An Existence Result for a Nonconvex Variational Problem via Regularity

Irene Fonseca, Nicola Fusco, Paolo Marcellini

Abstract

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x-dependence, explicitly appearing in the integrands, not only adds significant technical difficulties, but also yields some unexpected phenomena.

Keywords: nonconvex variational problems, uniform convexity, regularity, implicit differential equations.

1991 Mathematics Subject Classification: Primary 49J45, 49K20. Secondary 35F30, 35R70.

1 Introduction

In this paper we establish existence and regularity of minimizers of energy integrals of the type

∫_{Ω} f (x, Du(x)) dx ,  (1.1)

subject to Dirichlet boundary conditions. The main feature of our problem is the fact that the integrand f = f(x, ξ) is not convex with respect to the gradient variable ξ.

In recent years the study of non convex variational problems has undergone remarkable developments, motivated in part by advances in the study of material stability and instability. Contemporary issues such as phase transitions in certain alloys (see [3], [4]), nucleation [20], the onset of microstructure, and optimal design problems for thin films [18], require a good understanding of existence of (classical or generalized) equilibrium solutions for nonconvex energies. In addition, qualitative information on quasistatic solutions (e.g. regularity, hysteresis, oscillatory behavior) are needed in order to develop the evolutionary framework, and, in particular, to search for the dynamical evolution of phase boundaries. These issues have challenged traditional theories.

Within this setting, a relevant example has been considered by Ball-James [3], [4], who studied the two “potential wells” problem:

minimize ∫_{Ω} f (Du(x)) dx ,

where u : Ω ⊂ ℝ^{n} → ℝ^{n} is a vector-valued function, f : ℝ^{n×n} → [0, +∞) is identically zero on two distinct potential wells SO(n)ξ, SO(n)η and f > 0 elsewhere. Here ξ, η ∈ ℝ^{n×n} and SO(n) stands for the special orthogonal group. The existence of minimizers for the two potential wells problem has been obtained in two dimensions (i.e., n = 2) by Dacorogna–Marcellini [10] and by Müller–Sverak [26] (for the case n = 3 see also Dolzmann–Kirchheim–Müller– Sverak [15]). Nothing is known in higher dimension or for general integrands f as in (1.1).

In this paper we restrict ourselves to the scalar-valued case, as a starting point to approach the vectorial setting. Also, the scalar-valued case is still far from being completely understood, unless the integrand f depends only on the gradient variable ξ and some special assumptions are made on the boundary data (see the references quoted below). Here we consider general boundary data u_{0} ∈ W^{1,p} (Ω), p > 1, and we allow the nonconvex integrands f to explicitly depend on x as in (1.1). In the proofs of the attainment results presented below the x-dependence not only introduces substantial technical difficulties, but also seems to yield some unexpected phenomena.

The proof of the existence results for nonconvex variational problems considered in this paper hinges on the local Lipschitz continuity of minimizers of the relaxed problem associated to the bipolar f^{∗∗} of f. These regularity results are presented in Section 2, and they apply to minimizers of some integrals of the Calculus of Variations with integrands f^{∗∗}(x, ξ) convex with respect to ξ ∈ ℝ^{n}, but not everywhere uniformly convex; hence, we believe that the regularity results presented in Section 2 should be of interest by themselves.

In Section 3 we consider the variational problem

inf {∫_{Ω} f (x, Du(x)) dx : u ∈ u_{0} + W_{0}^{1,p} (Ω)} ,  (1.2)

where u_{0} ∈ W^{1,p} (Ω) is a given boundary datum and f = f(x, ξ) is a continuous function satisfying some growth conditions similar to the ones considered in the previous Section 2, so as to ensure Lipschitz continuity of minimizers of the relaxed problem. Here the most relevant fact is that f may be nonconvex with respect to the variable ξ ∈ ℝ^{n}.

It is known that the variational problem (1.2) may lack a minimizer (see Marcellini [22]; see also [6], [21], [12]). In the examples of nonexistence the following condition, expressed in terms of the bipolar f^{∗∗} of f, is violated: for every x ∈ Ω the function f^{∗∗}(x, ·) is affine on the set

A(x) = {ξ ∈ ℝ^{n} : f(x, ξ) > f^{∗∗}(x, ξ)} ,

i.e., there exist a continuous function q and a vector field m of class C^{1}, defined in the open set Ω_{A} := {x ∈ Ω : A(x) ≠ ∅}, such that

f^{∗∗}(x, ξ) = q(x) + ⟨m(x), ξ⟩ , ∀ x ∈ Ω_{A}, ξ ∈ A(x) .  (1.3)

We also assume that the boundary (more precisely, the part of the boundary in Ω) of the set

{x ∈ Ω_{A} : div m(x) = 0}  (1.4)

has zero (n-dimensional) measure.

In this paper we prove that (1.3), (1.4) (see also the more general assumptions made in Section 3.2) are sufficient conditions for existence of minimizers to the variational problem (1.2). We emphasize that we do not require any other condition on the vector field m other than (1.4); in particular, we do not assume that the vector field m has null divergence. We notice that, while condition (1.3) is necessary for guaranteeing the existence of minimizers (see [22], [6], [21]), we do not know whether condition (1.4) may be removed.

Existence theorems without convexity assumptions have been widely investigated in the one-dimensional case n = 1 (see [12] for an extensive list of references). Theorem 3.1 in Section 3 is specific to the case n ≥ 2, and it is an extension of some analogous results, obtained under more restrictive assumptions, by Marcellini [22], Mascolo-Schianchi [23], Cellina [7] and Friesecke [21]. In particular, Theorem 3.1 is an extension of related results recently proved by Sychev [28] and Zagatti [29] for integrands independent of x and under a strong assumption on the growth of f which ensures the almost everywhere differentiability of minimizers, i.e., p > n, by Celada-Perrotta [5] for p > 1, and by Dacorogna-Marcellini in [13], [12].

Finally we recall that Marcellini [22] pointed out the necessity of the condition of affinity (1.3) of the function f^{∗∗} on the set where f ≠ f^{∗∗} to guarantee existence of minimizers. Cellina [6], [7] and Friesecke [21] proved the necessity and sufficiency of the condition of affinity for linear boundary data u_{0}. The explicit dependence of the integrand on the variable x was first considered by Mascolo-Schianchi in [24], assuming that the divergence of the vector field m in (1.3) is identically equal to zero in Ω, in addition to other strong assumptions on the boundary data u_{0}. Also, in [27] Raymond studied a case where the divergence of the vector field m in (1.3) is always different from zero in Ω, and some type of explicit dependence on u is allowed.

2 Local Lipschitz Continuity

2.1 Preliminary results

Let f : ℝ^{n} → [0, +∞) be a continuous function such that

0 ≤ f(ξ) ≤ L(1 + |ξ|^{p}) ,  (2.1)

where L > 0, p > 1. We say that f is uniformly convex at infinity if there exist R, ν > 0 such that, if the segment with endpoints ξ_{1}, ξ_{2} (that we will denote by [ξ_{1}, ξ_{2}]) is contained in the complement of the closed ball \overline{B}_{R}, then

f (\frac{ξ_{1} + ξ_{2}}{2} ) ≤ \frac{1}{2}f(ξ_{1}) + \frac{1}{2}f(ξ_{2}) − ν(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2} . (2.2)

Note that (2.2) is equivalent to

f (\frac{ξ_{1} + ξ_{2}}{2} ) ≤ \frac{1}{2}f(ξ_{1}) + \frac{1}{2}f(ξ_{2}) − ν′(|ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2} ,

for some ν′ > 0, since |ξ_{1}| , |ξ_{2}| > R > 0.

If the above inequality (2.2) is satisfied for any ξ_{1}, ξ_{2} ∈ ℝ^{n}, then we say that f is uniformly convex in ℝ^{n}. A form of uniform convexity at infinity was also considered by Mascolo and Schianchi in [25].

The following result is proved in [1].

Lemma 2.1 If γ > −1/2 then there exist positive constants c_{1} = c_{1}(γ), c_{2} = c_{2}(γ) such that, for all ξ, η ∈ ℝ^{n},

c_{1}(1 + |ξ|^{2} + |η|^{2})^{γ} ≤ ∫_{0}^{1} t(1 + |tξ + (1 − t)η|^{2})^{γ} dt ≤ c_{2}(1 + |ξ|^{2} + |η|^{2})^{γ} .

The following result provides two conditions which are equivalent to uniform convexity in ℝ^{n}.

Proposition 2.2 Let f : ℝ^{n} → [0, +∞) be a continuous function satisfying (2.1). The following conditions are equivalent:

(i) f is uniformly convex in ℝ^{n};

(ii) f(ξ) = c_{1}ν(1 + |ξ|^{2})^{p/2} + g(ξ), for some c_{1} = c_{1}(p) > 0, where g(ξ) is a convex function such that 0 ≤ g(ξ) ≤ L(1 + |ξ|^{p}) for all ξ;

(iii) ∫_{Q}[f(ξ + Dφ(x)) − f(ξ)] dx ≥ c_{2}ν ∫_{Q}(1 + |ξ|^{2} + |Dφ|^{2})^{\frac{p−2}{2}} |Dφ(x)|^{2} dx, for all ξ ∈ ℝ^{n}, φ ∈ C_{0}^{1}(Q), where Q = (0, 1)^{n} and c_{2} = c_{2}(p) is a suitable constant.

Remark 2.3 The condition (iii) in Proposition 2.2 is related to the notion of uniform quasiconvexity, introduced by Evans [16] and later studied by Evans-Gariepy [17].

Proof of Proposition 2.2. (i) ⟹ (ii). We define g(ξ) := f(ξ) − c_{1}ν(1 + |ξ|^{2})^{p/2}, where c_{1} will be chosen later, and we show that g is convex. Given ξ_{1}, ξ_{2} ∈ ℝ^{n} we set ξ := (ξ_{1} + ξ_{2})/2. From (i) we easily get that

\frac{1}{2}g(ξ_{1}) + \frac{1}{2}g(ξ_{2}) ≥ g(ξ) + ν(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2}

+\frac{c_{1}ν}{2} [2(1 + |ξ|^{2})^{p/2} − (1 + |ξ_{1}|^{2})^{p/2} − (1 + |ξ_{2}|^{2})^{p/2}] .

Thus the assertion follows immediately from the fact that there exists a constant c = c(p) such that

(1 + |ξ_{1}|^{2})^{p/2} + (1 + |ξ_{2}|^{2})^{p/2} − 2(1 + |ξ|^{2})^{p/2} ≤ c(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2} ,

and by setting c_{1} := c^{−1}. To establish this inequality we write, for i = 1, 2,

h (ξ_{i}) = h (ξ) + ⟨Dh (ξ) , ξ_{i} − ξ⟩
+ ∫_{0}^{1}(1 − t) ⟨D^{2}h (ξ + t (ξ_{i} − ξ)) (ξ_{i} − ξ) , ξ_{i} − ξ⟩ dt

where h (ξ) := (1 + |ξ|^{2})^{p/2}, yielding

(1 + |ξ_{i}|^{2})^{p/2} − (1 + |ξ|^{2})^{p/2} ≤ p(1 + |ξ|^{2})^{\frac{p−2}{2}} ⟨ξ_{i} − ξ, ξ⟩
+c(p)|ξ_{i} − ξ|^{2} ∫_{0}^{1}(1 − t)(1 + |ξ + t(ξ_{i} − ξ)|^{2})^{\frac{p−2}{2}} dt .

It suffices now to sum the above inequalities for i = 1, 2 and apply Lemma 2.1.

(ii) ⟹ (iii). From Lemma 2.1 we easily get that (iii) holds for the function ξ ↦ (1+|ξ|^{2})^{p/2}. Hence the general case follows from Jensen’s inequality applied to g.

(iii) ⟹ (i). Given ξ_{1}, ξ_{2} ∈ ℝ^{n} we set ξ := (ξ_{1}+ξ_{2})/2, λ := ξ_{2}−ξ_{1} and denote by χ the characteristic function of the interval (0, 1/2) extended periodically to ℝ with periodicity 1. Define

u_{h}(x) := |λ| ∫_{0}^{⟨x,λ/|λ|⟩} (χ(ht) − 1/2) dt .

Then u_{h} → 0 uniformly in Q, as h → +∞, and since

Du_{h}(x) = (χ (⟨hx, λ/|λ|⟩) − 1/2) λ, ∥Du_{h}∥_{∞} = |λ|/2 ,

we have that

\frac{1}{2}f(ξ_{1}) + \frac{1}{2}f(ξ_{2}) = lim_{h→+∞} ∫_{Q} f(ξ + Du_{h}(x)) dx .  (2.3)

Fix t ∈ (0, 1/2), set Q_{t} = (t, 1 − t)^{n}, and let φ ∈ C_{0}^{1}(Q) be a function such that 0 ≤ φ(x) ≤ 1 for all x ∈ Q, φ(x) = 1 on Q_{t}. By (2.3), (2.1), since (iii) ensures the convexity of f, we have

\frac{1}{2}f(ξ_{1}) + \frac{1}{2}f(ξ_{2}) = lim_{h→+∞} ∫_{Q} f(ξ + Du_{h}(x)) dx

≥ lim_{h→+∞} [∫_{Q} f(ξ + D(φu_{h})) dx − ∫_{Q\Q_{t}} c(1 + |ξ_{1}|^{p} + |ξ_{2}|^{p} + |u_{h}Dφ|^{p})]

≥ f(ξ) + c_{2} ν lim inf_{h→+∞} ∫_{Q}(1 + |ξ|^{2} + |D(φu_{h})|^{2})^{\frac{p−2}{2}} |D(φu_{h})|^{2} dx − ct

≥ f(ξ) + c′_{2} ν |Q_{t}| (1 + |ξ|^{2} + |λ|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2} − ct .

The result now follows by letting t → 0.

Lemma 2.4 Let f : ℝ^{n} → [0, +∞) be a C^{2} function. Then f satisfies (2.2) if and only if there exists a constant c_{0} such that for all ξ ∈ ℝ^{n} \ \overline{B}_{R}

⟨D^{2}f(ξ)λ, λ⟩ ≥ c_{0} ν (1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2}  ∀λ ∈ ℝ^{n} .  (2.4)

Proof. If f satisfies (2.2) and |ξ| > R then, for any λ we have for t > 0 small enough,

\frac{1}{2}[f(ξ + tλ) − f(ξ)] + \frac{1}{2}[f(ξ − tλ) − f(ξ)] ≥ 4t^{2}ν (1 + |ξ + tλ|^{2} + |ξ − tλ|^{2})^{\frac{p−2}{2}} |λ|^{2} .

Hence, by Taylor’s formula, we get

\frac{t^{2}}{2} ⟨D^{2}f(ξ)λ, λ⟩ + o(t^{2}) ≥ 4t^{2}ν (1 + |ξ + tλ|^{2} + |ξ − tλ|^{2})^{\frac{p−2}{2}} |λ|^{2} ;

thus, dividing by t^{2} and letting t → 0, we obtain

⟨D^{2}f(ξ)λ, λ⟩ ≥ 8ν (1 + 2|ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} .

This inequality implies (2.4) with c_{0} = 4\sqrt{2}.

Conversely, if (2.4) holds and [ξ_{1}, ξ_{2}] ⊂ ℝ^{n} \ \overline{B}_{R}, setting ξ := (ξ_{1} + ξ_{2})/2 by Lemma 2.1 we have for i = 1, 2

f(ξ_{i}) = f(ξ) + ⟨Df(ξ), ξ_{i} − ξ⟩ + ∫_{0}^{1}(1 − t)⟨D^{2}f(ξ + t(ξ_{i} − ξ))(ξ_{i} − ξ), ξ_{i} − ξ⟩ dt
≥ f(ξ) + ⟨Df(ξ), ξ_{i} − ξ⟩ + c_{0}ν∫_{0}^{1}(1 − t)(1 + |ξ + t(ξ_{i} − ξ)|^{2})^{\frac{p−2}{2}} |ξ_{i} − ξ|^{2} dt
≥ f(ξ) + ⟨Df(ξ), ξ_{i} − ξ⟩ + c(p)c_{0}ν (1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2} .

The result then follows by adding up this inequality with i = 1, 2.

Lemma 2.5 Let f : ℝ^{n} → [0, +∞) be a continuous function satisfying (2.1) and (2.2). Then there exist R_{0}, ν_{0}, C_{0} > 0, depending only on R, ν and L, such that for all ξ ∈ ℝ^{n} \ \overline{B}_{R_{0}} there exists q_{ξ} ∈ ℝ^{n} such that |q_{ξ}| ≤ C_{0}(1 + |ξ|^{p−1}) and

f(η) ≥ f(ξ) + ⟨q_{ξ}, η − ξ⟩ + ν_{0}(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} ∀η ∈ ℝ^{n} . (2.5)

Moreover, if |ξ| > R_{0} then f^{∗∗}(ξ) = f(ξ).

Proof. For 0 < ε < 1 set f_{ε} := ρ_{ε} ∗ f, where ρ_{ε}(η) := ε^{−n}ρ(η/ε) and ρ(η) = ρ̂(|η|) is a positive radially symmetric mollifier with support equal to \overline{B}, with B := B_{1}, ρ(η) > 0 if |η| < 1 and ∫_{B} ρ(η)dη = 1. From (2.2) it follows easily that if [ξ_{1}, ξ_{2}] ⊂ ℝ^{n} \ \overline{B}_{R+1}, ξ := (ξ_{1} + ξ_{2})/2, then

\frac{1}{2}[f_{ε}(ξ_{1}) +f_{ε}(ξ_{2})] ≥ f_{ε}(ξ)+ν|ξ_{1}−ξ_{2}|^{2}∫_{B}ρ(η)(1+|ξ_{1}+εη|^{2}+|ξ_{2}+εη|^{2})^{\frac{p−2}{2}} dη .

The integral above can be estimated from below by

∫_{B∩\{⟨ξ,η⟩≥0\}} ρ(η)(1+|ξ_{1}|^{2}+|ξ_{2}|^{2}+2ε^{2}|η|^{2}+4ε⟨ξ, η⟩)^{\frac{p−2}{2}} dη ≥ \frac{1}{2}(1+|ξ_{1}|^{2}+|ξ_{2}|^{2})^{\frac{p−2}{2}}

if p ≥ 2, and by

∫_{B} ρ(η)(1 + 2|ξ_{1}|^{2} + 2|ξ_{2}|^{2} + 4ε^{2}|η|^{2})^{\frac{p−2}{2}} dη ≥ 5^{(p−2)/2}(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}}

when 1 < p < 2. In both cases

\frac{1}{2}[f_{ε}(ξ_{1}) + f_{ε}(ξ_{2})] ≥ f_{ε}(ξ) + cν(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2} ;

hence, by Lemma 2.4, if |ξ| ≥ R + 1, then

⟨D^{2}f_{ε}(ξ)λ, λ⟩ ≥ cν (1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2},  ∀ λ ∈ ℝ^{n}.  (2.6)

Moreover, it can be easily checked that 0 ≤ f_{ε}(ξ) ≤ C(L)(1 + |ξ|^{p}) for all ξ, and a simple argument based on the convexity of f_{ε} in ℝ^{n} \ \overline{B}_{R+1} shows that there exists a constant C_{1}(L, R) such that

|Df_{ε}(ξ)| ≤ C_{1}(1 + |ξ|^{p−1}),  ∀ ξ ∈ ℝ^{n} \ \overline{B}_{R+2} .  (2.7)

We claim that there exists R_{0} >> 1 such that, if |ξ| > R_{0} , then

f_{ε}(η) ≥ f_{ε}(ξ) + ⟨Df_{ε}(ξ), η − ξ⟩ + c(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}  (2.8)

for all η ∈ ℝ^{n}. Assume that (2.8) holds. Notice that, if |ξ| > R_{0} > R + 2, then by (2.7) there exists a sequence (ε_{h}) converging to 0 such that Df_{ε_{h}}(ξ) → q_{ξ} for some q_{ξ} ∈ ℝ^{n} such that |q_{ξ}| ≤ C_{1}(1 + |ξ|^{p−1}). Hence (2.5) follows from (2.8), letting ε go to 0^{+}. The equality f^{∗∗}(ξ) = f(ξ) for |ξ| > R_{0} then follows at once from (2.5).

The remaining of the proof concerns the assertion of the claim (2.8). Fix ξ such that |ξ| > R_{0}, with R_{0} > 2(R + 3) to be chosen later, and denote by C_{ξ} the open cone with vertex at ξ, tangent to the ball \overline{B}_{R+3}.

Case 1: If η ∈ ℝ^{n} \ C_{ξ} or η ∈ C_{ξ} \ B_{R+3} and ⟨η, ξ⟩ ≥ 0, then f_{ε} is convex along the line t → ξ + t(η − ξ) provided R_{0} is sufficiently large, and (2.8) follows from (2.6).

Case 2: If η ∈ B_{R+3} we consider \overline{ξ} := ξ(R + 3)/|ξ| and a constant M := C(L)(1 + (R + 3)^{p}) such that

0 ≤ f_{ε}(η) ≤ M

for all η ∈ B_{R+3} and all ε ∈ (0, 1); by Case 1 we may apply (2.8) to \overline{ξ} (notice that \overline{ξ} ∈ ∂B_{R+3}, ⟨\overline{ξ}, ξ⟩ > 0), thus getting

f_{ε}(η) = f_{ε}(\overline{ξ}) + f_{ε}(η) − f_{ε}(\overline{ξ})

≥ f_{ε}(ξ) + ⟨Df_{ε}(ξ), \overline{ξ} − ξ⟩ − M + c (1 + (R + 3)^{2} + |ξ|^{2})^{\frac{p−2}{2}} (|ξ| − R − 3)^{2}

≥ f_{ε}(ξ) + ⟨Df_{ε}(ξ), η − ξ⟩ − |Df_{ε}(ξ)||η − \overline{ξ}| − M + c̃(1 + |η|^{2} + |ξ|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}. The estimate (2.8) now follows for η and ξ from the previous inequalities, together with the estimate

|Df_{ε}(ξ)||η − \overline{ξ}| + M ≤ \frac{1}{2}c̃(1 + |η|^{2} + |ξ|^{2})^{\frac{p−2}{2}} |ξ − η|^{2},

and the latter holds by virtue of (2.7), and provided ξ > R_{0} and R_{0} > 2 (R + 3) sufficiently large.

Case 3: Finally, let us assume that η ∈ C_{ξ} \ B_{R+3}, ⟨η, ξ⟩ < 0. In this case we have |ξ − η|^{2} > |ξ|^{2} + |η|^{2} and, denoting by η̃ the projection of η on the boundary of the cone, and by α_{ξ} the half angle at the vertex of C_{ξ},

|η − η̃| ≤ |ξ − η| sin α_{ξ} = \frac{R + 3}{|ξ|}|ξ − η| ≤ \frac{1}{2}|ξ − η| .  (2.9)

Notice that, if R_{0} is sufficiently large, then [η, η̃] ⊂ ℝ^{n} \ \overline{B}_{R+2} ; therefore we may use (2.7) to estimate f_{ε}(η) − f_{ε}(η̃). This, together with (2.8) applied to η̃ ∈ ∂C_{ξ}, yields

f_{ε}(η) = f_{ε}(η̃) + f_{ε}(η) − f_{ε}(η̃)

≥ f_{ε}(ξ)+⟨Df_{ε}(ξ), η̃−ξ⟩−C_{1}(1+|η|^{p−1}+|η̃|^{p−1})|η−η̃|+c(1+|η̃|^{2}+|ξ|^{2})^{\frac{p−2}{2}} |ξ−η̃|^{2}. Since, by (2.9), |ξ − η| ≤ 2|ξ − η̃| ≤ 3|ξ − η|, for any p > 1 we have easily

c(1 + |η̃|^{2} + |ξ|^{2})^{\frac{p−2}{2}} |ξ − η̃|^{2} ≥ \overline{c}(p)(1 + |η|^{2} + |ξ|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} ,

and, using (2.7) once more, we obtain

f_{ε}(η) ≥ f_{ε}(ξ) + ⟨Df_{ε}(ξ), η − ξ⟩  (2.10)
−C_{1}|η − η̃| [(1 + |ξ|^{p−1}) + (1 + |η|^{p−1} + |η̃|^{p−1})]
+\overline{c}(p)(1 + |η|^{2} + |ξ|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} .

By virtue of (2.9), and recalling that |ξ|^{2} + |η|^{2} < |ξ − η|^{2}, we have

C_{1}|η − η̃| [(1 + |ξ|^{p−1}) + (1 + |η|^{p−1} + |η̃|^{p−1})]
≤ c\frac{R + 3}{|ξ|}|ξ − η|(1 + |η|^{p−1} + |ξ|^{p−1}) ≤ c′ \frac{R + 3}{|ξ|}|ξ − η|^{2}(1 + |η|^{2} + |ξ|^{2})^{\frac{p−2}{2}}
≤ (\overline{c}(p)/2)(1 + |η|^{2} + |ξ|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}

if |ξ| > R_{0}, with R_{0} large enough. This, together with (2.10), concludes the proof of (2.8).

Remark 2.6 Let f satisfy (2.1) and (2.2), and fix a point ξ_{0} such that R_{0} < |ξ_{0}| < 2R_{0}. Applying (2.5) with ξ = ξ_{0}, and recalling that |q_{ξ}| ≤ C_{0}(1 + |ξ|^{p−1}), for all η such that |η| > 2R_{0} it holds

f(η) ≥ c̃_{1}(R_{0}, ν_{0}, C_{0}) |η|^{p} − c̃_{2}(R_{0}, ν_{0}, C_{0}) .

Hence, f(ξ) ≥ c_{1}|ξ|^{p} − c_{2} for all ξ ∈ ℝ^{n}, with c_{1}, c_{2} depending only on R, ν, L.

2.2 A regularity result

In this section we assume that f : Ω × ℝ^{n} → [0, +∞) is a continuous function satisfying the growth condition

0 ≤ f(x, ξ) ≤ L(1 + |ξ|^{p}),  ∀ (x, ξ) ∈ Ω × ℝ^{n}  (2.11)

and for some L > 0. Let us denote by f^{∗∗} := f^{∗∗}(x, ξ) the bipolar of f, that is the convex envelope of f(x, ·). We assume that f^{∗∗} is continuous and that f is uniformly convex at infinity with respect to ξ (see (2.2)), i.e., there exist R, ν > 0 such that if the segment ξ_{1}, ξ_{2} is contained in the complement of the closed ball \overline{B}_{R}, then for all x ∈ Ω

f (x, \frac{ξ_{1} + ξ_{2}}{2} ) ≤ \frac{1}{2}f(x, ξ_{1})+ \frac{1}{2}f(x, ξ_{2})−ν(1+|ξ_{1}|^{2}+|ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1}−ξ_{2}|^{2}. (2.12)

We assume further that, if |ξ| > R then the vector field x ∈ Ω ↦ f_{ξ}(x, ξ) is weakly differentiable and

|D_{x}f_{ξ}(x, ξ)| ≤ L(1 + |ξ|^{p−1}), ∀ (x, ξ) ∈ Ω × (ℝ^{n} \ \overline{B}_{R}) .  (2.13)

If u ∈ W_{loc}^{1,p}(Ω) and A ⊂ Ω is open, then we set

F (u, A) := ∫_{A} f(x, Du(x)) dx .

The main result of this section is Theorem 2.7 below. We recall that u is said to be a local minimizer of F in Ω if

F (u, B_{R}(x_{0})) ≤ F (v, B_{R}(x_{0}))

whenever B_{R}(x_{0}) ⊂⊂ Ω and v ∈ u + W_{0}^{1,p}(B_{R}(x_{0})).

Theorem 2.7 Let f : Ω × ℝ^{n} → [0, +∞) be a continuous function satisfying (2.11), (2.12) and (2.13). If u ∈ W_{loc}^{1,p}(Ω) is a local minimizer of the functional F, then u is locally Lipschitz continuous in Ω. Moreover, there exists a constant C_{0}, depending on L, p, ν, R, such that, if B_{r}(x_{0}) ⊂ Ω then

sup_{B_{r/2}(x_{0})} |Du|^{p} ≤ C_{0} (1 + − ∫_{B_{r}(x_{0})} |Du|^{p} dx) .  (2.14)

We first show in Lemma 2.8 that, provided we know already that u is locally Lipschitz, (2.14) holds with a constant C_{0} depending only on L, p, ν, R. Once the a priori estimate (2.14) is established, the regularity result is obtained using an approximation argument.

Lemma 2.8 Let f satisfy the assumptions of Theorem 2.7. Assume, in addition, that f is C^{2} and that, for all x ∈ Ω, ξ, λ ∈ ℝ^{n},

D_{ξ_{i}ξ_{j}}f(x, ξ)λ_{i}λ_{j} ≥ ε_{0}(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2},  (2.15)

and that u ∈ W_{loc}^{1,p}(Ω) is a locally Lipschitz local minimizer of F in Ω. Then (2.14) holds with a constant depending only on L, p, ν, R and, in particular, independent on ε_{0}.

Proof. Step 1: From Lemma 2.4 we have that for every x ∈ Ω, ξ, λ ∈ ℝ^{n}, with |ξ| > R,

D_{ξ_{i}ξ_{j}}f(x, ξ)λ_{i}λ_{j} ≥ cν(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} .  (2.16)

Since ũ(y) := u(x_{0} + ry)/r is a local minimizer in (Ω − x_{0})/r of the functional F̃, where F̃(v) := ∫_{(Ω−x_{0})/r} f(x_{0} + ry, Dv(y))dy still satisfies the assumptions of Theorem 2.7, it is clear that in order to prove (2.14) we may always assume, with no loss of generality, that B := B_{1} ⊂ Ω. Since u satisfies the Euler equation for F,

∫_{B} D_{ξ_{i}}f(x, Du)D_{i}ϕ dx = 0 , ∀ ϕ ∈ C_{0}^{1}(B),

using (2.15) and the fact that D_{ξ_{i}ξ_{j}}f(x, Du(x)) are locally bounded in B (which follows from the C^{2} regularity of f, together with the fact that Du is locally bounded) we have that u ∈ W_{loc}^{2,2}(B), by a standard different quotient argument.

We fix s ∈ {1, …, n}, η ∈ C_{0}^{1}(B), 0 ≤ η ≤ 1, ψ ∈ C^{2}(B), and in the above Euler equation we take ϕ = η^{2}D_{s}ψ to obtain

∫_{B} D_{ξ_{i}}f(x, Du)D_{s}(D_{i}ψ)η^{2} dx = −2 ∫_{B} ηD_{ξ_{i}}f(x, Du)D_{s}ψD_{i}η dx .

Integrating by parts the first integral, we have

∫_{B} D_{ξ_{i}ξ_{j}}f(x, Du)D_{j}(D_{s}u)D_{i}ψη^{2} dx = 2 ∫_{B} ηD_{ξ_{i}}f(x, Du)D_{s}ψD_{i}η dx

−2 ∫_{B} D_{x_{s}ξ_{i}}f(x, Du)D_{i}ψη^{2} dx − 2 ∫_{B} ηD_{ξ_{i}}f(x, Du)D_{i}ψD_{s}η dx  (2.17)

for all functions ψ ∈ W^{1,2}(B). Set

V_{+}(x) := 1+R^{2}+∑_{h=1}^{n}[(D_{h}u(x)−R)^{+}]^{2}, V_{−}(x) := 1+R^{2}+∑_{h=1}^{n}[(D_{h}u(x)+R)^{−}]^{2},

and notice that there exist constants c_{1}, c_{2}, depending only on n, such that

c_{1}(V_{+}(x) + V_{−}(x)) ≤ 1 + R^{2} + |Du(x)|^{2} ≤ c_{2}(V_{+}(x) + V_{−}(x)) .  (2.18)

Let ψ := V_{+}^{β}(D_{s}u − R)^{+}, where β ≥ 0. By (2.11) and the convexity of f (x, ·), |D_{ξ}f(x, ξ)| ≤ c(1 + |ξ|^{2})^{\frac{p−1}{2}} , and (2.17) yields

∫_{B} D_{ξ_{i}ξ_{j}}f(x, Du)D_{j}(D_{s}u − R)^{+}D_{i}(D_{s}u − R)^{+}V_{+}^{β}η^{2} dx

+β∫_{B}D_{ξ_{i}ξ_{j}}f(x, Du)D_{j}(D_{s}u−R)^{+}(D_{s}u−R)^{+}D_{i} (∑_{h=1}^{n}[(D_{h}u − R)^{+}]^{2}) V_{+}^{β−1}η^{2} dx

≤ c∫_{B}η(η + |Dη|)(1 + |Du|^{p−1})|D(D_{s}u − R)^{+}|V_{+}^{β} dx

+cβ∫_{B}η(η + |Dη|)(1 + |Du|^{p−1}) |D (∑_{h=1}^{n}[(D_{h}u − R)^{+}]^{2})| (D_{s}u − R)^{+}V_{+}^{β−1} dx .

Since all the integrals are evaluated in the set where |Du| > R, summing up on s, using (2.16), the fact that (D_{s}u − R)^{+} ≤ V_{+}^{1/2} and Young’s inequality, it follows easily that

∫_{B}(1 + |Du|^{2})^{\frac{p−2}{2}} |D (∑_{h=1}^{n}[(D_{i}u − R)^{+}]^{2})|^{2} V_{+}^{β−1}η^{2} dx

≤ \frac{c}{ν^{2}} ∫_{B}(1 + |Du|^{p}) V_{+}^{β} (η^{2} + |Dη|^{2}) dx ,

where the constant c depends only on n, p, L. Since the integral on the left hand side is evaluated in the set where |Du| > R, in turn this last inequality is equivalent to the following one:

∫_{B}(1 + R^{2} + |Du|^{2})^{\frac{p−2}{2}} |D (∑_{h=1}^{n}[(D_{i}u − R)^{+}]^{2})|^{2} V_{+}^{β−1}η^{2} dx

≤ \frac{c}{ν^{2}} ∫_{B}(1 + R^{2} + |Du|^{2})^{p/2} V_{+}^{β} (η^{2} + |Dη|^{2}) dx .

Inserting ψ = V_{−}^{β}(D_{s}u + R)^{−} in (2.17), and using a similar argument, we get also

∫_{B}(1 + R^{2} + |Du|^{2})^{\frac{p−2}{2}} |D (∑_{h=1}^{n}[(D_{i}u + R)^{−}]^{2})|^{2} V_{−}^{β−1}η^{2} dx

≤ \frac{c}{ν^{2}} ∫_{B}(1 + R^{2} + |Du|^{2})^{p/2} V_{−}^{β}(η^{2} + |Dη|^{2}) dx .

Therefore, adding the last two inequalities and using (2.18), we arrive to

∫_{B} V ^{\frac{p−2}{2}} [|DV_{+}|^{2}V_{+}^{β−1} + |DV_{−}|^{2}V_{−}^{β−1}] η^{2} dx
≤ \frac{c}{ν^{2}} ∫_{B} V^{p/2+β}(η^{2} + |Dη|^{2}) dx ,

where V := max {V_{+}; V_{−}}.

Step 2: From the inequality above we deduce that

∫_{B} |DV^{p/4+β/2}|^{2}η^{2} dx ≤ c(p + 2β)^{2} ∫_{B} V^{p/2+β}(η^{2} + |Dη|^{2}) dx .

In turn, this implies that

∫_{B} |D(V^{p/4+β/2}η)|^{2} dx ≤ c(p + 2β)^{2} ∫_{B} V^{p/2+β}(η^{2} + |Dη|^{2}) dx ,

where the constant c depends only on L, p, n, R, ν. Setting γ := p/4 + β/2 ≥ p/4, using the Sobolev–Poincaré inequality, and the arbitrariness of β ≥ 0, we get that, for any γ ≥ p/4,

∥V^{γ}η∥_{L^{2χ}(B)} ≤ cγ ∥V^{γ} (η + |Dη|)∥_{L^{2}(B)} ,

where χ := n/(n − 2) if n ≥ 3, or any number > 1 if n = 2. Considering the sequence of radii r_{i} := 1/2 + 1/2^{i} for i = 1, …, we apply the inequality above to γ = γ_{i} := (p/4)χ^{i−1}, and choose η ∈ C_{0}^{1}(B) such that η = 1 on B_{r_{i+1}}, 0 ≤ η ≤ 1, |Dη| ≤ c2^{i}. We obtain

∥V ∥_{L^{2γ_{i+1}} (B_{r_{i+1}} )} ≤ (c2^{i}γ_{i})^{1/γ_{i}} ∥V ∥_{L^{2γ_{i}} (B_{r_{i}})} .

Iterating the above formula yields, for every i,

∥V ∥_{L^{2γ_{i+1}} (B_{1/2})} ≤ C ∥V ∥_{L^{p/2}(B)} ,

where C = ∏_{i=1}^{∞}(c2^{i}γ_{i})^{1/γ_{i}} < +∞. Therefore, letting i go to +∞ and using (2.18), we obtain (2.14).

Remark 2.9 It follows immediately from the proof that the estimate (2.14) may be generalized to read

sup_{B_{ρ}(x_{0})} |Du|^{p} ≤ C(ρ) (1 + − ∫_{B_{r}(x_{0})} |Du|^{p} dx) ,

for all 0 < ρ < r, where C(ρ) depends only on L, p, ν, R and ρ.

We are now in position to prove Theorem 2.7, by using the following approximation lemma.

Lemma 2.10 Let g : ℝ^{n} → [0, +∞) be a C^{2} convex function such that for all ξ ∈ ℝ^{n}

0 ≤ g(ξ) ≤ L(1 + |ξ|^{p}) ,

where p > 1, L > 0, and assume that there exist R, ν > 0, such that if |ξ| > R, λ ∈ ℝ^{n},

D_{ij}g(ξ)λ_{i}λ_{j} ≥ ν(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} .

Then there exists a constant c = c(n, p) and a sequence g_{h} of C^{2}(ℝ^{n}) convex functions such that

(a) 0 ≤ g_{h}(ξ) ≤ cL(1 + |ξ|^{p}) ∀ ξ ∈ ℝ^{n};

(b) for any h there exists ε_{h} > 0 such that, for all ξ, λ ∈ ℝ^{n},

ε_{h}(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} ≤ D_{ij}g_{h}(ξ)λ_{i}λ_{j} ≤ ε_{h}^{−1}(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2};

(c) D_{ij}g_{h}(ξ)λ_{i}λ_{j} ≥ cν(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2}, ∀ λ ∈ ℝ^{n}, |ξ| > R + 1;

(d) g_{h} → g uniformly on compact subset of ℝ^{n}.

Proof. The proof of this lemma can be obtained arguing as in the Step 2 and Step 3 of the proof of Lemma 3.4 of [19], with the obvious simple modifications needed in the present case. Therefore we omit the details.

Proof of Theorem 2.7. Notice that if u is a local minimizer F, then u is also a local minimizer of the relaxed functional v ↦ ∫_{Ω} f^{∗∗}(x, Dv)dx, where, for all x ∈ Ω , f^{∗∗}(x, ξ) is the bipolar of ξ ↦ f(x, ξ). Indeed, if v ∈ u + W_{0}^{1,p}(B_{r}(x_{0})), then

∫_{B_{r}(x_{0})} f^{∗∗}(x, Dv) dx

= inf {lim inf_{h→+∞} ∫_{B_{r}(x_{0})} f(x, Dv_{h}) dx : v_{h} ⇀ v in W_{0}^{1,p}(B_{r}(x_{0}))} .

Also, by virtue of Lemma 2.5, the function f^{∗∗} satisfies the assumptions of Theorem 2.7. Therefore, with no loss of generality, we may assume that f is convex in ξ.

Step 1: Let us assume that

f(x, ξ) = ∑_{i=1}^{N} a_{i}(x)g_{i}(ξ) ,

where, for i = 1, …, N, the function g_{i} ∈ C^{2}(ℝ^{n}) satisfies the assumptions of Lemma 2.10 for some L, R, ν > 0, and ⟨D^{2}g_{i}(ξ)λ, λ⟩ ≥ ε_{0}(1 + |ξ|^{2})^{(p−2)/2}|λ|^{2} for all ξ, λ ∈ ℝ^{n} and for some ε_{0} > 0. Moreover, let us assume that, for all i, the function a_{i} is a nonnegative C^{2} function such that ∥Da_{i}∥_{∞} ≤ M and that γ^{−1} < ∑_{i=1}^{N} a_{i}(x) < γ for all x ∈ Ω and for some positive constant γ.

For every i let us denote by g_{i,h} a sequence of C^{2}(ℝ^{n}) functions such that g_{i,h} → g_{i} uniformly on the compact sets of ℝ^{n}, satisfying the conditions (a), (b) and (c) of Lemma 2.10, and let us set for all (x, ξ) ∈ Ω × ℝ^{n}

f_{h}(x, ξ) := ∑_{i=1}^{N} a_{i}(x) g_{i,h}(ξ) .

From Remark 2.6 it follows easily that there exist constants c_{1}, c_{2}, depending only on L, R, ν, γ, such that, for all (x, ξ) and for any h,

f(x, ξ), f_{h}(x, ξ) ≥ c_{1}|ξ|^{p} − c_{2} .  (2.19)

Given B_{r}(x_{0}) ⊂⊂ Ω, we denote by u_{h} the solution of the problem

min {∫_{B_{r}(x_{0})} f_{h}(x, Dv) dx : v ∈ u + W_{0}^{1,p}(B_{r}(x_{0}))} .  (2.20)

Since the functions g_{i,h} satisfy condition (b) of Lemma 2.10, standard elliptic regularity theory implies that u_{h} ∈ C^{1,α}(Ω) ∩ W_{loc}^{2,2}(Ω) for any h. From the assumptions on f, from the approximation provided by Lemma 2.10, and from (2.19), it follows that the sequence u_{h} is bounded in W^{1,p}(B_{r}(x_{0})). Moreover, by Lemma 2.8 (see Remark 2.9), for all ρ < r we obtain

sup_{B_{ρ}(x_{0})} |Du_{h}|^{p} ≤ C (1 + − ∫_{B_{r}(x_{0})} |Du_{h}|^{p} dx) ,  (2.21)

where the constant C depends ultimately only on L, p, R, ν, γ, M and ρ, but not on h. Hence we may assume, up to a subsequence, that u_{h} ⇀ u_{∞} weakly* in W^{1,∞}(B_{ρ}(x_{0})) for any ρ < r. Since f_{h} → f uniformly on compact sets of ℝ^{n} and the integrand f is convex, for any ρ < r we have

∫_{B_{ρ}(x_{0})}f(x, Du_{∞}) dx ≤ lim inf_{h→+∞} ∫_{B_{ρ}(x_{0})}f(x, Du_{h}) dx = lim inf_{h→+∞} ∫_{B_{ρ}(x_{0})}f_{h}(x, Du_{h}) dx

≤ lim inf_{h→+∞} ∫_{B_{r}(x_{0})}f_{h}(x, Du) dx = ∫_{B_{r}(x_{0})} f(x, Du) dx ,

where we used the fact that u_{h} is a solution for (2.20). Letting ρ → r and recalling that u is a local minimizer of the functional F, we deduce that u_{∞} is also a minimizer of F in B_{r}(x_{0}). Since the functional F is strictly convex, we have that u = u_{∞}. Passing to the limit as h → +∞ in (2.21), we conclude that u is also locally Lipschitz. Indeed, using the minimality of u_{h} and (2.21), we get

sup_{B_{r/2}(x_{0})}|Du|^{p} ≤ lim inf_{h→+∞}( sup_{B_{r/2}(x_{0})}|Du_{h}|^{p})

≤ c lim inf_{h→+∞}(1 + − ∫_{B_{r}(x_{0})}|Du_{h}|^{p} dx) ≤ c lim inf_{h→+∞} (1 + − ∫_{B_{r}(x_{0})}f_{h}(x, Du_{h}) dx)

≤ c lim inf_{h→+∞} (1 + − ∫_{B_{r}(x_{0})}f_{h}(x, Du) dx) ≤ c (1 + − ∫_{B_{r}(x_{0})}|Du|^{p} dx) .

Step 2: Let us now assume that f ∈ C^{2}(Ω×ℝ^{n}) and that there exists ε_{0} > 0 such that D_{ξ_{i}ξ_{j}}f(x, ξ)λ_{i}λ_{j} ≥ ε_{0}(1 + |ξ|^{2})^{(p−2)/2}|λ|^{2} for all (x, ξ) ∈ Ω × ℝ^{n} and for any λ ∈ ℝ^{n}. Fix an open set A ⊂⊂ Ω and let us prove that (2.14) holds for any ball B_{r}(x_{0}) ⊂ A (with a constant C_{0} not depending on A).

To this aim let ψ ∈ C_{0}^{∞}(ℝ^{n}) be a cut-off function such that 0 ≤ ψ(x) ≤ 1 for all x, suppψ ⊂ (−3, 3)^{n} and such that ψ(x) ≡ 1 if x ∈ [−1, 1]^{n}. For any h ∈ ℕ we denote by Q_{i,h}(x_{i,h}) the standard covering of ℝ^{n} with closed cubes, centered at x_{i,h}, with sides parallel to the coordinates axes, side length equal to 2/h and having pairwise disjoint interiors. Then, for any i, h, we set ψ_{i,h}(x) := ψ (h (x − x_{i,h})) and

σ_{h}(x) := ∑_{i=1}^{∞} ψ_{i,h}(x), φ_{i,h}(x) := \frac{ψ_{i,h}(x)}{σ_{h}(x)} .

Finally, for all h such that 12\sqrt{n}/h <dist(A; ∂Ω), and for every x ∈ A, ξ ∈ ℝ^{n}, we set

f_{h}(x, ξ) := ∑_{i=1}^{∞} φ_{i,h}(x)f(x_{i,h}, ξ) .

Notice that the above sum is actually finite (indeed it consists of at most 3^{n} terms), and that each function f_{h} is of the type considered in Step 1. Moreover, we claim that the functions f_{h} satisfy the assumptions of Lemma 2.8 with suitable constants L, ε_{0}, R, ν not depending on h. The verification of our claim in the case of assumptions (2.11), (2.12) (or equivalently (2.16)) and (2.15) follows easily by the corresponding assumptions on f. We limit ourselves to show that for any h

|D_{xξ}f_{h}(x, ξ)| ≤ cL(1 + |ξ|^{p−1}), ∀ (x, ξ) ∈ A × (ℝ^{n} \ \overline{B}_{R}) ,  (2.22)

where L is the constant appearing in (2.13) (relative to f) and c is a constant depending only on n, ψ, but not on h. Let us fix x_{0} ∈ A and ξ ∈ ℝ^{n} \ \overline{B}_{R}. By construction there exist at most 3^{n} cubes, Q_{j_{1},h}(x_{j_{1},h}), …, Q_{j_{3^{n}} ,h}(x_{j_{3^{n}} ,h}), such that for any x in a suitable neighborhood U of x_{0}

f_{h}(x, ξ) = ∑_{l=1}^{3^{n}} φ_{j_{l},h}(x) f(x_{j_{l},h}, ξ),  ∑_{l=1}^{3^{n}} φ_{j_{l},h}(x) = 1 .

Therefore for all x ∈ U we have that

D_{xξ}f_{h}(x, ξ) = ∑_{l=1}^{3^{n}} D_{x}φ_{j_{l},h}(x) D_{ξ}f(x_{j_{l},h}, ξ)  (2.23)

= ∑_{l=2}^{3^{n}} D_{x}φ_{j_{l},h}(x) [D_{ξ}f(x_{j_{l},h}, ξ) − D_{ξ}f(x_{j_{1},h}, ξ)] .

In view of assumption (2.13), we have that for all l,

|D_{ξ}f(x_{j_{l},h}, ξ) − D_{ξ}f(x_{j_{1},h}, ξ)| ≤ \frac{c(n)L}{h}(1 + |ξ|^{p−1}) .

On the other hand, for any j, there exists a set of indices I_{j} containing j, with #(I_{j}) = 3^{n}, such that, for all x ∈ ℝ^{n},

D_{x}φ_{j,h}(x) = \frac{D_{x}ψ_{j,h}(x)}{σ_{h}(x)} − \frac{ψ_{j,h}(x)}{σ_{h}^{2}(x)} ∑_{k∈I_{j}} D_{x}ψ_{k,h}(x) .

Therefore, since by construction σ_{h}(x) ≥ 1 for all x, we have that, for all x ∈ ℝ^{n} and any j,

|D_{x}φ_{j,h}(x)| ≤ (3^{n} + 1)h(∥D_{x}ψ∥_{∞}) .

In view of the above estimates and from (2.23), we may conclude that for all (x, ξ) ∈ A × (ℝ^{n} \ \overline{B}_{R}) and for any h

|D_{xξ}f_{h}(x, ξ)| ≤ c(n)L ∥D_{x}ψ∥_{∞} (1 + |ξ|^{p−1}) ,

and thus (2.22) follows. Finally, notice that f_{h}(x, ξ) → f(x, ξ) uniformly in A × K for every K compact subset of ℝ^{n}. Hence the rest of the proof goes as in Step 1, since also in this case the function ξ → f(x, ξ) is strictly convex for all x ∈ A.

Step 3: Let f satisfy the assumptions of Theorem 2.7. Fix an open set A ⊂⊂ Ω, an infinitesimal sequence ε_{h} of positive numbers and a positive symmetric mollifier ρ. For h large enough we set for all (x, ξ) ∈ A × ℝ^{n}

f_{h}(x, ξ) := ∫_{B×B} ρ(y)ρ(η)f(x + ε_{h}y, ξ + ε_{h}η) dydη + ε_{h}(1 + |ξ|^{p}) ,

where B is the unit open ball in ℝ^{n}. Notice that each function f_{h} is of the type considered in Step 2 and that f_{h}(x, ξ) → f(x, ξ) uniformly on any set of the type A × K, with K ⊂ ℝ^{n} compact. Moreover, the functions f_{h} satisfy the assumptions of Theorem 2.7 with the corresponding constants L, R, ν bounded from above and away from zero.

As in Step 1, given a ball B_{r}(x_{0}) ⊂ A we denote by u_{h} the solution of the problem

min {∫_{B_{r}(x_{0})} f_{h}(x, Dv) dx : v ∈ u + W^{1,p}(B_{r}(x_{0}))} .

From the assumptions on f and the construction of the functions f_{h} it follows easily that the sequence u_{h} is bounded in W^{1,p}(B_{r}(x_{0})). Moreover, by Step 2, for all ρ < r we have

sup_{B_{ρ}(x_{0})} |Du_{h}|^{p} ≤ C (1 + − ∫_{B_{r}(x_{0})} |Du_{h}|^{p} dx) ,

where the constant C depends only on L, p, R, ν and ρ, but not on h. Hence we may assume that, up to a subsequence, u_{h} ⇀ u_{∞} weakly* in W^{1,∞}(B_{ρ}(x_{0})) for any ρ < r. As in Step 1 we have again that also u_{∞} is a minimizer of F in B_{r}(x_{0}). However, in the present case the functional F is not necessarily strictly convex, hence we may not conclude as before that u_{∞} = u in B_{r}(x_{0}).

Set E_{0} := {x ∈ B_{r}(x_{0}) : |Du_{∞}(x)+Du(x)| > 2R_{0}}, where R_{0} is the constant provided by Lemma 2.5. If E_{0} has positive measure, then from the convexity of f(x, ·) and Remark 2.6 we have, setting ũ := (u + u_{∞})/2,

∫_{B_{ρ}(x_{0})\E_{0}} f(x, Dũ) dx ≤ \frac{1}{2} ∫_{B_{ρ}(x_{0})\E_{0}} f(x, Du) dx+ \frac{1}{2} ∫_{B_{ρ}(x_{0})\E_{0}} f(x, Du_{∞}) dx .

(2.24)

Also, applying twice (2.5) in Lemma 2.5, first with ξ := Dũ and η := Du, then with ξ := Dũ and η := Du_{∞}, and adding up the corresponding two inequalities, yields

∫_{E_{0}∩B_{ρ}(x_{0})} f(x, Dũ) dx < \frac{1}{2} ∫_{E_{0}∩B_{ρ}(x_{0})} f(x, Du) dx+\frac{1}{2} ∫_{E_{0}∩B_{ρ}(x_{0})} f(x, Du_{∞}) dx .

(2.25)

Adding (2.24) and (2.25) we get a contradiction with the minimality of u and u_{∞}. Therefore E_{0} has zero measure. Applying Step 2 to the functions u_{h}, (2.5) to the functions f_{h}, and using the minimality of u_{h}, we deduce that

sup_{B_{r/2}(x_{0})}|Du_{∞}|^{p} ≤ lim inf_{h→+∞}( sup_{B_{r/2}(x_{0})}|Du_{h}|^{p})

≤ C̃ lim inf_{h→+∞}(1 + − ∫_{B_{r}(x_{0})}|Du_{h}|^{p} dx) ≤ c lim inf_{h→+∞}(1 + − ∫_{B_{r}(x_{0})}f_{h}(x, Du_{h}) dx)

≤ c lim inf_{h→+∞}(1 + − ∫_{B_{r}(x_{0})}f_{h}(x, Du_{∞}) dx) ≤ c (1 + − ∫_{B_{r}(x_{0})}|Du_{∞}|^{p} dx) .

Then the result follows, since |Du(x) + Du_{∞}(x)| ≤ 2R_{0} for a.e. x.

3 Attainment of minima for nonconvex problems

Here we give an existence result for the variational problem

inf {∫_{Ω} f (x, Du(x)) dx : u ∈ u_{0} + W_{0}^{1,p} (Ω)} ,  (3.1)

where Ω is a bounded open set of ℝ^{n} and u_{0} ∈ W^{1,p}(Ω), p > 1. Throughout this section we assume that f : Ω × ℝ^{n} → ℝ is a continuous function satisfying the growth condition

c_{1}|ξ|^{p} − c_{2} ≤ f(x, ξ) ≤ L(1 + |ξ|^{p}),  ∀ (x, ξ) ∈ Ω × ℝ^{n},  (3.2)

for some constants c_{1}, c_{2}, L > 0, and is uniformly convex at infinity with respect to ξ, i.e., there exist R, ν > 0 such that, if the segment [ξ_{1}, ξ_{2}] is contained in ℝ^{n} \ \overline{B}_{R}, then

f (x, \frac{ξ_{1} + ξ_{2}}{2} ) ≤ \frac{1}{2}f(x, ξ_{1}) + \frac{1}{2}f(x, ξ_{2}) − ν(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2} (3.3)

for every x ∈ Ω (see (2.2)). Notice that, if 0 ≤ f(x, ξ) ≤ L(1 + |ξ|^{p}), then condition (3.3) implies the coercivity inequality in the left hand side of (3.2) (see Remark 2.6). In addition, we assume that there exists the distributional derivative D_{xξ}f(x, ξ) and

|D_{xξ}f(x, ξ)| ≤ L(1 + |ξ|^{p−1}), ∀ x ∈ Ω, |ξ| > R,  (3.4)

provided L in (3.2) is chosen to be sufficiently large (see (2.13)). Let us denote by f^{∗∗} := f^{∗∗}(x, ξ) the bipolar of f, that is the convex envelope of f(x, ·) (i.e., the largest convex function in ξ which is less than or equal to f(x, ·) on ℝ^{n}). We assume that f^{∗∗} is continuous; hence, for any x ∈ Ω, the set

A(x) := {ξ ∈ ℝ^{n} : f(x, ξ) > f^{∗∗}(x, ξ)}  (3.5)

is open.

We shall prove the existence of a minimizer for the problem (3.1) under the main assumption that f^{∗∗}(x, ·) is affine on each connected component of A(x). However, in order to present the argument of the proof in a simplified setting, we shall treat first the case where f^{∗∗}(x, ·) is affine (with the same slope) on the whole set A(x). We will refer to this situation as the model case. The proof of this case contains all the ideas and technical tools which are needed to treat the general situation in which f^{∗∗}(x, ·) is affine (with possibly different slopes) on each connected component of A(x).

3.1 The model case

As before f : Ω × ℝ^{n} → ℝ is a continuous function satisfying (3.2)−(3.4). We assume that f^{∗∗} : Ω × ℝ^{n} → ℝ is continuous and we denote by A(x) the set defined in (3.5) and by Ω_{A} := {x ∈ Ω : A(x) ≠ ∅}. Since f and f^{∗∗} are continuous functions, Ω_{A} is open. Here we consider the case where f^{∗∗}(x, ·) is affine in A(x); more precisely, we assume that there exist a function q ∈ C^{0} (Ω_{A}) and a vector field m ∈ C^{1} (Ω_{A}; ℝ^{n}) such that

f^{∗∗}(x, ξ) = q(x) + ⟨m(x), ξ⟩ ,  ∀ x ∈ Ω_{A}, ξ ∈ A(x) .  (3.6)

We also assume that the boundary of the set where the divergence of m is equal to zero is negligible, i.e.,

meas (Ω ∩ ∂ {x ∈ Ω_{A} : div m(x) = 0}) = 0 .  (3.7)

Finally, for every x ∈ Ω_{A}, we set

E(x) := {ξ ∈ ℝ^{n} : f^{∗∗}(x, ξ) = q(x) + ⟨m(x), ξ⟩} ;

note that, by the growth conditions in (3.2) and by the assumption that f is uniformly convex at infinity with respect to ξ, then the set E(x) is bounded uniformly for x ∈ Ω_{A} (see also Lemma 2.5). We assume there exists an increasing function ω : [0, +∞) → [0, +∞), with ω(t) = 0 if and only if t = 0, such that, if x ∈ Ω_{A}, ξ ∈ ∂E(x), η ∈ ℝ^{n} \ E(x), then

f^{∗∗} (x, \frac{ξ + η}{2} ) ≤ \frac{1}{2}f^{∗∗}(x, ξ) + \frac{1}{2}f^{∗∗}(x, η) − ω (|ξ − η|) .  (3.8)

The main result of this section is the following existence theorem.

Theorem 3.1 Let f, f^{∗∗} : Ω × ℝ^{n} → ℝ be continuous functions (f not necessarily convex with respect to ξ ∈ ℝ^{n}). Under the above assumptions on f and f^{∗∗} ((3.2)−(3.4) and (3.6)−(3.8)), for any given boundary datum u_{0} ∈ W^{1,p} (Ω) the variational problem (3.1) attains its minimum. Moreover, any minimizers is of class W_{loc}^{1,∞}(Ω).

The proof of Theorem 3.1 is obtained using the same method as in Dacorogna– Marcellini [13], [12], who considered integrands independent of x. Our proof however follows from some new lemmas. The first one concerns the relaxed variational problem

inf {∫_{Ω} f^{∗∗} (x, Du(x)) dx : u ∈ u_{0} + W_{0}^{1,p} (Ω)} .  (3.9)

Lemma 3.2 The minimum of the relaxed variational problem (3.9) is attained. Moreover, there exist a minimizer v ∈ W_{loc}^{1,∞}(Ω) ∩ (u_{0} + W_{0}^{1,p}(Ω)) of (3.9) and an open set Ω′ ⊂ Ω (possibly empty) such that

\{ \begin{matrix}Dv(x) ∈ A(x) a.e. x ∈ Ω′ \\ Dv(x) ∉ A(x) a.e. x ∈ Ω \ Ω′ \end{matrix}  (3.10)

and div m = 0 in Ω′.

Remark 3.3 Formally, if Dv(x) ∈ A(x) then by (3.6)

f^{∗∗}(x, Dv(x)) = q(x) + ⟨m(x), Dv(x)⟩ .

Therefore, the Euler’s equation for v gives

∑_{s=1}^{n} \frac{∂}{∂x_{s}} f_{ξ_{s}}^{∗∗}(x, Dv(x)) = div m(x) = 0, a.e. x such that Dv(x) ∈ A(x) .

Thus (3.10) would follow if we could define Ω′ := {Dv(x) ∈ A(x)}. A striking feature of Lemma 3.2 is that the set Ω′ may be chosen to be open, and so Lemma 3.2 may be considered to be a strong form of Euler’s first variation for the minimizers. The property (3.10) is a regularity result, and in fact it follows from the regularity results obtained in Section 2.

The proof of Lemma 3.2 follows the argument by Dacorogna–Marcellini [12] in Theorem 10.9. Previous arguments related to Lemma 3.2 are due to De Blasi–Pianigiani [14], Sychev [28], and Zagatti [29].

Proof of Lemma 3.2. As before we denote by Ω_{A} the open subset of Ω consisting of those points x such that A(x) ≠ ∅. We split Ω_{A} into three sets (possibly empty),

Ω_{A}^{+} := {x ∈ Ω_{A} : div m(x) > 0} , Ω_{A}^{−} := {x ∈ Ω_{A} : div m(x) < 0} , (3.11)
Ω_{A}^{0} := {x ∈ Ω_{A} : div m(x) = 0} .  (3.12)

Since div m is a continuous function, Ω_{A}^{+} ∪ Ω_{A}^{−} ∪ int Ω_{A}^{0} is an open set and, by (3.7),

meas (Ω_{A}\ (Ω_{A}^{+} ∪ Ω_{A}^{−} ∪ int Ω_{A}^{0})) ≤ meas (Ω ∩ ∂Ω_{A}^{0}) = 0 .

By (3.2), (3.3), (3.4), Lemma 2.5, and Theorem 2.7 the relaxed variational problem (3.9) has a minimizer u^{∗∗} in the Sobolev class W_{loc}^{1,∞}(Ω) ∩ (u_{0} + W^{1,p}(Ω)). Thus, by Rademacher theorem (see, for example, Theorem 2.2.1 of [30], or Theorem 2.14 of [2]) u^{∗∗} is classically differentiable for almost every x ∈ Ω.

Let x_{0} be a point of Ω_{A} \ ∂Ω_{A}^{0} where u^{∗∗} is differentiable. Then

u^{∗∗}(x) = u^{∗∗}(x_{0}) + ⟨Du^{∗∗}(x_{0}), x − x_{0}⟩ + o (|x − x_{0}|) , x ∈ Ω.  (3.13)

Also, assume that

Du^{∗∗}(x_{0}) ∈ A(x_{0}) = {ξ ∈ ℝ^{n} : f(x_{0}, ξ) > f^{∗∗}(x_{0}, ξ)} .

Since Ω_{A} and A(x) are open sets, there exists γ ∈ (0, 1) (depending on u^{∗∗} and x_{0}) such that

x ∈ Ω_{A}, ξ ∈ A(x),  (3.14)

for all (x, ξ) ∈ Ω × ℝ^{n} such that

|x − x_{0}| ≤ γ, |ξ − Du^{∗∗}(x_{0})| ≤ 2γ .  (3.15)

Recall that x_{0} ∈ Ω_{A} \ ∂Ω_{A}^{0} ; thus we can also assume that γ is sufficiently small so that

\{ \begin{matrix}x_{0} ∈ Ω_{A}^{±} , |x − x_{0}| ≤ γ ⟹ x ∈ Ω_{A}^{±} , \\ x_{0} ∈ int Ω_{A}^{0}, |x − x_{0}| ≤ γ ⟹ x ∈ int Ω_{A}^{0}. \end{matrix}  (3.16)

Choose δ ∈ (0, γ] (depending on x_{0}) sufficiently small such that

\frac{|o (|x − x_{0}|)|}{|x − x_{0}|} ≤ γ,  ∀ x ∈ \overline{B}_{δ}(x_{0}), x ≠ x_{0} ,  (3.17)

and

δ ≤ \frac{γ}{2 |Du^{∗∗}(x_{0})| + 4γ} ,  \overline{B}_{δ}(x_{0}) ⊂ Ω .  (3.18)

By (3.16) and by the definition of Ω_{A}^{+}, Ω_{A}^{−}, Ω_{A}^{0} in (3.11), (3.12), we have

\{ \begin{matrix}x_{0} ∈ Ω_{A}^{±}  ⟹ div m(x) ≷ 0 ∀x ∈ \overline{B}_{δ}(x_{0}), \\ x_{0} ∈ int Ω_{A}^{0} ⟹ div m(x) = 0 ∀x ∈ \overline{B}_{δ}(x_{0}). \end{matrix}  (3.19)

For every r ∈ (0, δ], let us define in Ω the function v_{x_{0}}^{r} by

v_{x_{0}}^{r}(x) := u^{∗∗}(x_{0}) + ⟨Du^{∗∗}(x_{0}), x − x_{0}⟩ ± γ · (r − 2 |x − x_{0}|) , x ∈ Ω,

the sign + being chosen if x_{0} ∈ Ω_{A}^{+}, while the sign − is selected if x_{0} ∈ Ω_{A}^{−}. If x_{0} ∈ int Ω_{A}^{0} then any sign in the definition of v_{x_{0}}^{r}(x) is a good choice; in order to fix the ideas, we choose the sign + if x_{0} ∈ int Ω_{A}^{0}. Since |D |x|| = 1 for every x ∈ ℝ^{n} \ {0}, we obtain

|Dv_{x_{0}}^{r}(x) − Du^{∗∗}(x_{0})| = 2γ · |∓D |x − x_{0}|| = 2γ a.e. x ∈ Ω,

and thus by (3.14), (3.15),

Dv_{x_{0}}^{r}(x) ∈ A(x), a.e. x ∈ \overline{B}_{δ}(x_{0}), ∀ r ∈ (0, δ] .  (3.20)

If x_{0} ∈ Ω_{A}^{+} ∪ int Ω_{A}^{0} we set

G(x_{0}, r) := \{ x ∈ \overline{B}_{δ}(x_{0}) ⊂ Ω : v_{x_{0}}^{r}(x) ≥ u^{∗∗}(x)\} ,  (3.21)

and if x_{0} ∈ Ω_{A}^{−} we define

G(x_{0}, r) := \{ x ∈ \overline{B}_{δ}(x_{0}) ⊂ Ω : v_{x_{0}}^{r}(x) ≤ u^{∗∗}(x)\} .  (3.22)

We claim that G(x_{0}, r) is a closed set satisfying

\overline{B}_{r/3}(x_{0}) ⊂ G(x_{0}, r) ⊂ \overline{B}_{r}(x_{0}).  (3.23)

Let us verify (3.23) when x_{0} ∈ Ω_{A}^{+} ∪ int Ω_{A}^{0}. If x ∈ \overline{B}_{δ}(x_{0}) but x ∉ \overline{B}_{r}(x_{0}) (that is r < |x − x_{0}| ≤ δ) then by (3.13) and (3.17) we get

v_{x_{0}}^{r}(x) − u^{∗∗}(x) = γ · (r − 2 |x − x_{0}|) − o (|x − x_{0}|)

< −γ |x − x_{0}| − o (|x − x_{0}|) = − |x − x_{0}| · (γ + \frac{o (|x − x_{0}|)}{|x − x_{0}|}) ≤ 0.

Thus v_{x_{0}}^{r}(x) − u^{∗∗}(x) < 0 and x ∉ G(x_{0}, r). On the other hand if x ∈ \overline{B}_{r/3}(x_{0}), then r/3 ≥ |x − x_{0}|, and again by (3.13), (3.17), we obtain

v_{x_{0}}^{r}(x) − u^{∗∗}(x) = γ · (r − 2 |x − x_{0}|) − o (|x − x_{0}|) ≥

γ |x − x_{0}| − o (|x − x_{0}|) = |x − x_{0}| (γ − \frac{o (|x − x_{0}|)}{|x − x_{0}|}) ≥ 0

and x ∈ G(x_{0}, r). Thus (3.23) is proved.

By (3.21) and the continuity of u^{∗∗} and v_{x_{0}}^{r}

∂G(x_{0}, r) = \{ x ∈ \overline{B}_{δ}(x_{0}) : v_{x_{0}}^{r}(x) = u^{∗∗}(x)\} ,

and setting

w(x) := u^{∗∗}(x) − u^{∗∗}(x_{0}) − ⟨Du^{∗∗}(x_{0}), x − x_{0}⟩ ± 2γ |x − x_{0}| , x ∈ Ω,

then

∂G(x_{0}, r) = \{ x ∈ \overline{B}_{δ}(x_{0}) : w(x) = γr\} ,

i.e., for every r ∈ (0, δ) ∂G(x_{0}, r) is a level set of a W_{loc}^{1,∞}(Ω) function. Therefore we can always choose a sequence r_{h} of real numbers such that

\{ \begin{matrix}r_{h} → 0 as h → +∞, \\ 0 < r_{h} ≤ δ, ∀ h ∈ ℕ, \\ meas (∂G(x_{0}, r_{h})) = 0, ∀ h ∈ ℕ. \end{matrix}  (3.24)

Let us consider the measurable subset of Ω

M := {x_{0} ∈ Ω_{A}^{+} ∪ Ω_{A}^{−} ∪ int Ω_{A}^{0} : u^{∗∗} differentiable at x_{0}, Du^{∗∗}(x_{0}) ∈ A(x_{0})}

and consider the family of open sets

G := {int G(x, r_{h}) : x ∈ M, r_{h} as in (3.24)} .

Since G is a fine covering of M and each set G(x, r) ∈ G satisfies (3.23), by Vitali’s covering theorem (see, for example, Chapter 10 of [12]), there exists in G a (at most) countable subcollection G′ of sets with disjoint closures such that the open set

Ω′ = ⋃_{int G(x_{k},r_{k})∈G′} int G(x_{k}, r_{k}) ,  (3.25)

covers M up to a set of zero measure, i.e. meas (M \ Ω′) = 0. Let us define the function v in Ω by

v(x) := \{ \begin{matrix}u^{∗∗}(x) if x ∈ Ω \ Ω′, \\ v_{x_{k}}^{r_{k}}(x) if x ∈ G(x_{k}, r_{k}), \end{matrix}  (3.26)

and introduce the functions

u_{h}(x) := \{ \begin{matrix}u^{∗∗}(x) if x ∈ Ω \ ∪_{k=1}^{h}G(x_{k}, r_{k}), \\ v_{x_{k}}^{r_{k}}(x) if x ∈ G(x_{k}, r_{k}), for some 1 ≤ k ≤ h. \end{matrix}

By (3.21) and (3.22) the functions u_{h} belong to u^{∗∗} + W_{0}^{1,p}(Ω) ∩ W_{loc}^{1,∞}(Ω) for all h, since u_{h} is locally maximum or minimum of two W^{1,p}(Ω) ∩ W_{loc}^{1,∞}(Ω) functions. Moreover, we claim that each u_{h} is a minimizer of (3.9). To this aim, notice that by (3.19), (3.21), (3.22), we have for all h

div m(x) (u_{h}(x) − u^{∗∗}(x)) ≥ 0 a.e. x ∈ ∪_{k=1}^{h}G(x_{k}, r_{k}) .  (3.27)

Moreover, by (3.20)

Du_{h}(x) ∈ A(x), a.e. x ∈ ∪_{k=1}^{h}G(x_{k}, r_{k}) .  (3.28)

and Du_{h}(x) = Du^{∗∗}(x) a.e. in Ω \ ∪_{k=1}^{h}G(x_{k}, r_{k}). By the convexity inequality f^{∗∗}(x, ξ) ≥ q(x) + ⟨m(x), ξ⟩, valid for every x ∈ Ω_{A} and every ξ ∈ ℝ^{n}, we have

inf {∫_{Ω} f^{∗∗} (x, Du(x)) dx : u ∈ u_{0} + W_{0}^{1,p} (Ω)}

= ∫_{Ω\∪_{k=1}^{h}G(x_{k},r_{k})} f^{∗∗} (x, Du^{∗∗}(x)) dx + ∫_{∪_{k=1}^{h}G(x_{k},r_{k})} f^{∗∗} (x, Du^{∗∗}(x)) dx

≥ ∫_{Ω\∪_{k=1}^{h}G(x_{k},r_{k})} f^{∗∗} (x, Du^{∗∗}(x)) dx+∫_{∪_{k=1}^{h}G(x_{k},r_{k})} {q(x) + ⟨m(x), Du^{∗∗}(x)⟩} dx,

where, in view of (3.14), (3.15) and (3.23), we have used the fact that ∪_{k=1}^{h}G(x_{k}, r_{k}) ⊂ Ω_{A}.

Since u_{h} ∈ u^{∗∗} +W_{0}^{1,p}(int G(x_{k}, r_{k})) for all 1 ≤ k ≤ h, using (3.27) we obtain

∫_{G(x_{k},r_{k})} {⟨m(x), Du^{∗∗}(x)⟩ − ⟨m(x), Du_{h}(x)⟩} dx
= − ∫_{G(x_{k},r_{k})} div m(x) (u^{∗∗}(x) − u_{h}(x)) dx ≥ 0 .

Therefore we may conclude, from the inequality above and by (3.28), that

inf {∫_{Ω} f^{∗∗} (x, Du(x)) dx : u ∈ u_{0} + W_{0}^{1,p} (Ω)}

≥ ∫_{Ω\∪_{k=1}^{h}G(x_{k},r_{k})} f^{∗∗} (x, Du^{∗∗}(x)) dx+∫_{∪_{k=1}^{h}G(x_{k},r_{k})} {q(x) + ⟨m(x), Du_{h}(x)⟩} dx

= ∫_{Ω} f^{∗∗} (x, Du_{h}(x)) dx .

This proves the minimality of each u_{h}. By (3.2) the sequence u_{h} is bounded in W^{1,p}(Ω) and since u_{h}(x) converges to v(x) a.e. in Ω, it follows that u_{h} converges weakly to v in W^{1,p}(Ω). In particular, v ∈ u_{0} + W_{0}^{1,p}(Ω) and, by lower semicontinuity, v is also a minimizer of (3.9). Theorem 2.7 now yields v ∈ W_{loc}^{1,∞}(Ω). By (3.20) we have

Dv(x) ∈ A(x), a.e. x ∈ Ω′.

On the other hand, since v = u^{∗∗} in Ω \ Ω′, there exists a negligible set E_{0} such that, for all x ∈ (Ω \ Ω′) \ E_{0}, v and u^{∗∗} are differentiable at x and Dv(x) = Du^{∗∗}(x). Thus, if x ∉ E_{0} ∪ (M \ Ω′) ∪ (Ω ∩ ∂Ω_{A}^{0}), which is a set of measure zero, and if x ∉ Ω \ Ω′, then we have

Dv(x) ∉ A(x) .

Finally, let us prove that div m = 0 in Ω′. To this end notice that, since

f^{∗∗}(x, ξ) = q(x) + ⟨m(x), ξ⟩ , whenever x ∈ Ω_{A} and ξ ∈ A(x),

the function ξ → f^{∗∗}(x, ξ) is differentiable in A(x) and D_{ξ}f^{∗∗}(x, ξ) = m(x) for every x ∈ Ω′, ξ ∈ A(x). Therefore the standard argument used to derive the Euler equation of an integral functional still applies in the open set Ω′ and the Euler’s equation in weak form gives

∫_{Ω′} div m(x) φ(x) dx = 0 , ∀ φ ∈ C_{0}^{1} (Ω′) ,

which proves that div m = 0 in Ω′.

The second lemma that we consider in this section uses the notion of Kuratowski convergence, or convergence in the Hausdorff metric of a sequence of compact sets of ℝ^{n}. We recall that a sequence E_{h} of compact sets of ℝ^{n} converges, as h → +∞, in the sense of Kuratowski (or in the Hausdorff metric) to a compact set E ⊂ ℝ^{n} if, for every ε > 0, there exists h_{0} such that

E_{h} ⊂ I_{ε} (E) , E ⊂ I_{ε} (E_{h}) ∀ h > h_{0} ,

where I_{ε} (·) denotes the neighborhood of radius ε of the set under consideration. We recall that, if the sequence E_{h} is bounded in ℝ^{n} uniformly with respect to h ∈ ℕ, then E_{h} → E in the sense of Kuratowski if and only if the following two properties hold:

(i) for every ξ ∈ E there exists a sequence ξ_{h}, with ξ_{h} ∈ E_{h} for every h ∈ ℕ, such that ξ_{h} converges to ξ as h → +∞;

(ii) if a sequence ξ_{h}, with ξ_{h} ∈ E_{h} for every h ∈ ℕ, admits a subsequence ξ_{h_{k}} converging to a point ξ, then ξ ∈ E.

In the sequel we need some properties of the distance function to a convex set. For the statements and the proofs, we refer the reader to the Appendix.

Let E(x) ≠ ∅ be a compact subset of ℝ^{n} for every value of a parameter x in an open set Ω_{A} ⊂ ℝ^{n}. We say that the set function E(x) is continuous in the sense of Kuratowski if E(x_{k}) converges in the sense of Kuratowski to E(x) for every x, x_{k} ∈ Ω_{A} such that x_{k} → x.

Lemma 3.4 Let f : Ω_{A} × ℝ^{n} → ℝ be a continuous function such that the bipolar f^{∗∗} is also continuous and satisfies (3.6) and (3.8). Let E(x) ⊂ ℝ^{n} be defined by

E(x) := {ξ ∈ ℝ^{n} : f^{∗∗}(x, ξ) − [q(x) + ⟨m(x), ξ⟩] = 0} .

Then the map x ∈ Ω_{A} ↦ E(x) is continuous in the sense of Kuratowski.

Proof. Let x_{h} be a sequence in Ω_{A} converging to x ∈ Ω_{A}. We claim that, if ξ_{h_{k}} ∈ E(x_{h_{k}}) and ξ_{h_{k}} → ξ, then ξ ∈ E(x). In fact, since

f^{∗∗}(x_{h_{k}}, ξ_{h_{k}}) = q(x_{h_{k}}) + ⟨m(x_{h_{k}}), ξ_{h_{k}}⟩ ,

the continuity of f^{∗∗}, m, and q yields ξ ∈ E(x).

Let us now show that, given ξ ∈ E(x), there exist ξ_{h} ∈ E(x_{h}) such that ξ_{h} → ξ. Indeed, if this is not true, then we can find ε > 0 and a sequence h_{k} such that dist (ξ; E(x_{h_{k}})) > ε for all k ∈ ℕ. For every k ∈ ℕ choose ξ_{h_{k}} ∈ ∂E(x_{h_{k}}) such that |ξ − ξ_{h_{k}} | = dist (ξ; E(x_{h_{k}})) > ε. Extracting, if necessary, a further subsequence, we may assume that ξ_{h_{k}} → η, and η ∈ E(x) by the first part of the proof. Since ξ ∉E(x_{h_{k}}) and ξ_{h_{k}} ∈ ∂E(x_{h_{k}}), from (3.8) we get, for all k ∈ ℕ,

f^{∗∗} (x_{h_{k}}, \frac{ξ + ξ_{h_{k}}}{2} ) ≤ \frac{1}{2}f^{∗∗}(x_{h_{k}}, ξ) + \frac{1}{2}f^{∗∗}(x_{h_{k}}, ξ_{h_{k}}) − ω (|ξ − ξ_{h_{k}}|) .

Passing to the limit as k → +∞, we get

f^{∗∗} (x, \frac{ξ + η}{2} ) ≤ \frac{1}{2}f^{∗∗}(x, ξ) + \frac{1}{2}f^{∗∗}(x, η) − ω (ε) .

Since f^{∗∗}(x, ·) is affine on E(x) and ξ, η ∈ E(x), the inequality above implies ω (ε) ≤ 0, which gives a contradiction since ω(t) > 0 for all t > 0.

Lemma 3.5 Under the same assumptions of Lemma 3.4 the function F : Ω_{A} × ℝ^{n} → ℝ, defined by

F (x, ξ) = \{ \begin{matrix}dist (ξ; ∂E(x)) & if ξ ∈ ℝ^{n} \ E(x), \\ − dist (ξ; ∂E(x)) & if ξ ∈ E(x), \end{matrix}

is continuous in Ω_{A} × ℝ^{n} and convex with respect to ξ ∈ ℝ^{n}.

Proof. The convexity of F (x, ·) follows from Lemma 4.4 and the convexity of E(x). The continuity of F with respect to x ∈ Ω_{A} is a direct consequence of Lemma 4.2, asserting the continuity of the distance function with respect to the Kuratowski convergence, and of Lemma 3.4, yielding the continuity of the set function E(x) with respect to x. The continuity of F with respect to (x, ξ) then follows from the fact that for all x ∈ Ω_{A}, ξ, η ∈ ℝ^{n} we have that |F (x, ξ) − F (x, η)| ≤ |ξ − η|.

We are now ready to give the proof of Theorem 3.1. Proof of Theorem 3.1. By Lemma 3.2 the minimum of the relaxed variational problem (3.9) is attained and there exist a minimizer v and an open set Ω′ ⊂ Ω (possibly empty) such that

\{ \begin{matrix}Dv(x) ∈ A(x) a.e. x ∈ Ω′ \\ Dv(x) ∉ A(x) a.e. x ∈ Ω \ Ω′ , \end{matrix}

with div m = 0 in Ω′.

If Ω′ is an empty set then Dv(x) ∉ A(x) a.e. x ∈ Ω, and so f^{∗∗}(x, Dv(x)) = f(x, Dv(x)) a.e. x ∈ Ω, and v is also a minimizer for the original problem (3.1).

Otherwise, recalling (3.25), we consider for all k the Dirichlet problem for the implicit differential equation

\{ \begin{matrix}F (x, Du (x)) = 0, & a.e. x ∈ int G(x_{k}, r_{k}) , \\ u (x) = v (x) , & x ∈ ∂G(x_{k}, r_{k}) , \end{matrix}  (3.29)

where F : Ω_{A} × ℝ^{n} → ℝ is the signed distance function defined in Lemma 3.5. Notice that v is Lipschitz continuous on G(x_{k}, r_{k}), and since A(x) ⊂ E(x) we deduce that

F (x, Dv (x)) ≤ 0 a.e. x ∈ int G(x_{k}, r_{k}) .  (3.30)

Therefore, we may apply Theorem 2.3 by Dacorogna-Marcellini [12] (note that (3.30) is exactly the compatibility condition (2.6) required in Theorem 2.3 in [12]) to obtain the existence of a function v_{k} ∈ W^{1,∞} (int G(x_{k}, r_{k})) satisfying (3.29). Notice that F (x, ξ) = 0 if and only if ξ ∈ ∂E(x), and thus Dv_{k}(x) ∈ ∂E(x) ⊂ ℝ^{n} \ A(x) almost everywhere in int G(x_{k}, r_{k}). Therefore

f^{∗∗}(x, Dv_{k}(x)) = f(x, Dv_{k}(x)) a.e. x ∈ int G(x_{k}, r_{k}).  (3.31)

Let us now prove that the functions

w_{h}(x) := \{ \begin{matrix}v_{k} (x) & if x ∈ G(x_{k}, r_{k}), 1 ≤ k ≤ h , \\ v(x) & if x ∈ Ω \ ∪_{k=1}^{h}G(x_{k}, r_{k}) , \end{matrix}

are all minimizers in the class u_{0} + W_{0}^{1,p} (Ω) of the integral in (3.9). In fact, by the affinity assumption (3.6), for any h we get

∫_{Ω} f^{∗∗} (x, Dv(x)) dx − ∫_{Ω} f^{∗∗} (x, Dw_{h}(x)) dx  (3.32)

= ∑_{k=1}^{h} ∫_{G(x_{k},r_{k})} {f^{∗∗} (x, Dv(x)) − f^{∗∗} (x, Dv_{k}(x))} dx

= ∑_{k=1}^{h} ∫_{G(x_{k},r_{k})}⟨m(x), Dv(x) − Dv_{k}(x)⟩ dx = 0 ,

where we have used the fact that v = v_{k} on ∂G(x_{k}, r_{k}), together with the property that div m = 0 in Ω′. The minimality of the function w_{h} follows immediately.

The proof will be over once we show that w_{h} converges strongly in W^{1,p}(Ω) to a function w. Indeed, the limiting function w will then be the required minimizer because by (3.2), (3.32), and (3.31),

∫_{Ω} f (x, Dw(x)) dx = lim_{h→+∞} ∫_{Ω} f (x, Dw_{h}(x)) dx

= ∫_{Ω} f^{∗∗} (x, Dv(x)) dx+ lim_{h→+∞} [∫_{Ω} f (x, Dw_{h}(x)) dx − ∫_{Ω} f^{∗∗} (x, Dw_{h}(x)) dx]

= ∫_{Ω} f^{∗∗} (x, Dv(x)) dx+ lim_{h→+∞} ∫_{Ω′\(∪_{k≤h}G(x_{k},r_{k}))} [f (x, Dv(x)) − f^{∗∗} (x, Dv(x))] dx

= ∫_{Ω} f^{∗∗} (x, Dv(x)) dx ,

where we have used the fact that f(x, Dv(x)) = f^{∗∗}(x, Dv(x)) for a.e. x ∉ Ω′. In order to prove that w_{h} converges in W^{1,p}(Ω), it suffices to show that Dw_{h} is a Cauchy sequence in L^{p}. Since v_{k} minimizes the integral

∫_{G(x_{k},r_{k})} f^{∗∗}(x, Du(x)) dx

in the class v + W_{0}^{1,p}(int G(x_{k}, r_{k})), by (3.2) there exists a constant c such that for all k

∫_{G(x_{k},r_{k})} |Dv_{k}|^{p} dx ≤ c ∫_{G(x_{k},r_{k})}(1 + |Dv|^{p}) dx .

Hence, if h > k we get

∫_{Ω} |Dw_{h} − Dw_{k}|^{p} dx = ∑_{i=k+1}^{h} ∫_{G(x_{i},r_{i})} |Dv_{i} − Dv|^{p} dx

≤ c ∫_{Ω′\(∪_{i≤k}G(x_{i},r_{i}))}(1 + |Dv|^{p}) dx ,

and the integral on the right hand side converges to zero as k → ∞.

3.2 The general case

In the previous section we have proved Theorem 3.1 under the assumption that the bipolar function f^{∗∗} coincides with the affine function ⟨m(·), ξ⟩ + q(·) on the set A(x) = {ξ ∈ ℝ^{n} : f(x, ξ) > f^{∗∗}(x, ξ)} defined in (3.5). Here we consider the more general situation where the set A(x) can be split into an union of a (at most countable) family of pairwise disjoint open sets A_{j}(x), and that in each A_{j}(x) the function f^{∗∗}(x, ξ) coincides with an affine function ⟨m_{j}(x), ξ⟩ + q(x), where the slopes m_{j} and the functions q_{j} may vary with j.

Precisely, let f : Ω×ℝ^{n} → ℝ be a continuous function, satisfying the growth condition (3.2), the uniform convexity condition (3.3), and (3.4). As before, we assume that f^{∗∗} is continuous and we denote by Ω_{A} the set of points x ∈ Ω such that A(x) ≠ ∅. Moreover, we assume that, for all x ∈ Ω_{A}, there exists a sequence A_{j}(x) of pairwise disjoint open sets such that

A(x) = ∪_{j}A_{j}(x) ,  (3.33)

and that, for every j, the set Ω_{A_{j}} := {x ∈ Ω : A_{j}(x) ≠ ∅} is open. Further, assume that, for all j, there exist a function q_{j} ∈ C^{0} (Ω_{A_{j}}) and a vector field m_{j} ∈ C^{1} (Ω_{A_{j}}; ℝ^{n}) such that

f^{∗∗}(x, ξ) = q_{j}(x) + ⟨m_{j}(x), ξ⟩ ,  ∀ x ∈ Ω_{A_{j}} , ξ ∈ A_{j}(x) ,  (3.34)

and the boundary of the set where the divergence of m_{j} is equal to zero is negligible; i.e., for all j,

meas (Ω ∩ ∂ {x ∈ Ω_{A_{j}} : div m_{j}(x) = 0}) = 0 .

For all x ∈ Ω_{A_{j}} set

E_{j}(x) := {ξ ∈ ℝ^{n} : f^{∗∗}(x, ξ) = q_{j}(x) + ⟨m_{j}(x), ξ⟩}

and we assume that, for every j ∈ ℕ, there exists an increasing function ω_{j} : [0, +∞) → [0, +∞), with ω_{j}(t) = 0 if and only if t = 0, such that, if x ∈ Ω_{A}, ξ ∈ ∂E_{j}(x), η ∈ ℝ^{n} \ E_{j}(x), then

f^{∗∗} (x, \frac{ξ + η}{2} ) ≤ \frac{1}{2}f^{∗∗}(x, ξ) + \frac{1}{2}f^{∗∗}(x, η) − ω_{j} (|ξ − η|) .  (3.35)

As in the previous section we have the following existence result.

Theorem 3.6 Let f, f^{∗∗} : Ω × ℝ^{n} → ℝ be continuous functions (f not necessarily convex with respect to ξ ∈ ℝ^{n}). Under the above assumptions on f and f^{∗∗}, for any given boundary datum u_{0} ∈ W^{1,p} (Ω) the variational problem (3.1) attains its minimum. Moreover, any minimizers is of class W_{loc}^{1,∞}(Ω).

Remark 3.7 Notice that the assumptions (3.33) and (3.34) are equivalent to the following local assumption: for any (x_{0}, ξ_{0}) ∈ Ω × ℝ^{n} such that f(x_{0}, ξ_{0}) > f^{∗∗}(x_{0}, ξ_{0}) there exist δ > 0, a function q ∈ C^{0} (B_{δ}(x_{0})), and a vector field m ∈ C^{1} (B_{δ}(x_{0}); ℝ^{n}) such that, if |x − x_{0}| < δ, |ξ − ξ_{0}| < δ, then

f^{∗∗}(x, ξ) = q(x) + ⟨m(x), ξ⟩ .  (3.36)

In fact, set D := {(x, ξ) ∈ Ω × ℝ^{n} : f(x, ξ) > f^{∗∗}(x, ξ)}. The set D is open, hence we may consider its connected components D_{j}. For every x ∈ Ω and every j let us set A_{j}(x) := {ξ ∈ ℝ^{n} : (x, ξ) ∈ D_{j}}. For every x the family {A_{j}(x)} is a partition of A(x) := {ξ ∈ ℝ^{n} : f(x, ξ) > f^{∗∗}(x, ξ)}. Moreover, the set Ω_{A_{j}} := {x ∈ Ω : A_{j}(x) ≠ ∅} is open, since it is equal to the projection of D_{j} onto ℝ^{n}. Finally, the existence for every j of a function q_{j} ∈ C^{0}(Ω_{A_{j}}) and a vector field m_{j} ∈ C^{1}(Ω_{A_{j}}; ℝ^{n}) satisfying (3.34) is an immediate consequence of the local assumption (3.36) and of the connectedness of D_{j}.

To prove Theorem 3.6 one may follow exactly the same argument used in the proof of Theorem 3.1, with the obvious changes due to the fact that now we have to deal separately with the different affine representations (3.34) of f^{∗∗}. Therefore we shall limit ourselves to point out where the proof has to be modified.

The statement of Lemma 3.2 must be replaced by the following one.

Lemma 3.8 The minimum of the relaxed variational problem (3.9) is attained. Moreover, there exists a minimizer v ∈ W_{loc}^{1,∞}(Ω) ∩ (u_{0} + W^{1,p}(Ω)) of (3.9) and there exist pairwise disjoint open sets Ω′_{j} ⊂ Ω (possibly empty) such that

\{ \begin{matrix}Dv(x) ∈ A_{j}(x) a.e. x ∈ Ω′_{j} \\ Dv(x) ∉ A(x) a.e. x ∈ Ω \ ∪_{j}Ω′_{j} \end{matrix}

and div m_{j} = 0 in Ω′_{j}.

Proof. To proof this lemma we argue as in the proof of Lemma 3.2, splitting each open set Ω_{A_{j}} into the three open sets

Ω_{A_{j}}^{+} := {x ∈ Ω_{A_{j}} : div m_{j}(x) > 0} , Ω_{A_{j}}^{−} := {x ∈ Ω_{A_{j}} : div m_{j}(x) < 0} ,

Ω_{A_{j}}^{0} := {x ∈ Ω_{A_{j}} : div m_{j}(x) = 0} .

Taking a point x_{0} ∈ Ω_{A_{j}} \ ∂Ω_{A_{j}}^{0} such that u^{∗∗} is differentiable in x_{0} and Du^{∗∗}(x_{0}) ∈ A_{j}(x_{0}) for some j, we construct the function v_{x_{0}}^{r} as before, noticing that the parameter γ can be always chosen so small (see (3.20)) that

Dv_{x_{0}}^{r}(x) ∈ A_{j}(x) a.e. x ∈ \overline{B}_{δ}(x_{0}), ∀ r ∈ (0, δ] .  (3.37)

Arguing as in the model case, we get again a sequence G(x_{k}, r_{k}) of pairwise disjoint sets such that the open set

Ω′ := ⋃_{int G(x_{k},r_{k})∈G′} int G(x_{k}, r_{k})

covers, up to a set of measure zero, the set of points x_{0} where u^{∗∗} is differentiable and Du^{∗∗}(x_{0}) ∈ A_{j}(x_{0}) for some j. Therefore, if we define v as in (3.26) we have again that Dv(x) ∉ A(x) for a.e. x ∈ Ω \ Ω′ and, by (3.37) that for any k there exists j_{k} such that

Dv(x) ∈ A_{j_{k}}(x) for a.e. x ∈ G(x_{k}, r_{k}) .

In particular Ω′ can be written as the disjoint union of the open sets

Ω′_{j} = ⋃ {int G(x_{s}, r_{s}) : Dv_{x_{s}}^{r_{s}}(x) ∈ A_{j}(x) ∀ x ∈ G(x_{s}, r_{s})}

and Dv(x) ∈ A_{j}(x) a.e. x ∈ Ω′_{j}, for all j. The rest of the proof follows with the obvious adaption.

Proof of Theorem 3.6. We claim that, for all j, the set function x ∈ Ω_{A_{j}} ↦ E_{j}(x) is continuous in the sense of Kuratowski. It can be easily checked via the same argument used in Lemma 3.4, and by virtue of assumption (3.35) (this is the only point where this hypothesis is needed). Therefore, in view of Lemma 3.5, for every j the function F_{j} defined by

F_{j} (x, ξ) := \{ \begin{matrix}dist (ξ; ∂E_{j}(x)) & if ξ ∈ ℝ^{n} \ E_{j}(x) \\ − dist (ξ; ∂E_{j}(x)) & if ξ ∈ E_{j}(x), \end{matrix}

is continuous in Ω_{A_{j}} × ℝ^{n} and convex with respect to ξ. Then, replacing (3.20) with (3.37), the proof proceeds exactly as in the model case.

4 Appendix: some properties of the distance function

In this section we prove some properties of the signed distance function to a convex set, which has been used in Section 3 in order to establish Lemma 3.4 and Lemma 3.5. We recall that if E ⊂ ℝ^{n} then the signed distance function to E is a function d_{E} : ℝ^{n} → ℝ defined as follows:

d_{E}(ξ) := \{ \begin{matrix}dist (ξ; ∂E) if ξ ∉ E, \\ − dist (ξ; ∂E) if ξ ∈ E. \end{matrix}  (4.1)

Lemma 4.1 Let E ⊂ ℝ^{n} be a closed convex set. If ξ ∈ ℝ^{n} is such that

ξ ∈ I_{δ} (E) , dist (ξ; ∂I_{δ} (E)) > δ,

for some δ > 0, then ξ ∈ E (more precisely ξ ∈ int E).

Proof. If ξ ∉ E then we consider the projection ξ_{0} of ξ on E and a supporting hyperplane H to E through ξ_{0}. This hyperplane H separates ξ from E. Let us denote by η the point

η := ξ + δ \frac{ξ − ξ_{0}}{|ξ − ξ_{0}|};

clearly |η − ξ| = δ. Since by the assumptions \overline{B}_{δ}(ξ) ⊂ I_{δ} (E), in particular η ∈ I_{δ} (E). On the other hand, dist (η; E) ≥ dist (η; H) = |ξ − ξ_{0}| + δ > δ, which gives a contradiction.

Lemma 4.2 Let E_{h} and E be convex, compact sets of ℝ^{n}, such that E_{h} → E in the sense of Kuratowski. Then

d(ξ) = lim_{h→+∞} d_{h}(ξ), ∀ ξ ∈ ℝ^{n},

where d_{h} and d are, respectively, the signed distance functions to E_{h} and E.

Proof. Note that we use the assumption of convexity of the sets E_{h}, E only in Steps 2 and 3 below.

Step 1: Let us consider the case where ξ ∉ E. Then there exists ε > 0 such that dist (ξ; ∂E) > ε and, for large h ∈ ℕ, ξ ∉ E_{h}. Let ξ_{0} ∈ E be a point such that d(ξ) = |ξ − ξ_{0}|, and let ξ_{h} be a sequence of points in E_{h} converging to ξ_{0}. Then

d(ξ) = lim_{h→+∞} |ξ − ξ_{h}| ≥ lim sup_{h→+∞} d_{h}(ξ).

Conversely, let ξ_{h_{k}} ∈ E_{h_{k}} be such that

lim inf_{h→+∞} d_{h}(ξ) = lim_{k→+∞} |ξ − ξ_{h_{k}} | .

By extracting, if necessary, a further subsequence, we may assume that ξ_{h_{k}} → ξ_{0}, for some ξ_{0} ∈ E. Thus

lim inf_{h→+∞} d_{h}(ξ) = lim_{k→+∞} |ξ − ξ_{h_{k}} | = |ξ − ξ_{0}| ≥ d(ξ).

Step 2: Let ξ ∈ ∂E. There exists a sequence h_{k} such that

lim sup_{h→+∞} d_{h}(ξ) = lim_{k→+∞} d_{h_{k}}(ξ).

Let ξ_{h_{k}} ∈ E_{h_{k}} be such that ξ_{h_{k}} → ξ. Since |ξ − ξ_{h_{k}} | ≥ d_{h_{k}}(ξ), we obtain

lim sup_{h→+∞} d_{h}(ξ) ≤ 0 = d(ξ).

Suppose now that lim inf_{h→+∞} d_{h}(ξ) < 0. As before, there exists a sequence h_{k} such that lim inf_{h→+∞} d_{h}(ξ) = lim_{k→+∞} d_{h_{k}}(ξ). Fix ε > 0. For k large enough d_{h_{k}}(ξ) < 0 and ξ ∈ E_{h_{k}} ⊂ I_{ε} (E). Let us denote by ν a unit vector in ℝ^{n}, orthogonal to a supporting hyperplane to E at ξ, pointing to the exterior of E.

By the convexity of E, the vector ξ + 2εν ∉ I_{ε} (E), and so ξ + 2εν ∉ E_{h_{k}} for large k. Therefore, for such k, dist (ξ; ∂E_{h_{k}}) = |d_{h_{k}}(ξ)| < 2ε. This implies that lim inf_{h→+∞} d_{h}(ξ) ≥ 0.

Step 3: Let ξ ∈ int E and fix δ ∈ (0, dist (ξ; ∂E)) such that B_{δ}(ξ) ⊂ E. For every ε < δ there exists h_{ε} such that E ⊂ I_{ε} (E_{h}) for h > h_{ε}. If η ∈ B_{δ−ε}(ξ) ⊂ E ⊂ I_{ε} (E_{h}) then dist (η; ∂I_{ε} (E_{h})) > ε; therefore, by Lemma 4.1, we have that B_{δ−ε}(ξ) ⊂ E_{h} for h > h_{ε} (in particular ξ ∈ int E_{h}) and dist (ξ; ∂E_{h}) ≥ δ − ε. From this it follows that lim inf_{h→+∞} dist (ξ; ∂E_{h}) = lim inf_{h→+∞} [−d_{h}(ξ)] ≥ δ. Letting δ → dist (ξ; ∂E), we get

d(ξ) ≥ lim sup_{h→+∞} d_{h}(ξ).

Conversely, let ξ_{0} ∈ ∂E be such that |ξ − ξ_{0}| = dist (ξ; ∂E). From Step 2 it follows that there exists ξ_{h} ∈ ∂E_{h} such that |ξ_{h} − ξ_{0}| → 0. We also have

dist (ξ; ∂E_{h}) ≤ |ξ_{h} − ξ| ≤ |ξ_{h} − ξ_{0}| + dist (ξ; ∂E) .

Since ξ ∈ E_{h}, we conclude that

lim inf_{h→+∞} d_{h}(ξ) = lim inf_{h→+∞} [− dist (ξ; ∂E_{h})] ≥ − dist (ξ; ∂E) = d(ξ).

Lemma 4.3 Let E be a compact, convex set of ℝ^{n}. There exists a sequence E_{h} of compact, convex sets of ℝ^{n} such that E_{h} ⊃ E, ∂E_{h} ∈ C^{∞} for every h ∈ ℕ, and E_{h} → E in the sense of Kuratowski.

Proof. Set δ(ξ) := dist (ξ; E), and δ_{h}(ξ) := (ρ_{ε_{h}} ∗ δ) (ξ), where ε_{h} → 0 and ρ_{ε_{h}} is a sequence of smooth mollifiers. The functions δ_{h} are of class C^{∞}, convex, and converge to δ uniformly in ℝ^{n}. Therefore, with no loss of generality, we may assume that

|δ_{h}(ξ) − δ(ξ)| < \frac{1}{h} ∀ ξ ∈ ℝ^{n}.

By Sard’s theorem, for every h ∈ ℕ there exists t_{h} ∈ (1/h, 2/h) such that the level set {ξ ∈ ℝ^{n} : δ_{h}(ξ) = t_{h}} is a C^{∞}-hypersurface. Hence

E_{h} := {ξ ∈ ℝ^{n} : δ_{h}(ξ) ≤ t_{h}}

is a compact convex set with C^{∞} boundary ∂E_{h}. If ξ ∈ E then δ_{h}(ξ) = δ_{h}(ξ) − δ(ξ) < 1/h < t_{h} and so ξ ∈ E_{h}. This proves that E ⊂ E_{h}.

On the other hand, if ξ ∈ E_{h} then

δ(ξ) = δ(ξ) − δ_{h}(ξ) + δ_{h}(ξ) < \frac{1}{h} + t_{h} < \frac{3}{h} ,

ans so E_{h} ⊂ I_{3/h}(E); hence, given ε > 0, E_{h} ⊂ I_{ε}(E) for h sufficiently large.

We conclude this appendix by proving the convexity of the signed distance function to a convex set.

Lemma 4.4 If E is a compact, convex subset of ℝ^{n} then the signed distance d_{E} , defined in (4.1), is a convex function.

Proof. We may assume that ∂E is of class C^{2}, the general case being a consequence of Lemma 4.3 and Lemma 4.2. We recall that when the boundary ∂E is of class C^{2} there exists a tubular neighborhood U of ∂E where the projection on ∂E is unique, d_{E} ∈ C^{2}(U), Dd_{E}(ξ) = ν(ξ), the unit exterior normal vector to ∂E at ξ. To prove the convexity of d_{E} it is enough to show that

d_{E} (\frac{ξ_{1} + ξ_{2}}{2} ) ≤ \frac{1}{2}d_{E}(ξ_{1}) + \frac{1}{2}d_{E}(ξ_{2}) ∀ ξ_{1}, ξ_{2} ∈ ℝ^{n}.  (4.2)

We consider three cases.

Case 1: ξ_{1}, ξ_{2} ∈ ℝ^{n} \E. If the midpoint (ξ_{1} + ξ_{2}) /2 belongs to E, then (4.2) is trivial, since the left hand side is less than or equal to zero and the right hand side is nonnegative. If the midpoint (ξ_{1} + ξ_{2}) /2 does not belongs to E, then let η_{1}, η_{2} ∈ ∂E be such that d_{E}(ξ_{i}) = |ξ_{i} − η_{i}| for i = 1, 2. Since (η_{1} + η_{2}) /2 belongs to E, then

d_{E} (\frac{ξ_{1} + ξ_{2}}{2} ) ≤ |\frac{ξ_{1} + ξ_{2}}{2} − \frac{η_{1} + η_{2}}{2} |

≤ \frac{1}{2} |ξ_{1} − η_{1}| + \frac{1}{2} |ξ_{2} − η_{2}| = \frac{1}{2}d_{E}(ξ_{1}) + \frac{1}{2}d_{E}(ξ_{2}).

Case 2: ξ_{1}, ξ_{2} ∈ E. Set d_{i} := dist (ξ_{i}; ∂E) for i = 1, 2. Let us assume that d_{1}, d_{2} > 0, i.e., ξ_{1}, ξ_{2} ∈ int E, and let us fix ρ_{i} ∈ (0, d_{i}) for i = 1, 2, and a unit vector η. Since ξ_{i} + ρ_{i}η ∈ int E, then

\frac{ξ_{1} + ξ_{2}}{2} + \frac{ρ_{1} + ρ_{2}}{2} η ∈ int E,

and the arbitrariness of η yields

dist (\frac{ξ_{1} + ξ_{2}}{2} ; ∂E) ≥ \frac{ρ_{1} + ρ_{2}}{2} .

Letting ρ_{i} → d_{i} we obtain (4.2) when d_{1}, d_{2} > 0. If d_{1} > 0 and d_{2} = 0 it is enough to approximate ξ_{2} with interior points (which have positive distance from ∂E) and apply (4.2) with ξ_{2} replaced by the approximating points. If d_{1} = d_{2} = 0 then (4.2) is trivial.

Case 3: ξ_{1} ∈ E, ξ_{2} ∈ ℝ^{n} \ E (in fact, to avoid a trivial case, we assume that ξ_{1} ∈ int E). In this case the result follows from the fact that the function t ∈ [0, 1] ↦ φ (t) = d_{E} ((1 − t)ξ_{1} + tξ_{2}) is convex on the whole interval [0, 1]. In fact, from Case 1 and Case 2, we deduce that this function φ (t) is separately convex on the two intervals [0, t_{0}] and [t_{0}, 1], where t_{0} is the value of t such that the point (1 − t_{0})ξ_{1} + t_{0}ξ_{2} ∈ ∂E. Moreover, the differentiability of d_{E} at the boundary ∂E implies that φ is differentiable at t_{0}. Therefore φ (t) is convex on the whole interval.

Acknowledgment. The research of Irene Fonseca was partially supported by the National Science Foundation under Grant No. DMS-9731957 and through Center for Nonlinear Analysis (CNA). The research of Nicola Fusco and Paolo Marcellini was partially supported by the Italian Ministero dell’Università e della Ricerca Scientifica e Tecnologica (MURST).

References

[1] E. Acerbi, N. Fusco, Regularity for minimizers of nonquadratic functionals: the case 1 < p < 2, J. Math. Anal. Appl., 140 (1989), 115–135.

[2] L. Ambrosio, N. Fusco, D. Pallara, Special functions of bounded variation and free discontinuity problems, Oxford University Press, to appear.

[3] J. M. Ball, R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., 100 (1987), 15–52.

[4] J. M. Ball, R. D. James, Proposed experimental tests of a theory of fine microstructure and the two wells problem, Phil. Trans. Royal Soc. London, A 338 (1991), 389–450.

[5] P. Celada, S. Perrotta, Minimizing non convex, multiple integrals: a density result, Preprint SISSA, 1998.

[6] A. Cellina, On minima of a functional of the gradient: necessary conditions; Nonlinear Anal. Theory Methods Appl., 20 (1993), 337-341.

[7] A. Cellina, On minima of a functional of the gradient: sufficient conditions, Nonlinear Anal. Theory Methods Appl., 20 (1993), 343-347.

[8] B. Dacorogna, P. Marcellini, Existence of minimizers for non quasiconvex integrals, Arch. Rat. Mech. Anal., 131 (1995), 359-399.

[9] B. Dacorogna, P. Marcellini, Théorème d’existence dans le cas scalaire et vectoriel pour les équations de Hamilton-Jacobi, C.R. Acad. Sci. Paris, 322, Série I, (1996), 237-240.

[10] B. Dacorogna, P. Marcellini, Sur le problème de Cauchy-Dirichlet pour les systèmes d’équations non linéaires du premier ordre, C. R. Acad. Sci. Paris, 323, Série I, (1996), 599–602.

[11] B. Dacorogna, P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case, Acta Mathematica, 178 (1997), 1-37.

[12] B. Dacorogna, P. Marcellini, Implicit partial differential equations, Birkhäuser, Boston, 1999.

[13] B. Dacorogna, P. Marcellini, Attainment of minima and implicit partial differential equations, Ricerche di Matematica, 48 (1999), 311-346.

[14] F. S. De Blasi, G. Pianigiani, On the Dirichlet problem for first order partial differential equations. A Baire category approach, NoDEA Nonlinear Diff. Eq. Appl., 6 (1999), 13-34.

[15] G. Dolzmann, B. Kirchheim, S. Müller, V. Sverak, The two-well problem in three dimensions, Max-Planck Inst., Preprint 21, 1999.

[16] L. C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rat. Mech. Anal., 95 (1986), 227–252.

[17] L. C. Evans, R. F. Gariepy, Blowup, compactness and partial regularity in the calculus of variations, Ind. Univ. Math. J., 36 (1987), 361-371.

[18] I. Fonseca, G. Francfort, 3D −2D asymptotic analysis of an optimal design problem for thin films, J. Reine Angew. Math., 505 (1998), 173–202.

[19] I. Fonseca, N. Fusco, Regularity results for anisotropic image segmentation models, Ann. Sc. Norm. Sup. Pisa, 24 (1997), 463-499.

[20] I. Fonseca, G. Leoni, Bulk and contact energies: nucleation and relaxation, SIAM J. Math. Anal., 30 (1998), 190–219.

[21] G. Friesecke, A necessary and sufficient condition for non attainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Royal Soc. Edinburgh, 124A (1994), 437-471.

[22] P. Marcellini, A relation between existence of minima for nonconvex integrals and uniqueness for not strictly convex integrals of the calculus of variations, Math. Theories of Optimization, Cecconi J.P. and Zolezzi T. Ed., Lecture Notes in Math., Springer-Verlag, 979 (1983), 216-231.

[23] E. Mascolo, R. Schianchi, Existence theorems for nonconvex problems, J. Math. Pures Appl., 62 (1983), 349–359.

[24] E. Mascolo, R. Schianchi, Nonconvex problems in the calculus of variations, Nonlinear Analysis, Theory Meth. Appl., 9 (1985), 371–379.

[25] E. Mascolo, R. Schianchi, Existence theorems in the calculus of variations, J. Diff. Eq., 67 (1987), 185–198.

[26] S. Müller, V. Sverak, Attainment results for the two-well problem by convex integration, ed. Jost J., International Press, 1996, 239–251.

[27] J. P. Raymond, Existence of minimizers for vector problems without quasiconvexity conditions, Nonlinear Analysis, Theory Meth. Appl., 18 (1992), 815–828.

[28] M. A. Sychev, Characterization of homogeneous scalar variational problems solvable for all boundary data, Preprint CMU Pittsburgh, 97-203, October 1997.

[29] S. Zagatti, Minimization of functionals of the gradient by Baire’s theorem, Preprint SISSA, 71/97, May 1997.

[30] Ziemer W.P., Weakly differentiable functions, Graduate Texts in Math., Springer-Verlag, New York, 1989.

I. Fonseca: Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA (fonseca+@andrew.cmu.edu)

N. Fusco: Dipartimento di Matematica “R. Caccioppoli”, Università di Napoli, Via Cintia, 80126 Napoli, Italy (fusco@matna1.dma.unina.it)

P. Marcellini: Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67 A, 50134 Firenze, Italy (marcell@udini.math.unifi.it)