Approximation of Lipschitz Functions Preserving Boundary Values

Given an open subset of a Banach space and a Lipschitz real-valued function defined on its closure, we study whether it is possible to approximate this function uniformly by Lipschitz functions having the same Lipschitz constant and preserving the values of the initial function on the boundary of the open set, and which are k times continuously differentiable on the open. A consequence of our result is that every 1-Lipschitz function defined on the closure of an open subset of a finite-dimensional normed space of dimension greater than one, and such that the Lipschitz constant of its restriction to the boundary is less than 1, can be uniformly approximated by differentiable 1-Lipschitz functions preserving the values of the initial function on the boundary of the open set, and such that its derivative has norm one almost everywhere on the open. This result does not hold in general without assumption on the Lipschitz constant of the restriction of the initial function to the boundary.


Introduction
It is no doubt useful to be able to approximate Lipschitz functions by smooth Lipschitz functions preserving the Lipschitz constants as much as possible in Banach spaces. The Lasry-Lions method [1] provides uniform approximation of Lipschitz functions by differentiable Lipschitz functions with Lipschitz derivatives without increasing the Lipschitz constant of the initial functions, in Hilbert spaces. This result was extended to a wider class of functions in [2]. As a consequence of the main theorem of [3], one can obtain approximation of locally Lipschitz functions by smooth locally Lipschitz functions on open subsets of a finite-dimensional space, in which the approximation can be taken with locally Lipschitz constants arbitrarily close to the original locally Lipschitz constants, and such that the approximating function is continuously close to the initial function. In [4], it was proved a similar result for separable (possibly) infinite-dimensional Riemannian manifolds. In more general Banach space, the results in [5][6][7] yield Lipschitz and smooth approximation of Lipschitz functions, in which the Lipschitz constant of the approximating function is controlled by the Lipschitz constant of the initial function, up to factor which only depends on the space and is bigger than 1 in general.
In this paper, we study approximation of Lipschitz functions defined on open subsets of Banach spaces by smooth functions preserving both the Lipschitz constant and the boundary value of the initial function. In addition, we study approximation of Lipschitz functions defined on the closure of an open subset of a finite-dimensional space by almost classical solutions of the Eikonal equation introduced in [8], with a Lipschitz boundary value; i.e., 1-Lipschitz differentiable functions, which satisfy the Eikonal equation almost everywhere and coincide with a given 1-Lipschitz boundary data.

Description of the Main Results
Throughout this paper, for every metric space (E, d) and every function f : E → R, we will denote the Lipschitz constant or Lipschitz rate of f on E by Lip( f , E), that is, Also, if λ ≥ 0, we will say that f : E → R is λ-Lipschitz on E whenever | f (x) − f (y)| ≤ λd(x, y) for every x, y ∈ E. We will denote by B[x 0 , r ] the closed ball centered at x 0 and with radius r > 0 with respect to the metric on E. The closure of any subset A will be denoted by cl(A). Finally, for any Banach space X with norm · , the dual norm on X * will be denoted by · * . In this paper, we deal with the following problem. In finite-dimensional spaces, the integral convolution with mollifiers provides uniform approximation by C ∞ functions preserving the Lipschitz constant of the function to be approximated. However, this approximation does not necessarily preserve the value of u 0 on ∂Ω. On the other hand, it was proved in [3, Theorem 2.2] an approximation theorem for locally Lipschitz functions defined on open subsets of R n which implies that for any continuous function δ : Ω →]0, + ∞[, and any locally Lipschitz function u 0 there exists a function v of class C ∞ satisfying (among other properties) that Using the above result with δ(x) = min{ε, dist(x, ∂Ω)}, we get a smooth Lipschitz approximation v of u 0 that extends continuously to cl(Ω) by setting v = u 0 on ∂Ω. The function v has Lipschitz constant arbitrarily close to Lip(u 0 , cl(Ω)), but bigger than Lip(u 0 , cl(Ω)) in general. Thus, this does not yield any answer to Problem 2.1. In the infinite-dimensional case, it was proved in [4, Theorem 1] that every Lipschitz function defined on an open subset Ω of a separable Hilbert space (or even a separable infinite-dimensional Riemannian manifold) can be approximated in the C 0 -fine topology by C ∞ functions whose Lipschitz constant can be taken to be arbitrarily close to the Lipschitz constant of u 0 , i.e., for any given continuous function δ : One can find in [5][6][7] some results on approximation of Lipschitz functions by C ksmooth Lipschitz functions in more general Banach spaces. In these results, the approximating function preserves the Lipschitz constant of the original function up to a factor C 0 ≥ 1, which only depends on the space and is bigger than 1 in general. In this paper, we show that the answer to Problem 2.1 depends on the relation between Lip(u 0 , ∂Ω) and Lip(u 0 , cl(Ω)). Let us now state our main results in this direction. Theorem 2.1 Let X be a finite-dimensional normed space, or a separable Hilbert space or the space c 0 (Γ ), for an arbitrary set of indices Γ . Let Ω be an open subset of X and let u 0 : cl(Ω) → R be a Lipschitz function such that Lip(u 0 , ∂Ω) < Lip(u 0 , cl(Ω)).
For nonseparable Hilbert spaces, we have the following. Theorem 2.2 Let X be a Hilbert space. Let Ω be an open subset of X and let u 0 : cl(Ω) → R be a Lipschitz function such that Lip(u 0 , ∂Ω) < Lip(u 0 , cl(Ω)). Given ε > 0, there exists a function v : cl(Ω) → R such that v is of class C 1 (Ω), v is Lipschitz on cl(Ω) with Lip(v, cl(Ω)) ≤ Lip(u 0 , cl(Ω)), v = u 0 on ∂Ω and |u 0 − v| ≤ ε on cl(Ω). Theorems 2.1 and 2.2 give a positive answer to Problem 2.1 for the C 1 (Ω) or C ∞ (Ω) class, when Lip(u 0 , ∂Ω) < Lip(u 0 , cl(Ω)), in certain Banach spaces. These theorems will be proved by combining approximation techniques in the pertinent space with the following result. Theorem 2.3 Let k ∈ N ∪ {∞} and let X be a Banach space with the property that for every Lipschitz function f : X → R and every η > 0, there exists a function g : In Sect. 6, we will see an example on R 2 with the 1 norm showing that Problem 2.1 has a negative answer (even for the class of functions which are merely differentiable on Ω) if we allow Lip(u 0 , ∂Ω) = Lip(u 0 , cl(Ω)). Therefore, one can say that Theorem 2.1 is optimal (in the sense of Problem 2.1), at least in the setting of finite-dimensional normed spaces. We now consider a subproblem of Problem 2.1 when X is a finite-dimensional normed space. Observe that, if w = u 0 on ∂Ω and Lip(u 0 , ∂Ω) < 1, then the Mean Value Theorem yields the existence of x ∈ Ω such that Dw(x) * < 1. Therefore, the function w (if it exists) has no continuous derivative in this case. The following theorem gives a positive answer to Problem 2.2 when Lip(u 0 , ∂Ω) < 1.
In Sect. 6, we prove, using the theory of absolutely minimizing Lipschitz extensions, that if Ω is an open subset in a 2-dimensional Euclidean space and if u 0 : ∂Ω → R is a 1-Lipschitz function, then there exists a differentiable 1-Lipschitz function w : cl(Ω) → R such that Dw * = 1 almost everywhere on Ω and w = u 0 on ∂Ω. However, Example 6.1 shows that Problem 2.2 may have a negative answer if we drop the hypothesis Lip(u 0 , ∂Ω) < 1. Observe that Theorem 2.4 covers the case of homogeneous Dirichlet conditions. Also, we notice that the above theorem does not hold when X = R. Indeed, if u 0 : [0, 1] → R is 1-Lipschitz and differentiable on ]0, 1[, with |u 0 (1) − u 0 (0)| < 1, then a result of A. Denjoy [9] tells us that either {x : |u 0 (x)| < 1} is empty or else it has positive Lebesgue measure. But this subset is nonempty by the Mean Value Theorem. The contents of the paper are as follows. In Sect. 3, we show that in general metric spaces, one can approximate a Lipschitz function u 0 by a function which coincides with u 0 on a given subset and has, on bounded subsets, better Lipschitz constants. In Sect. 4, we will give the proof of Theorems 2.3, 2.1 and 2.2 with the decisive help of the above result. In Sect. 5, we use Theorem 2.1 and the results in [10] to prove Theorem 2.4. Finally, in Sect. 6, we consider the case Lip(u 0 , ∂Ω) = Lip(u 0 , cl(Ω)): although a partial positive result in the Euclidean setting can be obtained, we show that Problem 2.1 does not always have a positive answer in this limiting case.

Approximation by Functions with Smaller Lipschitz Constants
Throughout this section, all the sets involved are considered to be subsets of a metric space (X , d) and all the Lipschitz constants are taken with respect to the distance d.
The following result will be very useful in Sect. 4, and it is interesting in itself.

Theorem 3.1 Let E and F be two nonempty closed sets such that F
there exists a function u : E → R such that |u − u 0 | ≤ ε on E, u = u 0 on F and u has the property that Lip(u, B) < K for every bounded subset B of E.
A crucial step for proving the above theorem is the following lemma. For any two nonempty subsets A and B of X and for any x ∈ X , we will denote

Lemma 3.1 Let E and F be two nonempty closed subsets such that F ⊂ E and E\F is bounded. Let u
Proof In the case when E\F = ∅, we have that E = F and then it is enough to take u λ = u μ . From now on, we assume that E\F = ∅, we fix μ < λ < 1, and we denote ε λ = ε(λ, μ, E, F). We now define the strategy of proof of the lemma. We first show that the family is nonempty, and then we define the function u λ by: In order to prove that the function u λ is the required solution, it will be enough to check that u λ ∈ C λ and that u 0 ≤ u λ + δ + ε λ on E. 1. We now prove that the family C λ is nonempty. Consider the function and let us see In the case when In the case when which in turn implies Using first that u 0 is 1-Lipschitz on E and then (3) and (2), we obtain Hence, in both cases, we have that The function u λ belongs to C λ because a supremum of λ-Lipschitz functions is a λ-Lipschitz function, and because inequalities and equalities are preserved by taking supremum. Before proving the inequality Observe that, since u μ ≤ u 0 +δ on F, S λ and F are disjoint. Since u λ is λ-Lipschitz on E (and, in particular, on F ∪ S λ ), the function v λ is the greatest λ-Lipschitz extension of u λ from the set F ∪ S λ . Thus v λ = u λ on F ∪ S λ and u λ ≤ v λ on E. Hence, by (1), we will have that v λ = u λ as soon as we see that v λ ≤ u 0 + δ + ε λ on E. Let us define Assume that G λ = ∅. Since E\F is bounded, then v λ − u 0 is bounded on G λ and we can define We next define the function The function w λ is λ-Lipschitz on E and satisfies the following.
From the remarks (i), (ii) and (iii) above we obtain that It turns out that y belongs to S λ , which is a contradiction since G λ and S λ are disjoint subsets. This proves Claim 3.1.
3. We now show that u 0 (x) ≤ u λ (x) + δ + ε λ for every x ∈ E. Since u 0 ≤ u μ + δ = u λ + δ on F, we only need to consider the situation when x ∈ E\F. Let us fix η > 0. We can find a point Moreover, by (5), it is clear that there exists y η ∈ F ∪ S λ such that Suppose first that y η ∈ S λ . In particular y η ∈ E\F and u λ (y η ) ≥ u 0 (y η ) + δ + ε λ 2 . Using that u 0 is 1-Lipschitz together with (7) we obtain Suppose now that y η ∈ F. Using (7) and the fact that u λ is μ-Lipschitz on F, we can write which implies, taking into account (6), Bearing in mind that u λ + δ = u μ + δ ≥ u 0 on F and using (7) and (8) we obtain We have thus shown the inequality Proof of Theorem 3.1 Without loss of generality we may and do assume that K = 1. Let us fix a point p ∈ F and set E n = (E ∩ B[ p, n]) ∪ F and F n = E n−1 for every n ≥ 1, where F 1 = E 0 = F. It is clear that we can construct an increasing sequence of numbers {λ n } n≥1 with λ 0 < λ 1 and λ n < 1 for every n ≥ 1 such that for every n ≥ 1 such that E n \F n = ∅. Let us construct by induction a sequence of functions {u n } n≥1 such that each u n : E n → R is λ n -Lipschitz on E n and satisfy u n = u n−1 on E n−1 and |u n − u 0 | ≤ ε on E n for every n ≥ 1. (9). Observe that u 1 = u 0 on F. Now assume that we have constructed functions u 1 , . . . , u n , respectively, defined on E 1 , . . . , E n such that each u k is λ k -Lipschitz on E k , with u k = u k−1 on E k−1 = F k and for every 1 ≤ k ≤ n. Then we apply Lemma 3.1 with δ = ε/2 + · · · + ε/2 n , E n = F n+1 ⊂ E n+1 , μ = λ n , u μ = u n : E n → R and u 0 : E n+1 → R to obtain a λ n+1 -Lipschitz function u n+1 : E n+1 → R such that u n+1 = u 0 on E n and, thanks to (9), This proves the induction. We now define the function u : E → R as follows: given x ∈ E, we take a positive integer n with x ∈ E n and set u(x) := u n (x). Since E = n≥1 E n and each u n coincides with u n−1 on E n−1 , the function u is well defined. Because u = u n on each E n , we have that which implies that |u − u 0 | ≤ ε on E. Also, note that u = u 0 on F because u = u 1 on E 1 and u 1 = u 0 on F ⊂ E 1 . Finally, given a bounded subset B of E, we can find some natural n with B ⊂ E n . This implies that u = u n on B, where u n is λ n -Lipschitz and λ n < 1.

Approximation by Smooth Lipschitz Functions: Proof of Theorem 2.3
This section contains the proofs of Theorems 2.3, 2.1 and 2.2. Let us start with the proof of Theorem 2.3, so let us assume from now on that X is a Banach space satisfying the hypothesis of Theorem 2.3 for some k ∈ N ∪ {∞}. We will need the following two claims.
Proof By replacing ε with min{ε, 1 2 dist(·, ∂Ω)}, we may and do assume that ε ≤ The assumption on X implies in particular that there exists a constant C 0 ≥ 1 such that, for every Lipschitz function f : X → R and every η > 0, there exists a C k Lipschitz function g : If n > n x , then U x ∩ supp(ϕ n, p ) = ∅ for every p ∈ Ω.
If n ≤ n x , then U x ∩ supp(ϕ n, p ) = ∅ for at most one p ∈ Ω.
We can assume that u is extended to all of X with the same Lipschitz constant. Using the assumption on X , we can find a family of C k (X ) Lipschitz functions {v n, p } (n, p)∈N×Ω such that, for every (n, p) ∈ N × Ω, for every ball B[x 0 , r ] contained in Ω. We define the approximation v : By the properties of the partition {ϕ n, p } (n, p)∈N×Ω , the function v is well defined and is of class C k (Ω). Given x ∈ Ω, (11) implies Note that if p ∈ Ω is such that x ∈ B[ p, δ p ], then ε(x) ≥ ε( p)/2 ≥ 2δ p and we can write, by virtue of (12), that Therefore, we obtain This completes the proof of (b).   Then ρ is continuous and we can replace ε by min{1, ε, ρ, 1 2 dist(·, ∂Ω)} on Ω. In particular, this implies that B[x, ε(x)] ⊂ Ω for every x ∈ Ω. We thus have from Claim 4.1 that there exists v ∈ C k (Ω) such that

Proof
Hence, the last inequality leads us to for every x ∈ Ω. This shows that Dv(x) * < K on Ω.
We are now ready to prove Theorem 2.3.

Proof of Theorem 2.3
Assume that X satisfies the hypothesis of Theorem 2.3 for some k ∈ N ∪ {∞}. Let us denote by λ 0 and K the Lipschitz constants Lip(u 0 , ∂Ω) and Lip(u 0 , cl(Ω)) of u 0 on ∂Ω and cl(Ω), respectively. By Theorem 3.1, there exists a function u : cl(Ω) → R with and the Lipschitz constant of u on every bounded subset of cl(Ω) is strictly smaller than K . Now, applying Claim 4.2 for u, we can find a function v : Ω → R of class C k (Ω) such that If we extend v to the boundary ∂Ω of Ω by setting v = u on ∂Ω and we use the inequality (14), we obtain, for every x ∈ ∂Ω, y ∈ Ω, that This proves that the function v is continuous on cl(Ω). Therefore, the fact that v is K -Lipschitz on cl(Ω) is a consequence of the following well-known fact. It only remains to see that v is ε-close to u 0 . Indeed, by using (13) and (14) we obtain

Finite-Dimensional and Hilbert Spaces
We are now going to prove that if X is a finite-dimensional space or a Hilbert space, then X satisfies the assumption of Theorem 2.3 with k = ∞ in the separable case and with k = 1 in the nonseparable case.
In addition, f δ → f uniformly on R d as δ → 0. This proves the lemma in the finite-dimensional case. Now, let X be a Hilbert space and let us denote by · the norm on X . If g : X → R is a K -Lipschitz function, then the functions defined by for all x ∈ X and λ, μ > 0, are K -Lipschitz as well. Also, it is easy to see that the infimum/supremum defining g λ (x) and g μ (x) can be restricted to the ball B[x, 2λK ] and B[x, 2μK ], respectively. Let us now prove the following relation between the local Lipschitz constants of g and g λ : Indeed, let us fix a ball B[x 0 , r ], two points x, x ∈ B[x 0 , r ] and ε > 0. We can find y ∈ B[x , 2λK ] such that The points y and x − x + y belong to B[x 0 , r + 2λK ] and then we can write which easily implies (15). Similarly, we show that Now, we consider the Lasry-Lions sup-inf convolution formula for g, that is for all x ∈ X and 0 < μ < λ. By the preceding remarks, the function g μ λ is K -Lipschitz and satisfies that Moreover, in [1,2] it is proved that g μ λ is of class C 1 (X ) and g μ λ converges uniformly to g as 0 < μ < λ → 0. Now, given our K -Lipschitz function f : X → R and ε > 0, we can find 0 < μ < λ small enough so that the function f μ λ is K -Lipschitz and of class C 1 (X ), with | f μ λ − f | ≤ ε/2 on X and, by virtue of (17), If we further assume that X is separable, then we can use [11,Theorem 1] in order to obtain a function g ∈ C ∞ (X ) such that where · * denotes the dual norm of · . From the first inequality we see that | f − g| ≤ ε on X . The second one together with (18) shows that Combining Lemma 4.1 with Theorem 2.3, we obtain Theorems 2.2 and 2.1 when X is a separable Hilbert space or a finite-dimensional space.

Remark 4.1
In the case when the function to be approximated vanishes on the boundary, the proof of Theorem 2.1 for finite-dimensional spaces can be very much simplified as we do not need to use Theorem 3.1. Indeed, if R n is endowed with an arbitrary norm and u 0 : cl(Ω) → R is a Lipschitz function with u 0 = 0 on ∂Ω, given ε > 0, we define the function ϕ ε : R → R by We can assume that u 0 is extended to all of R n by putting u 0 = 0 on R n \ cl(Ω), preserving the Lipschitz constant. The function u = ϕ ε • u 0 defined on R n is Lipschitz because so are u 0 and ϕ ε , and Lip(u, R n ) ≤ Lip(u 0 , R n ). Also, since |ϕ ε (t)−t| ≤ ε/2 for every t ∈ R, it is clear that Using the preceding remarks together with the well-known properties of the integral convolution of Lipschitz functions with mollifiers, it is straightforward to check that, for δ > 0 small enough, v is the desired approximating function, i.e., v is of class

The Space c 0 ( )
Let us now prove that the space X = c 0 (Γ ) satisfies the hypothesis of Theorem 2.3 with k = ∞. In order to do this, we will use the construction given in [12, Theorem 1] and we will observe that the local Lipschitz constants are preserved.

Lemma 4.2
If Γ is an arbitrary subset, X = c 0 (Γ ) and f : X → R is a Lipschitz function, then, for every ε > 0, there exists a function g : Proof If K denotes the Lipschitz constant of f , let us consider 0 < η < ε 2(1+K ) . Let us define the function φ : and with the property that, for every x ∈ X , there exists a finite subset F of Γ such that whenever y, y ∈ B[x, η 2 ] and P F (y) = P F (y ) (here P F (z) = γ ∈F e * γ (z)e γ for every z ∈ X ) we have h(y) = h(y ). Moreover, we observe that if x, y ∈ B[x 0 , r ] ⊂ X , then φ(x), φ(y) ∈ B[x 0 , r + η] and therefore . Now we use the construction of [12,Lemma 6] to obtain the desired approximation g : let us define g as the limit of the net {g F } F∈Γ <ω , where each g F is defined by and θ is a even C ∞ smooth nonnegative function on R such that R θ = 1 and supp(θ ) ⊂ [−cε, cε], for a suitable small constant c > 0. It turns out that g is of class C ∞ (X ) with |g − h| ≤ ε 2 on X and with the property that, for every x ∈ X , there exists a finite subset F x of Γ such that g(x) = g H (x) for every finite subset H of Γ containing F x . See [12,Lemma 6] for details. In addition, we notice that if x, y ∈ B[x 0 , r ], and we consider finite subsets F x and F y of Γ with the above property, then for the set H = F x ∪ F y , we have that This shows that for every ball B[x 0 , r ] ⊂ X . This proves the lemma.
Combining Lemma 4.2 with Theorem 2.3, we obtain Theorem 2.1 in the case X = c 0 (Γ ).

Approximation by Almost Classical Solutions of the Eikonal Equation
Throughout this section, X will denote a finite-dimensional normed space with dim(X ) ≥ 2. At the end of the section we will complete the proof of Theorem 2.4. We need to recall the notion of almost classical solutions of stationary Hamilton-Jacobi equations with Dirichlet boundary condition. This concept was introduced in [8] for the Eikonal equation and was generalized in [10] as follows.
In [8, (ii) For every compact subset K of Ω there exist constants α K , M K > 0 such that for all x ∈ K , r ∈ [0, α K ] and x * ∈ X * with x * * ≥ M k we have F(r , x, x * ) > 0.
Then, given ε > 0, there exists a function u ≥ 0 on cl(Ω) such that |u| ≤ ε on cl(Ω) and u is an almost classical solution of the equation F(u(x), x, Du(x)) = 0 on Ω with Dirichlet condition u = 0 on ∂Ω. Moreover, the extensionũ of u defined byũ = 0 on X \Ω is differentiable on X .
Proof Although [10, Theorem 3.1] was originally stated when X = R n is endowed with the Euclidean norm, we can easily rewrite its statement (and its proof) for general finite-dimensional normed spaces by using the following proposition, which is an easy consequence of [8, Corollary 3.6].

Proposition 5.2
Suppose that B is a closed ball of X * . There exists a mapping t : B → S X * * such that if (σ n ) n ⊂ B is a sequence with t(σ n )(σ n+1 − σ n ) ≥ 0 for every n, then (σ n ) n converges.
In [10,Theorem 3.1], Ω is decomposed as Ω = j≥1 C j , where {C j } j≥1 is a locally finite family of closed cubes and the function u satisfies u = 0 on j≥1 ∂C j (because u is the sum of a series of functions all vanishing on this union). Moreover, it is possible to choose the covering {C j } j≥1 so that diam(C j ) ≤ ε for every j ≥ 1, and then, the Mean Value Theorem yields that |u| ≤ ε on Ω.

The Limiting Case
In this section, we are concerned about constructions of functions u 0 with prescribed values on the boundary of Ω such that u 0 is differentiable on Ω and Lip(u 0 , ∂Ω) = Lip(u 0 , Ω). The notion of Absolutely Minimizing Lipschitz (AML for short) function will be involved in the proof of the following proposition. Given an open subset Ω of R n , we say that a Lipschitz function u : Ω → R is an AML function provided that Lip(u, V ) = Lip(u, ∂V ) for every V open such that cl(V ) is a compact subset of Ω.
If, in addition, u agrees with a Lipschitz function u 0 : ∂Ω → R on ∂Ω, we say that u is an Absolutely Minimizing Lipschitz Extension (AMLE) of u 0 . The existence of these AMLE of a boundary data u 0 and its equivalence with infinity harmonic functions (that is, viscosity solutions of the Infinity-Laplace equation) was proved in [13], while the uniqueness was shown in [14]. The regularity of these solutions was studied in [15,16]. See [17] for a survey paper on the theory of absolutely minimizing Lipschitz functions.  [15] that the AMLE of u 0 to cl(Ω) is of class C 1 (Ω). In particular, there exists a 1-Lipschitz extension v : cl(Ω) → R of u 0 such that v ∈ C 1 (Ω). If we consider the problem and define F : Ω × R 2 → R by F(x, p) = |p + ∇v(x)|, x ∈ Ω, p ∈ R 2 , we have that F is a continuous function which is easily checked to satisfy the hypothesis of [10, which implies that u(x, 0) = |x| for every x ∈ [− 1, 1]. Therefore u is not differentiable at (0, 0).
The above example shows in particular that, if u 0 is extended to a 1-Lipschitz on cl(Ω) and ε > 0, then there is no 1-Lipschitz function v on cl(Ω) which is differentiable on Ω, v = u 0 on ∂Ω and |u 0 − v| ≤ ε on cl(Ω). Thus Problem 2.1 has a negative answer in the limiting case Lip(u 0 , ∂Ω) = Lip(u 0 , cl(Ω)). An example with the same properties can be obtained with the ∞ norm by means of the isometry T : (R 2 , · 1 ) → (R 2 , · ∞ ), defined by T (x, y) = (x + y, x − y).

Conclusions
We studied the problem of approximating Lipschitz functions defined on an open subset of a Banach space by differentiable Lipschitz functions preserving both the Lipschitz constant and the boundary value. In order to do that, we first obtained the following purely metric result, that can be of independent interest: Given a real-valued Lipschitz function on a metric space, such that its restriction to a given closed subset has a better Lipschitz constant, we can approximate it uniformly by a function with better Lipschitz constant on bounded sets, and which coincides with the initial function on the given closed subset. This intermediate result allowed us to give a positive answer to our problem, when the Lipschitz constant on the boundary of the function to be approximated is smaller than its global Lipschitz constant. The order of differentiability of the approximating functions depends on the regularity of the partitions of unity of the pertinent space, and then the approximations can be taken infinitely many times differentiable in finite-dimensional and Hilbert spaces. These results yield approximation of 1-Lipschitz functions by everywhere-differentiable functions, which satisfy the Eikonal equation almost everywhere and coincides in the boundary with the initial function, provided that the restriction of the function to the boundary has Lipschitz constant less than 1. We proved the optimality of these results by exhibiting an example of a 1-Lipschitz function defined on the boundary of an open subset of a two-dimensional normed space, which does not admit any 1-Lipschitz differentiable extension. The question whether the main problem of this paper, without restrictions on the boundary value of the function to be approximated, has a positive solution in a finite-dimensional Euclidean space remains open. A related question would be to find conditions on the norm of a finite-dimensional space for which our problem has a positive solution, whitout restrictions on the boundary value of the initial function.