2-Quasiconvexity versus 1-Quasiconvexity

Gianni Dal Maso Sissa Irene Fonseca CMU Giovanni Leoni CMU

Massimiliano Morini CMU

May 6, 2003

Abstract

In this paper it is shown that a smooth strictly 2-quasiconvex function with p-growth at infinity, p > 1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of classical first order theorems.

1 Introduction

In recent years there has been a renewed interest in higher order variational problems motivated by various mathematical models in engineering and materials science: in connection with the so-called gradient theories of phase transitions within elasticity regimes (see [7], [17], [21]); in the study of equilibria of micromagnetic materials where mastery of second order energies (here accounting for the exchange energy) is required (see [6], [9], [21], [24]); in the theory of second order structured deformations (SOSD) (see [23]), in the Blake-Zisserman model for image segmentation in computer vision (see [5]); etc..

In the study of lower semicontinuity, relaxation and Γ-convergence problems for second order functional the natural notion of convexity, 2-quasiconvexity, was introduced by Meyers in [20] (see also [3], [14]). We recall that a real valued function f, defined on the space 𝕄_{sym}^{n×n} of n × n symmetric matrices, is 2-quasiconvex if

∫_{Q} [f (A + ∇^{2}ϕ) − f (A)] dx ≥ 0

for every A ∈ 𝕄_{sym}^{n×n} and every ϕ ∈ C_{c}^{2} (Q), where Q := (0, 1)^{n} is the unit cube.

While lower semicontinuity properties of functionals depending uniquely on the second order derivatives can be proved easily, the case where lower order terms are present is significantly more difficult since the correct tools to handle the localization and truncation of gradients are still missing.

To bypass these difficulties one would be tempted to transform higher order into first order problems, where one uses the standard notion of quasiconvexity, called 1-quasiconvexity in this paper. We recall that a real valued function f, defined on the space 𝕄^{n×n} of n × n matrices, is 1-quasiconvex if

∫_{Q} [f (A + ∇φ) − f (A)] dx ≥ 0

for every A ∈ 𝕄^{n×n} and every φ ∈ C_{c}^{1} (Q; ℝ^{n}).

Thus we are led to the following question.

(Q) Is every 2-quasiconvex function the restriction of a 1-quasiconvex function to the space of symmetric matrices?

A good indication of the plausibility of an affirmative answer is that it holds true for polyconvex functions as noticed by Dacorogna and Fonseca. Indeed if

f (A) = g (M (A)) A ∈ 𝕄_{sym}^{n×n},

where g is a convex function and M (A) stands for the vector whose components are all the minors of A, then the function

F (A) := g (\frac{M (A) + M (A)^{t}}{2} ) A ∈ 𝕄^{n×n}

is a polyconvex extension of f to the whole space 𝕄^{n×n} of n × n matrices.

It is known that 2-gradient Young measures, i.e. Young measures generated by second order gradients, may be characterized by duality via Jensen’s inequality with respect to 2-quasiconvex functions (see [12]), just as gradient Young measures are characterized by duality with 1-quasiconvex functions (see [19]). Therefore, the understanding of the structure of 2-gradient Young measures helps the study of 2-quasiconvex functions, and, accordingly, the following result by Šverák in [25, Lemma 1] provides further evidence that 1-quasiconvexity and 2-quasiconvexity are somehow strictly linked: If a Young measure ν on 𝕄^{n×n} is generated by a sequence {∇u_{k}} of gradients, with {u_{k}} bounded in W^{1,p} (Ω; ℝ^{n}) for some p > 1, and supp ν_{x} ⊂ 𝕄_{sym}^{n×n} for L^{n} a.e. x ∈ Ω, then ν is generated also by a sequence {∇^{2}w_{k}}, with {w_{k}} bounded in W^{2,p} (Ω).

A partial answer to (Q) was given by Müller and Šverák (see [22]). Indeed, as an auxiliary result to construct a counter-example to regularity for elliptic systems, they proved that any smooth, strictly 2-quasiconvex function f : 𝕄_{sym}^{2×2} → ℝ, with bounded second derivatives, is the restriction of a 1quasiconvex function. The main purpose of this paper is to extend their result to any space dimension and to a larger class of strictly 2-quasiconvex functions with p-growth at infinity, with p > 1.

Theorem 1.1 Let f ∈ C^{1} (𝕄_{sym}^{n×n}) satisfy the following conditions for suitable constants p > 1, μ ≥ 0, L ≥ ν > 0:

(a) (strict 2-quasiconvexity)

∫_{Q} [f (A + ∇^{2}ϕ) − f (A)] dx ≥ ν ∫_{Q} (μ^{2} + |A|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx

for every A ∈ 𝕄_{sym}^{n×n} and every ϕ ∈ C_{c}^{2} (Q);

(b) (Lipschitz condition for gradients)

|∇f (A + B) − ∇f (A)| ≤ L (μ^{2} + |A|^{2} + |B|^{2})^{\frac{p−2}{2}} |B|  (1.1)

for every A, B ∈ 𝕄_{sym}^{n×n}.

Then there exists a 1-quasiconvex function F : 𝕄^{n×n} → ℝ such that

F (A) = f (A)  ∀A ∈ 𝕄_{sym}^{n×n},  (1.2)
|F (A)| ≤ c_{f} (1 + |A|^{p})  ∀A ∈ 𝕄^{n×n},  (1.3)

for a suitable constant c_{f} > 0 depending on f.

We remark that a 1-quasiconvex function F satisfying (1.2) and (1.3) is constructed explicitly if p ≥ 2 (see (3.8)), while in the case 1 < p < 2 it is defined as the quasiconvex envelope of a suitable extension of f to 𝕄^{n×n}.

The proof of the theorem relies on a Korn-type inequality for divergence-free vector fields and uses heavily the Lipschitz condition on the gradient of f. The use of Korn-type inequalities prevents us from obtaining a similar result for the case p = 1, which, if valid, will require a different treatment.

We do not know if the result continues to hold without assuming (1.1). However, when condition (1.1) is dropped we can still prove the following weaker version of Theorem 1.1.

Theorem 1.2 Let f : 𝕄_{sym}^{n×n} → ℝ satisfy the following conditions for suitable constants p > 1, μ ≥ 0, ν > 0, M > 0:

(a) (strict 2-quasiconvexity)

∫_{Q} [f (A + ∇^{2}ϕ) − f (A)] dx ≥ ν ∫_{Q} (μ^{2} + |A|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx

for every A ∈ 𝕄_{sym}^{n×n} and every ϕ ∈ C_{c}^{2} (Q);

(b) (growth condition)

|f (A)| ≤ M (1 + |A|^{p})  (1.4)

for every A ∈ 𝕄_{sym}^{n×n}.

Then there exists an increasing sequence {F_{k}} of 1-quasiconvex functions F_{k} : 𝕄^{n×n} → ℝ such that

lim_{k→∞} F_{k} (A) = f (A)  ∀A ∈ 𝕄_{sym}^{n×n},  (1.5)
|F_{k} (A)| ≤ c_{k} (1 + |A|^{p})  ∀A ∈ 𝕄^{n×n},  (1.6)

for a suitable sequence of constants {c_{k}} depending only on k and on the structural constants p, μ, ν, M, but not on the specific function f.

Theorem 1.2 allows us to reduce lower semicontinuity problems for 2-quasi convex normal integrands of the form f = f(x, u, ∇u, ∇^{2}u) to first order problems (see Section 4 for more details). Indeed, as a consequence of Theorem 1.2 we can prove the following result, which extends to the second order setting a lower semicontinuity property of 1-quasiconvex functions in SBV (Ω; ℝ^{d}) due to Ambrosio [2] and later generalized by Kristensen [18]. For the definition and properties of the space SBH(Ω) we refer to [4] and [5].

Theorem 1.3 Let Ω ⊂ ℝ^{n} be a bounded open set and let

f : Ω × ℝ × ℝ^{n} × 𝕄_{sym}^{n×n} → [0, +∞)

be an integrand which satisfies the following conditions:

(a) the function f(x, ·, ·, ·) is lower semicontinuous on ℝ × ℝ^{n} × 𝕄_{sym}^{n×n} for L^{n} a.e. x ∈ Ω;

(b) the function f(x, u, ξ, ·) is 2-quasiconvex on 𝕄_{sym}^{n×n} for L^{n} a.e. x ∈ Ω and every (u, ξ) ∈ ℝ × ℝ^{n};

(c) there exist a locally bounded function a : Ω × ℝ × ℝ^{n} → [0, +∞) and a constant p > 1 such that

0 ≤ f(x, u, ξ, A) ≤ a (x, u, ξ) (1 + |A|^{p})

for L^{N} a.e. x ∈ Ω and every (u, ξ, A) ∈ ℝ^{n} × ℝ^{n} × 𝕄_{sym}^{n×n}.

Then

∫_{Ω} f(x, u, ∇u, ∇^{2}u) dx ≤ lim inf_{j→∞} ∫_{Ω} f(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx

for every u ∈ SBH(Ω) and any sequence {u_{j}} ⊂ SBH(Ω) converging to u in W^{1,1}(Ω) and such that

sup_{j} (∥∇^{2}u_{j}∥_{L^{p}} + ∫_{S(∇u_{j})} θ(|[∇u_{j}]|) dH^{n−1}) < ∞,  (1.7)

where θ : [0, ∞) → [0, ∞) is a concave, nondecreasing function such that

lim_{t→0^{+}} \frac{θ (t)}{t} = ∞,

and [∇u_{j}] denotes the jump of ∇u_{j} on the jump set S(∇u_{j}).

An analogous result has been proved in [11] in the space BH(Ω) in the case where (1.7) is replaced by

sup_{j} ∥∇^{2}u_{j}∥_{L^{p}} < ∞  |D_{s}^{2}u_{j}| (Ω) → 0,

where D_{s}^{2}u_{j} is the singular part of the 𝕄_{sym}^{n×n}-valued measure D^{2}u_{j}. Note that condition (1.7) arises naturally in the context of free-discontinuity problems (see [2]).

As a corollary of Theorem 1.3 we have the following result.

Corollary 1.4 Let Ω and f be as in Theorem 1.3. Then

∫_{Ω} f(x, u, ∇u, ∇^{2}u) dx ≤ lim inf_{j→∞} ∫_{Ω} f(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx

for every u ∈ W^{2,p}(Ω) and any sequence {u_{j}} ⊂ W^{2,p}(Ω) weakly converging to u in W^{2,p}(Ω).

In this generality Corollary 1.4 was proved in [11] and under stronger hypotheses in [14], [15], and [20].

Remark 1.5 All of the above are still valid in a vectorial setting, i.e., u : Ω → ℝ^{d}, with 𝕄_{sym}^{n×n} replaced now by (𝕄_{sym}^{n×n})^{d}. The proofs are entirely similar to those presented in this paper for the case d = 1, and we leave the obvious adaptations to the reader.

The paper is organized as follows. In Section 2 we present some auxiliary results including the Korn-type inequality mentioned above. In Section 3 we prove Theorems 1.1 and 1.2, while Theorem 1.3 is addressed in the last section.

2 Auxiliary results

We begin with some results on the Helmholtz Decomposition and on Korn’s type inequalities.

A function w : ℝ^{n} → ℝ^{d} is said to be Q–periodic if w(x + e_{i}) = w(x) for a.e. x ∈ ℝ^{n} and every i = 1, …, n, where (e_{1}, …, e_{n}) is the canonical basis of ℝ^{n}. The spaces of Q-periodic functions of W_{loc}^{1,p} (ℝ^{n}; ℝ^{n}), W_{loc}^{2,p} (ℝ^{n}), and C^{∞}(ℝ^{n}; ℝ^{n}) are denoted by W_{per}^{1,p} (Q; ℝ^{n}), W_{per}^{2,p} (Q), and C_{per}^{∞}(Q; ℝ^{n}), respectively.

Lemma 2.1 (Helmholtz decomposition) For every p > 1 and every φ ∈ W_{per}^{1,p} (Q; ℝ^{n}) there exist two functions ϕ ∈ W_{per}^{2,p} (Q) and ψ ∈ W_{per}^{1,p} (Q; ℝ^{n}) such that

φ = ∇ϕ + ψ, div ψ = 0.

The function ψ is uniquely determined, while ϕ is determined up to an additive constant.

Proof. Since, by periodicity, div φ has zero average on Q, there exists a Qperiodic solution ϕ of the equation Δϕ = div φ, which is unique up to an additive constant. It is clear now that ψ := φ − ∇ϕ is Q-periodic and div ψ = 0.

Throughout the paper, for every A ∈ 𝕄^{n×n}, we denote the symmetric and antisymmetric parts of A by

A^{s} := \frac{A + A^{t}}{2} , A^{a} := \frac{A − A^{t}}{2} ,

where A^{t} is the transpose matrix of A.

If ψ : Ω ⊂ ℝ^{n} → ℝ^{d} is any function, then ∇ψ is a d × n matrix in 𝕄^{d×n}, with (∇ψ)_{ij} := \frac{∂ψ_{i}}{∂x_{j}} . Also, differential operators applied to matrix-valued fields are understood on a row-by-row basis, e.g., if ψ : Ω ⊂ ℝ^{n} → ℝ^{n}, then

div ∇ψ := (div ∇ψ_{1}, …, div ∇ψ_{n}) .

To simplify the notation, for any ψ : Ω ⊂ ℝ^{n} → ℝ^{n}, we set

∇ψ^{t} := (∇ψ)^{t} ,  ∇ψ^{s} := (∇ψ)^{s} ,  ∇ψ^{a} := (∇ψ)^{a} .

Lemma 2.2 For every p > 1 there exists a constant γ_{n,p} ≥ 1 such that

∫_{Q} |∇ψ|^{p} dx ≤ γ_{n,p} ∫_{Q} |∇ψ^{a}|^{p} dx  (2.1)

for every Q-periodic function ψ : ℝ^{n} → ℝ^{n} of class C^{∞} with div ψ = 0.

Proof. Since div ∇ψ^{t} = ∇ (div ψ) = 0 we have

Δψ = 2 div (\frac{∇ψ − ∇ψ^{t}}{2} ) = 2 div (∇ψ^{a}) .  (2.2)

Hence (2.1) follows from standard L^{p} estimates for periodic solutions of the Poisson equation (see [16]).

Next we study the behavior of auxiliary functions of the type

g (x) := (μ^{2} + |x|^{2})^{\frac{p}{2}}  (2.3)

defined on an arbitrary Hilbert space X.

Lemma 2.3 For every p > 1 there exist two constants κ_{p} and K_{p}, with 0 < κ_{p} ≤ 1 ≤ K_{p}, such that the following inequalities hold:

∫_{0}^{1} (μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} (1 − t) dt ≥ κ_{p} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} ,  (2.4)
∫_{0}^{1} (μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} dt ≤ K_{p} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}}  (2.5)

for every x, y ∈ X and every constant μ ≥ 0.

Proof. Let us prove (2.4). If 1 < p ≤ 2, then (2.4) holds with κ_{p} = 2^{\frac{p}{2} −2}, since

(μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} ≥ 2^{\frac{p−2}{2}} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} .

If p > 2, then we consider first the case where |y|^{2} ≤ 4 (μ^{2} + |x|^{2}). For 0 ≤ t ≤ 1/2 we have

(μ^{2} + |x + ty|^{2})^{\frac{1}{2}} ≥ (μ^{2} + |x|^{2})^{\frac{1}{2}} − t |y| ≥ (1 − 2t) (μ^{2} + |x|^{2})^{\frac{1}{2}}
≥ 5^{− \frac{1}{2}} (1 − 2t) (μ^{2} + |x|^{2} + |y|^{2})^{\frac{1}{2}} ,

hence

(μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} ≥ 5^{\frac{2−p}{2}} (1 − 2t)^{p−2} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} .

We deduce that (2.4) holds provided

κ_{p} ≤ 5^{\frac{2−p}{2}} ∫0^{\frac{1}{2}} (1 − 2t)^{p−2} (1 − t) dt = \frac{5^{\frac{2−p}{2}}}{4} \frac{2p − 1}{p (p − 1)}.

If p > 2 and |y|^{2} > 4 (μ^{2} + |x|^{2}), then for 1/2 ≤ t ≤ 1 we have

(μ^{2} + |x + ty|^{2})^{\frac{1}{2}} ≥ t |y| − (μ^{2} + |x|^{2})^{\frac{1}{2}} ≥ (t − \frac{1}{2}) |y|

≥ 2 · 5^{− \frac{1}{2}} (t − \frac{1}{2}) (μ^{2} + |x|^{2} + |y|^{2})^{\frac{1}{2}} .

Therefore

(μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} ≥ 5^{\frac{2−p}{2}} (2t − 1)^{p−2} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} .

We conclude that (2.4) is verified with

κ_{p} ≤ 5^{\frac{2−p}{2}} ∫ 1_{\frac{1}{2}} (2t − 1)^{p−2} (1 − t) dt = \frac{5^{\frac{2−p}{2}}}{4} \frac{1}{p (p − 1)}.

This concludes the proof of (2.4).

Let us prove (2.5). If p ≥ 2 then (2.5) holds with K_{p} = 2^{\frac{p−2}{2}} , since

(μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} ≤ 2^{\frac{p−2}{2}} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} .

In the case 1 < p < 2 we observe that

(μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} ≤ |(μ^{2} + |x|^{2})^{\frac{1}{2}} − t |y||^{p−2} .

Let a := (μ^{2} + |x|^{2})^{\frac{1}{2}} and b := |y|. Then

∫_{0}^{1} (μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} dt ≤ ∫_{0}^{1} |a − tb|^{p−2} dt

and (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} = (a^{2} + b^{2})^{\frac{p−2}{2}} .

If b ≤ a, then

∫_{0}^{1} |a − tb|^{p−2} dt = \frac{a^{p−1} − (a − b)^{p−1}}{(p − 1) b} ≤ \frac{a^{p−2}}{p − 1},

where the last inequality is obtained by comparing the difference quotients of the concave function t ↦ t^{p−1} in the intervals [a − b, a] and [0, a]. Therefore we have that

∫_{0}^{1} |a − tb|^{p−2} dt ≤ \frac{a^{p−2}}{p − 1} ≤ \frac{2^{\frac{2−p}{2}}}{p − 1} (a^{2} + b^{2})^{\frac{p−2}{2}} ,

and (2.5) is satisfied for K_{p} ≥ 2^{\frac{2−p}{2}} /(p − 1).

If a < b, then

∫_{0}^{1} |a − tb|^{p−2} dt = \frac{a^{p−1} + (b − a)^{p−1}}{(p − 1) b} ≤ \frac{2^{2−p}b^{p−2}}{p − 1} .

On the other hand, in this case we have b^{p−2} ≤ 2^{\frac{2−p}{2}} (a^{2} + b^{2})^{\frac{p−2}{2}} , so that (2.5) holds for K_{p} ≥ 2^{^{\frac{3}{2}} (2−p)}/(p − 1).

Lemma 2.4 For every p > 1, there exist two constants θ_{p} > 0 and Θ_{p} > 0 such that for every μ ≥ 0 the function g defined in (2.3) satisfies the following inequalities

θ_{p} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} ≤ g (x + y) − g (x) − ∇g (x) · y
≤ Θ_{p} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2}

for every x, y ∈ X.

Proof. By continuity it is enough to prove the statement when 0 does not belong to the segment joining x and x + y. In this case the function

h (t) := (μ^{2} + |x + ty|^{2})^{\frac{p}{2}}

belongs to C^{∞} ([0, 1]), and Taylor’s formula with integral remainder yields

h (1) − h (0) − h′ (0) = ∫_{0}^{1} h′′ (t) (1 − t) dt.

By direct computation we see that

p ((p − 1) ∧ 1) (μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} |y|^{2} ≤ h′′ (t)
≤ p ((p − 1) ∨ 1) (μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} |y|^{2} .

The conclusion follows from Lemma 2.3.

In the proof of Theorem 1.1 we will need the following extension of Lemma 2.4 to the family of functions

g_{β} (x, y) := (μ^{2} + |x|^{2} + β^{2} |y|^{2})^{\frac{p}{2}} , β ≥ 0,

defined on the product of two Hilbert spaces X and Y.

Lemma 2.5 Let p > 1, β ≥ 0 and μ ≥ 0. Then

g_{β} (x + ξ, y + η) − g_{β} (x, y) − ∇_{x}g_{β} (x, y) · ξ − ∇_{y}g_{β} (x, y) · η
≥θ_{p} (μ^{2} + |x|^{2} + |ξ|^{2} + β^{2} |y|^{2} + β^{2} |η|^{2})^{\frac{p−2}{2}} (|ξ|^{2} + β^{2} |η|^{2})

for every x, ξ ∈ X, y, η ∈ Y, where θ_{p} is the first constant in Lemma 2.4. Therefore, if p ≥ 2, we have

g_{β} (x + ξ, y + η) − g_{β} (x, y) − ∇_{x}g_{β} (x, y) · ξ − ∇_{y}g_{β} (x, y) · η

≥ θ_{p} (μ^{2} + |x|^{2} + |ξ|^{2})^{\frac{p−2}{2}} |ξ|^{2} + \frac{θ_{p}β^{2}}{2} (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |η|^{2} + \frac{θ_{p}β^{p}}{2} |η|^{p}

for every x, ξ ∈ X, y, η ∈ Y.

Proof. Observing that g_{β} (x, y) = g_{1} (x, βy), the inequality can be obtained by applying Lemma 2.4 to the Hilbert space X × Y.

We continue with some technical lemmas which are used in the proofs of Theorems 1.1 and 1.2.

Lemma 2.6 Let X be a Hilbert space and let 1 < p ≤ 2. Then for every μ ≥ 0 and every 0 < ε < 1 we have

(μ^{2} + |x + y|^{2})^{\frac{p−2}{2}} |x + y|^{2} ≤ 2 (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |x|^{2} + 2 (μ^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} ,
ε^{\frac{2−p}{2}} (μ^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} ≤ (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + ε (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |x|^{2}

for every x, y ∈ X.

Proof. Since the mapping t ↦ (μ^{2} + t)^{\frac{p−2}{2}} t is nondecreasing, while the mapping t ↦ (μ^{2} + t)^{\frac{p−2}{2}} is nonincreasing, we have

(μ^{2} + |x + y|^{2})^{\frac{p−2}{2}} |x + y|^{2} ≤ (μ^{2} + 2 |x|^{2} + 2 |y|^{2})^{\frac{p−2}{2}} (2 |x|^{2} + 2 |y|^{2})
≤ 2 (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |x|^{2} + 2 (μ^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} ,

which proves the first inequality. For the same reason, we have

ε^{\frac{2−p}{2}} (μ^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} ≤ ε^{\frac{2−p}{2}} (μ^{2} + ε |x|^{2} + |y|^{2})^{\frac{p−2}{2}} (ε |x|^{2} + |y|^{2})
≤ (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} (ε |x|^{2} + |y|^{2})

≤ ε (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |x|^{2} + (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} ,

and this concludes the proof.

Lemma 2.7 Let 1 < p ≤ 2. Then

b^{p} ≤ 8ε^{\frac{p−2}{p}} (μ^{2} + a^{2} + b^{2})^{\frac{p−2}{2}} b^{2} + εa^{p} + εμ^{p}  (2.6)

for every a ≥ 0, b ≥ 0, μ ≥ 0, and 0 < ε < 1. Proof. By Young’s inequality with exponents 2/p and 2/ (2 − p) we have that

b^{p} = (2^{\frac{2−p}{2}} ε^{\frac{p−2}{2}} (μ^{2} + b^{2})^{\frac{p(p−2)}{4}} b^{p}) (2^{\frac{p−2}{2}} ε^{\frac{2−p}{2}} (μ^{2} + b^{2})^{\frac{p(2−p)}{4}} )  (2.7)

≤ 2ε^{\frac{p−2}{p}} (μ^{2} + b^{2})^{\frac{p−2}{2}} b^{2} + \frac{ε}{2} (μ^{2} + b^{2})^{\frac{p}{2}} .

If a^{2} ≤ μ^{2} + b^{2}, we have

(μ^{2} + b^{2})^{\frac{p−2}{2}} ≤ 2 (μ^{2} + a^{2} + b^{2})^{\frac{p−2}{2}} ,

and so from (2.7) we obtain

b^{p} ≤ 4ε^{\frac{p−2}{p}} (μ^{2} + a^{2} + b^{2})^{\frac{p−2}{2}} b^{2} + \frac{ε}{2} (μ^{p} + b^{p}) .

Subtracting (ε/2) b^{p} to both sides we obtain (2.6).

On the other hand, if a^{2} > μ^{2} + b^{2}, then from (2.7) with (μ^{2} + b^{2}) replaced by (μ^{2} + a^{2} + b^{2}) we obtain

b^{p} ≤ 2ε^{\frac{p−2}{p}} (μ^{2} + a^{2} + b^{2})^{\frac{p−2}{2}} b^{2} + \frac{ε}{2} (μ^{2} + a^{2} + b^{2})^{\frac{p}{2}}
≤ 2ε^{\frac{p−2}{p}} (μ^{2} + a^{2} + b^{2})^{\frac{p−2}{2}} b^{2} + εa^{p},

which proves (2.6).

The next lemma shows that condition (1.1) in Theorem 1.1 can be obtained from a suitable bound on the second derivatives of f. This is trivial in the case p ≥ 2, but requires some work in the case 1 < p < 2.

Lemma 2.8 Let X be a Hilbert space, and let f ∈ C^{1} (X) ∩ C^{2} (X \ {0}) . Assume that there exist two constants p > 1, C > 0, and μ ≥ 0 such that

|∇^{2}f (x)| ≤ C (μ^{2} + |x|^{2})^{\frac{p−2}{2}}  (2.8)

for every x ∈ X \ {0}. Then

|∇f (x + y) − ∇f (x)| ≤ K_{p}C (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|  (2.9)

for every x, y ∈ X, where K_{p} is the second constant in Lemma 2.3.

Proof. By continuity it is enough to prove the statement when 0 does not belong to the segment joining x and x + y. In this case by (2.8) we have

|∇f (x + y) − ∇f (x)| ≤ C |y| ∫_{0}^{1} (μ^{2} + |x + ty|^{2})^{\frac{p−2}{2}} dt,

and the conclusion follows from Lemma 2.3.

The estimate given by the following lemma will be crucial in the proof of Theorem 1.1.

Lemma 2.9 Let X be a Hilbert space and let f ∈ C^{1} (X). Assume that there exist p > 1 and μ ≥ 0 such that

|∇f (x + y) − ∇f (x)| ≤ (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|  (2.10)

for every x, y ∈ X. If 1 < p ≤ 2, then for every ε > 0 there exists c_{ε,p} > 0, depending only on ε and p, such that

|f (x + y + z) − f (x + y) − ∇f (x) · z|  (2.11)
≤ ε (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + c_{ε,p} (μ^{2} + |z|^{2})^{\frac{p−2}{2}} |z|^{2}

for every x, y, z ∈ X.

If p ≥ 2, then for every ε > 0 there exists c_{ε,p} > 0, depending only on ε and p, such that

|f (x + y + z) − f (x + y) − ∇f (x) · z|  (2.12)
≤ ε (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + c_{ε,p} (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |z|^{2} + c_{ε,p} |z|^{p}

for every x, y, z ∈ X.

Proof. Let us consider first the case 1 < p ≤ 2. Clearly (2.10) implies that

|∇f (x + y) − ∇f (x)| ≤ (μ^{2} + |y|^{2})^{\frac{p−2}{2}} |y|

for every x, y ∈ X. By the Mean Value Theorem we have

|f (x + y) − f (x) − ∇f (x) · y| ≤ |∇f (x + ty) − ∇f (x)| |y|
≤ (μ^{2} + t^{2} |y|^{2})^{\frac{p−2}{2}} t |y|^{2}

for some t ∈ [0, 1]. Since the function t ↦ (μ^{2} + t^{2} |y|^{2})^{\frac{p−2}{2}} t is nondecreasing, we obtain

|f (x + y) − f (x) − ∇f (x) · y| ≤ (μ^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2}  (2.13)

for every x, y ∈ X.

By (2.10) and (2.13) we have

|f(x + y + z) − f (x + y) − ∇f (x) · z|
≤ |f (x + y + z) − f (x + y) − ∇f (x + y) · z|  (2.14)
+ |∇f (x + y) · z − ∇f (x) · z|
≤ (μ^{2} + |z|^{2})^{\frac{p−2}{2}} |z|^{2} + (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y| |z| .

We now estimate the last term. If |z| ≤ μ then for every ε > 0 we have

(μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y| |z| ≤ ε (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2}
+ \frac{1}{4ε} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |z|^{2}  (2.15)
≤ε (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + \frac{1}{4ε}μ^{p−2} |z|^{2}
≤ε (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + \frac{1}{ε} (μ^{2} + |z|^{2})^{\frac{p−2}{2}} |z|^{2} .

If |z| > μ, let q := p/ (p − 1) be the conjugate exponent of p. Since \frac{p−2}{2} + \frac{1}{2} − \frac{1}{q} = \frac{p−2}{2q} we have

(μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y| |z| = (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{1− \frac{2}{q}} |y|^{\frac{2}{q}} |z|
≤ (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2q}} |y|^{\frac{2}{q}} |z|  (2.16)

≤ ε (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + \frac{1}{p (qε)^{p−1}} |z|^{p}

≤ ε (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + k_{ε,p} (μ^{2} + |z|^{2})^{\frac{p−2}{2}} |z|^{2} for some constant k_{ε,p} depending only on ε and p.

Let us consider now the case p ≥ 2. By the Mean Value Theorem and by the Cauchy Inequality we have

|f(x + y + z) − f (x + y) − ∇f (x) · z|
≤ |f (x + y + z) − f (x + y) − ∇f (x + y) · z|

+ |∇f (x + y) · z − ∇f (x) · z|

≤ (μ^{2} + 2 |x|^{2} + 2 |y|^{2} + |z|^{2})^{\frac{p−2}{2}} |z|^{2} + (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y| |z|
≤6^{\frac{p−2}{2}} (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |z|^{2} + 6^{\frac{p−2}{2}} |y|^{p−2} |z|^{2} + 3^{\frac{p−2}{2}} |z|^{p}

+ \frac{ε}{2} (μ^{2} + |x|^{2} + |y|^{2})^{\frac{p−2}{2}} |y|^{2} + k_{ε,p} (μ^{2} + |x|^{2})^{\frac{p−2}{2}} |z|^{2} + k_{ε,p} |y|^{p−2} |z|^{2}

for some constant k_{ε,p} depending only on ε and p. The conclusion follows from Young’ inequality with exponents p/ (p − 2) and p/2.

The following lemma states an elementary property of strictly 2-quasiconvex functions.

Lemma 2.10 Assume that f : 𝕄_{sym}^{n×n} → ℝ satisfies the strict 2-quasiconvexity condition (a) of Theorem 1.1 for some constants p > 1, μ ≥ 0, ν > 0, and let g : 𝕄_{sym}^{n×n} → ℝ be the function defined by

g(A) := (μ^{2} + |A|^{2})^{\frac{p}{2}} .  (2.17)

Then the function f_{λ} := f − λg is 2-quasiconvex for λ ≤ ν/Θ_{p}, where Θ_{p} is the second constant in Lemma 2.4.

Proof. Let A ∈ 𝕄_{sym}^{n×n} and ϕ ∈ C_{c}^{2} (Q). Since, by periodicity,

∫_{Q} ∇g(A) · ∇^{2}ϕ dx = 0,

we have

∫Q [f_{λ} (A + ∇^{2}ϕ) − f_{λ} (A)] dx = ∫_{Q} [f (A + ∇^{2}ϕ) − f (A)] dx
− λ ∫_{Q} [g (A + ∇^{2}ϕ) − g (A) + ∇g(A) · ∇^{2}ϕ] dx

≥ (ν − λΘ_{p}) ∫_{Q} (μ^{2} + |A|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx ≥ 0,

which concludes the proof.

In the proof of Theorem 1.1 we need the following generalization of Lemma 2.2.

Lemma 2.11 For every p > 1 there exists a constant τ_{n,p} ≥ 1 such that

∫_{Q} (μ^{2} + |∇ψ^{s}|^{2})^{\frac{p−2}{2}} |∇ψ^{s}|^{2} dx ≤ τ_{n,p} ∫_{Q} (μ^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx

(2.18)

for every constant μ ≥ 0 and every Q-periodic function ψ : ℝ^{n} → ℝ^{n} of class C^{∞} with div ψ = 0.

Proof. Let μ and ψ be as in the statement of the lemma. In the case p ≥ 2 by Lemma 2.2 we have

∫_{Q} |∇ψ|^{p} dx ≤ γ_{n,p} ∫_{Q} |∇ψ^{a}|^{p} dx,

and so

∫Q (μ^{2} + |∇ψ|^{2})^{\frac{p−2}{2}} |∇ψ|^{2} dx
≤ 2^{\frac{p−2}{2}} \{μ^{p−2} ∫_{Q} |∇ψ|^{2} dx + ∫_{Q} |∇ψ|^{p} dx\}
≤ 2^{\frac{p−2}{2}} \{μ^{p−2}γ_{n,2} ∫_{Q} |∇ψ^{a}|^{2} dx + γ_{n,p} ∫_{Q} |∇ψ^{a}|^{p} dx\}

≤ τ_{n,p} ∫_{Q} (μ^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx

with τ_{n,p} := 2^{\frac{p−2}{2}} (γ_{n,2} + γ_{n,p},).

We consider now the case 1 < p < 2. Let E_{μ} := {|∇ψ^{a}| ≤ μ} and E^{μ} := {|∇ψ^{a}| > μ}, and let Ψ_{μ} := 1_{E_{μ}}∇ψ^{a} and Ψ^{μ} := 1_{E^{μ}}∇ψ^{a}, where 1_{E} is the characteristic function of the set E. Note that Ψ_{μ} and Ψ^{μ} are periodic vectorfields. Let ψ_{μ} and ψ^{μ} be periodic solutions of the equations

Δψ_{μ} = 2 div Ψ_{μ}  and  Δψ^{μ} = 2 div Ψ^{μ}.

From the first equation we get

∫_{Q} |∇ψ_{μ}|^{2} dx ≤ 4 ∫_{Q} |Ψ_{μ}|^{2} dx.  (2.19)

Standard L^{p} estimates for periodic solutions of the Poisson equation (see [16]) yield a constant γ̃_{n,p} ≥ 4 such that

∫_{Q} |∇ψ^{μ}|^{p} dx ≤ γ̃_{n,p} ∫_{Q} |Ψ^{μ}|^{p} dx.  (2.20)

From (2.19) we obtain

∫_{Q} (μ^{2} + |∇ψ_{μ}|^{2})^{\frac{p−2}{2}} |∇ψ_{μ}|^{2} dx ≤ μ^{p−2} ∫_{Q} |∇ψ_{μ}|^{2} dx
≤ 4μ^{p−2} ∫_{Q} |Ψ_{μ}|^{2} dx  (2.21)

≤ 8 ∫_{Q∩E_{μ}} (μ^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx,

where the last inequality follows from the fact that |Ψ_{μ}| = |∇ψ^{a}| 1_{E_{μ}} ≤ μ. From (2.20) we obtain

∫_{Q} (μ^{2} + |∇ψ^{μ}|^{2})^{\frac{p−2}{2}} |∇ψ^{μ}|^{2} dx ≤ ∫_{Q} |∇ψ^{μ}|^{p} dx
≤ γ̃_{n,p} ∫_{Q} |Ψ^{μ}|^{p} dx  (2.22)

≤ 2γ̃_{n,p} ∫_{Q∩E^{μ}} (μ^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx,

where the last inequality follows from the fact that |Ψ^{μ}| = |∇ψ^{a}| 1_{E^{μ}} ≥ μ1_{E^{μ}}. By (2.2) we have

Δ (ψ_{μ} + ψ^{μ}) = 2 div ∇ψ^{a} = Δψ,

and since ψ_{μ} + ψ^{μ} − ψ is a periodic function we deduce that ∇ψ = ∇ψ_{μ} + ∇ψ^{μ}. Finally, from Lemma 2.6 and using (2.21) and (2.22) we obtain

∫_{Q} (μ^{2} + |∇ψ|^{2})^{\frac{p−2}{2}} |∇ψ|^{2} dx ≤ τ_{n,p} ∫_{Q} (μ^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx,

with τ_{n,p} = 2γ̃_{n,p}. Since |∇ψ^{s}| ≤ |∇ψ| and the mapping t ↦ (μ^{2} + t)^{\frac{p−2}{2}} t is nondecreasing, inequality (2.18) follows.

3 Proofs

Proof of Theorem 1.1. We begin by observing that (1.1) gives

|f (A)| ≤ k_{f} (1 + |A|^{p})  ∀A ∈ 𝕄_{sym}^{n×n}  (3.1)

for a suitable constant k_{f} depending on f. Step 1: We first consider the case 1 < p < 2. Let g : 𝕄^{n×n} → ℝ be the function defined by

g(A) := (μ^{2} + |A^{a}|^{2})^{\frac{p}{2}} − μ^{p}.

Given a constant β > 0, to be chosen at the end of the proof, let G : 𝕄^{n×n} → ℝ be the function defined by

G(A) := f(A^{s}) + βg(A^{a}),  (3.2)

and let F be its 1-quasiconvexification, i.e., (see, e.g., [8])

F (A) = inf {∫_{Q} G(A + ∇φ(x)) dx : φ ∈ C_{per}^{∞}(Q; ℝ^{n})} ,  (3.3)

for all A ∈ 𝕄^{n×n}.

We want to prove that for every ε > 0 there exists β > 0 such that

∫_{Q} [G (A + ∇φ) − G (A)] dx ≥ −ε (μ^{2} + |A^{a}|^{2})^{\frac{p−2}{2}} |A^{a}|^{2}  (3.4)

for every A ∈ 𝕄^{n×n} and for every φ ∈ C_{per}^{∞}(Q; ℝ^{n}). In view of (3.3) this will imply that for every A ∈ 𝕄^{n×n} we have

G(A) − ε (μ^{2} + |A^{a}|^{2})^{\frac{p−2}{2}} |A^{a}|^{2} ≤ F (A) ≤ G(A)  (3.5)

which yields (1.2). Inequality (1.3) follows from (3.1), (3.2) and (3.5).

Let us prove (3.4). Fix φ ∈ C_{per}^{∞}(Q; ℝ^{n}) and consider the periodic Helmholtz decomposition

φ = ∇ϕ + ψ

given by Lemma 2.1. Following the argument used by Müller and Šverák in the proof of Lemma 4.2 in [22], we have

∫_{Q}[G (A + ∇φ) − G (A)] dx
= ∫_{Q} [f (A^{s} + ∇^{2}ϕ + ∇ψ^{s}) − f (A^{s} + ∇^{2}ϕ)] dx
+ ∫_{Q} [f (A^{s} + ∇^{2}ϕ) − f (A^{s})] dx  (3.6)
+ β ∫_{Q} [(μ^{2} + |A^{a} + ∇ψ^{a}|^{2})^{\frac{p}{2}} − (μ^{2} + |A^{a}|^{2})^{\frac{p}{2}} ] dx
=: I_{1} + I_{2} + I_{3}.

Since ∇f (A^{s}) is a symmetric matrix we have ∇f (A^{s}) · ∇ψ^{s} = ∇f (A^{s}) · ∇ψ, and therefore, by periodicity,

∫_{Q} ∇f (A^{s}) · ∇ψ^{s} dx = 0.

Hence

I_{1} = ∫_{Q} [f (A^{s} + ∇^{2}ϕ + ∇ψ^{s}) − f (A^{s} + ∇^{2}ϕ) − ∇f (A^{s}) · ∇ψ^{s}] dx.

By Lemma 2.9 we have

I_{1} ≥ −ν ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx

− c_{ν,p,L} ∫_{Q} (μ^{2} + |∇ψ^{s}|^{2})^{\frac{p−2}{2}} |∇ψ^{s}|^{2} dx,

while the strict 2-quasiconvexity of f (condition (a)) yields

I_{2} ≥ ν ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx,

and so, using Lemma 2.11, we obtain

I_{1} + I_{2} ≥ −c_{ν,p,L} ∫_{Q} (μ^{2} + |∇ψ^{s}|^{2})^{\frac{p−2}{2}} |∇ψ^{s}|^{2} dx  (3.7)

≥ −c_{ν,p,L} τ_{n,p} ∫_{Q} (μ^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx.

Since ∇g(A^{a}) is an antisymmetric matrix we have ∇g(A^{a})·∇ψ^{a} = ∇g(A^{a})·∇ψ, and therefore, by periodicity,

∫_{Q} ∇g(A^{a}) · ∇ψ^{a} dx = 0.

Hence, by Lemma 2.4 and Lemma 2.6, for every 0 < δ < 1 we obtain

I_{3} = β ∫_{Q} [g (A^{a} + ∇ψ^{a}) − g (A^{a}) − ∇g(A^{a}) · ∇ψ^{a}] dx

≥ βθ_{p} ∫_{Q} (μ^{2} + |A^{a}|^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx

≥ βθ_{p}δ^{\frac{2−p}{2}} ∫_{Q} (μ^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx − βθ_{p}δ (μ^{2} + |A^{a}|^{2})^{\frac{p−2}{2}} |A^{a}|^{2} .

Choosing β > 0 and 0 < δ < 1 so that

βθ_{p}δ^{\frac{2−p}{2}} ≥ c_{ν,p,L} τ_{n,p}, βθ_{p}δ ≤ ε

we obtain

I_{1} + I_{2} + I_{3} ≥ −ε (μ^{2} + |A^{a}|^{2})^{\frac{p−2}{2}} |A^{a}|^{2} ,

which, together with (3.6), yields (3.4). Step 2: Let us consider now the case p ≥ 2. Let λ := ν/Θ_{p}, where Θ_{p} is the second constant in Lemma 2.4. Given a constant β > 0, to be chosen at the end of the proof, let F : 𝕄^{n×n} → ℝ be the function defined by

F (A) := f(A^{s}) − λ (μ^{2} + |A^{s}|^{2})^{\frac{p}{2}} + λ (μ^{2} + |A^{s}|^{2} + β^{2} |A^{a}|^{2})^{\frac{p}{2}} .  (3.8)

It is clear that (1.2) holds, while (1.3) follows from (3.1).

It remains to prove that, for some β > 0, the function F is 1-quasiconvex, i.e.,

∫_{Q} [F (A + ∇φ) − F (A)] dx ≥ 0  (3.9)

for every A ∈ 𝕄^{n×n} and for every φ ∈ C_{per}^{∞}(Q; ℝ^{n}).

Let f_{λ} be the 2-quasiconvex function defined in Lemma 2.10, and let

g_{β}(A) = ĝ_{β} (A^{s}, A^{a}) := (μ^{2} + |A^{s}|^{2} + β^{2} |A^{a}|^{2})^{\frac{p}{2}} ,

so that

F (A) = f_{λ}(A^{s}) + λg_{β}(A).

Let us prove (3.9). Fix a Q-periodic function φ : ℝ^{n} → ℝ^{n} of class C^{∞} and consider the periodic Helmholtz decomposition

φ = ∇ϕ + ψ

given by Lemma 2.1. Then we have

∫_{Q}[F (A + ∇φ) − F (A)] dx
= ∫_{Q} [f_{λ} (A^{s} + ∇φ^{s}) − f_{λ} (A^{s} + ∇φ^{s} − ∇ψ^{s})] dx
+ ∫_{Q} [f_{λ} (A^{s} + ∇^{2}ϕ) − f_{λ} (A^{s})] dx  (3.10)
+ λ ∫_{Q} [g_{β} (A + ∇φ) − g_{β} (A)] dx
=: I_{1} + I_{2} + I_{3}.

Since ∇f_{λ} (A^{s}) is a symmetric matrix, we have ∇f_{λ} (A^{s})·∇ψ^{s} = ∇f_{λ} (A^{s})·∇ψ, and therefore, by periodicity,

∫_{Q} ∇f_{λ} (A^{s}) · ∇ψ^{s} dx = 0.

Hence

I_{1} = − ∫_{Q} [f_{λ} (A^{s} + ∇φ^{s} − ∇ψ^{s}) − f_{λ} (A^{s} + ∇φ^{s}) + ∇f_{λ} (A^{s}) · ∇ψ^{s}] dx.

Since the function g defined in (2.17) clearly satisfies condition (2.8) , by Lemma 2.8 and (1.1) it follows that (1.1) still holds for the function f_{λ} for a suitable constant M > 0 in place of L. We are now in position to apply Lemma 2.9 to obtain a constant σ = σ_{p,M} such that

I_{1} ≥ −λθ_{p} ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇φ^{s}|^{2})^{\frac{p−2}{2}} |∇φ^{s}|^{2} dx

− σ (μ^{2} + |A^{s}|^{2})^{\frac{p−2}{2}} ∫_{Q} |∇ψ^{s}|^{2} dx − σ ∫_{Q} |∇ψ^{s}|^{p} dx,

and so, using Lemma 2.2, we obtain

I_{1} ≥ −λθ_{p} ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇φ^{s}|^{2})^{\frac{p−2}{2}} |∇φ^{s}|^{2} dx  (3.11)

− σγ_{n,2} (μ^{2} + |A^{s}|^{2})^{\frac{p−2}{2}} ∫_{Q} |∇ψ^{a}|^{2} dx − σγ_{n,p} ∫_{Q} |∇ψ^{a}|^{p} dx.

On the other hand, the 2-quasiconvexity of f_{λ} yields

I_{2} ≥ 0.  (3.12)

Since, by periodicity,

∫_{Q} ∇g_{β}(A) · ∇φ dx = 0,

by Lemma 2.5 we have

I_{3} = λ ∫_{Q} [g_{β} (A + ∇φ) − g_{β} (A) − ∇g_{β}(A) · ∇φ] dx

≥ λθ_{p} ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇φ^{s}|^{2})^{\frac{p−2}{2}} |∇φ^{s}|^{2} dx  (3.13)

+ \frac{λθ_{p}β^{2}}{2} (μ^{2} + |A^{s}|^{2})^{\frac{p−2}{2}} ∫_{Q} |∇ψ^{a}|^{2} dx + \frac{λθ_{p}β^{p}}{2} ∫_{Q} |∇ψ^{a}|^{p} dx.

Choosing β > 0 so that

\frac{λθ_{p}β^{2}}{2} ≥ σγ_{n,2}, \frac{λθ_{p}β^{p}}{2} ≥ σγ_{n,p},

by (3.11), (3.12), and (3.13), we obtain

I_{1} + I_{2} + I_{3} ≥ 0,

which together with (3.10) yields (3.9).

Proof of Theorem 1.2. Since the function t ↦ f (A + ta ⊗ b + tb ⊗ a) is convex on ℝ for every A ∈ 𝕄_{sym}^{n×n} and every a, b ∈ ℝ^{n} (see, e.g., [[12]]), from the growth condition (b) it follows that there exists a constant L > 0 depending only on M and p such that

|f (A + B) − f (A)| ≤ L (1 + |A|^{p−1} + |B|^{p−1}) |B|  (3.14)

for every A, B ∈ 𝕄_{sym}^{n×n}. Given a constant β > 0, to be chosen at the end of the proof, let G : 𝕄^{n×n} → ℝ be the function defined by

G(A) := f(A^{s}) + β |A^{a}|^{p} ,  (3.15)

and let F be its 1-quasiconvexification.

We want to prove that there exist two increasing sequences of positive numbers {β_{k}} and {λ_{k}}, depending only on k, p, μ, ν, M, but not on the specific function f, such that the corresponding functions G_{k} satisfy

∫_{Q} [G_{k} (A + ∇φ) − G_{k} (A)] dx ≥ −\frac{1}{k} |A^{s}|^{p} − λ_{k} |A^{a}|^{p} − \frac{1}{k}  (3.16)

for every A ∈ 𝕄^{n×n} and for every Q-periodic function φ : ℝ^{n} → ℝ^{n} of class C^{∞}. This will imply (see, e.g., [8]) that for every A ∈ 𝕄^{n×n} we have

G_{k}(A) − \frac{1}{k} |A^{s}|^{p} − λ_{k} |A^{a}|^{p} − \frac{1}{k} ≤ F_{k}(A) ≤ G_{k}(A)  (3.17)

which yields (1.5) and (1.6) since G_{k}(A) = f (A) whenever A ∈ 𝕄_{sym}^{n×n}.

Let us prove (3.16). Fix a Q-periodic function φ : ℝ^{n} → ℝ^{n} of class C^{∞} and consider the periodic Helmholtz decomposition

φ = ∇ϕ + ψ

given by Lemma 2.1. Then we have

∫_{Q}[G (A + ∇φ) − G (A)] dx
= ∫_{Q} [f (A^{s} + ∇^{2}ϕ + ∇ψ^{s}) − f (A^{s} + ∇^{2}ϕ)] dx
+ ∫_{Q} [f (A^{s} + ∇^{2}ϕ) − f (A^{s})] dx  (3.18)
+ β ∫_{Q} [|A^{a} + ∇ψ^{a}|^{p} − |A^{a}|^{p}] dx
=: I_{1} + I_{2} + I_{3}.

By (3.14) and by Cauchy’s inequality, for every δ > 0 there exists a constant c_{δ,p,L} > 0 such that

I_{1} ≥ −L ∫_{Q} (1 + |A^{s} + ∇^{2}ϕ|^{p−1} + |∇ψ^{s}|^{p−1}) |∇ψ^{s}| dx
≥ −δ − δ |A^{s}|^{p} − δ ∫_{Q} |∇^{2}ϕ|^{p} dx − c_{δ,p,L} ∫_{Q} |∇ψ^{s}|^{p} dx.

Hence, using Lemma 2.2 we obtain

I_{1} ≥ −δ − δ |A^{s}|^{p} − δ ∫_{Q} |∇^{2}ϕ|^{p} dx − c_{δ,p,L}γ_{n,p} ∫_{Q} |∇ψ^{a}|^{p} dx.

If p ≥ 2 then we have

I_{1} ≥ −δ − δ |A^{s}|^{p} − δ ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx

− c_{δ,p,L}γ_{n,p} ∫_{Q} (|A^{a}|^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx.

If 1 < p ≤ 2 then by Lemma 2.7, and by Lemma 2.6 with μ = 0, we obtain for every 0 < ε < 1

I_{1} ≥ −δ (1 + εμ^{p}) − δ (1 + ε) |A^{s}|^{p}
− 8δε^{\frac{p−2}{p}} ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx

− c_{δ,p,L}γ_{n,p}ε^{\frac{p−2}{2}} ∫_{Q} (|A^{a}|^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx − c_{δ,p,L}γ_{n,p}ε^{\frac{p}{2}} |A^{a}|^{p} .

In both cases there exists a sequence of positive numbers {λ_{k}}, depending only on p, μ, ν, M, such that for every k

I_{1} ≥ −\frac{1}{k} − \frac{1}{k} |A^{s}|^{p} − ν ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx  (3.19)

− λ_{k} ∫_{Q} (|A^{a}|^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx − λ_{k} |A^{a}|^{p} .

The strict 2-quasiconvexity of f (condition (a)) yields

I_{2} ≥ ν ∫_{Q} (μ^{2} + |A^{s}|^{2} + |∇^{2}ϕ|^{2})^{\frac{p−2}{2}} |∇^{2}ϕ|^{2} dx.  (3.20)

Since, by periodicity,

∫_{Q} A^{a} · ∇ψ^{a} dx = 0,

by Lemma 2.4 with μ = 0, we have

I_{3} = β ∫_{Q} [|A^{a} + ∇ψ^{a}|^{p} − |A^{a}|^{p} − p |A^{a}|^{p−2} A^{a} · ∇ψ^{a}] dx  (3.21)

≥ βθ_{p} ∫_{Q} (|A^{a}|^{2} + |∇ψ^{a}|^{2})^{\frac{p−2}{2}} |∇ψ^{a}|^{2} dx.

Choosing β_{k} > 0 so that β_{k}θ_{p} ≥ λ_{k}, from (3.19), (3.20), and (3.21), we obtain

I_{1} + I_{2} + I_{3} ≥ −\frac{1}{k} − \frac{1}{k} |A^{s}|^{p} − λ_{k} |A^{a}|^{p} ,

which together with (3.18) gives (3.16).

Remark 3.1 It is clear from the proof of Theorem 1.2 that if f is nonnegative the we may take F_{k} to be also nonnegative.

4 Lower semicontinuity

The proof of Theorem 1.3 relies on the so-called Decomposition Lemma (see [13]).

Lemma 4.1 (Decomposition Lemma) Let Ω be a bounded open set in ℝ^{n}, let p > 1, and let {u_{k}} be a sequence weakly converging to a function u in W^{1,p} (Ω; ℝ^{n}). Then there exists a subsequence (not relabeled) and a sequence {v_{k}} weakly converging to u in W^{1,p} (Ω; ℝ^{n}) such that v_{k} = u in a neighborhood of ∂Ω, {|∇v_{k}|^{p}} is equi-integrable, and L^{n} ({u_{k} ≠ v_{k}}) → 0.

The following simple lemma may be found in [11], however we include its proof for the convenience of the reader.

Lemma 4.2 Let D ⊂ ℝ^{m} be an open set and let

f : D × 𝕄^{d×n} → ℝ

be a lower semicontinuous function such that for every v ∈ D the function f(v, ·) is continuous. Then for every v̄ ∈ D, ε > 0, and L > 0 there exists δ = δ(v̄, ε, L) ∈ (0, 1) such that

f(v̄, A) ≤ f(v, A) + ε

for every (v, A) ∈ D × 𝕄^{d×n}, with |v − v̄| ≤ δ and |A| ≤ L.

Proof. Assume, for contradiction, that there exist v̄ ∈ D, L > 0, ε̄ > 0, and a sequence

\{(v_{k}, A_{k})\} ⊂ D × \overline{B_{d×n}(0, L)},

such that

ε̄+ f(v_{k}, A_{k}) < f(v̄, A_{k})  (4.1)

and (v_{k}, A_{k}) → (v̄, Ā) as k → ∞, for some Ā ∈ \overline{B_{d×n}(0, L)}. Since the function f(v̄, ·) is continuous and f is lower semicontinuous, for any ε < \frac{1}{2} ε̄ there exists δ > 0 such that

|f(v̄, A) − f(v̄, Ā)| ≤ ε, f(v̄, Ā) ≤ f(v, A) + ε

for all (v, A) ∈ D × \overline{B_{d×n}(0, L)} with

|v − v̄| + |A − Ā| ≤ δ.

Thus for all n sufficiently large, also by (4.1), we have

ε̄+ f(v_{k}, A_{k}) < f(v̄, A_{k}) ≤ f(v̄, Ā) + ε ≤ f(v_{k}, A_{k}) + 2ε,

which is a contradiction.

Although the following lemma is well known to experts, its proof is not easy to find in the literature and so we present it below for the reader’s convenience.

Lemma 4.3 Let D ⊂ ℝ^{m} be an open set and let

f : D × 𝕄^{d×n} → ℝ

be a lower semicontinuous function which satisfies the following conditions:

(a) for every v ∈ D the function f(v, ·) is continuous in 𝕄^{d×n};

(b) there exist a locally bounded function a : D → [0, +∞), a lower semicontinuous function b : D → (0, +∞), a locally bounded function c : D → [0, +∞), and a constant p > 1 such that

b (v) |A|^{p} − c(v) ≤ f(v, A) ≤ a(v)(1 + |A|^{p})

for every (v, A) ∈ D × 𝕄^{d×n}.

For every v ∈ D, let Qf(v, ·) be the 1-quasiconvexification of the function f(v, ·). Then Qf is lower semicontinuous on D × 𝕄^{d×n}.

Proof. By conditions (a) and (b) for every (v, A) ∈ D×𝕄^{d×n}, we have (see [8])

Qf(v, A) = inf {∫_{Q} f(v, A + ∇φ(x)) dx : φ ∈ C_{c}^{1}(Q; ℝ^{d})} .  (4.2)

By replacing f(v, A) with f(v, A) + c̃(v), where c̃ is any continuous function with c̃ ≥ c, we may assume without loss of generality that f ≥ 0.

We begin by showing that for every fixed A ∈ 𝕄^{d×n} the function Qf(·, A) is lower semicontinuous. Without loss of generality we may assume that A = 0. Let {v_{k}} ⊂ D be a sequence converging to some v̄ ∈ D. If

lim inf_{k→∞} Qf(v_{k}, 0) = ∞,

then there is nothing to prove. Thus, without loss of generality, we may assume that

lim inf_{k→∞} Qf(v_{k}, 0) = lim_{k→∞} Qf(v_{k}, 0) < ∞,

and

C := sup_{k} Qf(v_{k}, 0) < ∞.  (4.3)

By (4.2) for every fixed 0 < ε < 1 and for every k ∈ ℕ there exists φ_{k} ∈ C_{c}^{1}(Q; ℝ^{d}) such that

Qf(v_{k}, 0) + ε ≥ ∫_{Q} f(v_{k}, ∇φ_{k}(x)) dx.  (4.4)

Hence, by condition (b) and (4.3), we have

C + 1 ≥ b (v_{k}) ∫_{Q} |∇φ_{k}(x)|^{p} dx − c(v_{k})
≥ b_{0} ∫_{Q} |∇φ_{k}(x)|^{p} dx − c_{0},

where b_{0} := inf_{k} b(v_{k}) > 0 and c_{0} := sup_{k} c(v_{k}) < ∞, since b is lower semicontinuous and c is locally bounded. By the Decomposition Lemma (see Lemma 4.1), there exists a subsequence of {φ_{k}} (not relabeled) and a sequence {w_{k}} weakly converging to some function w in W_{0}^{1,p} (Q; ℝ^{d}) such that {|∇w_{k}|^{p}} is equi-integrable, and L^{n} ({φ_{k} ≠ w_{k}}) → 0. Hence we may find L ≥ 1 and k̄ ∈ ℕ such that

L^{n} ({φ_{k} ≠ w_{k}}) + ∫_{{|∇w_{k}|≥L}} |∇w_{k}|^{p} dx ≤ ε,  (4.5)

for every k ≥ k̄. By Lemma 4.2 there exists δ = δ(v̄, L, ε) ∈ (0, 1) such that

f(v̄, B) ≤ f(v, B) + ε

for all (v, B) ∈ D × 𝕄^{d×n}, with |v − v̄| ≤ δ and all |B| ≤ L. Therefore

Qf(v_{k}, 0) + ε ≥ ∫_{{φ_{k}=w_{k}}∩{|∇w_{k}|<L}} f(v_{k}, ∇w_{k}(x)) dx
≥ ∫_{{φ_{k}=w_{k}}∩{|∇w_{k}|<L}} f(v̄, ∇w_{k}(x)) dx − ε  (4.6)
≥ ∫_{{φ_{k}=w_{k}}} f(v̄, ∇w_{k}(x)) dx − ε(2 + a(v̄)),

where in the last inequality we have used (4.5) and condition (b). Since {|∇w_{k}|^{p}} is equi-integrable and L^{n} ({φ_{k} ≠ w_{k}}) → 0, by condition (b) we have

lim inf_{k→∞} ∫_{{φ_{k}=w_{k}}} f(v̄, ∇w_{k}(x)) dx = lim inf_{k→∞} ∫_{Q} f(v̄, ∇w_{k}(x)) dx,

hence letting k → ∞ in (4.6) yields

lim inf_{k→∞} Qf(v_{k}, 0) + ε ≥ lim inf_{k→∞} ∫_{Q} f(v̄, ∇w_{k}(x)) dx − ε(2 + a(v̄))
≥ Qf(v̄, 0) − ε(2 + a(v̄)).

It is now sufficient to let ε → 0^{+} to conclude that v ↦ Qf(v, A) is lower semicontinuous.

Finally, we observe that the continuity of A ↦ Qf(v, A) is an immediate consequence of the quasiconvexity of Qf(v, ·). Since the coefficient a(v) in (b) is locally bounded, the functions A ↦ Qf(v, A) have the same local modulus of continuity when v varies in a compact subset of D. Since we have seen that v ↦ Qf(v, A) is lower semicontinuous on D, this implies that Qf is lower semicontinuous on D × 𝕄^{d×n}.

Proof of Theorem 1.3. Fix u ∈ W^{1,1}(Ω) and let {u_{j}} ⊂ SBH(Ω) be any sequence converging to u in W^{1,1}(Ω) and satisfying (1.7). Without loss of generality we may assume that

lim inf_{j→∞} ∫_{Ω} f(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx = lim_{j→∞} ∫_{Ω} f(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx.

Fix ε ∈ (0, 1) and let

f_{ε}(x, u, ξ, A) := f(x, u, ξ, A) + ε |A|^{p} .

The function f_{ε} satisfies all the conditions of Theorem 1.2, hence there exists an increasing sequence {F_{k,ε}} of functions F_{k,ε} : Ω×ℝ×ℝ^{n}×𝕄^{n×n} → [0, +∞) such that for L^{n} a.e. x ∈ Ω and for every (u, ξ) ∈ ℝ × ℝ^{n} the function F_{k,ε} (x, u, ξ, ·) is 1-quasiconvex,

lim_{k→∞} F_{k,ε} (x, u, ξ, A) = f_{ε} (x, u, ξ, A)  ∀A ∈ 𝕄_{sym}^{n×n},  (4.7)
0 ≤ F_{k,ε} (x, u, ξ, A) ≤ c_{k}(x, u, ξ) (1 + |A|^{p})  ∀A ∈ 𝕄^{n×n}.  (4.8)

Let C := sup_{j} ∥∇^{2}u_{j}∥_{L^{p}}^{p}. For every fixed ε ∈ (0, 1) and k ∈ ℕ, we have

lim_{j→∞} ∫_{Ω} f(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx ≥ lim inf_{j→∞} ∫_{Ω} f_{ε}(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx − εC (4.9)

≥ lim inf_{j→∞} ∫_{Ω} F_{k,ε}(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx − εC,

where we used the fact that f_{ε} ≥ F_{k,ε}. We note that, in view of the construction in the proof of Theorem 1.2, the function F_{k,ε} is defined as 1-quasiconvexification of f_{ε}(x, u, ξ, A^{s}) + β |A^{a}|^{p}, and so, by the previous lemma, we have that F_{k,ε} is a normal integrand. Define

G_{k,ε} : Ω × (ℝ × ℝ^{n}) × (ℝ^{n} × 𝕄^{n×n}) → [0, ∞)

as

G_{k,ε} (x, (w_{1}, w_{2}) , (ξ_{1}, ξ_{2})) := F_{k,ε} (x, w_{1}, w_{2}, ξ_{2}) .

Indeed, with w_{j} := (u_{j}, ∇u_{j}) then

∫_{Ω} F_{k,ε}(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx

reduces to

∫_{Ω} G_{k,ε}(x, w_{j}, ∇w_{j}) dx.

It is clear that {w_{j}} ⊂ SBV (Ω, ℝ × ℝ^{n}) and that

sup_{j} (∥∇w_{j}∥_{L^{p}} + ∫_{S(w_{j})} θ(|[w_{j}]|) dH^{n−1}) < ∞.

Hence we may apply Theorem 1.2 in [18] to obtain

lim_{j→∞} ∫_{Ω} f(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx ≥ lim inf_{j→∞} ∫_{Ω} G_{k,ε}(x, w_{j}, ∇w_{j}) dx − εC
≥ ∫_{Ω} G_{k,ε}(x, w, ∇w) dx − εC = ∫_{Ω} F_{k,ε}(x, u, ∇u, ∇^{2}u) dx − εC,

where we have used (4.9), and w := (u, ∇u).

By Lebesgue’s Monotone Convergence Theorem, letting k → ∞ in the previous inequality and using (4.7) gives

lim_{j→∞} ∫_{Ω} f(x, u_{j}, ∇u_{j}, ∇^{2}u_{j}) dx ≥ ∫_{Ω} f_{ε}(x, u, ∇u, ∇^{2}u) dx − εC.

It now suffices to let ε → 0^{+}.

Acknowledgments

The research of I. Fonseca was partially supported by the National Science Foundation under Grant No. DMS–0103799. The work of Gianni Dal Maso is part of the Project “Calculus of Variations” 2002, supported by the Italian Ministry of Education, University, and Research.

This work was undertaken when G. Dal Maso visited the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791), Carnegie Mellon University, Pittsburgh, PA, USA. The authors thank the Center for Nonlinear Analysis for its support during the preparation of this paper.

I. Fonseca was engaged in several stimulating discussions on the subject of this paper with B. Dacorogna during her visit to EPFL (Lausanne, Switzerland) in Fall 2001. G. Leoni wishes to thank J. Kristensen for pointing out reference [22].

The authors wish to thank D. Kinderlehrer for many interesting conversations on the subject of this paper.

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S.I.S.S.A, Trieste, Italy dalmaso@sissa.it

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA fonseca@andrew.cmu.edu giovanni@andrew.cmu.edu morini@andrew.cmu.edu