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Advances in Nonlinear Analysis

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On the Hölder continuity for a class of vectorial problems

Giovanni Cupini / Matteo Focardi
  • Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy
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/ Francesco Leonetti
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  • Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università di L’Aquila, Via Vetoio snc - Coppito, 67100, L’Aquila, Italy
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/ Elvira Mascolo
  • Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy
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Published Online: 2019-08-28 | DOI: https://doi.org/10.1515/anona-2020-0039

Abstract

In this paper we prove local Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands. The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the Hölder continuity. In the final section, we provide some non-trivial applications of our results.

Keywords: Holder; continuity; regularity; vectorial; minimizer; variational; integral

MSC 2010: Primary: 49N60; Secondary: 35J50

1 Introduction

In this paper we establish Hölder regularity for vector-valued minimizers of a class of integral functionals of the Calculus of Variations. We shall apply such results to minimizers of quasiconvex integrands, therefore satisfying the natural condition to ensure existence in the vectorial setting.

For equations and scalar integrals, such a topic is strictly related to the celebrated De Giorgi result in [1]. Several generalizations in the scalar case have then been given, let us mention the contribution of Giaquinta-Giusti [2], establishing Hölder regularity for minima of non differentiable scalar functionals.

The question whether the previous theory and results extend to systems and vectorial integrals was solved in [3] by De Giorgi himself constructing an example of a second order linear elliptic system with solution x|x|y, y > 1 (see the nice survey [4]; we also refer to the paper [5] for the most recent result and an up-to-date bibliography on the subject). Motivated by the above mentioned counterexamples, in the mathematical literature there are two different research directions in the study of the regularity in the vector-valued setting: partial regularity as introduced by Morrey in [6], i.e., smoothness of solutions up to a set of zero Lebesgue measure, and everywhere regularity following Uhlenbeck [7]. For more exhaustive lists of references on such topics see for example [8, 9, 10].

Let us now introduce our working assumptions. Given n, N ≥ 2, and a bounded open set Ω ⊆ ℝn, let f : Ω × ℝN×n → ℝ be a function such that there exist Carathéodory functions Fα : Ω × ℝn → ℝ, α ∈ {1, …, N}, and G : Ω × ℝN×n → ℝ, such that for all ξ ∈ ℝN×n and for 𝓛n-a.e. xΩ

f(x,ξ):=α=1NFα(x,ξα)+G(x,ξ).(1.1)

Here, we have adopted the notation

ξ=ξ1ξN(1.2)

where ξα ∈ ℝn, α ∈ {1, …, N}, is the α-th row of the matrix ξ.

Furthermore, we assume on each function Fα the following growth conditions: there exist an exponent p ∈ (1, n), constants k1, k2 > 0 and a non-negative function aLlocσ(Ω), σ > 1, such that for all α ∈ {1, …, N}, for all λ ∈ ℝn and for 𝓛n-a.e. xΩ

k1|λ|pa(x)Fα(x,λ)k2|λ|p+a(x).(1.3)

In addition, we assume that G is rank-one convex and satisfies for all ξ ∈ ℝN×n and for 𝓛n-a.e. xΩ

|G(x,ξ)|k2|ξ|q+b(x)(1.4)

for some q ∈ [1, p), and a non-negative function bLlocσ(Ω) (for the precise definition of rank-one convexity and other generalized convexity conditions see Section 2).

Consider the energy functional ℱ defined for every map uWloc1,p(Ω, ℝN) and for every measurable subset E ⊂ ⊂ Ω by

F(u;E):=Ef(x,Du(x))dx.

The main result of the paper concerns the regularity of local minimizers of the functional ℱ. We recall for convenience that a function uWloc1,p(Ω, ℝN) is a local minimizer of ℱ if for all φW1,p(Ω, ℝN) with supp φΩ

F(u;suppφ)F(u+φ;suppφ).

Theorem 1.1

Let f satisfy (1.1) and the growth conditions (1.3), (1.4) with p ∈ (1, n). Assume further that

1q<p2nandσ>np.(1.5)

Then the local minimizers uWloc1,p (Ω;ℝN) ofare locally Hölder continuous.

Existence of local minimizers for ℱ is not assured under the conditions of Theorem 1.1 since f might fail to be quasiconvex under the given assumptions. In the statement we have chosen to underline the only conditions needed to establish the regularity result. For the existence issue see [11, 12, 13]. Despite this, we shall give some non-trivial applications of Theorem 1.1 in Section 5. In particular, by using the function introduced by Zhang in [14], we construct examples of genuinely quasiconvex integrands f, which are not convex, and satisfying (1.1)-(1.4). Furthermore, by considering the well-know Šverák’s example [15], we exhibit an example of a convex energy density f satisfying the regularity assumptions with non-convex principal part F and with the perturbation G rank-one convex but not quasiconvex. For more examples see Section 5.

The special structure of the energy density f in (1.1) permits to prove Hölder regularity by applying the De Giorgi methods to each scalar component uα of the minimizer u. More precisely, inspired by [16], we show that each component uα satisfies a Caccioppoli type inequality, and then it is local Hölder continuous by applying the De Giorgi’s arguments; see [8, 17]. As regards the application of the techniques of De Giorgi in the vector-valued case but in a different framework we quote [18]; for related Hölder continuity results for systems we quote [19, 20, 21]. We remark that in [22] local y -Hölder continuity for every y ∈ (0, 1) has been proved for stationary points of similar variational integrals with rank-one convex lower order perturbations G differentiable at every point and with principal part F(ξ) = |ξ|p.

In Section 4 we consider the case of polyconvex integrands. Precisely, the Hölder continuity of local minimizers is obtained under the same structural assumptions on F and suitable polyconvex lower order perturbations G depending only on the (N − 1) × (N − 1) minors of the gradient, see Theorem 4.1. The more rigid structure of the energy density f allows to obtain regularity results under weaker assumptions on the exponents when compared to Theorem 1.1, see Remark 4.2 and Example 1 in Section 5. We notice that in the recent papers [16, 23] the local boundedness of minimizers has been established for more general energy functionals ℱ with polyconvex integrands and under less restrictive conditions on the growth exponents.

We remark that the assumption p < n is not restrictive. Indeed, it is well-known that the regularity results still hold true if pn, even without assuming the special structure of f in (1.1). This is a consequence of the p-growth satisfied by f, the Sobolev embedding, if p > n, together with the higher integrability of the gradient if p = n (see [8, Theorem 6.7]).

We finally resume the contents of the paper. In Section 2 we introduce the various convexity notions in the vectorial setting of the Calculus of Variations and we recall De Giorgi’s regularity result. Section 3 is devoted to the proof of Theorem 1.1. In Section 4 we deal with functionals with a polyconvex lower order term G. Finally, in Section 5 we provide several non trivial examples of application of our regularity results.

2 Preliminaries

2.1 Convexity conditions

Motivated by applications to nonlinear elasticity, J. Ball in 1977 pointed out in [11] that convexity of the energy density is an unrealistic assumption in the vectorial case. Indeed, it conflicts, for instance, with the natural requirement of frame-indifference for the elastic energy. Then, in the vector-valued setting N > 1, different convexity notions with respect to the gradient variable ξ play an important role. We recall all of them in what follows.

Definition 2.1

A function f = f(x, ξ) : Ω × ℝN×n → ℝ is said to be

  1. rank-one convex: if for all λ ∈ [0, 1] and for all ξ, η ∈ ℝN×n with rank(ξη) ≤ 1

    f(x,λξ+(1λ)η)λf(x,ξ)+(1λ)f(x,η)(2.1)

    for 𝓛n a.e. xΩ;

  2. quasiconvex (in Morrey’ sense): if f is Carathéodory, f(x, ⋅) is locally integrable, and

    Ln(Ω)f(x,ξ)Ωfx,ξ+Dφ(y)dy,(2.2)

    for every ξ ∈ ℝN×n, φCc (Ω, ℝN), and for 𝓛n a.e. xΩ;

  3. polyconvex: if there exists a function g : Ω × ℝτ → ℝ, with g(x, ⋅) convex for 𝓛n a.e. xΩ, such that

    f(x,ξ)=g(x,T(ξ)).(2.3)

In the last item we have adopted the standard notation

T(ξ)=(ξ,adj2ξ,,adjiξ,,adjNnξ).

for every matrix ξ ∈ ℝN×n, where adji ξ is the adjugate matrix of order i ∈ {2, …, Nn} of ξ, that is the Ni×ni matrix of all minors of order i of ξ. We will denote by (adji ξ)α the α-row of such a matrix. In particular, adj1 ξ := ξ if i = 1. Thus T(ξ) is a vector in ℝτ, with

τ=τ(N,n):=i=1NnNi

It is well-known that

f convex f polyconvex f quasiconvex f rank-one convex,

and that in the scalar case all these notions are equivalent (see for instance [13, Theorem 5.3]).

On the other hand, none of the previous implications can be reversed except for some particular cases. We refer to [13, Chapter 5] for several examples and counterexamples. In particular, in Section 5 we shall extensively deal with Šverák’s celebrated counterexample to the reverse of the last implication above.

2.2 De Giorgi classes

In this section we recall the well-known regularity result in the scalar case due to De Giorgi [1].

Definition 2.2

Let Ω ⊆ ℝn be a bounded, open set and v : Ω → ℝ. We say that vWloc1,p(Ω) belongs to the De Giorgi class DG+(Ω, p, y, y*, δ), p > 1, y and δ > 0, y* ≥ 0 if

Bσρ(x0)|D(vk)+|pdxy(1σ)pρpBρ(x0)(vk)+pdx+y(Ln({v>k}Bρ(x0)))1pn+pδ(2.4)

for all k ∈ ℝ, σ ∈ (0, 1) and all pair of balls Bσρ(x0) ⊂ Bρ(x0) ⊂ ⊂ Ω.

The De Giorgi class DG(Ω, p, y, y*, δ) is defined similary with (vk)+ replaced by (vk).

Finally, we set DG(Ω, p, y, y*, δ) = DG+(Ω, p, y, y*, δ) ∩ DG(Ω, p, y, y*, δ).

(2.4) is a Caccioppoli type inequality on super-/sub-level sets and contains several informations on the smoothness of the function u. Indeed, functions in the De Giorgi classes have remarkable regularity properties. In particular, they are locally bounded and locally Hölder continuous in Ω (see [17, Theorems 2.1 and 3.1] and [8, Chapter 7]).

Theorem 2.3

Let vDG(Ω, p, y, y*, δ) and τ ∈ (0, 1). There exists a constant C > 1 depending only upon the data and independent of v, such that for every pair of balls Bτρ(x0) ⊂ Bρ(x0) ⊂ ⊂ Ω

vL(Bτρ(x0))max{yρnδ;C(1τ)1δ(1Ln(Bρ(x0))Bρ(x0)|v|pdx)1p},

moreover, there exists α̃ ∈ (0, 1) depending only upon the data and independent of v, such that

osc(v,Bρ(x0))Cmax{yρnδ;ρRα~osc(v,BR(x0))}

where osc(v, Bρ(x0)) := ess supBρ(x0) v − ess infBρ(x0) v. Therefore, vCloc0,α~0(Ω) with α̃0 := α̃ ∧(n δ).

3 Proof of Theorem 1.1

The specific structure (1.1) of the energy density f yields a Caccioppoli inequality on every sub-/superlevel set for any component uα of u. To provide the precise statement we introduce the following notation: given x0Ω, 0 < t < dist(x0, ∂Ω), and with fixed k ∈ ℝ and α ∈ {1, …, N} set

Ak,t,x0α:={xBt(x0):uα(x)>k}andBk,t,x0α:={xBt(x0):uα(x)<k}.(3.1)

Proposition 3.1

(Caccioppoli inequality on sub-/superlevel sets). Let f be as in (1.1), satisfying the growth conditions (1.3), (1.4). Let uWloc1,p(Ω; ℝN) be a local minimizer of ℱ.

Then there exists c = c(k1, k2, p, q, n) > 0, such that for all x0Ω and for every 0 < ρ < R < R0 ∧ dist(x0, ∂Ω), with 𝓛n(BR0) ≤ 1, and α ∈ {1, …, N} we have

Ak,ρ,x0α|Duα|pdxcAk,R,x0α|uαk|Rρpdx+c(1+aLσ(BR(x0))+bLσ(BR(x0))+DuLp(BR(x0),RN×n)q)(Ln(Ak,R,x0α))ϑ(3.2)

where ϑ:=min{1qp,11σ}. The same inequality holds substituting Ak,R,x0αwithBk,R,x0α.

Proof

Without loss of generality we may assume α = 1. For the sake of notational convenience we drop the x0-dependence in the notation of the corresponding sub-/superlevel set. We start off with proving the inequality on the super-level sets. Given 0 < ρ < s < t < R < R0 ∧ dist(x0, ∂Ω), with 𝓛n(BR0) ≤ 1, consider a smooth cut-off function ηC0(Bt) satisfying

0η1,η1inBs(x0),|Dη|2ts.(3.3)

With fixed k ∈ ℝ, define wWloc1,p(Ω;ℝN) by

w1:=max(u1k,0),wα:=0α{2,,N}

and

φ:=ηpw.

We have φ = 0 𝓛n-a.e. in Ω ∖ ({η > 0} ∩ {u1 > k}), thus

Du+Dφ=DuLn-a.e. inΩ({η>0}{u1>k}).(3.4)

Set

A:=pη1(ku1)DηDu2DuN.(3.5)

Then notice that 𝓛n-a.e. in {η > 0} ∩ {u1 > k}

Du+Dφ=(1ηp)Du1+pηp1(ku1)DηDu2DuN=(1ηp)Du+ηpA.(3.6)

Thus, since Du-𝔸 is a rank-one matrix, the rank-one-convexity of G yields

G(x,Du+Dφ)(1ηp)G(x,Du)+ηpG(x,A)Ln-a.e. in{η>0}{u1>k}.(3.7)

By the local minimality of u, (3.4), (3.7) and taking into account that 𝓛n-a.e. in Ω

Fα(x,(Du+Dφ)α)=Fα(x,Duα)α{2,,N}

we have

Ak,t1{η>0}F1(x,Du1)+G(x,Du)dxAk,t1{η>0}F1(x,(Du+Dφ)1)+(1ηp)G(x,Du)+ηpG(x,A)dx.

The latter inequality and (3.4) imply that

Ak,t1F1(x,Du1)dxAk,t1F1(x,(Du+Dφ)1)dx+Ak,t1{η>0}ηpG(x,A)G(x,Du)dx.(3.8)

By (1.3), (3.6), the convexity of t ↦ |t|p and (3.3), we get

Ak,t1F1(x,(Du+Dφ)1)dxAk,t1k2|(Du+Dφ)1|p+a(x)dxAk,t1k2(1ηp)|Du1|p+k2|p(ku1)Dη|p+a(x)dxcAk,t1Ak,s1|Du1|pdx+cAk,t1((u1kts)p+a(x))dx

with c = c(k2, p). Therefore, (3.8) implies

Ak,t1F1(x,Du1)dxcAk,t1Ak,s1|Du1|pdx+cAk,t1((u1kts)p+a(x))dx+Ak,t1{η>0}ηpG(x,A)G(x,Du)dx.(3.9)

We now estimate the last integral at the right hand side. The growth condition in (1.4) for G, Hölder’s and Young’s inequalities imply, for some c = c(k2, p, q) > 0,

Ak,t1{η>0}ηp(Gx,AGx,Du)dxcAk,t1((u1kts)q+|Du|q+b(x))dxcAk,t1(u1kts)pdx+cLn(Ak,t1)+cDuLp(Bt,RN×n)q(Ln(Ak,t1))1qp+cbLσ(Bt)(Ln(Ak,t1))11σ.(3.10)

Hence, by taking into account estimates (1.3), (3.9) and (3.10), we obtain

k1Ak,s1|Du1|pdxcAk,t1Ak,s1|Du1|pdx+cAk,t1(u1kts)pdx+cLn(Ak,t1)+cDuLp(Bt,RN×n)q(Ln(Ak,t1))1qp+caLσ(Bt)+bLσ(Bt)(Ln(Ak,t1))11σ

for c = c(k2, p, q) > 0. By hole-filling, i.e. adding to both sides

cAk,s1|Du1|pdx,

we obtain

Ak,s1|Du1|pdxck1+cAk,t1|Du1|pdx+Ak,t1(u1kts)pdx+Ln(Ak,t1)+DuLp(Bt,RN×n)q(Ln(Ak,t1))1qp+aLσ(Bt)+bLσ(Bt)(Ln(Ak,t1))11σ

for c = c(k2, p, q) > 0. By Lemma 3.2 below we get

Ak,ρ1|Du1|pdxcAk,R1(u1kRρ)pdx+cLn(Ak,R1)+cDuLp(BR,RN×n)q(Ln(Ak,R1))1qp+caLσ(BR)+bLσ(BR)(Ln(Ak,R1))11σ,(3.11)

for c = c(k1, k2, p, q) > 0. Estimate (3.2) follows at once from (3.11), by taking into account that Ln(Ak,R1) ≤ 𝓛n(BR0) ≤ 1.

In conclusion, the analogous estimate with Bk,R1 in place of Ak,R1 follows from (3.2) itself since −u is a local minimizer of the integral functional with energy density (x, ξ) := f(x, − ξ). □

The following lemma finds an important application in the hole-filling method. The proof can be found for example in [8, Lemma 6.1].

Lemma 3.2

Let h : [r, R0] → ℝ be a non-negative bounded function and 0 < ϑ < 1, A, B ≥ 0 and β > 0. Assume that

h(s)ϑh(t)+A(ts)β+B,

for all rs > tR0. Then

h(r)cA(R0r)β+cB,

where c = c(ϑ, β) > 0.

We are now ready to prove the local Hölder continuity of local minimizers.

Proof of Theorem 1.1

We use Proposition 3.1 for u, with the exponents p, q satisfying (1.5) or, equivalently,

ϑ>1pn,(3.12)

recalling that ϑ=min{1qp,11σ}. Then inequality (2.4) holds for each uα, α ∈ {1, …, N}, i.e. uα belongs to a suitable De Giorgi’s class and Theorem 2.3 ensures that uα is locally Hölder continuous. □

4 The polyconvex case

In this section we deal with the case of a suitable class of polyconvex functions G. We will exploit their specific structure to obtain Hölder continuity results not included in Theorem 1.1. We shall use extensively the notation introduced in Section 2.1.

For uWloc1,p(Ω; ℝN) and E ⊂ ⊂ Ω a measurable set, we shall consider functionals

F(u;E):=Ef(x,Du(x))dx,

with Carathéodory integrands f : Ω × ℝN×n → ℝ, nN ≥ 3, satisfying

f(x,ξ):=α=1NFα(x,ξα)+G(x,ξ).(4.1)

We assume that the functions Fα are as in the previous section. In particular, we assume that

there exist p ∈ (1, n), k1, k2 > 0 and a non-negative function aLlocσ(Ω), σ > 1, such that

k1|λ|pa(x)Fα(x,λ)k2|λ|p+a(x)(4.2)

for all λ ∈ ℝn and for 𝓛n-a.e. xΩ

As far as G : Ω × ℝN×n → ℝ is concerned, G depends only on (N − 1) × (N − 1) minors of ξ as follows:

G(x,ξ):=α=1NGαx,(adjN1ξ)α.(4.3)

For every α ∈ {1, ⋯, N} we assume that Gα : Ω × ℝN → ℝ is a Carathéodory function, λGα(x, λ) convex, such that

there exist r ∈ [1, p) and a non-negative function bLlocσ(Ω), σ > 1, such that

0Gαx,λk2|λ|r+b(x)(4.4)

for all λ ∈ ℝN and for 𝓛n-a.e. xΩ.

Theorem 4.1

Let f be as in (4.1), and assume (4.2)(4.4). If

1r<p2(N2)n+pandσ>np,(4.5)

then the local minimizers uWloc1,p(Ω; ℝN) ofare locally Hölder continuous.

Remark 4.2

A comparison between Theorem 4.1 and Theorem 1.1 is in order. We do it in the case n = N = 3. By (4.3), the function

G(x,ξ):=α=13Gα(x,adj2ξα)

is a polyconvex function, satisfying

0G(x,ξ)c|ξ|2r+b(x)+1ξR3×3

for a positive costant c depending on p and k2.

By Theorem 1.1 we get that if σ>3p and

1r<p26,

then the Wloc1,p(Ω;ℝ3)-local minimizers ofare Hölder continuous.

The Hölder regularity of the local minimizers can be obtained by Theorem 4.1 under the following weaker condition on r

1r<p2p+3.

The key result to establish Theorem 4.1 is, as in the case of Theorem 1.1, the following Caccioppoli’s type inequality which improves Proposition 3.1. We state it only for the first component u1 of u. We recall that the super-(sub-)level sets are defined as in (3.1). Moreover, we use here the following notation:

u^:=(u2,u3,,uN).

For the sake of simplicity, in the Lebesgue norms we will avoid to indicate the target space of the functions involved.

Proposition 4.3

(Caccioppoli inequality on sub-/superlevel sets). Let f be as in (4.1), and assume that Fα and G satisfy (4.2)(4.4). Assume that

1r<pN1,σ>1.(4.6)

If uWloc1,p(Ω;ℝN) is a local minimizer of ℱ, then there exists c = c(n, N, p, k1, k2, r) > 0, such that for all x0Ω and for every 0 < ρ < R < R0 ∧ dist(x0, ∂Ω), with 𝓛n(BR0) ≤ 1, we have

Ak,ρ,x01|Du1|pdxcAk,R,x01|u1k|Rρpdx+ca+bLσ(BR(x0))+Du^Lp(BR)(N2)rppr(Ln(Ak,R,x01))ϑ,(4.7)

where

ϑ:=min1(N2)rpr,11σ.(4.8)

The same inequality holds substituting Ak,R,x01 with Bk,R,x01.

We limit ourselves to exhibit the proof of Proposition 4.3, given that Theorem 4.1 follows with the same lines of the proof of Theorem 1.1.

Proof of Proposition 4.3

The proof is the same of that of Proposition 3.1 up to inequality (3.9). By keeping the notation introduced there, (3.9) and the left inequality in (4.2) imply

k1Ak,t1{η>0}ηp|Du1|pdxcAk,t1Ak,s1|Du1|pdx+cAk,t1((u1kts)p+a(x))dx+Ak,t1{η>0}ηp(Gx,AGx,Du)dx,(4.9)

with c depending on p, k2. We exploit next the specific structure of G. Taking into account the definition of 𝔸, see (3.5), we have

G1(x,(adjN1A)1)=G1(x,(adjN1Du)1),

therefore

Gx,AGx,Du=α=2NGαx,(adjN1A)αGαx,(adjN1Du)α.

Using the growth condition (4.4), that in particular implies that Gα is non-negative, we get

α=2NGαx,(adjN1A)αGαx,(adjN1Du)αα=2NGαx,(adjN1A)αcα=2N(|(adjN1A)α|r+b(x))

with c depending on k2.

Denote û := (u2, u3, ⋯, uN). For every α ∈ {2, ⋯, N} we have

|(adjN1A)α|c|A1||adjN2Du^|

with c depending on n and N. Since r < p we can use the Young’s inequality with exponents pr and ppr, so we have, 𝓛n-a.e. in Ak,t1{η>0},

|A1||adjN2Du^|rc|A1|p+|adjN2Du^|)rpprcu1ktsp+|Du^|(N2)rppr,

with c = c(n, N, p, r).

Collecting the above inequalities, we get

Gx,AGx,Du)dxcAk,t1{η>0}u1ktspdx+cAk,t1|Du^|(N2)rpprdx+cAk,t1b(x)dx,(4.10)

with c = c(k2, n, N, p, r) > 0. By (4.6) (N2)rpr<1 therefore by Hölder’s inequality we get

Ak,t1|Du^|(N2)rpprdxBt|Du^|pdx(N2)rpr(Ln(Ak,t1))1(N2)rpr.

Analogously,

Ak,t1(a(x)+b(x))dxa+bLσ(Bt)(Ln(Ak,t1))11σ.

Therefore by (4.9) and (4.10) we get

k1Ak,s1|Du1|pdxcAk,t1Ak,s1|Du1|pdx+cAk,t1u1ktspdx+cDu^Lp(Bt)(N2)rppr(Ln(Ak,t1))1(N2)rpr+ca+bLσ(Bt)(Ln(Ak,t1))11σ,(4.11)

with c = c(n, N, p, k2, r) > 0.

We now proceed as in the proof of Proposition 3.2: adding to both sides of (4.11)

cAk,s1|Du1|pdx

and using Lemma 3.2 we obtain that

Ak,ρ1|Du1|pdxcAk,R1u1kRρpdx+cDu^Lp(BR)(N2)rppr+a+bLσ(BR)(Ln(Ak,R1))ϑ

with ϑ as in (4.8) and c = c(n, N, p, k1, k2, r) > 0.

In conclusion, the analogous estimate with Bk,t1 in place of Ak,t1 follows from (4.7) itself since −u is a local minimizer of the integral functional with energy density (x, ξ) := f(x, −ξ). □

5 Examples

We provide some non trivial applications of Theorems 1.1 and 4.1. In particular, we infer Hölder continuity of local minimizers to vectorial variational problems with quasiconvex integrands. The energy densities that we consider satisfy (1.1)-(1.4) and have neither radial nor quasi-diagonal structure. More in details, the integrands in Examples 1 and 2 are not convex, respectively they are polyconvex and quasiconvex, being F convex but G only polyconvex in the first case, and quasiconvex in the second. In Example 3 we construct a convex density though with non-convex principal part. Instead, the energy density f in Examples 4 and 5 is convex. In particular, in the first one F is convex and G is the rank-one convex non-quasiconvex function introduced by Šverák in [15]; in the second we construct a non-convex integrand by modifying F in Example 4, keeping the same G.

Being in all cases the resulting f quasiconvex, existence of local minimizers for the corresponding functional ℱ easily follows from the Direct Method of the Calculus of Variations.

Example 1

Let n = N = 3 and consider f : ℝ3×3 → ℝ,

f(ξ):=α=13|ξα|p+(1+|(adj2ξ)111|)r,

with p ≥ 1 and r ≥ 1. We recall that, for all ξ ∈ ℝ3×3, adj2 ξ ∈ ℝ3×3 denotes the adjugate matrix of ξ of order 2, whose components are

(adj2ξ)iy=(1)y+idetξkαξlαξkβξlβy,i{1,2,3},

where α, β ∈ {1, 2, 3} ∖ {y}, α < β, and k, l ∈ {1, 2, 3} ∖ {i}, k < l.

We claim that f is a polyconvex, non-convex function satisfying the structure condition (4.1) with suitable Fα and G satisfying the growth conditions (4.2) and (4.4), respectively.

As far as the stucture is concerned, it is easy to see that (1.1) holds, if we define, for all α ∈ {1, 2, 3} and λ ∈ ℝ3

Fα(λ)=F(λ):=|λ|p

and, for all ξ ∈ ℝ3×3,

G(ξ):=h((adj2ξ)11),

with

h(t):=(1+|t1|)r,tR.

The polyconvexity of f follows from the convexity of F and h (the latter holds true since r ≥ 1), see e.g. [13]. Let us now prove that f is not convex. Consider the matrices ξ1 := εξ̃ and ξ2 := –ξ1, ε > 0, where

ξ~:=000010001.

We shall prove that for ε > 0 sufficiently small

f(12(ξ1+ξ2))>12(f(ξ1)+f(ξ2)),(5.1)

thus establishing the claim. Indeed, on one hand the right hand side rewrites as

12(f(ξ1)+f(ξ2))=f(ξ1)=2εp+(1+|ε21|)r=:φ(ε),

while on the other hand the left hand side rewrites as

f(12(ξ1+ξ2))=f(0_)=φ(0).

Note that φC2((–1, 1)) since p > 2. Simple computations show that φ(0) = 0 and φ(0) = –r 2r < 0. Thus, for some δ ∈ (0, 1) and for all ε ∈ (0, δ) we have φ(ε) < φ(0) = 0. Thus φ(ε) < φ(0), and inequality (5.1) follows at once.

By using Theorem 4.1 we have that, if p(12(1+13),3) and r[1,p23+p), then the Wloc1,p-local minimizers of the corresponding functional ℱ are locally Hölder continuous.

We note that the arguments in [22, Theorem 1] do not apply since the function G is not differentiable.

Example 2

Let n, N ≥ 2. Given two matrices ξ1, ξ2 in ℝN×n such that rank(ξ1ξ2) > 1, define

K:={ξ1,ξ2}.

Denoting Qdist(⋅, K) the quasiconvex envelope of the distance function from K, we define, for q ≥ 1, the quasiconvex function G : ℝN×n → [0, +∞) by

G(ξ):=Qdist(ξ,K)(dist(ξ,coK))q,

where coK is the convex envelope of the set K. For all ϱ ∈ [0, 1] define the energy density fϱ : ℝN×n → ℝ,

fϱ(ξ):=ϱα=1n|ξα|p+(1ϱ)G(ξ),

and note that fϱ satisfies (1.1)-(1.4) and it is quasiconvex.

We claim that, fixed p ≥ 1, there exists ϱ0 > 0 such that, for every ϱ ∈ (0, ϱ0), fϱ is quasiconvex, but not convex. Given this for granted, by Theorem 1.1 we have that the Wloc1,p-local minimizers of the corresponding functional ℱ are locally Hölder continuous provided that 1q<p2n.

To prove the claim, we first observe that the function G is not convex, since G−1((–∞, 0]) turns out to be the set K, which is non-empty and non-convex. Indeed, by [14, Theorem 1.1, Example 4.3], the zero set of the quasiconvex function with linear growth ξQdist(ξ, K) is K. This implies G−1((–∞, 0]) = K.

Next we consider the set J := {ϱ ∈ [0, 1]: fϱ is convex} and note that J is non-empty, as 1 ∈ J, and closed, since convexity is stable under pointwise convergence. Since 0 ∉ J we can find ϱ0 > 0 such that [0, ϱ0)∩J = ∅. Hence, we conclude that fϱ is non-convex for ϱ ∈ [0, ϱ0).

Example 3

We give an example of an overall convex function f having non-convex principal part and convex lower order term.

Let 2 ≤ q < p < n, μ > 0, and B1 := {z ∈ ℝn : |z| < 1}. Given φCc(B1, [0, 1]) with φ(0) = 1 and D2φ(0) negative definite, let

F(ξ):=α=1NFα(ξα)

where Fα(λ)=h(λ):=(μ+|λ|2)p2 for α ∈ {2, …, N}, λ ∈ ℝn, F1(λ) := h(λ) + (λ), M > 0 to be chosen in what follows.

We claim that it is possible to find Mμ > 0 such that for every MMμ and for all η1 ∈ ℝn ∖ {0}

D2F1(0_)η1,η1<0.(5.2)

With this aim we first compute the Hessian matrices of Fα and F. Simple computations yield for all λ, ζ ∈ ℝn.

D2h(λ)ζ,ζ=p(μ+|λ|2)p22((μ+|λ|2)|ζ|2+(p2)λ,ζ2).(5.3)

Hence, if we set Fα(λ):=h(λ) and F(ξ):=α=1NFα(ξα)=α=1Nh(ξα), being p > 2, we get that

D2F(ξ)η,η=α=1ND2Fα(ξα)ηα,ηαpα=1N(μ+|ξα|2)p21|ηα|2pμp21|η|2.(5.4)

We are now ready to show that F1 is not convex. Indeed, we have

D2F1(0_)η1,η1=5.3pμp21|η1|2+MD2φ(0_)η1,η1pμp21|η1|2+MΛ|η1|2,(5.5)

where Λ < 0 is the maximum eigenvalue of D2φ(0). Hence, (5.2) follows at once provided that M > Mμ := pμp21|Λ|1.

In particular, the function F is not convex on ℝN×n, since it is not convex with respect to the variable ξ1. Indeed, if η̄ ∈ ℝN×n is such that η̄α = 0 for α ∈ {2, …, N} and |η̄1| > 0 we conclude that

D2F(0_)η¯,η¯=D2F1(0_)η¯1,η¯1<(5.2)0.

Let > 0 and set

G(ξ):=(+|ξ|2)q2,

and recall that for all ξ, η ∈ ℝN×n (cf. (5.4)) being q ≥ 2

D2G(ξ)η,η=q(+|ξ|2)q22((+|ξ|2)|η|2+(q2)ξ,η2)qq2|η|2.

To show that f := F + G is convex we compute its Hessian, being clearly fC2(ℝN×n). We have

D2f(ξ)η,η(5.4)pμp21|η|2+MD2φ(ξ1)η1,η1+qq2|η|2(pμp21+qq2)|η|2Msup|ξ1|1,|z|1|D2φ(ξ1)z,z||η1|2(pμp21+qq2Msup|ξ1|1,|z|1|D2φ(ξ1)z,z|)|η1|20,

if, for instance, >(q1Msup|ξ1|1,|z|1|D2φ(ξ1)z,z|)2q.

In conclusion, since f satisfies (1.1)-(1.4), its convexity assures the existence of Wloc1,p-local minimizers of the corresponding functional ℱ, which, in view of Theorem 1.1, are locally Hölder continuous.

Example 4

In what follows we construct an example of a convex energy density f satisfying (1.1)-(1.4) with G rank-one convex but not quasiconvex. With this aim we recall next the construction of Šverák’s celebrated example in [15] in some details, following the presentation given in the book [13]. With this aim consider

L:=ζR3×2:ζ=x00yzz wherex,y,zR,(5.6)

and let h : L → ℝ be given by

hx00yzz=xyz.

Let P : ℝ3×2L be defined as

P(ζ):=ζ1100ζ2212(ζ13+ζ23)12(ζ13+ζ23),

and set

gε,y(ζ):=h(P(ζ))+ε|ζ|2+ε|ζ|4+y|ζP(ζ)|2.

One can prove that there exists ε0 > 0 such that gε,y is not quasiconvex if ε ∈ (0, ε0) for every y ≥ 0 (cf. [13, Theorem 5.50, Step 3]). In addition, for every ε > 0 one can find y(ε) > 0 such that gε,y(ε) is rank-one convex (cf. [13, Theorem 5.50, Steps 4, 4’ and 4”]).

It is convenient to recall more details of the proof of the rank-one convexity of gε,y(ε). To begin with, since h is a homogeneous polynomial of degree three we have

D2h(P(ζ))z,zϑ|ζ||z|2(5.7)

for some ϑ > 0 and for all ζ, z ∈ ℝ3×2. It turns then out that

D2gε,y(ζ)z,z=D2h(P(ζ))z,z+2ε|z|2+4ε|ζ|2|z|2+8εζ,z2+2y|zP(z)|2.

for all ζ, z ∈ ℝ3×2. In particular, we conclude that for all ε > 0 and y ≥ 0

D2gε,y(ζ)z,z(4ε|ζ|ϑ)|ζ||z|2ϑ+14ε|z|2(5.8)

for all z ∈ ℝ3×2 provided that ζ ∈ ℝ3×2 is such that |ζ| ≥ ϑ+14ε. Note that the last inequality holds true independently of the fact that rank(z) = 1. Therefore, the uniform convexity of gε,y on ℝ3×2Bϑ+14ε follows (cf. [13, Theorem 5.50, Step 4’]). The appropriate choice of y(ε) establishes the rank-one convexity of gε,y(ε) on the bounded set Bϑ+14ε.

We set gε := gε,y(ε), for ε ∈ (0, ε0), in a way that gε is rank-one convex but not quasiconvex.

Let n ≥ 2 and N ≥ 3, let π : ℝN×n → ℝ3×2 be the projection

π(ξ)=ξ11ξ21ξ12ξ22ξ13ξ23,

and set

Gε(ξ):=gε(π(ξ))

where gε : ℝ3×2 → ℝ is defined above. Then Gε is rank-one convex and not quasiconvex (cf. [1.3, Theorem 5.50, Step 1]).

Let μ > 0 and Fα(λ):=(μ+|λ|2)p2 for all λ ∈ ℝn and α ∈ {1, …, N}. We claim that the function

f(ξ):=α=1NFα(ξα)+Gε(ξ)

is convex for μμε > 0 large enough. Given this for granted, f satisfies (1.1)-(1.4) with q = 4 and p ∈ (2n, n) if n ≥ 5. Therefore, we conclude in view of Theorem 1.1 that the Wloc1,p-local minimizers of the corresponding functional ℱ are locally Hölder continuous.

To prove the claim, since fC2(ℝN×n) we shall compute its Hessian. First note that F(ξ):=α=1NFα(ξα) is uniformly convex on ℝN×n in view of (5.4), which, together with (5.8), yields

D2f(ξ)η,η0(5.9)

for all ξ ∈ ℝN×n such that |π(ξ)| ≥ ϑ+14ε and for all η ∈ ℝN×n. In addition, using again that p > 2, by (5.7) we have

D2f(ξ)η,ηD2F(ξ)η,ηϑ|π(ξ)||π(η)|2(pμp21ϑ|π(ξ)|)|π(η)|2.(5.10)

Hence, the Hessian of f at ξ with |π(ξ)| < ϑ+14ε is non-negative provided that

μμε:=(ϑ(ϑ+1)4pε)2p2.(5.11)

Example 5

Finally, we give an example that exploits the full strength of the assumptions of Theorem 1.1 on the leading term. By keeping the notation introduced in Example 4, we shall modify F1 there to get a non-convex function so that the resulting principal term is non-convex. On the other hand, the sum + Gε turns out to be convex exploiting the uniform convexity of Gε on the subspace L for large values of the variable π(ξ) (cf. (5.6) and (5.8)).

Consider a function φ : ℝn → [0, 2], φCc(B3), with φ(0) = 2,

D2φ(0_)=2Idn×n(5.12)

and

supxRnsupzRn,|z|1|D2φ(x)z,z|=supzRn,|z|1|D2φ(0_)z,z|=2=|Λ|(5.13)

where Λ := –2 < 0 is the (unique) eigenvalue of D2φ(0) (see Lemma 5.1 below for the existence of such a function φ). Let

F~(ξ):=α=1NF~α(ξα)

where

F~α=Fαforα{2,,N},andF~1:=F1+Mφσ,

M > 0 to be chosen in what follows, and σ : ℝn → ℝn defined by

σ(ξ1):=(ξ11,0,,0).

Note that C2(ℝN×n). In particular, 1 = F1 for all ξ ∈ ℝN×nσ−1(suppφ), and for such points D21(ξ1) = D2F1(ξ1). Moreover, it is possible to find Mμ > 0 such that for every M > Mμ and for some η̄1 ∈ ℝn ∖ {0} (independent of M)

D2F~1(0_)η¯1,η¯1<0.(5.14)

Indeed, arguing as to obtain (5.5), and using (5.12), for all η̄1 ∈ ℝn such that |η̄1| = |σ(η̄1)| > 0 we get

D2F~1(0_)η¯1,η¯1(pμp21+ΛM)|σ(η¯1)|2<0,(5.15)

provided that

M>Mμ:=pμp21|Λ|1.(5.16)

In particular, the function is not convex on ℝN×n, since it is not convex with respect to the variable ξ1. Indeed, if η̄ ∈ ℝN×n is such that η̄α = 0 for α ∈ {2, …, N} and |η̄1| = |σ(η̄1)| > 0 we conclude that

D2F~(0_)η¯,η¯=D2F~1(0_)η¯1,η¯1<(5.15)0.

For fix ε ∈ (0, ε0) consider zε ∈ ℝn given by zε := (ϑ+14ε + 3, 0, …, 0).

Let Gε be the rank-one convex, non-quasiconvex function introduced in Example 4, we claim that the integrand

f~(ξ):=F~(ξ)+Gε(ξ1zε,ξ2,,ξN)=α=1N(μ+|ξα|2)p2+Mφ(σ(ξ1))+Gε(ξ1zε,ξ2,,ξN)

is convex for all μμε=(ϑ(ϑ+1)4pε)2p2 and for all MMμ + |Λ|−1 ϑ+14ε (the value of με has been introduced in (5.11)).

With this aim, it suffices to check the Hessian of being C2(ℝN×n). First note that coincides with a variant of the function f in Example 3 on the open set Σ := {ξ ∈ ℝN×n : σ(ξ1) ∉ suppφ}. More precisely, if ξΣ

f~(ξ)=F(ξ)+Gε(ξ1zε,ξ2,,ξN).

Then its convexity for all ξ ∈ ℝN×n such that |(ξ1zε, ξ2, …, ξN)| ≥ ϑ+14ε follows at once from (5.8) and (5.4). Instead, if |(ξ1zε, ξ2, …, ξN)| < ϑ+14ε, arguing as in (5.10), we get

D2f~(ξ)η,ηD2F(ξ)η,ηϑ|π(ξ1zε,ξ2,,ξN)||π(η)|2(5.4)(pμp21ϑϑ+14ε)|π(η)|20

thanks to the choice μμε.

On the other hand, by the convexity of each Fα with respect to ξα (cf. (5.3)) and by taking into account that σ(ξ1) ∈ suppφB3 yields |π(ξ1zε)||ξ11ϑ+14ε3|ϑ+14ε, we have

D2f~(ξ)η,η(5.4)pμp21|η|2+MD2(φσ)(ξ1)η1,η1+D2Gε(ξ1zε,ξ2,,ξN)η,η=pμp21|η|2+MD2φ(σ(ξ1))σ(η1),σ(η1)+D2Gε(ξ1zε,ξ2,,ξN)η,η(5.8)pμp21|η|2Msup|σ(ξ1)|3,|z|1|D2φ(σ(ξ1))z,z||σ(η1)|2+ϑ+14ε|π(η)|2(pμp21Msup|σ(ξ1)|3,|z|1|D2φ(σ(ξ1))z,z|+ϑ+14ε)|σ(η1)|2.

Thus, the Hessian of at such ξ’s is nonnegative provided that

pμp21+ϑ+14εMsup|σ(ξ1)|3,|z|1|D2φ(σ(ξ1))z,z|.

In conclusion, we have to ensure the following two inequalities

Mμ=pμp21|Λ|1<M(sup|σ(ξ1)|3,|z|1|D2φ(σ(ξ1))z,z|)1(pμp21+ϑ+14ε).(5.17)

Since by (5.13)

sup|σ(ξ1)|3,|z|1|D2φ(σ(ξ1))z,z|=|Λ|,

then (5.17) holds for every M such that Mμ < MMμ + |Λ|−1 ϑ+14ε.

In conclusion, since satisfies (1.1)-(1.4) with q = 4 and p ∈ (2n, n) if n ≥ 5, its convexity assures the existence of Wloc1,p-local minimizers of the corresponding functional ℱ͠, which, in view of Theorem 1.1, are locally Hölder continuous.

Lemma 5.1

There exists a function φ : ℝn → [0, 2], φCc(B3), with φ(0) = 2,

D2φ(0_)=2Idn×n

and

supxRnsupηRn,|η|1|D2φ(x)η,η|=supηRn,|η|1|D2φ(0_)η,η|=2.

Proof

Define ϕ : ℝ → [0, ∞),

ϕ(t):=(t+2)2ift[2,1]2t2ift(1,1)(t2)2ift[1,2]0elsewhere.

We have that ϕC1,1(ℝ), ϕC(ℝ ∖ {–2, –1, 1, 2}) and

max{|ϕ(t)|,|ϕ(t)t|}2tR{2,1,0,1,2}.(5.18)

Let us define Φ : ℝn → [0, 2], by Φ(x) := ϕ(|x|). Then ΦC2(ℝn ∖ {x : |x| ∈ {1, 2}}),

DΦ(0_)=0_,DΦ(x)=ϕ(|x|)x|x|if|x|0,

and

D2Φ(x)=ϕ(|x|)|x|Idn×n+ϕ(|x|)ϕ(|x|)|x|x|x|x|x|if|x|0,1,2

and D2Φ(0) = ϕ″(0)Idn×n.

In particular, ϕ″(0) is the only eigenvalue of D2Φ(0). Moreover, we claim that if |x| ≠ 0 then the eigenvalues of D2Φ(x) are ϕ(|x|)|x| and ϕ″(|x|). Indeed, if |x| ≠ 0,

D2Φ(x)v=ϕ(|x|)|x|v

for every vRn,vx|x|; moreover, if |x| ≠ 0, 1, 2,

D2Φ(x)x=ϕ(|x|)x

Therefore, using (5.18), if |x| ∉ {0, 1, 2}

supηRn,|η|1|D2Φ(x)η,η|max|ϕ(|x|)|,|ϕ(|x|)||x|2,

that, taking into account that sup|η|≤1| 〈D2Φ(0)η,η〉 | = | ϕ″(0) | = 2, implies

ess-supxRnsupηRn,|η|1|D2Φ(x)η,η|=2=|ϕ(0)|.

Let us now consider a family of positive radial symmetric mollifiers ρεCc(Bε), ε ∈ (0, 1), such that ∫n ρε(x) dx = 1. Consider Φε := Φ * ρε. It is easy to check that ΦεCc(B2+ε) and, since ΦC1,1(ℝn), for all x ∈ ℝn we get

D2Φε(x)=(D2Φρε)(x).

Moreover, for all x, η ∈ ℝn it holds

D2Φε(x)η,η=RnD2Φ(xy)η,ηρε(y)dy,

then we have

supηRn,|η|1|D2Φε(x)η,η|RnsupηRn,|η|1|D2Φ(xy)η,η|ρε(y)dy2.

Since D2Φ(x) = –2 Idn×n if |x| < 1, there exists ε0 ∈ (0, 1) small enough so that D2Φε0(0) = –2 Idn×n. Thus,

supxRnsupηRn,|η|1|D2Φε0(x)η,η|=supηRn,|η|1|D2Φε0(0_)η,η|=2.

The conclusion then follows on setting φ := Φε0.□

Acknowledgement

The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Cupini has been supported by University of Bologna, Leonetti by University of L’Aquila.

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About the article

Received: 2018-07-31

Accepted: 2019-06-26

Published Online: 2019-08-28

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 1008–1025, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0039.

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© 2020 G. Cupini et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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