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

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.


Introduction
In this paper we study regularity of weak solutions to the boundary value problem − div S(Du) = f in Ω, where Du := 1 2 (∇u + ∇u ⊤ ) denotes the symmetric part of the gradient ∇u and where 1 Ω ⊂ R 3 is a bounded domain with a C 2,1 boundary ∂Ω. Our interest in this system comes from the p-Stokes system − div S(Du) + ∇π = f in Ω, div u = 0 in Ω, (1. 2) In both problems the typical example for S we have in mind is where p ∈ (1, 2], δ ≥ 0, 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, the system (1.1) is obtained from (1.2) by dropping the divergence constraint and the resulting pressure gradient. Thus the 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 the system (1.1) is of 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 = (u 1 , u 2 , u 3 ) ⊤ , while the external body force f = (f 1 , f 2 , f 3 ) ⊤ is given. The stress tensor S, which is the tensor of small elasto-plastic deformations, 1 We restrict ourselves to the problem in three space dimensions, even if results can be easily transferred to the problem in R d for all d ≥ 2.
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 W 1,2 (Ω), where the nonlinear tensor-valued function F is defined in (2.8). 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) ∈ W 1,2 (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 R d , d ≥ 2, which follows immediately from interior estimates, i.e. F(Du) ∈ W 1,2 loc (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]), and (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) ∈ W 1,2 loc (Ω), tangential derivatives of F(Du) near the boundary belong to L 2 , while the normal derivative of F(Du) near the boundary belongs to some L q , 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 system, namely if (1.2) is considered with S depending on the full velocity gradient ∇u. In this case it is proved in [7] that u ∈ W 2,r (R 3 ) ∩ W 1,p 0 (R 3 ) for some r > 3, provided p < 2 is very close to 2.
In the present paper we extend to the general case of bounded sufficiently smooth domains and to possibly degenerate stress tensors, that is the case δ = 0, the optimal regularity result for (1.1) of Seregin and Shilkin [24] in the case of a flat boundary. The precise result we prove is the following: Theorem 1.3. 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 Ω ⊂ R 3 be a bounded domain with C 2,1 boundary and let f ∈ L p ′ (Ω). Then, the unique weak solution u ∈ W 1,p 0 (Ω) of the problem (1.1) satisfies where c denotes a positive function which is non-decreasing in f p ′ and δ, and which depends on the domain through its measure |Ω| and the C 2,1 -norms of the local description of ∂Ω. In particular, the above estimate implies that u ∈ W 2, 3p p+1 (Ω).

Preliminaries and main results
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 are independent of the crucial quantities. Moreover, we write f ∼ g if and only if there exists constants c, C > 0 such that c f ≤ g ≤ C f . 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.
We use standard Lebesgue spaces (L p (Ω), . p ) and Sobolev spaces (W k,p (Ω), . k,p ), where Ω ⊂ R 3 , is a sufficiently smooth bounded domain. The space W 1,p 0 (Ω) is the closure of the compactly supported, smooth functions C ∞ 0 (Ω) in W 1,p (Ω). Thanks to the Poincaré inequality we equip W 1,p 0 (Ω) with the gradient norm ∇ · p . When dealing with functions defined only on some open subset G ⊂ Ω, we denote the norm in L p (G) by . p,G . As usual we use the symbol ⇀ to denote weak convergence, and → to denote strong convergence. The symbol spt f 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, v ∈ R 3 we denote u Greek lower-case letters take only 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.
are satisfied for all P, Q ∈ R 3×3 with P sym = 0 and all i, j, k, l = 1, 2, 3. The constants κ 0 , κ 1 , and p are called the characteristics of S.
Remark 2.7. (i) Assume that S has (p, δ)-structure for some δ ∈ [0, δ 0 ]. Then, if not otherwise stated, the constants in the estimates depend only on the characteristics of S and on δ 0 , but are independent of δ.
(ii) An important example of a tensor field S having (p, δ)-structure is given by S(P) = ϕ ′ (|P sym |)|P sym | −1 P sym . In this case the characteristics of S, namely κ 0 and κ 1 , depend only on p and are independent of δ ≥ 0.
Proposition 2.9. Let S have (p, δ)-structure, and let F be defined in (2.8). Then 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], valid for all smooth enough symmetric tensor fields Q ∈ R 3×3 sym . The proof of this equivalence is based on Proposition 2.9. This Proposition and the theory of divided differences also imply (cf. [4, (2.26 for all smooth enough symmetric tensor fields Q ∈ R 3×3 sym . A crucial observation in [24] is that the quantities in (2.11) 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 (2. 13) Recall, that in the definition of P i (Q) there is no summation convention over the repeated Latin lower-case index i in ∂ i S(Q)·∂ i Q. Note that if S has (p, δ)-structure, then P i (v) ≥ 0, for i = 1, 2, 3. There hold the following important equivalences, first proved in [24]: Proposition 2.14. 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 with 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 (2.15) follow from (2.11), (2.13) and the fact that S has (p, δ)-structure. Furthermore, we have, using (2.15), which proves one inequality of (2.16). The other follows from where we used (2.6) and (2.15).

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 p ≤ 2.
We have the following very standard result: 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: 19. Let the tensor field S in (1.1) have (p, δ)-structure for some p ∈ (1, 2], and δ ∈ [0, ∞) and let f ∈ L p ′ (Ω) be given. Then, there exists a unique weak solution The solution u ε satisfies the estimate Remark 2.22. In fact, one could already prove more at this point. Namely, that for ε → 0, the unique solution u ε 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 (2.21) and the properties of S follows that Passing to the limit in the weak formulation of the perturbed problem, we get One can not show directly that lim ε→0 Ω εDu ε · (Du ε − Du) dx = 0, since Du belongs to L p (Ω) only. Instead one uses the Lipschitz truncation method (cf. [10,22]). Denoting by v ε,j the Lipschitz truncation of ξ(u ε − u), where ξ ∈ C ∞ 0 (Ω) is a localization, one can show, using the ideas from [10,22], that which implies Du ε → Du almost everywhere in Ω. Consequently, we have χ = S(Du), since weak and a.e. limits coincide.

Description and properties of the boundary.
We assume that the boundary ∂Ω is of class C 2,1 , that is for each point P ∈ ∂Ω there are local coordinates such that in these coordinates we have P = 0 and ∂Ω is locally described by a C 2,1 -function, i.e., there exist R P , R ′ P ∈ (0, ∞), r P ∈ (0, 1) and a C 2,1 -function open ball with center 0 and radius r > 0. Note that r P can be made arbitrarily small if we make R P small enough. In the sequel we will also use, for 0 < λ < 1, the following scaled open sets, λ Ω P ⊂ Ω P defined as follows To localize near to ∂Ω ∩ ∂Ω P , for P ∈ ∂Ω, we fix smooth functions ξ P : is the indicator function of the measurable set A. For the remaining interior estimate we localize by a smooth function 0 ≤ ξ 00 ≤ 1 with spt ξ 00 ⊂ Ω 00 , where Ω 00 ⊂ Ω is an open set such that dist(∂Ω 00 , ∂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 P ∈ ∂Ω, h ∈ (0, RP 16 ), and simply write ξ := ξ P , a := a P . We use the standard notation x = (x ′ , x 3 ) ⊤ and denote by e i , i = 1, 2, 3 the canonical orthonormal basis in R 3 . In the following lower-case Greek letters take values 1, 2. For a function g with spt g ⊂ spt ξ we define for α = 1, 2 and if ∆ + g := g τ − g, we define tangential divided differences by d + g := h −1 ∆ + g. It holds that, if g ∈ W 1,1 (Ω), then we have for α = 1, 2 almost everywhere in spt ξ, (cf. [18,Sec. 3]). Conversely uniform L q -bounds for d + g imply that ∂ τ g belongs to L q (spt ξ).
For simplicity we denote ∇a := (∂ 1 a, ∂ 2 a, 0) ⊤ . The following variant of integration per parts will be often used. Consequently, Ω f d + g dx = Ω (d − f )g dx. Moreover, if in addition f and g are smooth enough and at least one vanishes on ∂Ω, then

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 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 (2.20).
The first intermediate step is the following result for the approximate problem.

(3.2)
Here ξ 0 is a cut-off function with support in the interior of Ω, while for arbitrary P ∈ ∂Ω the function ξ P is a cut-off function with support near to the boundary ∂Ω, as defined in Sec. 2.4. The tangential derivative ∂ τ is defined locally in Ω P by (2.24). Moreover, there exists a constant C 1 > 0 such that provided that in the local description of the boundary there holds r P < C 1 in (b3).
The two estimates (3.2) are uniform with respect to ε and could be also proved directly for the problem (1.1). However, the third estimate (3.3) depends on ε but is needed to justify all subsequent steps, which will give the proof of an estimate uniformly in ε, by using a different technique.  [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 P ∈ ∂Ω and use in Ω where ξ := ξ P , a := a P , and h ∈ (0, RP 16 ), as a test function in the weak formulation of (2.20). This yields From the assumption on S, Proposition 2.9, and [4, Lemma 3.11] we have the following estimate This proves the second estimate in (3.2) by standard arguments. The first estimate in (3.2) 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 u ε ∈ W 2,2 loc (Ω) and that the system (2.20) is well-defined point-wise a.e. in Ω. To estimate the derivatives in the x 3 direction we use equation (2.20) 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 x 3 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 2.5). Denoting, for 3 i = 1, 2, 3, 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 ∂ 3 D i3 u ε as expected, but by ∂ 3 D i3 u ε , where D αβ u ε = 0, for α, β = 1, 2, D α3 u ε = D 3α u ε = 2D α3 u ε , for α = 1, 2, D 33 u ε = D 33 u ε . Summing over i = 1, 2, 3 we get, by using the symmetries in Remark 2.7 (iii), that in Ω .

By straightforward manipulations (cf. [4, Sections 3.2 and 4.2]) we can estimate the right-hand side as follows
Note that we can deduce from b information about b i := ∂ 2 33 u i ε , i = 1, 2, 3, because |b| ≥ 2| b| − |∂ τ ∇u ε | − ∇a ∞ |∇ 2 u ε | holds a.e. in Ω P . This and the last last two inequalities imply a.e. in Ω P Adding on both sides, for α = 1, 2 and i, k = 1, 2, 3 the term (ε + ϕ ′′ (|Du ε |)) |∂ α ∂ i u k ε | , and using on the right-hand side the definition of the tangential derivative (cf. (2.24)), we finally arrive at which is valid a.e. in Ω P . Note that the constant c only depends on the characteristics of S. Next, we can choose the open sets Ω P in such a way that ∇a P (x) ∞,ΩP is small enough, so that we can absorb the last term from the right hand side, which yields 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 The already proven results on tangential derivatives and Korn's inequality imply that the last integral from right-hand side is finite. Thus, the properties of the covering imply the last estimate in (3.2).
3.1. Improved estimates for normal derivatives. The proof of (3.3) used the system (2.20) and resulted in an estimate that is not uniform with respect to ε. In this section, by following the ideas in [24], we proceed differently and estimate P 3 in terms of quantities occurring in (3.2). The main technical step of the paper is the proof of the following result: Proposition 3.5. Let the same hypotheses as in Theorem 1.3 be satisfied with δ > 0 and let the local description a P of the boundary and the localization function ξ P satisfy (b1)-(b3) and (ℓ1) (cf. Section 2.4). Then, there exist a constant C 2 > 0 such that the weak solution u ε ∈ W 1,2 0 (Ω) of the approximate problem (2.20) satisfies for every P ∈ ∂Ω Proof. Let us fix an arbitrary point P ∈ ∂Ω and a local description a = a P of the boundary and the localization function ξ = ξ P satisfying (b1)-(b3) and (ℓ1). 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 2.14 there exists a constant C 0 , depending only on the characteristics of S, such that a.e. in Ω.
Thus, we get, using also the symmetry of Du ε and S, To estimate I 2 we multiply and divide by the quantity ϕ ′′ (|Du ε |) = 0, use Young's inequality and Proposition 2.14. This yields that for all λ > 0 there exists c λ > 0 such that Here and in the following we denote by c λ 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 I 3 we proceed as follows: We use the well-known algebraic identity, valid for smooth enough vectors v and i, j, k = 1, 2, 3, 6) and the equations (2.20) point-wise, which can be written for j = 1, 2, 3 as, This is possible due to Proposition 3.1. Hence, we obtain The right-hand side can be estimated similarly as I 2 . This yields that for all λ > 0 there exists c λ > 0 such that estimated by Observe that we used p ≤ 2 to estimate the term involving f . To estimate I 1 we employ the algebraic identity (3.6) to split the integral as follows The first term is estimated similarly as I 2 , yielding for all λ > 0 To estimate B we observe that by the definition of the tangential derivative we have and consequently the term B can be split into the following three terms: We estimate B 2 as follows The term B 3 is estimated similarly as I 2 , yielding for all λ > 0 Concerning the term B 1 , 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 B 1 we would like to use that This formula can be justified by using an appropriate approximation, that exists for u ε ∈ W 1,2 0 (Ω) ∩ W 2,2 (Ω) since ∂ τ u ε = 0 on ∂Ω. More precisely, to treat the term B 1 we use that the solution u ε of (2.20) belongs to W 1,2 0 (Ω) ∩ W 2,2 (Ω). Thus, ∂ τ u ε|Ω P = 0 on ∂Ω P ∩ ∂Ω, hence ξ P ∂ τ (u 3 ε ) = 0 on ∂Ω. This implies that we can find a sequence (S n , U n ) ∈ C ∞ (Ω) × C ∞ 0 (Ω) such that (S n , U n ) → (S ε , ∂ τ u ε ) in W 1,2 (Ω) × W 1,2 0 (Ω) and perform calculations with (S n , U n ), showing then that all formulas of integration by parts are valid. 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 This shows that To estimate B 1,1 , B 1,3 , B 1,4 , B 1,6 we observe that By using Young inequality, the growth properties of S in (2.10d) and (2.12) we get Similarly we get To estimate B 1,2 and B 15 we observe that, using the algebraic identity (3.6) and the defintion of the tangential derivative, Hence by substituting and again by the same inequalities as before we arrive to the following estimates Collecting all estimates and using that ∇a ∞ ≤ r P ≤ 1, we finally obtain . The quantities that are bounded uniformly in L 2 (Ω P ) are the tangential derivatives of ε Du ε and of F(Du ε ). By definition we have and if we substitute we obtain By choosing first λ > 0 small enough such that λ C < 4 −1 C 0 and then choosing in the local description of the boundary R = R P small enough such that c λ ∇a ∞ < 4 −1 C 0 , we can absorb the first two terms from the right-hand side into the left-hand side to obtain , where now c λ depends on the fixed paramater λ, the characteristics of S and on C 2 . 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 3.1-3.5 yield the following result: Theorem 3.8. Let the same hypotheses as in Theorem 1.3 with δ > 0 be satisfied. Then, it holds 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 = lim ε→0 u ε , 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 1.3). Let us firstly assume that δ > 0. From Proposition 2.19, Proposition 2.9 and Theorem 3.8 we know that F(Du ε ) is uniformly bounded with respect to ε in W 1,2 (Ω). This also implies (cf. [3,Lemma 4.4]) that u ε is uniformly bounded with respect to ε in W 2,p (Ω). The properties of S and Proposition 2.19 also yield that S(Du ε ) is uniformly bounded with respect to ε in L p ′ (Ω). Thus, there exists a subsequence {ε n } (which converges to 0 as n → +∞), u ∈ W 2,p (Ω), F ∈ W 1,2 (Ω), and χ ∈ L p ′ (Ω) such that u εn ⇀ u in W 2,p (Ω) ∩ W 1,p 0 (Ω) , Du εn → Du a.e. in 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) and χ = S(Du) .
These results enable us to pass to the limit in the weak formulation of the perturbed problem (2.20), which yields where we also used that lim εn→0 Ω ε n Du εn · Dv dx = 0. By density we thus know that u is the unique weak solution of problem (1.1). Finally the lower semicontinuity of the norm implies that Ω |∇F(Du)| 2 dx ≤ lim inf εn→0 Ω |∇F(Du εn )| 2 dx ≤ c, ending the proof in the case δ > 0.
Now we can finish the proof in the same way as in the case δ > 0.