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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Global regularity for systems with p-structure depending on the symmetric gradient

Luigi C. BerselliORCID iD: https://orcid.org/0000-0001-6208-9934 / Michael Růžička
Published Online: 2018-10-11 | DOI: https://doi.org/10.1515/anona-2018-0090

Abstract

In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having p-structure depending only on the symmetric part of the gradient.

Keywords: Regularity of weak solutions; symmetric gradient; boundary regularity; natural quantities

MSC 2010: 76A05; 35D35; 35Q35

1 Introduction

In this paper we study regularity of weak solutions to the boundary value problem

{-div𝐒(𝐃𝐮)=𝐟in Ω,𝐮=𝟎on Ω,(1.1)

where 𝐃𝐮:=12(𝐮+𝐮) denotes the symmetric part of the gradient 𝐮 and Ω3 is a bounded domain with a C2,1 boundary Ω.1 Our interest in this system comes from the p-Stokes system

{-div𝐒(𝐃𝐮)+π=𝐟in Ω,div𝐮=0in Ω,𝐮=𝟎on Ω.(1.2)

In both problems the typical example for 𝐒 we have in mind is

𝐒(𝐃𝐮)=μ(δ+|𝐃𝐮|)p-2𝐃𝐮,

where p(1,2], δ0 and μ>0. In previous investigations of (1.2), only suboptimal results for the regularity up to the boundary have been proved. Here we mean suboptimal in the sense that the results are weaker than the results known for p-Laplacian systems, cf. [1, 13, 14]. Clearly, system (1.1) is obtained from (1.2) by dropping the divergence constraint and the resulting pressure gradient. Thus, system (1.1) lies in between system (1.2) and p-Laplacian systems, which depend on the full gradient 𝐮.

We would like to stress that system (1.1) is of its own independent interest, since it is studied within plasticity theory, when formulated in the framework of deformation theory (cf. [11, 24]). In this context the unknown is the displacement vector field 𝐮=(u1,u2,u3), while the external body force 𝐟=(f1,f2,f3) is given. The stress tensor 𝐒, which is the tensor of small elasto-plastic deformations, depends only on 𝐃𝐮. Physical interpretation and discussion of both systems (1.1) and (1.2) and the underlying models can be found, e.g., in [5, 11, 15, 19, 20].

We study global regularity properties of weak solutions to (1.1) in sufficiently smooth and bounded domains Ω; we obtain, for all p(1,2], the optimal result, namely, that 𝐅(𝐃𝐮) belongs to W1,2(Ω), where the nonlinear tensor-valued function 𝐅 is defined in (2.5). This result has been proved near a flat boundary in [24] and is the same result as for p-Laplacian systems (cf. [1, 13, 14]). The situation is quite different for (1.2). There the optimal result, i.e., 𝐅(𝐃𝐮)W1,2(Ω), is only known for

  • (i)

    two-dimensional bounded domains (cf. [16], where even the p-Navier–Stokes system is treated);

  • (ii)

    the space-periodic problem in d, d2, which follows immediately from interior estimates, i.e., 𝐅(𝐃𝐮)Wloc1,2(Ω), which are known in all dimensions, and the periodicity of the solution;

  • (iii)

    if the no-slip boundary condition is replaced by perfect slip boundary conditions (cf. [17]);

  • (iv)

    in the case of small 𝐟 (cf. [6]).

We also observe that the above results for the p-Stokes system (apart those in the space periodic setting) require the stress tensor to be non-degenerate, that is, δ>0. In the case of homogeneous Dirichlet boundary conditions and three- and higher-dimensional bounded, sufficiently smooth domains only suboptimal results are known. To our knowledge, the state of the art for general data is that 𝐅(𝐃𝐮)Wloc1,2(Ω), tangential derivatives of 𝐅(𝐃𝐮) near the boundary belong to L2, while the normal derivative of 𝐅(𝐃𝐮) near the boundary belongs to some Lq, where q=q(p)<2 (cf. [2, 4] and the discussion therein). We would also like to mention a result for another system between (1.2) and p-Laplacian systems, namely, if (1.2) is considered with 𝐒 depending on the full velocity gradient 𝐮. In this case, it is proved in [7] that 𝐮W2,r(3)W01,p(3) for some r>3, provided p<2 is very close to 2.

In the present paper we extend the optimal regularity result for (1.1) of Seregin and Shilkin [24] in the case of a flat boundary to the general case of bounded sufficiently smooth domains and to possibly degenerate stress tensors, that is, the case δ=0. The precise result we prove is the following:

Theorem 1.1.

Let the tensor field S in (1.1) have (p,δ)-structure (cf. Definition 2.1) for some p(1,2] and δ[0,), and let F be the associated tensor field to S defined in (2.5). Let ΩR3 be a bounded domain with C2,1 boundary, and let fLp(Ω). Then the unique weak solution uW01,p(Ω) of problem (1.1) satisfies

Ω|𝐅(𝐃𝐮)|2d𝐱c,

where c denotes a positive function which is non-decreasing in fp and δ, and which depends on the domain through its measure |Ω| and the C2,1-norms of the local description of Ω. In particular, the above estimate implies that uW2,3pp+1(Ω).

2 Notations and preliminaries

In this section we introduce the notation we will use, state the precise assumptions on the extra stress tensor 𝐒, and formulate the main results of the paper.

2.1 Notation

We use c,C to denote generic constants which may change from line to line, but they are independent of the crucial quantities. Moreover, we write fg if and only if there exists constants c,C>0 such that cfgCf. In some cases we need to specify the dependence on certain parameters, and consequently we denote by c() a positive function which is non-decreasing with respect to all its arguments. In particular, we denote by c(δ-1) a possibly critical dependence on the parameter δ as δ0, while c(δ) only indicates that the constant c depends on δ and will satisfy c(δ)c(δ0) for all δδ0.

We use standard Lebesgue spaces (Lp(Ω),p) and Sobolev spaces (Wk,p(Ω), k,p), where Ω3 is a sufficiently smooth bounded domain. The space W01,p(Ω) is the closure of the compactly supported, smooth functions C0(Ω) in W1,p(Ω). Thanks to the Poincaré inequality, we equip W01,p(Ω) with the gradient norm p. When dealing with functions defined only on some open subset GΩ, we denote the norm in Lp(G) by p,G. As usual, we use the symbol to denote weak convergence, and to denote strong convergence. The symbol sptf denotes the support of the function f. We do not distinguish between scalar, vector-valued or tensor-valued function spaces. However, we denote vectors by boldface lower-case letter as, e.g., 𝐮 and tensors by boldface upper case letters as, e.g., 𝐒. For vectors 𝐮,𝐯3, we denote 𝐮𝑠𝐯:=12(𝐮𝐯+(𝐮𝐯)), where the standard tensor product 𝐮𝐯3×3 is defined as (𝐮𝐯)ij:=uivj. The scalar product of vectors is denoted by 𝐮𝐯=i=13uivi and the scalar product of tensors is denoted by 𝐀𝐁:=i,j=13AijBij.

Greek lower-case letters take only the values 1,2, while Latin lower-case ones take the values 1,2,3. We use the summation convention over repeated indices only for Greek lower-case letters, but not for Latin lower-case ones.

2.2 (p,δ)-structure

We now define what it means that a tensor field 𝐒 has (p,δ)-structure, see [8, 23]. For a tensor 𝐏3×3, we denote its symmetric part by 𝐏sym:=12(𝐏+𝐏)sym3×3:={𝐏3×3𝐏=𝐏}. We use the notation |𝐏|2=𝐏𝐏.

It is convenient to define for t0 a special N-function2 ϕ()=ϕp,δ(), for p(1,), δ0, by

ϕ(t):=0t(δ+s)p-2s𝑑s.(2.1)

The function ϕ satisfies, uniformly in t and independently of δ, the important equivalences

ϕ′′(t)tϕ(t),(2.2)ϕ(t)tϕ(t),tp+δpϕ(t)+δp.(2.3)

We use the convention that if ϕ′′(0) does not exist, the left-hand side in (2.2) is continuously extended by zero for t=0. We define the shifted N-functions {ϕa}a0 (cf. [8, 9, 23]), for t0, by

ϕa(t):=0tϕ(a+s)sa+s𝑑s

Note that the family {ϕa}a0 satisfies the Δ2-condition uniformly with respect to a0, i.e., ϕa(2t)c(p)ϕa(t) holds for all t0.

Definition 2.1 ((p,δ)-structure).

We say that a tensor field 𝐒:3×3sym3×3, belonging to C0(3×3,sym3×3)C1(3×3{𝟎},sym3×3), satisfying 𝐒(𝐏)=𝐒(𝐏sym) and 𝐒(𝟎)=𝟎, possesses (p,δ)-structure if for some p(1,), δ[0,), and the N-function ϕ=ϕp,δ (cf. (2.1)), there exist constants κ0,κ1>0 such that

i,j,k,l=13klSij(𝐏)QijQklκ0ϕ′′(|𝐏sym|)|𝐐sym|2,|klSij(𝐏)|κ1ϕ′′(|𝐏sym|)(2.4)

for all 𝐏,𝐐3×3, with 𝐏sym𝟎, and all i,j,k,l=1,2,3. The constants κ0, κ1 and p are called the characteristics of 𝐒.

Remark 2.2.

  • (i)

    Assume that 𝐒 has (p,δ)-structure for some δ[0,δ0]. Then, if not otherwise stated, the constants in the estimates depend only on the characteristics of 𝐒 and on δ0, but they are independent of δ.

  • (ii)

    An important example of a tensor field 𝐒 having (p,δ)-structure is given by 𝐒(𝐏)=ϕ(|𝐏sym|)|𝐏sym|-1𝐏sym. In this case, the characteristics of 𝐒, namely, κ0 and κ1, depend only on p and are independent of δ0.

  • (iii)

    For a tensor field 𝐒 with (p,δ)-structure, we have klSij(𝐏)=klSji(𝐏) for all i,j,k,l=1,2,3 and all 𝐏3×3, due to its symmetry. Moreover, from 𝐒(𝐏)=𝐒(𝐏sym), it follows that klSij(𝐏)=12klSij(𝐏sym)+12lkSij(𝐏sym) for all i,j,k,l=1,2,3 and all 𝐏3×3, and consequently klSij(𝐏)=lkSij(𝐏) for all i,j,k,l=1,2,3 and all 𝐏sym3×3.

To a tensor field 𝐒 with (p,δ)-structure, we associate the tensor field 𝐅:3×3sym3×3 defined through

𝐅(𝐏):=(δ+|𝐏sym|)p-22𝐏sym.(2.5)

The connection between 𝐒, 𝐅, and {ϕa}a0 is best explained in the following proposition (cf. [8, 23]).

Proposition 2.3.

Let S have (p,δ)-structure, and let F be as defined in (2.5). Then

(𝐒(𝐏)-𝐒(𝐐))(𝐏-𝐐)|𝐅(𝐏)-𝐅(𝐐)|2ϕ|𝐏sym|(|𝐏sym-𝐐sym|)ϕ′′(|𝐏sym|+|𝐏sym-𝐐sym|)|𝐏sym-𝐐sym|2,|𝐒(𝐏)-𝐒(𝐐)|ϕ|𝐏sym|(|𝐏sym-𝐐sym|)(2.6)

uniformly in P,QR3×3. Moreover, uniformly in QR3×3,

𝐒(𝐐)𝐐|𝐅(𝐐)|2ϕ(|𝐐sym|).

The constants depend only on the characteristics of S.

For a detailed discussion of the properties of 𝐒 and 𝐅 and their relation to Orlicz spaces and N-functions, we refer the reader to [23, 3]. Since in the following we shall insert into 𝐒 and 𝐅 only symmetric tensors, we can drop in the above formulas the superscript “sym” and restrict the admitted tensors to symmetric ones.

We recall that the following equivalence, which is proved in [3, Lemma 3.8], is valid for all smooth enough symmetric tensor fields 𝐐sym3×3:

|i𝐅(𝐐)|2ϕ′′(|𝐐|)|i𝐐|2.(2.7)

The proof of this equivalence is based on Proposition 2.3. This proposition and the theory of divided differences also imply (cf. [4, Equation (2.26)]) that

|τ𝐅(𝐐)|2ϕ′′(|𝐐|)|τ𝐐|2(2.8)

for all smooth enough symmetric tensor fields 𝐐sym3×3.

A crucial observation in [24] is that the quantities in (2.7) are also equivalent to several further quantities. To formulate this precisely, we introduce, for i=1,2,3 and for sufficiently smooth symmetric tensor fields 𝐐, the quantity

𝒫i(𝐐):=i𝐒(𝐐)i𝐐=k,l,m,n=13klSmn(𝐐)iQkliQmn.(2.9)

Recall that in the definition of 𝒫i(𝐐) there is no summation convention over the repeated Latin lower-case index i in i𝐒(𝐐)i𝐐. Note that if 𝐒 has (p,δ)-structure, then 𝒫i(𝐐)0 for i=1,2,3. The following important equivalences hold, first proved in [24].

Proposition 2.4.

Assume that S has (p,δ)-structure. Then the following equivalences are valid, for all smooth enough symmetric tensor fields Q and all i=1,2,3:

𝒫i(𝐐)ϕ′′(|𝐐|)|i𝐐|2|i𝐅(𝐐)|2,(2.10)𝒫i(𝐐)|i𝐒(𝐐)|2ϕ′′(|𝐐|),(2.11)

with the constants only depending on the characteristics of S.

Proof.

The assertions are proved in [24] using a different notation. For the convenience of the reader, we sketch the proof here. The equivalences in (2.10) follow from (2.7), (2.9) and the fact that 𝐒 has (p,δ)-structure. Furthermore, using (2.10), we have

|𝒫i(𝐐)|2|i𝐒(𝐐)|2|i𝐐|2c|i𝐒(𝐐)|2𝒫i(𝐐)ϕ′′(|𝐐|),

which proves one inequality of (2.11). The other one follows from

|i𝐒(𝐐)|2ck,l=13|kl𝐒(𝐐)iQkl|2c(ϕ′′(|𝐐|))2|i𝐐|2cϕ′′(|𝐐|)𝒫i(𝐐),

where we used (2.4) and (2.10). ∎

2.3 Existence of weak solutions

In this section we define weak solutions of (1.1), recall the main results of existence and uniqueness and discuss a perturbed problem, which is used to justify the computations that follow. From now on, we restrict ourselves to the case p2.

Definition 2.5.

We say that 𝐮W01,p(Ω) is a weak solution to (1.1) if for all 𝐯W01,p(Ω),

Ω𝐒(𝐃𝐮)𝐃𝐯𝑑𝐱=Ω𝐟𝐯𝑑𝐱.

We have the following standard result.

Proposition 2.6.

Let the tensor field S in (1.1) have (p,δ)-structure for some p(1,2] and δ[0,). Let ΩR3 be a bounded domain with C2,1 boundary, and let fLp(Ω). Then there exists a unique weak solution u to (1.1) such that

Ωϕ(|𝐃𝐮|)𝑑𝐱c(𝐟p,δ).

Proof.

The assertions follow directly from the assumptions by using the theory of monotone operators. ∎

In order to justify some of the following computations, we find it convenient to consider a perturbed problem, where we add to the tensor field 𝐒 with (p,δ)-structure a linear perturbation. Using again the theory of monotone operators one can easily prove the following proposition.

Proposition 2.7.

Let the tensor field S in (1.1) have (p,δ)-structure for some p(1,2] and δ[0,), and let fLp(Ω) be given. Then there exists a unique weak solution uεW01,2(Ω) of the problem

{-div𝐒ε(𝐃𝐮ε)=𝐟in Ω,𝐮ε=𝟎on Ω,(2.12)

where

𝐒ε(𝐐):=ε𝐐+𝐒(𝐐),with ε>0,

i.e., uε satisfies, for all vW01,2(Ω),

Ω𝐒ε(𝐃𝐮ε)𝐃𝐯𝑑𝐱=Ω𝐟𝐯𝑑𝐱.

The solution uε satisfies the estimate

Ωε|𝐮ε|2+ϕ(|𝐃𝐮ε|)d𝐱c(𝐟p,δ).(2.13)

Remark 2.8.

In fact, one could already prove more at this point. Namely, that for ε0, the unique solution 𝐮ε converges to the unique weak solution 𝐮 of the unperturbed problem (1.1). Let us sketch the argument only, since later we get the same result with different easier arguments. From (2.13) and the properties of 𝐒 follows that

𝐮ε𝐮in W01,p(Ω),𝐒(𝐃𝐮ε)𝝌in Lp(Ω).

Passing to the limit in the weak formulation of the perturbed problem, we get

Ω𝝌𝐃𝐯𝑑𝐱=Ω𝐟𝐯𝑑𝐱for all 𝐯W01,p(Ω).

One can not show directly that limε0Ωε𝐃𝐮ε(𝐃𝐮ε-𝐃𝐮)𝑑𝐱=0, since 𝐃𝐮 belongs to Lp(Ω) only. Instead one uses the Lipschitz truncation method (cf. [10, 22]). Denoting by 𝐯ε,j the Lipschitz truncation of ξ(𝐮ε-𝐮), where ξC0(Ω) is a localization, one can show, using the ideas from [10, 22], that

lim supε0|Ω(𝐒(𝐃𝐮ε)-𝐒(𝐃𝐮))𝐃𝐯ε,jd𝐱|=0,

which implies 𝐃𝐮ε𝐃𝐮 almost everywhere in Ω. Consequently, we have 𝝌=𝐒(𝐃𝐮), since weak and a.e. limits coincide.

2.4 Description and properties of the boundary

We assume that the boundary Ω is of class C2,1, that is, for each point PΩ, there are local coordinates such that, in these coordinates, we have P=0 and Ω is locally described by a C2,1-function, i.e., there exist RP,RP(0,),rP(0,1) and a C2,1-function aP:BRP2(0)BRP1(0) such that

  • (b1)

    𝐱Ω(BRP2(0)×BRP1(0))x3=aP(x1,x2),

  • (b2)

    ΩP:={(x,x3)x=(x1,x2)BRP2(0),aP(x)<x3<aP(x)+RP}Ω,

  • (b3)

    aP(0)=𝟎, and for all x=(x1,x2)BRP2(0), |aP(x)|<rP,

where Brk(0) denotes the k-dimensional open ball with center 0 and radius r>0. Note that rP can be made arbitrarily small if we make RP small enough. In the sequel, we will also use, for 0<λ<1, the following scaled open sets:

λΩP:={(x,x3)x=(x1,x2)BλRP2(0),aP(x)<x3<aP(x)+λRP}ΩP.

To localize near to ΩΩP for PΩ, we fix smooth functions ξP:3 such that

  • (A1)

    χ12ΩP(𝐱)ξP(𝐱)χ34ΩP(𝐱),

where χA(𝐱) is the indicator function of the measurable set A. For the remaining interior estimate, we localize by a smooth function 0ξ001, with sptξ00Ω00, where Ω00Ω is an open set such that dist(Ω00,Ω)>0. Since the boundary Ω is compact, we can use an appropriate finite sub-covering which, together with the interior estimate, yields the global estimate.

Let us introduce the tangential derivatives near the boundary. To simplify the notation, we fix PΩ, h(0,RP16), and simply write ξ:=ξP, a:=aP. We use the standard notation 𝐱=(x,x3) and denote by 𝐞i, i=1,2,3, the canonical orthonormal basis in 3. In the following, lower-case Greek letters take the values 1,2. For a function g, with sptgsptξ, we define, for α=1,2,

gτ(x,x3)=gτα(x,x3):=g(x+h𝐞α,x3+a(x+h𝐞α)-a(x)),

and if Δ+g:=gτ-g, we define tangential divided differences by d+g:=h-1Δ+g. It holds that, if gW1,1(Ω), then we have for α=1,2

d+gτg=ταg:=αg+αa3gas h0,(2.14)

almost everywhere in sptξ (cf. [18, Section 3]). Conversely, uniform Lq-bounds for d+g imply that τg belongs to Lq(sptξ). For simplicity, we denote a:=(1a,2a,0). The following variant of integration by parts will be often used.

Lemma 2.9.

Let sptgsptfsptξ, and let h be small enough. Then

Ωfg-τ𝑑𝐱=Ωfτg𝑑𝐱.

Consequently, Ωfd+g𝑑x=Ω(d-f)g𝑑x. Moreover, if in addition f and g are smooth enough and at least one vanishes on Ω, then

Ωfτgd𝐱=-Ω(τf)g𝑑𝐱.

3 Proof of the main result

In the proof of the main result we use finite differences to show estimates in the interior and in tangential directions near the boundary, and for calculations involving directly derivatives in “normal” directions near the boundary. In order to justify that all occurring quantities are well posed, we perform the estimate for the approximate system (2.12).

The first intermediate step is the following result for the approximate problem.

Proposition 3.1.

Let the tensor field S in (1.1) have (p,δ)-structure for some p(1,2] and δ(0,), and let F be the associated tensor field to S. Let ΩR3 be a bounded domain with C2,1 boundary and let fLp(Ω). Then the unique weak solution uεW01,2(Ω) of the approximate problem (2.12) satisfies3

{Ωεξ02|2𝐮ε|2+ξ02|𝐅(𝐃𝐮ε)|2d𝐱c(𝐟p,ξ02,,δ),ΩεξP2|τ𝐃𝐮ε|2+ξP2|τ𝐅(𝐃𝐮ε)|2d𝐱c(𝐟p,ξP2,,aPC2,1,δ).(3.1)

Here ξ0 is a cut-off function with support in the interior of Ω, while for arbitrary PΩ the function ξP is a cut-off function with support near to the boundary Ω, as defined in Section 2.4. The tangential derivative τ is defined locally in ΩP by (2.14). Moreover, there exists a constant C1>0 such that4

Ωεξ2|3𝐃𝐮ε|2+ξ2|3𝐅(𝐃𝐮ε)|2d𝐱c(𝐟p,ξ2,,aC2,1,δ-1,ε-1,C1),(3.2)

provided that in the local description of the boundary, we have rP<C1 in (b3).

In particular, these estimates imply that uεW2,2(Ω) and that (2.12) holds pointwise a.e. in Ω.

The two estimates in (3.1) are uniform with respect to ε and could be also proved directly for problem (1.1). However, the third estimate (3.2) depends on ε and it is needed to justify all subsequent steps, which will give the proof of an estimate uniformly in ε, by using a different technique.

Proof of Proposition 3.1.

The proof of estimate (3.1) is very similar, being in fact a simplification (due to the fact that there is no pressure term involved) to the proof of the results in [4, Theorems 2.28 and 2.29]. On the other hand, the proof of (3.2) is different from the one in [4] due to the missing divergence constraint. In fact, it adapts techniques known from nonlinear elliptic systems. For the convenience of the reader, we recall the main steps here.

Fix PΩ and use in ΩP

𝐯=d-(ξ2d+(𝐮ε|12ΩP)),

where ξ:=ξP, a:=aP, and h(0,RP16), as a test function in the weak formulation of (2.12). This yields

Ωξ2d+𝐒ε(𝐃𝐮ε)d+𝐃𝐮ε𝑑𝐱=-Ω𝐒ε(𝐃𝐮ε)(ξ2d+3𝐮ε-(ξ-τd-ξ+ξd-ξ)3𝐮ε)𝑠d-ad𝐱-Ω𝐒ε(𝐃𝐮ε)ξ2(3𝐮ε)τ𝑠d-d+a-𝐒ε(𝐃𝐮ε)d-(2ξξ𝑠d+𝐮ε)d𝐱+Ω𝐒ε((𝐃𝐮ε)τ)(2ξ3ξd+𝐮ε+ξ2d+3𝐮ε)𝑠d+ad𝐱+Ω𝐟d-(ξ2d+𝐮ε)d𝐱=:j=18Ij.

From the assumption on 𝐒, Proposition 2.3 and [4, Lemma 3.11], we have the following estimate:

Ωεξ2|d+𝐮ε|2+εξ2|d+𝐮ε|2+|d+𝐅(𝐃𝐮ε)|2+ϕ(ξ|d+𝐮|)+ϕ(ξ|d+𝐮|)d𝐱cΩξ2d+𝐒ε(𝐃𝐮ε)d+𝐃𝐮ε𝑑𝐱+c(ξ1,,aC1,1)Ωsptξϕ(|𝐮ε|)𝑑𝐱.

The terms I1I7 are estimated exactly as in [4, Equations (3.17)–(3.22)], while I8 is estimated as the term I15 in [4, (4.20)]. Thus, we get

Ωεξ2|d+𝐮ε|2+εξ2|d+𝐮ε|2+ξ2|d+𝐅(𝐃𝐮ε)|2+ϕ(ξ|d+𝐮ε|)+ϕ(ξ|d+𝐮ε|)d𝐱c(𝐟p,ξ2,,aC2,1,δ).

This proves the second estimate in (3.1) by standard arguments. The first estimate in (3.1) is proved in the same way with many simplifications, since we work in the interior where the method works for all directions. This estimate implies that 𝐮εWloc2,2(Ω) and that the system (2.12) is well-defined point-wise a.e. in Ω.

To estimate the derivatives in the x3 direction, we use equation (2.12) and it is at this point that we have changes with respect to the results in [4]. In fact, as usual in elliptic problems, we have to recover the partial derivatives with respect to x3 by using the information on the tangential ones. In this problem the main difficulty is that the leading order term is nonlinear and depends on the symmetric part of the gradient. Thus, we have to exploit the properties of (p,δ)-structure of the tensor 𝐒 (cf. Definition 2.1). Denoting, for i=1,2,3,5 𝔣i:=-fi-γσSi3(𝐃𝐮ε)3Dγσ𝐮ε-k,l=13klSiβ(𝐃𝐮ε)βDkl𝐮ε, we can re-write the equations in (2.12) as follows:

k=13k3Si3(𝐃𝐮ε)3Dk3𝐮ε+3αSi3(𝐃𝐮ε)3D3α𝐮ε=𝔣ia.e. in Ω.

Contrary to the corresponding equality [4, Equation (3.49)], here we use directly all the equations in (1.1), and not only the first two. Now we multiply these equations not by 3Di3𝐮ε as expected, but by 3D^i3𝐮ε, where D^αβ𝐮ε=0 for α,β=1,2, D^α3𝐮ε=D^3α𝐮ε=2Dα3𝐮ε for α=1,2, D^33𝐮ε=D33𝐮ε. Summing over i=1,2,3, we get, by using the symmetries in Remark 2.2 (iii), that

4α3Sβ3ε(𝐃𝐮ε)3Dα3𝐮ε3Dβ3𝐮ε+2α3S33ε(𝐃𝐮ε)3Dα3𝐮ε3D33𝐮ε+233Sβ3ε(𝐃𝐮ε)3D33𝐮ε3Dβ3𝐮ε+33S33ε(𝐃𝐮ε)3D33𝐮ε3D33𝐮ε=i=13𝔣i3D^i3𝐮εa.e. in Ω.(3.3)

To obtain a lower bound for the left-hand side, we observe that the terms on the left-hand side of (3.3) containing 𝐒 are equal to

i,j,k,l=13klSij(𝐃𝐮ε)QijQkl

if we choose 𝐐=3𝐃¯𝐮ε, where D¯αβ𝐮ε=0 for α,β=1,2, D¯α3𝐮ε=D¯3α𝐮ε=Dα3𝐮ε for α=1,2, and D¯33𝐮ε=D33𝐮ε. Thus, it follows from the coercivity estimate in (2.4) that these terms are bounded from below by κ0ϕ′′(|𝐃𝐮ε|)|3𝐃¯𝐮ε|2. Similarly, we see that the remaining terms on the left-hand side of (3.3) are equal to ε|3𝐃¯𝐮ε|2. Denoting 𝔟i:=3Di3𝐮ε, i=1,2,3, we see that |𝔟||𝐃^𝐮ε||𝐃¯𝐮ε|. Consequently, we get from (3.3) the estimate

(ε+ϕ′′(|𝐃𝐮ε|))|𝖇||𝖋|a.e. in Ω.

By straightforward manipulations (cf. [4, Sections 3.2 and 4.2]), we can estimate the right-hand side as follows:

|𝖋|c(|𝐟|+(ε+ϕ′′(|𝐃𝐮ε|))(|τ𝐮ε|+a|2𝐮ε|)).

Note that we can deduce from 𝖇 information about 𝔟~i:=332uεi, i=1,2,3, because

|𝖇|2|𝖇~|-|τ𝐮ε|-a|2𝐮ε|

holds a.e. in ΩP. This and the last two inequalities imply

(ε+ϕ′′(|𝐃𝐮ε|))|𝖇~|c(|𝐟|+(ε+ϕ′′(|𝐃𝐮ε|))(|τ𝐮ε|+a|2𝐮ε|))a.e. in ΩP.

Adding on both sides, for α=1,2 and i,k=1,2,3, the term

(ε+ϕ′′(|𝐃𝐮ε|))|αiuεk|,

and using on the right-hand side the definition of the tangential derivative (cf. (2.14)), we finally arrive at

(ε+ϕ′′(|𝐃𝐮ε|))|2𝐮ε|c(|𝐟|+(ε+ϕ′′(|𝐃𝐮ε|))(|τ𝐮ε|+a|2𝐮ε|)),

which is valid a.e. in ΩP. Note that the constant c only depends on the characteristics of 𝐒. Next, we can choose the open sets ΩP in such a way that aP(x),ΩP is small enough, so that we can absorb the last term from the right-hand side, which yields

(ε+ϕ′′(|𝐃𝐮ε|))|2𝐮ε|c(|𝐟|+(ε+ϕ′′(|𝐃𝐮ε|))|τ𝐮ε|)a.e. in ΩP,

where again the constant c only depends on the characteristics of 𝐒. By neglecting the second term on the left-hand side (which is non-negative), raising the remaining inequality to the power 2, and using that 𝐒 has (p,δ)-structure for p<2, we obtain

ΩεξP2|2𝐮ε|2d𝐱cΩ|𝐟|2d𝐱+(ε+δ2(p-2))ε(εΩξP2|τ𝐮ε|2d𝐱).

The already proven results on tangential derivatives and Korn’s inequality imply that the last integral from the right-hand side is finite. Thus, the properties of the covering imply the estimate in (3.2). ∎

3.1 Improved estimates for normal derivatives

In the proof of (3.2), we used system (2.12) and obtained an estimate that is not uniform with respect to ε. In this section, by following the ideas in [24], we proceed differently and estimate 𝒫3 in terms of the quantities occurring in (3.1). The main technical step of the paper is the proof of the following result.

Proposition 3.2.

Let the hypotheses in Theorem 1.1 be satisfied with δ>0, and let the local description aP of the boundary and the localization function ξP satisfy (b1)–(b3) and (1) (cf. Section 2.4). Then there exists a constant C2>0 such that the weak solution uεW01,2(Ω) of the approximate problem (2.12) satisfies,6 for every PΩ,

ΩεξP2|3𝐃𝐮ε|2+ξP2|3𝐅(𝐃𝐮ε)|2d𝐱C(𝐟p,ξP2,,aPC2,1,δ,C2),

provided rP<C2 in (b3).

Proof.

Let us fix an arbitrary point PΩ and a local description a=aP of the boundary and the localization function ξ=ξP satisfying (b1)–(b3) and (1). In the following we denote by C constants that depend only on the characteristics of 𝐒. First we observe that, by the results of Proposition 2.4, there exists a constant C0, depending only on the characteristics of 𝐒, such that

1C0|3𝐅(𝐃𝐮ε)|2𝒫3(𝐃𝐮ε)a.e. in Ω.

Thus, using also the symmetry of 𝐃𝐮ε and 𝐒, we get

Ωεξ2|3𝐃𝐮ε|2+1C0ξ2|3𝐅(𝐃𝐮ε)|2d𝐱Ωξ2(ε3𝐃𝐮ε+3𝐒(𝐃𝐮ε))3𝐃𝐮εd𝐱=Ωi,j=13ξ2(ε3Dij𝐮ε+3Sij(𝐃𝐮ε))3juid𝐱=Ωξ2(ε3Dαβ𝐮ε+3Sαβ(𝐃𝐮ε))3Dαβ𝐮εd𝐱+Ωξ2(ε3D3α𝐮ε+3S3α(𝐃𝐮ε))αD33𝐮εd𝐱+Ωj=13ξ23(εDj3𝐮ε+Sj3(𝐃𝐮ε))32uεjd𝐱=:I1+I2+I3.

To estimate I2, we multiply and divide by the quantity ϕ′′(|𝐃𝐮ε|)0, and use Young’s inequality and Proposition 2.4. This yields that, for all λ>0, there exists cλ>0 such that

|I2|α=12Ωξ2|3𝐒(𝐃𝐮ε)||α𝐃𝐮ε|ϕ′′(|𝐃𝐮ε|)ϕ′′(|𝐃𝐮ε|)d𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1α=12Ωεξ2|α𝐃𝐮ε|2𝑑𝐱λΩξ2|3𝐒(𝐃𝐮ε)|2ϕ′′(|𝐃𝐮ε|)𝑑𝐱+cλ-1α=12Ωξ2ϕ′′(|𝐃𝐮ε|)|α𝐃𝐮ε|2𝑑𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1α=12Ωεξ2|α𝐃𝐮ε|2𝑑𝐱CλΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱+cλ-1a=12Ωξ2|α𝐅(𝐃𝐮ε)|2𝑑𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1α=12Ωεξ2|α𝐃𝐮ε|2𝑑𝐱.

Here and in the following we denote by cλ-1 constants that may depend on the characteristics of 𝐒 and on λ-1, while C denotes constants that may depend on the characteristics of 𝐒 only.

To treat the third integral I3, we proceed as follows: We use the following well-known algebraic identity, valid for smooth enough vectors 𝐯 and i,j,k=1,2,3:

jkvi=jDik𝐯+kDij𝐯-iDjk𝐯,(3.4)

and equations (2.12) point-wise, which can be written, for j=1,2,3, as

3(εDj3𝐮ε+Sj3(𝐃𝐮ε))=-fj-β(εDjβ𝐮ε+Sjβ(𝐃𝐮ε))a.e. in Ω.

This is possible due to Proposition 3.1. Hence, we obtain

|I3|j=13|Ωξ2(-fj-βSjβ(𝐃𝐮ε)-εβDjβ𝐮ε)(23Dj3𝐮ε-jD33𝐮ε)d𝐱|.

The right-hand side can be estimated in a way similar to I2. This yields that, for all λ>0, there exists cλ>0 such that

|I3|Ωξ2(|𝐟|+β=12|β𝐒(𝐃𝐮ε)|)(2|3𝐃𝐮ε|+α=12|α𝐃𝐮ε|)ϕ′′(|𝐃𝐮ε|)ϕ′′(|𝐃𝐮ε|)d𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1β=12Ωεξ2|β𝐃𝐮ε|2𝑑𝐱λCΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱+cλ-1β=12Ωξ2|β𝐅(𝐃𝐮ε)|2𝑑𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1β=12Ωεξ2|β𝐃𝐮ε|2𝑑𝐱+cλ-1Ωξ2|𝐟|2ϕ′′(|𝐃𝐮ε|)𝑑𝐱λCΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱+cλ-1β=12Ωξ2|β𝐅(𝐃𝐮ε)|2𝑑𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1β=12Ωεξ2|β𝐃𝐮ε|2𝑑𝐱+cλ-1(𝐟pp+𝐃𝐮εpp+δp).

Observe that we used p2 to estimate the term involving 𝐟.

To estimate I1, we employ the algebraic identity (3.4) to split the integral as follows:

I1=Ωξ2(ε3Dαβ𝐮ε+3Sαβ(𝐃𝐮ε))(αD3β𝐮ε+βD3α𝐮ε)𝑑𝐱-Ωξ2(ε3Dαβ𝐮ε+3Sαβ(𝐃𝐮ε))βαuε3d𝐱=:A+B.

The first term is estimated in a way similar to I2, yielding, for all λ>0,

|A|CλΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱+cλ-1β=12Ωξ2|β𝐅(𝐃𝐮ε)|2𝑑𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1β=12Ωεξ2|β𝐃𝐮ε|2𝑑𝐱.

To estimate B we observe that by the definition of the tangential derivative, we have

αβuε3=ατβuε3-(αβa)D33𝐮ε-(βa)αD33𝐮ε,

and consequently the term B can be split into the following three terms:

-Ωξ2(ε3Dαβ𝐮ε+3Sαβ(𝐃𝐮ε))(ατβuε3-(αβa)D33𝐮ε-(βa)αD33𝐮ε)d𝐱=:B1+B2+B3.

We estimate B2 as follows:

|B2|Ωξ2|3𝐒(𝐃𝐮ε)||2a||𝐃𝐮ε|ϕ′′(|𝐃𝐮ε|)ϕ′′(|𝐃𝐮ε|)+εξ2|3𝐃𝐮ε||2a||𝐃𝐮ε|d𝐱λΩξ2|3𝐒(𝐃𝐮ε)|2ϕ′′(|𝐃𝐮ε|)𝑑𝐱+cλ-12a2Ωξ2|𝐃𝐮ε|2ϕ′′(|𝐃𝐮ε|)𝑑𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-12a2Ωεξ2|𝐃𝐮ε|2𝑑𝐱λCΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱+cλ-12a2ρϕ(|𝐃𝐮ε|)+18Ωεξ2|3𝐃𝐮ε|2𝑑𝐱+2ε2a2𝐃𝐮ε22.

The term B3 is estimated in a way similar to I2, yielding, for all λ>0,

|B3|λCΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱+cλ-1a2β=12Ωξ2|β𝐅(𝐃𝐮ε)|2𝑑𝐱+λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1|a2β=12β=12Ωεξ2|β𝐃𝐮ε|2𝑑𝐱.

Concerning the term B1, we would like to perform some integration by parts, which is one of the crucial observations we are adapting from [24]. Neglecting the localization ξ in B1, we would like to use that

Ω3Sαβε(𝐃𝐮ε)ατβuε3d𝐱=ΩαSαβε(𝐃𝐮ε)3τβuε3d𝐱.

This formula can be justified by using an appropriate approximation that exists for 𝐮εW01,2(Ω)W2,2(Ω), since τ𝐮ε=𝟎 on Ω. More precisely, to treat the term B1, we use that the solution 𝐮ε of (2.12) belongs to W01,2(Ω)W2,2(Ω). Thus, τ(𝐮ε|ΩP)=𝟎 on ΩPΩ, hence ξPτ(𝐮ε3)=𝟎 on Ω. This implies that we can find a sequence {(𝓢n,𝓤n)}C(Ω)×C0(Ω) such that (𝓢n,𝓤n)(𝐒ε,τ𝐮ε) in W1,2(Ω)×W01,2(Ω), and perform calculations with (𝓢n,𝓤n), showing then that all formulas of integration by parts are valid. The passage to the limit as n+ is done only in the last step. For simplicity, we drop the details of this well-known argument (sketched also in [24]), and we write directly formulas without this smooth approximation. Thus, performing several integrations by parts, we get

Ωξ23Sαβ(𝐃𝐮ε)ατβuε3d𝐱=Ω(αξ2)Sαβ(𝐃𝐮ε)3τβuε3d𝐱-Ω(3ξ2)Sαβ(𝐃𝐮ε)ατβuε3d𝐱+Ωξ2αSαβ(𝐃𝐮ε)3τβuε3d𝐱

and

εΩξ23Dαβ𝐮εατβuε3d𝐱=εΩ(αξ2)Dαβ𝐮ε3τβuε3d𝐱-εΩ(3ξ2)Dαβ𝐮εατβuε3d𝐱+εΩξ2αDαβ𝐮ε3τβuε3d𝐱.

This shows that

B1=Ω2ξαξSαβ(𝐃𝐮ε)3τβuε3d𝐱-Ω2ξ3ξSαβ(𝐃𝐮ε)ατβuε3d𝐱+Ωξ2αSαβ(𝐃𝐮ε)3τβuε3d𝐱+εΩ2ξαξDαβ𝐮ε3τβuε3d𝐱-εΩ2ξ3ξDαβ𝐮εατβuε3d𝐱+εΩξ2αDαβ𝐮ε3τβuε3d𝐱=:B1,1+B1,2+B1,3+B1,4+B1,5+B1,6.

To estimate B1,1,B1,3,B1,4,B1,6, we observe that

3τβuε3=τβ3uε3=τβD33𝐮ε.

By using Young’s inequality, the growth properties of 𝐒 in (2.6) and (2.8), we get

|B1,1|ξ2Ω|𝐒(𝐃𝐮ε)|2ϕ′′(|𝐃𝐮ε|)𝑑𝐱+Cβ=12Ωξ2ϕ′′(|𝐃𝐮ε|)|τβ𝐃𝐮ε|2𝑑𝐱ξ2ρϕ(|𝐃𝐮ε|)+Cβ=12Ωξ2|τβ𝐅(𝐃𝐮ε)|2𝑑𝐱

and

|B1,3|β=12Ωξ2|β𝐒αβ(𝐃𝐮ε)|2ϕ′′(|𝐃𝐮ε|)𝑑𝐱+β=12Ωξ2ϕ′′(|𝐃𝐮ε|)|τβ𝐃𝐮ε|2𝑑𝐱Cβ=12Ωξ2|β𝐅(𝐃𝐮ε)|2+ξ2|τβ𝐅(𝐃𝐮ε)|2d𝐱.

Similarly, we get

|B1,4|Cεξ2𝐃𝐮ε22+Cβ=12Ωεξ2|τβ𝐃𝐮ε|2𝑑𝐱

and

|B1,6|Cεβ=12Ωξ2|β𝐃𝐮ε|2+ξ2|τβ𝐃𝐮ε|2d𝐱.

To estimate B1,2 and B15, we observe that, using the algebraic identity (3.4) and the definition of the tangential derivative,

ατβuε3=α(β𝐮ε3+βa3uε3)=αβuε3+αβaD33𝐮ε+βaαD33𝐮ε=αDβ3𝐮ε+βDα3𝐮ε-3Dαβ𝐮ε+αβaD33𝐮ε+βaαD33𝐮ε.

Hence, by substituting and again the same inequalities as before, we arrive to the following estimates:

|B1,2|λCΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱+C(1+a2)β=12Ωξ2|β𝐅(𝐃𝐮ε)|2𝑑𝐱+cλ-1(1+2a)ξ2ρϕ(|𝐃𝐮ε|),|B1,5|λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+cλ-1(1+a2)β=12Ωεξ2|β𝐃𝐮ε|2𝑑𝐱+cλ-1(1+2a)ξ2ε𝐃𝐮ε22.

Collecting all estimates and using that arP1, we finally obtain

Ωεξ2|3𝐃𝐮ε|2+1C0ξ2|3𝐅(𝐃𝐮ε)|2d𝐱λΩεξ2|3𝐃𝐮ε|2𝑑𝐱+λCΩξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱   +cλ-1β=12Ωξ2|β𝐅(𝐃𝐮ε)|2+ξ2|τβ𝐅(𝐮ε)|2d𝐱+cλ-1β=12Ωεξ2|β𝐃𝐮ε|2𝑑𝐱   +cλ-1(1+2a2+(1+2a2)ξ2)(𝐟pp+ρϕ(|𝐃𝐮ε|)+ρϕ(δ))   +cλ-1(1+2a2+(1+2a2)ξ2)𝐃𝐮ε22.

The quantities that are bounded uniformly in L2(ΩP) are the tangential derivatives of ε𝐃𝐮ε and of 𝐅(𝐃𝐮ε). By definition, we have

β𝐃𝐮ε=τβ𝐃𝐮ε-βa3𝐃𝐮ε,β𝐅(𝐃𝐮ε)=τβ𝐅(𝐃𝐮ε)-βa3𝐅(𝐃𝐮ε),

and if we substitute, we obtain

Ωεξ2|3𝐃𝐮ε|2+1C0ξ2|3𝐅(𝐃𝐮ε)|2d𝐱(λ+4a2)Ωεξ2|3𝐃𝐮ε|2𝑑𝐱+(λC+cλ-1a2)Ωξ2|3𝐅(𝐃𝐮ε)|2𝑑𝐱   +cλ-1β=12Ωξ2|τβ𝐅(𝐮ε)|2𝑑𝐱+cλ-1β=12Ωεξ2|τβ𝐃𝐮ε|2𝑑𝐱   +cλ-1(1+2a2+(1+2a2)ξ2)(𝐟pp+ρϕ(|𝐃𝐮ε|)+ρϕ(δ))   +cλ-1(1+2a2+(1+2a2)ξ2)𝐃𝐮ε22.

By choosing first λ>0 small enough such that λC<4-1C0 and then choosing in the local description of the boundary R=RP small enough such that cλa<4-1C0, we can absorb the first two terms from the right-hand side into the left-hand side to obtain

Ωεξ2|3𝐃𝐮ε|2+1C0ξ2|3𝐅(𝐃𝐮ε)|2d𝐱cλ-1β=12Ωξ2|τβ𝐅(𝐮ε)|2𝑑𝐱+cλ-1β=12Ωεξ2|τβ𝐃𝐮ε|2𝑑𝐱   +cλ-1(1+2a2+(1+2a2)ξ2)(𝐟pp+ρϕ(|𝐃𝐮ε|)+ρϕ(δ))   +cλ-1(1+2a2+(1+2a2)ξ2)𝐃𝐮ε22,

where now cλ depends on the fixed paramater λ, the characteristics of 𝐒 and on C2. The right-hand side is bounded uniformly with respect to ε>0, due to Proposition 3.1, proving the assertion of the proposition. ∎

Choosing now an appropriate finite covering of the boundary (for the details, see also [4]), Propositions 3.13.2 yield the following result.

Theorem 3.3.

Let the hypotheses in Theorem 1.1 with δ>0 be satisfied. Then7

ε𝐃𝐮ε22+𝐅(𝐃𝐮ε)22C(𝐟p,δ,Ω).

3.2 Passage to the limit

Once this has been proved, by means of appropriate limiting process, we can show that the estimate is inherited by 𝐮=limε0𝐮ε, since 𝐮 is the unique solution to the boundary value problem (1.1). We can now give the proof of the main result.

Proof of Theorem 1.1.

Let us firstly assume that δ>0. From Propositions 2.3 and 2.7, and Theorem 3.3, we know that 𝐅(𝐃𝐮ε) is uniformly bounded with respect to ε in W1,2(Ω). This also implies (cf. [3, Lemma 4.4]) that 𝐮ε is uniformly bounded with respect to ε in W2,p(Ω). The properties of 𝐒 and Proposition 2.7 also yield that 𝐒(𝐃𝐮ε) is uniformly bounded with respect to ε in Lp(Ω). Thus, there exists a subsequence {εn} (which converges to 0 as n+), 𝐮W2,p(Ω), 𝐅~W1,2(Ω), and 𝝌Lp(Ω) such that

𝐮εn𝐮in W2,p(Ω)W01,p(Ω),𝐃𝐮εn𝐃𝐮a.e. in Ω,𝐅(𝐃𝐮εn)𝐅~in W1,2(Ω),𝐒(𝐃𝐮εn)𝝌in Lp(Ω).

The continuity of 𝐒 and 𝐅, and the classical result stating that the weak limit and the a.e. limit in Lebesgue spaces coincide (cf. [12]) imply that

𝐅~=𝐅(𝐃𝐮)and𝝌=𝐒(𝐃𝐮).

These results enable us to pass to the limit in the weak formulation of the perturbed problem (2.12), which yields

Ω𝐒(𝐃𝐮)𝐃𝐯𝑑𝐱=Ω𝐟𝐯𝑑𝐱for all 𝐯C0(Ω),

where we also used that limεn0Ωεn𝐃𝐮εn𝐃𝐯𝑑𝐱=0. By density, we thus know that 𝐮 is the unique weak solution of problem (1.1). Finally, the lower semi-continuity of the norm implies that

Ω|𝐅(𝐃𝐮)|2d𝐱lim infεn0Ω|𝐅(𝐃𝐮εn)|2d𝐱c.

Note that in [3, Section 4] it is shown that

𝐮W2,3pp+1(Ω)pc(𝐅(𝐃𝐮)22+δp),

which implies the Sobolev regularity stated in Theorem 1.1. This finishes the proof in the case δ>0.

Let us now assume that δ=0. Propositions 3.1 and 3.2 are valid only for δ>0 and thus cannot be used directly for the case that 𝐒 has (p,δ)-structure with δ=0. However, it is proved in [3, Section 3.1] that for any stress tensor with (p,0)-structure 𝐒, there exist stress tensors 𝐒κ, having (p,κ)-structure with κ>0, approximating 𝐒 in an appropriate way.8 Thus, we approximate (2.12) by the system

{-div𝐒ε,κ(𝐃𝐮ε,κ)=𝐟in Ω,𝐮=𝟎on Ω,

where

𝐒ε,κ(𝐐):=ε𝐐+𝐒κ(𝐐),with ε>0,κ(0,1).

For fixed κ>0, we can use the above theory and the fact that the estimates are uniformly in κ to pass to the limit as ε0. Thus, we obtain that for all κ(0,1), there exists a unique 𝐮κW01,p(Ω) satisfying, for all 𝐯W01,p(Ω),

Ω𝐒κ(𝐃𝐮κ)𝐃𝐯𝑑𝐱=Ω𝐟𝐯𝑑𝐱

and

Ω|𝐅κ(𝐃𝐮κ)|2+|𝐅κ(𝐃𝐮κ)|2d𝐱c(𝐟p,Ω),(3.5)

where the constant is independent of κ(0,1) and 𝐅κ:3×3sym3×3 is defined through

𝐅κ(𝐏):=(κ+|𝐏sym|)p-22𝐏sym.

Now we can proceed as in [3]. Indeed, from (3.5) and the properties of ϕp,κ (in particular (2.3)), it follows that 𝐅κ(𝐃𝐮κ) is uniformly bounded in W1,2(Ω), that 𝐮κ is uniformly bounded in W01,p(Ω) and that 𝐒κ(𝐃𝐮κ) is uniformly bounded in Lp(Ω). Thus, there exist 𝐀W1,2(Ω), 𝐮W01,p(Ω), 𝝌Lp(Ω), and a subsequence {κn}, with κn0, such that

𝐅(𝐃𝐮κn)𝐀in W1,2(Ω),𝐅κn(𝐃𝐮κn)𝐀in L2(Ω) and a.e. in Ω,𝐮κn𝐮in W01,p(Ω),𝐒κ(𝐃𝐮κ)𝝌in Lp(Ω).

Setting 𝐁:=(𝐅0)-1(𝐀), it follows from [3, Lemma 3.23] that

𝐃𝐮κn=(𝐅κn)-1(𝐅κn(𝐃𝐮κn))(𝐅0)-1(𝐀)=𝐁a.e. in Ω.

Since weak and a.e. limit coincide, we obtain that

𝐃𝐮κn𝐃𝐮=𝐁a.e. in Ω.

From [3, Lemma 3.16] and [3, Corollary 3.22], it now follows that

𝐅(𝐃𝐮κn)𝐅0(𝐃𝐮)in W1,2(Ω),𝐒κn(𝐃𝐮κn)𝐒(𝐃𝐮)a.e. in Ω.

Since weak and a.e. limit coincide, we obtain that

𝐃𝐮=𝝌a.e. in Ω.

Now we can finish the proof in the same way as in the case δ>0. ∎

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Footnotes

  • 1

    We restrict ourselves to the problem in three space dimensions, however the results can be easily transferred to the problem in d for all d2. 

  • 2

    For the general theory of N-functions and Orlicz spaces, we refer to [21]. 

  • 3

    Recall that c(δ) only indicates that the constant c depends on δ and will satisfy c(δ)c(δ0) for all δδ0. 

  • 4

    Recall that c(δ-1) indicates a possibly critical dependence on δ as δ0. 

  • 5

    Recall that we use the summation convention over repeated Greek lower-case letters from 1 to 2. 

  • 6

    Recall that c(δ) only indicates that the constant c depends on δ and will satisfy c(δ)c(δ0) for all δδ0. 

  • 7

    Recall that c(δ) only indicates that the constant c depends on δ and will satisfy c(δ)c(δ0) for all δδ0. 

  • 8

    The special case 𝐒(𝐃)=|𝐃|p-2𝐃 could be approximated by 𝐒δ=(δ+|𝐃|)p-2𝐃. However, for a general extra stress tensor 𝐒 having only (p,δ)-structure, this is not possible. 

About the article

Received: 2018-04-16

Revised: 2018-07-18

Accepted: 2018-07-18

Published Online: 2018-10-11

Published in Print: 2019-03-01


The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM.


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 176–192, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0090.

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