Higher Sobolev Regularity of Convex Integration Solutions in Elasticity
In this article we discuss quantitative properties of convex integration solutions arising in problems modeling shape-memory materials. For a two-dimensional, geometrically linearized model case, the hexagonal-to-rhombic phase transformation, we prove the existence of convex integration solutions with higher Sobolev regularity, i.e. there exists such that for , with . We also recall a construction, which shows that in situations with additional symmetry much better regularity properties hold.
Key words and phrases:Convex integration solutions, elasticity, solid-solid phase transformations, differential inclusion, higher Sobolev regularity
2010 Mathematics Subject Classification:Primary 35B36, 35B65, 32F32
In this article we are concerned with the detailed analysis of certain convex integration solutions, which arise in the modeling of solid-solid, diffusionless phase transformations in shape-memory materials. We seek to precisely analyze the regularity properties of these constructions in a simple, two-dimensional, geometrically linear model case.
Shape-memory materials undergo a solid-solid, diffusionless phase transition upon temperature change (see e.g. [Bha03] and the references given there): In the high temperature phase, the austenite phase, the materials form very symmetric lattices. Upon cooling down the material, the symmetry of the lattice is reduced, the material transforms into the martensitic phase. Due to the loss of symmetry, there are different variants of martensite, which make these materials very flexible at low temperature and give rise to a variety of different microstructures. Mathematically, it has proven very successful to model this behavior variationally in a continuum framework as the following minimization problem [BJ89]:
Here is the reference configuration of the undeformed material. The mapping describes the deformation of the material with respect to the reference configuration. It is assumed to be of a suitable Sobolev regularity. The function denotes the energy density of a given deformation gradient at a certain temperature . Due to frame indifference, is required to be invariant with respect to rotations, i.e.
Modeling the behavior of shape-memory materials, the energy density further reflects the physical properties of these materials. In particular, it is assumed that at high temperatures the energy density has a single minimum (modulo symmetry), which (upon normalization) we may assume to be given by the orbit of , where with (c.f. [Bal04]). This is the (austenite) energy well at temperature . Upon lowering the temperature below a critical temperature , the function displays a (discrete) multi-well behavior (modulo ): There exist finitely many matrices , , such that
The matrices represent the variants of martensite at temperature and are referred to as the (martensite) energy wells. At the critical temperature both the austenite and the martensite wells are energy minimizers.
In the sequel, we assume that is fixed, so that only the variants of martensite are energy minimizers. We seek to study the quantitative behavior of minimizers for energies of the type (1). Here we make the following simplifications:
Reduction to the -well problem. Instead of studying the full variational problem (1), we only focus on exact minimizers. Restricting to the low temperature regime, this implies that we seek solutions to the differential inclusion
for some .
Small deformation gradient case, geometric linearization. We further modify (2) and assume that is close to the identity. This allows us to linearize the problem around this constant value (c.f. Chapter 11 in [Bha03]). Instead of considering (2), we are thus lead to the inclusion problem
The symmetrized gradient represents the infinitesimal deformation strain associated with the displacement , which is defined as (with slight abuse of physical convention in the sequel we do not distinguish between the deformation and the displacement in our use of language and will simply refer to both as a “deformation”). The symmetric matrices are the exactly stress-free strains representing the variants of martensite. While this linearizes the geometry of the problem (by replacing the symmetry group by an invariance with respect to the linear space ), the differential inclusion (3) preserves the inherent physical nonlinearity, which arises from the multi-well structure of the problem.
Reduction to two dimensions and the hexagonal-to-rhombic phase transformation. In the sequel studying an as simple as possible model case, we restrict to two dimensions and a specific two-dimensional phase transformation, the hexagonal-to-rhombic phase transformation (this is for instance used in studying materials such as or Mg-Cd alloys undergoing a (three-dimensional) hexagonal-to-orthorhombic transformation, c.f. [CPL14], [KK91], and also for closely related materials such as , which undergo a (three-dimensional) hexagonal-to-monoclinic transformation, c.f. [MA80a], [MA80b], [CPL14]). From a microscopic point of view, the hexagonal-to-rhombic phase transformation occurs, if a hexagonal atomic lattice is transformed into a rhombic atomic lattice. From a continuum point of view, we model it as solutions to the differential inclusion
where is a bounded Lipschitz domain and
We note that all the matrices in are trace-free, which corresponds to the (infinitesimal) volume preservation of the transformation. We note that the set is “large” (its convex hull is a two-dimensional set in the three-dimensional ambient space of two-by-two, symmetric matrices, c.f. Lemma 2.7).
1.1. Main result
The geometrically linearized hexagonal-to-rhombic phase transformation is a very flexible transformation, which allows for numerous exact solutions to the associated three-well problem (4) with different types of boundary data. Here the simplest possible solutions are so-called simple laminates, for which the strain is a one-dimensional function for some vector and for which
i.e. only attains two values. The possible directions of these laminates, as given by the vector are (up to sign reversal) six discrete values, which arise as the symmetrized rank-one directions between the energy wells: For each with there exists (up to sign reversal and exchange of the roles of and and renormalization) exactly one pair with the property that
The possible vectors are collected in Lemma 2.13.
In addition to these “simple” constructions, there are further exact solutions to the three-well problem associated with the hexagonal-to-rhombic phase transformation, e.g. there are patterns involving all three variants as depicted in Figures 24 and 25 in the Appendix (Section 7).
In the sequel, we study solutions to the hexagonal-to-rhombic phase transformation with affine boundary conditions, i.e. we consider with such that
Here we investigate the rigidity/ non-rigidity of the problem by asking whether it has non-affine solutions:
Are there (non-affine) solutions to (6) with ?
Clearly, a necessary condition for this is that . Using the method of convex integration, Müller and Šverák [MŠ99] (c.f. also the Baire category arguments of [Dac07], [DM12]) constructed multiple solutions to related differential inclusions, displaying the existence of a variety of solutions to the problem. Noting that these techniques are applicable to our set-up of the three-well problem, ensures that for any with there exists a non-affine solution to (6).
In general these convex integration solutions are however very “wild” in the sense that they do not possess very strong regularity properties (c.f. [DM95b]). As our inclusion (6) is motivated by a physical problem, a natural question addresses the relevance of this multitude of solutions:
Are all the convex integration solutions physically relevant? Or are they only mathematical artifacts? Is there a mechanism distinguishing between the “only mathematical” and the “really physical” solutions?
Guided by the physical problem at hand and the literature on these problems,
natural criteria to consider are surface energy constraints and surface
energy regularizations. For our differential inclusion these translate into
regularity constraints and lead to the question, whether unphysical
convex integration solutions have a natural regularity threshold. Here an
immediate regularity property of solutions to (6) is that . With slightly more care, it is also possible to
obtain solutions with the property that . However, prior to
this work it was not known whether these solutions can enjoy more regularity, i.e.
whether for instance there are convex integration solutions with for some , .
Motivated by these questions, in this article, we study the regularity of a specific convex integration construction and obtain higher Sobolev regularity properties for the resulting solutions:
Let us comment on this result: To the best of our knowledge it represents the first higher regularity result for convex integration solutions arising in differential inclusions for shape-memory materials.
The given quantitative dependences for are certainly not optimal in the specific constants. While it is certainly possible to improve on these numeric values, a more interesting question deals with the qualitative expected dependences: Is it necessary that depends on ?
Since for with there are no non-affine solutions to (6), it is natural to expect that convex integration constructions deteriorate for matrices with approaching the boundary of . The precise dependence on the behavior towards the boundary however is less intuitive. In this context, it is interesting to note that the regularity threshold does not depend on the distance to the boundary of , but rather on the angle, which is formed between the initial matrix and the boundary of . This is in agreement with the intuition that the larger the angle is, the better the convex integration algorithm becomes, as it moves the values of the iterations, which are used to construct the displacement , further into the interior of . In the interior of it is possible to use larger length scales, which increases the regularity of solutions. Whether this dependence is necessary in the value of the product of or whether the product should be independent of this and only the value of the corresponding norm should deteriorate with a smaller angle, is an interesting open question.
We remark that in the special case of additional symmetries it is possible to construct much better solutions. An example is given in the appendix for the case (c.f. also [Pom10] and [CPL14]). It is an important and challenging open question, whether it is possible to exploit further symmetries and thus to construct further solutions with these much better regularity properties.
1.2. Literature and context
A fascinating problem in studying solid-solid, diffusionless phase transformations modeling shape-memory materials is the dichotomy between rigidity and non-rigidity. Since the work of Müller and Šverák [MŠ99], who adapted the convex integration method of Gromov [Gro73], [EM02] and Nash-Kuiper [Nas54], [Kui55] to the situation of solid-solid phase transformations, and the work of Dacorogna and Marcellini [Dac07], [DM12], it is known that under suitable conditions on the convex hulls of the energy wells, there is a very large set of possible minimizers to (1) (c.f. also [SJ12] and [Kir03] for a comparison of these two methods).
More precisely, the set of minimizers forms a residual set (in the Baire sense) in the associated function spaces. However, in general convex integration solutions are “wild”; they do not enjoy very good regularity properties.
This has rigorously been proven for the case of the geometrically nonlinear two-well problem [DM95b], [DM95a], the geometrically nonlinear three-well problem in three dimensions (the “cubic-to-tetragonal phase transformation”) [Kir98], [CDK07] and (under additional assumptions) for the geometrically linear six-well problem (the “cubic-to-orthorhombic phase transformation”) [Rül16]. In these works it has been shown that on the one hand convex integration solutions exist, if the deformation gradient is only assumed to be regular. If on the other hand, the deformation gradient is regular (or a replacement of this), then solutions are very rigid and for most constant matrices the analogue of (6) does not possess a solution.
Thus, convex integration solutions cannot exist at regularity for the deformation gradient; at this regularity solutions are rigid. At regularity they are however flexible and a multitude of solutions exist. Similarly as in the related (though much more complicated) situation of the Onsager conjecture for Euler’s equations [SJ12], [DLSJ16] or the situation of isometric embeddings [CDLSJ12], it is hence natural to ask whether there is a regularity threshold, which distinguishes between the rigid and the flexible regime.
It is the purpose of this article to make a first, very modest step into the understanding of this dichotomy by analyzing the regularity of a (known) convex integration scheme in an as simple as possible model case.
1.3. Main ideas
In our construction of solutions to the differential inclusion (6) we follow the ideas of Müller and Šverák [MŠ99] (in the version of [Ott12]) and argue by an iterative convex integration algorithm. For the hexagonal-to-rhombic transformation this is particularly simple, since the laminar convex hull equals the convex hull of the wells and since all matrices in the convex hull are symmetrized rank-one-connected with the wells (c.f. Lemma 2.7). As a consequence it is possible to construct piecewise affine solutions (in the language of [Kir03], Chapter 4). This simplifies the convergence of the iterative construction drastically. It is one of the reasons for studying the hexagonal-to-rhombic phase transformation as a model problem.
Yet, in spite of the (relative) simplicity of obtaining convergence of the iterative construction to a solution of (6) and hence of showing existence, substantially more care is needed in addressing regularity. In this context we argue by an interpolation result (c.f. Theorem 2 and Proposition 5.5): While our approximating deformations are such that the norms of the iterations increase (exponentially), the norm of their difference decreases exponentially. If the threshold is chosen appropriately, the norm for is controlled by an interpolation of the and the norms, which can be balanced to be uniformly bounded. To ensure this, we have to make the iterative algorithm quantitative in several ways:
Tracking the error in strain space. In order to iterate the convex integration construction, it is crucial not to leave the interior of the convex hull of in the iterative modification steps. In qualitative convex integration algorithms, it suffices to use errors, which become arbitrarily small and to invoke the openness of . As the admissible error in strain space is however coupled to the length scales of the convex integration constructions (c.f. Lemma 3.3) and as these in turn are directly reflected in the solutions’ regularity properties, in our quantitative algorithm we have to keep track of the errors in strain space very carefully. Here we seek to maximize the possible length scales (and hence the error) without leaving in each iteration step. This leads to the distinction of various possible cases (the “stagnant”, the “push-out”, the “parallel” and the “rotated” case, c.f. Notation 3.6, Definition 3.10 and Algorithm 3.8). In these we quantitatively prescribe the admissible error according to the given geometry in strain space.
Controlling the skew part without destroying the structure of (i). Seeking to construct solutions, we have to control the skew part of our construction. Due to the results of Kirchheim, it is known that this is generically possible (c.f. [Kir03], Chapter 3). However, in our quantitative construction, we cannot afford to arbitrarily change the direction of the rank-one connection, which is chosen in the convex integration algorithms, at an arbitrary iteration step. This would entail bounds, which could not be compensated by the exponentially decreasing bounds in the interpolation argument. Hence we have to devise a detailed description of controlling the skew part (c.f. Algorithm 3.11).
Precise covering construction. In order to carry out our convex integration scheme we have to prescribe an iterative covering of our domain by constructions, which successively modify a given gradient. As our construction in Lemma 3.3 relies on triangles, we have to ensure that there is a class of triangles, which can be used for these purposes (c.f. Section 4). In particular, we have to quantitatively control the overall perimeter (which can be viewed as a measure of the BV norm of ) of the covering at a given iteration step of the convex integration algorithm. This crucially depends on the specific case (“rotated” or “parallel”), in which we are in.
1.4. Organization of the article
The remainder of the article is organized as follows: After briefly collecting preliminary results in the next section (interpolation results, results on the convex hull of the hexagonal-to-rhombic phase transition), in Section 3 we begin by describing the convex integration scheme, which we employ. Here we first recall the main ingredients of the qualitative scheme (Section 3.1) and then introduce our more quantitative algorithms in Sections 3.2-3.3.2. As this algorithm crucially relies on the existence of an appropriate covering, we present an explicit construction of this in Section 4. Here we also address quantitative covering estimates for the perimeter and the volume. The ingredients from Sections 3 and 4 are then combined in Section 5, where we prove Theorem 1 for a specific class of domains. In Section 6 we explain how this can be generalized to arbitrary Lipschitz domains. Finally, in the Appendix, Section 7, we recall a symmetry based construction for a solution to (6) with with much better regularity properties.
In this section we collect preliminary results, which will be relevant in the sequel. We begin by stating the interpolation results of [CDDD03], on which our bounds rely. Next, in Section 2.2 we recall general facts on matrix space geometry and in particular apply this to the hexagonal-to-rhombic phase transformation and its convex hulls.
2.1. An interpolation inequality and Sickel’s result
Seeking to show higher Sobolev regularity for convex integration solutions, we rely on the characterization of Sobolev functions. Here we recall the following two results on an interpolation characterization [CDDD03] and on a geometric characterization of the regularity of characteristic functions [Sic99]:
Theorem 2 (Interpolation with BV, [Cddd03]).
We have the following interpolation results:
Let and assume that for some . Then
Let and let for some . Let further be such that
for some , where denotes an arbitrary positive number slightly less than . Then,
Before proceeding to the proof of Theorem 2, we present an immediate corollary of it: For functions, which are “essentially” characteristic functions, we obtain the following unified result:
Let be a function, such that
Then, for any we have that
where and .
Proof of Corollary 2.1.
By virtue of Theorem 2 (i) and equation (7), it suffices to consider the regime, in which . In this case the statement follows from a combination of equation (8) and the fact that for functions satisfying (9) we have
for , and . We postpone a proof of (11) to the end of this proof, and observe first that it indeed suffices to show (11) to conclude the claim of (10). To this end, we note that the exponents in (8) obey the relation
This in turn is a consequence of the three identitites
where for abbreviation we have set . With this we infer
This concludes the argument. ∎
After this discussion, we come to the proof of Theorem 2:
Proof of Theorem 2.
If , the interpolation result is a special case of Theorem 1.4 in [CDDD03] (where in the notation of [CDDD03] we have chosen , ): Indeed, for and satisfying with being the dual exponent of , the estimate in Theorem 1.4 from [CDDD03] reads
We note that in the setting of Theorem 2 the estimate (12) is applicable, as in the notation of [CDDD03] and with dimension we have that , which implies the validity of (7). The simplification from (12) to (7) is then a consequence of the facts that
and for the embedding is valid (Theorem 2.41 in [BCD11]).
This concludes the argument for (i).
To obtain (ii), we combine (i) with an additional interpolation inequality, which becomes necessary, as the inclusion is no longer valid for . Hence, we rely on the following interpolation estimate (c.f. Lemma 3 in [BM01])
which is valid for , , , with
Here the spaces denote the (modified) Triebel-Lizorkin spaces from [BM01]. The main advantage of the estimate (13), which goes back to Oru [Oru98], is that there are no conditions on the relations between in this estimate. In particular, we can choose , and . Using that
for , and that for this range ,
for , , ,
we can simplify (13) to yield
which is valid for , , with
We apply with , , and lying on the boundary of the interpolation region from (i) (c.f. the blue region in Figure 1), i.e.
In particular, these equations uniquely determine . Hence, we obtain
We conclude the proof of (ii) by noting that and that
As an alternative to the interpolation approach, a more geometric criterion for regularity is given by Sickel:
Theorem 3 (Sickel, [Sic99]).
Let , and let be a bounded set satisfying
Although this theorem provides good geometric intuition and could have been used as an alternative means of proving Theorem 1, we do not pursue this further in the sequel, but postpone its discussion to future work.
We note that the estimate (15) in Theorem 3 yields a condition on the product , while, at first sight, Theorem 2 and Corollary 2.1 pose a restriction on individually. As we are dealing with bounded (or even characteristic) functions, we however observe that it is also possible to obtain an analogous condition on the product in Theorem 2 and Corollary 2.1: Indeed, assume that is such that for some the product
is bounded. Then, we claim that for
also the product
is bounded. To derive this, we first observe that the bound for allows us to infer that for any
As a consequence, we deduce that
Here we have made use of the specific choices of exponents from (17) and the boundedness of , which allowed us to invoke (18). This concludes the argument for the claim.
Thus, relying on the bound (19), we infer that given a bound on (16), we obtain that for all exponents from (17)
Here we applied Theorem 2 (or Corollary 2.1), for which we noted that the respective exponents are admissible. On the one hand, this is the desired analogue of the condition from Theorem 3 and allows us to obtain a whole family of bounds for , where . On the other hand, it shows that although is not admissible in Theorem 2 and Corollary 2.1, for our purposes, it still suffices to consider the case and to prove a control for (16), which then gives the full range of expected exponents in the form of the estimate (20).
Remark 2.3 (Fractal packing dimension).
Following Sickel [Sic99], Proposition 3.3 (c.f. also [JM96], Theorem 2.2) we remark that for a characteristic function its regularity has direct consequences on the packing dimension (c.f. [JM96], [Mat99]), which we denote by , of its boundary: If for some set its characteristic function satisfies for some and , then
and denotes the complement of .
2.2. Matrix space geometry
Before discussing our convex integration scheme, we recall some basic notions and properties of the hexagonal-to-rhombic phase transformation, which we will use in the sequel.
We begin by introducing notation for the symmetric and antisymmetric part of two matrices.
Definition 2.4 (Symmetric and antisymmetric parts).
Let . We denote the uniquely determined symmetric and antisymmetric parts of by
2.2.1. Lamination convexity notions
Relying on the notation from Definition 2.4, in the sequel we discuss the different notions of lamination convexity. Here we distinguish between the usual lamination convex hull (defined by successive rank-one iterations) and the symmetrized lamination convex hull (defined by successive symmetrized rank-one iterations):
Definition 2.5 (Lamination convex hull, symmetrized lamination convex hull).
We define the following notions of lamination convex hulls:
Let . Then we set
We refer to as the laminar convex hull of and to as the laminates of order at most .
Let . Then we define
Here . We refer to as the symmetrized laminar convex hull of and to as the symmetrized laminates of order at most .
We denote the convex hull of a set by .
We note that if or is (relatively) open, then also or is (relatively) open.
Lemma 2.7 (Convex hull = laminar convex hull).
Let be as in (5). Then
Moreover, each element is symmetrized rank-one connected with each element in .
The following lemma establishes a relation between rank-one connectedness and symmetrized rank-one connectedness. It in particular shows that in two dimensions all symmetric trace-free matrices are pairwise symmetrized rank-one connected.
Lemma 2.8 (Rank-one vs symmetrized rank-one connectedness).
Let with . Then the following statements are equivalent:
There exist vectors such that
There exist matrices and vectors such that
We refer to [Rül16], Lemma 9 for a proof of this statement. ∎
This lemma allows us to view symmetrized rank-one connectedness essentially as equivalent to rank-one connectedness.
2.2.2. Skew parts
We discuss some properties of the associated skew symmetric parts of rank-one connections, which occur between points in the interior of . To this end, we introduce the following identification:
Notation 2.9 (Skew symmetric matrices).
As the two dimensional skew symmetric matrices are all of the form
we use the mapping to identify with . We define an ordering on by the corresponding ordering on , i.e.
We begin by estimating the symmetric and skew-symmetric parts of a symmetrized rank-one connection:
Let with . Then
Here denotes the spectral norm, i.e. , where denotes the norm.
Since , we obtain that
As forms an orthonormal basis, this shows that .
Similarly, we obtain that
and hence . ∎
Using the previous result, we can control the size of the skew part which occurs in rank-one connections with :
For all matrices with and with being rank-one connected with a matrix it holds
For each there are exactly two matrices such that
Let for some . Then, is explicitly given by . Thus, Lemma 2.10 implies
since and by the trace-free condition. As is a compact set, is uniformly bounded. Moreover, the diameter of is less than five, which yields the desired bound. ∎
2.2.3. Geometry of the hexagonal-to-rhombic phase transformation
In this subsection, we discuss the specific matrix space geometry of the hexagonal-to-rhombic phase transformation. To this end we decompose each matrix of the form into a component in and a component in direction.
With this notation we make the following observations:
Let be as above. Then,
Furthermore, we have that