REGULARITY RESULTS VIA DUALITY FOR MINIMIZERS OF DEGENERATE FUNCTIONALS

Menita Carozza* Gioconda Moscariello** Antonia Passarelli di Napoli***

*Università del Sannio, Piazza Arechi II- Palazzo De Simone - 82100 Benevento

e-mail:carozza@unisannio.it

**Dipartimento di Matematica e Applicazioni ”R. Caccioppoli” Università di Napoli “Federico II”, via Cintia - 80126 Napoli

e-mail:gmoscari@unina.it

***Dipartimento di Matematica e Applicazioni ”R. Caccioppoli” Università di Napoli “Federico II”, via Cintia - 80126 Napoli

e-mail: antonia.passarelli@unina.it (Corresponding author)

ABSTRACT. We prove a regularity result for local minimizers of degenerate variational integrals, whose model arises in the study of mappings with finite distortion.

The degeneracy function K(x) lies in the exponential class, i.e. exp(λK(x)) is integrable for some λ > 0.

The right space of the gradient of a local minimizer u turns out to be the Zygmund class L^{p} log^{−1} L. Our result states that if λ is sufficiently large, then Du belongs to the Zygmund space L^{p} log^{α} L, α ≥ 1 and α encreases with λ.

AMS Classifications. 49N15; 49N60; 49N99.

Key words. Degenerate variational integrals; Duality Theory; Mappings with finite distortion.

1 Introduction

In this paper we establish a regularity result for local minimizers of variational integrals satisfying a degenerate ellipticity condition.

The variational integrals we treat here are of the form

I(Ω, v) = ∫_{Ω} f(x, Dv)dx  (1.1)

where Ω is an open bounded subset of IR^{N}, v : Ω ⊂ IR^{N} → IR and f : Ω × IR^{N} → IR is a Caratheodory function.

We assume that there exits a function 1 ≤ k(x) < ∞ such that

(H1) \frac{1}{k(x)}|ξ|^{p} ≤ f(x, ξ) ≤ k(x)|ξ|^{p} for 1 < p < ∞, almost every x ∈ Ω and all ξ, η ∈ IR^{N} and

(H2) ξ → f(x, ξ) is strictly convex a. e. x ∈ Ω.

Remark that, even in the case p = 2, we are dealing with genuine degenerate functionals since the ratio between the eigenvalues is unbounded.

Definition 1.1. A function u ∈ W_{loc}^{1,1}(Ω) is a local minimizer of functional (1.1) if I(Ω′, u) < ∞ for every Ω′ ⊂⊂ Ω and

∫_{suppφ} f(x, Du)dx ≤ ∫_{suppφ} f(x, Du + Dφ)dx  (1.2)

for every function φ ∈ C_{0}^{∞}(Ω).

Our basic assumption is that the growth coefficient k(x), which measures the degree of degeneracy of our variational integral, belongs to the Orlicz space EXP (Ω). Precisely, setting K(x) = max\{k(x), k^{\frac{1}{p−1}} (x)\}, we assume that

∫_{Ω} e^{λK(x)}dx < ∞  (1.3)

for some λ > 0.

Remark 1.1. By Definition 1.1 a minimizer u satisfy

I(Ω′, u) = ∫_{Ω′} f(x, Du)dx < ∞

for every Ω′ ⊂⊂ Ω. Then assumption (H1) and relation (1.3) immediately imply, via Hölder inequality, that

∫_{Ω′} |Du|^{p} log^{−1}(e + |Du|)

is finite, i.e. the gradient of u belongs to the Zygmund class L^{p} log^{−1} L_{loc}(Ω).

Roughly speaking our result shows that the integrability of Du increases when λ gets larger. Indeed, we have the following

Theorem 1.1. Let u be a local minimizer of the variational integral (1.1) such that Du ∈ L^{p} log^{−1} L_{loc}(Ω). Assume that f(x, ξ) verifies assumptions (H1) and (H2). For every α > 0 there exists λ_{α} = λ_{α}(N) such that whenever K(x) satifies (1.3) with λ > λ_{α} then Du ∈ L^{p} log^{α} L_{loc}(Ω) and

||Du||_{L^{p} log^{α} L(Ω′′)} ≤ C_{α}(N, Ω′) ∫_{Ω′} f(x, Du)dx

for every domain Ω′′ ⊂ Ω′ ⊂⊂ Ω.

From the proof of Theorem 1.1 one can easily deduce that λ_{α} increases with α.

Our main idea to get this result is to approximate the integrand f by more regular ones. More precisely, we shall approximate f by strictly convex and uniformly elliptic functions f_{ε}, ε > 0, growing as ξ^{p} in the gradient. To every minimizer h_{ε} of such regular problems, according to the duality theory in the sense of Ekeland and Temam (see section 2), we can associate a suitable divergence free vector field denoted by B_{ε}. For the pair (Dh_{ε}, B_{ε}) we are entitled to apply an a priori estimate proved in [12], which is the basic tool in our proof.

More important, these a priori estimates are preserved in passing to the limit and thus the minimizers of the regular integrals converge to a minimizer of our original problem.

We also obtain the convergence in the weak topology of L^{q} log^{α} L_{loc}(Ω) of the solutions of the dual problems B_{ε} ∈ ∂f(x, Dh_{ε}). In the following we will refer to B_{ε} as “momenta” of h_{ε}.

We can summarize our result as follows: i) Dh_{ε} converges weakly in L^{p} log^{α} L_{loc}(Ω) to Du

ii)∫_{Ω′} f_{ε}(x, Dh_{ε})dx converges to ∫_{Ω′} f(x, Du)dx for every Ω′ ⊂⊂ Ω

iii) B_{ε} converges weakly in L^{q} log^{α} L_{loc}(Ω) to B, such that B ∈ ∂f(x, Du), for a.e. x ∈ Ω.

When f is a Gateaux differentiable integrand the minimizers of (1.1) are solutions to the corresponding Euler - Lagrange equation. Note that in this case the divergence free vector field B coincides with the differential of f in the sense of Gateaux. In any case, when f degenerates, the associated Euler Lagrange equation degenerates too. In connection with the study of mappings with finite distortion the regularity for solutions to equations in which the ellipticity bounds degenerate has been extensively studied ([4], [10], [12], [13], [16]).

In particular in [10] and [16] higher integrability results for finite energy solutions of equation

div(A(x)|Du|^{p−2}Du) = 0  (1.4)

have been established in the cases p = n and 2 ≤ p < n respectively.

It is worth pointing out that here we do not assume differentiability of the integrand f. Therefore our regularity result cannot be deduced by the above mentioned papers.

As an application of Theorem 1.1 we conclude with a continuity result for minizers of (1.1), when p is sufficiently close N (see section 6).

Here we deal with question of regularity for Du, but it is worth pointing out that under assumptions (H1) and (H2) a minimizer u of I(Ω, v) exists in W^{1,r}, 1 ≤ r < p (see [8]).

2 p-Harmonic couples

Let Ω be an open subset of IR^{N}. A pair [B, E] of vector fields such that

divB(x) = 0 curlE(x) = 0

will be called a p-harmonic couple, 1 < p < ∞, of distortion K(x) ≥ 1 if

\frac{|E|^{p}}{p} + \frac{|B|^{q}}{q} ≤ K(x)⟨B, E⟩ in Ω  (2.1)

We will be interested in the distortion K = K(x) of the exponential class EXP (Ω). Precisely, we shall assume that

∫_{Ω} e^{λK(x)}dx < ∞  (2.2)

for some λ > 0.

It has to be noted (see [13]) that the distortion function K(x) admits a BMO-majorant. Precisely we can majorize K(x) point-wise by a function K(x) ∈ BMO(IR^{N}) such that

e^{λ(K−K_{0})} − 1 ∈ L^{1}(IR^{N})

for some K_{0} ∈ L^{∞}(IR^{N}), with 1 ≤ K_{0}(x) ≤ K(x). Moreover, we can also ensure a bound for the BMO-norm of K(x) in terms of the exponent λ, i.e.

||K||_{BMO} ≤ \frac{c(N)}{λ}.  (2.3)

The following quantity will be used in the sequel

[K] = ||K_{0}||_{∞} + \frac{1}{λ} ∫_{IR^{N}} (e^{λ(K−K_{0})} − 1)  (2.4)

The right spaces for p-harmonic couples of exponential integrable distortion are the Orlicz-Zygmund spaces L^{s} log^{α} L, 1 ≤ s < ∞, α ∈ IR, equipped by the Luxemburg norm ([17]).

Let us recall that for α ≥ 0 the non-linear functional

[f]_{s,α} = [∫ |f|^{s} log^{α} (e + \frac{|f|}{∥f∥_{s}} )]^{\frac{1}{s}}

is comparable with the Luxemburg norm in the sense that

||f||_{L^{s} log^{α} L} ≤ [f]_{s,α} ≤ 2||f||_{L^{s} log^{α} L},  (2.5)

see [12].

If Ω is a measurable subset of IR^{N}, with positive and finite measure, if s > 1 the following relations are straightforward

L^{s} log^{−1} L(Ω) ⊂ L^{s−ε}(Ω) ∀ε > 0

and

f ∈ L^{s} log^{−1} L(Ω) implies lim_{ε→0} ε ∫_{Ω} |f|^{s−ε}dx = 0

A technical but useful estimate is contained in the following lemma proved in [12].

Lemma 2.1. For a function F ∈ L^{p} log^{α} L, α ≥ 1 and K ∈ BMO satisfying (2.3) we have

||KF ||_{L^{p} log^{α−p} L} ≤ \frac{C}{λ} ||F ||_{L^{p} log^{α} L} + C([K])||F ||_{L^{p} log^{α−1} L}

with C = C(N) and [K] is the quantity defined at (2.4).

Let us recall now some results about div-curl couples, that we shall use in the following. We begin with the Sobolev-Poincaré lemma ([13])

Lemma 2.2. Let F = [B, E] a div-curl couple in L_{loc}^{s}(Ω, IR^{N} × IR^{N}), 1 < s < ∞ and φ ∈ C_{0}^{∞}(Ω). Then there exists a div-curl couple F_{0} = (B_{0}, E_{0}) ∈ L^{s}(IR^{N}, IR^{N} × IR^{N}) such that

∥D(φB − B_{0})∥_{s} ≤ c(n, s)∥|∇φ| |F |∥_{s}

∥D(φE − E_{0})∥_{s} ≤ c(n, s)∥|∇φ| |F |∥_{s}.

Concerning integrability properties of p-harmonic couples in IR^{N}, the following basic a priori estimate was proved in [12].

Theorem 2.3. Let F = [B, E] be a p-harmonic couple of distortion K with E ∈ L^{p} log^{α} L and B ∈ L^{q} log^{α} L, where 1 < p, q < ∞, p + q = pq. For any α ≥ 0 there exists λ_{α} ≥ 1 such that, whenever K satisfies (2.2) with λ ≥ λ_{α}, we have

∥ |E|^{p} + |B|^{q}∥_{L log^{α} L(IR^{N})} ≤ c([K])∥ |E|^{p} + |B|^{q}∥_{L log^{α−1} L(IR^{N})}.

Combining Lemma 2.1 and Theorem 2.3, one gets the following a priori estimate for the curl free vector field E of a p-harmonic couple.

Corollary 2.4. Let F = [B, E] be a p-harmonic couple of distortion K with E ∈ L^{p} log^{α} L and B ∈ L^{q} log^{α} L, where 1 < p, q < ∞, p + q = pq. For any α ≥ p + 1 there exists λ_{α} ≥ 1 such that, whenever K satisfies (2.2) with λ ≥ λ_{α}, we have

∥ |E|^{p}∥_{L log^{α} L} ≤ c(K)∥ |E|^{p}∥_{L log^{α−1} L}.

Proof: By distortion inequality (2.1) there exists a positive constant c_{p} such that

|B|^{q} ≤ c_{p}|KE|^{p} ≤ c_{p}|KE|^{p}

where K is a BMO-majorant of K verifying (2.3). Fix α ≥ p + 1 and an integer p ≤ n ≤ p + 1. We apply Theorem 2.3 n-times, getting

∥|E|^{p} + |B|^{q}∥_{L log^{α} L} ≤ [c(K)]^{n}∥|E|^{p} + |B|^{q}∥_{L log^{α−n} L}
≤ 2[c(K)]^{n}∥ |KE|^{p}∥_{L log^{α−n} L}  (2.6)

Last term in (2.6) can be estimate using Lemma 2.1 obtaining

∥|KE|^{p}∥_{L log^{α−n} L} ≤ \frac{c}{λ}∥ |E|^{p}∥_{L log^{α−n+p} L} + C_{K}∥ |E|^{p}∥_{L log^{α−n+p−1} L} (2.7)

Putting together estimates (2.6) and (2.7) and using the assumption p−n ≤ 0 we get

∥|E|^{p}∥_{L log^{α} L} ≤ \frac{2[c(K)]^{n}}{λ} ∥ |E|^{p}∥_{L log^{α} L} + C_{K}∥ |E|^{p}∥_{L log^{α−1} L}

The proof of the Corollary is completed by choosing λ > 2[c(K)]^{n}.

In what follows we shall use a local version of Corollary 2.4.

Theorem 2.5. For any α ≥ p + 1 there exists λ_{α} ≥ 1 such that, whenever K satisfies (2.2) with λ ≥ λ_{α}, we have

∥ |φ^{q}E|^{p}∥_{L log^{α} L} ≤ c([K])∥ |Dφ^{q}||E|^{p} ∥_{L^{\frac{N}{N−1}}}  (2.8)

for any function φ ∈ C_{0}^{∞}(Ω).

The proof of estimate (2.8) can be easily achieved using Corollary 2.4 and Lemma 2.2.

3 The dual problem

Let V, V^{∗} two locally convex topological vector spaces (l.c.t.v.s) in duality and let (·, ·) denote the duality between them. For any G : V → IR we indicate by G^{∗} its Young-Fenchel transform, i.e.

G^{∗}(v^{∗}) = sup_{v∈V}{(v, v^{∗}) − G(v)} v^{∗} ∈ V^{∗}

In case of integral functionals the expression of the Young-Fenchel transform can be obtained by using the following (see [6])

Proposition 3.1. If g(x, z) : Ω × IR^{N} → IR is a Caratheodory function and G(u) = ∫_{Ω} g(x, u(x))dx , u ∈ L^{s}(Ω), is finite on some u_{o} ∈ L^{s}(Ω) then

G^{∗}(u^{∗}) = ∫_{Ω} g^{∗}(x, u^{∗}(x))dx

where g^{∗} denote the Young-Fenchel transform of g and is defined by

g^{∗}(x, η) = sup_{ξ∈IR^{N}}{⟨η, ξ⟩ − g(x, ξ)}

where ⟨·, ·⟩ is the scalar product between vectors in IR^{N}.

Let W, W^{∗} be other two l.c.t.v.s in duality and let Λ : V → W be a linear continuous operator. We shall denote by Λ^{∗} the adjoint operator of Λ.

Consider the problem

(P)  inf_{v∈V} {F (v) + G(Λv)}

where F : V → (−∞, +∞] and G : W → (−∞, +∞],

G(Λu) = ∫_{Ω} g(x, Λu(x))dx

are convex and lower semicontinuous functionals.

The dual problem to P is

(P^{∗})  sup_{p∈W^{∗}}{−F^{∗}(Λ^{∗}p) − G^{∗}(−p)}

where F^{∗}, G^{∗} are the Young-Fenchel transforms of F, G respectively. The relationship between P and P^{∗} is given (see [6]) by the following Theorem 3.2. Let V be a reflexive Banach space. If G(Λv) is finite for every v ∈ V then problem P^{∗} admits a solution p and

inf P = sup P^{∗}

Moreover if the problem P admits a solution u the following extremality relations

F (u) + F^{∗}(Λ^{∗}p) = (Λ^{∗}p, u)  (3.1)
g(x, Λu) + g^{∗}(x, −p) = ⟨u, −p⟩  (3.2)

hold for a.e. x ∈ Ω.

If Φ = t^{p} log^{p−1}(e+t), p > 1, we will consider the following Orlicz-Sobolev spaces:

W^{1,Φ}(Ω) = {v ∈ W^{1,1}(Ω) : Dv ∈ L^{Φ}(Ω)}
W_{o}^{1,Φ}(Ω) = {v ∈ W_{o}^{1,1}(Ω) : Dv ∈ L^{Φ}(Ω)}

equipped with the seminorm

||v||_{W^{1,Φ}(Ω)} = ||Dv||_{L^{Φ}(Ω)}

.

Let us fix a function ψ ∈ W^{1,Φ}(Ω) and consider the following problem

(P_{Φ})  inf_{v∈ψ+W_{o}^{1,Φ}(Ω)} ∫_{Ω} f(x, Dv(x))dx

where the integrand f(x, ξ) is a Caratheodory function verifying assumption (H1). Following the arguments of [5], we have

Lemma 3.3. Let Φ̃ = t^{q} log^{−1}(e + t), \frac{1}{p} + \frac{1}{q} = 1. The dual problem to P_{Φ} is given by

(P_{Φ}^{∗})  sup_{p∈LΦ̃(Ω) div p=0} {∫_{Ω}⟨p, Dψ⟩dx − ∫_{Ω} f^{∗}(x, p)dx}

and admits a solution p̄ ∈ LΦ̃(Ω). Moreover

inf P_{Φ} = sup P_{Φ}^{∗}

and if ū is a solution of problem P_{Φ} the following extremality relation

f(x, Dū) + f^{∗}(x, p̄) = ⟨Dū, p̄⟩  (3.3)

holds for a.e. x ∈ Ω.

Proof: Let us take

V = W^{1,Φ}(Ω) V^{∗} = (W^{1,Φ}(Ω))^{∗}

W = L^{Φ}(Ω) W^{∗} = LΦ̃(Ω)

and let Λ : V → W the operator defined by Λv = Dv. Moreover, we set G(w) = ∫_{Ω} f(x, w(x))dx ∀w ∈ L^{Φ}(Ω)

F (v) = \{ \begin{matrix}0 & if v ∈ ψ + W_{o}^{1,Φ}(Ω) \\ +∞ & otherwise \end{matrix}

The dual problem to P_{Φ} is

sup_{p∈LΦ̃(Ω)}{−F^{∗}(Λ^{∗}p) − G^{∗}(−p)}.

By definition we have that

F^{∗}(Λ^{∗}p) = sup_{v∈ψ+W_{o}^{1,Φ}(Ω)} (∫_{Ω}⟨p, Dv⟩dx) = ∫_{Ω}⟨p, Dψ⟩dx+ sup_{v∈W_{o}^{1,Φ}(Ω)} (∫_{Ω}⟨p, Dv⟩dx)

But

sup_{v∈W_{o}^{1,Φ}(Ω)} (∫_{Ω}⟨p, Dv⟩dx) = sup_{v∈D(Ω)}{(−divp, v)_{D(Ω)}}

and then

F^{∗}(Λ^{∗}p) = \{ \begin{matrix}∫_{Ω}⟨p, Dψ⟩dx & if divp = 0 \\ +∞ & otherwise \end{matrix}

On the other hand by proposition 3.1 we have

G^{∗}(p) = ∫_{Ω} f^{∗}(x, p)

We observe that

G(Dv) = ∫_{Ω} f(x, Dv)

is finite for every v ∈ W^{1,Φ}(Ω), thanks to the growth assumption (H1) and Hölder inequality in Orlicz spaces. Moreover, the convexity of f tells us that P_{Φ}^{∗} has a solution p̄ for which the extremality relation (3.3) holds (see [6], Propositions III 2.2, III 2.3 and III 2.4).

Remark 3.5 The growth and coercitivity assumptions on f(x, ξ) imply that the f^{∗} satisfies the following

(H3)  c_{p} \frac{1}{k^{\frac{1}{p−1}} (x)}|η|^{q} ≤ f^{∗}(x, η) ≤ c_{p}k^{\frac{1}{p−1}} (x)|η|^{q}

for almost every x ∈ Ω and all η ∈ IR^{N} with p, q Hölder conjugate exponents.

To every solution ū of P_{Φ} correspond two vector fields, the first one B(x), solution of P_{Φ}^{∗}, is divergence free, whereas the second E(x) = Dū(x) is curl free. The growth and coercitivity assumptions on f(x, ξ) and on f^{∗}(x, η) expressed by (H1) and (H3) imply that B(x) and E(x) are coupled by the inequality

\frac{|E|^{p}}{p} + \frac{|B|^{q}}{q} ≤ K(x)⟨B, E⟩ in Ω

One can easily check that the function K(x) coincide with k(x), in case p ≥ 2 and with k^{\frac{1}{p−1}} (x), in case 1 < p < 2.

This inequality tells us that a solution of P_{Φ} give rise to a p-harmonic couple.

4 Proof of Theorem 1.1

Let us fix a ball Ω′ compactly contained in Ω. We divide the proof in two steps. In the first one we construct a map h : Ω′ → R^{N}, whose gradient lies in L^{p} log^{α} L_{loc}(Ω′), for all α > 0, while in the second one we show that the map h has the same energy of our minimizer u on the ball Ω′. Thus we conclude that u = h a.e. using strict convexity of f.

More precisely, we have the following:

Step 1 The approximation

Let ε denote a sequence of positive real numbers converging to zero. Define u_{ε} as the ε-mollification of u through φ_{ε}, where {φ_{ε}}_{ε>0} is a family of smooth mollifiers. Then Du_{ε} → Du in L^{p} log^{−1} L(Ω′)

Then, for δ > 0, we set

f_{ε,δ}(x, ξ) = \frac{1 + εδ}{1 + εδ + εk(x)}f(x, ξ) + \frac{εk(x)}{1 + εδ + εk(x)}|ξ|^{p}

One can easily check that

\frac{1}{k(x)}|ξ|^{p} ≤ f_{ε,δ}(x, ξ) ≤ k(x)|ξ|^{p}  (4.1)
\frac{ε}{1 + ε + εδ} |ξ|^{p} ≤ f_{ε,δ}(x, ξ) ≤ \frac{1 + ε + εδ}{ε} |ξ|^{p}  (4.2)

Next, consider the variational integrals

I_{ε,δ}[h] = ∫_{Ω′} f_{ε,δ}(x, Dh)dx

and the problems

(P_{ε,δ})  inf_{h∈u_{ε}+W_{0}^{1,p}(Ω′)} ∫_{Ω′} f_{ε,δ}(x, Dh)dx

that are well defined since u_{ε} ∈ W^{1,p}(Ω′) for every ε > 0 . Since f_{ε,δ} is strictly convex by definition and it is coercive thanks to (4.2), each problem P_{ε,δ} admits a unique solution h_{ε} = h_{ε,δ} ∈ W^{1,p}(Ω′). Setting

V = W^{1,p}(Ω′) V^{∗} = (W^{1,p}(Ω′))^{∗}
W = L^{p}(Ω′) W^{∗} = L^{q}(Ω′)

and Λu = Du in Theorem 3.2, we find that the dual problem to P_{ε} is given by

(P_{ε,δ}^{∗})  sup_{B∈L^{q}, divB=0} {∫_{Ω′}⟨B, Du_{ε}⟩dx − ∫_{Ω′} f_{ε,δ}^{∗}(x, B)dx}

and has a solution denoted by B_{ε} = B_{ε,δ}, which , by the extremality relations of Theorem 3.2, satisfies

⟨Dh_{ε,δ}, B_{ε,δ}⟩ = f_{ε,δ}(x, Dh_{ε,δ}) + f_{ε,δ}^{∗}(x, B_{ε,δ})  (4.3)

The div-curl couple (Dh_{ε,δ}, B_{ε,δ}) is a p-harmonic couple in Ω′ of distortion K(x), thanks to (4.1).

Moreover, (4.2) implies that, for every ε, (Dh_{ε,δ}, B_{ε,δ}) is also a p-harmonic couple of bounded distortion and then

|B_{ε,δ}| ≤ C_{ε,δ}|Dh_{ε,δ}|^{p−1}  (4.4)

Classical higher integrability results for minimizers of strictly elliptic variational integrals tell us that h_{ε} belongs to W_{loc}^{1,r}(Ω′) for some r > p and then Dh_{ε} ∈ L^{p} log^{α} L_{loc}(Ω′), for every α > 0 (see [7], [15]). By (4.4), B_{ε} ∈ L^{q} log^{α} L_{loc}(Ω′), for every α > 0 where, as usual, q is the Hölder conjugate exponent of p.

Therefore, we are legitimate to apply Theorem 2.5 to find , for every Ω′′ compactly contained in Ω′, that

||Dh_{ε}||_{L^{p} log^{α} L(Ω′′)}^{p} ≤ C ∫_{Ω′} f_{ε,δ}(x, Dh_{ε}(x))dx
≤ C ∫_{Ω′} f_{ε,δ}(x, Du_{ε}(x))dx  (4.5)

where we also used (4.1) and the minimality of h_{ε}. Here and in what follows C denotes a constant depending on α, N and Ω′′.

Now, we choose δ related to ε via the equality

δ = δ_{ε} := (∫_{Ω′} k(x)|Du_{ε}|^{p}dx)^{2}  (4.6)

Using the definition of f_{ε,δ(ε)}, Jensen inequality, dominated convergence Theorem and (4.6) we get

||Dh_{ε}||_{L^{p} log^{α} L(Ω′′)}^{p} ≤ C ∫_{Ω′} f_{ε,δ(ε)}(x, Du_{ε}(x))dx

≤ C[ ∫_{Ω′} f(x, Du_{ε}(x))dx + ∫_{Ω′} \frac{εk}{1 + εδ + εk} |Du_{ε}|^{p}dx]

≤ C[ ∫_{Ω′} f(x, Du(x))dx + o(ε)]

A routine diagonal argument gives us the sequence h_{j} = h_{ε_{j}} weakly converging. The gradient of limit mapping h, Dh, actually belongs to L^{p} log^{α} L_{loc}(Ω′), by the lower semicontinuity of the norms.

Step 2 The uniqueness

Thank to the strict convexity of f, to complete the proof we need to show that

∫_{Ω′} f(x, Du) = ∫_{Ω′} f(x, Dh)  (4.7)

Fix an arbitrary compact subset Ω′′ of Ω′ and a number 0 < s < 1. Observe that

lim_{s→1} f^{s}(x, Dh_{ϵ}(x)) = f(x, Dh_{ϵ}(x))

and

f^{s}(x, Dh_{ϵ}(x)) ≤ k^{s}|Dh_{ϵ}(x)|^{sp}

Since

∫_{Ω′} k^{s}|Dh_{ϵ}(x)|^{sp}dx ≤ ||k||_{exp}^{s}||Dh_{ϵ}||_{L^{p} log^{m} L}^{sp} ≤ C (∫_{Ω′} f(x, Du)dx)^{s}

we are legitimate to apply dominated convergence Theorem and obtain

lim_{s→1} ∫_{Ω′} f^{s}(x, Dh_{ϵ}(x))dx = ∫_{Ω′} f(x, Dh_{ϵ}(x))dx  (4.8)

The lower semicontinuity of the integral, the definition of f_{ε,δ} and (4.8) imply, up to a not relabelled subquence, that

∫_{Ω′′} f(x, Dh)dx ≤ lim_{ε→0} ∫_{Ω′} f(x, Dh_{ε})dx

= lim_{ε→0} lim_{s→1} ∫_{Ω′} f(x, Dh_{ε})^{s}dx
≤ lim_{ε→0} lim_{s→1} ∫_{Ω′} f_{ε,δ}(x, Dh_{ε})^{s}(\frac{1 + εδ + εk}{1 + εδ} )^{s}  (4.9)
≤ lim_{ε→0} lim_{s→1} ( ∫_{Ω′} f_{ε,δ}(x, Dh_{ε}))^{s}[ ∫_{Ω′} (\frac{1 + εδ + εk}{1 + εδ} )^{\frac{s}{1−s}} ]^{1−s}
≤ lim_{ε→0} [ ∫_{Ω′} f_{ε,δ}(x, Dh_{ε}) lim_{s→1} (|Ω′|^{1−s} + ε||k||_{exp}^{s})]
= lim_{ε→0} ∫_{Ω′} f_{ε,δ}(x, Dh_{ε})

where we used Hölder inequality. Now, using in (4.9) the minimality of h_{ε}, choosing δ given by (4.6), again by Jensen inequality and dominated convergence Theorem we get

∫_{Ω′′} f(x, Dh)dx
≤ lim_{ε→0} ∫_{Ω′} f_{ε,δ}(x, Dh_{ε})
≤ lim_{ε→0} ∫_{Ω′} f_{ε,δ}(x, Du_{ε})
≤ ∫_{Ω′} f(x, Du)

Since Ω′′ is an arbitrary subset of Ω′ previous inequality yields that

∫_{Ω′} f(x, Dh(x))dx ≤ ∫_{Ω′} f(x, Du(x))dx

The opposite inequality follows by the minimality of u, once we observe that the limit map h coincides with u on the boundary of Ω′.

5 The convergence of the momenta

Theorem 1.1 tells us that u is a solution of the problem P_{Φ} defined in Section

3. Namely, if Ω′ ⊂⊂ Ω, we have

∫_{Ω′} f(x, Du(x))dx = inf_{v∈u+W_{o}^{1,1}(Ω′)} ∫_{Ω′} f(x, Dv(x))dx = min_{v∈u+W_{o}^{1,Φ}(Ω′)} ∫_{Ω′} f(x, Dv(x))dx

and also

inf P_{Φ} = sup P_{Φ}^{∗}

We have the following Theorem 5.1. Let the assumptions of Theorem 1.1 hold. For every α > 0 there exists λ_{α} = λ_{α}(N) such that whenever K(x) satisfies (1.3) with λ > λ_{α} then P_{Φ}^{∗} admits a solution B ∈ L^{q} log^{α} L_{loc}(Ω, IR^{N}) , i.e.

f^{∗}(x, B) + f(x, Du) = ⟨Du, B⟩

for a.e. x ∈ Ω.

Proof. Observe that, by the extremality relations (4.3), we have that the sequence {B_{ε}} of the solutions of problems P_{ε}^{∗} is bounded in L^{q} log^{α} L(Ω′), for every α > 0 and every Ω′ compactly contained in Ω.

Then, there exists a divergence free vector field B such that

B_{ε} → B weakly in L^{q} log^{α} L(Ω′)

The definition of f_{ε} implies that

f^{∗}(x, η) ≤ \frac{1 + εδ + εk(x)}{1 + εδ}f_{ε}^{∗}(x, η) + c_{p} \frac{εk(x)}{1 + εδ} |η|^{q}  (5.1)

The lower semicontinuity of ∫_{Ω′} f^{∗}(x, η), (5.1) and the extremality relations (4.3), up to a not relabelled subsequence, yield

∫_{Ω′} f^{∗}(x, B)dx ≤ lim_{ε→0} ∫_{Ω′} f^{∗}(x, B_{ε})dx

≤ lim_{ε→0} ∫_{Ω′} (\frac{1 + εδ + εk(x)}{1 + εδ})f_{ε}^{∗}(x, B_{ε})dx + c_{p} ∫_{Ω′} (\frac{εk(x)}{1 + εδ} )|B_{ε}|^{q}
= lim_{ε→0} ∫_{Ω′} f_{ε}^{∗}(x, B_{ε})dx  (5.2)
= lim_{ε→0} ∫_{Ω′}(⟨Dh_{ε}, B_{ε}⟩ − f_{ε}(x, Dh_{ε})dx
= ∫_{Ω′}(⟨Du, B⟩ − f(x, Du))dx

where the last equality follows by compensated compactness theorem (see [18]). Inequality (5.2) implies

∫_{Ω′}(f^{∗}(x, B) + f(x, Du))dx ≤ ∫_{Ω′}⟨Du, B⟩

Opposite inequality follows by the definition of Young-Fenchel transform.

6 The continuity

When p > n − 1, Theorem 1 implies continuity of finite energy local minimizers of the variational integral (1.1). We begin by proving that every finite energy local minimizers of the variational integral (1.1) is weakly monotone in the sense of Manfredi (see [14]). To this aim let us give the following

Definition 6.1. A real valued function u ∈ W_{loc}^{1,p}(Ω) is said to be weakly monotone if for every relatively compact subdomain Ω′ of Ω and all constants m ≤ M such that

(m − u)^{+} ∈ W_{0}^{1,p}(Ω′)

and

(u − M)^{+} ∈ W_{0}^{1,p}(Ω′)

we have

m ≤ u(x) ≤ M.

Corollary 6.1. Let u be a finite energy local minimizer of integral (1.1). There exists λ_{0} = λ_{0}(n, p) such that if K(x) satisfies (1.3) for some λ ≥ λ_{0}, then u is weakly monotone.

Proof. Fix α > 0. From Theorem 1.1, there exists λ_{α} = λ_{α}(n, p) such that if K satisfies (1.3) for λ ≥ λ_{α} then Du ∈ L^{p} log^{α} L(Ω′) for every Ω′ ⊂⊂ Ω. Then, if m ≤ M are two constants such that (u − M)^{+} and (m − u)^{+} are in the space W_{0}^{1,p}(Ω′), we obtain

∫_{Ω′} f(x, Du)dx ≤ ∫_{Ω′} f(x, Du − D(u − M)^{+})dx = ∫_{Ω′∩{x: u(x)≤M}} f(x, Du)dx

This inequality yields

∫_{Ω′∩{x: u(x)>M}} f(x, Du)dx = 0

Then we have that Du(x) = 0 for a.e. x ∈ Ω′ ∩{x : u(x) > M}. This implies that u ≤ M for a.e. x ∈ Ω′. Arguing in the same way we prove that m ≤ u for a.e. x ∈ Ω′ and then the result.

Now, we are in position to apply a result of [9] to obtain continuity of local minimizer of integral (1.1), when n − 1 < p ≤ n. Namely, we have

Corollary 6.2. Let u be a finite energy local minimizer of integral (1.1). There exists λ_{0} = λ_{0}(n, p) such that if K(x) satisfies (1.3) for some λ ≥ λ_{0}, then u is locally bounded and continuous outside a set Z with H^{n−p}(Z) = 0. Moreover, u is differentiable a.e. .

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