Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers

Ronald Gariepy^{1}, Changyou Wang^{2,∗}, Yifeng Yu^{3}

1,2 Department of Mathematics, University of Kentucky, Lexington, KY 40506 3 Department of Mathematics, University of California, Berkeley, CA 94720

Abstract. For any nonnegative, quasi-convex Hamiltonian H ∈ C^{2}(R^{n}), we consider the L^{∞}-functional F (u, ·) = ∥H(∇u)∥_{L^{∞}(·)} over Wloc^{1,∞}(Ω). We introduce the notion of comparison with generalized cones (abbreviated CGC) and prove that CGC, viscosity solutions of the Aronsson equation, and absolute minimizers of F (·) are equivalent. This extends an earlier result by Crandall-Evans-Gariepy [CEG].

§1. Introduction

The calculus of variations in the sup-norm was initiated by Aronsson [A1,2,3,4,5,6] in 1960’s. The past decade has seen great interest in this highly nonconventional variational problem (see, e.g., [J] [CEG] [CE] [C] [BJW] [BJ] [CP] [FS1,2] [S] [Y] etc), because of both its mathematical difficulties and its many potential applications in other fields (see for instance the survey article by Barron [B] and the very recent excellent article by Aronsson-Crandall-Juutinen [AJL] and the references therein). For a bounded Lipschitz domain Ω ⊂ R^{n}, the simplest such model problem is the pointwise functional

I[u] = ∥∇u∥_{L^{∞}(Ω)}, u ∈ W_{g}^{1,∞}(Ω) = {u ∈ W^{1,∞}(Ω), u = g on ∂Ω}  (1.1)

where g : ∂Ω → R is a given Lipschitz continuous function. A basic question has been to understand minimization problem involving this functional. A central discovery of Aronsson [A4,5] was that any minimizer u ∈ C^{2}(Ω), properly interpreted, of the functional I[·] satisfies a highly nonlinear, degenerate elliptic PDE, called the ∞-Laplace equation:

Δ_{∞}u := ∇u ⊗ ∇u : ∇^{2}u = 0, in Ω.  (1.2)

However, Aronsson [A4,5] also showed that (1.2) may not have C^{2} solutions. Jensen [J] proved the first basic theorems about solvability of the Dirichlet problem for (1.2): equation (1.2) has a unique viscosity solution u ∈ C(Ω̄) satisfying u = g on ∂Ω, and this solution is characterized by the following absolutely minimizing Lipschitz extension(abbreviated AML) of g: for any open subset U ⊂⊂ Ω and for any v ∈ W^{1,∞}(U), one has

if u = v on ∂U, then ∥∇u∥_{L^{∞}(U)} ≤ ∥∇v∥_{L^{∞}(U)}.  (1.3)

∗ Partially supported by NSF grant 0400718 Another important contribution on both equation (1.2) and AML property was developed by Crandall-Evans-Gariepy [CEG] where the authors showed that both AML and viscosity solutions of (1.2) are equivalent to the property comparison with cones: for any U ⊂ Ω, x_{0} ∈ Ω, and any a, b ∈ R,

if u(x) ≤ (≥)a + b|x − x_{0}| on ∂(U \ {x_{0}}, then u(x) ≤ (≥)a + b|x − x_{0}| in U. (1.4)

The goal of this paper is to extend the main theorem of [CEG] to a class of general Hamiltonian functions H = H(p) : R^{n} → R. In order to state our results, let’s first recall a few definitions. Definition 1. For a given H ∈ C(R^{n} × R × R^{n}), a function u ∈ Wloc^{1,∞}(Ω) is called an absolute minimizer (or AM) of the functional F (u, ·) = ∥H(x, u(x), ∇u(x))∥_{L^{∞}(·)} over Wloc^{1,∞}(Ω) if for any open subset U ⊂⊂ Ω and v ∈ W^{1,∞}(U) with v|_{∂U} = u|_{∂U}, then

F (u, U) = ∥H(x, u(x), ∇u(x))∥_{L^{∞}(U)} ≤ F (v, U) = ∥H(x, v(x), ∇v(x))∥_{L^{∞}(U)}. (1.5)

The existence of absolute minimizers of F (·) has been obtained by Barron-Jensen-Wang [BJW1] for H = H(x, z, p) ∈ C(R^{n}×R×R^{n}) which is quasi-convex in the p-variable and satisfies suitable growth conditions on the p-variable. Moreover, it was proved by [BJW1] and Crandall [C] that any absolute minimizer u ∈ Wloc^{1,∞}(Ω) is a viscosity solution of the Aronsson equation:

⟨∇_{x}(H(x, u(x), ∇u(x))), H_{p}(x, u(x), ∇u(x))⟩ = 0 in Ω.  (1.6)

Very recently, Yu in [Y] proved that if H(p, x) ∈ C^{2}(R^{n} × Ω) is convex p and

lim_{|p|→+∞} H(p, x) = +∞ uniformly in x ∈ Ω̄,

then any viscosity solution of the Aronsson equation (1.6) is an absolute minimizers. Partly motivated by the results of [Y], we prove in this paper that the comparison principle with cones for AML by [CEG] can be extended to AMs of a class of general L^{∞}-functionals.

The notation of viscosity solutions of 2nd order degenerate elliptic PDEs can be found in Crandall-Isshi-Lions [CIL]. For the readers’ convenience, we recall it here. Definition 2. For a degenerate elliptic operator A = A(x, z, p, X) ∈ C(R^{n} × R × R^{n} × R^{n×n}), a function u ∈ USC(Ω) is a viscosity subsolution of

A(x, w(x), ∇w(x), ∇^{2}w(x)) = 0 in Ω  (1.7)

if for any (x_{0}, ϕ) ∈ Ω × C^{2}(Ω)

0 = u(x_{0}) − ϕ(x_{0}) ≥ u(x) − ϕ(x) for all x ∈ Ω

then

A(x_{0}, ϕ(x_{0}), ∇ϕ(x_{0}), ∇^{2}ϕ(x_{0})) ≥ 0.  (1.8)

Similarly, a function v ∈ LSC(Ω) is a viscosity supersolution of eqn. (1.7) if −v is a viscosity supersolution of (1.7) with A replaced by Ā(x, z, p, X) ≡ −A(x, −z, −p, −X) for (x, z, p, X) ∈ R^{n} × R × R^{n} × R^{n×n}. A function w ∈ C(Ω) is a viscosity solution of (1.7) if it is both a viscosity subsolution and a viscosity supersolution of (1.7).

In this paper, we will consider Hamiltonian functions H = H(p) : R^{n} → R that are independent of (x, z)-variable. Throughout this paper, we make the following three assumptions on the Hamilton H : R^{n} → R: H1. H ∈ C^{2}(R^{n}), H(0) = 0, H(p) ≥ 0 for any p ∈ R^{n}. H2. H is quasi-convex:

H(tp + (1 − t)q) ≤ max{H(p), H(q)}, ∀p, q ∈ R^{n}, 0 ≤ t ≤ 1.  (1.9)

H3. H is coercive:

lim_{|p|→+∞} H(p) = +∞.  (1.10)

Our main theorem shows that under the assumptions H1-H3, any viscosity solution of the Aronsson equation (1.6) is an absolute minimizer for H. When H has x dependence and H(·, x) satisfies H2 and H3 for each x ∈ Ω, an example given by Yu [Y] indicated that a viscosity solution of the Aronsson equation (1.6) may not be an absolute minimizer of the associated L^{∞}-functional. Considering other examples given in [Y], the results in this paper and [Y] are sharp for the sufficiency of the Aronsson equations.

We would like to remark that the quasiconvexity condition (1.9) for the Hamiltonian function H is an essential condition for the existence of absolute minimizers. Since the direct method is one of the standard approachs to obtain absoulte minimizers, which is based on showing any minimizing sequence has a convergent subsequence in appropriate spaces. This in turn requires to prove the sequential lower semicontinuity of F (·, U) with respect to (W^{1,∞})^{∗} convergence. For this, Barron-Jensen-Wang [BJW2] have proved that the quasiconvexity condition (1.9), modular suitable growth and coercivity conditions, is both the necessary and sufficient condition for the functional F (u, U) to have lower semicontinuity.

Typical examples of quasiconvex functions are compositions of monotone functions and convex functions, i.e, H = g ∘ f, where g is monotonically nondecreasing or nonincreasing, and f is convex. In fact, it is an interesting open question whether a quasiconvex function has this form. See [F] for some results related to this question.

Even if H = g ∘ f for some smooth monotone function g and convex function f, an absolute minimizer for H may not be an absolute minimizer for f. Hence it is impossible to reduce the quasiconvex case to the convex case which was treated by [Y]. Hence it is both interesting and necessary to work directly with quasiconvexity Hamiltonian functions H.

To do this, we will use the idea of comparison with cones by [CEG] and develop similar machinery that may also serve as preparations for further studies of Aronsson equations (see [WY] for the detail). More precisely, we define the notion of generalized cones. Definition 3. For any k ≥ 0, a generalized cone centered at the origin with slope k, C_{k}^{H}(x), is defined by

C_{k}^{H}(x) = max_{{H(p)=k}}⟨x, p⟩.  (1.11)

Definition 4. A function u ∈ USC(Ω) enjoys comparison with generalized cones from above in Ω (or abbreviated u ∈ CGCA(Ω)) if for any V ⊂⊂ Ω, x_{0} ∈ Ω, and k ≥ 0,

u(x) ≤ u(x_{0}) + C_{k}^{H}(x − x_{0}) ∀x ∈ ∂(V \ {x_{0}}),  (1.12)

then

u(x) ≤ u(x_{0}) + C_{k}^{H}(x − x_{0}) ∀x ∈ V.  (1.13)

A v ∈ LSC(Ω) enjoys comparison with generalized cones from below in Ω (or abbreviated v ∈ CGCB(Ω)) if −v enjoys comparison with generalized cones from above in Ω with respect to Ĥ, here Ĥ(p) ≡ H(−p) for all p ∈ R^{n}. A w ∈ C(Ω) enjoys comparison with generalized cones in Ω (or abbreviated w ∈ CGC(Ω)) if w ∈ CGCA(Ω) ∩ CGCB(Ω).

Now we are ready to state our main results.

Theorem A. Under the assumptions H1-H3. The following three statements are equivalent:

(a) u ∈ Wloc^{1,∞}(Ω) is an absolute minimizer of F (v, ·) = ∥H(∇v)∥_{L^{∞}(·)}.

(b) u ∈ C(Ω) is in CGC(Ω).

(c) u ∈ C(Ω) is a viscosity solution of the Aronsson equation:

H_{p}(∇u(x)) ⊗ H_{p}(∇u(x)) : ∇^{2}u(x) = 0 in Ω.  (1.14)

A direct consequence of theorem A is the following regularity result.

Corollary B. Under the assumptions H1-H3. Any viscosity solution u ∈ C(Ω) of the Aronsson equation (1.14) is in Wloc^{1,∞}(Ω).

The main difficulty in the proof of Theorem A is to show that (c) implies (b). According to Crandall [C] [ACJ], it is not difficult to see that C_{k}^{H} is a viscosity supersolution of the equation (1.14). Since we only assume that H is quasiconvex, the cone function C_{k}^{H} which is a viscosity solution of H(∇C_{k}^{H}) = k in V \{x_{0}} lacks enough regularities. To get the comparison between u and C_{k}^{H}, we need to consider the inf-convolution, which is semiconcave and hence twice differentiable almost everywhere (see [J]),

C_{k,ϵ}^{H}(x) = inf_{y∈V̄}\{C_{k}^{H}(y) + \frac{1}{ϵ}|x − y|^{2}\}.

Moreover, we need to show that there exists proper k such that H_{p}(∇C_{k,ϵ}^{H}) · ∇C_{k,ϵ}^{H} ≠ 0, which is essentially needed to construct a strict viscosity supersolution of the equation (1.14). Such a difficulty is overcomed by the key observation (see Lemma 2.2 in §2): if H ∈ C^{2}(R^{n}) is quasiconvex and v is a viscosity solution of the Hamilton-Jacobi equation:

H(∇v) = k in U,

then for δ > 0 and ϵ > 0 sufficiently small, the inf-convolution of v, v_{ϵ} satisfies

H(∇v_{ϵ}) = k a.e. in U_{δ} = {x ∈ U| d(x, ∂U) > δ}.  (1.15)

In particular, by [ACJ], C_{k,ϵ}^{H} is a viscosity supersolution of (1.14).

Note that Sard’s theorem and H3 imply that regular values of H are dense. Therefore we can choose a regular value k of H, i.e. H_{p}(∇C_{k,ϵ}^{H}) ≠ 0. This, combined with H1 and Proposition 3.2 of [BJW1], implies

H_{p}(∇C_{k,ϵ}^{H}) · ∇C_{k,ϵ}^{H} > 0.  (1.16)

It is readily seen that, with the help of (1.16), C_{k,ϵ}^{H} is a strict viscosity supersolution of (1.14). Now the comparision between u and C_{k}^{H} follows from sligh modifications of the arguments by [ACJ] [J] (see §5 below for more details).

An interesting question we want to study in the future is: under what conditions, viscosity solutions of the Aronsson equation (1.14) are unique. The comparison principle with generalized cones for viscosity solutions of equation (1.14) has been very useful to study the C^{1}-regularity issue for equation (1.14) (see Savin [S] and [WY] for more details). Notation. |x| the Euclidean norm of x and by ⟨x, y⟩ = ∑_{i=1}^{n} x_{i}y_{i} the Euclidean inner-product. B_{r}(x_{0}) = {y ∈ R^{n} : |y − x_{0}| < r}, S_{r}(x_{0}) = ∂B_{r}(x_{0}). X ⊗ Y : B = ∑_{i,j=1}^{n} x_{i}y_{j}b_{ij} for X = (x_{1}, ⋯, x_{n}), Y = (y_{1}, ⋯, y_{n}) ∈ R^{n}, B = (b_{ij}) ∈ R^{n×n}.

For any subset V ⊂ R^{n}, we put

USC(V) = {u : V → R : u is upper semicontinuous},
LSC(V) = {u : V → R : u is lower semicontinuous}.

The paper is written as follows. In §2 we collect some basic properties of C_{k}^{H}(x, x_{0}). In §3 we study functions in the space CGC(Ω). In §4 we prove the equivalence between (a) and (b) of theorem A. In §5 we give two different proofs of theorem A: (c) implies (b).

§2. Some basic properties of C_{k}^{H}(x)

In this section, we gather together some basic properties of generalized cone functions C_{k}^{H}(x).

Proposition 2.1. Under the assumptions H1-H3. For any k ≥ 0 and x_{0} ∈ R^{n}, C_{k}^{H}(x) satisfies:

(a) C_{k}^{H}(x) ∈ W^{1,∞}(R^{n}) is convex and positively homogeneous of degree one, i.e.

C_{k}^{H}(λx) = λC_{k}^{H}(x), ∀λ > 0, x ∈ R^{n}.  (2.1)

(b) C_{k}^{H}(x) is monotonically nondecreasing with respect to k ≥ 0. Moreover, for any k_{0} ≥ 0, C_{k}^{H}(x) ↓ C_{k_{0}}^{H}(x) uniformly on any compact set K ⊂ R^{n} as k ↓ k_{0} ≥ 0.

(c) C_{k}^{H}(0) = 0, C_{k}^{H}(x) ≥ 0 for any x ∈ R^{n}, and

C_{k}^{H}(x + y) ≤ C_{k}^{H}(x) + C_{k}^{H}(y), for any x, y ∈ R^{n}.  (2.2)

(d) C_{k}^{H}(x) is a viscosity solution of the Hamilton-Jacobi equation:

H(∇u) = k, in R^{n} \ {0}.  (2.3)

Proof. (a) Since C_{k}^{H}(x) = sup_{{H(p)=k}}⟨x, p⟩ is the supremum of a family of convex functions, C_{k}^{H}(x) is convex. Moreover, H3 implies

∥∇C_{k}^{H}(x)∥_{L^{∞}(R^{n})} ≤ max_{{H(p)=k}} |p| < ∞.

For any λ > 0, we have

C_{k}^{H}(λx) = max_{{H(p)=k}}⟨λx, p⟩ = λ max_{{H(p)=k}}⟨x, p⟩ = λC_{k}^{H}(x).

(b) As H is quasiconvex, {p : H(p) ≤ k} is convex for any k ≥ 0. Therefore, by the linear programming principle, we have

C_{k}^{H}(x) = max_{{H(p)≤k}}⟨x, p⟩  (2.4)

and C_{k}^{H}(x) is monotonically nondecreasing with respect to k ≥ 0. For any k_{0} ≥ 0, H3 implies that {p : H(p) ≤ k_{0} + 1} ⊂ R^{n} is compact. Therefore, for any x ∈ R^{n} and k ≤ k_{0} + 1, there exist p_{k}(x) ∈ {p : H(p) ≤ k} such that

C_{k}^{H}(x) = ⟨x, p_{k}(x)⟩.

For k ↓ k_{0}, we can assume that there exists p_{0} ∈ {H(p) ≤ k_{0}} such that p_{k}(x) → p_{0} and

lim_{k↓k_{0}} C_{k}^{H}(x) = ⟨x, p_{0}⟩ ≤ max_{{H(p)≤k_{0}}}⟨x, p⟩ = C_{k_{0}}^{H}(x).

Therefore lim_{k↓k_{0}} C_{k}^{H}(x) = C_{k_{0}}^{H}(x) for any x ∈ R^{n}. This, combined with the bound of ∥∇C_{k}^{H}(x)∥_{L^{∞}(R^{n})}, implies the uniform convergence.

(c) This follows from the definition of C_{k}^{H}(x) and 0 ∈ {p : H(p) ≤ k} for any k ≥ 0.

(d) We first prove that C_{k}^{H}(x) is a viscosity subsolution of (2.3) in R^{n} \ {0} by

Claim. If C_{k}^{H}(x) is differentiable at x_{1} ∈ R^{n} \ {0}, then

H(∇C_{k}^{H}(x_{1})) ≤ k.  (2.5)

For any q ∈ R^{n}, let x(t) = x_{1} + tq for t ∈ R with t > 0 small. Then it follows the convexity of C_{k}^{H}(x) that

C_{k}^{H}(x(t)) − C_{k}^{H}(x_{1}) ≥ ⟨∇C_{k}^{H}(x_{1}), tq⟩.

On the other hand, by (2.2), we have

C_{k}^{H}(x(t)) − C_{k}^{H}(x_{1}) ≤ C_{k}^{H}(x(t))

= max_{{H(p)≤k}}⟨x(t), p⟩ = t max_{{H(p)≤k}}⟨q, p⟩.

Therefore we have

⟨∇C_{k}^{H}(x_{1}), q⟩ ≤ max_{{H(p)≤k}}⟨p, q⟩, ∀q ∈ R^{n}.

It follows from the convexity of {H(p) ≤ k} that ∇C_{k}^{H}(x_{1}) ∈ {p : H(p) ≤ k}, i.e. H(∇C_{k}^{H}(x_{1})) ≤ k.

Since C_{k}^{H}(x) is Lipschitz continuous, C_{k}^{H}(x) is differentiable for a.e. x ∈ R^{n}. Therefore we have that for a.e. x ∈ R^{n} \ {0}, H(∇C_{k}^{H}(x)) ≤ k. This, combined with H2, standard mollification and the stability of viscosity subsolutions of (2.3), implies that C_{k}^{H}(x) is a viscosity subsolution of (2.3) on R^{n} \ {0} (see also §3 below).

To see that C_{k}^{H}(x) is also a viscosity supersolution of (2.3). Let x_{1} ≠ 0 and ϕ ∈ C^{1}(R^{n}) touch C_{k}^{H}(x) at x_{1} from above. Then, for any 0 < t < 1, we have

ϕ(x_{1}) − ϕ(tx_{1} + (1 − t)x_{0}) ≥ C_{k}^{H}(x_{1}) − C_{k}^{H}(tx_{1} + (1 − t)x_{0})

= (1 − t) max_{{H(p)≤k}}⟨x_{1}, p⟩.

Therefore we have

⟨∇ϕ(x_{1}), x_{1}⟩ ≥ max_{{H(p)≤k}}⟨x_{1}, p⟩.

This implies that ∇ϕ(x_{1}) ∉ {H(p) < k}, i.e. H(∇ϕ(x_{1})) ≥ k.

We end this section with an important fact about the inf-convolution of C_{k}^{H}(x). Recall that, for any ψ ∈ C(Ω), the inf-convolution ψ_{ϵ} is defined by

ψ_{ϵ}(x) = inf_{y∈\overline{Ω}}\{ψ(y) + \frac{1}{2ϵ}|x − y|^{2}\}, ∀x ∈ Ω.

It is well-known that ψ_{ϵ} is semi-concave (i.e. ∇^{2}ψ_{ϵ} ≤ Cϵ^{−2}I_{n} in the sense of distribution) (cf. [CIL]). In particular, ψ_{ϵ} is Lipschitz continuous in Ω.

Lemma 2.2. Denote w(x) = C_{k}^{H}(x) and let w_{ϵ} : Ω → R be its inf-convolution. If w_{ϵ} is differentiable at x_{1} ∈ Ω, then

H(∇w_{ϵ}(x_{1})) = k.  (2.6)

In particular, for a.e. x ∈ Ω, we have H(∇w_{ϵ}(x)) = k.

Proof. Set p_{ϵ} = ∇w_{ϵ}(x_{1}). Then for any δ > 0 there exists r_{0} > 0 such that

w_{ϵ}(x) > w_{ϵ}(x_{1}) + ⟨p_{ϵ}, x − x_{1}⟩ − δ(\sqrt{2} − 1)r_{0}  (2.7)

for any |x − x_{1}| ≤ r_{0}.

Let y_{1} ∈ Ω be such that w_{ϵ}(x_{1}) = w(y_{1}) + \frac{1}{2ϵ}|x_{1} − y_{1}|^{2}. Then for any x ∈ ∂B_{r_{0}}(x_{1}) and y ∈ Ω, we have

w(y) + \frac{1}{2ϵ}|x − y|^{2} > w(y_{1}) + \frac{1}{2ϵ}|x_{1} − y_{1}|^{2} + ⟨p_{ϵ}, x − x_{1}⟩ − δ(\sqrt{2} − 1)r_{0}.  (2.8)

For any y ∈ ∂B_{r_{0}}(y_{1}), by substituting x = x_{1} − y_{1} + y ∈ ∂B_{r_{0}}(x_{1}) into (2.7), we have

w(y) > ϕ_{ϵ}(y) ≡ w(y_{1}) + ⟨p_{ϵ}, y − y_{1}⟩ − δ(\sqrt{r_{0}^{2} + |y − y_{1}|^{2}} − r_{0}), ∀y ∈ ∂B_{r_{0}}(y_{1}). (2.9)

On the other hand, we have w(y_{1}) = ϕ_{ϵ}(y_{1}). Therefore, there exists y_{∗} ∈ B_{r_{0}}(y_{1}) such that

ϕ_{ϵ}(y_{∗}) − w(y_{∗}) = max_{y∈B_{r_{0}} (y_{1})}(ϕ_{ϵ}(y) − w(y)).  (2.10)

Since w is a viscosity supersolution of (2.3), we have

H(∇ϕ_{ϵ}(y_{∗})) ≥ k.

Direct calculations imply that

∇ϕ_{ϵ}(y_{∗}) = p_{ϵ} − \frac{δ(y_{∗} − y_{1})}{\sqrt{r_{0}^{2} + |y_{∗} − y_{1}|^{2}}} → ∇w_{ϵ}(x_{1}), as δ → 0.  (2.11)

Therefore we have H(∇w_{ϵ}(x_{1})) ≥ k.

To see the other direction, observe that (2.10) and (2.2) imply that for 0 < r_{1} < r_{0} − |y_{∗} − y_{1}| we have

ϕ_{ϵ}(y) − ϕ_{ϵ}(y_{∗}) ≤ w(y) − w(y_{∗}) ≤ C_{k}^{H}(y, y_{∗}), ∀y ∈ B_{r_{1}}(y_{∗}).  (2.12)

For any q ∈ R^{n} and t > 0 sufficiently small, we have y = y_{∗} + tq ∈ B_{r_{1}}(y_{∗}) and obtain

\frac{ϕ_{ϵ}(y_{∗} + tq) − ϕ_{ϵ}(y_{∗})}{t} ≤ max_{\{H(p)≤k\}}⟨p, q⟩.

By taking t into zero, this implies

⟨∇ϕ_{ϵ}(y_{∗}), q⟩ ≤ max_{{H(p)≤k}}⟨p, q⟩, ∀q ∈ R^{n}.

Similar to the proof of the Claim in the proposition 2.1, this implies H(∇w_{ϵ}(y_{∗})) ≤ k. Therefore, by (2.11), we obtain H(∇w_{ϵ}(x_{1})) ≤ k. The proof is complete.

§3. Some basic properties of functions in CGC(Ω)

In this section, we establish some important properties of functions in CGC(Ω).

Proposition 3.1. Under the assumptions H1-H3. If u ∈ C(Ω) ∩ CGCA(Ω), then u ∈ Wloc^{1,∞}(Ω), and for any x_{0} ∈ Ω and 0 < r < d(x_{0}, ∂Ω),

S_{r}^{+}(H, u, x_{0}) = inf{k ≥ 0 | u(x) ≤ u(x_{0}) + C_{k}^{H}(x − x_{0}), ∀x ∈ ∂B_{r}(x_{0})}  (3.1)

exists and is monotonically nondecreasing with respect to r > 0. In particular,

S^{+}(H, u, x_{0}) = lim_{r→0^{+}} S_{r}^{+}(H, u, x_{0})  (3.2)

exists for any x_{0} ∈ Ω and is upper semicontinuous.

Proof. First we claim that there exists 0 < k_{0} < ∞ such that

u(x) ≤ u(x_{0}) + C_{k_{0}}^{H}(x − x_{0}), ∀x ∈ ∂B_{r}(x_{0}).

Let k_{x_{0},r} = max_{\{|p|≤max_{x∈Br (x_{0})} \frac{|u(x)−u(x_{0})|}{r} \}} H(p). Then 0 < k_{x_{0},r} < ∞. Moreover, for any x ∈ ∂B_{r}(x_{0}), we have

u(x) − u(x_{0}) ≤ max_{z∈∂B_{r}(x_{0})}⟨x − x_{0}, \frac{u(z) − u(x_{0})}{r^{2}}(z − x_{0})⟩

≤ max_{{H(p)≤k_{x_{0},r}}}⟨x − x_{0}, p⟩ = C_{k_{x_{0},r}}^{H}(x − x_{0}).

This implies 0 ≤ S_{r}^{+}(H, u, x_{0}) ≤ k_{x_{0},r} < +∞. If u ∈ CGCA(Ω), then for any x ∈ B_{r}(x_{0}), we have

u(x) ≤ u(x_{0}) + C_{k}^{H}(x − x_{0}), with k = S_{r}^{+}(H, u, x_{0}).

This implies that for any 0 < r_{1} < r and x ∈ ∂B_{r_{1}}(x_{0}), we have

u(x) ≤ u(x_{0}) + C_{k}^{H}(x − x_{0}).

Therefore S_{r_{1}}^{+}(H, u, x_{0}) ≤ S_{r}^{+}(H, u, x_{0}) and S_{r}^{+}(H, u, x_{0}) is monotonically nondecreasing. In particular, S^{+}(H, u, x_{0}) exists for any x_{0} ∈ Ω and is upper semicontinuous.

For u ∈ C(Ω), we know that k_{x,r} is bounded from above for x ∈ Ω and 0 < r < d(x, ∂Ω). Therefore we have

|u(x) − u(y)| ≤ max{ max_{{H(p)≤k_{x,r}}} |p|, max_{{H(p)≤k_{y,r}}} |p|}|x − y|,

for |x − y| ≤ r and max(d(x, ∂Ω), d(y, ∂Ω)) < r. This clearly implies that u is locally Lipschitz continuous.

From now on, we denote Ĥ(p) = H(−p) for any p ∈ R^{n}. For function in CGCB(Ω), we have

Proposition 3.2. Under the assumptions H1-H3. If u ∈ C(Ω) ∩ CGCB(Ω), then u ∈ Wloc^{1,∞}(Ω), and for any x_{0} ∈ Ω and 0 < r < d(x_{0}, ∂Ω),

S_{r}^{−}(H, u, x_{0}) := S_{r}^{+}(Ĥ, −u, x_{0})
= inf{k ≥ 0 | − u(x) ≤ −u(x_{0}) + CĤ ; k(x − x_{0}), ∀x ∈ ∂B_{r}(x_{0})}  (3.3)

exists and is monotonically nondecreasing with respect to r. In particular,

S^{−}(H, u, x_{0}) = lim_{r→0^{+}} S_{r}^{−}(H, u, x_{0})  (3.4)

exists for any x_{0} ∈ Ω and is upper semicontinuous.

Proof. It is identical to that of Proposition 3.1. We omit it here.

Proposition 3.3. Under the assumptions H1-H3. If u ∈ C(Ω) ∩ CGC(Ω), then u ∈ Wloc^{1,∞}(Ω). Moreover, for any x_{0} ∈ Ω, if ∇u(x_{0}) exists, then we have

H(∇u(x_{0})) = S^{+}(H, u, x_{0}) = S^{−}(H, u, x_{0}).  (3.5)

Proof. Denote S_{r}^{+}(H, u, x_{0}) (or S^{+}(H, u, x_{0}), resp.) by S_{r}^{+} (or S^{+} resp.). For any 0 < r < d(x_{0}, ∂Ω), Proposition 3.1 implies

u(x) ≤ u(x_{0}) + C_{S_{r}^{+}}^{H}(x − x_{0}), ∀x ∈ B_{r}(x_{0}).

For any q ∈ R^{n} and 0 < t < 1 small, let x(t) = x_{0} + tq. Then x(t) ∈ B_{r}(x_{0}) and

u(x(t)) − u(x_{0}) ≤ C_{S_{r}^{+}}^{H}(x(t) − x_{0}) = max_{{H(p)≤S_{r}^{+}}}⟨x(t) − x_{0}, p⟩ = t max_{{H(p)≤S_{r}^{+}}}⟨p, q⟩.

Therefore we have

⟨∇u(x_{0}), q⟩ = lim_{t→0^{+}} \frac{u(x(t)) − u(x_{0})}{t} ≤ max_{\{H(p)≤S_{r}^{+}\}}⟨p, q⟩.  (3.6)

Since {H(p) ≤ S_{r}^{+}} is convex, this implies ∇u(x_{0}) ∈ {H(p) ≤ S_{r}^{+}} i.e. H(∇u(x_{0})) ≤ S_{r}^{+}. For r > 0 is arbitrary, we conclude H(∇u(x_{0})) ≤ S^{+}.

On the other hand, for any h > 0 small, there exists x_{h} ∈ ∂B_{h}(x_{0}) such that

u(x_{h}) − u(x_{0}) = C_{S_{h}^{+}}^{H}(x_{h} − x_{0}) ≥ C_{S^{+}}^{H}(x_{h} − x_{0}) = max_{{H(p)≤S^{+}}}⟨x_{h} − x_{0}, p⟩.  (3.7)

By taking a subsequence, we may assume that there is |q_{0}| = 1 so that lim_{h→0^{+}} \frac{x_{h}−x_{0}}{h} = q_{0}. Then (3.7) implies

⟨∇u(x_{0}, q_{0}⟩ ≥ max_{{H(p)≤S^{+}}}⟨p, q_{0}⟩.  (3.8)

This, combined with the convexity of {H(p) ≤ S^{+}} implies ∇u(x_{0}) ∉ {H(p) < S^{+}}, i.e.H(∇u(x_{0})) ≥ S^{+}.

Since −u ∈ CGCA(Ω) with respect to Ĥ, the same argument gives

S^{−}(H, u, x_{0}) = Ĥ(−∇u(x_{0})) = H(∇u(x_{0})).  (3.9)

This completes the proof of Proposition 3.3.

Finally, we need Proposition 3.4. Under the assumptions H1-H3. Assume u ∈ C(Ω) ∩ CGCA(Ω). For any x_{0} ∈ Ω and 0 < r < d(x_{0}, ∂Ω), if x_{r} ∈ ∂B_{r}(x_{0}) satisfies

u(x_{r}) − u(x_{0}) = C_{S_{r}^{+}(H,u,x_{0})}^{H}(x_{r} − x_{0})  (3.10)

then for any 0 < R < d(x_{r}, ∂Ω) we have

S_{R}^{+}(H, u, x_{r}) ≥ S_{r}^{+}(H, u, x_{0}).  (3.11)

Proof. Let S_{r}^{+}(x) = S_{r}^{+}(H, u, x). For θ ∈ (0, 1), let x^{θ} = θx_{0} + (1 − θ)x_{r}. Then we have

C_{S_{r}^{+}(x_{0})}^{H}(x^{θ} − x_{0}) = max_{{H(p)≤S_{r}^{+}(x_{0})}}⟨x^{θ} − x_{0}, p⟩

= (1 − θ) max_{{H(p)≤S_{r}^{+}(x_{0})}}⟨x_{r} − x_{0}, p⟩

= (1 − θ)C_{S_{r}^{+}(x_{0})}^{H}(x_{r} − x_{0}).

Hence, by (3.10), we have

u(x^{θ}) ≤ u(x_{0}) + C_{S_{r}^{+}(x_{0})}^{H}(x^{θ} − x_{0})
= u(x_{0}) + (1 − θ)C_{S_{r}^{+}(x_{0})}^{H}(x_{r} − x_{0})
= u(x_{r}) − θC_{S_{r}^{+}(x_{0})}^{H}(x_{r} − x_{0}).  (3.12)

Let θ > 0 be such that R > 2θr. Since

u(y) ≤ u(x^{θ}) + C_{S_{R−θr}^{+}(x^{θ})}^{H}(y − x^{θ}), ∀y ∈ B_{R−θr}(x^{θ}),

and |x_{r} − x^{θ}| = θr < R − θr, we have

u(x_{r}) ≤ u(x^{θ}) + C_{S_{R−θr}^{+}(x^{θ})}^{H}(x_{r} − x^{θ}).  (3.13)

Therefore (3.12) and (3.13) imply

C_{S_{r}^{+}(x_{0})}^{H}(x_{r} − x_{0}) ≤ C_{S_{R−θr}^{+}(x^{θ})}^{H}(x_{r} − x_{0}).

This implies

S_{r}^{+}(x_{0}) ≤ S_{R−θr}^{+}(x^{θ}).  (3.14)

It is easy to see

lim_{θ→0^{+}} S_{R−θr}^{+}(x^{θ}) ≤ S_{R}^{+}(x_{r}).  (3.15)

This implies (3.11) and the proof is complete.

§4. Equivalence between CGC(Ω) and absolute minimizers

This section is devoted to the proof of theorem A: (a) is equivalent to (b).

Lemma 4.1. Under the assumptions H1-H3. Any absolute minimizer u ∈ Wloc^{1,∞}(Ω) of F (v, ·) = ∥H(∇u)∥_{L^{∞}(·)} is in CGC(Ω).

Proof. Suppose that the conclusion were false. Then there exist x_{0} ∈ Ω, an open subset U ⊂⊂ Ω with x_{0} ∉ U, and k ≥ 0 such that

u(x) = u(x_{0}) + C_{k}^{H}(x − x_{0}) ∀x ∈ ∂U; u(x) > u(x_{0}) + C_{k}^{H}(x − x_{0}), ∀x ∈ U. (4.1)

By the absolute minimality of u and Proposition 2.1, we have

∥H(∇u)∥_{L^{∞}(U)} ≤ ∥H(∇C_{k}^{H}(·, x_{0}))∥_{L^{∞}(U)} = k.  (4.2)

Claim. u is a viscosity subsolution of H(∇ϕ) = k, in U.

To see this, let ϵ > 0 be small and u_{ϵ} ∈ C^{∞}(U_{ϵ}) be the standard ϵ-mollification of u, here U_{ϵ} = {x ∈ U : dist(x, ∂U) ≥ ϵ}. Then u_{ϵ} converges uniformly to u on U_{ϵ}. Moreover, by H2, we have

H(∇u_{ϵ}(x)) ≤ ∥H(∇u)∥_{L^{∞}(U)} ≤ k, ∀x ∈ U_{ϵ}.  (4.3)

This implies that u_{ϵ} is a viscosity subsolution of H(∇ϕ) = k in U_{ϵ}. Therefore the stabiliity theorem of viscosity subsolutions (cf. CIL]) yields that u is a viscosity subsolution of H(∇ϕ) = k in U.

Since Proposition 2.1 implies that C_{k+δ}^{H}(x − x_{0}) converges uniformly to C_{k}^{H}(x − x_{0}) on U as δ ↓ 0, there exist δ_{0} > 0 and x̄ ∈ U such that

u(x) ≤ u(x_{0}) + C_{k+δ_{0}}^{H}(x − x_{0}), ∀x ∈ ∂U  (4.4)
u(x̄) > u(x_{0}) + C_{k+δ_{0}}^{H}(x̄ − x_{0}).  (4.5)

It also follows from Proposition 2.1 that C_{k+δ_{0}}^{H}(x − x_{0}) is a viscosity solution of

H(∇ϕ) = k + δ_{0} in U.  (4.6)

In particular, C_{k+δ_{0}}^{H}(x − x_{0}) is a strict, supersolution of H(∇ϕ) = k in U. Therefore, the comparison principle (cf. [CIL]) implies that

u(x) ≤ C_{k+δ_{0}}^{H}(x − x_{0}), ∀x ∈ U.  (4.7)

This contradicts (4.5) and the proof is complete.

Now we want to show

Lemma 4.2. Under the assumptions H1-H3. Suppose that u ∈ C(Ω) ∩ CGC(Ω). Then u ∈ Wloc^{1,∞}(Ω) is an absolute minimizer of F (v, ·) = ∥H(∇v)∥_{L^{∞}(·)}.

Proof. Suppose that it were false. Then there exist an open subset V ⊂⊂ Ω, v ∈ W_{u}^{1,∞}(V), and δ > 0 such that

∥H(∇v)∥_{L^{∞}(V)} + 2δ ≤ ∥H(∇u)∥_{L^{∞}(V)}.  (4.8)

Therefore there exists x_{0} ∈ V such that ∇u(x_{0}) exists, and

H(∇u(x_{0})) ≥ ∥H(∇v)∥_{L^{∞}(V)} + δ.  (4.9)

Denote S_{r}^{+}(H, u, x_{0}) (or S^{−}r(H, u, x_{0}) resp.) by S_{r}^{+}(x_{0}) (or S_{r}^{−}(x_{0}) resp.). It follows from Proposition 3.5 that for any 0 < r < d(x_{0}, ∂V), we have

S_{r}^{+}(x_{0}) ≥ ∥H(∇v)∥_{L^{∞}(V)} + δ, S_{r}^{−}(x_{0}) ≥ ∥H(∇v)∥_{L^{∞}(V)} + δ.  (4.10)

Similar to [CEG] theorem 3.2, we iteratively define a doubly infinite sequence {x_{j}}_{−∞}^{∞} ⊂ V such that

|x_{j} − x_{j−1}| = d(x_{j−1}, ∂V)

and u(x_{j}) − u(x_{j−1}) = C_{S_{|x_{j} −x_{j−1}|}^{+}}^{H}(x_{j} − x_{j−1}) ≥ C_{S^{+}(x_{0})}^{H})(x_{j} − x_{j−1}) ∀j ≥ 1,

u(x_{j}) − u(x_{j−1}) = CĤ ; S_{|x_{j} −x_{j−1}|}^{−}(x_{j} − x_{j−1}) ≥ CĤ ; S^{−}(x_{0})(x_{j} − x_{j−1}) ∀j ≤ 0.

Therefore we have

u(x_{l}) − u(x_{−l}) ≥ ∑_{j=0}^{l−1} C_{S^{+}(x_{0})}^{H}(x_{j+1} − x_{j}) + ∑_{j=−l}^{−1} CĤ ; S^{−}(x_{0})(x_{j+1} − x_{j}).

It follows that there are x_{+∞}, x_{−∞} ∈ ∂V such that

lim_{l→+∞} x_{l} = x_{+∞}, lim_{l→+∞} x_{−l} = x_{−∞}.

Then we have

v(x_{+∞}) − v(x_{−∞}) = u(x_{+∞}) − u(x_{−∞})

≥ ∑_{j=0}^{+∞} C_{S^{+}(x_{0})}^{H}(x_{j+1} − x_{j}) + ∑_{j=−∞}^{−1} CĤ ; S^{−}(x_{0})(x_{j+1} − x_{j}).

On the other hand, we have

v(x_{+∞}) − v(x_{−∞}) = ∑_{j=0}^{+∞}(v(x_{j+1}) − v(x_{j})) + ∑_{j=−∞}^{−1}(v(x_{j+1}) − v(x_{j}))

≤ ∑_{j=0}^{+∞} C_{∥H(∇v)∥_{L^{∞}(V )}}^{H}(x_{j+1} − x_{j}) + ∑_{j=−∞}^{−1} CĤ ; ∥H(∇v)∥_{L^{∞}(V )}(x_{j+1} − x_{j}).

Hence we have used the fact: for any x, y ∈ V if {tx + (1 − t)y : 0 ≤ t ≤ 1} ⊂ V then

v(x) − v(y) = ⟨∫_{0}^{1} ∇v(tx + (1 − t)y), x − y⟩ ≤ ∫_{0}^{1} max_{{H(p)≤∥H(∇v)∥_{L^{∞}(V )}}}⟨p, x − y⟩
= C_{∥H(∇v)∥_{L^{∞}(V )}}^{H}(x − y).  (4.11)

Therefore we may assume that there is a nonnegative j_{0} such that

C_{S^{+}(x_{0})}^{H}(x_{j_{0}+1} − x_{j_{0}}) ≤ C_{∥H(∇v)∥_{L^{∞}(V )}}^{H}(x_{j_{0}+1} − x_{j_{0}}).  (4.12)

This implies S^{+}(x_{0}) = H(∇u(x_{0})) ≤ ∥H(∇v)∥_{L^{∞}(V)}, which contradicts with (4.9). The proof is complete.

§4. Viscosity solutions of the Aronsson equation are in CGC(Ω)

In this section, we give two different proofs of the part of theorem A: (c) implies (b). The first proof is based on the observation that, under some additional assumptions on H, generalized cones C_{k}^{H}(x) are C^{2}(R^{n} \ {0}) and thus we can use small perturbations of C_{k}^{H}(x) as test functions in the definition of viscosity solutions. For a general H, we don’t have this C^{2}-regularity of C_{k}^{H}(x) and our second proof is based on several nontrivial modifications of the original arguments of [J] (see also [ACJ] and [Y]). The first proof is outlined in the following proposition.

Proposition 5.1. If, in addition to H1-H3, H ∈ C^{3}(R^{n}) is uniformly convex. Then any viscosity subsolution u ∈ C(Ω) to the Aronsson equation (1.14) is in CGCA(Ω).

Proof. Suppose that the conclusion were false. Then, similar to the proof of Lemma 4.1, there exist k > 0 and an open subset V ⊂⊂ Ω such that (i) k is a regular value of H, (ii) u(x) = u(x_{0}) + C_{k}^{H}(x − x_{0}) for all x ∈ ∂V, and (iii)

u(x) > u(x_{0}) + C_{k}^{H}(x − x_{0}) ∀x ∈ V.  (5.1)

Claim 1. Σ_{k} = {p ∈ R^{n} : H(p) = k} is a closed, strictly convex C^{2} hypersurface in R^{n}.

In fact, S_{k} = {p ∈ R^{n} : H(p) ≤ k} is a compact and strictly convex subset of R^{n} and Σ_{k} = ∂S_{k} is closed and strictly convex. Moreover, (i) implies

inf_{{p∈Σ_{k}}} |∇H(p)| ≥ δ_{k} > 0  (5.2)

and, by the implicit function theorem, we have that Σ_{k} is a C^{2} hypersurface.

Let ν(p) = \frac{∇H(p)}{|∇H(p)|} : Σ_{k} → S^{n−1} be the Gauss map. Then ν is 1-1 and onto. Hence the inverse Gauss map Y (x) = ν^{−1}(x) : S^{n−1} → Σ_{k} is defined and continuous. Moreover, it follows from the uniform convexity of H that there exists a δ_{0} > 0 such that ∇^{2}H(p) ≥ δ_{0}I_{n} for any p ∈ R^{n}. Using this, one can check that Y ∈ C^{2}(S^{n−1}, R^{n}).

Without loss of generality, we assume that x_{0} = 0 and u(0) = 0. Claim 2. C_{k}^{H} ∈ C^{2}(S^{n−1}).

In fact, for x ∈ S^{n−1}, C_{k}^{H}(x) = x · Y (x): by the Lagrange multiplier theorem, we have C_{k}^{H}(x) = x · p_{k}, where p_{k} satisfies H(p_{k}) = k and x = \frac{∇H(p_{k})}{|∇H(p_{k})|}, i.e. ν(p_{k}) = x or p_{k} = Y (x). Since Y ∈ C^{2}(S^{n−1}), this implies C_{k}^{H}(x) = x · Y (x) : S^{n−1} → R_{+} is in C^{2}(S^{n−1}).

It follows from (ii) that there exists a x_{1} ≠ 0 ∈ V such that

u(x_{1}) − C_{k}^{H}(x_{1}) = max_{x∈V}{u(x) − C_{k}^{H}(x)} > 0  (5.3)

Let R > 0 be sufficiently large so that V ⊂ B_{R}. Then for any small ϵ > 0, we have

u(x) ≤ C_{k}^{H}(x) + ϵ(R^{2} − |x|^{2}), ∀x ∈ ∂V,  (5.4)

and

u(x_{1}) > C_{k}^{H}(x_{1}) + ϵ(R^{2} − |x_{1}|^{2}).  (5.5)

Therefore, there exists x_{ϵ} ∈ V with |x_{ϵ} − x_{1}| ≤ \frac{1}{2}|x_{1}|, such that

u(x_{ϵ}) − Φ_{ϵ}(x_{ϵ}) = max_{x∈V}(u(x) − Φ_{ϵ}(x)) > 0  (5.6)

where Φ_{ϵ}(x) ≡ C_{k}^{H}(x) + ϵ(R^{2} − |x|^{2}). Since Φ_{ϵ} ∈ C^{2}(V), it follows from the definition of viscosity subsolutions that

H_{p}(∇Φ_{ϵ}) ⊗ H_{p}(∇Φ_{ϵ}) : ∇^{2}Φ_{ϵ}|_{x=x_{ϵ}} ≥ 0.  (5.7)

Direct calculations imply that at x = x_{ϵ} we have

∇Φ_{ϵ}(x) = ∇C_{k}^{H}(x) − 2ϵx,  (5.8)

∇_{ij}^{2}Φ_{ϵ}(x) = ∇_{ij}^{2}C_{k}^{H}(x) − 2ϵδ_{ij}, 1 ≤ i, j ≤ n.  (5.9)

Proposition 2.1 implies

H(∇C_{k}^{H}(x, x_{0})) = k, ∀x ≠ x_{0}.  (5.10)

Taking derivatives of (5.10), we get

∑_{i} H_{p_{i}}(∇C_{k}^{H}(x_{ϵ}, x_{0}))∇_{ij}^{2}C_{k}^{H}(x_{ϵ}) = 0, 1 ≤ j ≤ n.  (5.11)

Observe

|H_{p}(∇Φ_{ϵ}(x_{ϵ})) − H_{p}(∇C_{k}^{H}(x_{ϵ}))| ≤ 2ϵ|x_{ϵ}| max_{{H(p)≤k+1}} |∇^{2}H(p)| ≤ Cϵ.  (5.12)

For |x_{ϵ}| ≥ \frac{1}{2}|x_{1}|, there exists C > 0 independent of ϵ such that

|∇^{2}C_{k}^{H}(x_{ϵ})| ≤ C.  (5.13)

This, combined with (5.2), implies that for ϵ > 0 sufficiently small,

|H_{p}(∇C_{k}^{H}(x_{ϵ}) − 2ϵx_{ϵ})| ≥ \frac{δ}{2} > 0.  (5.14)

Putting these estimates together, we have

0 ≤ H_{p}(∇Φ_{ϵ}(x_{ϵ})) ⊗ H_{p}(∇Φ_{ϵ}(x_{ϵ})) : (∇^{2}C_{k}^{H}(x_{ϵ}) − 2ϵI_{n})
≤ −2ϵ|H_{p}(∇C_{k}^{H}(x_{ϵ}) − 2ϵx_{ϵ})|^{2}
+ (H_{p}(∇C_{k}^{H}(x_{ϵ})) + O(ϵ)) ⊗ H_{p}(∇C_{k}^{H}(x_{ϵ})) + O(ϵ)) : ∇^{2}C_{k}^{H}(x_{ϵ})
≤ −2ϵ|H_{p}(∇C_{k}^{H}(x_{ϵ}) − 2ϵx_{ϵ})|^{2} + O(ϵ^{2})
≤ (−δ + O(ϵ))ϵ < 0.  (5.15)

This yields the desired contradiction.

For the general case, we outline our second proof in the following proposition.

Proposition 5.2. Under the assumptions H1-H3. Any viscosity subsolution u ∈ C(Ω) of the Aronsson equation (1.14) is in CGCA(Ω).

Proof. The proof is inspired by that of [Y]. Suppose that the conclusion were false. Then, similar to the proof of proposition 5.1, there exist k_{0} > 0, x_{0} ∈ Ω, and U ⊂⊂ Ω with x_{0} ∉ U such that (i) k_{0} is a regular value of H, (ii) u(x) ≤ u(x_{0}) + C_{k_{0}}^{H}(x − x_{0}) for any x ∈ ∂U, and (iii) u(x) > u(x_{0}) + C_{k_{0}}^{H}(x − x_{0}) for any x ∈ U.

Claim 3. There exists a δ_{1} > 0 such that

inf_{{H(p)=k_{0}}}⟨∇H(p), p⟩ ≥ δ_{1}.  (5.16)

In fact, for k_{0} > 0 and H(0) = 0, there exists r_{0} > 0 such that B_{r_{0}}(0) ⊂ {H(p) < k_{0}}, i.e. max_{p∈B_{r_{0}} (0)} H(p) < k_{0}. Then, by (H2), we have that for any q ∈ B_{r_{0}}(0) and H(p) = k_{0},

H(tq + (1 − t)p) ≤ max{H(q), H(p)} = H(p), 0 ≤ t ≤ 1,

and hence

⟨−∇H(p), q − p⟩ = lim_{t↓0} \frac{H(p) − H(tq + (1 − t)p)}{t} ≥ 0.

Set q = r_{0} \frac{∇H(p)}{|∇H(p)|} ∈ B_{r_{0}}(0), we get

⟨∇H(p), p⟩ ≥ r_{0}|∇H(p)| ≥ r_{0} inf_{{H(v)=k_{0}}} |∇H(v)| ≥ r_{0}δ_{0} = δ_{1} > 0

where we have used (5.2) in the last inequality.

Denote w(x) = C_{k_{0}}^{H}(x − x_{0}) for x ∈ U. As x_{0} ∉ U, Proposition 2.1 implies that w is a viscosity solution of H(∇w) = k_{0} on U. For any ϵ > 0, let w_{ϵ} ∈ W^{1,∞}(U) be the inf-convolution of w on U. Then one can check that w_{ϵ} is both convex and semiconcave, i.e.

0 ≤ ∇^{2}w_{ϵ} ≤ 4ϵ^{−2}I_{n}  (5.17)

in the sense of distributions. In particular, for a.e. x̂ ∈ U, ∇^{2}w_{ϵ}(x̂) exists (cf. Evans-Gariepy [EG]). As ∇w_{ϵ} is continuous at x = x̂, Lemma 2.2 implies H(∇w_{ϵ}(x)) = k_{0} for any x close to x̂. Hence we conclude that

H_{p}(∇w_{ϵ}(x)) ⊗ H_{p}(∇w_{ϵ}(x)) : ∇^{2}w_{ϵ}(x) = 0, for a.e. x ∈ U.  (5.18)

In fact, it follows from the remark 6.7 of [ACJ] that w_{ϵ} is a viscosity supersolution of the Aronsson equation (1.14) in U.

For ϵ > 0, let u^{ϵ}(x) = sup_{y∈Ω}\{u(y) − \frac{1}{2ϵ}|x − y|^{2}\}, x ∈ Ω, be the sup-convolution of u. Then it is well-known (cf. [CIL]) that u^{ϵ} ∈ W^{1,∞}(Ω) is semiconvex, i.e.

∇^{2}u^{ϵ} ≥ −\frac{1}{4ϵ^{2}} I_{n}  (5.19)

in the sense of distributions. Moreover, u_{ϵ} → u uniformly on Ω as ϵ → 0. Since the Aronsson equation (1.14) has no x-dependence, it follows from [J] [CIL] [ACJ] that u^{ϵ} is again a viscosity subsolution of the Aronsson equation (1.14).

Now we want to show

sup_{x∈U}(u^{ϵ} − w_{ϵ})^{+} ≤ sup_{x∈∂U}(u^{ϵ} − w_{ϵ})^{+}.  (5.20)

Observe that for any constant c ∈ R, u^{ϵ}+c (or w_{ϵ}+c, resp.) is again a viscosity subsolution (or supersolution, resp.) of the equation (1.14), and (5.20) is unchanged with u^{ϵ} (or w_{ϵ}, resp.) replaced by u^{ϵ} + c or (w_{ϵ} + c, resp.). Therefore, without loss of generality, we may assume that w_{ϵ}(x) ≥ 1 for x ∈ U.

To prove (5.20), let m > 0 be sufficiently large and define

ϕ_{m}(t) = t + \frac{1}{m}e^{−mt}, t > 0

Then it is easy to see

ϕ′_{m}(t) = 1 − e^{−mt}, ϕ′′_{m}(t) = me^{−mt}, t > 0.

This implies that there exists a unique w_{m} : U → R such that

w_{ϵ}(x) = ϕ_{m}(w_{m}(x)), x ∈ U.

Direct calculations imply

∇^{2}w_{ϵ} = ϕ′_{m}(w_{m})∇^{2}w_{m} + ϕ′′_{m}(w_{m})∇w_{m} ⊗ ∇w_{m}
≥ ϕ′_{m}(w_{m})∇^{2}w_{m}.

Hence we have that for m > 0 sufficiently large w_{m} is semiconcave and satisfies

∇^{2}w_{m} ≤ 8ϵ^{−2}I_{n}, in U  (5.21)

in the sense of distributions.

Now we claim

sup_{x∈U}(u^{ϵ} − w_{m})^{+} ≤ sup_{x∈∂U}(u^{ϵ} − w_{m})^{+}.  (5.22)

As w_{m} → w_{ϵ} uniformly on U as m → ∞, it is clear that (5.22) implies (5.20).

We prove (5.22) by contradiction. Suppose that it were false. Then there exists x_{1} ∈ U such that

(u^{ϵ} − w_{m})(x_{1}) = max_{x∈Ū}(u^{ϵ} − w_{m}(x)).

As u^{ϵ} − w_{m} is semiconvex on U, it is well-known (cf. [J] [CIL] [ACJ]) that both ∇u^{ϵ}(x_{1}) and ∇w_{m}(x_{1}) exist, and

∇u^{ϵ}(x_{1}) = ∇w_{m}(x_{1}),

and there exist {p_{l}} ⊂ R^{n} with |p_{l}| ≤ l^{−1} and x_{l} ∈ U such that u^{ϵ}(x) − w_{m}(x) − ⟨p_{l}, x⟩ attains its local maximum at x_{l}. Furthermore, both ∇^{2}u^{ϵ}(x_{l}) and ∇^{2}w_{m}(x_{l}) exist. In particular, we have

∇u^{ϵ}(x_{l}) = ∇w_{m}(x_{l}) + p_{l}, ∇^{2}u^{ϵ}(x_{l}) ≤ ∇w_{m}^{2}(x_{l}).

Therefore we have

H_{p}(∇w_{m} + p_{l}) ⊗ H_{p}(∇w_{m} + p_{l}) : ∇^{2}w_{m}|_{x=x_{l}} ≥ 0.  (5.23)

On the other hand, since w_{ϵ} is viscosity supersolution of (1.14), we have at x_{l}

H_{p}(∇w_{ϵ}) ⊗ H_{p}(∇w_{ϵ}) : ∇^{2}w_{m} ≤ −\frac{ϕ′′_{m}(w_{m})}{(ϕ′_{m}(w_{m}))^{3}} ⟨H_{p}(∇w_{ϵ}), ∇w_{ϵ}⟩^{2}

≤ −\frac{me^{−mw_{m}}}{1 − e^{−mw_{m}}} δ_{1}.

Moreover we have

−4ϵ^{−2}I_{n} ≤ ∇^{2}w_{m}(x_{l}) ≤ 8ϵ^{−2}I_{n}.  (5.24)

At x_{l}, we also have

|H_{p}(∇w_{ϵ}) − H_{p}(∇w_{m} + p_{l})| ≤ C|(ϕ′_{m}(w_{m}) − 1)∇w_{m}(x_{l}) − p_{l}|
≤ C(l^{−1} + e^{−mw_{m}})

where we have used

|∇w_{m}(x_{l})| ≤ C|∇w_{ϵ}(x_{l})| ≤ C max{|p| : H(p) = k}.

Therefore putting all these estimates together we have

C(l^{−1} + e^{−mw_{m}})(x_{l}) ≥ me^{−mw_{m}}(x_{l})δ_{1}.

Sending l to infinity, this implies

mδ_{1} ≤ C.

This is impossible if m is sufficiently large. We get the desired contradiction and the proof is therefore complete.

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