Regularity for scalar integrals without structure conditions

Michela Eleuteri; Paolo Marcellini; Elvira Mascolo

doi:10.1515/acv-2017-0037

Open Access Published by De Gruyter March 16, 2018

Regularity for scalar integrals without structure conditions

Michela Eleuteri , Paolo Marcellini and Elvira Mascolo

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2017-0037

Abstract

Integrals of the Calculus of Variations with p,q-growth may have not smooth minimizers, not even bounded, for general p,q exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f⁢(x,ξ) with dependence on the modulus of the gradient, i.e. f⁢(x,ξ)=g⁢(x,|ξ|). Without imposing structure conditions, we prove that if qp is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.

Keywords: Elliptic equations; local minimizers; local Lipschitz continuity; general growthconditions

MSC 2010: 35J60; 35B65; 49N60; 35J70; 35B45

1 Introduction

The fundamental classical problem of the Calculus of Variations in the scalar case usually is formulated as finding a function u assuming a given value u0 at the boundary ∂⁡Ω of an open bounded set Ω⊂ℝn which minimizes the integral

(1.1)∫Ωf⁢(x,D⁢v)⁢𝑑x

among all functions v:Ω→ℝ, assuming the same boundary value u0 as u. The precise functional space where to look for solutions depends on the growth conditions of f=f⁢(x,ξ) as ξ∈ℝn grows in modulus to +∞. Usually this growth is stated in terms of an inequality of the type

(1.2)f⁢(x,ξ)≥M1⁢|ξ|p

for a.e. x∈Ω, ξ∈ℝn and for some positive constant M1. Here p=1 is associated to the BV⁢(Ω) space of functions with bounded variation, while p>1 is related to the Sobolev space W1,p⁢(Ω). Usually the condition p>1 and the strict convexity of f⁢(x,ξ) with respect to ξ are sufficient conditions for the existence and uniqueness of minimizers.

A different problem is the regularity of minimizers. A large literature is known about regularity (see for instance [26, 28, 30]) partly based on the nowadays classical well-known Hölder continuity result by De Giorgi [16]. To this aim it seems necessary to impose also a growth condition from above, to be associated to the growth condition from below in (1.2), of the type

(1.3)f⁢(x,ξ)≤M2⁢(1+|ξ|q)

for a.e. x∈Ω, ξ∈ℝn, for q≥p and for some positive constant M2. The so-called “natural growth conditions” appear if q=p, while the more general assumption q>p allows us to consider a much larger class of integrals of the Calculus of Variations, such as for example

(1.4)f⁢(ξ)=|ξ|p⁢log⁡(1+|ξ|)

(1.5)f⁢(x,ξ)=|ξ|p⁢(x) or f⁢(x,ξ)=(1+|ξ|2)p⁢(x)2.

We recall also the integrands recently considered in [12, 11, 20, 19], see also [2, 3, 4],

(1.6)f⁢(x,ξ)=a⁢(x)⁢|ξ|p+b⁢(x)⁢|ξ|q,

where a⁢(x),b⁢(x)≥0 and possibly zero on some part of Ω, being at least one of the two coefficients positive at almost every x∈Ω. The above examples (1.4), (1.5), (1.6) enter in the theory presented in this paper. However, here we study the more general case with f=f⁢(x,ξ) without a structure, i.e. not necessarily depending on the modulus of ξ of the type f⁢(x,ξ)=g⁢(x,|ξ|).

We assume that f:Ω×ℝn→[0,+∞) is a convex function with respect to the gradient variable and it is strictly convex only at infinity. More precisely, there exists M0>0 such that fξ⁢ξ, fξ⁢x are Carathéodory functions satisfying

(1.7){M1⁢|ξ|p-2⁢|λ|2≤∑i,jfξi⁢ξj⁢(x,ξ)⁢λi⁢λj,|fξ⁢ξ⁢(x,ξ)|≤M2⁢|ξ|q-2,|fξ⁢x⁢(x,ξ)|≤h⁢(x)⁢|ξ|p+q-22

for a.e. x∈Ω and for all λ,ξ∈ℝn, with |ξ|≥M0 and for positive constants M1,M2. Here 1<p≤q and h∈Lr⁢(Ω) for some r>n.

Model integrands satisfying condition (1.7) are, for instance, the function f⁢(x,ξ) in (1.6) and also

(1.8)f⁢(x,ξ)=|ξ|p+c⁢(x)⁢|ξ|s+|ξn|q,

ξn being the last component (or any other component) of the vector ξ=(ξ1,ξ2,…,ξn), when

s≤p+q2.

For instance, when s=p and q≥p, we are considering energy integrals with integrand of the type (we denote here a⁢(x)=1+c⁢(x) a generic positive coefficient)

(1.9)f⁢(x,ξ)=a⁢(x)⁢|ξ|p+|ξn|q.

Note that the cases (1.8) and (1.9) can be handled with Theorem 1.1. On the other hand example (1.10) below enters in Theorem 1.2:

(1.10)f⁢(x,ξ)=|ξ|p+b⁢(x)⁢|ξ|q.

The main regularity result that we prove here is the following a-priori estimate.

Theorem 1.1 (A-priori estimate).

Let u∈W1,p⁢(Ω) be a smooth local minimizer of the integral functional (1.1) with exponents p,q fulfilling

(1.11)qp<1+2⁢(1n-1r).

Under the growth assumption (1.7), there exist positive constants C,β,γ depending on n,r,p,q, M0,M1,M2 such that, for every 0<ρ<R≤ρ+1,

(1.12)∥D⁢u∥L∞⁢(Bρ;ℝn)≤C⁢(∥1+h∥Lr⁢(Ω)R-ρ)β⁢γ⁢(∫BR{1+|D⁢u|p}⁢𝑑x)γp.

Note that to get regularity of solutions it is natural, and also necessary, to assume that the gap q-p is small or that qp is close to 1, because of the known counterexamples [27, 31, 33].

The L∞-bound of the gradient is obtained through several steps. The first step of the a-priori estimate is Lemma 2.3 below, where on the right-hand side of the a-priori estimate there is the norm of the minimizer u in W1,q⁢m⁢(Ω) (m=rr-2), and the exponents p,q are related by the condition

(1.13)qp<1+2⁢nn-2⁢(1n-1r)

with n≥3. Note that if r=+∞ in (1.11) and (1.13), we recover the bounds in [33, Theorems 2.1 and 3.1]. An interpolation method allows us to obtain (1.12).

The mathematical literature on the regularity under p,q-growth is now very large; we refer to [35, 34, 33, 32] and to [36] for a complete survey on the subject. A new impulse to the subject has been given by the recent articles already cited [12, 11, 15, 20] for the case of elliptic equations and by [6, 7, 8] for the case of parabolic equations and systems under p,q-growth. We observe that here the ellipticity and growth assumptions hold only for large values of the gradient variable, i.e. we consider functionals which are uniformly convex only at infinity. In this context see [10, 25, 14] and recently [20, 19, 13]. The Sobolev dependence on x has recently been considered in [37, 1] and for obstacle problems in [21].

The previous a-priori estimate, more precisely Theorem 2.6, under assumptions (1.7) where the last condition is replaced by

(1.14)|fξ⁢x⁢(x,ξ)|≤h⁢(x)⁢|ξ|q-1

for a.e. x∈Ω, with |ξ|≥M0, allows us to obtain the following existence and regularity result.

Theorem 1.2 (Existence and regularity).

Assume that f satisfies (1.7) and (1.14) with 1<p≤q and

qp<1+1n-1r.

The Dirichlet problem min⁡{F⁢(u):u∈W01,p⁢(Ω)+u0}, with F defined in (1.15) below and u0∈W1,q⁢(Ω), has at least one locally Lipschitz continuous solution.

Here we emphasize the definition (i.e. the precise meaning) of the integral F⁢(u) to be minimized; in fact, the integral in (1.1) is well defined if u∈Wloc1,q⁢(Ω), due to the growth assumption in (1.3), but a-priori it is not uniquely defined if u∈W1,p⁢(Ω)∖Wloc1,q⁢(Ω). In this context of x-dependence, we cannot a-priori exclude the Lavrentiev phenomenon; however, note that in Section 5 we assume a special form of f to rule out this possibility.

For the gap in the Lavrentiev phenomenon we refer to [39, 9] and recently [24, 22, 23] for related results.

For the functional F we adopt the classical definition which refers to the pioneering research by Serrin [38] (see also [29]), which is related to the Γ-convergence theory by De Giorgi [17]. Precisely, for all u∈W1,p⁢(Ω),

(1.15)F⁢(u)=inf⁡{lim infk⁡∫Ωf⁢(x,D⁢uk)⁢𝑑x:uk∈W01,q⁢(Ω)+u0,uk⁢⇀𝑤⁢u⁢ in W1,p⁢(Ω)}.

We discuss more in details in Section 3 the definition of F in (1.15), while in Section 2 we give the proof of the a-priori estimate. Finally, in Section 4 we give the proof of Theorem 1.2.

2 A-priori estimates

Let us start with two technical lemmas.

Lemma 2.1.

The inequality

(2.1)(1+t)β≤cβ⁢(1+∫0t(1+s)β-2⁢s⁢𝑑s)

holds for every t∈[0,+∞) and every β∈(0,+∞), where

(2.2)cβ=β1-(1+β⁢(β-1))11-β

if β≠1, while (by continuity)

(2.3)c1=limβ→1⁡cβ=ee-1.

Proof.

In order to prove inequality (2.1) we first consider the case β=1.

Step 1 (β=1). We compute the integral on the right-hand side of (2.1):

∫0t(1+s)-1⁢s⁢𝑑s=∫0t(1-(1+s)-1)⁢𝑑s=t-log⁡(1+t)

and inequality (2.1) becomes

1+t≤c1⁢(1+t-log⁡(1+t)),

which is equivalent to

log⁡(1+t)1+t≤c1-1c1.

A computation shows that g(t)=:log⁡(1+t)1+t is positive for t∈(0,+∞) and has a maximum at t=e-1, thus

g(t)=:log⁡(1+t)1+t≤g(e-1)=1e;

with the position c1-1c1=:1e we find (2.3).

Step 2 (β≠1). We compute the integral on the right-hand side of (2.1) under the condition β≠1 and with the notation r=:1+s:

∫0t(1+s)β-2⁢s⁢𝑑s=∫1t+1rβ-2⁢(r-1)⁢𝑑r

=∫1t+1rβ-1⁢𝑑r-∫1t+1rβ-2⁢𝑑r

=[rββ]r=1r=t+1-[rβ-1β-1]r=1r=t+1

=(t+1)ββ-(t+1)β-1β-1+1β⁢(β-1).

Inequality (2.1) takes then the form

1cβ⁢(1+t)β≤1+(t+1)ββ-(t+1)β-1β-1+1β⁢(β-1).

We can write it equivalently

(2.4)g⁢(t)≤1+1β⁢(β-1),

where

g(t)=:(t+1)β-1β-1-(1β-1cβ)(t+1)β.

We can compute the maximum of g⁢(t) when t∈[0,+∞). We find that the derivative g′⁢(t) is equal to zero if t=βcβ-β and, since cβ>β, we obtain

max⁡{g⁢(t):t∈[0,+∞)}=g⁢(βcβ-β)=(cβcβ-β)β-1⁢1β⁢(β-1).

Therefore inequality (2.4) holds if we choose cβ to satisfy the condition

(cβcβ-β)β-1⁢1β⁢(β-1)=1+1β⁢(β-1).

A further computation gives for cβ the explicit expression in (2.2). Note that cβ→c1 as β→1. ∎

In the sequel we apply the previous lemma to get the a-priori estimates in particular to deal with the left-hand side of (2.26), with β=γ2+p2, for γ≥0; thus β≥p2. In the next result in fact we consider β∈[β0,+∞) for some fixed β0>0.

Lemma 2.2.

Let β0>0. There exist constants c′ and c′′, depending on β0 but independent of β≥β0 and of t≥0, such that

(2.5)(1+t)β≤c′⁢β2log⁡(1+β)⁢(1+∫0t(1+s)β-2⁢s⁢𝑑s),

(2.6)(1+t)β≤c′′⁢β2⁢(1+∫0t(1+s)β-2⁢s⁢𝑑s)

for every β∈[β0,+∞) and every t∈[0,+∞).

Proof.

First we show that the constant cβ is bounded independently of β≤1 if β∈[β0,1] (here we assume that β0∈(0,1), otherwise nothing to be proved at this step). Precisely, we show that

(2.7)cβ≤β1-e-β0 for all ⁢β∈[β0,1].

In fact, by the inequality log⁡t≤t-1, valid for all t>0, by posing t=1+β⁢(β-1) if β<1, we obtain

(1+β⁢(β-1))11-β=elog⁡(1+β⁢(β-1))1-β≤eβ⁢(β-1)1-β=e-β

and (2.7) follows if β∈[β0,1), since

cβ=β1-(1+β⁢(β-1))11-β≤β1-e-β≤β1-e-β0.

Finally, if β=1, then c1=ee-1<11-e-β0 holds, since it is equivalent to 1<e1-β0.

We now consider the case β>1. By Taylor’s formula we get

(1+β⁢(β-1))11-β=elog⁡(1+β⁢(β-1))1-β=1+log⁡(1+β⁢(β-1))1-β+o⁢(log⁡(1+β⁢(β-1))1-β)

and thus the quantity

cβ⁢log⁡(1+β)β2=log⁡(1+β)β⁢[log⁡(1+β⁢(β-1))β-1+o⁢(log⁡(1+β⁢(β-1))1-β)]

has a finite limit as β→+∞ (equal to 12) and it is a bounded function for β∈[1,+∞), let us say bounded by c′. This proves (2.5). The other inequality (2.6) is a direct consequence of (2.5). ∎

Let now Ω be an open bounded subset of ℝn for n≥2 and assume that f satisfies (1.7). We observe that we can transform f⁢(x,ξ) into f⁢(x,M0⁢ξ), which satisfies the same assumptions for |ξ|≥1 (with different constants depending on M0). Then it is sufficient to obtain the a-priori bound and the regularity results for v=1M0⁢u. Therefore, for clarity of exposition and without loss of generality, we assume M0=1. Throughout the paper we will denote by Bρ and BR balls of radii, respectively, ρ and R (with ρ<R) compactly contained in Ω and with the same center, let us say, x0∈Ω.

In this section we assume the following supplementary assumptions on f. Assume that f∈𝒞2⁢(Ω×ℝn) and there exist two positive constants k and K such that for all ξ∈ℝn and all x∈Ω,

(2.8){k⁢(1+|ξ|2)q-22⁢|λ|2≤∑i,jfξi⁢ξj⁢(x,ξ)⁢λi⁢λj,|fξ⁢ξ⁢(x,ξ)|≤K⁢(1+|ξ|2)q-22,|fξ⁢x⁢(x,ξ)|≤K⁢(1+|ξ|2)q-12.

In the next lemma, we obtain an a-priori estimate for the L∞-norm of the gradient of u which is independent of k and K.

Lemma 2.3.

Let u be a local minimizer of the integral functional (1.1) with f satisfying (1.7) and (2.8) with

(2.9)qp<1+2⁢αn-2 with ⁢α=1-nr

if n≥3 and p<q if n=2. Then there exists a positive constants C depending only on n,r,p,q,M1,M2 (depending also on |Ω| when n=2) such that

(2.10)∥D⁢u∥L∞⁢(Bρ;ℝn)≤C⁢[∥1+h∥Lr⁢(Ω)(R-ρ)]θ⁢β~⁢(∫BR{1+|D⁢u|q⁢m}⁢𝑑x)θq⁢m

for every 0<ρ<R≤ρ+1, where

(2.11)β~:=2*p⁢2*2-q⁢m,θ:=q⁢m⁢(2*2⁢m-1)p⁢2*2-q⁢m,m:=rr-2.

Remark 2.4.

We observe that

(2.12)1≤m:=rr-2<nn-2=2*2,since r>n;

the last inequality holds for n>2, while we set 2* equal to any fixed real number greater than 2 if n=2. Moreover, we also have

(2.13)12⁢m-12*=r-22⁢r-n-22⁢n=n⁢(r-2)-r⁢(n-2)2⁢n⁢r=r-nn⁢r=1n-1r=αn,

therefore (2.9) can be equivalently expressed as

(2.14)qp<2*2⁢m

because

1+2⁢αn-2=1+2*⁢αn⁢=(2.13)⁢1+2*⁢(12⁢m-12*)=2*2⁢m.

Therefore, due to (2.14), in (2.11) we have β~>0 and θ>1.

Remark 2.5.

The result obtained is sharp in the sense that if m=1 (r=+∞), then the relation between p and q reduces to the analogous one in [33, Theorem 2.1], i.e. qp<nn-2.

Proof.

Let u∈W1,q⁢(Ω) be a local minimizer of (1.1). Then u satisfies the Euler first variation

∫Ω∑i=1nfξi⁢(x,D⁢u)⁢φxi⁢(x)⁢d⁢x=0 for all ⁢φ∈W01,q⁢(Ω).

By (2.8), the technique of the difference quotients (see [30, 18], in particular [28, Chapter 8, Sections 8.1 and 8.2]) gives

(2.15)u∈Wloc1,∞⁢(Ω)∩Wloc2,min⁡(2,q)⁢(Ω) and (1+|D⁢u|2)q-22⁢|D2⁢u|2∈Lloc1⁢(Ω).

Let η∈C01⁢(Ω) and for any fixed s∈{1,…,n} define

φ=η2⁢uxs⁢Φ⁢((|D⁢u|-1)+)

for Φ:[0,+∞)→[0,+∞) increasing, locally Lipschitz continuous function, with Φ and Φ′ bounded on [0,+∞), such that Φ⁢(0)=Φ′⁢(0)=0 and

(2.16)Φ′⁢(s)⁢s≤cΦ⁢Φ⁢(s)

for a suitable constant cΦ≥1. Here (a)+ denotes the positive part of a∈ℝ; in the following we denote

Φ⁢((|D⁢u|-1)+)=Φ⁢(|D⁢u|-1)+.

We have then

(2.17)φxi=2⁢η⁢ηxi⁢uxs⁢Φ⁢(|D⁢u|-1)++η2⁢uxs⁢xi⁢Φ⁢(|D⁢u|-1)++η2⁢uxs⁢Φ′⁢(|D⁢u|-1)+⁢[(|D⁢u|-1)+]xi.

Let q≥2. By (2.15) we have that |D2⁢u|2∈Lloc1⁢(Ω) . Otherwise if 1<q<2, we use the fact that u∈Wloc1,∞⁢(Ω) to infer that there exists M=M⁢(supp⁡φ) such that

|D⁢u⁢(x)|≤M for a.e. ⁢x∈supp⁡φ.

Now since q-2<0, we have

(1+M2)q-22⁢|D2⁢u|2≤(1+|D⁢u|2)q-22⁢|D2⁢u|2,

and by (2.15) we again get |D2⁢u|2∈L1⁢(supp⁡φ). Therefore we can insert φxi in the following second variation,

∫Ω{∑i,j=1nfξi⁢ξj⁢(x,D⁢u)⁢uxj⁢xs⁢φxi+∑i=1nfξi⁢xs⁢(x,D⁢u)⁢φxi}⁢𝑑x=0 for all ⁢s=1,…,n,

and we obtain

0=∑s[∫Ω2ηΦ(|Du|-1)+∑i,jfξi⁢ξj(x,Du)ηxiuxsuxs⁢xjdx

+∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢∑i,jfξi⁢ξj⁢(x,D⁢u)⁢uxs⁢xi⁢uxs⁢xj⁢d⁢x

+∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢∑i,jfξi⁢ξj⁢(x,D⁢u)⁢uxs⁢uxs⁢xj⁢[(|D⁢u|-1)+]xi⁢d⁢x

+∫Ω2⁢η⁢Φ⁢(|D⁢u|-1)+⁢∑ifξi⁢xs⁢(x,D⁢u)⁢ηxi⁢uxs⁢d⁢x

+∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢∑ifξi⁢xs⁢(x,D⁢u)⁢uxs⁢xi⁢d⁢x

+∫Ωη2Φ′(|Du|-1)+∑ifξi⁢xs(x,Du)uxs[(|Du|-1)+]xidx]

(2.18)=:∑s(I1s+I2s+I3s+I4s+I5s+I6s).

In the following, constants will be denoted by C, regardless of their actual value.

Let us start with the estimate of the first integral in (2.18). By the Cauchy–Schwarz inequality, the Young inequality and (1.7), we have

|∑sI1s|=|∫Ω2ηΦ(|Du|-1)+∑i,j,sfξi⁢ξj(x,Du)ηxiuxsuxs⁢xjdx|

≤∫Ω2⁢η⁢Φ⁢(|D⁢u|-1)+⁢{∑i,j,sfξi⁢ξj⁢(x,D⁢u)⁢ηxi⁢uxs⁢ηxj⁢uxs}12⁢{∑i,j,sfξi⁢ξj⁢(x,D⁢u)⁢uxs⁢xi⁢uxs⁢xj}12⁢𝑑x

≤C⁢∫Ω|D⁢η|2⁢Φ⁢(|D⁢u|-1)+⁢|D⁢u|q⁢𝑑x+12⁢∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢∑i,j,sfξi⁢ξj⁢(x,D⁢u)⁢uxs⁢xi⁢uxs⁢xj⁢d⁢x.

Let us consider the third integral in (2.18). First of all we observe that

[(|D⁢u|-1)+]xi⁢∑suxs⁢uxs⁢xj=[(|D⁢u|-1)+]xi⁢|D⁢u|⁢[(|D⁢u|-1)+]xj.

This entails using (1.7)

∑sI3s=∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢∑i,j,sfξi⁢ξj⁢(x,D⁢u)⁢uxs⁢uxs⁢xj⁢[(|D⁢u-1|+)]xi⁢d⁢x

≥M1⁢∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢|D⁢u|p-1⁢|D⁢(|D⁢u|-1)+|2⁢𝑑x≥0.

We now deal with the fourth integral in (2.18). We have

|∑sI4s|=|∫Ω2ηΦ(|Du|-1)+∑i,sfξi⁢xs(x,Du)ηxiuxsdx|

≤(1.7)⁢∫Ω2⁢η⁢Φ⁢(|D⁢u|-1)+⁢h⁢(x)⁢|D⁢u|p+q-22⁢∑i,s|ηxi⁢uxs|⁢d⁢x

≤C⁢∫Ω(η2+|D⁢η|2)⁢h⁢(x)⁢Φ⁢(|D⁢u|-1)+⁢|D⁢u|q⁢𝑑x.

Consider now the fifth integral in (2.18). We have

|∑sI5s|=|∫Ωη2Φ(|Du|-1)+∑i,sfξi⁢xs(x,Du)uxs⁢xjdx|

≤(1.7)⁢∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢h⁢(x)⁢|D⁢u|p+q-22⁢|D2⁢u|⁢𝑑x

≤∫Ω[η2⁢Φ⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2]12⁢[η2⁢Φ⁢(|D⁢u|-1)+⁢|h⁢(x)|2⁢|D⁢u|q]12⁢𝑑x

≤ε⁢∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x+Cε⁢∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢|h⁢(x)|2⁢|D⁢u|q⁢𝑑x,

where in the last line we used the Young inequality. Finally, for any 0<δ<1,

|∑sI6s|=|∫Ωη2∑i,sfξi⁢xs(x,Du)uxsΦ′(|Du|-1)+[(|Du|-1)+]xidx|

≤(1.7)⁢∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢h⁢(x)⁢|D⁢u|p+q-22⁢|D⁢u|⁢|D⁢(|D⁢u|-1)+|⁢𝑑x

≤∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢h⁢(x)⁢|D⁢u|p+q2⁢|D2⁢u|⁢𝑑x

=∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢h⁢(x)⁢[(|D⁢u|-1)++δ]⁢[(|D⁢u|-1)++δ]-1⁢|D⁢u|p+q2⁢|D2⁢u|⁢𝑑x

≤∫Ωη2⁢{1cΦ⁢Φ′⁢(|D⁢u|-1)+⁢[(|D⁢u|-1)++δ]⁢|D⁢u|p-2⁢|D2⁢u|2}12

×{cΦ⁢Φ′⁢(|D⁢u|-1)+⁢|h⁢(x)|2⁢|D⁢u|q+2⁢[(|D⁢u|-1)++δ]-1}12⁢d⁢x

≤Cε⁢cΦ⁢∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢|h⁢(x)|2⁢|D⁢u|q+2⁢[(|D⁢u|-1)++δ]-1⁢𝑑x

+εcΦ⁢∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢[(|D⁢u|-1)++δ]⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x.

Since Ω={x:|D⁢u⁢(x)|≥2}∪{x:|D⁢u⁢(x)|<2} and (|D⁢u|-1)+≥1 in {x:|D⁢u⁢(x)|≥2}, we also have

(2.19)(|D⁢u|-1)++δ≤2⁢(|D⁢u|-1)+

as long as we have chosen δ<1. Therefore, using (2.16), we can estimate the last integral as

∫Ωη2⁢Φ′⁢(|D⁢u|-1)+⁢[(|D⁢u|-1)++δ]⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

=∫|D⁢u|≥2η2⁢Φ′⁢(|D⁢u|-1)+⁢[(|D⁢u|-1)++δ]⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

+∫1<|D⁢u|<2η2⁢Φ′⁢(|D⁢u|-1)+⁢[(|D⁢u|-1)++δ]⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

⁢≤(2.19)⁢2⁢∫|D⁢u|≥2η2⁢Φ′⁢(|D⁢u|-1)+⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

+∫1<|D⁢u|<2η2⁢Φ′⁢(|D⁢u|-1)+⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

+δ⁢∫1<|D⁢u|<2η2⁢Φ′⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

⁢≤(2.16)⁢2⁢cΦ⁢∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x+δ⁢∫1<|D⁢u|<2η2⁢Φ′⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x.

Therefore we finally have

|∑sI6s|≤CεcΦ∫Ωη2Φ′(|Du|-1)+|h(x)|2|Du|q+2[(|Du|-1)++δ]-1dx

+2⁢ε⁢∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x+δ⁢ε⁢∫1<|D⁢u|<2η2⁢Φ′⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x.

Now, for ε sufficiently small and putting together all the previous estimates, we deduce that there exists a constant C depending on n,r,p,q,M1 such that

∫Ωη2⁢Φ⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

≤C⁢cΦ⁢∫Ω(η2+|D⁢η|2)⁢(1+h⁢(x))2⁢|D⁢u|q⁢[Φ⁢(|D⁢u|-1)++Φ′⁢(|D⁢u|-1)+⁢|D⁢u|2⁢[(|D⁢u|-1)++δ]-1]⁢𝑑x

(2.20)+δ⁢∫1<|D⁢u|<2η2⁢Φ′⁢(|D⁢u|-1)+⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x.

Let us now set

Φ⁢(s):=(1+s)γ-2⁢s2,γ≥0,

with

Φ′⁢(s)=(γ⁢s+2)⁢s⁢(1+s)γ-3.

It is easy to check that Φ satisfies (2.16) with cΦ=2⁢(1+γ).

We now approximate this function Φ by a sequence of functions Φh, each of them being equal to Φ in the interval [0,h], and then extended to [h,+∞) with the constant value Φ⁢(h). Since Φh and Φh′ converge monotonically to Φ and Φ′, by inserting Φh in (2.20), it is possible to pass to the limit as h→+∞ by the Monotone Convergence Theorem.

Therefore, for every 0<δ<1, since

(|D⁢u|-1)+(|D⁢u|-1)++δ≤1 for all ⁢δ>0

and Φ′⁢(t-1)+≤C⁢(γ) when 1<t<2, we obtain

∫Ωη2⁢(1+(|D⁢u|-1)+)γ-2⁢(|D⁢u|-1)+2⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x

(2.21) ≤C⁢(1+γ)2⁢∫Ω(η2+|D⁢η|2)⁢(1+h⁢(x))2⁢(1+(|D⁢u|-1)+)γ+q⁢𝑑x+δ⁢C⁢(γ)⁢∫1<|D⁢u|<2η2⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x.

Using [20, formula (3.51) of Lemma 3.3], namely the fact that |D⁢u|p-2≤C⁢(p)⁢(1+|D⁢u|2)p-22 when |D⁢u|>1, we have

∫1<|D⁢u|<2η2⁢|D⁢u|p-2⁢|D2⁢u|2⁢𝑑x≤C⁢∫1<|D⁢u|<2η2⁢(1+|D⁢u|2)p-22⁢|D2⁢u|2⁢𝑑x<+∞

by (2.15), and for δ→0 and the last term in the previous inequality vanishes.

Since h∈Lr⁢(Ω), by the Hölder inequality, since 1m+2r=1, by (2.21) we have

∫Ωη2⁢(1+(|D⁢u|-1)+)γ-2⁢(|D⁢u|-1)+2⁢|D⁢u|p-2⁢|D⁢((|D⁢u|-1)+)|2⁢𝑑x

(2.22) ≤C⁢(1+γ)2⁢∥1+h∥Lr⁢(Ω)2⁢[∫Ω(η2+|D⁢η|2)m⁢(1+(|D⁢u|-1)+)(γ+q)⁢m⁢𝑑x]1m.

Let us introduce

(2.23)G⁢(t)=1+∫0t(1+s)γ2+p2-2⁢s⁢𝑑s.

We obtain

(2.24)[G⁢(t)]2≤4⁢(1+t)γ+p≤4⁢(1+t)γ+q,

where we used the fact that p≤q. On the other hand

(2.25)G′⁢(t)=(1+t)γ2+p2-2⁢t,

which in turn allows us to give the following estimate for the gradient of the function w=η⁢G⁢((|D⁢u|-1)+):

∫Ω|D⁢(η⁢G⁢((|D⁢u|-1)+))|2⁢𝑑x

≤2⁢∫Ω|D⁢η|2⁢|G⁢((|D⁢u|-1)+)|2⁢𝑑x+2⁢∫Ωη2⁢[Gt⁢((|D⁢u|-1)+)]2⁢[D⁢((|D⁢u|-1)+)]2⁢𝑑x

≤C⁢(1+γ)2⁢∥1+h∥Lr⁢(Ω)2⁢[∫Ω(η2+|D⁢η|2)m⁢[1+(|D⁢u|-1)+](γ+q)⁢m⁢𝑑x]1m,

the second inequality by (2.22), (2.24), (2.25). By Sobolev’s inequality there exists a constant C (depending also on |Ω| when n=2) such that

{∫Ω[η⁢G⁢((|D⁢u|-1)+)]2∗⁢𝑑x}22∗≤C⁢∫Ω|D⁢(η⁢G⁢((|D⁢u|-1)+))|2⁢𝑑x

and by the previous inequality we get (for a different constant)

(2.26){∫Ω[η⁢G⁢((|D⁢u|-1)+)]2∗⁢𝑑x}22∗≤C⁢(1+γ)2⁢∥1+h∥Lr⁢(Ω)2⁢[∫Ω(η2+|D⁢η|2)m⁢[1+(|D⁢u|-1)+](γ+q)⁢m⁢𝑑x]1m.

We take into account the definition of G⁢(t) in (2.23) and we use Lemma 2.2, and in particular formula (2.6) with β=γ+p2. Being γ≥0, we have β≥β0:=p2>0 and

(1+t)γ+p2≤c′′⁢(γ+p2)2⁢(1+∫0t(1+s)γ+p2-2⁢s⁢𝑑s)

for every γ≥0 and every t∈[0,+∞). In terms of G⁢(t)=1+∫0t(1+s)γ2+p2-2⁢s⁢𝑑s equivalently

(1+t)γ+p2≤c′′⁢(γ+p2)2⁢G⁢(t) for all ⁢γ≥0⁢ and all ⁢t≥0.

Therefore, if t:=(|D⁢u|-1)+,

(1+(|D⁢u|-1)+)γ+p2⁢2∗≤(c′′)2∗⁢(γ+p2)2⋅2∗⁢[G⁢((|D⁢u|-1)+)]2∗ for all ⁢γ≥0,

and by (2.26) we finally get

{∫Ωη2∗⁢[1+(|D⁢u|-1)+]γ+p2⁢2∗⁢𝑑x}22∗≤(c′′)2⁢(γ+p2)4⁢{∫Ω[η⁢G⁢((|D⁢u|-1)+)]2∗⁢𝑑x}22∗

≤C⁢(γ+1)6⁢∥1+h∥Lr⁢(Ω)2⁢[∫Ω(η2+|D⁢η|2)m⁢[1+(|D⁢u|-1)+](γ+q)⁢m⁢𝑑x]1m

with a new constant C and for every γ≥0.

As usual we consider a test function η equal to 1 in a ball Bρ, with supp⁡η⊂BR and such that |D⁢η|≤2(R-ρ). We get

(2.27)[∫Bρ[1+(|D⁢u|-1)+][(γ+p)⁢m]⁢2∗2⁢m⁢𝑑x]2⁢m2∗≤C0⁢∥1+h∥Lr⁢(Ω)2⁢m⁢(γ+q)6⁢m(R-ρ)2⁢m⁢∫BR[1+(|D⁢u|-1)+](γ+q)⁢m⁢𝑑x,

where the constant C0 only depends on n,r,p,q,M1,M2 but is independent of γ.

Fixed 0<ρ0<R0≤ρ0+1, we define the following decreasing sequence of radii {ρk}k≥1:

ρk=ρ0+R0-ρ02k for all ⁢k≥1.

We define recursively a sequence αk in the following way:

(2.28)α1:=0,αk+1:=(αk+p⁢m)⁢2*2⁢m-q⁢m.

Then we have the following representation formula for αk which can easily be proved by induction:

(2.29)αk=(p⁢2*2-q⁢m)⁢[(2*2⁢m)k-1-1]2*2⁢m-1.

We rewrite (2.27) with R=ρk, ρ=ρk+1, γ=αkm and observe that

R-ρ:=ρk-ρk+1=R0-ρ02k+1.

Set, for all k≥1,

Ak:=(∫Bρk[1+(|D⁢u|-1)+]αk+q⁢m⁢𝑑x)1αk+q⁢m,

Ck:=C0⁢∥1+h∥Lr⁢(Ω)2⁢m⁢((αk+q⁢m)3⁢2k+1R0-ρ0)2⁢m;

we obtain for every k≥1,

(2.30)Ak+1≤Ck1αk+p⁢m⁢Akαk+q⁢mαk+p⁢m.

Let θ be defined by

(2.31)θ:=∏i=1∞αi+q⁢mαi+p⁢m.

We show that θ is finite and is given by

θ=q⁢m⁢(2*2⁢m-1)p⁢2*2-q⁢m.

Indeed by (2.31), the recursive definition of αk, i.e. (2.28) and the representation formula for αk, namely (2.29), we have

θk:=∏i=1kαi+q⁢mαi+p⁢m⁢=(2.28)⁢q⁢mαk+p⁢m⁢(2*2⁢m)k-1⁢=(2.29)⁢q⁢m⁢(2*2⁢m)k-1(p⁢2*2-q⁢m)⁢[(2*2⁢m)k-1-1]2*2⁢m-1+p⁢m

=q⁢m⁢(2*2⁢m)k-1⁢(2*2⁢m-1)p⁢m⁢(2*2⁢m-1)+(p⁢2*2-q⁢m)⁢[(2*2⁢m)k-1-1],

which yields (2.31) once we pass to the limit as k→∞ as long as 2*2⁢m>1 in view of (2.12). Note that θ makes sense due to (2.12) and the bound (2.14) in Remark 2.4.

Iterating (2.30), we deduce

(2.32)Ak+1≤C~⁢([∥1+h∥Lr⁢(Ω)(R0-ρ0)]β~⁢A1)θk,k≥1,

where

C~:=C0β~⁢θ⁢exp⁡[θ⁢∑i=1∞log⁡[(αi+q⁢m)6⁢m⁢22⁢m⁢(i+1)]αi+p⁢m]<+∞,

which is finite because the series is convergent (αi from the representation formula (2.29) grows exponentially) and

∑i=1∞2⁢mαi+p⁢m⁢⁢=(2.29)⁢∑i=1∞2⁢m(p⁢2*2-q⁢m)⁢[(2*2)i-1-1]2*2⁢m-1+p⁢m≤∑i=1∞2⁢m(p⁢2*2-q⁢m)⁢(2*2⁢m)i-12*2⁢m-1

≤2⁢m⁢(2*2⁢m-1)p⁢2*2-q⁢m∑i=0∞(2⁢m2*)i=2⁢m⁢(2*2⁢m-1)p⁢2*2-q⁢m1(1-2⁢m2*)=2*p⁢2*2-q⁢m=:β~,

where in the first inequality we used the fact that

(p⁢2*2-q⁢m)⁢[(2*2⁢m)i-1-1]2*2⁢m-1+p⁢m≥(p⁢2*2-q⁢m)⁢(2*2⁢m)i-12*2⁢m-1⇔-p⁢2*2-q⁢m2*2⁢m-1+p⁢m≥0⇔q≥p.

By letting k→+∞ in (2.32), we have (2.10). Therefore, the proof of Lemma 2.3 is complete. ∎

The a-priori estimate (1.12) in Theorem 1.1 follows by the classical interpolation inequality

(2.33)∥v∥Ls⁢(Bρ)≤∥v∥Lp⁢(Bρ)ps⁢∥v∥L∞⁢(Bρ)1-ps

for any s≥p, which permits to estimate the essential supremum of the gradient of the local minimizer in terms of its Lp-norm.

Proof of Theorem 1.1.

Let us set

V⁢(x):=1+(|D⁢u|⁢(x)-1)+

then estimate (2.10) becomes

(2.34)supx∈Bρ⁡|V⁢(x)|≤C⁢([∥1+h∥Lr⁢(Ω)R-ρ]β~⁢∥V∥Lq⁢m⁢(BR))θ

for every ρ,R such that 0<ρ<R≤ρ+1 and where C=C⁢(n,r,p,q,M1,M2).

In the following we denote

s:=q⁢m.

At this point, (2.34) and (2.33) give

(2.35)∥V∥Ls⁢(Bρ)≤C1-ps⁢∥V∥Lp⁢(Bρ)ps⁢([∥1+h∥Lr⁢(Ω)R-ρ]β~⁢∥V∥Ls⁢(BR))θ⁢(1-pq⁢m).

We observe that

(2.36)τ:=θ⁢(1-pq⁢m)<1,

because

θ⁢(1-pq⁢m)<1⇔q⁢2*2-q⁢m-p⁢2*2⁢m+pp⁢2*2-q⁢m<1⇔q<p⁢(1-22*+1m)⁢=(2.13)⁢p⁢(1+2⁢αn).

For 0<ρ<R and for every k≥0, let us define ρk:=R-(R-ρ)⁢2-k. By inserting in (2.35) ρ=ρk and R=ρk+1, (so that R-ρ=(R-ρ)⁢2-(k+1)) we have, for every k≥0,

(2.37)∥V∥Ls⁢(Bρk)≤C1-ps⁢∥V∥Lp⁢(Bρk)ps⁢(2β~⁢(k+1)⁢[∥1+h∥Lr⁢(Ω)(R-ρ)]β~⁢∥V∥Ls⁢(Bρk+1))τ.

By iteration of (2.37), we deduce for k≥0,

(2.38)∥V∥Ls⁢(Bρ0)≤(C1-ps⁢[∥1+h∥Lr⁢(Ω)(R-ρ)]β~⁢τ⁢∥V∥Lp⁢(Bρk)ps)∑i=0kτi⁢2β~⁢∑i=0k+1i⁢τi⁢(∥V∥Ls⁢(Bρk+1))τk+1.

By (2.36), the series appearing in (2.38) are convergent. Since

∥V∥Ls⁢(Bρk)≤∥V∥Ls⁢(BR),

we can pass to the limit as k→+∞ and we obtain for every 0<ρ<R with a constant C=C⁢(n,r,p,q,M1,M2) independent of k,

(2.39)∥V∥Ls⁢(Bρ)≤C⁢([∥1+h∥Lr⁢(Ω)(R-ρ)]β~⁢τ⁢∥V∥Lp⁢(BR)ps)11-τ.

Combining (2.34) and (2.39), by setting ρ′=(R+ρ)2 we have

∥V∥L∞⁢(Bρ)≤C⁢([∥1+h∥Lr⁢(Ω)(ρ′-ρ)]β~⁢∥V∥Ls⁢(Bρ′))θ

≤C⁢([∥1+h∥Lr⁢(Ω)(ρ′-ρ)]β~⁢(1-τ)⁢[1+∥h∥Lr⁢(Ω)(R-ρ′)]β~⁢τ⁢∥V∥Lp⁢(BR)ps)θ1-τ;

now, since

(ρ′-ρ)=(R-ρ′)=R-ρ2,

we get

∥D⁢u∥L∞⁢(Bρ;ℝn)≤C⁢([∥1+h∥Lr⁢(Ω)(R-ρ)]β⁢(∫BR{1+|D⁢u|p}⁢𝑑x)1p)γ,

where

β:=β~⁢q⁢mp=2*⁢qp⁢mp⁢2*2-q⁢m,γ:=θ⁢pq⁢m⁢(1-θ⁢(1-pq⁢m)),θ:=q⁢m⁢(2*2⁢m-1)p⁢2*2-q⁢m,

so (1.12) follows. ∎

Let now f satisfy (1.7) and (1.14). Under these assumptions on f, we obtain the following result.

Theorem 2.6.

Let u∈W1,p⁢(Ω) be a local minimizer of the integral functional (1.1). Assume that f=f⁢(x,ξ) in (1.1) satisfies (1.7), (1.14) and (2.8), with

(2.40)qp<1+1n-1r.

Then there exist positive constants C,β^,γ^ depending on n,r,p,q,M1,M2,ρ,R such that

(2.41)∥D⁢u∥L∞⁢(Bρ;ℝn)≤C⁢[∥1+h∥Lr⁢(Ω)]β^⁢γ^⁢(∫BR{1+f⁢(x,D⁢u)}⁢𝑑x)γ^p

for every 0<ρ<R≤ρ+1.

Proof.

Let t:=2⁢q-p. Then (1.14) can be written explicitly in the form

|fξ⁢x⁢(x,ξ)|≤h⁢(x)⁢|ξ|t+p-22,|ξ|≥1.

Moreover, (2.40) in terms of p and t is equivalent to

tp<1+2⁢αn with ⁢α=1-nr.

Thus all the assumptions of Theorem 1.1 are satisfied with q replaced by t. In particular, the second inequality in (1.7) holds with q replaced by t since t≥q. Then the conclusion of Theorem 1.1 holds with q=t which corresponds to (2.41) with

β^:=2*p⁢(2⁢q-p)⁢mp⁢2*2-(2⁢q-p)⁢m,γ^:=θ⁢p(2⁢q-p)⁢m⁢(1-θ)+p⁢θ,

since f⁢(x,ξ)≥C⁢|ξ|p for every |ξ|≥1. ∎

3 Extension of the integral energy

Let f:Ω×ℝn→[0,+∞) be a continuous function, convex in ξ such that

(3.1)|ξ|p≤f⁢(x,ξ)≤C⁢(1+|ξ|q) for a.e. x∈Ω and all ξ∈ℝn.

For u0∈W1,q⁢(Ω), we define the extension to W1,p⁢(Ω) of the integral functional ∫Ωf⁢(x,D⁢u)⁢𝑑x, i.e.

(3.2)F⁢(u)=inf⁡{lim infk⁡∫Ωf⁢(x,D⁢uk)⁢𝑑x:uk∈W01,q⁢(Ω)+u0,uk⁢⇀𝑤⁢u⁢ in W1,p⁢(Ω)}

with

∫Ωf⁢(x,D⁢u0)⁢𝑑x<+∞.

It is easy to check that

F⁢(u)=∫Ωf⁢(x,D⁢u)⁢𝑑x for u∈W01,q⁢(Ω)+u0.

In fact, for uk=u for all k,

F⁢(u)≤∫Ωf⁢(x,D⁢u)⁢𝑑x.

On the other hand, by the semicontinuity of ∫Ωf⁢(x,D⁢u)⁢𝑑x with respect to the weak topology of W1,p, the inverse inequality also holds.

Lemma 3.1.

For each v∈W01,p⁢(Ω)+u0, there exists a sequence vk∈W01,q⁢(Ω)+u0 such that vk⇀v weakly in W1,p⁢(Ω) and

F⁢(v)=limk→+∞⁡∫Ωf⁢(x,D⁢vk)⁢𝑑x.

Proof.

The proof follows similarly as in [5]. We give the sketch of the proof.

Let v∈W01,p⁢(Ω)+u0 such that F⁢(v)<∞. Then, for all k, there exists vh(k)∈W01,q⁢(Ω)+u0 such that vh(k)⁢⇀𝑤⁢v, as h→+∞, weakly in W1,p⁢(Ω) and

F⁢(v)≤limh→+∞⁡∫Ωf⁢(x,D⁢vh(k))⁢𝑑x≤F⁢(v)+1k.

Moreover, by the weak convergence of vh(k) in W1,p⁢(Ω) we get

limh→+∞⁡∥vh(k)-v∥Lp⁢(Ω)=0

and for h sufficiently large,

∫Ω|D⁢vh(k)|p⁢𝑑x≤∫Ωf⁢(x,D⁢vh(k))⁢𝑑x≤F⁢(v)+1.

Then for all k there exists hk such that for all h≥hk,

∥vh(k)-v∥Lp⁢(Ω)<1k

and for h=hk, by denoting wk=vhk(k), we have

∥wk-v∥Lp⁢(Ω)<1k and ∫Ω|D⁢wk|p⁢𝑑x≤C;

then wk⁢⇀𝑤⁢v as k→+∞ in the weak topology of W1,p⁢(Ω) and

F⁢(v)≤∫Ωf⁢(x,D⁢wk)⁢𝑑x≤F⁢(v)+1k,

i.e.

limk→+∞⁡∫Ωf⁢(x,D⁢wk)⁢𝑑x=F⁢(v).∎

4 Existence and regularity

First of all we prove an approximation theorem for f through a suitable sequence of regular functions.

Proposition 4.1.

Let f be satisfying the growth conditions (3.1), fξ⁢ξ and fξ⁢x Carathéodory functions, satisfying (1.7) and (1.14) with M0=1 and f strictly convex at infinity. Then there exists a sequence of C2-functions fℓ⁢k:Ω×Rn→[0,+∞), fℓ⁢k convex in the last variable and strictly convex at infinity, such that fℓ⁢k converges to f as ℓ→∞ and k→∞ for a.e. x∈Ω, for all ξ∈Rn and uniformly in Ω0×K, where Ω0⊂⊂Ω and K being a compact set of Rn. Moreover:

there exists C~, independently of k,ℓ, such that
(4.1)|ξ|p≤fℓ⁢k⁢(x,ξ)≤C~⁢(1+|ξ|q) for a.e. x∈Ω and all ξ∈ℝn,
there exists M~1>0 such that for |ξ|>2 and a.e. x∈Ω,
(4.2)M~1⁢|ξ|p-2⁢|λ|2≤∑i,jfξi⁢ξjℓ⁢k⁢(x,ξ)⁢λi⁢λj,λ∈ℝn,
there exists c⁢(k)>0 such that for all (x,ξ)∈Ω×ℝn and λ∈ℝn,
(4.3)c⁢(k)⁢(1+|ξ|2)q-22⁢|λ|2≤∑i,jfξi⁢ξjℓ⁢k⁢(x,ξ)⁢λi⁢λj,
there exists M~2>0 such that for |ξ|>2 and a.e. x∈Ω,
(4.4)|fξ⁢ξℓ⁢k⁢(x,ξ)|≤M~⁢|ξ|q-2,
there exists C⁢(k) such that for a.e. x∈Ω and ξ∈ℝn,
(4.5)|fξ⁢ξℓ⁢k⁢(x,ξ)|≤C⁢(k)⁢(1+|ξ|2)q-22,
there exists a constant C>0 such that for a.e. x∈Ω and |ξ|>2,
(4.6)|fξ⁢xℓ⁢k⁢(x,ξ)|≤C⁢hℓ⁢(x)⁢|ξ|q-1,
where hℓ∈𝒞∞⁢(Ω) is the regularized function of h which converges to h in Lr⁢(Ω),
for Ω0⊂⊂Ω, there exists a constant C such that for x∈Ω0 and ξ∈ℝn,
(4.7)|fξ⁢xℓ⁢k⁢(x,ξ)|≤C⁢(k,ℓ,Ω0)⁢(1+|ξ|2)q-12.

Proof.

We argue as in the proof of [25, Theorem 2.7 (Step 3)] and [20, Lemma 4.3]. For the sake of completeness, we give a sketch of the arguments of the proof.

Let B be the unit ball of ℝn centered in the origin and consider a positive decreasing sequence εℓ→0. We introduce

fℓ⁢(x,ξ)=∫B×Bρ⁢(y)⁢ρ⁢(η)⁢f⁢(x+εℓ⁢y,ξ+εℓ⁢η)⁢𝑑η⁢𝑑y,

where ρ is a positive symmetric mollifier, and

(4.8)fℓ⁢k⁢(x,ξ)=fℓ⁢(x,ξ)+1k⁢(1+|ξ|2)q2.

It is easy to check that the sequence fℓ⁢k satisfies conditions (4.1), (4.2), (4.3), (4.4), (4.5), (4.7). Let us verify (4.6). For |ξ|>2 we have

where

hℓ⁢(x)=∫Bρ⁢(y)⁢h⁢(x+εℓ⁢y)⁢𝑑y,

hℓ is a smooth function and it converges to h in Lr⁢(Ω). Moreover,

|fξ⁢xℓ⁢k⁢(x,ξ)|≤C⁢(k,Ω0)⁢[∥1+hℓ∥L∞⁢(Ω0)]⁢(1+|ξ|2)q-12.

This concludes the proof. ∎

Proof of Theorem 1.2.

For u0∈W1,q⁢(Ω), let us consider the variational problems

(4.9)inf⁡{∫Ωfℓ⁢k⁢(x,D⁢v)⁢𝑑x:v∈W01,q⁢(Ω)+u0},

where fℓ⁢k are defined in (4.8). By semicontinuity arguments, there exists vℓ⁢k∈u0+W01,q⁢(Ω), a solution to (4.9). By the growth conditions and the minimality of vℓ⁢k, we get

∫Ω|D⁢vℓ⁢k|p⁢𝑑x≤∫Ωfℓ⁢k⁢(x,D⁢vℓ⁢k)⁢𝑑x

≤∫Ωfℓ⁢k⁢(x,D⁢u0)⁢𝑑x

=∫Ωfℓ⁢(x,D⁢u0)⁢𝑑x+1k⁢∫Ω(1+|D⁢u0|2)q2⁢𝑑x.

Moreover, the properties of the convolutions imply that

fℓ⁢(x,D⁢u0)→ℓ→∞f⁢(x,D⁢u0) a.e. in Ω,

and since

∫Ωfℓ⁢(x,D⁢u0)⁢𝑑x≤C⁢∫Ω(1+|D⁢u0|2)q2⁢𝑑x,

by the Lebesgue Dominated Convergence Theorem we deduce therefore

limℓ→∞⁡∫Ω|D⁢vℓ⁢k|p⁢𝑑x≤limℓ→∞⁡∫Ωfℓ⁢(x,D⁢u0)⁢𝑑x+1k⁢∫Ω(1+|D⁢u0|2)q2⁢𝑑x

=∫Ωf⁢(x,D⁢u0)⁢𝑑x+1k⁢∫Ω(1+|D⁢u0|2)q2⁢𝑑x.

By Proposition 4.1, the functions fℓ⁢k satisfy (1.7), (1.14) and (2.8), so we can apply the a-priori estimate (2.41) to vℓ⁢k and obtain, by standard covering arguments for all Ω′⊂⊂Ω,

∥D⁢vℓ⁢k∥L∞⁢(Ω′;ℝn)≤C⁢(Ω′)⁢[∥1+hℓ∥Lr⁢(Ω)]β^⁢γ^⁢[∫Ω(1+fℓ⁢k⁢(x,D⁢vℓ⁢k))⁢𝑑x]γ^p.

Since ∥1+hℓ∥Lr⁢(Ω)=∥(1+h)ℓ∥Lr⁢(Ω)≤∥1+h∥Lr⁢(Ω), we obtain

∥D⁢vℓ⁢k∥L∞⁢(Ω′;ℝn)≤C⁢(Ω′)⁢[∥1+h∥Lr⁢(Ω)]β^⁢γ^⁢[∫Ω(1+fℓ⁢k⁢(x,D⁢vℓ⁢k))⁢𝑑x]γ^p

≤C⁢(Ω′)⁢[∥1+h∥Lr⁢(Ω)]β^⁢γ^⁢[∫Ω1+fℓ⁢(x,D⁢u0)+1k⁢(1+|D⁢u0|2)q2⁢d⁢x]γ^p,

where C,γ^,β^ depend on n,r,p,q,M1,M2,ρ,R but are independent of ℓ,k. Therefore we conclude that

vℓ⁢k→ℓ→∞vk weakly in W01,p⁢(Ω)+u0,

vℓ⁢k→ℓ→∞vk weakly star in Wloc1,∞⁢(Ω),

and by the previous estimates

∥D⁢vk∥Lp⁢(Ω;ℝn)≤lim infℓ→∞⁡∥D⁢vℓ⁢k∥Lp⁢(Ω;ℝn)

≤∫Ωf⁢(x,D⁢u0)⁢𝑑x+∫Ω(1+|D⁢u0|2)q2⁢𝑑x

and

∥D⁢vk∥L∞⁢(Ω′;ℝn)≤lim infℓ→∞⁡∥D⁢vℓ⁢k∥L∞⁢(Ω′;ℝn)

≤C⁢(Ω′)⁢[∥1+h∥Lr⁢(Ω)]β^⁢γ^⁢[∫Ω1+f⁢(x,D⁢u0)⁢d⁢x+∫Ω(1+|D⁢u0|2)q2⁢𝑑x]γ^p.

Thus we can deduce that there exists, up to subsequences, u¯∈u0+W01,p⁢(Ω) such that

vk→u¯ weakly in W01,p⁢(Ω)+u0,

vk→u¯ weakly star in Wloc1,∞⁢(Ω).

Now, for any fixed k∈ℕ, using the uniform convergence of fℓ to f in Ω0×K (for any K compact subset of ℝn) and the minimality of vℓ⁢k, we get for all v∈W01,q⁢(Ω)+u0,

∫Ω0f⁢(x,D⁢vk)⁢𝑑x≤lim infℓ→∞⁡∫Ω0f⁢(x,D⁢vℓ⁢k)⁢𝑑x

=lim infℓ→∞⁡∫Ω0fℓ⁢(x,D⁢vℓ⁢k)⁢𝑑x

≤lim infℓ→∞⁡∫Ω0fℓ⁢(x,D⁢vℓ⁢k)⁢𝑑x+1k⁢∫Ω(1+|D⁢vℓ⁢k|2)q2⁢𝑑x

≤lim infℓ→∞⁡∫Ωfℓ⁢(x,D⁢vℓ⁢k)⁢𝑑x+1k⁢∫Ω(1+|D⁢vℓ⁢k|2)q2⁢𝑑x

≤lim infℓ→∞⁡∫Ωfℓ⁢(x,D⁢v)⁢𝑑x+1k⁢∫Ω(1+|D⁢v|2)q2⁢𝑑x.

Then, for Ω0→Ω,

∫Ωf⁢(x,D⁢vk)⁢𝑑x≤∫Ωf⁢(x,D⁢v)⁢𝑑x+1k⁢∫Ω(1+|D⁢v|2)q2⁢𝑑x.

By definition (3.2), we have

(4.10)F⁢(u¯)≤lim infk→∞⁡∫Ωf⁢(x,D⁢vk)⁢𝑑x≤∫Ωf⁢(x,D⁢v)⁢𝑑x for all ⁢v∈W01,q⁢(Ω)+u0.

Let w∈W01,p⁢(Ω)+u0. By Lemma 3.1, there exists vk∈W01,q⁢(Ω)+u0 such that vk⇀w weakly in W1,p⁢(Ω) and

limk→∞⁡∫Ωf⁢(x,D⁢vk)⁢𝑑x=F⁢(w).

By (4.10),

F⁢(u¯)≤∫Ωf⁢(x,D⁢vk)⁢𝑑x,

and we can conclude that

F⁢(u¯)≤limk→∞⁡∫Ωf⁢(x,D⁢vk)⁢𝑑x=F⁢(w) for all ⁢w∈W01,p⁢(Ω)+u0.

Then u¯∈Wloc1,∞⁢(Ω) is a solution to the problem min⁡{F⁢(u):u∈W01,p⁢(Ω)+u0}. ∎

5 Regularity of local minimizers in a special case

Let us consider now the case of a special form of integrand

(5.1)f⁢(x,ξ)=∑i=1Nai⁢(x)⁢gi⁢(ξ)

with ai⁢(x)>0 a.e. in Ω, ai∈W1,r⁢(Ω), r>n, gi:ℝn→[0,+∞) convex in ξ and strictly convex for ξ such that |ξ|≥M0. The following regularity result holds.

Theorem 5.1.

Assume that f=f⁢(x,ξ) as in (5.1) satisfies the assumptions of Theorem 2.6. Then every local minimizer u∈W1,p⁢(Ω) of the integral functional

(5.2)∫Ωf⁢(x,D⁢v)⁢𝑑x=∫Ω∑i=1Nai⁢(x)⁢gi⁢(D⁢u)⁢d⁢x

is locally Lipschitz continuous in Ω.

Proof.

Let u∈W1,p⁢(Ω) be a local minimizer of the integral functional (5.2). For a suitable φσ mollifier, consider uσ=u∗φσ∈Wloc1,q⁢(Ω). Consider the following sequence of problems in BR⊂⊂Ω:

(5.3)inf⁡{∫BRfℓ⁢k⁢(x,D⁢v)⁢𝑑x:v∈W01,q⁢(BR)+uσ},

where fℓ⁢k are defined in Proposition 4.1.

For fixed σ,ℓ,k, problem (5.3) has a unique solution vσℓ⁢k∈W01,q⁢(BR)+uσ. By proceeding as in the previous theorem, we have that for each fixed σ, by the minimality of vσℓ⁢k,

vσℓ⁢k→ℓ→∞vσk weakly in W01,p⁢(BR)+uσ,

vσℓ⁢k→ℓ→∞vσk weakly star in Wloc1,∞⁢(BR).

We also have

vσk→k→∞vσ weakly in W01,p⁢(BR)+uσ,

vσk→k→∞vσ weakly star in Wloc1,∞⁢(BR)

and

∥D⁢vσ∥L∞⁢(Bρ;ℝn)≤C⁢lim infk→∞⁡[1+∫BRf⁢(x,D⁢uσ)⁢𝑑x+1k⁢∫BR(1+|D⁢uσ|2)q2⁢𝑑x]γ^p

(5.4)=C⁢lim infk→∞⁡[1+∫BRf⁢(x,D⁢uσ)⁢𝑑x]γ^p

for any 0<ρ<R and where C is independent of k,σ. For fixed k, by proceeding as in the previous theorem we have

∫BRf⁢(x,D⁢vσk)⁢𝑑x≤∫BRf⁢(x,D⁢uσ)⁢𝑑x+1k⁢∫BR(1+|D⁢uσ|2)q2⁢𝑑x.

Then, by semicontinuity,

∫BRf⁢(x,D⁢vσ)⁢𝑑x≤lim infk→∞⁡∫BRf⁢(x,D⁢uσ)⁢𝑑x+1k⁢∫BR(1+|D⁢uσ|2)q2⁢𝑑x

(5.5)≤∫BRf⁢(x,D⁢uσ)⁢𝑑x.

Now we claim that, by the particular form of f, we may deduce

(5.6)lim infσ→0⁡∫BRf⁢(x,D⁢uσ)⁢𝑑x≤∫BRf⁢(x,D⁢u)⁢𝑑x.

Since gi is convex, for i=1,…,N, Jensen’s inequality (applied to each gi) yields

∫BRai⁢(x)⁢gi⁢(D⁢uσ)⁢𝑑x=∫BRai⁢(x)⁢gi⁢(∫BσD⁢u⁢(y)⁢φσ⁢(x-y)⁢𝑑y)⁢𝑑x

≤∫BRai⁢(x)⁢∫Bσgi⁢(D⁢u⁢(y))⁢φσ⁢(x-y)⁢𝑑y⁢𝑑x

=∫BR∫Bσai⁢(x)⁢φσ⁢(x-y)⁢𝑑y⁢gi⁢(D⁢u⁢(y))⁢𝑑x

≤∫BR+σ(ai)σ⁢(y)⁢gi⁢(D⁢u⁢(y))⁢𝑑y.

Then

∑i=1N∫BRai⁢(x)⁢gi⁢(D⁢uσ)⁢𝑑x≤∑i=1N∫BR+σ(ai)σ⁢(x)⁢gi⁢(D⁢u)⁢𝑑x

so that passing to the limit as σ→0,

lim infσ→0⁡∑i=1N∫BRai⁢(x)⁢gi⁢(D⁢uσ)⁢𝑑x≤∑i=1N∫BRai⁢(x)⁢gi⁢(D⁢u)⁢𝑑x

because (ai)σ→ai in L∞⁢(BR), gi⁢(D⁢u)∈L1⁢(BR), the Dominated Convergence Theorem may be applied, and (5.6) holds.

By collecting (5.5) and (5.6),

(5.7)lim infσ→0⁡∫BRf⁢(x,D⁢vσ)⁢𝑑x≤∫BRf⁢(x,D⁢u)⁢𝑑x.

On the other hand, the growth assumption on f yields, since u is a local minimizer of (5.2),

lim infσ→0⁡∫BR|D⁢vσ|p⁢𝑑x≤lim infσ→0⁡∫BRf⁢(x,D⁢vσ)⁢𝑑x⁢≤(5.7)⁢∫BRf⁢(x,D⁢u)⁢𝑑x<+∞.

Thus there exists v¯∈u+W01,p⁢(BR) such that, up to a subsequence,

vσ⇀v¯ weakly in W1,p⁢(BR).

By the semicontinuity of the functional, using (5.5) and (5.7),

(5.8)∫BRf⁢(x,D⁢v¯)⁢𝑑x≤lim infσ→0⁡∫BRf⁢(x,D⁢vσ)⁢𝑑x≤∫BRf⁢(x,D⁢u)⁢𝑑x.

Moreover, since (5.4) holds, D⁢vσ converges to D⁢v¯ as σ→0 in the weak star topology of L∞ and there exists a constant C such that, for any 0<ρ<R,

∥D⁢v¯∥L∞⁢(Bρ;ℝn)≤C⁢[1+∫BRf⁢(x,D⁢u)⁢𝑑x]γ^p.

Consider the following problem in BR⊂⊂Ω:

(5.9)inf⁡{∫BRf⁢(x,D⁢v)⁢𝑑x:v∈W01,p⁢(BR)+u}.

Then (5.8) implies that v¯ and u are solutions to (5.9) and v¯∈Wloc1,∞⁢(BR).

In the present case the functional is not strictly convex; we proceed as in [20, Theorem 2.1] (see also [25]) and we have that u∈Wloc1,∞⁢(BR). Indeed, set

E0:={x∈BR:|D⁢u⁢(x)+D⁢v¯⁢(x)2|>M0,D⁢u⁢(x)≠D⁢v¯⁢(x)} and w:=u+v¯2.

If E0 has positive measure, then from the convexity of f⁢(x,⋅) we have

(5.10)∫BR∖E0f⁢(x,D⁢w)⁢𝑑x≤12⁢∫BR∖E0f⁢(x,D⁢u)⁢𝑑x+12⁢∫BR∖E0f⁢(x,D⁢v¯)⁢𝑑x.

Now, by the strict convexity of f⁢(x,ξ) for ξ such that |ξ|≥M0 and applying two times the inequality

f⁢(x,η)>f⁢(x,ξ)+〈fξ⁢(x,ξ),η-ξ〉 for ξ such that |ξ|≥M0

first with ξ=D⁢w and η=D⁢u, then for ξ=D⁢w and η=D⁢v¯, finally by adding up the two inequalities obtained, we have

(5.11)∫BR∩E0f⁢(x,D⁢w)⁢𝑑x<12⁢∫BR∩E0f⁢(x,D⁢u)⁢𝑑x+12⁢∫BR∩E0f⁢(x,D⁢v)⁢𝑑x.

Adding (5.10) and (5.11), we get a contradiction with the minimality of u and v¯. Therefore the set E0 has zero measure, which implies that

and this yields the thesis. ∎

Communicated by Juha Kinnunen

Funding statement: The authors are members of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).

Acknowledgements

The authors wish to express their gratitude to the referee for carefully reading the manuscript providing useful comments and remarks. In particular, the referee pointed out a technical problem which has been modified correctly, which however did not influenced the method and the details in the proof of the a-priori estimates.

References

[1] A. L. Baisón, A. Clop, R. Giova, J. Orobitg and A. Passarelli di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal. 46 (2017), no. 3, 403–430. 10.1007/s11118-016-9585-7Search in Google Scholar

[2] P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206–222. 10.1016/j.na.2014.11.001Search in Google Scholar

[3] P. Baroni, M. Colombo and G. Mingione, Nonautonomous functionals, borderline cases and related function classes, Algebra i Analiz 27 (2015), no. 3, 6–50. 10.1090/spmj/1392Search in Google Scholar

[4] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, preprint (2017), https://arxiv.org/abs/1708.09147. 10.1007/s00526-018-1332-zSearch in Google Scholar

[5] L. Boccardo and P. Marcellini, Sulla convergenza delle soluzioni di disequazioni variazionali, Ann. Mat. Pura Appl. (4) 110 (1976), 137–159. 10.1007/BF02418003Search in Google Scholar

[6] V. Bögelein, F. Duzaar and P. Marcellini, Parabolic systems with p,q-growth: A variational approach, Arch. Ration. Mech. Anal. 210 (2013), no. 1, 219–267. 10.1007/s00205-013-0646-4Search in Google Scholar

[7] V. Bögelein, F. Duzaar and P. Marcellini, Existence of evolutionary variational solutions via the calculus of variations, J. Differential Equations 256 (2014), no. 12, 3912–3942. 10.1016/j.jde.2014.03.005Search in Google Scholar

[8] V. Bögelein, F. Duzaar and P. Marcellini, A time dependent variational approach to image restoration, SIAM J. Imaging Sci. 8 (2015), no. 2, 968–1006. 10.1137/140992771Search in Google Scholar

[9] G. Buttazzo and M. Belloni, A survey on old and recent results about the gap phenomenon in the calculus of variations, Recent Developments in Well-Posed Variational Problems, Math. Appl. 331, Kluwer Academic Publisher, Dordrecht (1995), 1–27. 10.1007/978-94-015-8472-2_1Search in Google Scholar

[10] M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 3–4, 291–303. 10.1017/S0308210500026378Search in Google Scholar

[11] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273. 10.1007/s00205-015-0859-9Search in Google Scholar

[12] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443–496. 10.1007/s00205-014-0785-2Search in Google Scholar

[13] G. Cupini, F. Giannetti, R. Giova and A. Passarelli di Napoli, Higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients, Nonlinear Anal. 154 (2017), 7–24. 10.1016/j.na.2016.02.017Search in Google Scholar

[14] G. Cupini, M. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with p-q growth, Nonlinear Anal. 54 (2003), no. 4, 591–616. 10.1016/S0362-546X(03)00087-7Search in Google Scholar

[15] G. Cupini, P. Marcellini and E. Mascolo, Existence and regularity for elliptic equations under p,q-growth, Adv. Differential Equations 19 (2014), no. 7–8, 693–724. 10.57262/ade/1399395723Search in Google Scholar

[16] E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43. Search in Google Scholar

[17] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), no. 6, 842–850. Search in Google Scholar

[18] E. DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. 10.1016/0362-546X(83)90061-5Search in Google Scholar

[19] M. Eleuteri, P. Marcellini and E. Mascolo, Lipschitz continuity for energy integrals with variable exponents, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016), no. 1, 61–87. 10.4171/RLM/723Search in Google Scholar

[20] M. Eleuteri, P. Marcellini and E. Mascolo, Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura Appl. (4) 195 (2016), no. 5, 1575–1603. 10.1007/s10231-015-0529-4Search in Google Scholar

[21] M. Eleuteri and A. Passarelli di Napoli, Higher differentiability for solutions to a class of obstacle problems, preprint (2017). 10.1007/s00526-018-1387-xSearch in Google Scholar

[22] A. Esposito, F. Leonetti and P. V. Petricca, Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth, Adv. Nonlinear Anal. (2017), 10.1515/anona-2016-0198. 10.1515/anona-2016-0198Search in Google Scholar

[23] L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with (p,q) growth, Forum Math. 14 (2002), no. 2, 245–272. 10.1515/form.2002.011Search in Google Scholar

[24] L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with (p,q) growth, J. Differential Equations 204 (2004), no. 1, 5–55. 10.1016/j.jde.2003.11.007Search in Google Scholar

[25] I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM Control Optim. Calc. Var. 7 (2002), 69–95. 10.1051/cocv:2002004Search in Google Scholar

[26] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. 105, Princeton University Press, Princeton, 1983. 10.1515/9781400881628Search in Google Scholar

[27] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), no. 2, 245–248. 10.1007/BF01158049Search in Google Scholar

[28] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing, River Edge, 2003. 10.1142/5002Search in Google Scholar

[29] A. D. Ioffe and V. M. Tihomirov, Extension of variational problems (in Russian), Trudy Moskov. Mat. Obšč. 18 (1968), 187–246. Search in Google Scholar

[30] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Search in Google Scholar

[31] P. Marcellini, Un example de solution discontinue d’un problème variationnel dans le cas scalaire, Preprint 11, Istituto Matematico “U.Dini”, Università di Firenze, 1987. Search in Google Scholar

[32] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), no. 3, 267–284. 10.1007/BF00251503Search in Google Scholar

[33] P. Marcellini, Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations 90 (1991), no. 1, 1–30. 10.1016/0022-0396(91)90158-6Search in Google Scholar

[34] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993), no. 2, 296–333. 10.1006/jdeq.1993.1091Search in Google Scholar

[35] P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl. 90 (1996), no. 1, 161–181. 10.1007/BF02192251Search in Google Scholar

[36] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), no. 4, 355–426. 10.1007/s10778-006-0110-3Search in Google Scholar

[37] A. Passarelli di Napoli, Higher differentiability of minimizers of variational integrals with Sobolev coefficients, Adv. Calc. Var. 7 (2014), no. 1, 59–89. 10.1515/acv-2012-0006Search in Google Scholar

[38] J. Serrin, A new definition of the integral for nonparametric problems in the calculus of variations, Acta Math. 102 (1959), 23–32. 10.1007/BF02559566Search in Google Scholar

[39] V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3 (1995), no. 2, 249–269. Search in Google Scholar

Received: 2017-07-02

Revised: 2017-11-01

Accepted: 2018-02-06

Published Online: 2018-03-16

Published in Print: 2020-07-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Regularity for scalar integrals without structure conditions

Abstract

1 Introduction

Theorem 1.1 (A-priori estimate).

Theorem 1.2 (Existence and regularity).

2 A-priori estimates

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Lemma 2.3.

Remark 2.4.

Remark 2.5.

Proof.

Proof of Theorem 1.1.

Theorem 2.6.

Proof.

3 Extension of the integral energy

Lemma 3.1.

Proof.

4 Existence and regularity

Proposition 4.1.

Proof.

Proof of Theorem 1.2.

5 Regularity of local minimizers in a special case

Theorem 5.1.

Proof.

Acknowledgements

References

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