε} implies

inf ; u∈W^{1,p}(Ω),∫_{Ω} u=0 {∫_{Ω} |Du|^{p} − ∫_{∂Ω} gudσ} ≥ lim inf_{ε→0} ∫_{Ω} |Du_{p,ε}|^{p}−∫_{ω_{ε}} g_{ε}u_{p,ε}.

And by (2.1) we obtain

inf ; u∈W^{1,p}(Ω), ∫_{Ω} u=0 {∫_{Ω} |Du|^{p} − ∫_{∂Ω} gu dσ} = ∫_{Ω} |Dv|^{p} − ∫_{∂Ω} gv dσ,

and therefore all possible limits v = u_{p,0} satisfy the extremal property (1.2).

Now, let us prove that u_{∞,ε} converges to u_{∞,0}, a maximizer of (1.1). Recall that u_{∞,ε} is a solution to the problem

M_{ε} = max {∫_{ω_{ε}} wg_{ε}: w ∈ W^{1,∞}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{∞} ≤ 1} .

That is,

M_{ε} = ∫_{ω_{ε}} u_{∞,ε}g_{ε}.

Therefore u_{∞,ε} is bounded in W^{1,∞}(Ω) and then there exists a subsequence (that we still denote by u_{∞,ε}) such that,

(2.2)

u_{∞,ε} ⇀^{∗} v weakly-* in W^{1,∞}(Ω) and
u_{∞,ε} → v uniformly in \overline{Ω},

LIMITS FOR MASS TRANSPORT PROBLEMS 9

as ε → 0. Hence

lim_{ε→0} ∫_{ω_{ε}} u_{∞,ε}g_{ε} = ∫_{∂Ω} vg dσ.

On the other hand, for every z ∈ C^{1}(\overline{Ω}) it holds that

lim_{ε→0} ∫_{ω_{ε}} g_{ε}z = ∫_{∂Ω} gz dσ.

Hence, if we call

(2.3) M = max {∫_{∂Ω} wg dσ: w ∈ W^{1,∞}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{∞} ≤ 1} ,

we obtain, from (2.2),

M ≤ lim inf_{ε→0} M_{ε} = ∫_{∂Ω} vg dσ.

Therefore v = u_{∞,0} is a maximizer of (2.3), as we wanted to prove. Finally, let us prove that u_{p,ε} → u_{∞,ε}. Recall that ∫_{ω_{ε}} u_{p,ε}g_{ε} = max {∫_{ω_{ε}} wg_{ε}: w ∈ W^{1,p}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{L^{p}(Ω)} ≤ 1} . Therefore, for any q < p

∥Du_{p,ε}∥_{L^{q}(Ω)} ≤ (∥Du_{p,ε}∥_{L^{p}(Ω)}|Ω|^{\frac{p−q}{p}} )^{1/q} ≤ |Ω|^{\frac{p−q}{pq}} .

Hence, we can extract a subsequence (still denoted by u_{p,ε}) such that,

u_{p,ε} → u, uniformly in \overline{Ω},

as p → ∞ with

∥Du∥_{L^{∞}(Ω)} ≤ 1.

Then

∫_{ω_{ε}} u_{p,ε}g_{ε} → ∫_{ω_{ε}} g_{ε}, as p → ∞.

This limit u verifies that ∫_{ω_{ε}} ug_{ε} ≤ max {∫_{ω_{ε}} wg_{ε}: w ∈ W^{1,∞}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{L^{∞}(Ω)} ≤ 1} . Let us prove that we have an equality here. If not, there exists a function v such that v ∈ W^{1,∞}(Ω), ∫_{Ω} v = 0, ∥Dv∥_{L^{∞}(Ω)} ≤ 1 with

∫_{ω_{ε}} ug_{ε} < ∫_{ω_{ε}} vg_{ε}.

If we normalize, taking φ = v/|Ω|^{1/p}, we obtain a function in W^{1,p}(Ω) with ∫_{Ω} φ = 0, ∥Dφ∥_{L^{p}(Ω)} ≤ 1 and such that

lim_{p→∞} ∫_{ω_{ε}} u_{p,ε}g_{ε} = ∫_{ω_{ε}} ug_{ε} < ∫_{ω_{ε}} vg_{ε} = lim_{p→∞} |Ω|^{1/p} ∫_{ω_{ε}} φg_{ε}.

This contradiction proves that

∫_{ω_{ε}} ug_{ε} = max {∫_{ω_{ε}} wg_{ε}: w ∈ W^{1,∞}(Ω), ∫_{Ω} w = 0, ∥Dw∥_{L^{∞}(Ω)} ≤ 1} .

This ends the proof. □

As a byproduct of the discussion before, the extremal characterization implies the following estimate of the gradients:

Lemma 2.1. The extremal functions in Theorem 1 satisfy:

|Du_{∞,ε}| ≤ 1, a.e.

and

|Du_{∞,0}| ≤ 1, a.e.

Let us close this section with the following remark. Remark. The limits of the solutions to the maximization problems (1.2) and (1.3) coincide with the limits of the solutions to the corresponding PDEs when p → ∞.

In fact, the unique maximizer of (1.2), u_{p}, is a weak solution to

(2.4)  \{ \begin{matrix}−Δ_{p}u_{p} = 0 & in Ω, \\ |Du_{p}|^{p−2} \frac{∂u_{p}}{∂ν} = λ_{p} g & on ∂Ω. \end{matrix}

Here λ_{p} is a Lagrange multiplier. If we take

ũ_{p} = (λ_{p})^{1/(p−1)}u_{p}

we get a solution to (1.6), that is,

\{ \begin{matrix}−Δ_{p}ũ_{p} = 0 & in Ω, \\ |Dũ_{p}|^{p−2} \frac{∂ũ_{p}}{∂ν} = g & on ∂Ω. \end{matrix}

From the weak form of (2.4) and our previous results we get

lim_{p→∞} λ_{p} = lim_{p→∞} (∫_{∂Ω} gu_{p} dσ)^{−1} = (∫_{∂Ω} gu_{∞} dσ)^{−1} ≠ 0.

Therefore,

lim_{p→∞} ũ_{p} = lim_{p→∞}(λ_{p})^{1/(p−1)}u_{p} = lim_{p→∞} u_{p}.

In a completely analogous way it can be proved that the limits as p → ∞ of the solutions to the maximization problems (1.3) and solutions to the PDEs (1.8) coincide.

3. Proofs of Theorems 2 and 3

In this section we deal with the PDE version of the commutative diagram (1.5). To this end it is natural to consider solutions in the viscosity sense.

Following [4] let us recall the definition of viscosity solution for elliptic problems with general boundary conditions. Assume

F : \overline{Ω} × ℝ^{N} × 𝕊^{N×N} → ℝ

a continuous function. The associated equation

F (x, Du, D^{2}u) = 0

is called (degenerate) elliptic if

F (x, ξ, X) ≤ F (x, ξ, Y) if X ≥ Y.

Definition 3.1. Consider the boundary value problem

\{ \begin{matrix}F (x, Du, D^{2}u) = 0 & in Ω, \\ B(x, u, Du) = 0 & on ∂Ω. \end{matrix}

(1) A lower semi-continuous function u is a viscosity supersolution if for every ϕ ∈ C^{2}(\overline{Ω}) such that u − ϕ has a strict minimum at the point x_{0} ∈ \overline{Ω} with u(x_{0}) = ϕ(x_{0}) we have: If x_{0} ∈ ∂Ω, we have the inequality

max{B(x_{0}, ϕ(x_{0}), Dϕ(x_{0})), F (x_{0}, Dϕ(x_{0}), D^{2}ϕ(x_{0}))} ≥ 0

and if x_{0} ∈ Ω then we require

F (x_{0}, Dϕ(x_{0}), D^{2}ϕ(x_{0})) ≥ 0.

(2) An upper semi-continuous function u is a subsolution if for every ψ ∈ C^{2}(\overline{Ω}) such that u − ψ has a strict maximum at the point x_{0} ∈ \overline{Ω} with u(x_{0}) = ψ(x_{0}) we have: If x_{0} ∈ ∂Ω the inequality

min{B(x_{0}, ψ(x_{0}), Dψ(x_{0})), F (x_{0}, Dψ(x_{0}), D^{2}ψ(x_{0}))} ≤ 0

holds, and if x_{0} ∈ Ω then we require

F (x_{0}, Dψ(x_{0}), D^{2}ψ(x_{0})) ≤ 0.

(3) Finally, u is a viscosity solution if it is a super and a subsolution.

In the sequel, we will use the same notation as in the definition: ϕ stands for the test functions touching from below the graph of u, and ψ stands for the test functions touching from above the graph of u.

First, we point out that the arguments in [5] could be used to prove the viscous counterpart of Lemma 2.1.

Lemma 3.2. The extremal functions in Theorem 1 satisfy in the viscosity sense:

|Du_{∞,ε}| ≤ 1 ; |Du_{∞,0}| ≤ 1 ; −|Du_{∞,ε}| ≥ −1 ; −|Du_{∞,0}| ≥ −1 .

On the other hand, at level p we can pass from weak solutions to solutions in the sense of viscosity:

Lemma 3.3. Let u_{p,0} be a continuous weak solution of (1.6) for p > 2. Then u_{p,0} is a viscosity solution to

\{ \begin{matrix}Δ_{p}u_{p,0} = 0 & in Ω, \\ |Du_{p,0}|^{p−2} \frac{∂u_{p,0}}{∂ν} = g & on ∂Ω. \end{matrix}

Proof. See [9], Lemma 2.3. □

Proof of Theorem 2. We decompose the proof in several steps.

Step 1. First, assume that a sequence of viscosity solutions to (1.6), u_{p,0} converge, as p → ∞, uniformly in Ω to a limit u_{∞,0}, then it is proved in [9] that u_{∞,0} is a viscosity solution to (1.7).

Step 2. As we have mentioned in the introduction at this part of the proof we have to deal with weak solutions since the right hand side of the problem (1.8) is not uniformly bounded in ε.

Assume that we have a sequence of weak solutions to (1.8), u_{p,ε} that converge, as ε → 0, uniformly in Ω to a limit u_{p,0} then let us prove that u_{p,0} is a weak (and a viscosity) solution of (1.6).

As it was proved in the previous section, we can pass to the limit in the variational formulation (that is equivalent to the weak formulation) of (1.8) to obtain that every limit of u_{p,ε} is a variational solution (and hence a weak solution) to (1.6).

To finish we just have to observe that a continuous weak solution to (1.6) is in fact a viscosity solution, thanks to Lemma 3.3. □

Next, we pass to the proof of Theorem 3.

Proof of Theorem 3. We recall that, thanks to lemma 3.2, we have the estimates |Du_{∞,ε}| ≤ 1 ; |Du_{∞,0}| ≤ 1 , −|Du_{∞,ε}| ≥ −1 ; −|Du_{∞,0}| ≥ −1, in the sense of viscosity, in all the domain Ω.

Next, to look for the equation that u_{∞,ε} satisfies in the viscosity sense, assume that u_{∞,ε}−ϕ has a strict minimum at x_{0} ∈ Ω. Depending on the location of the point x_{0} we have different cases. First, suppose that x_{0} ∈ Ω\ω_{ε}. By the uniform convergence of u_{p_{i},ε} to u_{∞,ε} there exists points x_{p_{i}} such that u_{p_{i},ε} − ϕ has a minimum at x_{p_{i}} with x_{p_{i}} ∈ Ω \ ω_{ε} for p_{i} large. Using that u_{p_{i}} is a viscosity solution to (1.8) we obtain

−Δ_{p}ϕ(x_{p_{i}}) = −div(|Dϕ|^{p_{i}−2}Dϕ)(x_{p_{i}}) ≥ 0.

Therefore

−(p_{i} − 2)|Dϕ|^{p_{i}−4}Δ_{∞}ϕ(x_{p_{i}}) − |Dϕ|^{p_{i}−2}Δϕ(x_{p_{i}}) ≥ 0.

If Dϕ(x_{0}) = 0 we get −Δ_{∞}ϕ(x_{0}) = 0. If this is not the case, we have that Dϕ(x_{p_{i}}) ≠ 0 for large i and then

−Δ_{∞}ϕ(x_{p_{i}}) ≥ \frac{1}{p_{i} − 2}|Dϕ|^{2}Δϕ(x_{p_{i}}) → 0, as p_{i} → ∞.

We conclude that

(3.1)  −Δ_{∞}ϕ(x_{0}) ≤ 0.

That is u_{∞,ε} is a viscosity subsolution of −Δ_{∞}u_{∞,ε} = 0 in Ω \ ω.

The fact that it is a viscosity supersolution of −Δ_{∞}u_{∞,ε} = 0 in Ω\ω is completely analogous.

Next suppose that x_{0} lies on the region where g_{ε} > 0.

Assume that we have a test function ϕ touching from below the graph of u_{∞,ε}, that is, u_{∞,ε} − ϕ has a strict minimum at the point x_{0} . Then, for p_{i} large, there exist points x_{p_{i}} such that u_{p_{i},ε} − ϕ has a minimum at x_{p_{i}} with g_{ε}(x_{p_{i}}) > 0. Using that u_{p_{i}} is a viscosity solution to (1.8) we obtain

−(p_{i} − 2)|Dϕ|^{p_{i}−4}Δ_{∞}ϕ(x_{p_{i}}) − |Dϕ|^{p_{i}−2}Δϕ(x_{p_{i}}) ≥ g_{ε}(x_{p_{i}}) > 0.

In particular |Dϕ(x_{p_{i}})| ≠ 0 and therefore

−Δ_{∞}ϕ(x_{p_{i}}) ≥ \frac{1}{p_{i} − 2}|Dϕ|^{2}Δϕ(x_{p_{i}}) + \frac{g_{ε}(x_{p_{i}})}{(p − 2)|Dϕ(x_{p_{i}})|^{p−4}} .

Since g_{ε}(x_{p_{i}}) → g(x_{0}) > 0, this means that |Dϕ(x_{0})| cannot be smaller than 1 (in this case the right hand side of the inequality tends to infinity). Therefore, we conclude that |Dϕ(x_{0})| ≥ 1.

Assume now that we have a test function ψ that touches from above the graph of u_{∞,ε}. Then, for p_{i} large, there exists points x_{p_{i}} such that u_{p_{i},ε} − ψ has a minimum at x_{p_{i}} with g_{ε}(x_{p_{i}}) > 0. Using that u_{p_{i}} is a viscosity solution to (1.8) we obtain

−(p_{i} − 2)|Dψ|^{p_{i}−4}Δ_{∞}ψ(x_{p_{i}}) − |Dψ|^{p_{i}−2}Δψ(x_{p_{i}}) ≤ g_{ε}(x_{p_{i}})(> 0).

Then, if |Dψ(x_{0})| > 1 , it follows that −Δ_{∞}ψ(x_{0}) ≤ 0. Therefore, the condition on ψ reads

(3.2)  min{|Dψ(x_{0})| − 1, −Δ_{∞}ψ(x_{0})} ≤ 0.

But notice that this condition is always satisfied, since we know that |Du_{∞,ε}| ≤ 1 in the sense of viscosity. Therefore, (3.1) and (3.2) imply that, if g_{ε}(x_{0}) > 0, then |Du_{∞,ε}(x_{0})| = 1.

Similar computations give us that if we look at a point x_{0} such that g_{ε}(x_{0}) < 0 then −|Du_{∞,ε}(x_{0})| = −1.

The next case to consider, is when g_{ε}(x_{0}) = 0 and the point x_{0} can be reached as a limit of points x_{p_{i}} that could be contained in the region {g_{ε} > 0} or in the region {g_{ε} = 0}. In other words, x_{0} ∈ Ω ∩ ∂{g_{ε} > 0} ∩ (∂{g_{ε} < 0})^{C}.

In this case, if we consider a test function ϕ touching from below the graph of u_{∞,ε} at x_{0}, then we get a sequence {x_{p_{i}}} converging to x_{0}, such that u_{p_{i},ε} − ϕ has a strict minimum at x_{p_{i}}. Passing to a subsequence if necessary, we have two possibilities: either g_{ε}(x_{p_{i}}) = 0, or g_{ε}(x_{p_{i}}) > 0. If we assume g_{ε}(x_{p_{i}}) = 0, then

−(p_{i} − 2)|Dϕ|^{p_{i}−4}Δ_{∞}ϕ(x_{p_{i}}) − |Dϕ|^{p_{i}−2}Δϕ(x_{p_{i}}) ≥ 0.

Then, if |Dϕ(x_{p_{i}})| ≠ 0 it follows that −Δ_{∞}ϕ(x_{0}) ≥ 0. On the other hand, if |Dϕ(x_{p_{i}})| = 0 for infinitely many indexes, then −Δ_{∞}ϕ(x_{0}) = 0.

If we assume g_{ε}(x_{p_{i}}) > 0, then |Dϕ(x_{p_{i}})| ≠ 0 and therefore passing to the limit we get −Δ_{∞}ϕ(x_{0}) ≥ 0.

Concerning the test functions touching from above ψ, if g_{ε}(x_{p_{i}}) = 0, then

−(p_{i} − 2)|Dψ|^{p_{i}−4}Δ_{∞}ψ(x_{p_{i}}) − |Dψ|^{p_{i}−2}Δψ(x_{p_{i}}) ≤ 0.

This implies that −Δ_{∞}ψ(x_{0}) ≤ 0. But if g_{ε}(x_{p_{i}}) > 0, then, as in a previous case, we get that min{|Dψ(x_{0})| − 1, −Δ_{∞}ψ(x_{0})} ≤ 0, and this condition is always satisfied because |Du_{∞,ε}| ≤ 1 .

As a conclusion, if x_{0} ∈ Ω∩∂{g_{ε} > 0}∩(∂{g_{ε} < 0})^{C}, we have in the sense of viscosity that −Δ_{∞}u_{∞,ε} ≥ 0 (jointly with the general viscosity estimates on the gradient, valid in all Ω).

In an analogous way, if x_{0} ∈ Ω ∩ (∂{g_{ε} > 0})^{C} ∩ ∂{g_{ε} < 0}, we have in the sense of viscosity that −Δ_{∞}u_{∞,ε} ≤ 0 (jointly with the general viscosity estimates on the gradient, valid in the whole domain Ω).

The next region consists on the points x_{0} ∈ Ω that can be reached as limits of sequences contained either in {g_{ε} > 0}, either in {g_{ε} = 0}, either in {g_{ε} < 0}. That is, x_{0} ∈ Ω ∩ ∂{g_{ε} > 0} ∩ ∂{g_{ε} < 0}. The same arguments as before give us that in this set the equation satisfied in the sense of viscosity is simply |Du_{∞,ε}| ≤ 1 and −|Du_{∞,ε}| ≥ −1.

Finally, the boundary condition satisfied by u_{∞,ε} in the sense of viscosity is

\frac{∂u_{∞,ε}}{∂ν} = 0.

To see this fact, we use that the p-Laplacian satisfies hypothesis of the ”strict monotonicity in the direction of the normal” as it is stated in [4]. Then, for instance, the boundary condition at level p reads simply

|Dϕ(x_{p})|^{p−2} \frac{∂ϕ}{∂ν}(x_{p}) ≥ 0,

for any test function ϕ touching the graph of u_{p,ε} from below at a point x_{p} ∈ ∂Ω. Test functions touching the graph from above can be handled in a similar way.

The last step of the proof consists of taking limits on the sequence {u_{∞,ε}}. Notice that these functions, as limits of the sequence {u_{p,ε}} as p → ∞, satisfy the extremal property (1.4). Therefore, by Theorem 1, the limit u_{∞,0} is an extremal of (1.1). From hypothesis {g = 0}^{o} = ∅, we obtain a uniqueness result for u_{∞,0} (see [9]), and then it must be the same function that we reach as the limit of the sequence {u_{p,0}} when p → ∞, and, as it was proved in [9], it satisfies (1.7). □

Acknowledgements. JGA and IP supported by project MTM2004-02223, M.C.Y.T. Spain, JJM supported in part by NSF award DMS-0100107, JDR supported by Fundacion Antorchas, CONICET and AN-PCyT PICT 05009.

References

[1] L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical aspects of evolving interfaces (Funchal, 2000), 1–52, Lecture Notes in Math., 1812, Springer, Berlin, 2003. Available at CVGMT preprint server.

[2] G. Aronsson, Extensions of functions satisfiying Lipschitz conditions. Ark. Math., 6 (1967), 551–561.

[3] G. Aronsson, M.G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc., 41 (2004), 439–505.

[4] G. Barles, Fully nonlinear Neumann type conditions for second-order elliptic and parabolic equations. J. Differential Equations, 106 (1993), 90–106.

[5] T. Bhattacharya, E. Di Benedetto and J. Manfredi. Limits as p → ∞ of Δ_{p}u_{p} = f and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15–68.

[6] G. Bouchitte, G. Buttazzo and L. De Pasquale. A p−laplacian approximation for some mass optimization problems. J. Optim. Theory Appl., 118(1), (2003), 1–25.

[7] M.G. Crandall, H. Ishii and P.L. Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27 (1992), 1–67.

[8] L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137 (1999), no. 653.

[9] J. García-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, The Neumann problem for the ∞-Laplacian and the Monge-Kantorovich mass transfer problem. Nonlinear Analysis TMA to appear (avalaible on line).

[10] J. García-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, Steklov eigenvalues for the ∞-Laplacian. Rend. Lincei Mat. Appl., 17(3) (2006), 199–210.

[11] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 (1993), 51–74.

[12] P. Juutinen, P. Lindqvist and J. J. Manfredi, The ∞−eigenvalue problem. Arch. Rational Mech. Anal., 148 (1999), 89–105.

[13] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation. SIAM J. Math. Anal., 33(3) (2001), 699–717.

[14] C. Villani, Topics in Optimal Transportation. Amer. Math. Soc. Graduate Studies in Mathematics, vol. 58, 2003.

Jesus Garcia-Azorero and Ireneo Peral Departamento de Matemáticas, U. Autonoma de Madrid, 28049 Madrid, Spain.

E-mail address: jesus.azorero@uam.es, ireneo.peral@uam.es

Juan J. Manfredi Department of Mathematics, University of Pittsburgh. Pittsburgh, Pennsylvania 15260.

E-mail address: manfredi@math.pitt.edu

Julio D. Rossi Consejo Superior de Investigaciones Científicas (CSIC), Serrano 117, Madrid, Spain, on leave from Departamento de Matemática, FCEyN UBA (1428) Buenos Aires, Argentina.

E-mail address: jrossi@dm.uba.ar