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Advances in Nonlinear Analysis

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Regularity for minimizers for functionals of double phase with variable exponents

Maria Alessandra Ragusa
  • Corresponding author
  • Dipartimento di Matematica e Informatica, Viale Andrea Doria, 6-95125, Catania, Italy
  • RUDN University”, 6 Miklukho - Maklay St, Moscow, 117198, Russia
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/ Atsushi Tachikawa
  • Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan
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Published Online: 2019-07-20 | DOI: https://doi.org/10.1515/anona-2020-0022

Abstract

The functionals of double phase type

H(u):=|Du|p+a(x)|Du|qdx,(q>p>1,a(x)0)

are introduced in the epoch-making paper by Colombo-Mingione [1] for constants p and q, and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of x and partly generalize their regularity results.

1 Introduction and main theorem

The main goal of this paper is to provide a regularity theorem for minimizers of a class of integral functionals of the calculus of variations called of double phase type with variable exponents defined for uW1,1(Ω; ℝN) (Ω ∈ ℝn, n, N ≥ 2) as

F(u,Ω):=Ω|Du|p(x)+a(x)|Du|q(x)dx,q(x)p(x)>1,a(x)0,

where p(x), q(x) and a(x) are assumed to be Hölder continuous. They do not only have strongly non-uniform ellipticity but also discontinuity of growth order at points where a(x) = 0. The above functional is provided by the following type of functionals with variable exponent growth

ug(x,Du)dx,λ|z|p(x)g(x,z)Λ(1+|z|)p(x),Λλ>0,

which are called of p(x)-growth. These p(x)-growth functionals have been introduced by Zhikov [2] (in this article α(x) is used as variable exponents) in the setting of Homogenization theory. He showed higher integrability for minimizers and, on the other hand, he gave an example of discontinuous exponent p(x) for which the Lavrentiev phenomenon occurs ([3, 4]).

Such functionals provide a useful prototype for describing the behaviour of strongly inhomogeneous materials whose strengthening properties, connected to the exponent dominating the growth of the gradient variable, significantly change with the point. In [3], Zhikov pointed out the relationship between p(x)-growth functionals and some physical problems including thermistor. As another application, the theory of electrorheological materials and fluids is known. About these objects see, for example, [5, 6, 7, 8].

These kind of functionals have been the object of intensive investigation over the last years, starting with the inspiring papers by Marcellini [9, 10, 11], where he introduced so-called (p, q)- or nonstandard growth functionals:

uf(x,u,Du)dx,λ|z|pf(x,u,z)Λ(1+|z|)q,qp1,Λλ>0.

About general (p, q)-growth functionals, see for example [3, 4, 12, 13, 14, 15, 16, 17, 18, 19] and the survey [20].

For the continuous variable exponent case, nowadays many results on the regularity for minimizer are known, see [21, 22, 23, 24]. Further results in this direction can be, for instance, found in [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] for partial regularity results for p(x)-energy type functionals:

uAijαβ(x,u)Dαui(x)Dβuj(x)p(x)dx,Aijαβ(x,u)zαizβjλ|z|2

In 2015 a new class of functional so-called functionals of double phase are introduced by Colombo-Mingione [1]. In the primary model they have in mind are

uH(u;Ω):=H(x,Du)dx,H(x,z):=|z|p+a(x)|z|q,

where p and q are constants with qp > 1 and a(⋅) is a Hölder continuous non-negative function. By Colombo-Mingione [1, 42, 43] and Baroni-Colombo-Mingione [44, 45, 46] many sharp results are given about the regularity of local minimizers of the functional defined as

uG(u;Ω):=ΩG(x,u,Du)dx,(1.1)

where G(x, u, z) : Ω × ℝ × ℝnR is a Carathéodory function satisfying the following growth condition for some constants Λλ > 0 besides several natural assumptions:

λH(x,z)G(x,u,z)ΛH(x,z).

For the scalar valued case, in [46] regularity results are given comprehensively. Under the conditions

a()C0,α(Ω),α(0,1]andqp1+αn,(1.2)

or

uL(Ω),a()C0,α(Ω),α(0,1]andqp1+αp,(1.3)

they showed that a local minimizer of 𝓖 defined as (1.1) is in the class C1,β for some β ∈ (0, 1).

For the scaler valued case, see also [47]. They proved Harnack’s inequality and the Hölde continuity for quasiminimizer of the functional fo type

φ(x,|Du|)dx,

where φ is the so-called Φ-function. We mention that Harnack’s inequality is not valid in the vector valued cases which we are considering in the present paper.

On the other hand, for vector valued case, in [1], under the condition

a()C0,α(Ω),α(0,1]andqp<1+αn,(1.4)

C1,β-regularity, for some β ∈ (0, 1), of local minimizers is given.

Zhikov has given in [3, 4] examples of functionals with discontinuous growth order for which Lavrentiev phenomenon occurs. So, in general settings, we can not expect regularity of minimizers for such functionals which change their growth order discontinuously. So, conditions (1.2), (1.3) and (1.4), which guarantee the regularity of minimizers, are very significant.

In this paper we deal with a typical type of functionals of double phase with variable exponents and show a regularity result for minimizers.

In our opinion these results present new and interesting features from the point of view of regularity theory.

Let Ω ⊂ ℝn be a bounded domain, p(x), q(x) and a(x) functions on Ω satisfying

p,qC0,σ(Ω),q(x)p(x)p0>1,forallxΩ(1.5)

where p0 is a fixed constant strictly larger than one and

aC0,α(Ω),a(x)0,(1.6)

for α, σ ∈ (0, 1]. Moreover, we assume that p(x) and q(x) satisfy

supxΩq(x)p(x)<1+βn,β=min{α,σ},(1.7)

at every xΩ (compare these conditions with (1.2)). Let F : Ω × ℝnN → [0, ∞) be a function defined by

F(x,z):=|z|p(x)+a(x)|z|q(x).(1.8)

We consider the functional with double phase and variable exponents defined for u : Ω → ℝN and DΩ as

F(u,D)=DF(x,Du)dx.(1.9)

For a bounded open set Ω ⊂ ℝn and a function p : Ω → [1, +∞), we define Lp(x)(Ω; ℝN) and W1,p(x)(Ω; ℝN) as follows:

Lp(x)(Ω;RN):={uL1(Ω;RN);Ω|u|p(x)dx<+}.W1,p(x)(Ω;RN):={uLp(x)W1,1(Ω;RN);DuLp(x)(Ω;RnN)}.

In what follows we omit the target space ℝN. We also define Llocp(x)(Ω) and Wloc1,p(x)(Ω) similarly. As mentioned in [48], if p(x) is uniformly continuous and ∂Ω satisfies uniform cone property, then

W1,p(x)(Ω)={uW1,1(Ω);DuLp(x)(Ω)}.

Let us define local minimizers of 𝓕 as follows:

Definition 1.1

A function uW1,1Ω) is called to be a local minimizer of 𝓕 if F(x, Du) ∈ L1(Ω) and satisfies

F(u;suppφ)F(u+φ;suppφ),

for any φWloc1,p(x)(Ω) with compact support in Ω.

The main result of this paper is the following:

Theorem 1.2

Assume that the conditions (1.5), (1.6) and (1.7) are fulfilled. Let uW1,1(Ω) be a local minimizer of 𝓕. Then uCloc1,y(Ω) for some y ∈ (0, 1).

Remark 1.3

(About the symbols for Hölder spaces) If we follow the standard textbooks, Dacorogna [49], Evans [50], Gilberg-Trudinger [51], etc., for k ∈ ℕ, 0 < α ≤ 1, Ck,α(Ω) mean the subspaces of Ck(Ω) consisting of functions whose k-th order partial derivatives are locally Hölder continuous. However, recently many authors (especially ones who study regularity problems) write them as Clock,α(Ω), and they use Ck,α(Ω) for Ck,α(Ω̄) (namely, for uniformly Hölder continuous cases). Anyway, with “loc" there is no doubt of misunderstanding. So, in this paper we follow their usage for Hölder spaces.

In order to prove the above theorem, we employ a freezing argument; namely we consider a frozen functional which is given by freezing the exponents, and compare a minimizer of the original functional under consideration with that of frozen one.

2 Preliminary results

In what follows, we use C as generic constants, which may change from line to line, but does not depend on the crucial quantities. When we need to specify a constant, we use small letter c with index.

For double phase functional with constant exponents, namely for

H(u,D):=DH(x,Du)dx,H(x,z)=|z|p+a(x)|z|q,(2.1)

we prepare the following Sobolev-Poincaré inequality which is a slightly generalised version of [1, Theorem 1.6] due to Colombo-Mingione.

Theorem 2.1

Let a(x) ∈ C0,β(Ω) for some β ∈ (0, 1) and 1 < p < q constants satisfying

qp<1+βn,

and let ωL(ℝn) with ω ≥ 0 andBR ω dx = 1 for BRΩ with R ∈ (0, 1). Then, there exists a constant C depending only on n, p, q, [a]0,β, RnωL andDwLp(BR) and exponents d1 > 1 > d2 depending only on n, p, q, β such that

BRHx,uuωRd1dx1d1CBRHx,Dud2dx1d2(2.2)

holds whenever uW1,p(BR), where

uω:=BRu(x)ω(x)dx.

Note that for the special choice ω = |BR|–1 χBR we have

uω=BRu(x)dx.

Proof

We can proceed exactly as in the proof of [1, Theorem 1.6] only replacing (3.11) of [1] by

|u(x)uω|RCRBR|Du(y)||xy|n1dy,

which is shown by [52, Lemma 1.50] (see also the proof of [53, Theorem 7]).□

From the above theorem, we have the following corollary.

Corollary 2.2

Assume that all conditions of Theorem 2.1 are satisfied, and let D be a subset of BR with positive measure. Then, there exists a constant C depending only on n, p, q, [a]0,β, Rn/|D| andDuLp(BR) and exponents d1 > 1 > d2 depending only on n, p, q, β such that the following inequality holds whenever uW1,p(x)(BR) satifies u ≡ 0 on D:

BRHx,uRd1dx1d1CBRHx,Dud2dx1d2.(2.3)

Proof

Choosing ω so that

ω(x)=0xBRD1|D|xD

and applying Theorem 2.1, we get the assertion.□

Remark 2.3

In [1, Theorem 6.1], and therefore also in the above theorem and corollary, the exponent d2 ∈ (0, 1) is chosen so that the following conditions hold:

qp<1+βd2n(2.4)

pq(n1)+1>1d2.(2.5)

In fact, in [1], they choose a constant y ∈ (1, p) so that

qp<1+αynandp+q(n1)yq(n1)>1,

(see [1, (3.6), (3.14)]), and put d2 = 1/y. Let us mention the that if d2 satisfies (2.4) and (2.5) for some q = q0 and p = p0, then the same d2 satisfies these inequalities for any q and p with q/pq0/p0.

For any yΩ and R > 0 with BR(x) ⊂ Ω let us put

p2(y,R):=supBR(y)p(x),p1(y,R):=infBR(y)p(x),(2.6)

q2(y,R):=supBR(y)q(x),q1(y,R):=infBR(y)q(x).(2.7)

We prove interior higher integrability of the gradient of a minimizer, similar results are contained in [54].

Proposition 2.4

Let uWloc1,p(x)(Ω) be a local minimizer of 𝓕. Then, for any compact subset KΩ, F(x, Du) ∈ L1+δ0(K) and there exists a positive constant δ0 and C depending only on the given data and K such that

BR/2(y)F(x,Du)1+δ0dx11+δ0C+CBR(y)F(x,Du)dx(2.8)

holds for any BR(y) ⋐ K.

Proof

Let KΩ be a compact subset and R0 ∈ (0, dist(K, ∂Ω)) a constant such that

0<R0σp021+σ[q]0,σ1+βnsupxΩq(x)p(x).(2.9)

For any x0K, put

κ0:=141+βnsupxBR(x0)q(x)p(x)>0.(2.10)

Then, letting xR0(x0) be a such that p(x) = p1(x0, R0), we have

q2(x0,R0)p1(x0,R0)=q(x)+q2(x0,R0)q(x)p1(x0,R0)supxBR0(x0)q(x)p(x)+2σ[q]0,σR0σp0supxBR0(x0)q(x)p(x)+121+βnsupxBR0(x0)q(x)p(x)=121+βn+supxBR0(x0)q(x)p(x)1+βn2κ0(2.11)

The above estimate (2.11) implies that

q2(x0,R0)<(p1(x0,R0))=np1(x0,R0)np1(x0,R0).(2.12)

For any BR(y) ⊂ BR0(x0) with 0 < R < 1, and 0 < tsR, let η be a cut-off function such that η ≡ 1 on Bt(y), η ≡ 0 outside Bs(y) and |Dη|2st. Put w := uη(uuR), where uR = BR(y) udx. Since

Dw=(1η)Du+(uuR)Dη,

we have

F(x,Dw)c0[((1η)|Du|)p(x)+|uuR||Dη|p(x)+a(x)((1η)|Du|)q(x)+|uuR||Dη|q(x)],

where c0 is a constant depending only on maxK q(x). On the other hand, since F(x, Du) ∈ L1, we have

uW1,p(x)W1,p1(x0,R0)Lp1(x0,R0)Lp2(x0,R0)Lq(x),

on BR0(x0). Thus, mentioning also that w = u outside Bs(y), we see that F(x, Dw) ∈ L1 (K), namely w is an admissible function. In the following part of the proof, let us abbreviate

pi:=pi(y,R),qi:=qi(y,R)(i=1,2).

Then, we have

Bs(y)F(x,Du)dxBs(y)F(x,Dw)dxc0Bs(y)(1η)p(x)(|Du|p(x)+a(x)|Du|q(x))dx+c0Bs(y)uuRstp(x)+a(x)uuRstq(x)dxc0Bs(y)Bt(y)F(x,Du)dx+c0(st)p2Bs(y)|uuR|p(x)+c0(st)q2Bs(y)a(x)|uuR|q(x)dx(2.13)

We can use hole-filling method. Add c0Bs(y)∖Bt(y) F(x, Du) dx to the both side and divide them by c0 + 1, then we get

Bt(y)F(x,Du)dxc0c0+1Bs(y)F(x,Du)dx+1(st)p2Bs(y)|uuR|p(x)dx+1(st)q2Bs(y)a(x)|uuR|q(x)dx.(2.14)

Using an iteration lemma [55, Lemma 6.1], we see, for some constant C = C(c0, p2, q2), that

Bt(y)F(x,Du)dxC(st)p2Bs(y)|uuR|p(x)+C(st)q2Bs(y)a(x)|uuR|q(x)dx.

Putting s = R and t = R/2, we have

BR2(y)F(x,Du)dxCRp2BR(y)|uuR|p(x)+CRq2BR(y)a(x)|uuR|q(x)dxCRp1p2BR(y)uuRRp(x)dx+CRq1q2BR(y)a(x)uuRRq(x)dxCRp1p2BR(y)1+uuRRp2dx+CRq1q2BR(y)1+a(x)1q(x)uuRRq2dx.(2.15)

Since Rp1p2 and Rq1q2 are bounded because of the Hölder continuity of exponents p(x) and q(x), putting

a~(x):=a(x)q2q(x),

from (2.15), we obtain the estimate

BR2(y)F(x,Du)dxCRn+CRnBR(y)uuRRp2dx+a~(x)uuRRq2dx=:I+II.(2.16)

In order to get the boundedness of Rp1p2 and Rq1q2 the so-called “log-Hölder continuity" (see [56, section 4.1]) is sufficient. On the other hand by virtue of the Hölder continuity of q(⋅), we have that ãC0,β (β = min{α, σ}). Let d2 ∈ (0, 1) be a constant satisfying (2.4) and (2.5) for β = min{α, σ}, q = q2(x0, R0) and p = p1(x0, R0). Then, for any BR(y) ⊂ BR0(x0), this d2 satisfy (2.4) and (2.5) with q = q2(y, R) and p = p2(y, R).

By Theorem 2.1, we can estimate II as follows.

IICRnBR(y)|Du|p2+a~(x)|Du|q2d2dx1d2CRnBR(y)|Du|d2p2dx1d2+CRnBR(y)a(x)1q(x)|Du|d2q2dx1d2.(2.17)

As mentioned above, (2.17) holds for for any BR(y) ⊂ BR0(x0) with same d2. Now, take R > 0 sufficiently small so that

d2p2(y,R)<p1(y,R)andd2q2(y,R)<q1(y,R),

and let θ ∈ (d2, 1) be a constant satisfying

d2p2(y,R)<θp1(y,R)andd2q2(y,R)<θq1(y,R).(2.18)

Then, using Hölder inequality, we can estimate the first term of the right hand side of (2.17) as follows.

BR(y)|Du|d2p2dx1d2BR(y)|Du|θp1dxp2θp1=BR(y)|Du|θp1dxp2p1θp1BR(y)|Du|θp1dx1θBR(y)(1+|Du|p(x))dxp2p1θp1BR(y)1+|Du|θp1dx1θ.(2.19)

Since,

BR(y)|Du|p(x)dxF(u,BR(y))F(u,K)

and u locally minimizes 𝓕, ∫BR(y) |Du|p(x) dx is bounded. On the other hand, as mentioned after (2.15), R–(p2p1) is bounded. So, there exists a constant c1 = c1 (𝓕(u, K), p(x), d2, n, θ)

BR(y)|Du|p(x)dxp2p1θp1(ωnRn)(p2p1)θp1F(u,K)p2p1θp1c1(F(u,K),p(x),d2,n,θ),

where ωn denotes the volume of a n-dimensional unit ball. Thus, from (2.19) we obtain for some positive constant c2 = c2(c1, θ)

BR(y)|Du|d2p2dx1d2c2+c2BR(y)|Du|θp(x)dx1θ.(2.20)

Similarly, we can estimate the second term of the left hand side of (2.17) as follows.

BR(y)a(x)1q(x)|Du|d2q2dx1d2BR(y)a(x)1q(x)|Du|θq1dxq2θq1BR(y)a(x)1q(x)|Du|θq1dxq2q1θq1BR(y)a(x)1q(x)|Du|θq1dx1θBR(y)1+a(x)1q(x)|Du|q(x)dxq2q1θq1BR(y)1+a(x)1q(x)|Du|θq(x)dx1θ.(2.21)

As above, using local minimality of u and the fact that R–(q2q1) is bounded, we have for a positive constant c3 = c3(𝓕(u, K), q(x), d2, n, θ)

BR(y)a(x)1q(x)|Du|d2q2dxq2q1θq1c3(F(u,K),q(x),d2,n,θ).(2.22)

Thus, we obtain for some positive constant c4 = c4(c3, θ)

BR(y)a(x)1q(x)|Du|d2q2dx1d2c4+c4BR(y)a(x)1q(x)|Du|θq(x)dx1θ.(2.23)

Combining (2.16), (2.17), (2.20) and (2.23), we see that there exists a constant C depending on the given data and 𝓕(u, K) such that

BR2(y)F(x,Du)dxC+CBR(y)F(x,Du)θdx1θ(2.24)

for any BR(y) ⊂ BR0KΩ. Now, by virtue of the reverse Hölder inequality with increasing domain due to Giaquinta-Modica [57], we get the assertion.□

For δ0 determined in Proposition 2.4, in what follows, we always take R > 0 sufficiently small so that

1+δ02p2(y,R)(1+δ0)p1(y,R)and1+δ02q2(y,R)(1+δ0)q1(y,R).(2.25)

We need also higher integrability results on the neighborhood of the boundary. Let us use the following notation: for T > 0 we put

BT:=BT(0),BT+:={xRn;|x|<T,xn>0},ΓT:={xRn;|x|<T,xn=0},

We say “f = g on ΓT" when for any ηC0(BT) we have (fg)ηW01,1(BT+). For yBT, we write

Ωr:=Br(y)BT+.

Then, we have the following proposition on the higher integrability near the boundary, independently proved in [58, Lemma 5], see also [59, Lemma 5] for the manifold constrained case.

Proposition 2.5

Let a(x), q and p satisfy the same conditions in Theorem 2.1 and let for ABT+

H(w,A):=AH(x,w)dx,H(x,z):=|z|p+a(x)|z|q.

uW1,p(BT+) be a given function with

BT+|Du|p+a(x)|Du|q1+δ0dx<,

for some δ0 >. Assume that vW1,p (B+(T)) be a local minimizer of 𝓗 in the class

{wW1,p(BT+);u=wonΓT}

Then, for any S ∈ (0, T), there exists a constants δ ∈ (0, δ0) and C > 0 such that for any yBS+ and R ∈ (0, TS) we have

ΩR/2H(x,Dv)1+δdx11+δCΩRH(x,Dv)dx+CΩRH(x,Du)1+δdx11+δ.

Proof

For convenience, we extend u, v, Du, Dv to be zero in BTBT+. Of course, because extended u, v may have discontinuity on ΓT, they are not always in Wloc1,p (BT), and therefore Du, Dv do not necessarily coincide with distributional derivatives of u, v on B(T). On the other hand, since u = v on Γ(T), uv is in the class W1,p(B(S)) and DuDv can be regarded as the weak derivatives of uv on B(S) for any S < T.

Let R be a positive constant satisfying R ≤ (TS)/2. For x0BS+, we treat the two cases x0n34R and x0n>34R separately.

  • Case 1

    Suppose that x0n34R. Take radii s, t so that 0 < R/2 ≤ t < sR and choose a ηC0(BT) such that 0 ≤ η ≤ 1, η ≡ 1 on Bt, supp ηBs and || ≤ 2/(st). Defining

    φ:=η(vu),

    we see that φW01,1(BT+) with supp φBs, and that

    D(vφ)=(1η)Dv(vu)Dη+ηDu.

    Then, by virtue of the minimality of v, for a positive constant c4 depending only on q, we have

    ΩtH(x,Dv)dxΩsH(x,Dv)dxΩsH(x,D(vφ))dx=Ωs|D(vφ)|p+a(x)|D(vφ)|qdxc4ΩsΩt|Dv|p+a(x)|Dv|qdx+c4Ωs|Du|p+a(x)|Du|qdx+c4Ωs2stp|vu|p+a(x)2stq|vu|qdxc4ΩsΩt|Dv|p+a(x)|Dv|qdx+c4Ωs|Du|p+a(x)|Du|qdx+c42stpΩs|vu|pdx+c42stqΩsa(x)|vu|qdx.

    Now, we use the hole filling method as in the proof of Proposition 2.4. Namely, adding

    c4Ωt|Dv|p+a(x)|Dv|qdx

    and dividing both side by c4 + 1, we obtain

    ΩtH(x,Dv)dxc4c4+1ΩsH(x,Dv)dx+ΩsH(x,Du)dx+1(st)pΩs|vu|pdx+1(st)qΩsa(x)|vu|qdx,

    Using the iteration lemma [55, Lemma 6.1], we get for some constant C = C(c4, p, q)

    ΩtH(x,Dv)dxCΩsH(x,Du)dx+C(st)pΩs|vu|pdx+C(st)qΩsa(x)|vu|qdx.

    Putting t = R/2 and s = R, we have

    ΩR/2H(x,Dv)dxCΩRHx,vuRdx+CΩRH(x,Du)dx.

    Let us now consider the mean integral in all the terms, we obtain

    ΩR/2H(x,Dv)dxCΩRH(x,Du)dx+CΩRHx,vuRdx.

    Since we are assuming that x0n34R we can apply Corollary 2.2 with a constant independent on R for the last term in the right hand side and get

    ΩR/2H(x,Dv)dxCΩRH(x,Du)dx+CΩR(H(x,D(vu)))d2dx1d2.

    Taking into consideration that d2 < 1 we share in the last term Dv and Du, apply Hölder inequality for the integral of H(x, Du)d2, and obtain

    ΩR/2H(x,Dv)dxCΩRH(x,Du)dx+CΩRH(x,Dv)d2dx1d2.(2.26)

  • Case 2

    Let us deal with the case that x0n>34R. In this case, since B3R/4(x0) ⋐ BT+, we can proceed as in [1, 9. Proof of Theorem 1.1:(1.8)], slightly modifying the radii, to get

    ΩR/2H(x,Dv)dx=BR/2H(x,Dv)dxCB3R/4H(x,Dv)d2dx1d2CΩRH(x,Dv)d2dx1d2.(2.27)

    Thus, we see that (2.26) holds for every 0 < R < (ST)/2. Now, the reverse Hölder inequality allows us to obtain

    ΩRH(x,Dv)1+δdx11+δCΩR2H(x,Dv)dx+CΩRH(x,Du)1+δdx11+δ.

By virtue of [1, Theorem 1.1] and Proposition 2.5, we have the following global higher integrability for functions which minimize 𝓗 with Dirichlet boundary condition.

Corollary 2.6

Let a(x), q and p satisfy the same conditions in Theorem 2.1 and δ2 ∈ (0, 1) be a some constant. Assume that uW1,(1+δ1)p(BR(y)) be a given function with

BR(y)H(x,Du)1+δ1dx:=BR(y)|Du|p+a(x)|Dv|q1+δ1dxC

for some constant C > 0. Let vW1,p(BR(y)) be a minimizer of

H(w,BR(y):=BR(y)H(x,Dw)dx

in the class

u+W01,p(BR(y))={wW1,p(BR(y));uwW01,p(BR(x0))}.

Then, for some δ2 ∈ (0, δ1) and for any δ3 ∈ (0, δ2), we have H(x, Dv) ∈ L1+δ(BR(y)) and

BRH(x,Dv)1+δ3dxCBRH(x,Du)1+δ3dx.(2.28)

Proof

From [1, Theorem 1.1], Proposition 2.5 and covering argument, we have

BRH(x,Dv)1+δdx11+δCBRH(x,Dv)dx+CBRH(x,Du)1+δdx11+δ

and then, by the minimality of v,

BRH(x,Dv)1+δdx11+δCBRH(x,Du)dx+CBRH(x,Du)1+δdx11+δ

Once again we use the Hölder inequality for the first term of the right-hand side that gives us the assertion.□

3 Proof of the main theorem

In this section we prove Theorem 1.2. We employ the so-called direct approach, namely we consider a frozen functional for which the regularity theory has been established in [1] and compare a local minimizer of the frozen functional with u under consideration.

For a constant p > 1, let us define the auxiliary vector field Vp : ℝn → ℝn as

Vp(z):=|z|p2z.(3.1)

Let mention that Vp satisfies

|Vp(z)|2=|z|pand|Vp(z1)Vp(z2)|(|z1|+|z2|)p22|z1z2|.(3.2)

Proof of Theorem 1.2. We divide the proof into two parts. We prove the Hölder continuity of u in Part 1, and of the gradient Du in Part 2.

Part 1. Let K and BR0(x0), are as in the Proposition 2.4. For BR(y) ⊂ B2R(y) ⊂ BR0(x0), let us define pi and qi as in the Proposition 2.4. We define a frozen functional 𝓕0 as

F0(x,z):=|z|p2+a(x)q2q(x)|z|q2(3.3)

F0(w,D)=BR(y)F0(x,Dw)dx.(3.4)

In what follows, let us abbreviate a~(x)=a(x)q2q(x) as in the proof of Proposition 2.4.

Let vWp2(BR(y)) be a minimizer of 𝓕0 in the class

u+W0p2(BR(y)):={wWp2(BR(y));wuW0p2(BR(y))}.

Then, by [1, Theorem1.3], for any y ∈ (0, 1) there exists a constant C > 0 dependent on n, p2, q2, λ, Λ, [ã]0,β, ∥ã, ∥DvLp2(BR(y)) and y such that

Bρ(y)F0(x,Dv)dxCρRnyBR(y)F0(x,Dv)dxCρRnyBR(y)F0(x,Du)dx,(3.5)

where we used the minimality of v. Here, we mention that by the coercivity of the functional and the minimality of v we have the following:

DvLp2(BR(y))p2F0(v,BR(y))F0(u,BR(y)).(3.6)

On the other hand, since we are taking R > 0 sufficiently small so that (2.25) holds, there exists a constant C(p2, q2) > 0 such that

F0(x,ξ)C(p2,q2)(1+F(x,ξ))1+δ0(3.7)

holds for any (x, ξ) ∈ BR(y) × ℝnN. Now, by virtue of above 2 estimates and Proposition 2.4, we can see, for a constant C > 0 depending only on the given data on the functional, that

DvLp2(BR(y))p2F0(v,BR(y))C1+F(u,K)1+δ.(3.8)

Because of the local minimality of u, the last quantity is finite. Consequently, we can regard the constant in (3.5) is a constant depending only on given data and 𝓕(u, K).

For further convenience, let us mention that from (3.5), is nothing to see that

Bρ(y)(1+F0(x,Dv))dxCρRnyBR(y)(1+F0(x,Dv))dxCρRnyBR(y)(1+F0(x,Du))dx.(3.9)

Let us compare Du and Dv. Mentioning the elementary equality for a twice differentiable function

f(1)f(0)=f(0)+01(1t)f(t)dt,

as [21, (9)], and using the fact that v satisfies the Euler-Lagrange equation of 𝓕0, we can see that

F0(u)F0(v)=BR(y)ddtF0(x,tDu(1t)Dv)|t=0dx+BR(y)dx01(1t)d2dt2F0(x,tDu+(1t)Dv)dt=BR(y)DzF0(x,Dv)(DuDv)+BR(y)dx01(1t)DzDzF0(x,tDu+(1t)Dv)(DuDv)(DuDv)dtCBR(y)dx01(1t)|tDu+(1t)Dv|p22+a~(x)|tDu+(1t)Dv|q22|DuDv|2dtCBR(y)|Du|p22+|Dv|p22|DuDv|2dx+BR(y)a~(x)|Du|q22+|Dv|q22|DuDv|2dx.(3.10)

On the other hand, by the minimality of v, we have

F0(u)F0(v)F0(u)F(u,BR(y))+F(v,BR(y))F0(v).(3.11)

Since we are assuming p(x), q(x) ∈ C0,σ, using the inequality [21, (7)], we can see that, for any ε ∈ (0, 1), there exists a positive constant C such that

F0(u)F(u,BR(y))BR(y)|Du|p2|Du|p(x)+a(x)1q(x)|Du|q2a(x)1q(x)|Du|q(x)dxC(ε)RσBR(y)1+|Du|(1+ε)p2dx+C(ε)RσBR(y)1+a(x)1q(x)|Du|(1+ε)q2dxCRn+σ+C(ε)RσBR(y)1+|Du|p2(1+ε)+1+a~(x)|Du|q21+εdxCRn+σ+C(ε)RσBR(y)F0(x,Du)1+εdx(3.12)

Similarly we have

F(v,BR(y))F0(v)BR(y)|Dv|p2|Dv|p(x)+a(x)1q(x)|Dv|q2a(x)1q(x)|Dv|q(x)dxC(ε)RσBR(y)1+|Dv|(1+ε)p2dx+C(ε)RσBR(y)1+a(x)1q(x)|Dv|(1+ε)q2dxCRn+σ+C(ε)RσBR(y)1+|Dv|p2(1+ε)+1+a~(x)|Dv|q21+εdxCRn+σ+C(ε)RσBR(y)F0(x,Dv)1+εdx.(3.13)

Now, for δ0 of Proposition 2.4, choose δ3 > 0 so that (2.28) of Corollary 2.6 holds, and let us take ε so that ε ∈ (0, min{δ0/2, δ3}/2). Since we are choosing R so that (2.25) holds, we have

F0(x,)1+ε(1+F0(x,))1+min{δ0/2,δ3}C(1+F(x,))1+δ0.(3.14)

By Proposition 2.4 and (3.14), we deduce from (3.12) that

F0(u)F(u,BR(y))CRn+σ+C(ε)RσBR(y)1+F(x,Du)1+δ0dxCRn+σ+CRσBR(y)F(x,Du)1+δ0dxCRn+σ+CRσnεB2R(y)F(x,Du)dx1+δ0CRn+σ+CRσnεB2R(y)F(x,Du)dx,(3.15)

where we used the fact that

B2R(y)F(x,Du)dxKF(x,Du)dxM0

for some constant M0. The existence of M0 guaranteed by the local minimality of u.

For (3.13) we use Proposition 2.6, Proposition 2.4 and (3.14), to get

F(v,BR(y))F0(v)CRn+σ+C(ε)RσBR(y)F0(x,Du)1+εdxCRn+σ+CRσnεB2R(y)F(x,Du)dx.(3.16)

On the other hand, by the definition of F0, we have

F(x,Du)C1+F0(x,Du).

So we have, combining (3.10), (3.11), (3.15) and (3.16), that

BR(y)|Du|p22+|Dv|p22|DuDv|2dx+BR(y)a~(x)|Du|q22+|Dv|q22|DuDv|2dxF0(u)F0(v)CRn+σ+CRσnεB2R(y)(1+F0(x,Du))dx.(3.17)

By virtue of (3.2) and (3.9), we can see that

Bρ(y)(1+F0(x,Du))dx=Bρ(y)(1+F0(x,Dv))dx+Bρ(y)F0(x,Du)F0(x,Dv)dxCρRnyB(y)(1+F0(x,Dv))dx+Bρ(y)|Vp2(Du)|2+a~(x)|Vq2(Du)|2|Vp2(Dv)|2+a~(x)|Vq2(Dv)|2dxCρRnyB(y)(1+F0(x,Dv))dx+BR(y)|Vp2(Du)|2|Vp2(Dv)|2+a~(x)|Vq2(Du)|2|Vq2(Dv)|2dxCρRnyB(y)(1+F0(x,Dv))dx+BR(y)|Vp2(Du)Vp2(Dv)|2dx+BR(y)a~(x)|Vq2(Du)Vq2(Dv)|2dxCρRnyB(y)(1+F0(x,Dv))dx+BR(y)|Du|p22+|Dv|p22|DuDv|2dx+BR(y)a~(x)|Du|q22+|Dv|q22|DuDv|2dxCρRnyBR(y)(1+F0(x,Dv))dx+CRn+σ+CRσnεB2R(y)(1+F0(x,Du))dxCρRny+RσnεB2R(y)(1+F0(x,Du))dx+CRn+σ.(3.18)

Using well-known lemma (see for example [1, Lemma 5.13]), for sufficiently small R > 0, we can see that for any y ∈ (y, 1) there exists a constant C depending given data and ζ such that

Bρ(y)F0(x,Du)dxCρRnyB2R(y)F0(x,Du)dx+Cρny(3.19)

hold for any ρ ∈ (0, R). Now, since (3.9) holds for any y ∈ (0, 1), we can choose y ∈ (0, 1) arbitrarily in (3.19). On the other hand, since we are supposing that p(x) ≥ p0 > 1, for any ζ ∈ (0, 1), choosing y ∈ (0, 1) so that yp0(1 – ζ), we see that there exists a positive constant C dependent on the given data, KΩ and 𝓕(u, K) such that

Bρ(y)|Du|p0dxCρnp0(1ζ)

holds for any Bρ(y) with 4ρ ≤ dist (K, Ω). So, we conclude that uCloc0,ζ(Ω) for any ζ ∈ (0, 1) by virtue of Morrey’s theorem.

Part 2. Now, we are going to show the Hölder continuity of the gradient Du. For yK let R1 ∈ (0, R0) be a constant such that BR1(y) ⊂ K, and for 0 < R < R1/4 let v be as in Part 1. Then, by the estimate given by Colombo-Mingione at [1, p.484, l.-6], we see that there exist constants C > 0, dependent on n, p2, q2, λ, Λ, ∥ ã, dist(K, ∂Ω), 𝓕0(v, BR(y)) and α̃ ∈ (0, 1)

Bρ(y)|Dv(Dv)ρ|p2dxCρα~β64n,(3.20)

holds for any ρR/2. Here, as in Part 1, let us mention that 𝓕0(v, BR(y)) can be controlled by 𝓕(u, K) as (3.8). So, we can choose the above constant in (3.20) to be dependent only on the given data of the functional, the local minimizer u under consideration and K.

In what follows, let us abbreviate

α¯:=α~β64n.

By virtue of (3.20), for ρ and R as above, we get

Bρ(y)|Du(Du)ρ|p2dxCBρ(y)|Du(Dv)ρ|p2dxCBρ(y)Dv(Dv)ρp2dx+CBρ(y)DuDvp2dxCρn+α¯+CBR(y)DuDvp2dx.(3.21)

For the case that p2 ≥ 2, since there exists a constant such that

|z1z2|p2C|z1|p22+|z2|p22|z1z2|2

for any z1, z2 ∈ ℝn, using (3.17), we can estimate the last term of the right hand side of (3.21) as

BR(y)|DuDv|p2dxCRn+σ+CRσnεB2R(y)F0(x,Du)dx.(3.22)

We use (3.19) replacing ρ by 2R and R by R0 to see that

B2R(y)F0(x,Du)dxCRnζR0ζBR0F0(x,Du)dx+CRnζ.

Since R0 is determined in the beginning of the proof, we can regard R0ζBR0F0(x,Du)dx as a constant. So, we get

B2R(y)F0(x,Du)dxCRnζ.(3.23)

By (3.22) and (3.23), we obtain

BR(y)|DuDv|p2dxCRn+σ+CRnζ+σnεCRnζ+σnε.(3.24)

When 1 < p2 < 2, using Hölder’s inequality, (3.2) and (3.17), we can see that

BR(y)|DuDv|p2dxCBR(y)Vp2(Du)Vp2(Dv)p2(|Du|+|Dv|)p2(2p2)2dxCBR(y)Vp2(Du)Vp2(Dv)2dxp22BR(y)(|Du|+|Dv|)p22dx2p22BR(y)(|Du|+|Dv|)p22|DuDv|2dxp2BR(y)F0(x,Du)dx2p22CRn+σ+CRσnεB2R(y)F0(x,Du)dxp22B2R(y)F0(x,Du)dx2p22CR(n+σ)p22B2R(y)F0(x,Du)dx2p22+CR(σnε)p22B2R(y)F0(x,Du)dx.(3.25)

By (3.25) and (3.23), we obtain

BR(y)|DuDv|p2dxCRp2(n+σ)2R(2p2)(nζ)2+CR(σnε)p22Rnζ=CRnζ+p2(σ+ζ)2+CRnζ+p2(σnε)22CRnζ+p2(σnε)22CRnζ+(σnε)2.(3.26)

For the last inequality we used the following facts:

0<R1,0<σnε,p2>1.

Mentioning the above facts again and comparing (3.24) and (3.26), we see that, for p2 > 2, the estimate (3.26) holds. Now, combining (3.21) and (3.26), we obtain

Bρ(y)|Du(Du)ρ|p2dxCρn+α¯+Rnζ+σnε2.

This holds for any 0 < ρ < R/2 ≤ R0/8. For k > 1, let us put ρ = Rk/2 (bearing in mind that Rk/2 ≤ R/2 holds for k > 1), then

ρn+α¯+Rnζ+σnε2=ρn+α¯+(2ρ)2n2ζ+σnε2k.

So, we have

Bρ(y)|Du(Du)ρ|p2dxρn+α¯+(2ρ)2n2ζ+σnε2k.(3.27)

Since

α¯=α~64nβ=α~64nmin{α,σ}σ64,

we can take ε sufficiently small so that < (σ)/2 then, for sufficiently small ζ,

nζ+σnε2>n+α¯

holds. Now, for such a choice of ε and ζ, putting

k=2n2ζ+σnε2(n+α¯)(>1)

in (3.27), we get

Bρ(y)|Du(Du)ρ|p2dxCρn+α¯,

and therefore we obtain the Hölder continuity of Du by virtue of the Campanato’s theorem.

Acknowledgement

The authors are deeply grateful to Giuseppe Mingione for interesting them in the problem. This paper was partly prepared while the authors visited in Pisa the Centro di Ricerca Matematica Ennio De Giorgi – Scuola Normale Superiore in September 2016. The hospitality of the center is greatly acknowledged.

The first author is partially supported by PRIN 2017 and the Ministry of Education and Science of the Russian Federation (5-100 program of the Russian Ministry of Education). The second author is partially supported by Japan Society for the Promotion of Science KAKENHI Grant Number 17K05337.

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About the article

Received: 2018-11-12

Accepted: 2019-03-02

Published Online: 2019-07-20

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 710–728, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0022.

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© 2020 Maria Alessandra Ragusa and Atsushi Tachikawa, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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