Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians


We consider the classical functional of the Calculus of Variations of the form

$$\begin{array}{}
\displaystyle
I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx,
\end{array}$$
where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + 
$\begin{array}{}
\displaystyle
W^{1,p}_0
\end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.


Introduction
In 1927, Lavrentiev [1] provided an example of the fact, later called the Lavrentiev phenomenon, that the in mum, over the set of absolutely continuous functions, of a one-dimensional functional of the calculus of variations may be strictly lower than the in mum of the same functional over the set of Lipschitz functions satisfying the same boundary conditions: the example was re ned by Manià in [2] and, more recently, by Ball-Mizel in [3]. Finding the conditions that ensure the non-occurrence of the Lavrentiev phenomenon has some interest, if just for ensuring to catch the in mum of the functional via standard numerical methods. We point out that some authors refer to as the Lavrentiev phenomenon just the fact that the in ma among the two aforementioned classes of functions di er without taking care the boundary datum. If one allows the boundary datum to vary, things change dramatically: in Lavrentiev's celebrated example itself the in ma among Lipschitz/absolutely functions are the same if one allows one boundary datum to be just arbitrarily close to the initial one. Some recent results concerning the study of this kind of "local" Lavrentiev phenomenon have been recently obtained in [4,Theorem 4].
Alberti and Serra Cassano proved in [5] that, when the integration set is an interval in R, the phenomenon does not occur for autonomous Lagrangians. For scalar problems, where the domain is multi-dimensional, few results appeared in the literature. Of course, the problem becomes much easier if one imposes growth conditions of the Lagrangian from above, since approximations are facilitated by Lebesgue's dominated convergence. A two-dimensional analogue of Manià's example was provided in [6]. There are examples in [7] of functionals of the form F(x, ∇u) depending on the independent variable x of the space and on the gradient ∇u of the admissible functions that exhibit the Lavrentiev phenomenon; there are also cases in which the Lavrentiev phenomenon does not occur (see [8]). No example is known to the authors for scalar problems when the Lagrangian is autonomous.
It was conjectured by Buttazzo-Belloni in [9] that the phenomenon should not occur when the Lagrangian F(u, ∇u) is autonomous and convex in both variables, a fact that they proved in the case of a (strongly) star-shaped domain under the hidden growth assumption that F(u, ) is summable and zero as a boundary datum. Other results that appeared aimed to prove the conjecture: we mention Ekeland-Temam who proved in [10] its validity for functionals of the gradient on a Lipschitz domain for a zero boundary datum; Bonfanti and Cellina in [6,11] considered autonomous Lagrangians that are sum of a radial function of the gradient ∇u and a function of the variable u, under some smoothness assumptions on the boundary and on the boundary datum. A complete answer to the conjecture was given by Bousquet-Mariconda-Treu in [12], where they showed that whenever F(u, ∇u) is convex, given u ∈ W , (Ω) with a Lipschitz boundary datum and nite energy (i.e., F(u, ∇u) ∈ L (Ω)), there is no Lavrentiev gap at u: there exists a sequence (u k ) k of Lipschitz functions that share the same boundary datum and converge to u both in W , and in energy, no matter if u is a minimizer.
We consider here a convex nonautonomous Lagrangian F(x, u, ∇u), and establish a su cient condition under which no Lavrentiev gap occurs at any admissible function. As a byproduct it turns out that the Lavrentiev phenomenon does not occur if the Lagrangian is of the form with f (·, ) of class C (Ω) and h(·, ) of class C (Ω). The methods used here are mainly based on [12][13][14]: we show that we can approximate a function (both in W , and in energy) with a sequence of bounded functions that are Lipschitz in a neighbourhood of the boundary of the domain. A partial motivation for studying these kind of functionals comes from minimization problems in the Heisenberg group where one wants to consider functionals that generalize those studied in [15] and in references therein.
We do not consider here the vectorial case for which, when the Lagrangian depends only on the gradient, there are both examples of the occurrence of the Lavrentiev phenomenon and cases where it does not occur [7].
The authors are grateful to Pierre Bousquet for his useful comments and to both the referees for having carefully read the manuscript.
-For E ⊆ R n , |E| is the n-dimensional Lebesgue measure of E.
-E is the indicator function of a set E.

. Assumptions
-Ω ⊂ R n is an open and bounded set.
The following structure condition will be used, in alternative to the boundedness of the reference function u, in our main result.
1. There are positive sequences (τ k ) k and (σ k ) k such that: There is a measurable and bounded selection q(x) of the subgradient ∂s F(x, , ) of F at (x, , ) and, for all k ∈ N, Indeed the monotonicity of the subdi erential implies that, for a.e. x ∈ Ω, x, Hypothesis (H) reduces to Condition 1, namely that there are increasing , divergent sequences (τ k ) k and (σ k ) k such that Here are some Lagrangians that satisfy Hypothesis (H).

Proposition 2.2 (Validity of Hypothesis (H)
). Assume that the map (s, ξ ) → F(x, s, ξ ) is convex for a.e. x and that Then F ful lls Hypothesis (H).
(i) Since ζτ k , ζσ k do not depend on x it turns out that their divergence is zero. Similarly, Point 2 of Hypothesis (H) is ful lled since any q ∈ ∂s F( , ) does not depend on x and F is convex.
(ii) Assume now that F(x, s, ξ ) = f (x, s) + h(x, ξ ). Then, for each k, and div ∇ ξ h(x, ) is continuous, thus bounded on Ω. Moreover, any element of ∂s F(x, , ) is an element of ∂x f (x, ) and is thus bounded. Condition (2.2) follows from the convexity of s → F(x, s, ξ ), proving the validity of Hypothesis (H).

Approximation lemmas
In this section, we establish two preliminary results that will be used in the proof of Theorem 4.2.
As a rst step, we give a su cient condition under which there is no Lavrentiev gap between W ,p ϕ (Ω) and W ,p ϕ (Ω) ∩ L ∞ (Ω). We have de ned the space W ,p ϕ (Ω) as the set of those functions u ∈ W ,p (Ω) such that the extension of u by ϕ on R n \ Ω belongs to W ,p loc (R n ). We still denote by u this extension. In particular, (u − ϕ) belongs to W ,p (R n ) and has compact support. Proof. Let (τ k ) k and (σ k ) k satisfy the conditions formulated in Hypothesis (H). For k large enough such that both τ k > |ϕ| L ∞ (Ω) and σ k > |ϕ| L ∞ (Ω) , we de ne u k by It is clear that u k ∈ W ,p ϕ (Ω) ∩ L ∞ (Ω) and that u k converges to u in W ,p ϕ (Ω). Moreover, Let q(x) and (qτ k (x), ζτ k (x)) be as in Hypothesis (H). We have

Therefore, Hypothesis (H) yields
Analogously we get It follows from (3.2), (3.3) and (3.4) that We now prove that there is no Lavrentiev gap at u ∈ W ,p ϕ (Ω) if u is Lipschitz continuous on a neighborhood of ∂Ω. Lemma 3.2. Assume that F : Ω × R × R n → R is convex with respect to its three variables. Let u in W ,p ϕ (Ω) be such that F(x, u, ∇u) ∈ L (Ω). Assume, moreover, that either u is bounded or that F ful ls Hypothesis (H). If u is Lipschitz continuous on a neighborhood of ∂Ω, then there exists a sequence (u k ) k in Lip ϕ (Ω) such that (u k ) k converges to u in W ,p ϕ (Ω) and lim k→+∞ I(u k ) = I(u).
Moreover, if u is bounded in L ∞ (Ω), then the sequence (u k ) k may be taken to be bounded in L ∞ (Ω).
Proof. From Lemma 3.1 it is not restrictive to assume that u is bounded. We may consider u as extended by ϕ out of Ω. By assumption, there exists an open set V ⊂ R n such that ∂Ω ⊂ V and u is Lipschitz continuous on V ∩ Ω. In particular u and ∇u are in L ∞ (V ∩ Ω).
Let ρ ∈ C ∞ c (B , R + ) be even, R n ρ dx = and for k = , , ..., (ρ k ) k be the sequence of molli ers de ned by ρ k (x) := k n ρ(kx). Let also θ ∈ C ∞ c (Ω, [ , ]) be such that θ = on a neighborhood of Ω \ V. We then de ne For this purpose, we decompose I(u k ) = Ω F(x, u k , ∇u k ) dx as the sum On the set { ≤ θ < } ⊂ V ∩ Ω, ∇u ∈ L ∞ (V ∩ Ω) and Let k be such that Then, for k ≥ k and x ∈ { ≤ θ < } we have which in turn means that under the above assumptions both u k and ∇u k are bounded by a constant that does not depend on k. Since (u k ) k converges to u in W ,p (Ω) we may assume, by taking a subsequence, that (u k , ∇u k ) k converges a.e. to (u, ∇u). Now, since F is bounded on bounded sets, by Lebesgue's Theorem we have lim It remains to show that lim sup afterwards, in view of (3.7) and (3.8), we get (3.6). Notice that, since ρ is even, By Jensen's inequality, Since F(x, u, ∇u) ∈ L (Ω), we get (3.9).

Main result
We consider here domains Ω that are locally strongly star-shaped in the sense of [12, De nition 2.9]. These include Lipschitz ones and allow even some cusps at some boundary points.  . Assume that Ω is locally strongly star-shaped and that F : Ω × R × R n → R is convex. Let u ∈ W ,p ϕ (Ω) be such that F(x, u, ∇u) ∈ L (Ω). Assume, moreover, that either u is bounded or that F satis es the structure Hypothesis (H). Then the Lavrentiev gap for I does not occur at u, i.e. there exists a sequence (u k ) k in Lip ϕ (Ω) converging to u in W ,p (Ω) and such that lim k→+∞ I(u k ) = I(u). (4.1) Moreover, if u is bounded in L ∞ (Ω), the sequence (u k ) k may be taken to be bounded in L ∞ (Ω).
Proof. In view of Lemma 3.2 it is enough to provide a sequence (u k ) k in W ,p ϕ (Ω) satisfying the conditions of the claim with the exception that it is just Lipschitz continuous in a neighborhood of ∂Ω (instead of Lipschitz on Ω). We may consider u to be extended by ϕ out of Ω. Also, in view of Lemma 3.1, it is not restrictive to assume that u is bounded.
Without loss of generality, we can assume that F ≥ . Indeed, since F is convex with respect to its variables, if x ∈ Ω and (a, q, ζ ) ∈ ∂F(x , , ) then Moreover additive a ne terms do not perturb our convergence results: if a sequence (u k ) k converges to u in W ,p (Ω) then I(u k ) k converges to I(u) if and only if Consider rst the case where Ω is strongly star-shaped with respect to the origin, i.e., for every h ∈ [ , [, hΩ is relatively compact in Ω. Given λ, h ∈] / , [, set Notice that u λ h converges to u in W , as λ, h → and that u λ h = ϕ on R n \ hΩ. We then write as a convex combination in λ, namely The convexity of F yields Fix λ ∈] / , [; we then study the second term of the right hand side of (4.3). Since h ≥ / , we have The function F being bounded on bounded sets, by means of the dominated convergence theorem we get

Remark 4.3.
Assume that F(x, u, ∇u) = f (x, u) + h(∇u) with f convex and superlinear. In Theorem 4.2 the (alternative) assumption that u is bounded is satis ed if, for instance, for every constant boundary datum k ∈ Z, the minimizers of I among the functions that are equal to k in the boundary of Ω are bounded. Indeed, the fact that ϕ is bounded and the comparison principles of [16,17] show that u is bounded too.