Journal of Convex Analysis Volume 3 (1996), No. 2, 329–347

Numerical Approximation of Relaxed Variational Problems

T. Roubíček

Mathematical Institute, Charles University, Sokolovská 83, CZ-186 00 Praha 8,

and Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic,

Pod vodárenskou věží 4, CZ-182 08 Praha 8, Czech Republic. e-mail: roubicek@karlin.mff.cuni.cz

Received January 8, 1995 Revised manuscript received July 3, 1995

Non-quasiconvex variational problems require, in general, a relaxation to ensure existence of their solutions. Here a relaxation by (generalized) Young functionals is used and then a finite-element method for a direct numerical approximation of the relaxed problems is developed. Beside a mere convergence, also some error estimates are analysed and a comparison with other methods is discussed.

Keywords : Vectorial variational problem, relaxation, generalized Young functionals, finite-element approximation.

1991 Mathematics Subject Classification: 65N30, 49M99, 73V25.

1. Introduction

In this paper we will treat the multidimensional vectorial variational problem

minimize Φ(u) = ∫_{Ω} φ(x, u(x), ∇u(x)) dx for u ∈ W^{1,p}(Ω; IR^{m}) ,  (P)

where Ω ⊂ IR^{n} is a bounded Lipschitz domain, φ : Ω × (IR^{m} × IR^{mn}) → IR : (x, u, A) ↦ φ(x, u, A) a Carathéodory function, min(m, n) < p < +∞ , with m, n ≥ 1, and W^{1,p}(Ω; IR^{m}) the Sobolev space of functions u : Ω → IR^{m} with the norm ∥u∥_{W^{1,p}(Ω;IR^{m})} = ∥u∥_{L^{p}(Ω;IR^{m})} + ∥∇u∥_{L^{p}(Ω;IR^{mn})}. Throughout the paper, we will use the standard notation concerning function spaces C, C^{0}, C^{0,α}, and L^{p}, standing respectively for continuous, continuous bounded, α-Hölder continuous, and p-integrable functions on the domains indicated. Moreover, W^{α,p} will denote the Sobolev (if α is an integer) or Sobolev-Slobodecki (if α is an non-integer) space. We are especially interested in the case when φ(x, u, ·) : IR^{mn} → IR is not quasiconvex (in the sense of Morrey [19]) so that the minimum in (P) is generally not achieved. In other words, (P) may have no solution in W^{1,p}(Ω; IR^{m}) and a need of its relaxation immediately follows. Here we will consider the relaxation not by weakly l.s.c. (=lower semicontinuous) envelope of Φ (which requires quasiconvexification of φ) but the relaxation by continuity,

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330 T. Roubíček / Numerical approximation of relaxed variational problems

using Young measures; cf. [2, 3, 9]. This relaxation has the advantages that it can avoid the generally difficult quasiconvexification and keeps a certain “limit information” about oscillations of the minimizing sequences for (P). Here we will use rather so-called generalized Young functionals, which are more appropriate especially in the case min(m, n) ≥ 2, and make the relaxation according to [25]. This is done in Sect. 2. The aim of this paper is to propose a direct numerical approximation of the relaxed problem and to compare it with the usual approximation which uses standard finiteelement discretization of the original problem (P) converging eventually to a solution of the relaxed problem, cf. [6, 7, 8, 11, 12, 13, 14, 17, 21]. The formulation of the approximate problem as well as the convergence analysis are performed in Sect. 3, while a more detailed analysis supporting the actual implementation is made in Sect. 4. Finally, some error estimates are obtained in Sect. 5 and a comparison is discussed in Sect. 6. Another approximation involving element-wise homogeneous Young measures has been recently proposed by Nicolaides and Walkington [20] but without any analysis. In view of some results by Dacorogna [15], the scheme from [20] is relevant if the quasiconvexification of φ(x, u, ·) coincides with rank-1 convexification, while our scheme will basically apply if the quasiconvexification of φ(x, u, ·) is polyconvex. In particular, for scalar (m = 1) or one-dimensional (n = 1) problems, the approximation theory presented here is fairly complete. The requirement of the polyconvex quasiconvexification is certainly very artificial and thus, in the case min(m, n) > 1, our approximation theory is rather only a first step towards a very complicated problem; nevertheless, sometimes it is possible to verify experimentally that the polyconvex and the quasiconvex envelope do not differ much from each other, cf. [16].

2. Generalized Young functionals and relaxation of (P)

Let us introduce briefly some notation and some results from [25]. Following the original idea by L.C. Young [28], the generalized Young measures (or, more precisely, functionals) will be considered as certain linear continuous functionals on a suitable space H of Carathéodory integrands Ω × IR^{mn} → IR, cf. also [26, 27]. Though there is quite a large freedom in the choice of H, we will take one particular and enough large space, namely

H = {h_{0} + ∑_{s=0}^{min(m,n)} g_{s} ⊗ adj_{s}; h_{0} ∈ G ⊗ V, g_{s} ∈ L^{p/(p−s)}(Ω; IR^{σ(s)})} ,  (2.1)

where G ⊃ C^{0}(Ω̄) is a separable subspace of L^{∞}(Ω) closed under point-wise multiplication, and adj_{s} : IR^{mn} → IR^{σ(s)} assigns each matrix A ∈ IR^{mn} its cofactors of order s (i.e. the determinants of all s × s-submatrices) with the convention σ(0) = 1 and adj_{0} = 1; clearly

σ(s) = (_{s}^{m})(_{s}^{n}) = \frac{m!}{s!(m−s)!} \frac{n!}{s!(n−s)!}. Of course, g_{s} ⊗ adj_{s} abbreviates ∑_{l=1}^{σ(s)}[g_{s}]_{l} ⊗ [adj_{s}]_{l} with “⊗” denoting the usual tensorial product of functions, i.e. [g ⊗ v](x, A) = g(x)v(A). Obviously, adj_{1} ≅ id : IR^{mn} → IR^{mn} is the identity on IR^{mn}. Furthermore, V in (2.1) is supposed to satisfy

V a linear subspace of C(IR^{mn}),  (2.2a)

∀v ∈ V : sup_{A∈IR^{mn}} |v(A)|/(1 + |A|^{p}) < +∞,  (2.2b)
∀ 1 ≤ s ≤ min(m, n) : adj_{s} ∈ V^{σ(s)},  (2.2c)
{A ↦ v(A)/(1 + |A|^{p}); v ∈ V } is separable,  (2.2d)
∀v ∈ V : v^{#} ≠ −∞ ⇒ v^{#} is polyconvex,  (2.2e)

where v^{#}(A) = inf_{u∈W_{0}^{1,∞}(Ω,IR^{n})} ∫_{Ω} v(A + ∇u(x)) dx is the quasiconvex envelope of v.

Recall still that v is called polyconvex [1] if it can be expressed as a convex function of (adj_{s} A)_{s=1}^{min(m,n)}. For examples of V satisfying (2.2) we refer to [24]. Note that the only nontrivial condition (2.2e) is void if min(m, n) = 1. We can introduce a (possibly only semi-) norm on H from (2.1) by

∥h∥_{H} = inf ; h_{0}+∑_{s} g_{s}⊗adj_{s}=h (ess ; sup_{x∈Ω} sup_{A∈IR^{mn}} \frac{|h_{0}(x, A)|}{1 + |A|^{p}} + ∑_{s=0}^{min(m,n)} ∥g_{s}∥_{L^{p/(p−s)}(Ω;IR^{σ(s)})}) .

(2.3)

This norm makes H separable because of (2.2d) and of the separability of G. Therefore the weak* topology of the dual space H^{∗} is metrizable when restricted to bounded subsets. Furthermore, let us suppose that H contains a coercive integrand in the sense

∃h_{c} ∈ H ∀(x, A) ∈ Ω × IR^{mn} : h_{c}(x, A) ≥ |A|^{p},  (2.4)

and imbed L^{p}(Ω; IR^{mn}) (strong,weak*)-continuously into H^{∗} via the imbedding i_{H} : L^{p}(Ω; IR^{mn}) → H^{∗} defined by ⟨i_{H}(y), h⟩ = ∫_{Ω} h(x, y(x)) dx with h ∈ H. The elements of the set

Y_{H}^{p}(Ω; IR^{mn}) = w^{∗}-cl_{H^{∗}}i_{H}(L^{p}(Ω; IR^{mn}))  (2.5)

will be addressed as generalized Young functionals; it referes to the fact that for H = L^{1}(0, T; C_{0}(IR^{mn})) the elements of Y_{H}^{p}(Ω; IR^{mn}) attainable by bounded sequences can be identified with the classical Young measures. It was proved in [26, 27] that Y_{H}^{p}(Ω; IR^{mn}) is a convex, closed, weakly* σ-compact, locally compact, and also locally sequentially compact subset of H^{∗}. Let us also denote the set of all so-called gradient generalized Young functionals by

G_{H}^{p}(Ω; IR^{mn}) = {η ∈ Y_{H}^{p}(Ω; IR^{mn});  (2.6)

∃ a sequence {u_{k}}_{k∈IN} ⊂ W^{1,p}(Ω; IR^{m}) : w^{∗} - lim_{k→∞} i_{H}(∇u_{k}) = η} .

Let us remark that G_{H}^{p}(Ω; IR^{mn}) is not convex when min(m, n) ≥ 2. Indeed, this follows from [25, Corollary 2.1] when one realizes that, e.g., for h = 1 ⊗ v ∈ H with v quasiaffine but nonconvex, the functional Φ(u) = ∫_{Ω} h(x, ∇u(x)) dx is weakly l.s.c. on W^{1,p}(Ω; IR^{m}) but nonconvex, which can be seen if one makes a convex combination of u_{1}, u_{2} ∈ W^{1,p}(Ω; IR^{m}) in the form u_{i}(x) = A_{i}x with suitable A_{i} such that Rank(A_{1} − A_{2}) ≥ 2. For k ∈ IN we define the bilinear mapping (h, η) ↦ h^{•}η : H^{k} ×H^{∗} → [G^{∗}]^{k} by ⟨h^{•}η, g⟩ =

⟨η, g · h⟩ for any g ∈ G^{k}. We can understand h^{•}η as a substitution of η into an IR^{k}-valued

integrand h. It can be shown [26, 27] that η ↦ h^{•}η is an affine continuous extension of the Nemytskii mapping y ↦ h ∘ y. The important property of a particular η ∈ Y_{H}^{p}(Ω; IR^{mn}) is its possible “non-concen-tration”: we say that η ∈ Y_{H}^{p}(Ω; IR^{mn}) is p-nonconcentrating if there is a sequence {y_{k}}_{k∈IN} such that η =w*-lim_{k→∞}i_{H}(y_{k}) and that the set {|y_{k}|^{p}; k ∈ IN} is relatively weakly compact in L^{1}(Ω). Every such η admits a representation in terms of the classical Young measures, that means there is a (generally not uniquely determined) weakly* measurable mapping ν : x ↦ ν_{x}, called a Young measure, from Ω to regular probability measures on IR^{mn} such that

∀h ∈ H : h^{•}η = ⟨ν, h⟩ in the sense of L^{1}(Ω) ,  (2.7)

where ⟨ν, h⟩ abbreviates, as usual, the function x ↦ ⟨ν_{x}, h(x, ·)⟩ = ∫_{IR}^{mn} h(x, A)ν_{x}(dA); cf. [25, Remark 4.2]. We will denote the extended potential by Φ̄ : W^{1,p}(Ω; IR^{m}) × G_{H}^{p}(Ω; IR^{mn}) → IR and always suppose that

∀(u, η) ∈ w^{∗}-cl i(W^{1,p}(Ω; IR^{m})) : Φ̄(u, η) = lim inf_{i(u)→(u,η) weakly*} Φ(u) ,  (2.8)

where i : W^{1,p}(Ω; IR^{m}) → W^{1,p}(Ω; IR^{m}) × H^{∗} is defined by i(u) = (u, i_{H}(∇u)). Then we can define the relaxed variational problem:

\begin{matrix}minimize & Φ̄(u, η) \\ subject to & u ∈ W^{1,p}(Ω; IR^{m}), η ∈ G_{H}^{p}(Ω; IR^{mn}) , \\ & ∇u = (1 ⊗ id)^{•}η . \end{matrix}  \}  (RP)

For all our considerations we will also need the coercivity of (P), this means

∃a ∈ L^{1}(Ω) ∃b, c > 0 : φ(x, u, A) ≥ a(x) + b|u|^{p} + c|A|^{p} .  (2.9)

Proposition 2.1. Under the conditions (2.1), (2.2a-d), (2.3), (2.4), (2.8) and (2.9), (RP) is a proper relaxation of (P) in the sense that

a) (RP) has always a solution,

b) min(RP) = inf(P),

c) every cluster point of every minimizing sequence of (P) solves (RP), and

d) every solution of (RP) can be attained by a sequence minimizing (P).

Proof. See [25, Proposition 2.2 with Example 4.1] and realize that H is here supposed separable, which allows us to work in terms of sequences in Y_{H}^{p}(Ω; IR^{mn}).

3. Approximation of (RP)

We want to approximate (RP) directly. For this reason, we make a discretization of the admissible subset of W^{1,p}(Ω; IR^{m}) × G_{H}^{p}(Ω; IR^{mn}) on which (RP) is defined and change the structure of the constraints to make them readily implementable. We will use the finite-element technique, supposing, for simplicity, that Ω is polyhedral. For any d > 0, let T_{d} be a triangulation of Ω consisting of elements (=simplexes) of the diameter not exceeding d. For d ≥ d′ > 0, we suppose that T_{d} ⊂ T_{d′}, this means T_{d′} is a refinement of T_{d}. Furthermore, we suppose that the family {T_{d}}_{d>0} is regular in the sense of [10], i.e. there is ε > 0 such that each element of each T_{d} contains a ball of radius εd. Our aim is to approximate W^{1,p}(Ω; IR^{m}) by element-wise affine functions and Y_{H}^{p}(Ω; IR^{mn}) by element-wise constant (=homogeneous) generalized Young functionals. As for the former discretization, we put simply

U_{d} = {u ∈ W^{1,p}(Ω; IR^{m}); ∀E ∈ T_{d} : u|_{E} affine} .  (3.1)

The latter discretization is performed by the projection operator P_{d} : H → H defined by

[P_{d}h](x, y) = \frac{1}{meas(E)} ∫_{E} h(x̃, y) dx̃ if x ∈ E ∈ T_{d} .  (3.2)

Note that this projector is continuous with respect to the norm (2.3), and so is the adjoint projector P_{d}^{∗} : H^{∗} → H^{∗}. We put

Y_{d} = P_{d}^{∗}Y_{H}^{p}(Ω; IR^{mn}) .  (3.3)

Proposition 3.1. Y_{d} is convex, weakly* σ-compact, weakly* locally compact subset of Y_{H}^{p}(Ω; IR^{mn}). If V is also finite-dimensional, then Y_{d} is strongly locally compact. Moreover, for all d ≥ d′ > 0, we have Y_{d} ⊂ Y_{d′}.

Proof. The (quite nontrivial) inclusion Y_{d} ⊂ Y_{H}^{p}(Ω; IR^{mn}) was proved in [24, Lemma 3.2 with Sect. 4.1]. For the convexity and σ-compactness of Y_{d} we refer to [26, Section 3.5]. For the local compactness we refer to [27, proof of Theorem 2.1], if one realizes that ∫_{Ω} h_{c}(x, y(x)) dx → +∞ for ∥y∥_{L^{p}(Ω;IR^{mn})} → +∞ for h_{c} from (2.4). Also note that P_{d}H with H from (2.1) and P_{d} from (3.2) is finite-dimensional provided V is so. The monotonicity of the sequence {Y_{d}} follows, as in [26, Section 3.5], from P_{d} = P_{d} ∘ P_{d′} which holds whenever d ≥ d′ > 0 as a consequence of T_{d} ⊂ T_{d′}.

Supposing, for simplicity, that the extended potential Φ̄(u, η) can be evaluated exactly for every u and η which are element-wise affine and constant, respectively, we can define our approximate problem:

\begin{matrix}minimize & Φ̄(u, η) \\ subject to & u ∈ U_{d}, η ∈ Y_{d}, \\ & adj_{s} ∇u = (1 ⊗ adj_{s})^{•}η for all 1 ≤ s ≤ min(m, n). \end{matrix}  \} (RP_{d})

For the convergence analysis, we must inevitably strenghten (2.8). The realistic (cf. Remark 3.3 below) though a bit restrictive assumption seems to be the following:

∀r ∈ IR^{+} : Φ̄ : B_{r} × G_{H}^{p}(Ω; IR^{mn}) → IR is weakly×weakly* continuous ,  (3.4)

where B_{r} = {u ∈ W^{1,p}(Ω; IR^{m}); ∥u∥_{W^{1,p}(Ω;IR^{m})} ≤ r}. Note that (3.4) with (2.8) and (2.9) make Φ̄ the extension of Φ in the sense that Φ̄(u, i_{H}(∇u)) = Φ(u) for every u ∈ W^{1,p}(Ω; IR^{m}).

Proposition 3.2. Under the conditions (2.1)–(2.4), (2.8), and (2.9), (RP_{d}) has a solution (u_{d}, η_{d}). Moreover, there is a subsequence of {(u_{d}, η_{d})}_{d>0} converging weakly* with d→0 and, if (3.4) is also valid, the limit of each such subsequence solves (RP).

Proof. First, let us demonstrate that every (u, η) ∈ U_{d} × Y_{d} satisfying adj_{s} ∇u = (1 ⊗ adj_{s})^{•}η for all s = 1, ..., min(m, n) is admissible for (RP). Indeed, it suffices to show

η ∈ G_{H}^{p}(Ω; IR^{mn}), which, however, follows from [24, Corollary 4.], using the fact that η ∈ Y_{d} is p-nonconcentrating thanks to the coercivity (2.9), cf. [25]. Therefore, we have proved inf(RP_{d}) ≥ min(RP). The coercivity (2.9) of φ implies that the set M_{d}(c) = {(u, η) ∈ U_{d} × Y_{d}; Φ̄(u, η) ≤ c & adj_{s} ∇u = (1 ⊗ adj_{s})^{•}η, s = 1, ..., min(m, n)} is contained in w*-cl i(B_{r}) with B_{r} a

sufficiently large ball in W^{1,p}(Ω; IR^{m}). Then M_{d}(c) is weakly* compact because w*-cl i(B_{r}) is weakly* compact, U_{d} × Y_{d} is closed in w*-cl i(B_{r}), Φ̄ is weakly* l.s.c., and the mapping (u, η) ↦ adj_{s} ∇u − (1 ⊗ adj_{s})^{•}η : W^{1,p}(Ω; IR^{m}) × Y_{H}^{p}(Ω; IR^{mn}) → L^{p/s}(Ω; IR^{σ(s)})

is weakly* continuous; cf. [1, 22]. Also, M_{d}(c) is surely non-empty for c sufficiently large, e.g. for c ≥ Φ(0) because (0, i_{H}(0)) is certainly admissible for (RP_{d}). As Φ̄ is weakly* l.s.c., the existence of a solution to (RP_{d}) as well as the existence of a weakly* converging subsequence of these solutions follow by the standard compactness arguments. Now we want to prove that every (u, η) ∈ W^{1,p}(Ω; IR^{m}) × G_{H}^{p}(Ω; IR^{mn}) satisfying ∇u = (1⊗id)^{•}η can be approximated by suitable admissible pairs for (RP_{d}) when d → 0. First,

there is a bounded sequence {u_{k}} ⊂ W^{1,p}(Ω; IR^{m}) such that i_{H}(∇u_{k}) → η weakly* in H^{∗}. Also, {u_{k}} converges weakly (possibly as a subsequence only) to some ũ ∈ W^{1,p}(Ω; IR^{m}). Then i(u_{k} − ũ + u) → (u, η). By mollifying (if necessary) each u_{k}, we can even assume {u_{k}} ∈ C^{∞}(Ω̄). Let Π_{d}u_{k} ∈ U_{d} be the linear interpolant of u_{k} on the triangulation T_{d}. For k fixed and d → 0, we have Π_{d}u_{k} → u_{k} strongly in W^{1,p}(Ω; IR^{m}) because of the regularity of u_{k}. Therefore i_{H}(∇Π_{d}u_{k}) → i_{H}(∇u_{k}) weakly* in H^{∗} because ∫_{Ω} [h(x, ∇Π_{d}u_{k}(x)) − h(x, ∇u_{k}(x))] dx → 0 for any h ∈ H as a consequence of the continuity of the Nemytskii mapping y ↦ h ∘ y : L^{p}(Ω; IR^{mn}) → L^{1}(Ω). At the same time, the pair i(Π_{d}u_{k}) = (Π_{d}u_{k}, i_{H}(∇Π_{d}u_{k})) is admissible for (RP_{d}) because (1 ⊗ adj_{s})^{•}i_{H}(y) = adj_{s} y for any y ∈ L^{p}(Ω; IR^{mn}) and, in particular, also for y = ∇Π_{d}u_{k},

and because Π_{d}u_{k} ∈ U_{d} hence i_{H}(∇Π_{d}u_{k}) ∈ Y_{d}. Without any loss of generality, the sequence {u_{k}} can be considered bounded in W^{1,p}(Ω; IR^{m}). Then also {Π_{d}u_{k}} ⊂ B_{r} for r large enough and {i_{H}(∇Π_{d}u_{k})} is bounded in H^{∗}. Realizing the metrizability of the weak* topology on i(B_{r}) (recall that H is separable), we can select a subsequence such that Π_{d}u_{k} → u weakly in W^{1,p}(Ω; IR^{m}) and i_{H}(∇Π_{d}u_{k}) → η weakly* in H^{∗}. Let us suppose that (u, η) is a solution to (RP). We have Φ̄(Π_{d}u_{k}, i_{H}(∇Π_{d}u_{k})) ≥ min(RP_{d}) ≥ min(RP). As Φ̄ : B_{r} × G_{H}^{p}(Ω; IR^{mn}) → IR is weakly×weakly* continuous, we obtain Φ̄(Π_{d}u_{k}, i_{H}(∇Π_{d}u_{k})) → Φ̄(u, η) = min(RP), which shows min(RP_{d}) → min(RP) for d → 0. The rest of the assertion follows immediately by the standard compactness arguments, taking into account the coercivity of the problem.

An example for Φ̄ satisfying (3.4) is

Φ̄(u, η) = ⟨η, φ ∘ u⟩ ,  (3.5)

where [φ ∘ u](x, A) = φ(x, u(x), A), provided

∀u ∈ W^{1,p}(Ω; IR^{m}) : φ ∘ u ∈ H &
∀r ∈ IR^{+} : u ↦ φ ∘ u : (B_{r}, weak) → (H, norm) is continuous .

(3.6)

Remark 3.3. Sometimes it may happen that for {Π_{d}u_{k}}, constructed in the proof of Proposition 3.2, i_{H}(∇Π_{d}u_{k}) converges to η not only weakly* in H^{∗} but even in the standard dual norm ∥ · ∥_{H^{∗}}. Then Proposition 3.2 remains valid if (3.4) is weakened, so that Φ is to be continuous only from (B_{r},weak) × (G_{H}^{p}(Ω; IR^{mn}),norm) → IR. This appears in a trivial but important case V = {0} and min(m, n) = 1. Then Y_{H}^{p}(Ω; IR^{mn}) ≅ L^{p}(Ω; IR^{mn}) and our theory basically coincides with the standard finite-element approximation of a convexified problem: minimize Φ^{∗∗}(u) = ∫_{Ω} φ^{∗∗}(x, u(x), ∇u(x)) dx with φ^{∗∗}(x, u, ·) being a convex envelope of φ(x, u, ·). Typically φ^{∗∗} ∘ u ∉ H and Φ̄(u, y) = ∫_{Ω} φ^{∗∗}(x, u(x), y(x)) dx is not weakly continuous but only norm continuous. More generally, the same effect is for H = L^{p/(p−1)}(Ω; IR^{mn}) ⊗ {id} with min(m, n) arbitrary. Then Φ̄(u, y) = ∫_{Ω} φ^{#}(x, u(x), y(x)) dx. However, e.g. for the classical L^{p}-Young measures (i.e. for H = L^{1}(Ω; C_{0}(IR^{mn}))) it is easy to see that the remainder Y_{H}^{p}(Ω; IR^{mn}) \ i_{H} (L^{p}(Ω; IR^{mn})) cannot be reached from L^{p}(Ω; IR^{mn}) with respect to the norm topology of H^{∗} and therefore (3.4) cannot be weakened in general.

4. Implementation of a solution to (RP_{d}).

In principle, V can be finite-dimensional and then P_{d}^{∗}(H^{∗}) is also finite-dimensional and therefore (RP_{d}), being defined on a subset of the finite-dimensional linear space U_{d} × P_{d}^{∗}(H^{∗}), can be implemented directly on digital computers. In the opposite case when V is infinite-dimensional, we must (and may) use special properties of the solutions of (RP_{d}) because our task is not to implement every admissible pair (u, η) for (RP_{d}) but only a subset of admissible pairs which contains (some of) the solutions to (RP_{d}). The basic tool here consists in enough selective and informative optimality conditions for (RP_{d}). Such conditions, of the Euler-Weierstrass type, were generalized for relaxed vectorial variational problems in [25]. Here we will adapt them for our (semi)discretized problem (RP_{d}). Throughout this section d > 0 will be fixed.

336 T. Roubíček / Numerical approximation of relaxed variational problems

Let us introduce the “discrete” Hamiltonian H_{u,λ}^{d} ∈ H_{d} ≡ P_{d}(H) defined, for λ = (λ_{1}, ..., λ_{min(m,n)}) with λ_{s} ∈ L^{∞}(Ω; IR^{σ(s)}) element-wise constant, by

[H_{u,λ}^{d}] (x, y) = −P_{d}(φ ∘ u) + ∑_{s=1}^{min(m,n)} λ_{s} ⊗ adj_{s} .  (4.1)

Moreover, we will assume that

∀u ∈ W^{1,p}(Ω; IR^{m}) : \{[\frac{∂φ}{∂u} ∘ u] · ũ; ũ ∈ B_{1}\} ⊂ H bounded ,  (4.2a)
u ↦ [\frac{∂φ}{∂u} ∘ u] · ũ : (W^{1,p}(Ω; IR^{m}), weak) → (H, norm) equi-continuous  (4.2b)

with respect to ũ from the unit ball B_{1} of W^{1,p}(Ω; IR^{m}). Proposition 4.1. Let p > min(m, n), (2.9), (3.6) and (4.2) be valid, and (u, η) be a solution to (RP_{d}). Then there exists λ = (λ_{1}, ..., λ_{min(m,n)}) with λ_{s} ∈ L^{∞}(Ω; IR^{σ(s)}) element-wise constant such that the following integral identity is valid:

∫_{Ω} ( ∑_{s=1}^{min(m,n)} λ_{s}(x) · \frac{∂ adj_{s}}{∂A} (∇u(x)) · ∇ũ(x) − [(\frac{∂φ}{∂u} ∘ u)^{•}η](x) ũ(x)) dx = 0 (4.3)

for all ũ ∈ U_{d} and the following “Weierstrass-type” maximum principle holds:

[H_{u,λ}^{d} ^{•}η] (x) = max_{A∈IR^{mn}} H_{u,λ}^{d}(x, A) for a.a. x ∈ Ω .  (4.4)

Sketch of the proof. First, let us notice that (RP_{d}) has the form

\begin{matrix}minimize & Φ̄(u, η) \\ subject to & u ∈ U_{d}, η ∈ Y_{d} ⊂ P_{d}^{∗}H^{∗}, \\ & N(u) = Lη, \end{matrix}  \}  (RP′_{d})

with U_{d} and P_{d}^{∗}H^{∗} Banach spaces, Y_{d} closed convex, N : U_{d} → ∏_{s=1}^{min(m,n)} L_{d}(Ω; IR^{σ(s)}) : u ↦ (adj_{s} ∇u)_{s=1}^{min(m,n)} smooth and (for min(m, n) ≥ 2) nonlinear with L_{d}(Ω; IR^{σ(s)}) ⊂ L^{∞}(Ω; IR^{σ(s)}) denoting the space of element-wise constant functions Ω → IR^{σ(s)}, and L : P_{d}^{∗}(H^{∗}) → ∏_{s=1}^{min(m,n)} L_{d}(Ω; IR^{σ(s)}) : η ↦ ((1 ⊗ adj_{s})^{•}η)_{s=1}^{min(m,n)} linear and continuous. Also, (4.2) make Φ̄ continuously Fréchet differentiable with Φ̄′(u, η) = ((\frac{∂φ}{∂u} ∘ u)^{•}η, φ ∘ u) ∈ U_{d}^{∗} × H. Moreover, as in [25, Lemma 3.3], we can prove that L is surjective on Y_{d}. Identifying L_{d}(Ω; IR^{σ(s)}) with its own dual, we can use the theorem by Zowe and Kurczyusz [29], which gives the following necessary optimality conditions for (RP′_{d}): there is λ = (λ_{1}, ..., λ_{min(m,n)}) with λ_{s} ∈ L_{d}(Ω; IR^{σ(s)}) such that

[N′(u)]^{∗}λ + Φ̄′_{u}(u, η) = 0 ∈ U_{d}^{∗} ,  (4.5)
∀η̃ ∈ Y_{d} : ⟨L^{∗}λ − Φ̄′_{η}(u, η), η − η̃⟩ ≥ 0 .  (4.6)

After routine calculations, (4.5) gives (4.3). As for (4.6), notice that L^{∗}λ − Φ̄′_{η}(u, η) =

∑_{s=1}^{min(m,n)} λ_{s} ⊗ adj_{s} −φ ∘ u and η = P_{d}^{∗}η and η̃ = P_{d}^{∗}η̃. Then ⟨L^{∗}λ − Φ̄′_{η}(u, η), η − η̃⟩ =

⟨η − η̃, P_{d}(∑_{s=1}^{min(m,n)} λ_{s} ⊗ adj_{s} −φ ∘ u)⟩ = ⟨η − η̃, H_{u,λ}^{d}⟩. In other words, (4.6) means

just ⟨η, H_{u,λ}^{d}⟩ = max_{η̃∈Y_{d}} ⟨η̃, H_{u,λ}^{d}⟩. The Weierstrass-type maximum principle (4.4) fol-

lows from this “integral” maximum principle by a usual localization technique, using also the assumption p > min(m, n); cf. [25, Theorem 3.2].

So far we have the linear combination ∑_{k=1}^{K} θ_{k}η_{k} defined for η_{k} ∈ H^{∗} and θ_{k} ∈ IR only. However, we can generalize it naturally for θ_{k} ∈ G ⊂ L^{∞}(Ω) by means of the identity:

∀h ∈ H : ⟨∑_{k=1}^{K} θ_{k}η_{k} , h⟩ = ∑_{k=1}^{K}⟨h^{•}η_{k}, θ_{k}⟩ .  (4.7)

Of course, the left-hand-side duality is between H^{∗} and H while the right-hand-side one is between G^{∗} and G. Note that this extended definition has the previous meaning provided θ_{k} are constant on Ω.

Corollary 4.2. Let the assumptions of Proposition 4.1 be satisfied and, for every x ∈ Ω,

λ = (λ_{1}, ...λ_{min(m,n)}) and u ∈ U_{d}, let [H_{u,λ}^{d}] (x, ·) : IR^{mn} → IR attain the maximum at

no more than K points in IR^{mn}, and let (u, η) solve (RP_{d}). Then η can be written in the form

η = ∑_{k=1}^{K} θ_{k}i_{H}(y_{k}) , θ_{k} ≥ 0 , ∑_{k=1}^{K} θ_{k} = 1 ,  (4.8)

with θ_{k} ∈ L^{∞}(Ω) and y_{k} ∈ L^{∞}(Ω; IR^{mn}) element-wise constant. In other words, η admits a Young measure representation (2.7) with ν element-wise constant and ν_{x} being a convex combination of at most K Dirac measures.

Proof. Let (u, η) be a solution to (RP_{d}). By the coercivity of the problem, η is p-noncon-centrating (cf. [25]), and thus it admits a Young-measure representation η ≅ ν = {ν_{x}}_{x∈Ω}, cf. (2.7), such that x ↦ ν_{x} is element-wise constant. From the maximum principle (4.4) we can see that ∫_{IR}^{mn} H_{u,λ}^{d}(x, A)ν_{x}( dA) = max_{A∈IR^{mn}} H_{u,λ}^{d}(x, A). In particular,

the probability measure ν_{x} must be supported at the set on which H_{u,λ}^{d}(x, ·) attains it maximum over IR^{mn} which is supposed to consist from no more than K points, let us denote them by y_{k}(x) ∈ IR^{mn}, k = 1, ..., K. Moreover, these points are independent of x ∈ E for each particular element E ∈ T_{d} because H_{u,λ}^{d}(·, A) is element-wise constant.

In other words, ν_{x} = ∑_{k=1}^{K} θ_{k}(x)δ_{y_{k}(x)} and both θ_{k} and y_{k} can be assumed elementwise constant, where δ_{A} denotes the Dirac measure supported at A ∈ IR^{mn}. As ν_{x} is a probability measure, θ_{k} ≥ 0 and ∑_{k=1}^{K} θ_{k} = 1. Then (4.8) follows easily from the formulas (2.7) and (4.7) via the identity: h^{•}η = ⟨ν, h⟩ = ∑_{k=1}^{K} θ_{k}h(y_{k}) = ∑_{k=1}^{K} θ_{k}(h^{•}i_{H}(y_{k})) =

h^{•} (∑_{k=1}^{K} θ_{k}i_{H}(y_{k})), which holds for any h ∈ H.

Under the assumptions of Corollary 4.2 it is clear that every solution to (RP_{d}) can be implemented on digital computers because the number of elements in T_{d} as well as the maximal number of Dirac measures needed on each of them are finite. For an example of the implementation in the scalar case we refer to [23, Sect. 3]. However, it should be remarked that, especially in the case min(m, n) ≥ 2, the condition in Corollary 4.2 is not easy to be verified. On the other hand, mostly it is quite satisfactory to implement at least one of the solutions to (RP_{d}) because, even if we are actually able to implement all solutions, the nonlinear-programming algorithms we have eventually to use can normally find only one of them. From this standpoint, the following assertion makes a satisfactory basis for implementation of the problem (RP_{d}).

Corollary 4.3. Let the assumptions of Proposition 4.1 be satisfied. Then there always exists at least one solution (u, η) to (RP_{d}) such that η is in the form (4.8) with θ_{k} ∈ L^{∞}(Ω) and y_{k} ∈ L^{∞}(Ω; IR^{mn}) element-wise constant and

K = 1 + ∑_{s=1}^{min(m,n)} σ(s) .  (4.9)

Sketch of the proof. Take (u_{0}, η_{0}) to be a solution to (RP_{d}) and λ = (λ_{1}, ..., λ_{min(m,n)}) the corresponding Lagrange multipliers. Every pair (u_{0}, η) ∈ U_{d} × Y_{d} satisfying (1 ⊗ adj_{s})^{•}η = adj_{s} ∇u_{0} for s = 1, ..., min(m, n) and the maximum principle (4.4) with u = u_{0}

solves (RP_{d}). Indeed, such (u_{0}, η) is obviously an admissible pair for (RP_{d}) and also Φ̄(u_{0}, η) = min(RP_{d}) because

⟨η, φ ∘ u_{0}⟩ = ⟨P_{d}^{∗}η, φ ∘ u_{0}⟩ = ⟨η, P_{d}(φ ∘ u_{0})⟩

= ⟨η, ∑_{s=1}^{min(m,n)} λ_{s} ⊗ adj_{s} −H_{u_{0},λ}^{d}⟩ = ⟨η_{0}, ∑_{s=1}^{min(m,n)} λ_{s} ⊗ adj_{s} −H_{u_{0},λ}^{d}⟩

= ⟨η_{0}, P_{d}(φ ∘ u_{0})⟩ = ⟨P_{d}^{∗}η_{0}, φ ∘ u_{0}⟩ = ⟨η_{0}, φ ∘ u_{0}⟩ .

For this identity, we also used that

⟨η, λ_{s} ⊗ adj_{s}⟩ = ⟨(1 ⊗ adj_{s})^{•}η, λ_{s}⟩ = ⟨adj_{s} ∇u_{0}, λ_{s}⟩ = ⟨(1 ⊗ adj_{s})^{•}η_{0}, λ_{s}⟩ = ⟨η_{0}, λ_{s} ⊗ adj_{s}⟩

and

⟨η, H_{u_{0},λ}^{d}⟩ = ∫_{Ω} H_{u_{0},λ}^{d} ^{•}η dx = ∫_{Ω} max_{A∈IR^{mn}} H_{u_{0},λ}^{d}(x, A) dx = ∫_{Ω} H_{u_{0},λ}^{d} ^{•}η_{0} dx
= ⟨η_{0}, H_{u_{0},λ}^{d}⟩.

As a result, η should only satisfy ∑_{s=1}^{min(m,n)} σ(s) conditions on the momenta (1⊗adj_{s})^{•}η,

and also one additional momentum H_{u_{0},λ}^{d} ^{•}η which is located however at the maximum of

H_{u_{0},λ}^{d}, so that actually the investigated momenta range a (∑_{s=1}^{min(m,n)} σ(s))-dimensional manifold. Then the assertion follows by the Carathéodory theorem applied to a Youngmeasure representation of η; see [26, Proposition 4.3.9] for details.

Let us remark that the estimate (4.9) cannot be improved in general; in other words, there

exist relaxed problems that have no solutions of the form (4.8) with K ≤ ∑_{s=1}^{min(m,n)} σ(s).

E.g., for m = n = 1, one has obviously ∑_{s=1}^{min(m,n)} σ(s) = 1 but a one-dimensional scalar problem with a double well potential investigated in [11, 12, 18], having a unique twoatomic solution, has apparently no one-atomic solution.

5. Some error estimates.

The error estimates we are able to establish here differ from the usual results for elliptic equations mainly because the problem (RP) is not uniformly convex even if min(m, n) = 1; for min(m, n) ≥ 2 it is in general nonconvex. This fact implies that we are able to establish only error estimates concerning | min(RP) − min(RP_{d})|, which can be considered however as a significant indicator of the efficiency of the particular method. In the literature one can occasionally find error estimates also for the solutions (in appropriate dual norms on H^{∗} weaker than ∥ · ∥_{H^{∗}}) but it is always for special scalar (i.e. m = 1) problems possessing a unique solution; cf. [11, 12, 17] for n = 1 or [7] for n ≥ 1. Besides, except [20, 23], the discretization described in the literature concerns the original problem (P), hence the discrete problem looks as

minimize Φ(u) = ∫_{Ω} φ(x, u(x), ∇u(x)) dx for u ∈ U_{d} .  (P_{d})

The first comparison of the standard discretization (P_{d}) with our discretization (RP_{d}) can be based on the simple observation that

inf (P) = min(RP) ≤ min(RP_{d}) ≤ min(P_{d}) ,  (5.1)

which follows from the fact that every i(u) = (u, i_{H}(∇u)) with u ∈ U_{d} is admissible for (RP_{d}). Therefore our method cannot be worse than the standard discretization as far as the rate of convergence of the minimum of the discrete problem towards the minimum of the original problem concerns. Both methods lead to nonconvex mathematicalprogramming problems, our method having slightly greater dimensionality but of the same order O(d^{−n}) as the standard method provided Corollary 4.2 or 4.3 can be applied. The theoretical effort behind its implementation and slightly greater dimensionality are compensated by the fact that sometimes our method can give much faster convergence than the standard method; cf. Sect. 6 below. Our aim is to get the convergence min(RP_{d}) → min(RP) of the order O(d^{α}) with some α > 0. We will consider a subspace H̃ ⊂ H equiped with a norm stronger than the norm of H, so that the imbedding H̃ → H is continuous. Thus H̃^{∗} ⊃ H^{∗} and we can estimate the generalized Young functionals from Y_{H}^{p}(Ω; IR^{mn}) in the standard dual norm of H̃^{∗}. Our error analysis will rely on the following general pattern:

Proposition 5.1. Let (2.9) be valid, let the mapping u ↦ φ ∘ u map every ball B_{r} in W^{1,p}(Ω; IR^{m}) into a bounded subset of H̃, i.e.

∀r ∈ IR^{+} ∃K_{r} ∈ IR^{+} ∀u ∈ B_{r} : ∥φ ∘ u∥H̃ ≤ K_{r} ,  (5.2)

and let it be (W^{1−α,p}(Ω; IR^{m}), H)-Lipschitz continuous on each B_{r}, i.e.

∀r ∈ IR^{+} ∃L_{r} ∈ IR^{+} ∀u, ũ ∈ B_{r} : ∥φ ∘ u − φ ∘ ũ∥_{H} ≤ L_{r}∥u − ũ∥_{W^{1−α,p}(Ω;IR^{m})} . (5.3)

Also, let us assume that, for every (u, η) ∈ W^{1,p}(Ω; IR^{m}) × Y_{H}^{p}(Ω; IR^{mn}) and every d > 0, there exists an admissible pair (u_{d}, η_{d}) for (RP_{d}) such that the sequence {u_{d}}_{d>0} is bounded in W^{1,p}(Ω; IR^{m}) and the sequence {(u_{d}, η_{d})}_{d>0} has the following approximation property

∥u − u_{d}∥_{W^{1−α,p}(Ω;IR^{m})} ≤ C_{1}d^{α}∥u∥_{W^{1,p}(Ω;IR^{m})} ,  (5.4a)
∥η − η_{d}∥H̃^{∗} ≤ C_{2}d^{α}∥η∥_{H^{∗}} .  (5.4b)

Then | min(RP) − min(RP_{d})| = O(d^{α}).

Proof. First, let us note that, thanks to (5.3), Φ̄(u, η) = ⟨η, φ ∘ u⟩ actually defines the extension of Φ which is weakly* continuous on B_{r} × Y_{H}^{p}(Ω; IR^{mn}) for any r ∈ IR^{+}. As always min(RP_{d}) ≥ min(RP), we have only to estimate min(RP_{d}) − min(RP) from above. First we will show that Φ̄ from (3.5) is (W^{1−α,p}(Ω; IR^{m}) × H̃^{∗}, IR)-Lipschitz continuous on each B_{r} × H^{∗} at any (u, η) ∈ B_{r} × H^{∗} in the sense

|Φ̄(u, η) − Φ̄(ũ, η̃)| ≤ L_{r,η} (∥u − ũ∥_{W^{1−α,p}(Ω;IR^{m})} + ∥η − η̃∥H̃^{∗})  (5.5)

for any (ũ, η̃) ∈ B_{r} × H^{∗} and some L_{r,η} ∈ IR^{+} depending possibly on r and η. Indeed, for u, ũ ∈ B_{r} and η, η̃ ∈ H^{∗} we can estimate

|Φ̄(u, η) − Φ̄(ũ, η̃)| = |⟨η, φ ∘ u⟩ − ⟨η̃, φ ∘ ũ⟩| ≤ |⟨η, φ ∘ u − φ ∘ ũ⟩|
+ |⟨η − η̃, φ ∘ ũ⟩| ≤ ∥η∥_{H^{∗}}∥φ ∘ u − φ ∘ ũ∥_{H} + ∥η − η̃∥H̃^{∗}∥φ ∘ ũ∥H̃
≤ L_{r}∥η∥_{H^{∗}}∥u − ũ∥_{W^{1−α,p}(Ω;IR^{m})} + K_{r}∥η − η̃∥H̃^{∗} ,

which gives (5.5) with L_{r,η} = max(L_{r}∥η∥_{H^{∗}}, K_{r}). Eventually, let us take (u, η) a solution to (RP) and (u_{d}, η_{d}) as assumed. Putting r = sup_{d>0} ∥u_{d}∥_{W^{1,p}(Ω;IR^{m})} and using (5.4) and (5.5), we can estimate:

min(RP_{d}) − min(RP) ≤ Φ̄(u_{d}, η_{d}) − Φ̄(u, η) ≤ L_{r,η}∥u − u_{d}∥_{W^{1−α,p}(Ω;IR^{m})}
+ L_{r,η}∥η − η_{d}∥H̃^{∗} ≤ L_{r,η}C_{1}d^{α}∥u∥_{W^{1,p}(Ω;IR^{m})} + L_{r,η}C_{2}d^{α}∥η∥_{H^{∗}} = O(d^{α}) .

Thus we have reduced the error-estimation problem to the verification of (5.2)–(5.4); in fact, it suffices to verify (5.4) only for some solution to (RP). Let us first treat the one-dimensional case, which is especially simple. Recall that throughout this paper we suppose p > 1, hence we have now W^{1,p}(Ω; IR^{m}) ⊂ C^{0}(Ω; IR^{m}) for Ω ⊂ IR^{1} and we can speak about an element-wise affine interpolation of a given u ∈ W^{1,p}(Ω; IR^{m}), denoted by Π_{d}u as usual. In this one-dimensional case, it suffices to control the spatial smoothness only, using the parameter 0 < α ≤ 1. Thus we define naturaly the subspace H_{α} ⊂ H by

H_{α} = {h_{0} + ∑_{s=0}^{min(m,n)} g_{s} ⊗ adj_{s} ∈ H; h_{0} ∈ C^{0,α}(Ω) ⊗ V, g_{s} ∈ W^{α,p/(p−s)}(Ω; IR^{σ(s)})} ,

(5.6)

and endow it with the norm

∥h∥_{H_{α}} = inf ; h_{0}+∑_{s} g_{s}⊗adj_{s}=h (∥R_{p}h_{0}∥_{C^{α,0}(Ω;C^{0}(IR^{mn}))} + ∑_{s=0}^{min(m,n)} ∥g_{s}∥_{W^{α,p/(p−s)}(Ω;IR^{σ(s)})}) ,

(5.7)

where R_{p} : H → L^{∞}(Ω × IR^{mn}) is defined by [R_{p}h](x, A) = h(x, A)/(1 + |A|^{p}).

Lemma 5.2. Let n = 1 and let (u, η) ∈ W^{1,p}(Ω; IR^{m})×H^{∗} be such that (1⊗id)^{•}η = ∇u. Putting u_{d} = Π_{d}u and η_{d} = P_{d}^{∗}η, we have ∥u_{d}∥_{W^{1,p}(Ω;IR^{m})} ≤ ∥u∥_{W^{1,p}(Ω;IR^{m})} and (5.4) fulfilled for H̃ = H_{α}. Moreover, (1 ⊗ id)^{•}η_{d} = ∇u_{d}, hence each pair (u_{d}, η_{d}) is admissible for (RP_{d}).

Proof. The identity (1 ⊗ id)^{•}η_{d} = ∇u_{d} follows from (1 ⊗ id)^{•}P_{d}^{∗}η = A_{d}((1 ⊗ id)^{•}η) and ∇u_{d} = A_{d}∇u, where A_{d} denotes the operator L^{p}(Ω; IR^{m}) → L^{∞}(Ω; IR^{m}) which takes the mean over each element (here an interval, since n = 1). The estimate ∥u_{d}∥_{W^{1,p}(Ω;IR^{m})} ≤ ∥u∥_{W^{1,p}(Ω;IR^{m})} follows by easy calculations, while (5.4a) can be obtained by interpolation from standard results from the finite-element method; cf. [10]. The estimate (5.4b) follows from

∥h − P_{d}h∥_{H} ≤ C_{2}d^{α}∥h∥_{H_{α}}  (5.8)

by transposition as in [26, Section 3.5]. In view of (2.3) and (5.7), the estimate (5.8) follows from ∥g − A_{d}g∥_{C^{0}(Ω)} ≤ Cd^{α}∥g∥_{C^{0,α}(Ω)} and from the estimates ∥g − A_{d}g∥_{L^{p/(p−s)}(Ω)} ≤ Cd^{α}∥g∥_{W^{α,p/(p−s)}(Ω)} for s = 0, 1. The last estimate can be obtained by interpolation from the four estimates created by α = 0 or 1 and p/(p − s) = 1 or ∞, which are obvious.

Lemma 5.3. Let n = 1, φ have the form φ(x, u, A) = φ_{0}(x, u) + φ_{1}(x, A) with φ_{1} ∈ H_{α} for some 0 < α ≤ 1 and with φ_{0}(x, ·) : IR^{m} → IR Lipschitz continuous on bounded subsets of IR^{m} in the sense

\begin{matrix}∀c ∈ IR^{+} ∃L_{c} ∈ L^{p/(p−1)}(Ω) ∀x ∈ Ω ∀u, ũ ∈ IR^{m} : \\ |u| ≤ c, |ũ| ≤ c ⇒ |φ_{0}(x, u) − φ_{0}(x, ũ)| ≤ L_{c}(x)|u − ũ| \end{matrix}  (5.9)

and φ_{0}(·, 0) ∈ L^{1}(Ω). Then (5.2) with H̃ = H_{α} and (5.3) are satisfied.

Proof. For a given r ∈ IR^{+} there is c = c(r) ∈ IR^{+} such that any u ∈ B_{r} = {u ∈ W^{1,p}(Ω; IR^{m}); ∥u∥_{W^{1,p}(Ω;IR^{m})} ≤ r} take values bounded (in absolute values) by c(r), hence φ_{0} can be supposed Lipschitz with a Lipschitz constant L_{c(r)}, cf. (5.9). As for (5.2), it follows from the estimates ∥φ ∘ u∥_{H_{α}} ≤ ∥φ_{0} ∘ u∥_{W^{α,1}(Ω)} + ∥φ_{1}∥_{H_{α}} and

∥φ_{0} ∘ u∥_{W^{α,1}(Ω)} ≤ ∥φ_{0} ∘ u∥_{W^{1,1}(Ω)} = ∫_{Ω} (|φ_{0}(x, u(x))| + |\frac{d}{dx}φ_{0}(x, u(x))|) dx

≤ ∫_{Ω} (|φ_{0}(x, 0)| + L_{c(r)}(x)|u(x)| + |\frac{∂}{∂u}φ_{0}(x, u(x))\frac{d}{dx}u(x)|) dx

≤ ∥φ_{0}(x, 0)∥_{L^{1}(Ω)} + ∥L_{c(r)}∥_{L^{p/(p−1)}(Ω)}∥u∥_{W^{1,p}(Ω)} .

Furthermore, we can estimate

∥φ ∘ u − φ ∘ ũ∥_{H} ≤ ∥φ_{0} ∘ u − φ_{0} ∘ ũ∥_{L^{1}(Ω)} ≤ ∥L_{c(r)}∥_{L^{p/(p−1)}(Ω)}∥u − ũ∥_{L^{p}(Ω)} ,

which already implies (5.3) thanks to the continuous imbedding W^{1−α,p}(Ω) ⊂ L^{p}(Ω).

Therefore, for n = 1, Proposition 5.1 together with Lemmas 5.2 and 5.3 yields the desired estimate | min(RP) − min(RP_{d})| = O(d^{α}). In the general case, this result is likely not to be much improvable because the data qualifications in Lemma 5.3 are actually very weak; e.g. we admit φ(x, u, A) = g(x)v(A) with v ∈ V only continuous and coercive. On the other hand, in special cases this guaranteed rate of error can be still pessimistic; sometimes even the convergence of the order O(d^{∞}) can be achieved, cf. Section 6. In the multidimensional case (i.e. n ≥ 2) the construction η_{d} = P_{d}^{∗}η unfortunately does not work because (1 ⊗ id)^{•}P_{d}^{∗}η ∈ L^{p}(Ω, IR^{mn}) need not be the gradient of any function

from W^{1,p}(Ω; IR^{m}) and, if also m ≥ 2, P_{d}^{∗}η need not be any gradient Young measure even locally on each element E ∈ T_{d}, this means P_{d}η|_{E} ∉ G_{H}^{p}(E; IR^{mn}) in general. Nevertheless, in the scalar case (i.e. m = 1) a suitable η_{d} can be constructed by a more sophisticated way, using a shift operator T_{y} : H → H defined by [T_{y}h](x, A) = h(x, A + y(x)) with y ∈ L^{p}(Ω; IR^{mn}). Obviously, T_{y} makes a “vertical shift” of the integrands from H and, for y ∈ L^{∞}(Ω; IR^{mn}), it is a linear continuous operator with the continuous inverse T_{y}^{−1} = T_{−y}; in particular, in view of (2.3) it is easy to verify

∀r ∈ IR^{+} ∃N_{r} ∈ IR^{+} ∀h ∈ H ∀y ∈ L^{∞}(Ω; IR^{mn}) :

∥y∥_{L^{∞}(Ω;IR^{mn})} ≤ r ⇒ ∥T_{y}h∥_{H} ≤ N_{r}∥h∥_{H} . (5.10)

The adjoint operator T_{y}^{∗} : H^{∗} → H^{∗} maps Y_{H}^{p}(Ω; IR^{mn}) onto Y_{H}^{p}(Ω; IR^{mn}), making a “vertical shift” of the corresponding Young functionals. The philosophy of our construction of η_{d} is the following: first shift η in such a way that the first momentum of the shifted Young measure vanishes, then apply the operator P_{d}^{∗} without falling out of the class G_{H}^{p}(Ω; IR^{mn}), and eventually shift the resulted Young measure “almost” back; cf. (5.11) below. To realize this “almost”, we will have to suppose a W^{1+α,∞}-regularity of a solution and a stronger data qualification, requiring a certain weight-Lipschitz continuity of φ(x, u, ·) weaker than the usual Lipschitz continuity, which would not be a realistic requirement.

T. Roubíček / Numerical approximation of relaxed variational problems 343

The corresponding subspace of H may be chosen as H_{Lip} = {h ∈ H; ∥h∥_{H_{Lip}} < +∞} with the norm

∥h∥_{H_{Lip}} = ess ; sup_{x∈Ω} sup_{A,Ã∈IR^{mn}} \frac{|h(x, A) − h(x, Ã)|}{|A − Ã|(1 + |A|^{p} + |Ã|^{p})} .

We will use Proposition 5.1 with H̃ = H_{α} ∩ H_{Lip} equiped with the norm ∥h∥_{H_{α}∩H_{Lip}} = ∥h∥_{H_{α}} + ∥h∥_{H_{Lip}}.

Lemma 5.4. Let m = 1, 0 < α ≤ 1, and let (u, η) ∈ W^{1+α,∞}(Ω) × Y_{H}^{p}(Ω; IR^{n}) be such that (1 ⊗ id)^{•}η = ∇u. Putting

u_{d} = Π_{d}u and η_{d} = T_{∇u_{d}}^{∗}P_{d}^{∗}T_{−∇u}^{∗}η ,  (5.11)

we have got ∥u_{d}∥_{W^{1,p}(Ω)} ≤ ∥u∥_{W^{1,p}(Ω)} and (5.4) with H̃ = H_{α} ∩ H_{Lip} fulfilled. Moreover, η_{d} ∈ Y_{d} and (1 ⊗ id)^{•}η_{d} = ∇u_{d}, hence each pair (u_{d}, η_{d}) is admissible for (RP_{d}).

Proof. First, (1⊗id)^{•}η_{d} = (1⊗id)^{•}T_{∇u_{d}}^{∗} (P_{d}^{∗}T_{−∇u}^{∗}η) = ∇u_{d} + (1⊗id)^{•}P_{d}^{∗}T_{−∇u}^{∗}η = ∇u_{d} because (1⊗id)^{•}T_{−∇u}^{∗}η = −∇u+(1⊗id)^{•}η = 0, and therefore also (1⊗id)^{•}P_{d}^{∗}T_{−∇u}^{∗}η = 0. Secondly, let us notice that T_{y} and P_{d} commute with each other provided y is elementwise constant on T_{d}. In particular, T_{∇u_{d}}P_{d} = P_{d}T_{∇u_{d}} and we can write alternatively η_{d} = P_{d}^{∗}T_{∇u_{d}}^{∗}T_{−∇u}^{∗}η = P_{d}^{∗}T_{∇(u_{d}−u)}^{∗}η. It shows that η_{d} ∈ Y_{d}. The estimate ∥u_{d}∥_{W^{1,p}(Ω)} ≤ ∥u∥_{W^{1,p}(Ω)} is obvious. The estimate (5.4a) follows by interpolation from the standard estimates ∥u − u_{d}∥_{W^{1,p}(Ω)} ≤ 2∥u∥_{W^{1,p}(Ω)} and ∥u − u_{d}∥_{L^{p}(Ω)} ≤ Cd∥u∥_{W^{1,p}(Ω)}, cf. [10, Theorem 16.1]. As for (5.4b), we will estimate it as

∥η − η_{d}∥_{[H_{α}∩H_{Lip}]^{∗}} ≤ ∥η − T_{∇(u_{d}−u)}^{∗}η∥_{[H_{α}∩H_{Lip}]^{∗}}
+ ∥T_{∇(u_{d}−u)}^{∗}η − P_{d}^{∗}T_{∇(u_{d}−u)}^{∗}η∥_{[H_{α}∩H_{Lip}]^{∗}} ≡ I_{1} + I_{2}.

Let us estimate I_{1}. For any r ∈ IR^{+} there is C_{r} such that

|h(x, A) − h(x, A + Ã)| ≤ |Ã|(1 + |A|^{p} + |A + Ã|^{p})∥h∥_{H_{Lip}} ≤ C_{r}|Ã|(1 + |A|^{p})∥h∥_{H_{Lip}}

for any h ∈ H_{Lip} and A, Ã ∈ IR^{n} provided |Ã| ≤ r. This gives the estimate

∥h − T_{y}h∥_{H} ≤ sup_{x∈Ω} sup_{A∈IR^{n}} \frac{|h(x, A) − h(x, A + y(x))|}{1 + |A|^{p}} ≤ C_{r}∥y∥_{L^{∞}(Ω;IR^{n})}∥h∥_{H_{Lip}}

for any h ∈ H_{Lip} and y ∈ L^{∞}(Ω; IR^{n}) with ∥y∥_{L^{∞}(Ω;IR^{n})} ≤ r. From this, for y = ∇(u_{d}−u), we get by transposition I_{1} ≤ ∥η−T_{∇(u_{d}−u)}^{∗}η∥_{H_{Lip}^{∗}} ≤ C_{r}∥∇(u_{d}−u)∥_{L^{∞}(Ω;IR^{n})}∥h∥_{H^{∗}} provided ∇(u_{d} − u) is small enough, namely ∥∇(u_{d} − u)∥_{L^{∞}(Ω;IR^{n})} ≤ r. Afterwards we can use the estimate ∥∇(u_{d} − u)∥_{L^{∞}(Ω;IR^{n})} ≤ Cd^{α}∥u∥_{W^{1+α,∞}(Ω)}, which follows by interpolation from [10, Theorem 16.1]. This shows I_{1} = O(d^{α}).

By transposition of (5.10), we get ∥T_{y}^{∗}η∥_{H^{∗}} ≤ N_{r}∥η∥_{H^{∗}}. Using (5.8), we can eventually estimate the term I_{2} as follows:

I_{2} ≤ ∥T_{∇(u_{d}−u)}^{∗}η − P_{d}^{∗}T_{∇(u_{d}−u)}^{∗}η∥_{H_{α}^{∗}} ≤ C_{2}d^{α}∥T_{∇(u_{d}−u)}^{∗}η∥_{H^{∗}} ≤ C_{2}N_{r}d^{α}∥η∥_{H^{∗}}

provided ∥∇(u_{d} − u)∥_{L^{∞}(Ω)} ≤ r. It gives I_{2} = O(d^{α}) whenever u ∈ W^{1,∞}(Ω); recall that even u ∈ W^{1+α,∞}(Ω) had to be supposed.

As for (5.2) and (5.3), we can use readily Lemma 5.3 also for n > 1 with H_{α} replaced by H_{α} ∩ H_{Lip} if p > n, which ensures W^{1,p}(Ω) ⊂ L^{∞}(Ω). Then, for m = 1, Proposition 5.1 together with Lemma 5.3 (modified as outlined) and Lemma 5.4 enables us to get | min(RP) − min(RP_{d})| = O(d^{α}) if (RP) possesses at least one solution (u, η) with u ∈ W^{1+α,∞}(Ω) and if φ fulfils the data qualification from Lemma 5.3 with H_{α} replaced by H_{α} ∩ H_{Lip}. If p ≤ n, then (5.2) with H̃ = H_{α} ∩ H_{Lip} and (5.3) must be verified for each particular case. E.g., if φ(x, u, A) = φ_{1}(x, A), then they reduce to the only requirement φ_{1} ∈ H_{α} ∩ H_{Lip}. Let us also remark that, supposing additionally (3.6), requirement (5.3) can be weakened to ∥φ∘u−φ∘ũ∥_{H} ≤ L_{r}∥u−ũ∥_{W^{1,p}(Ω;IR^{m})} because, due to the regularity assumption u ∈ W^{1+α,∞}(Ω), we can use ∥u − u_{d}∥_{W^{1,p}(Ω)} ≤ C_{1}d^{α}∥u∥_{W^{1+α,p}(Ω)} instead of (5.4a). The error estimates for the case min(m, n) ≥ 2 remains open because no effective formula to construct an admissible pair (u_{d}, η_{d}) sufficiently near to (u, η) is known; however, in very special cases some error estimates for (P_{d}) have been obtained by Chipot, Collins and Kinderlehrer [8], which yields via (5.1) certain (probably rather pessimistic) error estimates for (RP_{d}), too.

6. A comparison and concluding remarks.

Let us compare our approximation scheme (RP_{d}) with the standard scheme (P_{d}). As already mentioned, we have always | min(RP)−min(RP_{d})| ≤ | min(RP)−min(P_{d})|, hence our method cannot be worse than (P_{d}) even in case min(m, n) ≥ 2. Sometimes, it might be even considerably better. The only analysis of (P_{d}) in multidimensional scalar case under data qualification similarly weak as in Section 5 was performed by Chipot and Collins, see [6, 7]. For m = 1, Dirichlet boundary conditions, and φ(x, u, A) = φ_{1}(A)+φ_{0}(u) with either φ_{0} ≡ 0 in [6] or some special φ_{0} having the growth at most O(|u|^{q}) in [7], the error | min(RP) − min(P_{d})| is of the order either O(d^{1/2}) if q ≥ 1 or O(d^{q/(q+1)}) if 0 < q < 1. A similar result, namely the rate O(d^{1/2}) for φ_{0} ≡ 0, was obtained also by Brighi and Chipot [5] in a bit different context. Let us remind that, in contrast to [5, 6, 7], we admitted φ to be spatially dependent and our guaranteed rate was O(d^{α}) with 0 < α ≤ 1 depending on the spatial smoothness of φ and, if n ≥ 2, also on W^{1+α,∞}-regularity of at least one solution to (RP). Moreover, there are several results concerning the one-dimensional scalar case n = m = 1, see [4, 11, 12, 13, 17, 21]. For φ(x, u, A) = φ_{1}(A) + φ_{0}(x, u) with a special φ_{0} and with φ_{1} having two wells (=local minima) of the same value and a quadratic growth in neighbourhoods of these wells, the rate of | min(RP)−min(P_{d})| was shown to be O(d^{2}). It is better than our guaranteed general rate, but (5.1) ensures in this special case the order also at least O(d^{2}). Moreover, the problems in [11, 12, 13, 17] are so special that even min(RP_{d}) = min(RP) because (RP) possesses in these special cases the solution (u, η) with u affine and η homogeneous, which is apparently admissible also for our approximate problems (RP_{d}) for any d > 0; therefore, in fact, even the rate O(d^{∞}) was achieved. A similar effect can happen even for multidimensional vectorial problems, where no rateof-error analysis is disposable for (RP_{d}), neither for (P_{d}). We can demonstrate it on the potential proposed by Ericksen and James and used by Collins, Luskin and Riordan [13, 14] for numerical experiments in the case n = m = 2:

φ(x, u, A) = φ_{1}(A) = k_{1}(b_{11} + b_{22} − 2)^{2} + k_{2}b_{12}^{2} + k_{3} ((\frac{b_{11} − b_{22}}{2} )^{2} − ε^{2})^{2} , (6.1)

where B = A^{T}A = [b_{ij}] and k_{1}, k_{2}, k_{3}, ε are positive, ε ≤ 1. It is known, cf. [13, 14], that the relaxed problem completed by suitable Dirichlet boundary conditions possesses the solution u(x) = (\frac{1}{2}A_{1} + \frac{1}{2}A_{2})x and η homogeneous having a classical-Young-measure representation as a two-atomic Young measure \frac{1}{2}δ_{A_{1}} + \frac{1}{2}δ_{A_{2}} with δ_{A} denoting the Dirac distribution at A ∈ IR^{2×2} and

A_{1} = ( \begin{matrix}\sqrt{1 − ε} & 0 \\ 0 & \sqrt{1 + ε} \end{matrix} ) , A_{2} = ( \begin{matrix}(1 + ε)\sqrt{1 − ε} & ε\sqrt{1 − ε} \\ −ε\sqrt{1 + ε} & (1 − ε)\sqrt{1 + ε} \end{matrix} )

Obviously, Rank(A_{1} − A_{2}) = 1 and (1 ⊗ id)^{•}η = ∇u. It implies that (u, η) is actually

admissible for (RP) and obviously also for (RP_{d}) with any d > 0, as well. Therefore we have got again min(RP) = min(RP_{d}) = 0, hence the rate O(d^{∞}) was achieved. One important remark should be however made to (RP_{d}) with φ from (6.1) because it is not known whether φ_{1} from (6.1) has a polyconvex quasiconvexification, thus φ∘u = 1⊗φ_{1} need not be contained in any H from (2.1) with V satisfying (2.2). Taking V satisfying only (2.2a-d) and containing φ_{1}, the extension by (3.5) is made possible but (RP_{d}) in general does not approximate (RP) but only the following auxiliary problem

\begin{matrix}minimize & Φ̄(u, η) \\ subject to & u ∈ W^{1,p}(Ω; IR^{m}), η ∈ Y_{H}^{p}(Ω; IR^{mn}), \\ & adj_{s} ∇u = (1 ⊗ adj_{s})^{•}η for all 1 ≤ s ≤ min(m, n). \end{matrix}  \}  (AP)

It may happen that min(AP) < min(RP) because not every η admissible for (AP) belongs to G_{H}^{p}(Ω; IR^{mn}) if (2.2e) is not satisfied. However, by the results from [15] it can be shown that, if (AP) possesses a solution (u, η) such that the poly- and the quasi-convex envelopes of φ_{1} from (6.1) coincides with each other at every point A = ∇u(x) with x ∈ Ω, then min(AP) = min(RP). This actually applies here because both the poly- and the quasiconvex envelopes of φ_{1} from (6.1) are zero at A = \frac{1}{2}A_{1} + \frac{1}{2}A_{2} = ∇u. Additionally, the exponents in (6.1) should be better modified a bit because the present form of φ_{1} ensures the coercivity in L^{4}(Ω; IR^{2×2}) via the first term while the third term can take values +∞ on this space.

Acknowledgements. The author is thankful to Professors Karl-Heinz Hoffmann and Stefan Müller for many discussions. This work has been carried out during author’s stay at the Institute for Applied Mathematics and Statistics, TU München.

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