BY 4.0 license Open Access Pre-published online by De Gruyter September 16, 2021

Lipschitz regularity for degenerate elliptic integrals with 𝑝, 𝑞-growth

Giovanni Cupini ORCID logo, Paolo Marcellini ORCID logo, Elvira Mascolo ORCID logo and Antonia Passarelli di Napoli ORCID logo

Abstract

We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the 𝑥-variable.

MSC 2010: 49N60; 35J50

1 Introduction

The paper deals with the regularity of minimizers of integral functionals of the Calculus of Variations of the form

(1.1)F(u)=Ωf(x,Du)dx,

where ΩRn, n2, is a bounded open set, u:ΩRN, N1, is a Sobolev map. The main feature of (1.1) is the possible degeneracy of the lagrangian f(x,ξ) with respect to the 𝑥-variable. We assume that the Carathéodory function f=f(x,ξ) is convex and of class C2 with respect to ξRN×n, with fξξ(x,ξ), fξx(x,ξ) also Carathéodory functions and f(,0)L1(Ω). We emphasize that the N×n matrix of the second derivatives fξξ(x,ξ) not necessarily is uniformly elliptic and it may degenerate at some xΩ.

In the vector-valued case N>1, minimizers of functionals with general structure may lack regularity, see [19, 49, 43], and it is natural to assume a modulus-gradient dependence for the energy density, i.e. that there exists g=g(x,t):Ω×[0,+)[0,+) such that

(1.2)f(x,ξ)=g(x,|ξ|).

Without loss of generality, we can assume g(x,0)=0; indeed, the minimizers of 𝐹 are minimizers of

uΩ(f(x,Du)-f(x,0))dx

too. Moreover, by (1.2) and the convexity of 𝑓, g(x,t) is a nonnegative, convex and increasing function of t[0,+).

As far as the growth and the ellipticity assumptions are concerned, we assume that there exist exponents p,q, nonnegative measurable functions a(x),k(x) and a constant L>0 such that

(1.3){a(x)(1+|ξ|2)p-22|λ|2fξξ(x,ξ)λ,λL(1+|ξ|2)q-22|λ|2,2pq,|fξx(x,ξ)|k(x)(1+|ξ|2)q-12

for a.e. xΩ and for every ξ,λRN×n. We allow the coefficient a(x) to be zero so that (1.3)1 is a not uniform ellipticity condition. As proved in Lemma 2.3, (1.3)1 implies the following possibly degenerate p,q-growth conditions for 𝑓:

(1.4)a(x)p(p-1)(1+|ξ|2)p-22|ξ|2f(x,ξ)L2(1+|ξ|2)q-22|ξ|2,a.e.xΩand for allξRN×n.

Our main result concerns the local Lipschitz regularity and the higher differentiability of the local minimizers of 𝐹.

Theorem 1.1

Let the functional 𝐹 in (1.1) satisfy (1.2) and (1.3). Assume moreover that

(1.5)1aLlocs(Ω),kLlocr(Ω),

with r,s>n and

(1.6)qp<ss+1(1+1n-1r).

If uWloc1,1(Ω) is a local minimizer of 𝐹, then for every ball BR0Ω, the estimates

(1.7)DuL(BR0/2)CKR0ϑ(BR0(1+f(x,Du))dx)ϑ,
(1.8)BR0/2a(x)(1+|Du|2)p-22|D2u|2dxCKR0ϑ(BR0(1+f(x,Du))dx)ϑ,
hold with the exponent 𝜗 depending on the data, the constant 𝐶 also depending on R0 and where

KR0=1+a-1Ls(BR0)kLr(BR0)2+aL(BR0).

It is well known that, to get regularity under p,q-growth, the exponents 𝑞 and 𝑝 cannot be too far apart; usually, the gap between 𝑝 and 𝑞 is described by a condition relating p,q and the dimension 𝑛. In our case, we take into account the possible degeneracy of a(x) and condition (1.3)2 on the mixed derivatives fξx in terms of a possibly unbounded coefficient k(x); then we deduce that the gap depends on 𝑠, the summability exponent of a-1 that “measures” how much 𝑎 is degenerate, and the exponent 𝑟 that tells us how far k(x) is from being bounded. If s=r=, then (1.6) reduces to qp<1+1n that is what one expects; see [12] and for instance [40]. Moreover, if s= and n<r+, then (1.6) reduces to qp<1+1n-1r, and we recover the result of [23].

Motivated by applications to the theory of elasticity, recently, Colombo and Mingione [9, 8] (see also [2, 24, 17, 18]) studied the so-called double phase integrals

Ω|Du|p+b(x)|Du|qdx,1<p<q.

When applied to this case, Theorem 1.1 gives the local Lipschitz continuity of the minimizers if b(x)Wloc1,r, for some r>n, and qp<1+1n-1r ([23]).

As it is well known, weak solutions to the elliptic equation in divergence form of the type

-div(A(x,Du))=0inΩ

are locally Lipschitz continuous provided the vector field A:Ω×RnRn is differentiable with respect to 𝜉 and satisfies the uniform ellipticity conditions

Λ1(1+|ξ|2)p-22|λ|2Aξ(x,ξ)λ,λΛ2(1+|ξ|2)p-22|λ|2.

Trudinger [50] started the study of the interior regularity of solutions to linear elliptic equations of the form

(1.9)i,j=1nxi(aij(x)uxj(x))=0,xΩRn,

where the measurable coefficients aij satisfy the non-uniform condition

λ(x)|ξ|2i,j=1naij(x)ξiξjn2μ(x)|ξ|2

for a.e. xΩ and every ξRn. Here λ(x) is the minimum eigenvalue of the symmetric matrix A(x)=(aij(x)) and μ(x):=supij|aij|. Trudinger proved that any weak solution of (1.9) is locally bounded in Ω, under the following integrability assumptions on 𝜆 and 𝜇:

λ-1Llocr(Ω)andμ1=λ-1μ2Llocσ(Ω)with1r+1σ<2n.

Equation (1.9) is usually called degenerate when λ-1L(Ω), whereas it is called singular when μL(Ω). These names in this case refer to the degenerate and the singular cases with respect to the 𝑥-variable, but in the mathematical literature, these names often refer to the gradient variable; this happens for instance with the 𝑝-Laplacian operator-div(|Du|p-2Du). We do not study in this paper the degenerate case with respect to the gradient variable, but we refer for instance to the analysis made by Duzaar and Mingione [21], who studied an L-gradient bound for solutions to non-homogeneous 𝑝-Laplacian type systems and equations; see also Cianchi and Maz’ya [7] and the references therein for the rich literature on the subject.

The result by Trudinger was extended in many settings and directions: firstly, by Trudinger himself in [51] and later by Fabes, Kenig and Serapioni in [29]; Pingen in [48] dealt with systems. More recently, for the regularity of solutions and minimizers, we refer to [3, 4, 10, 15, 16, 32]. For the higher integrability of the gradient, we refer to [33] (see also [6]). Very recently, Calderon–Zygmund’s estimates for the 𝑝-Laplace operator with degenerate weights have been established in [1]. The literature concerning non-uniformly elliptic problems is extensive, and we refer the interested reader to the references therein.

The study of the Lipschitz regularity in the p,q-growth context started with the papers by Marcellini [35, 36], and since then, many and various contributions to the subject have been provided; see the references in [42, 40]. The vectorial homogeneous framework was considered in [37, 41] and by Esposito, Leonetti and Mingione [27, 28]. Condition (1.3)2 for general non-autonomous integrands f=f(x,Du) has been first introduced in [23, 22, 24]. It is worth to highlight that, due to the 𝑥-dependence, the study of regularity is significantly harder and the techniques more complex. The research on this subject is intense, as confirmed by the many articles recently published; see e.g. [11, 13, 14, 20, 25, 30, 39, 38, 40, 45, 46, 47].

Let us briefly sketch the tools to get our regularity result. First, for Lipschitz and higher differentiable minimizers, we prove a weighted summability result for the second-order derivatives of minimizers of functionals with possibly degenerate energy densities; see Proposition 3.2. Next, in Theorem 3.3, we get an a priori estimate for the L-norm of the gradient. To establish the a priori estimate, we use Moser’s iteration method [44] for the gradient and the ideas of Trudinger [50]. An approximation procedure allows us to conclude. Actually, if 𝑢 is a local minimizer of (1.1), we construct a sequence of suitable variational problems in a ball BRΩ with boundary value data 𝑢. In order to apply the a priori estimate to the minimizers of the approximating functionals, we prove a higher differentiability result, see Theorem 3.3, for minimizers of the class of functionals with p,q-growth studied in [23], where only the Lipschitz continuity was proved. By applying the previous a priori estimate to the sequence of the solutions, we obtain a uniform control in L of the gradient which allows to transfer the local Lipschitz continuity property to the original minimizer 𝑢.

Another difficulty due to the 𝑥-dependence of the energy density is that the Lavrentiev phenomenon may occur. A local minimizer of 𝐹 is a function uWloc1,1(Ω) such that f(x,Du)Lloc1(Ω) and

Ωf(x,Du)dxΩf(x,Du+Dφ)dx

for every φC01(Ω). If 𝑢 is a local minimizer of the functional 𝐹, by virtue of (1.4), we have a(x)|Du|pLloc1(Ω) and, by (1.5), uWloc1,pss+1(Ω) since

(1.10)BR|Du|pss+1dx(BRa|Du|pdx)ss+1(BR1asdx)1s+1<+

for every ball BRΩ. Therefore, in our context, a priori, the presence of the Lavrentiev phenomenon cannot be excluded. Indeed, due to the growth assumptions on the energy density, the integral in (1.1) is well defined if uW1,q, but a priori, this is not the case if uW1,pss+1(Ω)Wloc1,q(Ω). However, as a consequence of Theorem 1.1, under the stated assumptions (1.2), (1.3), (1.5), (1.6), the Lavrentiev phenomenon for the integral functional 𝐹 in (1.1) cannot occur. For the gap in the Lavrentiev phenomenon, we refer to [52, 5, 28, 26].

We conclude this introduction by observing that, even in the one-dimensional case, the Lipschitz continuity of minimizers for non-uniformly elliptic integrals is not obvious. Indeed, if we consider a local minimizer 𝑢 to the one-dimensional integral

(1.11)F(u)=-11a(x)(1+|u(x)|2)p2dx,p>1,

then Euler’s first variation takes the form

-11a(x)p(1+|u(x)|2)p-22u(x)φ(x)dx=0for allφC01(-1,1).

This implies that the quantity a(x)(1+|u(x)|2)p-22u(x) is constant in (-1,1); it is a nonzero constant, unless u(x) itself is constant in (-1,1), a trivial case that we do not consider here. In particular, the sign of u(x) is constant, and we get

(1+|u(x)|2)p-22|u(x)|=ca(x),a.e.x(-1,1).

Therefore, if a(x) vanishes somewhere in (-1,1), then |u(x)| is unbounded (and vice versa), independently of the exponent p>1. Thus, for n=1, the local Lipschitz regularity of the minimizers does not hold in general if the coefficient a(x) vanishes somewhere.

We can compare this one-dimensional fact with the general conditions considered in the Theorem 1.1. In the case a(x)=|x|α for some α(0,1), then, taking into account the assumptions in (1.5), for the integral in (1.11), we have k(x)=a(x)=α|x|α-2x and

{1aLlocs(-1,1)1-αs>0α<1s,k=aLlocr(-1,1)r(α-1)>-1α>1-1r.

These conditions are compatible if and only if 1-1r<1s. Therefore, also in the one-dimensional case, we have a counterexample to the L-gradient bound in (1.7) if

(1.12)1r+1s>1.

This is a condition that can be easily compared with assumption (1.6) for the validity of L-gradient bound (1.7) in the general 𝑛-dimensional case. In fact, since 1qp, (1.6) implies

1<ss+1(1+1n-1r)1r+1s<1n,

which essentially is the complementary condition to (1.12) when n=1.

The plan of the paper is the following. In Section 2, we list some definitions and preliminary results. In Section 3, we prove a priori estimates of the L-norm of the gradient of local minimizers and a higher differentiability result; see Theorem 3.3. In the last section, we complete the proof of Theorem 1.1.

2 Preliminary results

We shall denote by 𝐶 or 𝑐 a general positive constant that may vary on different occasions, even within the same line of estimates. Relevant dependencies will be suitably emphasized using parentheses or subscripts. In what follows, B(x,r)=Br(x)={yRn:|y-x|<r} will denote the ball centered at 𝑥 of radius 𝑟. We shall omit the dependence on the center and on the radius when no confusion arises.

To prove our higher differentiability result (see Theorem 3.3 below), we use the finite difference operator. For a function u:ΩRk, Ω open subset of Rn, given {1,,n}, we define

τ,hu(x):=u(x+he)-u(x),xΩ|h|,

where e is the unit vector in the x direction, hR and

Ω|h|:={xΩ:dist(x,Ω)>|h|}.

We now list the main properties of this operator.

  1. If uW1,t(Ω), 1t, then τ,huW1,t(Ω|h|) and

    Di(τ,hu)=τ,h(Diu).
  2. If 𝑓 or 𝑔 has support in Ω|h|, then

    Ωfτ,hgdx=Ωgτ,-hfdx.
  3. If u,uxLt(BR), 1t<, and 0<ρ<R, then for every ℎ, |h|R-ρ,

    Bρ|τ,hu(x)|tdx|h|tBR|ux(x)|tdx.
  4. If uLt(BR), 1<t<, and for 0<ρ<R, there exists K>0 such that, for every ℎ, |h|<R-ρ,

    =1nBρ|τ,hu(x)|tdxK|h|t,

    then letting ℎ go to 0, there exists DuLt(Bρ) and uxLt(Bρ)K for every {1,,n}.

We recall the following estimate for the auxiliary function:

Vp(ξ):=(1+|ξ|2)p-24ξ,

which is a convex function since p2 (see [34, Step 2] and the proof of [31, Lemma 8.3]).

Lemma 2.1

Let 1<p<. There exists a constant c=c(n,p)>0 such that

c-1(1+|ξ|2+|η|2)p-22|Vp(ξ)-Vp(η)|2|ξ-η|2c(1+|ξ|2+|η|2)p-22

for any 𝜉, ηRn, ξη.

Remark 2.2

Note that if vWloc1,1(Ω) is such that Vp(Dv)Wloc1,2(Ω), then there exists a constant c(p) such that

c(p)-1|D2v|2(μ2+|Dv|2)p-22|DVp(Dv)|2c(p)|D2v|2(μ2+|Dv|2)p-22.

In the next lemma, we prove that (1.3)1 implies the, possibly degenerate, p,q-growth condition stated in (1.4).

Lemma 2.3

Let f=f(x,ξ)=g(x,|ξ|) be of class C2 with respect to the 𝜉-variable. Let us assume that the quadratic form of the second derivatives Dξξf(x,ξ) of 𝑓 satisfies the conditions

(2.1)a(x)(1+|ξ|2)p-22|λ|2Dξξf(x,ξ)λ,λb(x)(1+|ξ|2)q-22|λ|2

for some exponents 1<pq and nonnegative functions a,b and for all ξ,λRn. Then 𝑓 satisfies the following p,q-growth condition:

(2.2)c1a(x)(1+|ξ|2)p-22|ξ|2f(x,ξ)-f(x,0)c2b(x)(1+|ξ|2)q-22|ξ|2,

where c1=min{1(p-1)p;12} and c2=max{1(q-1)q;12}.

Remark 2.4

As noticed by an anonymous referee, that we thank for her/his careful reading of the manuscript, in Lemma 2.3, it is sufficient to assume the validity of (2.1) only for every λ,ξRn with 𝜆 proportional to 𝜉 (see formulas (2.5), (2.6) below in the proof); in this case, (2.1) simplifies into

(2.3)a(x)(1+t2)p-22gtt(x,t)b(x)(1+t2)q-22.

Therefore, alternatively, in Lemma 2.3, it is sufficient to assume directly (2.3) instead of (2.1).

Proof

For xΩ and sR, let us set φ(s)=g(x,st), where we recall that 𝑔 is linked to 𝑓 by (1.2). The assumptions on 𝑓 imply that φC2(R) with gt(x,0)=0. Since

φ(s)=gt(x,st)t,φ′′(s)=gtt(x,st)t2,

the Taylor expansion formula in integral form yields

φ(1)=φ(0)+φ(0)+01(1-r)φ′′(r)dr.

Recalling the definition of 𝜑, since φ(1)=g(x,t) and φ(0)=gt(x,0)t=0, we get

(2.4)g(x,t)=g(x,0)+01(1-r)gtt(x,rt)t2dr.

We recall [41, formula (3.3)],

(2.5)min{gtt(x,|ξ|);gt(x,|ξ|)|ξ|}Dξξf(x,ξ)λ,λ|λ|2max{gtt(x,|ξ|);gt(x,|ξ|)|ξ|},

which holds with equality when 𝜆 is proportional to 𝜉, when (2.5) simplifies to

Dξξf(x,ξ)λ,λ|λ|2|λproportional toξ=gtt(x,|ξ|).

Therefore, in the particular case with |λ|=|ξ|=t, assumption (2.1) translates into

(2.6)a(x)(1+t2)p-22gtt(x,t)b(x)(1+t2)q-22.

Inserting (2.6) in (2.4), we obtain

(2.7)a(x)t201(1-r)[1+(rt)2]p-22drg(x,t)-g(x,0)b(x)t201(1-r)[1+(rt)2]q-22dr.

Assume p2. For the integral in the left-hand side of (2.7), since r[0,1] and p2, we obtain

(2.8)01(1-r)[1+(rt)2]p-22dr01(1-r)[r2+(rt)2]p-22dr=(1+t2)p-2201(1-r)rp-2dr=(1+t2)p-22[rp-1p-1-rpp]r=0r=1=(1+t2)p-22(p-1)p.

Similarly (and simpler) for the integral in the right-hand side of (2.7),

(2.9)01(1-r)[1+(rt)2]q-22dr(1+t2)q-2201(1-r)dr=12(1+t2)q-22.

Combining the last two estimates and recalling in (2.7) that f(x,ξ)=g(x,|ξ|)=g(x,t), we get

a(x)(p-1)p(1+|ξ|2)p-22|ξ|2f(x,ξ)-f(x,0)b(x)2(1+|ξ|2)q-22|ξ|2(whenqp2).

If p<2, then we modify (2.8) into

01(1-r)[1+(rt)2]p-22dr01(1-r)(1+t2)p-22dr=12(1+t2)p-22;

then, if also 1<q<2, we modify (2.9) into

01(1-r)[1+(rt)2]q-22dr01(1-r)[r2+(rt)2]q-22dr=(1+t2)q-22(q-1)q.

In this case, we obtain

a(x)2(1+|ξ|2)p-22|ξ|2f(x,ξ)-f(x,0)b(x)(q-1)q(1+|ξ|2)q-22|ξ|2(when 1<pq2).

The conclusion (2.2) follows by combining all these cases. ∎

We end this preliminary section with two well-known properties. The first lemma has important applications in the so-called hole-filling method. Its proof can be found for example in [31, Lemma 6.1].

Lemma 2.5

Let h:[r,R0]R be a nonnegative bounded function and 0<ϑ<1, A,B0 and β>0. Assume that

h(s)ϑh(t)+A(t-s)β+B

for all rs<tR0. Then

h(r)cA(R0-r)β+cB,

where c=c(ϑ,β)>0.

We will also use the following (see e.g. [31]).

Lemma 2.6

Let ξ,ηRn. For every s>-1 and r>0, there exist positive constants c1(r,s) and c2(r,s) such that

c1(r,s)(1+|ξ|2+|η|2)s201(1-t)r(1+|(1-t)ξ+tη|s2dtc2(r,s)(1+|ξ|2+|η|2)s2.

We end this section with a remark that we will use when considering the summability of 𝑘.

Lemma 2.7

Let ΩRn, n2. If r,s>n and

1<ss+1(1+1n-1r),

then kLlocr(Ω) implies kLloc2ss-1(Ω).

Proof

Since kLlocr(Ω), we need to prove that 2ss-1r that is equivalent to 2r+1s1. This holds true because

(2.10)ss+1(1+1n-1r)>1nr+ns<1,

and we conclude because n2. ∎

3 The a priori estimate

The main result in this section is an a priori estimate of the L-norm of the gradient, and a higher differentiability result, of local minimizers of the functional 𝐹 in (1.1) satisfying assumptions (1.2), (1.3), (1.5) and (1.6), with exponents 𝑝 and 𝑞 satisfying 1<pq and not the more restrictive condition 2pq, assumed in Theorem 1.1.

We use the following weighted Sobolev type inequality, whose proof relies on the Hölder’s inequality; see e.g. [16].

Lemma 3.1

Let p>1, s1 and wW01,pss+1(Ω;RN) (wW01,p(Ω;RN) if s=), with 1<pss+1<n, and let λ:Ω[0,+) be a measurable function such that λ-1Ls(Ω). Then there exists a constant c=c(n) such that

(Ω|w|σ*dx)pσ*c(n)λ-1Ls(Ω)Ωλ|Dw|pdx,

where σ=pss+1 (σ=p if s=+).

In establishing the a priori estimate, we need to deal with quantities that involve the L2-norm of the second derivatives of the minimizer weighted with the function a(x). The next result tells that local Lipschitz continuous minimizers 𝑢 of 𝐹 possess weak second derivatives such that

Vp(Du)Wloc1,2ss+1(Ω)anda(x)|D(Vp(Du))|2Lloc1(Ω).

More precisely, we have the following proposition.

Proposition 3.2

Consider the functional 𝐹 in (1.1) satisfying assumptions (1.3) with 1<pq, and

(3.1)1aLlocs(Ω),kLloc2ss-1(Ω),

with s>1. If uWloc1,(Ω) is a local minimizer of 𝐹, then

Vp(Du)Wloc1,2ss+1(Ω)anda(x)|D(Vp(Du))|2Lloc1(Ω).

Proof

Since 𝑢 is a local minimizer of the functional 𝐹, then 𝑢 satisfies the Euler system

(3.2)Ωi,αfξiα(x,Du)φxiα(x)dx=0for allφC0(Ω;RN).

Fix x0Ω, BR0(x0)Ω, 0<ρ<R<R0. Consider a cut-off function ηC0(Ω), 0η1, η1 in Bρ, suppηBR and |h|1. Assume |Dη|2R-ρ.

Fix =1,,n, and consider the function

φα:=τ,-h(η2τ,huα),α=1,,N.

Thanks to the assumption uWloc1,(Ω), we can use 𝜑 as test function in equation (3.2), thus getting

(3.3)Ωτ,h(i,αfξiα(x,Du))(η2τ,huxiα+2ηηxiτ,huα)dx=0,

where we used property (ii) of the finite difference operator. Now, we observe that

τ,hfξiα(x,Du)=fξiα(x+he,Du(x+he))-fξiα(x+he,Du(x))+fξiα(x+he,Du(x))-fξiα(x,Du(x))=01j,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dtτ,huxjβ+01fξiαx(x+the,Du(x))dth,

and so (3.3) can be written as follows:

0=Ω2η(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)ηxiτ,huατ,huxjβdx+Ωη2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdx+Ω2η(01i,αfξiαx(x+the,Du(x))dt)hηxiτ,huαdx+Ωη2(01i,αfξiαx(x+the,Du(x))dt)hτ,huxiαdx=:J1+J2+J3+J4.

By the use of Cauchy–Schwarz and Young’s inequalities and by virtue of the second inequality of (1.3), we can estimate the integral J1. If q>2, we have

|J1|2Ω{(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)ηxi2τ,huαηxjτ,huβ}12{η2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβ}12dxCΩ|Dη|2(1+|Du(x)|2+|Du(x+he)|2)q-22|τ,hu|2dx+12Ωη2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdxC(Ω|Dη|2qq-2(1+|Du(x)|2+|Du(x+he)|2)q2dx)q-2q(suppη|τ,hu|qdx)2q+12Ωη2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdxC(R-ρ)21+|Du|L(BR0)q|h|2+12Ωη2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdx,

where we used in turn Lemma 2.6, property (iii) of the finite difference operator and the assumption uWloc1,(Ω).

If q2, a similar estimate of J1 holds true, and it can be obtained in a simpler way. Indeed, in this case,

Ω|Dη|2(1+|Du(x)|2+|Du(x+he)|2)q-22|τ,hu|2dx

can be estimated as follows:

Ω|Dη|2(1+|Du(x)|2+|Du(x+he)|2)q-22|τ,hu|2dxΩ|Dη|2|τ,hu|2dxC(R-ρ)2suppη|τ,hu|2dxC(R-ρ)2|h|2|Du|L(BR0)2.

At the end, we obtain

|J1|C(R-ρ)21+|Du|L(BR0)max{2,q}|h|2+12Ωη2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdx.

By the last inequality in (1.3), Hölder’s inequality, again by property (iii) of the finite difference operator and the assumption uWloc1,(Ω), we obtain

|J3|2|h|Ωη(01k(x+the)dt)(1+|Du|2)q-12i,α|ηxiτ,huα|dxC1+|Du|L(BR0)q-1|h|(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12si,α(Ωη|τ,huα|2ss+1)s+12sC1+|Du|L(BR0)q|h|2(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12s,

and also, for every ε>0, there exists Cε>0 such that

|J4|c|h|Ωη201k(x+the)dt(1+|Du|2)q-12|τ,hDu|dxεΩη2a(x)(1+|Du|2+|Du(x+h)|2)p-22|τ,hDu|2dx+Cε|h|2Ωη21a(x)(01k(x+the)dt)2(1+|Du|2)2q-p2dxεΩη2a(x)(1+|Du|2+|Du(x+h)|2)p-22|τ,hDu|2dx+Cε1+|Du|L(BR0)2q-p|h|2Ωη21a(x)(01k(x+the)dt)2dx.

Therefore, we get

Ωη2(01i,j,,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdx12Ωη2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdx+C(R-ρ)2|h|2+C|h|2(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12s+εΩη2a(x)(1+|Du|2+|Du(x+h)|2)p-22|τ,hDu|2dx+Cε|h|2Ωη21a(x)(01k(x+the)dt)2dx,

where all the constants in the previous estimate depend also on DuL(BR0). Reabsorbing the first integral in the right-hand side by the left-hand side, we obtain

(3.4)Ωη2(01i,j,α,βfξiαξjβ(x+he,Du(x)+tτ,hDu)dt)τ,huxiατ,huxjβdx2εΩη2a(x)(1+|Du|2+|Du(x+h)|2)p-22|τ,hDu|2dx+C(R-ρ)2|h|2+c|h|2(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12s+Cε|h|2Ωη21a(x)(01k(x+the)dt)2dx.

By the ellipticity assumption in (1.3) and by Lemma 2.6, we get that, for some c1(p)>0,

c1(p)Ωη2a(x)(1+|Du|2+|Du(x+h)|2)p-22|τ,hDu|2dx2εΩη2a(x)(1+|Du|2+|Du(x+h)|2)p-22|τ,hDu|2dx+C(R-ρ)2|h|2+c|h|2(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12s+Cε|h|2Ωη21a(x)(01k(x+the)dt)2dx.

Choosing ε=c1(p)4, we can reabsorb the first integral in the right-hand side by the left-hand side, thus getting

Ωη2a(x)(1+|Du|2+|Du(x+h)|2)p-22|τ,hDu|2dxC(R-ρ)2|h|2+c|h|2(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12s+cε|h|2Ωη21a(x)(01k(x+the)dt)2dx.

Using Lemma 2.1 to control the left-hand side of the previous estimate from below, we get

Ωη2a(x)|τ,hVp(Du)|2dxC(R-ρ)2|h|2+c|h|2(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12s+cε|h|2Ωη21a(x)(01k(x+the)dt)2dx,

and so, by the assumption 1aLlocs(Ω) and Hölder’s inequality,

(Ωη2|τ,hVp(Du)|2ss+1dx)s+1s(Ωη21as(x)dx)1sΩη2a(x)|τ,hVp(Du)|2dx{C(R-ρ)2|h|2+c|h|2(Ω|Dη|2ss-1(01k(x+the)dt)2ss-1dx)s-12s+cε|h|2Ωη21a(x)(01k(x+the)dt)2dx}a-1Ls(BR0).

Dividing both sides of the previous inequality by |h|2 and letting h0, and using (3.1), we obtain

lim|h|0(Ωη2(|τ,hVp(Du)||h|)2ss+1dx)s+1slim|h|01|h|2Ωη2a(x)|τ,hVp(Du)|2dxa-1Ls(BR0){C(R-ρ)2+c(Ω|Dη|2ss-1k2ss-1(x)dx)s-12s+cεΩη2k2(x)a(x)dx}a-1Ls(BR0)a-1Ls(BR0){C(R-ρ)2+c(Ω|Dη|2ss-1k2ss-1(x)dx)s-12s+c(Ωη21as(x)dx)1s(Ωη2k2ss-1(x)dx)s-1s}K,

with 𝐾 positive constant depending on kL2ss-1(BR0) and a-1Ls(BR0). Therefore, by property (iv) of the finite difference operator, we get

Vp(Du)Wloc1,2ss+1(Ω),

and for a sequence hn converging to 0, the sequence |τ,hnVp(Du)|/|hn| strongly converges to |D(Vp(Du))| in Lloc2ss+1(Ω) and also a.e. up to a subsequence. Hence, by Fatou’s lemma, we also have

Ωη2a(x)|DVp(Du)|2dxlim|h|01|h|2Ωη2a(x)|τ,hVp(Du)|2dxKa-1Ls(BR0).

We are now ready to establish the main result of this section, that holds true with the condition 1<pq and not only for 2pq.

Theorem 3.3

Consider the functional 𝐹 in (1.1) satisfying assumptions (1.2), (1.3), (1.5) and (1.6), with 1<pq. If uWloc1,(Ω) is a local minimizer of 𝐹, then for every ball BR0Ω,

(3.5)DuL(BR0/2)CKR0ϑ(BR0(1+f(x,Du))dx)ϑ,
(3.6)Bρa(x)(1+|Du|2)p-22|D2u|2dxc(BR0(1+f(x,Du))dx)ϑ
hold for any ρ<R2. Here

KR0=1+a-1Ls(BR0)kLr(BR0)2+aLrs2s+r(BR0),
ϑ>0 depends on the data, 𝐶 depends also on R0 and 𝑐 depends also on 𝜌 and KR0.

Proof

We begin by noting that, by Lemma 2.7 and Proposition 3.2, if uWloc1,(Ω) is a local minimizer of 𝐹, then

Vp(Du)Wloc1,2ss+1(Ω)anda(x)|D(Vp(Du))|2Lloc1(Ω).

Moreover, 𝑢 satisfies the Euler system

Ωi,αfξiα(x,Du)ψxiα(x)dx=0for allψC0(Ω;RN).

Fix =1,,n. By considering in the equality above ψ=φx with φC0(Ω;RN), we get

(3.7)Ω{i,j,α,βfξiαξjβ(x,Du)φxiαuxxjβ+i,αfξiαx(x,Du)φxiα}dx=0for allφC0(Ω;RN).

Fix a cut-off function ηC0(Ω), and define for any γ0 the function

φα:=η4uxα(1+|Du|2)γ2,α=1,,N.

One can easily check that

φxiα=4η3ηxiuxα(1+|Du|2)γ2+η4uxxiα(1+|Du|2)γ2+γη4uxα(1+|Du|2)γ-22|Du|(|Du|)xi.

By the a priori regularity properties of 𝑢, due to Proposition 3.2, and by observing that, by inequality (3.4), it can be easily deduced that

xi,j,α,βfξiαξjβ(x,Du)uxxjβuxxiα

is in Lloc1(Ω), through a density argument, we can use 𝜑 as test function in equation (3.7), thus getting

(3.8)0=Ω4η3(1+|Du|2)γ2i,j,,α,βfξiαξjβ(x,Du)ηxiuxαuxxjβdx+Ωη4(1+|Du|2)γ2i,j,,α,βfξiαξjβ(x,Du)uxxiαuxxjβdx+γΩη4(1+|Du|2)γ-22|Du|i,j,,α,βfξiαξjβ(x,Du)uxαuxxjβ(|Du|)xidx+Ω4η3(1+|Du|2)γ2i,,αfξiαx(x,Du)ηxiuxαdx+Ωη4(1+|Du|2)γ2i,,αfξiαx(x,Du)uxxiαdx+γΩη4(1+|Du|2)γ-22|Du|i,,αfξiαx(x,Du)uxα(|Du|)xidx=:I1+I2+I3+I4+I5+I6.

Estimate of I1. By the use of Cauchy–Schwarz and Young’s inequalities and by virtue of the second inequality in (1.3), we can estimate the integral I1 as follows:

(3.9)|I1|4Ω(1+|Du|2)γ2{η2i,j,,α,βfξiαξjβ(x,Du)ηxiuxαηxjuxβ}12{η4i,j,,α,βfξiαξjβ(x,Du)uxxiαuxxjβ}12C(ε,L)Ωη2|Dη|2(1+|Du|2)q+γ2dx+εΩη4(1+|Du|2)γ2i,j,,α,βfξiαξjβ(x,Du)uxxiαuxxjβdx,

where ε>0 will be chosen later.

Estimate of I3. Since

fξiαξjβ(x,ξ)=(gtt(x,|ξ|)|ξ|2-gt(x,|ξ|)|ξ|3)ξiαξjβ+gt(x,|ξ|)|ξ|δijδαβ

and

(3.10)(|Du|)xi=1|Du|α,uxixαuxα,

then

i,j,,α,βfξiαξjβ(x,Du)uxαuxxjβ(|Du|)xi=(gtt(x,|Du|)|Du|2-gt(x,|Du|)|Du|3)i,j,,α,βuxαuxxjβuxiαuxjβ(|Du|)xi+gt(x,|Du|)|Du|i,,αuxαuxxiα(|Du|)xi=(gtt(x,|Du|)|Du|-gt(x,|Du|)|Du|2)α(iuxiα(|Du|)xi)2+gt(x,|Du|)|D(|Du|)|2.

Thus,

I3=γΩη4(1+|Du|2)γ-22|Du|{(gtt(x,|Du|)|Du|-gt(x,|Du|)|Du|2)α(iuxiα(|Du|)xi)2+gt(x,|Du|)|D(|Du|)|2}dx.

Using the Cauchy–Schwarz inequality, i.e.

α(iuxiα(|Du|)xi)2|Du|2|D(|Du|)|2

and observing that gt(x,|Du|)0, we conclude

(3.11)I3γΩη4(1+|Du|2)γ-22|Du|gtt(x,|Du|)|Du|α(iuxiα(|Du|)xi)2dx0.

Estimate of I4. By using the last inequality in (1.3), we obtain

(3.12)|I4|4Ωη3k(x)(1+|Du|2)q-1+γ2i,,α|ηxiuxα|dx4Ωη3|Dη|k(x)(1+|Du|2)q+γ2dx.

Estimate of I5. Using the last inequality in (1.3) and Young’s inequality, we have that

(3.13)|I5|Ωη4k(x)(1+|Du|2)q-1+γ2|D2u|dxσΩη4a(x)(1+|Du|2)p-2+γ2|D2u|2dx+CσΩη4k2(x)a(x)(1+|Du|2)2q-p+γ2dx,

where σ(0,1) will be chosen later and 𝑎 is the function appearing in (1.3).

Estimate of I6. Using the last inequality in (1.3) and (3.10), we get

(3.14)|I6|γΩη4k(x)(1+|Du|2)q-1+γ2|D(|Du|)|dxγΩη4(1+|Du|2)q-1+γ2k(x)|D2u|dxσΩη4a(x)(1+|Du|2)p-2+γ2|D2u|2dx+Cσγ2Ωη4k2(x)a(x)(1+|Du|2)2q-p+γ2dx,

where we used Young’s inequality again. Since equality (3.8) can be written as

I2+I3=-I1-I4-I5-I6,

by virtue of (3.11), we get

I2|I1|+|I4|+|I5|+|I6|,

and therefore, recalling estimates (3.9), (3.12), (3.13) and (3.14), we obtain

Ωη4(1+|Du|2)γ2i,j,,α,βfξiαξjβ(x,Du)uxxiαuxxjβdxεΩη4(1+|Du|2)γ2i,j,,α,βfξiαξjβ(x,Du)uxxiαuxxjβdx+4Ωη3|Dη|k(x)(1+|Du|2)q+γ2dx+2σΩη4a(x)(1+|Du|2)p-2+γ2|D2u|2dx+Cσ(1+γ2)Ωη4k2(x)a(x)(1+|Du|2)2q-p+γ2dx+CεΩη2|Dη|2(1+|Du|2)q+γ2dx.

Choosing ε=12, we can reabsorb the first integral in the right-hand side by the left-hand side, thus getting

(3.15)Ωη4(1+|Du|2)γ2i,j,,α,βfξiαξjβ(x,Du)uxxiαuxxjβdx4σΩη4a(x)(1+|Du|2)p-2+γ2|D2u|2dx+CΩη3|Dη|k(x)(1+|Du|2)q+γ2dx+Cσ(1+γ2)Ωη4k2(x)a(x)(1+|Du|2)2q-p+γ2dx+CΩη2|Dη|2(1+|Du|2)q+γ2dx.

Now, using the ellipticity condition in (1.3) to estimate the left-hand side of (3.15), we get

c2Ωη4a(x)(1+|Du|2)p-2+γ2|D2u|2dx4σΩη4a(x)(1+|Du|2)p-2+γ2|D2u|2dx+CΩη3|Dη|k(x)(1+|Du|2)q+γ2dx+Cσ(1+γ2)Ωη4k2(x)a(x)(1+|Du|2)2q-p+γ2dx+CΩη2|Dη|2(1+|Du|2)q+γ2dx.

By uWloc1,(Ω) and Proposition 3.2, the first integral in the right-hand side of previous estimate is finite.

By choosing σ=c28, we can reabsorb the first integral in the right-hand side by the left-hand side, thus getting

(3.16)Ωη4a(x)(1+|Du|2)p-2+γ2|D2u|2dxCΩη3|Dη|k(x)(1+|Du|2)q+γ2dx+C(1+γ2)Ωη4k2(x)a(x)(1+|Du|2)2q-p+γ2dx+CΩη2|Dη|2(1+|Du|2)q+γ2dxCΩ(η2|Dη|2+|Dη|4)a(x)(1+|Du|2)p+γ2dx+C(γ+1)2Ωη4k2(x)a(x)(1+|Du|2)2q-p+γ2dx,

where we used Young’s inequality again. Now, we note that

η4a(x)|D((1+|Du|2)p+γ4)|2c(p+γ)2a(x)η4(1+|Du|2)p-2+γ2|D2u|2,

and so, fixing R02ρ<t<t<R<R0 with R0 such that BR0Ω and choosing ηC0(Bt) a cut-off function between Bt and Bt, by the assumption a-1Llocs(Ω), we can use the Sobolev type inequality of Lemma 3.1 with w=η2(1+|Du|2)p+γ4, λ=a and p=2, thus obtaining

(Bt(η2(1+|Du|2)p+γ4)(2ss+1)*dx)2(2ss+1)*c(t-t)2Bta(x)(1+|Du|2)p+γ2dx+c(p+γ)2Btη4a(x)(1+|Du|2)p-2+γ2|D2u|2dx,

with a constant 𝑐 depending on 𝑛 and a-1Ls(Bt).

Using (3.16) to estimate the last integral in the previous inequality, we obtain

(3.17)(Btη4nsn(s+1)-2s(1+|Du|2)(p+γ)ns2(n(s+1)-2s)dx)n(s+1)-2snsc((p+γ)2(t-t)2+(p+γ)2(t-t)4)Bta(x)(1+|Du|2)p+γ2dx+c(p+γ)4Btk2(x)a(x)(1+|Du|2)2q-p+γ2dxc((p+γ)2(t-t)2+(p+γ)2(t-t)4)Bta(x)(1+|Du|2)p+γ2dx+c(p+γ)4(Bt1asdx)1s(Btkrdx)2r(Bt(1+|Du|2)(2q-p+γ)rs2(rs-2s-r)dx)rs-2s-rrs,

where we used assumption (1.5) and Hölder’s inequality with exponents 𝑠, r2 and rsrs-2s-r.

Using the properties of 𝜂, we obtain

(Bt(1+|Du|2)(p+γ)ns2(n(s+1)-2s)dx)n(s+1)-2snsc(p+γ)4a-1Ls(BR0)kLr(BR0)2(Bt(1+|Du|2)(2q-p+γ)rs2(rs-2s-r)dx)rs-2s-rrs+c((p+γ)2(t-t)2+(p+γ)2(t-t)4)Bta(x)(1+|Du|2)p+γ2dxc(p+γ)4a-1Ls(BR0)kLr(BR0)2(Bt(1+|Du|2)(2q-p+γ)rs2(rs-2s-r)dx)rs-2s-rrs+c((p+γ)2(t-t)2+(p+γ)2(t-t)4)aLrs2s+r(BR0)(Bt(1+|Du|2)(p+γ)rs2(rs-2s-r)dx)rs-2s-rrs,

where we used that, by assumption (1.3), aLloc(Ω). Setting

(3.18)KR0=1+a-1Ls(BR0)kLr(BR0)2+aLrs2s+r(BR0)

and assuming without loss of generality that t-t<1, we can write the previous estimate as follows:

(Bt(1+|Du|2)(p+γ)ns2(n(s+1)-2s)dx)n(s+1)-2snsc(p+γ)4KR0(Bt(1+|Du|2)(2q-p+γ)rs2(rs-2s-r)dx)rs-2s-rrs+c(p+γ)2KR0(t-t)4(Bt(1+|Du|2)(p+γ)rs2(rs-2s-r)dx)rs-2s-rrs,

and using the a priori assumption uWloc1,(Ω), we get

(3.19)(Bt(1+|Du|2)(p+γ)ns2(n(s+1)-2s)dx)n(s+1)-2snsc(p+γ)4KR0(DuL(BR)2(q-p)+1(t-t)4)(Bt(1+|Du|2)(p+γ)rs2(rs-2s-r)dx)rs-2s-rrs.

Setting now

(3.20)m:=rsrs-2s-r

and noting that

(3.21)nsn(s+1)-2s=12(2ss+1)*=:2s*2,

we can write (3.19) as follows:

(3.22)(Bt((1+|Du|2)(p+γ)m2)2s*2mdx)2m2s*c(p+γ)4mKR0mDuL(BR)2(q-p)m(t-t)4mBt(1+|Du|2)(p+γ)m2dx,

where, without loss of generality, we supposed DuL(BR)2(q-p)m1.

By (2.10) and definitions (3.20) and (3.21), we have

2s*2m>1.

Define the increasing sequence of exponents

p0=pm,pi=pi-1(2s*2m)=p0(2s*2m)i

and the decreasing sequence of radii by setting

ρi=ρ+R-ρ2i.

As we will prove (see (3.30) below) the right-hand side of (3.22) is finite for γ=0. Then, for every ρ<ρi+1<ρi<R, we may iterate it on the concentric balls Bρi with exponents pi, thus obtaining

(3.23)(Bρi+1(1+|Du|2)pi+12dx)1pi+1j=0i(CmKR0mpj4mDuL(BR)2m(q-p)(ρj-ρj+1)4m)1pj(BR(1+|Du|2)p02dx)1p0=j=0i(CmKR0m4jmpj4mDuL(BR)2(q-p)m(R-ρ)4m)1pj(BR(1+|Du|2)p02dx)1p0=j=0i(4jmpj4m)1pjj=0i(CmKR0mDuL(BR)2(q-p)m(R-ρ)4m)1pj(BR(1+|Du|2)p02dx)1p0.

We have that

j=0i(4jmpj4m)1pj=exp(j=0i1pjlog(4jmpj4m))exp(j=01pjlog(4jmpj4m))c(n,r,s)

and

limi+j=0i(CmKR0mDuL(BR)2(q-p)m(R-ρ)4m)1pj=limi+(CKR0DuL(BR)2(q-p)(R-ρ)4)j=0impj=(CKR0DuL(BR)2(q-p)(R-ρ)4)j=0mpj=(CKR0DuL(BR)2(q-p)(R-ρ)4)2s*p(2s*-2m),

where, recalling that p0=pm, we used in the last equality that

j=0mpj=mp011-2m2s*=2s*p(2s*-2m).

Therefore, we can let i in (3.23), thus getting

DuL(Bρ)C(n,r,p)(KR0(R-ρ)4)2s*p(2s*-2m)DuL(BR)2(q-p)2s*p(2s*-2m)(BR(1+|Du|2)pm2dx)1pm.

Since assumption (1.6) implies that

2(q-p)2s*p(2s*-2m)<1,

we can use Young’s inequality with exponents

p(2s*-2m)2(q-p)2s*>1andp(2s*-2m)p(2s*-2m)-2(q-p)2s*

to deduce that

(3.24)DuL(Bρ)12DuL(BR)+C(n,r,p,s)(KR0(R-ρ)4)ϑ(BR(1+|Du|2)pm2dx)ς,

with ϑ=ϑ(p,q,n,r,s) and ς=ς(p,q,n,r,s).

We now estimate the last integral. By definition of 𝑚 and by the assumption on 𝑠, that implies s>nrr-n, we get

m<nsn(s+1)-2s.

Thus, by Hölder’s inequality,

(3.25)BR(1+|Du|2)pm2dxc(R0,n,r,s)(BR(1+|Du|2)pns2(n(s+1)-2s)dx)r(n(s+1)-2s)n(rs-2s-r).

This last integral can be estimated by using (3.17) with γ=0. Indeed, let us re-define t,t and 𝜂 as follows: consider Rt<t2R-ρR0 and 𝜂 a cut-off function, η1 on Bt and suppηBt. By (3.19) with γ=0,

(3.26)(Bt(1+|Du|2)pns2(n(s+1)-2s)dx)n(s+1)-2snsc(p2(t-t)2+p2(t-t)4)BR0a(x)(1+|Du|2)p2dx+cp4KBR0(Bt(1+|Du|2)(2q-p)rs2(rs-2s-r)dx)rs-2s-rrs.

If we denote

τ:=(2q-p)rsrs-2s-r,τ1:=npsn(s+1)-2s,τ2:=pss+1,

by (1.6) and s>rnr-n, we get

ττ1<1<ττ2.

Therefore, there exists θ(0,1) such that

1=θττ1+(1-θ)ττ2.

The precise value of 𝜃 is

(3.27)θ=ns(qr-pr+p)+qrnrs(2q-p).

By Hölder’s inequality with exponents τ1θτ and τ2(1-θ)τ, we get

(Bt(1+|Du|2)(2q-p)rs2(rs-2s-r)dx)rs-2s-rrs=(Bt(1+|Du|2)θτ2+(1-θ)τ2dx)2q-pτ(Bt(1+|Du|2)τ12dx)(2q-p)θτ1(Bt(1+|Du|2)τ22dx)(2q-p)(1-θ)τ2.

Hence, we can use the inequality above to estimate the last integral of (3.26) to deduce that

(3.28)(Bt(1+|Du|2)pns2(n(s+1)-2s)dx)n(s+1)-2snsc(p2(t-t)2+p2(t-t)4)BR0a(x)(1+|Du|2)p2dx+CKBR0(Bt(1+|Du|2)ps2(s+1)dx)(1-θ)(2q-p)(s+1)ps×(Bt(1+|Du|2)pns2(n(s+1)-2s)dx)θ(ns+n-2s)(2q-p)nps.

Note that, again by (1.6) and (3.27), we have

θ(2q-p)p<1.

We can use Young’s inequality in the last term of (3.28) with exponents pp-θ(2q-p) and pθ(2q-p) to obtain that, for every σ<1,

(Bt(1+|Du|2)pns2(n(s+1)-2s)dx)n(s+1)-2snsC(p2(t-t)2+p2(t-t)4)BR0a(x)(1+|Du|2)p2dx+CσKBR0pp-θ(2q-p)(Bt(1+|Du|2)ps2(s+1)dx)(1-θ)(2q-p)p-θ(2q-p)s+1s+σ(Bt(1+|Du|2)pns2(n(s+1)-2s)dx)n(s+1)-2sns.

By applying Lemma 2.5, and noting that 2R-ρ-R=R-ρ, we conclude that

(3.29)(BR(1+|Du|2)pns2(n(s+1)-2s)dx)n(s+1)-2snsC(p2(R-ρ)2+p2(R-ρ)4)BR0a(x)(1+|Du|2)p2dx+CKBR0pp-θ(2q-p)(BR0(1+|Du|2)ps2(s+1)dx)(1-θ)(2q-p)p-θ(2q-p)s+1s.

Collecting (3.25) and (3.29), we obtain

(3.30)BR(1+|Du|2)pm2dxC(p2(R-ρ)2+p2(R-ρ)4)BR0a(x)(1+|Du|2)p2dx+CKBR0pp-θ(2q-p)(BR0(1+|Du|2)ps2(s+1)dx)(1-θ)(2q-p)p-θ(2q-p)s+1s.

Notice that the right-hand side is finite because 𝑢 is a local minimizer and (1.4) and (1.10) hold. This inequality, together with (3.24), implies

DuL(Bρ)12DuL(BR)+C(KBR0(R-ρ)8)θ(BR0a(x)(1+|Du|2)p2dx)θ~+C(KBR0(R-ρ)8)θ(BR0(1+|Du|2)ps2(s+1)dx)ς~

with the constant 𝐶 depending on the data. Applying Lemma 2.5, we conclude the proof of estimate (3.5). Now, we write estimate (3.16) for γ=0 and for a cut-off function ηC0(BR2), η=1 on Bρ for some ρ<R2. This yields

Bρa(x)(1+|Du|2)p-22|D2u|2dxC(R)BR2a(x)(1+|Du|2)p2dx+CBR2k2(x)a(x)(1+|Du|2)2q-p2dxC(R)BR2f(x,Du)dx+C1+|Du|L(BR2)2q-pBR2k2(x)a(x)dxC(R)BR2f(x,Du)dx+C(R)1+|Du|L(BR2)2q-p(BR2kr(x)dx)2r(BR21as(x)dx)1s,

where we used Hölder’s inequality since 1s+2r<1 by assumptions. Using (3.5) to estimate the L norm of |Du| and recalling the definition of KR0 at (3.18), we get

Bρa(x)(1+|Du|2)p-22|D2u|2dxc(BR1+f(x,Du)dx)ϱ~,

i.e. (3.6), with 𝑐 depending on p,r,s,n,ρ,R,KR0. ∎

4 Proof of Theorem 1.1

Using the previous results and an approximation procedure, we can prove of our main result.

Proof of Theorem 1.1

For f(x,ξ) satisfying assumptions (1.3)–(1.6), let us introduce the sequence

fh(x,ξ)=f(x,ξ)+1h(1+|ξ|2)ps2(s+1).

Note that fh(x,ξ) satisfies the following set of conditions:

(4.1)1h(1+|ξ|2)ps2(s+1)fh(x,ξ)(1+L)(1+|ξ|2)q2,
(4.2)c1h(1+|ξ|2)ps2(s+1)-2|λ|2Dξξfh(x,ξ)λ,λ,
(4.3)|Dξξfh(x,ξ)|c2(1+L)(1+|ξ|2)q-22,
(4.4)|Dξxfh(x,ξ)|k(x)(1+|ξ|2)q-12
for some constants c1,c2>0, for a.e. xΩ and for every ξRN×n.

Now, fix a ball BRΩ, and let vhW1,pss+1(BR,RN) be the unique solution to the problem

min{BRfh(x,Dv)dx:vhu+W01,pss+1(BR,RN)}.

Since fh(x,ξ) satisfies (4.1), (4.2), (4.3) (4.4) with kLr, r>n, and (1.6) holds, then by the result in [23], we have vhWloc1,(BR). Therefore, it is legitimate to apply Proposition 3.2 to obtain that Vp(Dvh)Wloc1,2ss+1(BR) and a(x)|D(Vp(Dvh))|2Lloc1(BR).

Since fh(x,ξ) satisfies (4.1), by the minimality of vh, we get

BR|Dvh|pss+1dxcsBRa(x)|Dvh|p+csBR1as(x)dxcsBRfh(x,Dvh)dx+csBR1as(x)dxcsBRfh(x,Du)dx+csBR1as(x)dxcsBRf(x,Du)dx+cshBR(1+|Du|)pss+1dx+csBR1as(x)dxcsBRf(x,Du)dx+csBR(1+|Du|)pss+1dx+csBR1as(x)dx.

Therefore, the sequence vh is bounded in W1,pss+1(BR), so there exists vu+W01,pss+1(BR) such that, up to subsequences,

vhvweakly inW1,pss+1(BR).

On the other hand, we can apply Theorem 3.3 to fh(x,ξ) since the assumptions are satisfied, with 𝐿 replaced by 1+L. Thus, it is legitimate to apply estimates (3.5) and (3.6) to the solutions vh to obtain

(4.5)DvhL(Bρ)CKRϑ~(BR(1+fh(x,Dvh))dx)ς~CKRϑ~(BR(1+fh(x,Du))dx)ς~=CKRϑ~(BR(1+f(x,Du)+1h(1+|Du|2)ps2(s+1))dx)ς~CKRϑ~(BR(1+f(x,Du)+(1+|Du|2)ps2(s+1))dx)ς~,

with C,ϑ~,ς~ independent of ℎ and 0<ρ<R. Therefore, up to subsequences,

(4.6)vhvweakly* inW1,(Bρ).

Our next aim is to show that v=u. The lower semicontinuity of uBRf(x,Du) and the minimality of vh imply

BRf(x,Dv)dxlim infhBRf(x,Dvh)dxlim infhBRfh(x,Dvh)dxlim infhBRfh(x,Du)dx=lim infhBR(f(x,Du)+1h(1+|Du|2)ps2(s+1))dx=BRf(x,Du)dx.

The strict convexity of 𝑓 yields that u=v. Therefore, passing to the limit as h in (4.5), we get

DuL(Bρ)CKRϑ~(BR(1+f(x,Du)+(1+|Du|2)ps2(s+1))dx)ς~,

i.e. (1.7). Moreover, it is legitimate to apply estimate (3.6) to each vh, thus getting

Bρa(x)(1+|Dvh|2)p-22|D2vh|2dxc(BR(1+fh(x,Dvh))dx)ϱ~c(BR(1+fh(x,Du))dx)ϱ~=c(BR(1+f(x,Du)+1h(1+|Du|2)ps2(s+1))dx)ϱ~,

where we used the minimality of vh and the definition of fh(x,ξ). By Remark 2.2, the above estimate implies that the sequence xa(x)D(Vp(Dvh)) is bounded in the L2(Bρ)-norm. Using also (4.6) and that v=u, we get that, up to subsequences, this sequence converges in the weak topology of L2(Bρ)-norm to a(x)D(Vp(Du)). By the lower semicontinuity of the L2-norm with respect to the weak convergence, we conclude that

Bρa(x)|D(Vp(Du))|2dxlim infhBρa(x)(1+|Dvh|2)p-22|D2vh|2dxclim infh(BR(1+f(x,Du)+1h(1+|Du|2)ps2(s+1))dx)ϱ~=(BR(1+f(x,Du))dx)ϱ~,

i.e. (1.8). ∎

Funding source: Università degli Studi di Napoli Federico II

Award Identifier / Grant number: FRA 000022–ALTRI_CDA_75_2021_FRA_PASSARELLI

Funding statement: A. Passarelli di Napoli has been supported by Università di Napoli “Federico II” through the project FRA 000022–ALTRI_CDA_75_2021_FRA_PASSARELLI.

Acknowledgements

We thank the referee for carefully reading the manuscript, for the valuable remarks and suggestions. 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), whose support is warmly acknowledged.

  1. Communicated by: Frank Duzaar

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Received: 2020-12-16
Revised: 2021-07-22
Accepted: 2021-07-29
Published Online: 2021-09-16

© 2021 Cupini, Marcellini, Mascolo and Passarelli di Napoli, published by De Gruyter

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