Journal of Convex Analysis Volume 1 (1994), No.1, 31–45

Gap Phenomenon for Some Autonomous Functionals

Giovanni Alberti

Istituto di Matematiche Applicate, Universitá di Pisa, via Bonanno 25/B, 56126 Pisa, Italy e-mail: alberti@sns.it

Pietro Majer

Dipartimento di Matematica, Universitá di Pisa, via Buonarroti 2, 56127 Pisa, Italy e-mail: majer@dm.unipi.it

Received 15 November 1993 Revised manuscript received 25 May 1994

We give an example of an autonomous functional F (u) = ∫Ω f(u, Du)dx (Ω open subset of IR2, u : Ω → IR2 in the Sobolev space W 1,1) which is sequentially weakly lower semicontinuous in W^{1,p} for every p ≥ 1 but does not agree with the relaxation of the same functional restricted to smooth functions when p < 2. A Lavrentiev phenomenon occurs for a related boundary problem.

1991 Mathematics Subject Classification: 49J45 (28A75, 49K10)

1. Introduction

When dealing with variational problems involving integral functionals, a very surprising and interesting phenomenon is the so-called Lavrentiev phenomenon; let T be a space of weakly differentiable functions, X a dense subset of smooth functions and F a (weakly) lower semicontinuous integral functional on T : then it may happen that the infimum of F on T is strictly lower than the infimum on X (the first example of such behaviour was given in [18], many examples and references may be found in [6], [10] and [9], section 3.4.3).

For example Laverentiev phenomenon may occur with T = {u ∈ W^{1,2}(0, 1) : u(0) = 0, u(1) = 1}, X = {u ∈ T : u is Lipschitz} and F (u) = ∫_{0}^{1} f(x, u, u′)dx with f continuous, convex in the third variable and satisfying f(x, u, s) ≥ |s|^{2} (cf. [2]). Notice that in this particular case F is coercive on T (endowed with the weak topology) and then it attains a ‘natural’ minimum on T. Following [6], we consider Lavrentiev phenomenon in a more abstract framework: let T be a topological space, X a dense subset of T, F a lower semicontinuous function on T, and set for all u ∈ T

F (X, T)(u) := inf(lim inf_{n→∞} F (u_{n})) : u_{n} ∈ X and u_{n} → u.  (1.1)

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag Berlin

32 G. Alberti, P. Majer / Gap phenomenon for some autonomous functionals

Then we have that F (X, T) ≥ F but it may happen that equality does not hold: in this case we say that F has a gap from X to T (see Definition 2.1). Notice that when there is no gap, then there is no Lavrentiev phenomenon (indeed the infima of F on T and X are the same). In particular we have that F (X, T)(u) = F (u) if and only if there exists a sequence (u_{n}) ⊂ X such that u_{n} → u and F (u_{n}) → F (u); hence there is no gap when F is continuous with respect to any topology τ finer than the topology of T and such that X is still τ-dense in T (for example, this happens when T is a Sobolev space W^{1,p} endowed with the weak topology, X includes C^{∞} functions, and F is strongly W^{1,p} continuous).

A lot of result are available for integral functional on Sobolev spaces, giving examples of gap and Lavrentiev phenomenon or showing that under some assumption they never occur (see references in [6], [7] and [10]). An interesting case is the one of autonomous integral functionals, i.e., T = W^{1,p}(Ω, IR^{n}), X is the subset of all smooth functions (Lipschitz or C^{1}) and F (u) = ∫_{Ω} f(u, Du)dx. When Ω is one-dimensional and the integrand f is continuous, it may be proved that gap never occurs (see [1] for a direct proof, even if this can be obtained as a corollary of the regularity of minima of variational integrals proved in [8]). In this paper we show that a gap may occur when T = W^{1,p}(Ω, IR^{2}) with p < 2 and Ω two-dimensional. In particular, if we take

F (u) = ∫_{Ω} (|u|^{2} − 1)^{2}|Du|^{2}dx  (1.2)

and u_{0}(x) = (x − x_{0})/|x − x_{0}| where x_{0} ∈ Ω, then F (u_{0}) = 0 but lim inf F (u_{n}) ≥ 2π/3 whenever u_{n} are smooth functions which weakly converge to u_{0} in W^{1,1} (see Theorem 2.3). In Theorem 2.12 we give an example of Lavrentiev phenomenon for a related boundary value problem. The heuristic idea of the proof is the following: recalling that |Du|^{2} ≥ 2| det Du|, by area formula we have that for all smooth functions u

F (u) ≥ 2 ∫_{Ω} (|u|^{2} − 1)^{2}| det Du| dx ≥ 2 ∫_{u(Ω)} (|y|^{2} − 1)^{2}dy .  (1.3)

Now F (u_{0}) = 0 because the image of u_{0} is included in the unit circle in IR^{2} but since the functions u_{n} are smooth and converge to u_{0}, their images have to ”cover” the unit ball B of IR^{2} and then (1.3) yields

lim inf_{n→∞} F (u_{n}) ≥ 2 ∫_{B} (|y|^{2} − 1)^{2}dy = \frac{2π}{3} .

Notice that if we replace weak convergence in W^{1,1} by a weaker one (for example, L^{p} convergence), this argument does not work; indeed we can approximate u_{0} by smooth functions u_{n} with images included in the unit circle, and then F (u_{n}) = 0 for all n (see Remark 2.8). Another important remark is that the occurrence of a gap in this example is tightly connected with the structure of the singularity of u_{0}. In particular we cannot extend this example to Sobolev spaces W^{1,p} with p ≥ 2 or to the scalar case W^{1,p}(Ω, IR), even taking functionals with different growth. We think it should be very interesting to understand what happens in these cases. Finally, we want to point out the analogy of our example with the problem of relaxation of Dirichlet energy F (u) = ∫ |Du|^{2}dx for mappings which takes values into the sphere (see [3], [4], [15] and [16]), and indeed the function (|u|^{2} − 1)^{2} in the integrand in (1.2) plays a role which is very close to the geometrical constraints of this problem.

2. Statements and proofs of the results

In the following (unless differently stated) Ω is a bounded open subset of IR^{2}, u is a function of Ω into IR^{2} and we denote a point of the domain Ω as x = (x_{1}, x_{2}) and a point of the codomain IR^{2} as y = (y_{1}, y_{2}). B is the unit ball in IR^{2} and S = ∂B is the unit circle in IR^{2}. If A = (A_{ij}) is a matrix in IR^{2×2}, then the norm of A is given as usual by |A| := (∑_{i,j} A_{ij}^{2})^{1/2}. Notice that |A| ≥ 2| det A|. I denotes both the 2 × 2 identity matrix and

the identity map on IR^{2}. If F is a function on the set T and X is a subset of T, then F X denotes the restriction of F to X. If u : Ω → IR^{2} is a continuous function, A is an open set relatively compact in Ω and y belongs to IR^{2} \ u(∂A), then deg(u, A, y) is the degree of u restricted to A over the point y. For the basic results about degree we refer essentially to [11], chapters 1-3.

W^{1,p}(Ω, IR^{2}) (with 1 ≤ p ≤ ∞), Lip(Ω, IR^{2}) and C^{k}(Ω, IR^{2}) are the usual spaces of Sobolev, Lipschitz and C^{k} functions from Ω into IR^{2}. They are usually endowed with their strong topologies; we write W_{w}^{1,p} to indicate the Sobolev space W^{1,p} endowed with its weak topology. For every function u ∈ W^{1,2}(Ω, IR^{2}) we denote by Ju(x) the determinant of the matrix Du(x). If T and V are topological spaces, we write T ↪ V when T is included in V with continuous inclusion (i.e., when the topology of T is finer than the topology induced on T by V) and sequentially dense image.

Definition 2.1. Let T be a Hausdorff topological space and let F be a sequentially lower semicontinuous function on T. We say that a sequence {u_{n}} ⊂ T approximates u in energy if u_{n} → u and F (u_{n}) → F (u). Let X be a sequentially dense subset of T (i.e., every point of T is limit of some sequences of points of X). Then we define the following functions on T:

(i) F_{1}(X, T)(u) := inf(lim inf_{n} F (u_{n})) with u_{n} ∈ X and u_{n} → u, (ii) F_{2}(X, T) is the sequential relaxation of F X on T (namely, the maximal sequen-

tially lower semicontinuous function on T which is less than or equal to F on X).

Then F_{1}(X, T) ≥ F_{2}(X, T) ≥ F. When T is first countable (i.e., every point has a countable neighborhood basis) then F_{1}(X, T) and F_{2}(X, T) are the same function, that is, the relaxation of F X on T. If we drop the first countability assumption (for example when T is a Sobolev space endowed with the weak topology instead of the norm topology) we have to be more careful; in particular F_{1}(X, T) may be not sequentially l.s.c. and it may not agree with F_{2}(X, T) (and also F_{2}(X, T) may be not l.s.c.). Hence we have two slightly different problems: the approximation problem, that is determine whether F_{1}(X, T)(u) = F (u) (in other words, whether u can be approximated in energy by functions in X) and the relaxation problem, that is determine whether F_{2}(X, T)(u) = F (u). Equality F_{1}(X, T) = F may not hold in general: in this case we say that F has a gap from X to T. Notice that this equality hold when F is sequentially continuous with respect to some topology τ finer than the topology of T and such that X is τ dense in T.

Definition 2.2. For all u ∈ W^{1,1}(Ω, IR^{2}), set

F (u) := ∫_{Ω} f(u)|Du|^{2}dx  (2.1)

where f : IR^{2} → [0, ∞[ is a continuous map such that f(y) = 0 whenever |y| = 1, and set

C := 2 ∫_{B} f(y) dy .  (2.2)

Take x_{0} ∈ Ω and let u_{0} be the function given by

u_{0}(x) := (x − x_{0})/|x − x_{0}| .  (2.3)

Then u_{0} belongs to W^{1,p}(Ω, IR^{2}) for all p ∈ [1, 2[ and the following facts hold:

Theorem 2.3. Let F and u_{0} be given as in (2.1) and (2.2), and assume that C > 0. Then

(i) F is sequentially lower semicontinuous in the L^{1} topology;

(ii) F is (strongly) continuous in W^{1,p} for all p ≥ 2;

(iii) F_{1}(X, W^{1,p}) = F_{2}(X, W^{1,p}) = F W^{1,p} whenever p ≥ 2, and then F has no gap from X to W^{1,p};

(iv) F_{1}(X, T) = F_{1}(W^{1,2}, T) and F_{2}(X, T) = F_{2}(W^{1,2}, T) whenever X is a dense subset of W^{1,2} and W^{1,2} ↪ T;

(v) F_{1}(X, W_{w}^{1,p}) ≥ F_{2}(X, W_{w}^{1,p}) ≠ F whenever p ∈ [1, 2[ and X is a dense subset of W^{1,2}, and then F has a gap from X to W_{w}^{1,p}.

In particular, F (u_{0}) = 0 but F_{1}(X, W_{w}^{1,p})(u_{0}) = F_{2}(X, W_{w}^{1,p})(u_{0}) = C.

Remark 2.4. Statement (iii) means that every u ∈ W^{1,p} with p ≥ 2 may be approximated in energy in the W^{1,p} topology by smooth functions (Lipschitz or even C^{∞}). Statement (v) shows that this is false when p < 2 and in particular we can prove that no sequences in W^{1,2} approximates u_{0} in energy. More precisely, we have that F_{1}(X, W_{w}^{1,p})(u_{0}) = F_{2}(X, W_{w}^{1,p})(u_{0}) = C.

The difficult part of this theorem is to prove inequalities F_{1}(X, W_{w}^{1,p})(u_{0}) ≥ C and F_{2}(X, W_{w}^{1,p})(u_{0}) ≥ C: the proof of the second inequality relies on a result on currents we develope in section 3. Of course the first inequality may be deduced directly from the other (recalling that there always holds F_{1}(X, T) ≥ F_{2}(X, T)), but since this is the key lemma of this paper, we prefer to give also another indipendent proof which relies on simple properties of degree.

Proof of Theorem 2.3.

(i). Using Corollary 4.1 in [14], we can prove the lower semicontinuity with respect to the L^{1} topology of all functionals of the form

u ↦ ∫_{Ω} g(u)h(Du) dx, u ∈ W^{1,1}(Ω, IR^{2}),

where g is bounded and continuous and h is a convex function with linear growth. Since F is (trivially) the limit of an increasing sequence of such functionals, also F is lower semicontinuous. (ii) is well-known (cf. [9], chapter 3, and [5], chapter 2). (iii) and (iv) follow from (ii). (v). We may assume with no loss in generality that x_{0} = 0.

Taking into account (iv), inequalities C ≥ F_{1}(X, W_{w}^{1,p}) ≥ F_{2}(X, W_{w}^{1,p}) are proved if there exists a sequence of Lipschitz functions (v_{n}) which converges to u_{0} in W^{1,p} for all p ∈ [1, 2[ and verifies lim_{n} F (v_{n}) = C. Take indeed ρ_{n} ↓ 0 so that ρ_{n}B ⊂ Ω for all n and set

v_{n} := \{ \begin{matrix}x/ρ_{n} & if x ∈ ρ_{n}B, \\ x/|x| & otherwise. \end{matrix}  (2.4)

The sequence (v_{n}) converges to u_{0} in W^{1,p} for all p ∈ [1, 2[ and recalling that f(y) = 0 when |y| = 1,

lim_{n} F (v_{n}) = lim_{n} ∫_{ρ_{n}B} f(x/ρ_{n})\frac{2}{ρ_{n}^{2}} dx = 2 ∫_{B} f(y) dy = C .

Taking into account (iv), it is enough to prove inequalities F_{1}(X, W_{w}^{1,p})(u_{0}) ≥ C and F_{2}(X, W_{w}^{1,p})(u_{0}) ≥ C when X is C^{1}. This follows from Lemmas 2.5 and 2.6.

Lemma 2.5. Let F, C and u_{0} be as in (2.1), (2.2) and (2.3) respectively, and let u_{n} be a sequence of functions in C^{1}(Ω, IR^{2}) which converges to u_{0} in the weak topology of W^{1,1}. Then

lim inf_{n→∞} F (u_{n}) ≥ 2 ∫_{B} f(y) dy = C .  (2.5)

In other words, F_{1}(C^{1}, W_{w}^{1,1})(u_{0}) ≥ C.

Proof. First of all, notice that it is enough to prove that for every sequence (u_{n}) satisfying the hypothesis of Lemma 2.5 there exists a subsequence which satisfies (2.5). Now, let (u_{n}) be fixed and take r > 0 so that rB ⊂ Ω. By Lemma 3.9 below, we may find ρ ∈]0, r[ and a subsequence (u_{k}) such that u_{k} ρS converge to u_{0} ρS uniformly. We claim that this subsequence satisfies inequality (2.5).

For every integer k, set g_{k}(y) := card(u_{k}^{−1}(y)) for all y ∈ IR^{2}.

Since |A|^{2} ≥ 2| det A| for every matrix A ∈ IR^{2×2}, recalling area formula we obtain

∫_{Ω} f(u_{k})|Du_{k}|^{2}dx ≥ 2 ∫_{Ω} f(u_{k})|Ju_{k}| dx = 2 ∫_{IR_{2}} f(y)g_{k}(y) dy .

Thus conclusion follows from Fatou’s Lemma once we show that lim inf g_{k}(y) ≥ 1 for almost all y in B. To this end, define for all k

v_{k}(x) := \frac{|x|}{ρ} u_{k}(\frac{ρx}{|x|}) ∀x ∈ Ω .

Clearly each v_{k} is Lipschitz (and C^{1} in Ω \ {0}) and v_{k} ρS = u_{k} ρS. Hence deg(v_{k}, ρB, y) = deg(u_{k}, ρB, y) for all y ∈ IR^{2} \ v_{k}(ρS) (cf. [11], Theorem 3.1). Moreover, by the choice of ρ, v_{k} converges uniformly to I/ρ on Ω. Hence for all y ∈ B there holds definitively y ∉ v_{k}(ρS) and deg(v_{k}, ρB, y) = deg(I/ρ, ρB, y) (cf. [11], Theorem 3.1) and then

g_{k}(y) ≥ deg(u_{k}, ρB, y) = deg(v_{k}, ρB, y) = deg(I/ρ, ρB, y) = 1 ,  (2.6)

which ends the proof.

Lemma 2.6. Take F, C and u_{0} as in (2.1), (2.2) and (2.3) respectively. Then

F_{2}(C^{1}, W_{w}^{1,1})(u_{0}) ≥ 2 ∫_{B} f(y)dy = C .

Proof. Let ε > 0 be fixed. Since f is continuous we may find a smooth map ϕ on Ω×IR^{n} with compact support so that 0 ≤ ϕ(x, y) ≤ f(y) for all x, y and

∫ ϕ(0, y) dy ≥ ∫_{B} f(y) dy − ε = C/2 − ε .  (2.7)

Recalling that |A|^{2} ≥ 2(det A) for every matrix A, for all v ∈ C^{1}(Ω, IR^{2}) we get

F (v) = ∫_{Ω} f(v)|Dv|^{2}dx ≥ 2 ∫_{Ω} ϕ(x, v)Jv dx = 2⟨T v; ω⟩  (2.8)

where ω = ϕ dy^{1} ∧dy^{2} is a 2-form in Ω×IR^{2} and T v is the 2-dimensional current in Ω×IR^{2} associated to the graph of v as in Definition 3.1. Since the map v ↦ ⟨T v; ω⟩ is sequentially weakly continuous on W^{1,1} (Theorem 3.2), by (2.8) and Definition 2.1 we get

F_{2}(C^{1}, W_{w}^{1,1})(u_{0}) ≥ ⟨T u_{0}; ω⟩

and we claim that ⟨T u_{0}; ω⟩ = 2 ∫_{B} ϕ(0, y)dy: taking into account (2.7) and recalling that ε is arbitrarily taken, this would end the proof. To prove our claim, we take indeed v_{n} as in formula (2.4). The functions v_{n} converge to u_{0} in W^{1,1} and then

⟨T u_{0}; ω⟩ = lim_{n} ⟨T v_{n}; ω⟩ = lim_{n} ∫_{Ω} ϕ(x, v_{n})Jv_{n}dx

= lim_{n} ∫_{ρ_{n}B} ϕ(x, x/ρ_{n})ρ_{n}^{−2}dx = lim_{n} ∫_{B} ϕ(ρ_{n}y, y) dy

= ∫_{B} ϕ(0, y)dy

Remark 2.7. Lemma 2.6 may be proved without GMT framework using a recent result in [12] about functions with given Jacobien determinant. Take a smooth function ϕ : Ω → [0, 1] with compact support such that ϕ(0) = 1 and let g : IR^{2} → [0, ∞] be the continuous function which agrees with f in B and is 0 elsewhere.

Following [12], we may find a Lipschitz function G : IR^{2} → \overline{B} so that det DG = g everywhere and then, for every u ∈ C^{1}, g(u) Ju = J(G(u)), and

F (u) = ∫_{Ω} f(u)|Du|^{2}dx ≥ ∫_{Ω} g(u) Ju ϕ dx = ∫_{Ω} J(G(u)) ϕ dx .

One can show that the map u → ⟨ DetD(G(u)); ϕ⟩ is sequentially weakly continuous on W^{1,1} (cf. [17], and [9], section 4.2). Hence

F_{2}(C^{1}, W_{w}^{1,1})(u_{0}) ≥ ⟨ DetD(G(u_{0})); ϕ⟩ .

Moreover we may compute explicitly the right term of this inequality taking suitable v_{n} converging to u_{0} (e.g., as in formula (2.4)), and we get

F_{2}(C^{1}, W_{w}^{1,1})(u_{0}) ≥ 2 ∫_{B} g(y)dy = C .

Remark 2.8. It is important to notice the following fact: Lemmas 2.5 and 2.6 do not hold if we replace the W^{1,1} weak topology with the BV topology or any other weaker topology (we recall that a sequence in W^{1,1} converges in the BV topolgy if it converges in the L^{1} norm and is bounded in the W^{1,1} norm). In particular we can find a sequence of C^{∞} functions u_{n} which converges to u_{0} in the BV topology and F (u_{n}) = 0 for all n (and then lim_{n} F (u_{n}) = F (u_{0}) = 0). As usual, we assume x_{0} = 0 and then u_{0}(x) = x/|x|. Take r > 0 such that rB ⊃ Ω and set K := [0, 2π] × [0, r], and for all integers n, let ϕ_{n} : K → [0, 1] be a C^{∞} function s.t.:

(i) ϕ_{n} = 0 in a neighborhood of ∂K, (ii) ϕ_{n} = 1 out of a set A_{n} with measure less than 1/n, (iii) ∥Dϕ_{n}∥_{1} are uniformly bounded.

Now let u_{n} be the function which takes each point x = (ρ cos t, ρ sin t) with (t, ρ) ∈ K into

u_{n}(x) := ( cos(ϕ_{n}(t, ρ) t), sin(ϕ_{n}(t, ρ) t)) .

Recalling (i), it is not difficult to verify that each function u_{n} is a well-defined C^{∞} function from rB into S and then F (u_{n}) = 0. By (ii), u_{n}(x) = x/|x| = u_{0}(x) out of the set A′_{n} of all x = (ρ cos t, ρ sin t) with (t, ρ) ∈ A_{n} and since the measures of A_{n} converge to 0, the measures of A′_{n} converge to 0, and then u_{n} converge to u_{0} in L^{1}(Ω, IR^{2}). Using (iii), a simple computation shows that the derivatives of u_{n} are bounded in the L^{1} norm, and then we have convergence in the BV topology.

Notice that the proof of Lemma 2.5 does not work in the BV topology because Lemma 3.8 fails, and this happens essentially because the BV convergence in W^{1,1}(0, 1) does not imply uniform convergence (but the W^{1,1} weak convergence does). The proof of Lemma 2.6 fails because Theorem 3.2 fails (cf. Remark 3.4).

Remark 2.9. Let T be W^{1,1} endowed with the BV topology (or any weaker topology) and take u_{0} as in Theorem 2.3: in the previous remark, we have showed that F_{1}(C^{∞}, T)(u_{0}) = F_{2}(C^{∞}, T)(u_{0}) = F (u_{0}) = 0 (and then, taking into account statement (iv) of Theorem 2.3, the same hold if we replace C^{∞} with any dense subset of W^{1,2}). Thus the following question arises: is there a gap (for F) from C^{∞} to T?

Remark 2.10. (The higher dimension case)

Let N > 2 be fixed. Let Ω be a bounded open subset of IR^{N} and for all u ∈ W^{1,1}(Ω, IR^{N}) set

F (u) := ∫_{Ω} f(u)|Du|^{N}dx

where f : IR^{N} → [0, ∞[ is a continuous function such that f(y) = 0 whenever |y| = 1. Let B be the unit ball of IR^{N} and assume that C := N^{N/2} ∫_{B} f(y)dy > 0. Take x_{0} ∈ Ω and set u_{0}(x) := (x − x_{0})/|x − x_{0}| for all x ∈ Ω (u_{0} belongs to W^{1,p} for all p < N).

Then the essential statements of Theorem 2.3 may be generalized as follows (without essential modifications in the proofs): F is lower semicontinuous on W^{1,1} with respect to the L^{1} topology, and is continuous on W^{1,N} with respect to the norm topology. Then F_{1}(X, W^{1,p}) = F_{2}(X, W^{1,p}) = F whenever p ≥ N and X is a dense subset of W^{1,p} (and then there is no gap from X to W^{1,p}).

If p ∈]N − 1, N[, F_{1}(X, W_{w}^{1,p})(u_{0}) = F_{2}(X, W_{w}^{1,p})(u_{0}) = C while F (u_{0}) = 0 (X is a dense subset of W^{1,N}) and then there is a gap from X to W_{w}^{1,p}.

If p ∈ [1, N − 1], then F_{1}(X, W_{w}^{1,p})(u_{0}) = F_{2}(X, W_{w}^{1,p})(u_{0}) = F (u_{0}) = 0 whenever X is a dense subset of W^{1,N}: in particular we may find a sequence of C^{∞} functions u_{n} which weakly converge to u_{0} in W^{1,N−1} such that F (u_{n}) = 0 for all n (cf. Remark 2.8); in this case we are not able to say whether there is gap or not (cf. Remark 2.9).

G. Alberti, P. Majer / Gap phenomenon for some autonomous functionals 39

We end this section with an example of Lavrentiev phenomenon for some boundary value problems.

Definition 2.11. Let ε > 0 and p ∈]1, 2[ be given, and for all u in W^{1,p}(B, IR^{2}), set

G(u) := ∫_{B} (f(u)|Du|^{2} + ε|Du − I|^{p})dx  (2.9)

where f : IR^{2} → [0, ∞[ is taken as in Definition 2.2 (i.e., f is a continuous function such that f(y) = 0 whenever |y| = 1) and assume C := 2 ∫_{B} f(y)dy > 0. As usual we set u_{0}(x) := x/|x| for all x ∈ B. Let C^{1}(\overline{B}, IR^{2}) denote the class of all functions in C^{1}(B, IR^{2}) which admit a continuous extension to \overline{B}.

Thus we have the following result.

Theorem 2.12. G is a weakly lower semicontinuous functional on W^{1,p} with p-growth, there exist

M_{1} := min {G(u) : u ∈ W^{1,p}(B, IR^{2}), u = I on ∂B} ,  (2.10)
M_{2} := min \{G(u) : u ∈ C^{1}(\overline{B}, IR^{2}), u = I on ∂B, \} ,  (2.11)

and we have M_{1} ≤ G(u_{0}) ≤ \frac{5πε}{2 − p}, M_{2} = G(I) = C. Hence M_{1} < M_{2} if ε is small

enough.

Proof. G is weakly lower semicontinuous by well-known theorems. Since it has p-growth, it is coercive on the affine space of all functions u ∈ W^{1,p}(B, IR^{2}) such that u = I on ∂B, endowed with the weak topology of W^{1,p}. Hence G attains a minimum M_{1} and

M_{1} ≤ G(u_{0}) = ε ∫_{B} |Du_{0} − I|^{p}dx ≤ \frac{5πε}{2 − p} .

On the other side, G(I) = ∫_{B} f(y) 2dy = C, and then it is enough to prove that G(u) ≥ C for all u ∈ C^{1}(\overline{B}, IR^{2}) such that u = I on ∂B. To this end, notice that for all such u,

deg(u, B, y) = deg(I, B, y) = \{ \begin{matrix}1 & if y ∈ B, \\ 0 & otherwise \end{matrix}

(cf. Theorem 3.1 in [11]) and then, using area formula,

\begin{aligned}G(u) & = ∫_{B} f(u)|Du|^{2}dx ≥ 2 ∫_{B} f(u)Ju dx \\ & = 2 ∫_{IR_{2}} f(y) deg(u, B, y) dy ≥ 2 ∫_{B} f(y)dy = C \end{aligned}

3. Appendix

In this appendix we prove two results used above (Theorem 3.2 and Lemma 3.9). The first one is a theorem in current theory; D^{k}(A) is the (locally convex) space of all smooth k-forms with compact support in the open set A, and D_{k}(A) is the dual of D^{k}(A), i.e., the space of all k-dimensional currents on A. D_{k}(A) is always endowed with the dual topology. In this notation D^{0}(A) is the space of smooth functions on A with compact support and D_{0}(A) the space of Schwartz distributions. For the basic results and notation in current theory, we refer to [19].

Definition 3.1. When u : Ω → IR^{2} is a C^{1} function, then its graph is a 2-dimensional submanifold of class C^{1} of Ω × IR^{2} without boundary and we may orient it so that the orientation induced on Ω agrees with the standard orientation of IR^{2}. Thus we may see it as a current in Ω × IR^{2} (more precisely, a 2-dimensional current without boundary) which we denote by T u.

As usual, the standard basis B of the space of 2-covectors in IR^{2}×IR^{2} is given by dx^{1}∧dx^{2}, dx^{i} ∧dy^{j} (with i = 1, 2 and j = 1, 2) and dy^{1} ∧dy^{2}, and then every 2-form ω ∈ D^{2}(Ω×IR^{2}) may be written as

ω = ∑_{α∈B} ω_{α}α  (3.1)

with ω_{α} ∈ D^{0}(Ω × IR^{2}). Moreover, for every u ∈ C^{1}(Ω, IR^{2}) and every ϕ ∈ D^{0}(Ω × IR^{n}), we have

⟨T u; ϕ dx^{1} ∧ dx^{2}⟩ = ∫_{Ω} ϕ(x, u) dx  (3.2)

⟨T u; ϕ dx^{i} ∧ dy^{j}⟩ = ∫_{Ω} ϕ(x, u)(−1)^{i} \frac{∂u^{j}}{∂xî}dx  (3.3)
⟨T u; ϕ dy^{1} ∧ dy^{2}⟩ = ∫_{Ω} ϕ(x, u)Ju dx  (3.4)

where i = 1, 2, j = 1, 2 and î = 1 if i = 2, î = 2 if i = 1.

Let T : C^{1}(Ω, IR^{2}) → D_{2}(Ω × IR^{2}) be the map which takes each u in T u: we claim that T admits a continuous extension to the space W_{loc}^{1,1}(Ω, IR^{2}). More precisely, we have the following theorem.

Theorem 3.2. Let T be the function given before. Then there exists a unique sequentially weakly continuous map (which we still denote as T) from W_{loc}^{1,1}(Ω, IR^{2}) into D_{2}(Ω × IR^{2}) which extends T.

Remark 3.3. Uniqueness is obvious, because C^{1} is dense in W_{loc}^{1,1}. To prove existence, it is enough to show that for every 2-form ω ∈ D^{2}(Ω, IR^{2}) the map which takes each u ∈ C^{1} into ⟨T u; ω⟩ admits a continuous extension to W_{loc}^{1,1}. In fact, we shall prove that this extension exists for all 2-forms ω with compact support of class C^{1}, and not only C^{∞}, and taking into account the decomposition (3.1) and formulas (3.2), …, (3.4), it is enough to extend to W_{loc}^{1,1} the map

u ↦ ∫_{Ω} ϕ(x, u) Mu dx  (3.5)

whenever ϕ ∈ C_{c}^{1}(Ω × IR^{2}) and Mu is a minor of the matrix Du.

It is well-known how extend (3.5) to the space W_{loc}^{1,2}, because the same formula makes sense for all functions in this space and gives a sequentially weakly continuous functional (see [9], section 4.2), and moreover in this case T u is an integer multiplicity current without boundary by a well-known compactness theorem (see [19], Theorem 27.3). In the general case, formula (3.5) does not make sense because the minor of order 2 of Du, namely the Jacobian determinant, is not given for W^{1,1} functions. In this case T u may be not a locally rectifiable current and may have not locally bounded mass (but still it has no boundary); more precisely we show that for all u ∈ W_{loc}^{1,1}, T u actually belongs to the dual of all 2-forms of class C^{1} (and not only C^{∞}) with compact support, but by no means to the dual of all continuous 2-forms with compact support.

Remark 3.4. The (sequential) continuity of the extension does not hold if we replace in W_{loc}^{1,1} the weak topology with the BV_{loc} topology (cf. Remark 2.8) or any other weaker topology. With no loss in generality we may assume that 0 belongs to Ω and then set u_{0} = x/|x| for all x ∈ Ω. Now take u_{n} as in Remark 2.8 and v_{n} as in formula (2.4): they are both sequences of Lipschitz funtions which converge to u_{0} in the BV topology, but T u_{n} and T v_{n} do not converge to the same current.

Indeed, if we take ω := ϕ dy^{1} ∧ dy^{2} where ϕ : Ω × IR^{2} → IR is a smooth functions with compact support such that

(i) ϕ(x, y) = 1 in a neighborhood of \{0\} × \frac{1}{2}B,

(ii) ϕ(x, y) = 0 whenever |y| = 1,

an explicit computation shows that ⟨T u_{n}; ω⟩ = 0 for all n and ⟨T v_{n}; ω⟩ ≥ π/4 definitively.

Remark 3.5. Of course, the result of Theorem 3.2 can be extended (without any essential modifications in the proof) to arbitrary dimensions: if Ω is an open subset of IR^{N}, and T is the map which takes each function u ∈ C^{1}(Ω, IR^{M}) in its (oriented) graph, considered as an N-dimensional current of Ω × IR^{M} (N, M positive integers greater than 1), then we may find a unique continuos extension of T to the space W_{loc}^{1,p}(Ω, IR^{M}) for all p ≥ N ∧ M − 1. This extension is (sequentially) weakly continuous unless p = N ∧ M − 1 and N ∧ M > 2; in the last case we have the strong continuity only. If N ∧ M = 2 and p = 1, we have (sequential) continuity with respect to the W_{loc}^{1,1} weak topology but not with respect to the BV topology (or any other weaker topology). If p < N ∧ M − 1, there is no continuous extension.

Proof of Theorem 3.2.

Taking into account Remark 3.3, we shall prove that the map u ↦ ∫_{Ω} ϕ(x, u)Mu dx may

be extended to W_{loc}^{1,1} whenever ϕ ∈ C_{c}^{1}(Ω × IR^{2}) and Mu is a minor of Du.

When we consider minors of order 1 or 0, it is enough to recall that the map

u ↦ ∫_{Ω} ϕ(x, u)Mu dx u ∈ W_{loc}^{1,1}(Ω, IR^{2})  (3.6)

is a (well-defined) continuous map for every ϕ ∈ C_{c}(Ω × IR^{2}) (in fact, for every bounded continuous ϕ with support included in K ×IR^{2} for some compact K ⊂ Ω, see Lemma 3.6). When we consider the minor of order 2 (i.e., the Jacobian determinant) there is no trivial extension of the function because Ju is not given in general for functions in W^{1,1}. Using Lemma 3.8 below, for all ϕ ∈ C_{c}^{1}(Ω × IR^{2}) we may find a bounded Rϕ ∈ C(Ω × IR^{2}, IR^{2}), with support included in K × IR^{2} for some compact K ⊂ Ω, such that

∫_{Ω} ϕ(x, u)Ju dx = ∫_{Ω} Rϕ(x, u) · Du dx ∀u ∈ C^{1}(Ω, IR^{2})  (3.7)

and the right term of this equality is a function which can be easily extended to the whole space W_{loc}^{1,1} using Lemma 3.6.

Lemma 3.6. The following facts hold.

(i) For every bounded continuous function ϕ on Ω × IR^{2}, the mapping

u ↦ ∫_{Ω} ϕ(x, u) dx u ∈ W_{loc}^{1,1}(Ω, IR^{2})

is sequentially weakly continuous.

(ii) For every bounded continuous function ϕ on Ω × IR^{2} with support included in K × IR^{2} for some compact K ⊂ Ω, the mapping

u ↦ ∫_{Ω} ϕ(x, u)Du dx u ∈ W_{loc}^{1,1}(Ω, IR^{2})

is sequentially weakly continuous.

Proof. (i). Let u_{n} be a sequence in W_{loc}^{1,1} which weakly converges to u. Then it strongly converges in L_{loc}^{1} and hence, eventually taking a subsequence, we may assume that u_{n} converges to u a.e. Thus ϕ(x, u_{n}(x)) converges to ϕ(x, u(x)) for a.a. x because ϕ is continuous, and it is enough to apply Lebesgue theorem.

(ii). Let u_{n} be a sequence in W_{loc}^{1,1} which weakly converges to u and let A be a relatively compact open subset of Ω which includes K and set v_{n}(x) = ϕ(x, u_{n}(x)), v(x) = ϕ(x, u(x)) for all x and n. As before, we may assume that v_{n} converges a.e. to v. Moreover the functions v_{n} are uniformly bounded and 0 outside A. Since Du_{n} weakly converges to Du in L_{loc}^{1}, it is well-known that v_{n}Du_{n} weakly converges to v Du.

G. Alberti, P. Majer / Gap phenomenon for some autonomous functionals 43

Lemma 3.7. Let α be a function in C^{1}(Ω × IR^{2}) with support included in K × IR^{2} for some compact K ⊂ Ω, and let u = (u^{1}, u^{2}) be a function in C^{1}(Ω, IR^{2}). Then we have the following identity:

∫_{Ω} [α(x, u) + u^{2} \frac{∂α}{∂y_{2}} (x, u)]Ju dx = ∫_{Ω} (u^{2} \frac{∂α}{∂x_{2}} (x, u), −u^{2} \frac{∂α}{∂x_{1}} (x, u)) · Du dx . (3.8)

Proof. A simple density argument shows that we may assume α and u of class C^{∞}. Now, writing Ju as a divergence (cf. [17], and [9], section 4.2)

Ju = \frac{∂}{∂x_{1}} ( − u^{2} \frac{∂u^{1}}{∂x_{2}} ) + \frac{∂}{∂x_{2}} (u^{2} \frac{∂u^{1}}{∂x_{1}} )  (3.9)

and integrating by parts α(x, u)Ju we obtain (with some computations)

∫_{Ω} α(x, u)Ju dx = ∫_{Ω} [\frac{∂}{∂x_{1}} (α(x, u)) u^{2} \frac{∂u^{1}}{∂x_{2}} − \frac{∂}{∂x_{2}} (α(x, u)) u^{2} \frac{∂u^{1}}{∂x_{1}} ] dx

= ∫_{Ω} (u^{2} \frac{∂α}{∂x_{2}} (x, u), −u^{2} \frac{∂α}{∂x_{1}} (x, u)) · Du dx−

− ∫_{Ω} [u^{2} \frac{∂α}{∂y_{2}} (x, u)]Ju dx .

Lemma 3.8. For every ϕ ∈ C_{c}^{1}(Ω × IR^{2}) there exists a bounded Rϕ ∈ C(Ω × IR^{2}, IR^{2}) with support included in K × IR^{2} for some compact K ⊂ Ω which satisfies

∫_{Ω} ϕ(x, u)Ju dx = ∫_{Ω} Rϕ(x, u) · Du dx ∀u ∈ C^{1}(Ω, IR^{2}) .  (3.10)

Proof. Let K be the projection on Ω of the support of ϕ and set

α(x, y) := ∫_{0}^{1} ϕ(x, y_{1}, ty_{2})dt ∀(x, y) ∈ Ω × IR^{2} .  (3.11)

A simple computation shows that α is a function of class C^{1} with support included in K × IR^{2} which satisfies the equation ϕ = α + y_{2} \frac{∂α}{∂y_{2}} . Hence formula (3.8) yields

∫_{Ω} ϕ(x, u)Ju dx = ∫_{Ω} (u^{2} \frac{∂α}{∂x_{2}} (x, u), −u^{2} \frac{∂α}{∂x_{1}} (x, u)) · Du dx ,  (3.12)

and then it is enough to take

Rϕ := (y_{2} \frac{∂α}{∂x_{2}} , −y_{2} \frac{∂α}{∂x_{1}} ) .  (3.13)

44 G. Alberti, P. Majer / Gap phenomenon for some autonomous functionals

Formulas (3.11) and (3.13) immediately yield that Rϕ is bounded and continuous, with support included in K × IR^{2}.

Lemma 3.9. Let (u_{n}) be a sequence of functions in Lip(Ω, IR^{2}) which weakly converges to u in W^{1,1}, and take r such that rB ⊂ Ω. Then for almost all ρ ∈]0, r[, (u_{n} ρS) admits a subsequence which converges to u ρS uniformly.

Proof. Since the functions u_{n} weakly converge to u in W^{1,1}(Ω, IR^{2}), they converge also in L^{1} and the functions Du_{n} are uniformly integrable in L^{1} (by Dunford-Pettis theorem). Hence there exist a finite constant C and a convex function f : [0, +∞] → [0, +∞] such that lim_{t→∞} f(t)/t = +∞ and

∫_{Ω} f(|Du_{n}|)dx ≤ C ∀n  (3.14)

(cf. [13], chapter II, section 2). For all n and all ρ ∈]0, r[ set

\begin{aligned}u_{n}^{ρ} & := u_{n} ρS, u^{ρ} := u ρS \\ g_{n}(ρ) & := ∫_{ρS} |u_{n} − u| dH_{1} \\ h_{n}(ρ) & := ∫_{ρS} f(|Du_{n}|) dH_{1} . \end{aligned}

Then ∫_{0}^{r} g_{n}(ρ) dρ ≤ ∥u_{n} −u∥_{1} and g_{n} converges to 0 in L^{1}(0, r). Hence, eventually passing to a subsequence, we may assume that, for a.a. ρ, g_{n}(ρ) converges to 0, i.e., u_{n}^{ρ} converge to u^{ρ} in L^{1}(ρS, IR^{2}). Moreover (3.14) yields

∫_{0}^{r} h_{n}(ρ) dρ ≤ ∫_{Ω} f(|Du_{n}|)dx ≤ C ∀n

and then Fatou’s lemma yields

∫_{0}^{r} lim inf_{n→∞} h_{n}(ρ) dρ ≤ lim inf_{n→∞} ∫_{0}^{r} h_{n}(ρ) dρ ≤ C .

Thus, for a.a. ρ, u_{n}^{ρ} → u^{ρ}, lim inf_{n} h_{n}(ρ) < ∞, and we may find a subsequence n_{k} so that h_{n_{k}}(ρ) is bounded. Hence u_{n_{k}}^{ρ} is a sequence of functions in C^{1}(ρS, IR^{2}) which converges to u^{ρ} in L^{1} with uniformly integrable derivatives (cf [13], chapter II, section 2) and so we have convergence in the weak topology of W^{1,1}, which yields uniform convergence.

Acknowledgements. We wish to thank Giuseppe Buttazzo, who suggested the problem and helped us with many useful conversations.

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