The best approximation of a given function in L ²-norm by Lipschitz functions with gradient constraint

1 Introduction

Let us assume that Ω ⊂ ℝ N , N ≥ 1 , is a bounded open set with smooth boundary, say C 1 , γ , for some γ ∈ ( 0 , 1 ) . The main goal of this work is to study the optimal approximation in the L 2 -norm of a given function f ∈ L 2 ⁢ ( Ω ) by functions in W 1 , ∞ ⁢ ( Ω ) with a constraint on the gradient. Namely, we consider the minimization problem

where the set 𝕏 ⁢ ( Ω ) is given by

In order to obtain a minimizer to (1.1), one can argue by direct methods (taking a minimizing sequence), or rather noticing that such a problem appears naturally as Γ-limit, for n → ∞ , of the following classical energy functional

where p n is a (non-decreasing) sequence of numbers that diverges to + ∞ . Setting u n the minimizer of J n , the heuristic of the limiting process is that the measure of the set { | ∇ ⁡ u n | > 1 } has to go to zero in order to keep the energy bounded (notice that J n ⁢ ( u n ) ≤ J n ⁢ ( 0 ) = ∥ f ∥ L 2 ⁢ ( Ω ) 2 ). Therefore, we expect the sequence { u n } to converge to the unique minimizer u ∈ 𝕏 ⁢ ( Ω ) of J ∞ (see Proposition 2.2 below).

An interesting feature of the minimizer of (1.1) that we find here, is that it satisfies a kind of representation formula by cones inside the region { u ≠ f } .

Figure 1

Figure 2

Before stating our first result, in order to gain some intuition, let us consider three concrete examples in the special case Ω = ( - 1 , 1 ) .

1. Let f 1 ⁢ ( x ) = k ⁢ χ ( - r , r ) with r ∈ ( 0 , 1 ] and k ∈ ℝ . Observe that if r = 1 , then f 1 ∈ 𝕏 ⁢ ( - 1 , 1 ) and so u 1 ≡ f 1 is the minimizer. Otherwise, it is not hard to see that

   u 1 = min ⁡ { ( r + k 2 - | x | ) + , k }

   is the minimizer. In Figure 1 the case r > k 2 is represented.

2. Let f 2 ⁢ ( x ) = 2 ⁢ | x | . Then the minimizer is given by

   u 2 ⁢ ( x ) = | x | + 1 2 .

   See Figure 2.

3. Let f 3 ⁢ ( x ) = | x | . Then it is not hard to see that the minimizer is

   u 3 ⁢ ( x ) = { | x | + 2 9 for  ⁢ | x | ≤ 4 9 , | x | for  ⁢ | x | ≥ 4 9 .

   See Figure 3.

In the three examples above (see Section 6 for the explicit computations) the behavior of the solution u i , i = 1 , 2 , 3 follows the same idea: any u i tries to be as close as possible to the datum f i as long as f i is smooth and has gradient bounded by one. Otherwise, the best that the solution can do, in order to minimize the L 2 norm of the difference, is to grow as much as it is allowed, that is, as a line with slope + 1 or -1.

Figure 3

More in general we have the following result.

After considering the problem from a variational viewpoint, we want to address it from a PDE perspective. Since it is not clear a priori what equation the minimizer of (1.1) solves, we go back to the approximating sequence of minimizers of J n (see (1.2)). The minimizers of (1.2) are solutions to

where Δ p ⁢ v = div ⁡ ( ∇ ⁡ v ⁢ | ∇ ⁡ v | p - 2 ) , with p > 1 , is the p-Laplacian and we denote by ν the external unit normal to ∂ ⁡ Ω . As usual, we consider such solutions in the weak sense (see Definition 2.1). Our aim is, now, to pass to the limit in (1.5) to find the PDE solved by the limit function, u.

Typically, the limit as p → ∞ of solutions to equations related to the p-Laplacian leads to equations that involve the infinity Laplacian, namely,

In particular, we refer to the pioneer paper by T. Bhattacharya, E. Di Benedetto and J. Manfredi (see [9]) that first studied such type of problem. We recall that the infinity Laplacian is a second order differential operator (in nondivergence form) that appears in many contexts. For instance, infinity harmonic functions (i.e., solutions to - Δ ∞ ⁢ u = 0 ) appear naturally as limits of p-harmonic functions (i.e., solutions to - Δ p ⁢ u = - div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u ) = 0 ) and they have applications to optimal transport problems, image processing, etc. See, among the others, [4, 9, 24] and references therein. Moreover, the infinity Laplacian plays a fundamental role in the calculus of variations of L ∞ functionals, see e.g. [3, 6, 7, 17, 18, 19, 20, 26] and the survey [4]. Notice that such operator is degenerate elliptic (that is non-degenerate only in the direction of the gradient).

Before stating our next result, let us introduce some notations: here and in the rest of the paper for u , f ∈ C ⁢ ( Ω ¯ ) we denote

and analogously,

For the sake of clarity, notice that in Theorem 1.2 f is continuous and the open set Ω u + coincides with the interior of A + .

Now we are ready to write the problem solved by u.

Quite surprisingly, in the limit the second order operator is somehow lost: indeed there is no trace of the infinity Laplacian and, in fact, u satisfies a first order equation. We will provide an explanation for the absence of the second order operator in (1.6) in Proposition 1.6 below. Let us stress here that the two equations in (1.6) are not equivalent from the point of view of viscosity solutions and, indeed, they are the proper ones to reflect the information carried out by (1.3): in the region { u - f < 0 } the function u looks like a cone pointing upwards and therefore it solves the eikonal equation; analogously, in { u - f > 0 } the cone points downwards and it solves the eikonal equation with the reverse sign. The same double structure is recognizable in the boundary conditions (1.7).

As far as the boundary conditions are concerned, it is natural to wonder if it would be possible to extend the equation in (1.6) up to Ω ¯ . However the answer is negative, and, to get convinced that (1.7) are the natural conditions to impose on ∂ ⁡ Ω , it is enough to look at the second example above. Indeed, observe that in this case ∂ Ω = { ± 1 } ⊂ { u 2 - f < 0 } , so we have to focus on the second pair of conditions in (1.7). If x 0 = ± 1 , we have that u 2 satisfies (in viscosity sense) | u 2 ′ ⁢ ( x 0 ) | - 1 ≥ 0 (when u 2 is touched from below by any smooth function φ, it has to be | φ ′ ⁢ ( x 0 ) | ≥ 1 ). However, when we test the behavior of u 2 touching it from above by a smooth function φ, it is easy to check that it is not necessarily true that | φ ′ ⁢ ( x 0 ) | ≤ 1 ; but if | φ ′ ⁢ ( x 0 ) | > 1 then ∂ ν ⁡ φ ⁢ ( x 0 ) < 0 . This is exactly the condition prescribed by (1.7).

It is also worth to mention explicitly that the regions Ω u + and Ω u - (and their boundary counterparts) depend on the solution u itself. A consequence of this fact is that, even if the minimizer of problem (1.1) is unique, problem (1.6)–(1.7) does not posses a unique solution. Indeed, any affine function with modulus of the gradient equal to 1 that lies above or below f solves (1.6)–(1.7). Clearly, this type of solutions are not interesting for our study. However, it is still possible to construct a nontrivial solution to (1.6)–(1.7), that does not coincide with the minimizer of J ∞ , via a suitable obstacle problem.

Observe that u ¯ is not a solution of the minimization problem (1.1) (unless f ∈ 𝕏 ⁢ ( Ω ) ). Indeed, as the intuition suggests, since the minimizer u has to be close to f in L 2 , then u - f has to change sign in Ω (see Figure 4 and Proposition 2.4 for more details). Considering the reverse inequalities in the obstacle problem above, via a similar procedure, it is possible to construct another nonvariational solution u ¯ ∈ 𝕏 ⁢ ( Ω ) , such that u ¯ ≤ u ≤ u ¯ .

Figure 4

The limit procedure showed above can be, in fact, generalized to a larger class of differential equations that involve lower order terms that have a polynomial growth with respect to the gradient. More precisely, we consider the following problem:

where the Hamiltonian term H n ∈ C ⁢ ( Ω ¯ × ℝ N ) satisfies the following growth conditions:

and

Among all the possible choices of H n that satisfy (1.11), let us mention two important examples:

These two types of lower order terms, accounting for the “linear ( p - 1 ) growth” and “natural growth”, are thoroughly studied in literature (for fixed p). Typically, the presence of the Hamiltonian affects the coercivity of the operator and some specific techniques are required to obtain a priori estimates: the first case can be dealt with symmetrization procedures [8], slicing techniques [15] or arguing by contradiction [10]. For the case of natural growth we mention the exponential test function approach developed in [11, 12, 13], again symmetrization in [1], and an argument based on Schauder fixed point Theorem in [29].

Let us notice that, under assumption (1.11), for a smooth f and a fixed n, the existence of a solution v n ∈ W 1 , p n ⁢ ( Ω ) ∩ L ∞ ⁢ ( Ω ) to problem (1.10) is a straightforward consequence of the results contained in [13] (see Theorem 3.1 below for more details). We also explicitly point out that the presence of the zeroth lower order term allows us to solve (1.10) without any smallness assumption on the datum f.

The main difficulty when one wants to pass to the limit as p diverges in (1.10) is to obtain a priori estimates for v n that are stable with respect to n. Up to our knowledge, the only related result for quasilinear equations with gradient lower order terms is contained in [16], where a large solution problem is considered.

Our main result concerning problem (1.10) is the following.

At the first glance we notice that the structure of (1.14) is more involved then the one in (1.6). First of all v does not belong in general to 𝕏 ⁢ ( Ω ) since the bound on ∇ ⁡ v depends also on the Hamiltonian (see Section 3). Moreover, as expected, the presence of the Hamiltonian term changes both the limit equation and the boundary conditions.

Let us now go back to the minimizer u ∈ 𝕏 ⁢ ( Ω ) of (1.1) and notice that it also solves equation (1.14) with H ∞ ≡ 0 . Apparently, this provides some more information about - Δ ∞ ⁢ u that is missing in (1.6). However, we have the following result.

Roughly speaking this is due to the bound | ∇ ⁡ w | ≤ 1 and the fact that cones are solutions to the eikonal equation and are also infinity harmonic away from their singular points.

Finally, we include some considerations about a very well-known family of functionals that allow free discontinuities. Unfortunately, we are able to provide only partial results in this case and we left a more complete analysis of the limiting behavior of such functionals for p → ∞ for future research. Let us consider the functional ℐ p : SBV ⁢ ( Ω ) → ℝ ∪ { + ∞ } defined as

with r ≥ 1 . Here SBV ⁢ ( Ω ) is the space of Special Functions of Bounded Variation, ∇ ⁡ v is the absolutely continuous part of Dv with respect to the Lebesgue measure, S v is the singular set of v and ℋ N - 1 is the ( N - 1 ) -dimensional Hausdorff measure (more details and formal definitions can be found in Section 5). We set ℐ p ⁢ ( v ) = + ∞ when ∇ ⁡ v ∉ L p ⁢ ( Ω ) .

The functional ℐ p ⁢ ( v ) was introduced in the seminal paper by De Giorgi, Carriero and Leaci ([22]) to provide a weak formulation of the Mumford–Shah image segmentation problem, namely, minimizing

among all K ⊂ Ω ¯ closed sets, with v ∈ W 1 , p ⁢ ( Ω ∖ K ) . An in-depth overview about the history of these functionals, their relation with applications, and the huge impact that their study has had in the field of Calculus of Variations is definitively out of reach for our contribution. We refer the interested reader to the original paper [22], the classical monograph [2], and to the nice review [25]. For our aims, we simply recall that the main difficulty in dealing directly with (1.18) is that the Hausdorff measure is not lower semicontinuous with respect to any reasonable metric the collection of closed set of Ω can be equipped with. Therefore, the strategy of [22] was to find a minimizer u ∈ SBV ⁢ ( Ω ) of ℐ p and then show that ( S u ¯ , u ) was also a minimizer to (1.18). The most delicate step in their argument was to prove that ℋ N - 1 ⁢ ( S u ¯ ∩ Ω ) = ℋ N - 1 ⁢ ( S u ) and this crucial step was achieved by a lower bound on the density of S u , namely

(see [2, Theorems 7.15, 7.21 and 7.22]).

At this point it is natural to wonder what happens with the limiting minimization problem relative to (1.18) as n → ∞ . The answer is given by the following proposition.

The next step would be to prove that a minimizer of (1.20) is also a minimizer to

with K ⊂ Ω ¯ a closed set, u ∈ W 1 , ∞ ⁢ ( Ω ∖ K ) and | ∇ ⁡ u | ≤ 1 . Unfortunately, we are not able to show such a property since we do not know how to obtain a lower density bound, as in (1.19), for the limit function u. We leave this as an interesting open problem.

2 The best Lipschitz approximation with gradient constraint

Let p n be an increasing sequence such that p n → ∞ as n → ∞ . Let us set, for any f ∈ L 2 ⁢ ( Ω ) , the following functional

For any fixed n, it is not hard to see that there exists a unique (exploiting the convexity of (2.1)) minimizer u n ∈ W 1 , p n ⁢ ( Ω ) to (2.1). Thus, from direct variational arguments, the minimizer, u n , turns out to be a weak solution to

Let us recall here such a formulation.

The aim of this section is to deal with the limiting behavior of these three objects – the minimizer, the functional, and the equation – as p n diverges.

We start with the following proposition that provides a limit for the sequence of minimizers together with a limiting minimization problem associated to (2.1).

In other words, the minimizer of (2.4) is the best approximation of f among all the functions in 𝕏 ⁢ ( Ω ) . The gradient constraint imposes on the minimizer a specific geometric structure. The goal of the following results is to clarify such a structure.