In this paper we study regularity of weak solutions to the boundary value problem
where denotes the symmetric part of the gradient and is a bounded domain with a boundary .1 Our interest in this system comes from the p-Stokes system
In both problems the typical example for we have in mind is
where , and . 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 .
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 , while the external body force is given. The stress tensor , which is the tensor of small elasto-plastic deformations, depends only on . 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 , the optimal result, namely, that belongs to , where the nonlinear tensor-valued function is defined in (2.5). This result has been proved near a flat boundary in  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., , is only known for
two-dimensional bounded domains (cf. , where even the p-Navier–Stokes system is treated);
the space-periodic problem in , , which follows immediately from interior estimates, i.e., , which are known in all dimensions, and the periodicity of the solution;
if the no-slip boundary condition is replaced by perfect slip boundary conditions (cf. );
in the case of small (cf. ).
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, . 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 , tangential derivatives of near the boundary belong to , while the normal derivative of near the boundary belongs to some , where (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 depending on the full velocity gradient . In this case, it is proved in  that for some , provided is very close to 2.
In the present paper we extend the optimal regularity result for (1.1) of Seregin and Shilkin  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 . The precise result we prove is the following:
Let the tensor field in (1.1) have -structure (cf. Definition 2.1) for some and , and let be the associated tensor field to defined in (2.5). Let be a bounded domain with boundary, and let . Then the unique weak solution of problem (1.1) satisfies
where c denotes a positive function which is non-decreasing in and δ, and which depends on the domain through its measure and the -norms of the local description of . In particular, the above estimate implies that .
2 Notations and preliminaries
In this section we introduce the notation we will use, state the precise assumptions on the extra stress tensor , and formulate the main results of the paper.
We use to denote generic constants which may change from line to line, but they are independent of the crucial quantities. Moreover, we write if and only if there exists constants such that . In some cases we need to specify the dependence on certain parameters, and consequently we denote by a positive function which is non-decreasing with respect to all its arguments. In particular, we denote by a possibly critical dependence on the parameter δ as , while only indicates that the constant c depends on δ and will satisfy for all .
We use standard Lebesgue spaces and Sobolev spaces , , where is a sufficiently smooth bounded domain. The space is the closure of the compactly supported, smooth functions in . Thanks to the Poincaré inequality, we equip with the gradient norm . When dealing with functions defined only on some open subset , we denote the norm in by . As usual, we use the symbol to denote weak convergence, and to denote strong convergence. The symbol 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., and tensors by boldface upper case letters as, e.g., . For vectors , we denote , where the standard tensor product is defined as . The scalar product of vectors is denoted by and the scalar product of tensors is denoted by .
Greek lower-case letters take only the values , while Latin lower-case ones take the values . We use the summation convention over repeated indices only for Greek lower-case letters, but not for Latin lower-case ones.
It is convenient to define for a special N-function2 , for , , by
The function ϕ satisfies, uniformly in t and independently of δ, the important equivalences
Note that the family satisfies the -condition uniformly with respect to , i.e., holds for all .
Definition 2.1 (-structure).
We say that a tensor field , belonging to , satisfying and , possesses -structure if for some , , and the N-function (cf. (2.1)), there exist constants such that
for all , with , and all . The constants , and p are called the characteristics of .
Assume that has -structure for some . Then, if not otherwise stated, the constants in the estimates depend only on the characteristics of and on , but they are independent of δ.
An important example of a tensor field having -structure is given by . In this case, the characteristics of , namely, and , depend only on p and are independent of .
For a tensor field with -structure, we have for all and all , due to its symmetry. Moreover, from , it follows that for all and all , and consequently for all and all .
To a tensor field with -structure, we associate the tensor field defined through
Let have -structure, and let be as defined in (2.5). Then
uniformly in . Moreover, uniformly in ,
The constants depend only on the characteristics of .
For a detailed discussion of the properties of and and their relation to Orlicz spaces and N-functions, we refer the reader to [23, 3]. Since in the following we shall insert into and only symmetric tensors, we can drop in the above formulas the superscript “” 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 :
for all smooth enough symmetric tensor fields .
A crucial observation in  is that the quantities in (2.7) are also equivalent to several further quantities. To formulate this precisely, we introduce, for and for sufficiently smooth symmetric tensor fields , the quantity
Recall that in the definition of there is no summation convention over the repeated Latin lower-case index i in . Note that if has -structure, then for . The following important equivalences hold, first proved in .
Assume that has -structure. Then the following equivalences are valid, for all smooth enough symmetric tensor fields and all :
with the constants only depending on the characteristics of .
The assertions are proved in  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 has -structure. Furthermore, using (2.10), we have
which proves one inequality of (2.11). The other one follows from
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 .
We say that is a weak solution to (1.1) if for all ,
We have the following standard result.
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 with -structure a linear perturbation. Using again the theory of monotone operators one can easily prove the following proposition.
Let the tensor field in (1.1) have -structure for some and , and let be given. Then there exists a unique weak solution of the problem
i.e., satisfies, for all ,
The solution satisfies the estimate
In fact, one could already prove more at this point. Namely, that for , the unique solution converges to the unique weak solution 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 follows that
Passing to the limit in the weak formulation of the perturbed problem, we get
One can not show directly that , since belongs to only. Instead one uses the Lipschitz truncation method (cf. [10, 22]). Denoting by the Lipschitz truncation of , where is a localization, one can show, using the ideas from [10, 22], that
which implies almost everywhere in Ω. Consequently, we have , since weak and a.e. limits coincide.
2.4 Description and properties of the boundary
We assume that the boundary is of class , that is, for each point , there are local coordinates such that, in these coordinates, we have and is locally described by a -function, i.e., there exist and a -function such that
, and for all , ,
where denotes the k-dimensional open ball with center 0 and radius . Note that can be made arbitrarily small if we make small enough. In the sequel, we will also use, for , the following scaled open sets:
To localize near to for , we fix smooth functions such that
where is the indicator function of the measurable set A. For the remaining interior estimate, we localize by a smooth function , with , where is an open set such that . Since the boundary 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 , , and simply write , . We use the standard notation and denote by , , the canonical orthonormal basis in . In the following, lower-case Greek letters take the values . For a function g, with , we define, for ,
and if , we define tangential divided differences by . It holds that, if , then we have for
almost everywhere in (cf. [18, Section 3]). Conversely, uniform -bounds for imply that belongs to . For simplicity, we denote . The following variant of integration by parts will be often used.
Let , and let h be small enough. Then
Consequently, . Moreover, if in addition f and g are smooth enough and at least one vanishes on , then
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.
Let the tensor field in (1.1) have -structure for some and , and let be the associated tensor field to . Let be a bounded domain with boundary and let . Then the unique weak solution of the approximate problem (2.12) satisfies3
Here is a cut-off function with support in the interior of Ω, while for arbitrary the function is a cut-off function with support near to the boundary , as defined in Section 2.4. The tangential derivative is defined locally in by (2.14). Moreover, there exists a constant such that4
provided that in the local description of the boundary, we have in (b3).
In particular, these estimates imply that and that (2.12) holds pointwise a.e. in Ω.
The two estimates in (3.1) are uniform with respect to and could be also proved directly for problem (1.1). However, the third estimate (3.2) depends on and it is needed to justify all subsequent steps, which will give the proof of an estimate uniformly in , 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  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 and use in
where , , and , as a test function in the weak formulation of (2.12). This yields
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 and that the system (2.12) is well-defined point-wise a.e. in Ω.
To estimate the derivatives in the direction, we use equation (2.12) and it is at this point that we have changes with respect to the results in . In fact, as usual in elliptic problems, we have to recover the partial derivatives with respect to 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 -structure of the tensor (cf. Definition 2.1). Denoting, for ,5 , we can re-write the equations in (2.12) as follows:
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 as expected, but by , where for , for , . Summing over , we get, by using the symmetries in Remark 2.2 (iii), that
To obtain a lower bound for the left-hand side, we observe that the terms on the left-hand side of (3.3) containing are equal to
if we choose , where for , for , and . Thus, it follows from the coercivity estimate in (2.4) that these terms are bounded from below by . Similarly, we see that the remaining terms on the left-hand side of (3.3) are equal to . Denoting , , we see that . Consequently, we get from (3.3) the estimate
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 information about , , because
holds a.e. in . This and the last two inequalities imply
Adding on both sides, for and , the term
and using on the right-hand side the definition of the tangential derivative (cf. (2.14)), we finally arrive at
which is valid a.e. in . Note that the constant c only depends on the characteristics of . Next, we can choose the open sets in such a way that 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 . 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 has -structure for , we obtain
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 . In this section, by following the ideas in , we proceed differently and estimate in terms of the quantities occurring in (3.1). The main technical step of the paper is the proof of the following result.
Let the hypotheses in Theorem 1.1 be satisfied with , and let the local description of the boundary and the localization function satisfy (b1)–(b3) and (1) (cf. Section 2.4). Then there exists a constant such that the weak solution of the approximate problem (2.12) satisfies,6 for every ,
provided in (b3).
Let us fix an arbitrary point and a local description of the boundary and the localization function satisfying (b1)–(b3) and (1). In the following we denote by C constants that depend only on the characteristics of . First we observe that, by the results of Proposition 2.4, there exists a constant , depending only on the characteristics of , such that
Thus, using also the symmetry of and , we get
To estimate , we multiply and divide by the quantity , and use Young’s inequality and Proposition 2.4. This yields that, for all , there exists such that
Here and in the following we denote by constants that may depend on the characteristics of and on , while C denotes constants that may depend on the characteristics of only.
To treat the third integral , we proceed as follows: We use the following well-known algebraic identity, valid for smooth enough vectors and :
and equations (2.12) point-wise, which can be written, for , as
This is possible due to Proposition 3.1. Hence, we obtain
The right-hand side can be estimated in a way similar to . This yields that, for all , there exists such that
Observe that we used to estimate the term involving .
To estimate , we employ the algebraic identity (3.4) to split the integral as follows:
The first term is estimated in a way similar to , yielding, for all ,
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 as follows:
The term is estimated in a way similar to , yielding, for all ,
Concerning the term , we would like to perform some integration by parts, which is one of the crucial observations we are adapting from . Neglecting the localization ξ in , we would like to use that
This formula can be justified by using an appropriate approximation that exists for , since on . More precisely, to treat the term , we use that the solution of (2.12) belongs to . Thus, on , hence on . This implies that we can find a sequence such that in , and perform calculations with , showing then that all formulas of integration by parts are valid. The passage to the limit as is done only in the last step. For simplicity, we drop the details of this well-known argument (sketched also in ), and we write directly formulas without this smooth approximation. Thus, performing several integrations by parts, we get
This shows that
To estimate , we observe that
Similarly, we get
To estimate and , we observe that, using the algebraic identity (3.4) and the definition of the tangential derivative,
Hence, by substituting and again the same inequalities as before, we arrive to the following estimates:
Collecting all estimates and using that , we finally obtain
The quantities that are bounded uniformly in are the tangential derivatives of and of . By definition, we have
and if we substitute, we obtain
By choosing first small enough such that and then choosing in the local description of the boundary small enough such that , we can absorb the first two terms from the right-hand side into the left-hand side to obtain
where now depends on the fixed paramater λ, the characteristics of and on . The right-hand side is bounded uniformly with respect to , due to Proposition 3.1, proving the assertion of the proposition. ∎
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 , since 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 . From Propositions 2.3 and 2.7, and Theorem 3.3, we know that is uniformly bounded with respect to in . This also implies (cf. [3, Lemma 4.4]) that is uniformly bounded with respect to in . The properties of and Proposition 2.7 also yield that is uniformly bounded with respect to in . Thus, there exists a subsequence (which converges to 0 as , , , and such that
The continuity of and , and the classical result stating that the weak limit and the a.e. limit in Lebesgue spaces coincide (cf. ) imply that
These results enable us to pass to the limit in the weak formulation of the perturbed problem (2.12), which yields
where we also used that . By density, we thus know that is the unique weak solution of problem (1.1). Finally, the lower semi-continuity of the norm implies that
Note that in [3, Section 4] it is shown that
which implies the Sobolev regularity stated in Theorem 1.1. This finishes the proof in the case .
Let us now assume that . Propositions 3.1 and 3.2 are valid only for and thus cannot be used directly for the case that has -structure with . However, it is proved in [3, Section 3.1] that for any stress tensor with -structure , there exist stress tensors , having -structure with , approximating in an appropriate way.8 Thus, we approximate (2.12) by the system
For fixed , we can use the above theory and the fact that the estimates are uniformly in κ to pass to the limit as . Thus, we obtain that for all , there exists a unique satisfying, for all ,
where the constant is independent of and is defined through
Now we can proceed as in . Indeed, from (3.5) and the properties of (in particular (2.3)), it follows that is uniformly bounded in , that is uniformly bounded in and that is uniformly bounded in . Thus, there exist , , , and a subsequence , with , such that
Setting , it follows from [3, Lemma 3.23] that
Since weak and a.e. limit coincide, we obtain that
Since weak and a.e. limit coincide, we obtain that
Now we can finish the proof in the same way as in the case . ∎
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About the article
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.
© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0