Morrey’s Conjecture for the Planar Volumetric-Isochoric Split: Least Rank-One Convex Energy Functions

We consider Morrey’s open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies W:GL+(n)→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W{:}{\text {GL}}^+(n)\rightarrow \mathbb {R}$$\end{document} with an additive volumetric-isochoric split, i.e. W(F)=Wiso(F)+Wvol(detF)=W~iso(FdetF)+Wvol(detF),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} W(F)=W_{\mathrm{iso}}(F)+W_{\mathrm{vol}}(\det F)={\widetilde{W}}_{\mathrm{iso}}\bigg (\frac{F}{\sqrt{\det F}}\bigg )+W_{\mathrm{vol}}(\det F)\,, \end{aligned}$$\end{document}which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of “least” rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function Wmagic+(F)=λmaxλmin-logλmaxλmin+logdetF=λmaxλmin+2logλmin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} W_{\mathrm{magic}}^{+}(F)=\frac{\lambda _{\mathrm{max}}}{\lambda _{\mathrm{min}}}-\log \frac{\lambda _{\mathrm{max}}}{\lambda _{\mathrm{min}}}+\log \det F=\frac{\lambda _{\mathrm{max}}}{\lambda _{\mathrm{min}}}+2\log \lambda _{\mathrm{min}} \end{aligned}$$\end{document}is quasiconvex. In addition, we demonstrate that under affine boundary conditions, Wmagic+(F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{\mathrm{magic}}^+(F)$$\end{document} allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of Burkholder and Iwaniec in the field of complex analysis.

which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of onedimensional coupled differential inequalities. We identify a number of "least" rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function The open question is whether or not rank-one convexity implies quasiconvexity. Morrey (1952Morrey ( , 2009) conjectured that this is not the case in general and indeed the result by Šverák (1992b) settles the question for the case n ≥ 3. On the other hand, in the one-dimensional case, all four notions of convexity are equivalent. For the last nearly thirty years, impressive progress on the problem has been made (Kawohl and Sweers 1990;Harris et al. 2018;Faraco and Szekelyhidi 2008;Pedregal and Sverak 1998;Sebestyen and Szekelyhidi 2015;Müller 1999) without, however, leading to a full solution. While researchers coming from the area of nonlinear elasticity in general seem to believe that Morrey's conjecture is true also in the planar case, i.e. that rank-one convexity does not imply quasiconvexity, researchers with a background in conformal and quasiconformal analysis (Astala et al. 2008) tend to believe in the truth of the opposite implication (Iwaniec 1982). In conformal analysis, this conclusion would imply a number of far-reaching conjectures which are themselves deemed reasonable at present. Nonlinear elasticity is concerned with energies W defined on the group of matrices with positive determinant GL + (n) = F ∈ R n×n | det F > 0 , while the general case of Morrey's question considers scalar functions W defined on the linear space R n×n . It is conceivable that the answer to the planar Morrey's conjecture depends on whether one focuses on GL + (2) or R 2×2 . Assuming that an example for Morrey's conjecture in GL + (2) can be found, it is also not immediately clear how such an example might be generalized to the R 2×2 -case (although it is tempting to believe that this is always possible). The precise statement of Morrey's conjecture also depends on the invariance properties one requires for the energy W and the domain of definition. Different versions of Morrey's conjecture include the following, all of which currently remain open: Morrey I: Let W : R n×n → R (with no further invariance requirement). If W is rank-one convex, then W is not necessarily quasiconvex. For n ≥ 3 (W being continuous), this statement on the class of all real-valued functions as originally considered by Morrey (1952) has been conclusively settled by Sverak's famous counterexample (Šverák 1992b).
Morrey II: Let W : R 2×2 → R be continuous and left/right-SO(2) invariant (objective and hemitropic). If W is rank-one convex, then W is not necessarily quasiconvex. Morrey III: Let W : GL + (2) → R be left-SO(2) invariant (objective, frameindifferent) and right-O(2) invariant (isotropic). If W is rank-one convex, then W is not necessarily quasiconvex. Morrey IV: Let W : R 2×2 → R be left-SO(2) invariant (objective, frame-indifferent) and right-O(2) invariant (isotropic). If W is rank-one convex, then W is not necessarily quasiconvex. Morrey V: Let W : GL + (2) → R and W (F) → ∞ as det F → 0 be left/right-SO (2) invariant (objective and hemitropic). If W is rank-one convex, then W is not necessarily quasiconvex. Astala et al. (2012) and their groups try to disprove Morrey II to prove various conjectures in the context of complex analysis. It is conceivable, however, that Morrey II is false, while Morrey III is true. Similarly, Morrey III may also be false with Morrey II being true. There is no clear connection between the two statements: in Morrey III the domain of definition is smaller, but then rank-one convexity needs only to hold true on a smaller set as compared to Morrey II. The role of invariance requirements (isotropic versus hemitropic) or general anisotropy is also open. The restriction to GL + (2) and isotropy in Morrey III is advantageous because the check of rank-one convexity can be based on a version of Knowles/Sternberg conditions Sternberg 1976, 1978) operating only on the representation of the elastic energy in terms of singular values. The rank-one convexity condition on R 2×2 was also completely settled by Hamburger (1987), Aubert (1995). Nevertheless, it seems that one has to choose a favourable representation first and then use an adapted criterion. Quasiconvexity on the other hand is notoriously difficult to prove or disprove. A straightforward sufficient condition is Sir John Ball's polyconvexity condition (Ball 1976). One of the advantages of polyconvexity is that it is both suitable for GL + (2) and R 2×2 and that there are sharp characterizations of polyconvexity in the GL + (2) and isotropic case (Dacorogna 2008;Šilhavý 1997;Mielke 2005). Apparently, even for n ≥ 3, there is no isotropic example to Morrey I to be found in the literature. A common method to approach Morrey's conjecture is to discuss a specific family of energies, e.g. purely volumetric or isochoric energies (cf. Sects. 2.1 and 2.2), to find an often simple relation between the different notions of convexity. We will extend these findings to a much more involved class of energies with an additive volumetric-isochoric split (2.1) and follow the concept of "least" rank-one convex energies, i.e. functions that are as weakly rank-one convex as possible, to serve as canonical candidates. Focusing on energies with a concave volumetric part we show that the rank-one convex function W + magic (F) = λ max λ min − log λ max λ min + log det F = λ max λ min + 2 log λ min with λ max ≥ λ min > 0 as the ordered singular values of the deformation gradient F = ∇ϕ suffices as the sole candidate for this class of functions. The question of quasiconvexity for W + magic (F) closes Morrey's conjecture for all volumetric-isochorically split energies with a concave volumetric part, cf. Theorem 4.1, giving the following alternatives: • If W + magic is not quasiconvex, then Morrey's conjecture III is true. • If W + magic is quasiconvex, then no example of Morrey's conjecture III can be found in the class of volumetric-isochoric split energies with a concave volumetric part, i.e. in this class, rank-one convexity equals quasiconvexity. 1 In a sequel of this work (Voss et al. 2021a) we will numerically investigate the quasiconvexity of W + magic (F).

Basic Definitions of Convexity Properties
for any domain ⊂ R n with Lebesgue measure | |. The energy is strictly quasiconvex if inequality (1.1) is strict.
While quasiconvexity is sufficient to ensure weak lower semi-continuity of the energy functional, it has the simple disadvantage of being almost impossible to prove or disprove. This led to the introduction of sufficient and necessary conditions for quasiconvexity.
1 Since we show that W + magic is not polyconvex, rank-one convexity does not equal polyconvexity in this case, in contrast to the classes of purely volumetric (Dacorogna 2008) and purely isochoric (Martin et al. 2017) energies and for functions defined on SL(2) (Ghiba et al. 2018).
Whereas polyconvexity provides a sufficient criterion for quasiconvexity, rank-one convexity is introduced as a necessary condition.
If the energy is two times differentiable, global rank-one convexity is equivalent to the Legendre-Hadamard ellipticity condition which expresses the ellipticity of the Euler-Lagrange equations Div DW (∇ϕ) = 0 associated to the variational problem The energy is strictly rank-one convex if inequality (1.3) is strict.
Overall, for any W : R n×n → R ∪ {+∞} we have the well known hierarchy (Ball 1976;Dacorogna 2008) polyconvexity ⇒ quasiconvexity ⇒ rank-one convexity. (1.6) In nonlinear elasticity theory, energy functions are more naturally defined on GL + (n) instead of R n×n , since a non-positive determinant of the deformation gradient F = ∇ϕ would correspond either to an infinite compression (det F = 0) or to a change of orientation (det F < 0) within the modelled physical body, both of which are deemed mechanically implausible.

The Additive Volumetric-Isochoric Split
Next, we discuss the general family of planar energies W : GL + (2) → R ∪ {∞} with additive combinations of conformally invariant (isochoric) energies on the one hand, and area-dependent (volumetric) energies (also called the additive volumetricisochoric split) on the other hand: . (2.1) It is already known that for purely isochoric energies (cf. Sect. 2.2) as well as purely volumetric energies (cf. Sect. 2.1) rank-one convexity already implies polyconvexity (and thereby quasiconvexity). Thus, it is suggestive to inquire whether the same conclusion can be reached for additive combinations or not. The additive volumetric-isochoric split itself is a natural finite extension of isotropic linear elasticity, i.e.
The right-hand side of the energy is automatically additively separated into pure infinitesimal shape change and infinitesimal volume change of the displacement gradient ∇u, with a similar additive split of the linear Cauchy stress tensor into a deviatoric part and a spherical part, depending only on the shape change and volumetric change, respectively. This type of energy functions is often used for modeling slightly incompressible material behaviour (Ciarlet 1988;Hartmann and Neff 2003;Favrie et al. 2014;Ogden 1978;Charrier et al. 1988) or when otherwise no detailed information on the actual response of the material is available (Simo 1988;Gavrilyuk et al. 2016;Ndanou et al. 2017). In the nonlinear regime, this split is not a fundamental law of nature for isotropic bodies (as it is in the linear case) but rather introduces a convenient form of the representation of the energy. Formally, this split can also be generalized to the anisotropic case, where it shows, however, some severe deficiencies (Federico 2010;Murphy and Rogerson 2018) from a modeling point of view. 2 A primary application of the volumetric-isochoric split is the adaptation of incompressible hyperelastic models to (slightly) compressible materials in arbitrary dimension n ≥ 2: For any given isotropic function 3 W inc : SL(n) → R, an elastic energy potential W of the form (2.1) can be constructed by simply setting with an appropriate function f : R + → R, where I 1 , . . . , I n−1 denote the principal invariants of F n √ det F . Common examples of such energy functions include the compressible Neo-Hooke and the compressible Mooney-Rivlin model (Ogden 1983). A typical example of the volumetric-isochoric format is the geometrically nonlinear quadratic Hencky energy Hencky 1928;Sendova and Walton 2005;Martin et al. 2019) as well as its physically nonlinear extension, the exponentiated Hencky-model Neff and Ghiba 2016) which has been used for stable computation of the inversion of tubes (Nedjar et al. 2018), the modeling of tire-derived material (Montella et al. 2016) or applications in bio-mechanics (Schroder et al. 2018). It is well known that (2.3) and (2.4) are overall not rank-one convex ) and indeed there does not exist any elastic energy depending on dev n log V 2 , n ≥ 3 that is rank-one convex ). The situation is, surprisingly, completely different in the planar case: For n = 2, the energy in Eq. (2.4) is not only rank-one convex, but even polyconvex Ghiba et al. 2015b) if k ≥ 1 8 .

Purely Volumetric Energies
For some types of energies W : R 2×2 → R, precise conditions are known under which rank-one convexity implies polyconvexity and therefore quasiconvexity (Müller 1999;Ball 2002). One specific family of energies for which the relation between rank-one convexity and polyconvexity is already known is that of purely volumetric energies W (F) = f (det F). This class describes fluid-like material behaviour, i.e. shape change is not penalized at all.
The following are equivalent: The same holds for the so-called Hadamard material (Dacorogna 2008;Ball and Murat 1984;Grabovsky and Truskinovsky 2019;Dantas 2006): The following are equivalent:

Purely Isochoric Energies
In a recent contribution (Martin et al. 2017), we present a similar statement for isochoric energies, also called conformally invariant energies, i.e. functions W : denotes the conformal special orthogonal group. This requirement can equivalently be expressed as i.e. left-and right-invariance under the special orthogonal group SO(2) and invariance under scaling. An energy W satisfying W (a F) = W (F) is more commonly known as isochoric. Following Astala et al. (2008), we introduce the (nonlinear) distortion function or outer distortion where . denotes the Frobenius matrix norm with F 2 = 2 i, j=1 F 2 i j as well as the linear distortion or (large) dilatation where |F| = sup ξ =1 Fξ R 2 = λ max denotes the operator norm (i.e. the largest singular value) of F.
Note that the isotropy requirement (2.11) implies that h is already uniquely determined by its values on [1, ∞). While we do distinguish between λ 1 , λ 2 as the non-ordered singular values and λ max ≥ λ min as the ordered singular values, we will interchange h and h to simplify some notations here because h(t) = h(t) for all t ≥ 1.

Least Rank-One Convex Candidates
We aim to discuss Morrey's conjecture in this setting, i.e.: • Is every rank-one convex energy with an additive volumetric-isochoric split in 2D already quasiconvex?
Our approach to this question is based on identifying volumetric-isochorically split energies which are, in a cetain sense, extremal (Fenchel and Blackett 1953;Šverák 1992a) among the rank-one convex functions. To this end, we consider the set of coupled one-dimensional inequalities from Theorem 3.2, which are equivalent to rankone convexity, and find those functions for which equality holds in at least one case; due to this extremal property, we will refer to these functions as "least" rank-one convex. 4 In Sect. 4, this allows us to reduce the above question to a single function for the case of a convex volumetric part, while mutliple least rank-one convex energies with a non-convex volumetric part are identified in Sect. 5. A summary of our approach is shown in Fig. 2 on p. 21.

Rank-One Convexity
For objective and isotropic planar energies with additive volumetric-isochoric split, there exists a unique representation (Martin et al. 2017, Lemma 3.1) where λ 1 , λ 2 > 0 denote the singular values of F and h, f are given real-valued functions. In view of Lemmas 2.2 and 2.4 it is now tempting to believe that rank-one convexity conditions on W iso and W vol also allow for a sort of split and can be reduced to separate statements on h and f . However, this is not even the case in planar linear elasticity, where W lin from Eq. (2.2) is rank-one convex in the displacement gradient ∇u if and only if μ ≥ 0 and μ + κ ≥ 0. This means that rank-one convexity of W lin implies that W lin iso is rank-one convex, whereas W lin vol might not be rank-one convex, since e.g. κ < 0 is allowed. We are therefore prepared to expect some coupling in the conditions for h and f . For rank-one convexity on GL + (2), Sternberg 1976, 1978) (see also Šilhavý 1997, p. 308) established the following important and useful criterion.
Lemma 3.1 (Knowles and Sternberg 1976, 1978, cf. Šilhavý 1997Parry 1995;Dacorogna 2001;Šilhavý 1999Aubert and Tahraoui 1987;Aubert 1995) Let W : GL + (2) → R be an objective-isotropic function of class C 2 with the representation in terms of the singular values of the deformation gradient F given by . Then W is rank-one convex if and only if the following five conditions are satisfied: (i) g x x ≥ 0 and g yy ≥ 0 for all x, y ∈ (0, ∞) , (separate convexity) Furthermore, if all the above inequalities are strict, then W is strictly rank-one convex.
In a recent contribution (Voss et al. 2021b) we applied the criterion by Knowles and Sternberg to volumetric-isochorically split energies: using weakened regularity assumptions (Martin et al. 2020) the following conditions can be established.

The Volumetric Part f(z)
Theorem 3.2 yields: Proof The separate convexity condition 1) in Theorem 3.2 implies Thus for arbitrary s ∈ (0, 1) the monotonicity condition 2) in Theorem 3.2 implies On the other hand, if h and f are both convex, the energy is already polyconvex as the sum of two individual polyconvex energies (cf. Lemmas 2.1 and 2.4). Therefore, for our purpose it is sufficient to consider the two subclasses and where either f or h are convex. The conditions in Theorem 3.2 contain several different expressions related to the isochoric part h(t), while f 0 = inf z∈(0,∞) z 2 f (z) occurs exclusively for the volumetric term. We utilize this observation to reduce the volumetric parts of the sets M + and M − to a single function for all least rank-one convex representatives: that is rank-one convex but not quasiconvex. Then there exists a constant c ∈ R so that the energy is rank-one convex but not quasiconvex, as well.
Proof Both energies have the same isochoric part, but different volumetric parts. We set 5 Thus the conditions in Theorem 3.2 are identical for both energies and they share the same rank-one convexity behaviour. Therefore, W 0 (F) is rank-one convex. Regarding the quasiconvexity, we consider the difference between the two energies of two individual quasiconvex functions is quasiconvex as well, which contradicts the initial assumption that W (F) is not quasiconvex. Equation (3.9) shows that every rank-one convex function W (F) with a volumetricisochoric split can be rewritten as the sum of two individual rank-one convex energies (3.10) The second part W * (F) = f * (det F) is a purely volumetric energy, i.e. rank-one convexity already implies quasiconvexity, while the first part W 0 (F) maintains the isochoric part h(t). In other words, Lemma 3.4 states that if there exists a volumetricisochorically split energy W (F) that is rank-one convex but not quasiconvex, then there exists a function W 0 (F) = h λ 1 λ 2 + c log(λ 1 λ 2 ) that is rank-one convex but not quasiconvex as well.
Since multiplying with an arbitrary constant α ∈ R + has no effect on the convexity behaviour of the energy function, we can also set c = ±1 without loss of generality. We reduce the two subclasses M + and M − of volumetric-isochorically split energies to two new sets

Subclass M + : Convex Isochoric Part
We focus on volumetric-isochorically split energies which have a convex isochoric part h(t) together with a non-convex volumetric part f (det F) = log det F, i.e. W ∈ M * + . It is possible to reduce the whole class M * + and thereby M + , to a single rank-one convex representative W + magic (F) (cf. Theorem 4.1) as the only candidate to check for quasiconvexity to answer Morrey's questions for all energies with a volumetricisochoric split and a convex isochoric part. This candidate is found by searching for energy functions that satisfy inequality conditions of Theorem 3.2 by equality. We start with the first inequality which is the most important condition due to its equivalence to separate convexity: (4.1) We consider the case of equality for all t ≥ 1. For every W ∈ M * + , the volumetric part is given by f (z) = log z, which implies and we compute The constant b ∈ R is negligible regarding any convexity property and thus we set b = 0. We determine the other constant a ∈ R by testing the remaining convexity conditions and choose the constant a so that these inequalities are satisfied as nonstrictly as possible to obtain our first least rank-one convex candidate. In this way, the monotonicity condition 2) of Theorem 3.2 is given by Next, condition 3a) of Theorem 3.2, is satisfied for all a ≥ 1 while condition 4a) of Theorem 3.2, is satisfied for a ≥ −1. The two remaining conditions 3b) and 4b) of Theorem 3.2 are satisfied for a ≥ 1 and a ≥ 3 − 2 √ 2, respectively. Putting all conditions together, we conclude that the case a = 1 should be considered optimal and results in a least rank-one convex candidate. Overall, our canonical candidate is h(t) = t − log t, which corresponds to the energy function The energy W + magic (F) satisfies the conditions 1) and 3a) by equality for all t ≥ 1, conditions 2) and 3b) for t = 1. We show that this function is the single representative for the energy class M + regarding the question of quasiconvexity.
Theorem 4.1 The following are equivalent: such that h is convex, i.e. W ∈ M + , and W is rank-one convex but W is not quasiconvex.
with singular values λ max ≥ λ min of F is not quasiconvex.
Proof Using Theorem 3.2, we can directly prove the rank-one convexity of W + magic . We calculate Starting with W (F) from (i), we use Lemma 3.4 to reduce the class of energies from M + to M * + . Thus there exists a constant c > 0 so that the energy is rank-one convex but not quasiconvex. Without loss of generality, we set c = 1, because multiplying with an arbitrary positive constant has no effect on the convexity behaviour of the energy, and change h accordingly. We consider the difference which is a purely isochoric energy. We show that W * (F) itself is quasiconvex. For this, we study the first two conditions of Theorem 3.2 for the rank-one convex energy Thus h(t) is convex for all t ≥ 1 and non-decreasing at t = 1. Both together imply is quasiconvex. The sum of two individual quasiconvex functions is again quasiconvex; hence the non- Remark 4.2 Theorem 4.1 can also be stated as the equivalence between the quasiconvexity of W + magic and the implication i.e. for all energies with an additive volumetric-isochoric split and a convex isochoric part.
Due to the use of the ordered singular values λ max ≥ λ min instead of the unsorted singular values λ 1 , λ 2 of F, the representation h : R + → R of the isochoric part is not smooth at t = 1 in general. Nevertheless, for the candidate W + magic we have Although Theorem 4.1 reduces the question of whether M + contains a rank-one convex, non-quasiconvex function to the very particular question of whether W + magic (F) is quasiconvex, the latter still remains open at this point (cf. Sect. 6). We note, however, that the stronger condition of polyconvexity is indeed not satisfied for W + magic (F) Proof Every polyconvex function W : GL + (2) → R must grow at least linear for det F → 0 (Dacorogna 2008, Cor. 5.9). This can be motivated by the following: If W (F) is polyconvex, there exists a (not necessarily unique) representation P : R 2×2 × R + → R with W (F) = P(F, det F) which is convex. We consider the hyperplane attached to P at (1, 1). Convexity in R 2×2 × R + implies that P is bounded below from its hyperplane. On the contrary, To summarize, the question whether an energy W ∈ M + exists which is rank-one convex but not quasiconvex or not, i.e. rank-one convexity always implies quasiconvexity, can be reduced to the specific question of quasiconvexity of the single energy function W + magic (F). In this sense, our candidate represents an extreme point of M + , i.e. it is the single least rank-one convex function of this subclass. Moreover, our energy W + magic (F) is non-polyconvex, which is necessary to serve as a possible example for Morrey's conjecture.
i.e. W ∈ M * − . This time, the question of whether an energy W ∈ M − exists which is rank-one convex but not quasiconvex is much more involved and cannot be reduced to the matter of quasiconvexity of a single extreme representative. Nevertheless, we give a few candidates which are analogous to W + magic ∈ M + . Again, following the concept of least rank-one convex functions, we are looking for energies that cannot be split additively into two non-trivial functions which are both individually rank-one convex. For this, we take the four inequalities of Theorem 3.2 characterizing rank-one convexity and consider the corresponding differential equations, i.e. energies which only fulfill these inequalities conditions by equality. However, a function does not have to satisfy all four conditions non-strictly to be a least rank-one convex candidate; recall that the energy W + magic (F) as the single representative of M + satisfies conditions 1) and 3a) by equality for all t ≥ 1 and condition 2) for t = 1 but is not optimal for the remaining inequalities. As a method to find candidates for least rank-one convex functions of M − , we examine each condition characterizing rank-one convexity separately and search for equality for all t ≥ 1. Thus we are capable of finding very interesting energies, e.g. a function which is nowhere strictly separately convex or satisfies another inequality condition by equality. However, due to their dependence on t ≥ 1 this approach does not cover all possible extreme cases of M − . An energy can already be a least rank-one convex candidate if it satisfies one or more inequality conditions non-strictly for a single t ≥ 1 and not all t ≥ 1 as described above (cf. Sect. 5.2). Besides, conditions 3) and 4) allow for even more involved combinations because of their internal case-by-case distinction.

Obtaining W − magic
We start with the first inequality of Theorem 3.2, which is the most important condition due to its equivalence to separate convexity: (5.1) We consider the case of equality for all t ≥ 1. For every W ∈ M * − , the volumetric part is given by f (z) = − log z, which implies and we compute The constant b ∈ R is negligible regarding any convexity property and thus we set b = 0. We determine the other constant a ∈ R by testing the remaining convexity conditions and choose the constant a so that these inequalities are satisfied as nonstrictly as possible to obtain our first least rank-one convex candidate. In this way, the monotonicity condition 2) of Theorem 3.2 is given by Next, condition 3a) of Theorem 3.2, is satisfied for all a ≥ −1 while condition 4a) of Theorem 3.2, is satisfied for a ≥ 1. The two remaining conditions 3b) and 4b) of Theorem 3.2 are never satisfied for a = 1. Putting all conditions together, we conclude that the case a = 1 yields a least rank-one convex function. Overall, our canonical candidate regarding equality in condition 1) of Theorem 3.2 for all t ≥ 1 is h(t) = t + log t, which corresponds to the energy function an analogous candidate to W + magic ∈ M + . The energy W − magic (F) satisfies condition 1), 4a) and 4b) non-strict for all t ≥ 1, conditions 2) in the limit t → ∞ but is not optimal for the remaining inequalities. Contrary to the discussion in the last section, where we showed that W + magic is as the single least rank-one convex candidate for the energy class M + , the same is not true for W − magic (F), i.e. it exists several distinct least rank-one convex energies for the class M − .

Obtaining W smooth
We continue with the second condition of Theorem 3.2, which is equivalent to the Baker-Ericksen inequalities, and consider the limit case The only solution to this differential equation is the constant function h(t) = c which provides the purely volumetric energy W (F) = c − log det F. Thus we have no (nontrivial) least rank-one convex candidate corresponding to the monotonicity condition 2). Before we continue to work with the more involved conditions 3) and 4), we will consider an alternative variant to produce more least rank-one convex energies utilizing the fact that an energy function can be least rank-one convex without satisfying one single condition non-strict for all t ∈ (1, ∞). Starting with W − magic (F), we can weaken the monotonicity condition 2) by waiving the requirement of being nowhere strictly separately convex. However, we have to ensure that the energy remains non-strictly separately convex for one t ∈ [1, ∞]. Thus we are looking for a new function h(t) which is less monotone but slightly more convex then the expression t + log t. We choose the ansatz and with the first two conditions of Theorem 3.2 we compute We are interested in the limit case b = a + 1 so that h (1) = 0, i.e. the function is not strictly monotone for t = 1. The second limit case a = 0 would reestablish W − magic (F). Note that the separate convexity condition is only non-strict in the limit case t → ∞. We evaluate the remaining conditions and choose the remaining constant a ≥ 0 accordingly. Then which is satisfied for a ≥ −1 which is already ensured by conditions 2). Lastly, we compute which is fulfilled for a ≥ 1. For the case 0 ≤ a ≤ 1 the remaining condition is satisfied for a ≥ 1 3 . Overall, a = 1 3 yields the least rank-one convex candidate with the isochoric term h(t) = t 3 + 4 3t + log t which generates The energy W smooth (F) is strictly rank one convex for all t ∈ (1, ∞), but unlike W − magic (F), it is only a least rank-one candidate in the sense that it is non-strictly separately convex for t → ∞ and satisfies condition 2) of Theorem 3.2 by equality for t = 1.

Obtaining W 3b
We generate a third candidate using our original method with focus on the last two inequalities of Theorem 3.2 and start with condition We solve the differential equation by adding a particular solution to the associated homogeneous equation: (5.14) However, the monotonicity condition for all t > 1 (5.15) cannot be satisfied in the limit t → 1. Thus there does not exist a rank-one convex candidate which satisfies condition 3a) by equality for all t > 1. We continue with condition (5.16) Again, we solve the differential equation by adding a particular solution to the associated homogeneous equation: We set b = 0 and determine c ∈ R by evaluating the remaining rank-one convexity conditions. The separate convexity condition 1) yields In the case c ≥ 1, the infimum is attained at t = 1 and For the case c ≤ 1, the infimum is at t → ∞ and Thus c ≤ 1. The monotonicity condition is satified for c ≥ 1 2 . The remaining condition 3a) leads to 2t t − 1 Thus we can consider the two borderline cases (5.23) However, the first function is already polyconvex and therefore not suitable to provide an example for Morrey's question with respect to volumetric-isochorically split energies. The second part h(t) = t + log t is already known as the isochoric part of W − magic (F). We consider the condition With the support of Mathematica, the only solution that also satisfies the remaining inequalities turns out to be with an arbitrary constant c > 0. The condition 1) is non-strict in the limit t → ∞ and condition 2) is non-strict for t = 1 which is a similar behaviour to that of W smooth (F).
In the case of c = 1, condition 4a) is non-strict in the limit t → ∞ and we obtain h(t) = 1 2t 1 + t 2 + (1 + t) 1 + 6t + t 2 − log t + log 1 + 4t + t 2 + (1 + t) 1 + 6t + t 2 (5.27) as our new least rank-one convex candidate. Note that the relation implies that h is smooth at t = 1. Together with the volumetric part f (z) = − log z we arrive at which is equivalent to For the last differential equation of Theorem 3.2, we use Mathematica again, but this time the already known candidate W − magic (F) is the only solution which also satisfies the remaining conditions as a least rank-one convex candidate.

Comparison
Overall, we have obtained three energy function W − magic (F), W smooth (F) and W 3b (F) which satisfy different conditions of Theorem 3.2 by equality (cf. Table 1, Fig. 1). We use them as least rank-one convex representatives of M − .
Unfortunately, we cannot show yet that we identified all least rank-one convex energies in the class M − . Similar to Sect. 5.2, it is possible to construct more least rankone convex energies, in the sense that they satisfy at least one inequality condition of Theorem 3.2 non-strictly for an arbitrary t ∈ [1, ∞) or in the limit t → ∞ by choosing a different ansatz than equation (5.8). Therefore, it is not sufficient to test these three Here, represents that the inequality condition is strictly satisfied (and not by equality) while ✗ states that condition is not satisfied for all t ∈ (0, ∞). Only one of the two conditions 3a) and 3b) and one of the two conditions 4a) and 4b) have to be fulfilled candidates for quasiconvexity and conclude the quasiconvexity behaviour for the whole energy class M − . This has the following reasons: First, we only considered the case where the isochoric part satisfies one of the conditions 3a) and 4a) for all t ≥ 1 or 3b) and 4b) for all t ≥ 1. This is a more restrictive constraint than needed for rankone convexity, e.g. it would be sufficient if an energy satisfies condition 3a) for some interval I ⊂ R + and condition 3b) for the remaining part R + \ I . It is only necessary that for every t ≥ 1 one of the two conditions 3a) or 3b) (and 4a) or 4b) respectively) is satisfied. It is not clear how to decide which of our three energies W − magic , W smooth , W 3b is the best candidate for Morrey's conjecture because each energy is already least rank-one convex in the sense of satisfying one inequality condition of Theorem 3.2 non-strictly. Thus the difference between two least rank-one convex candidates is neither a rankone convex nor a rank-one concave function, e.g. the transition from W − magic to W smooth is done by adding the isochoric part h(t) = 4 3 1 t − 2 3 t which is convex but monotone decreasing. An energy function can also be non-rank-one convex and non-concave simultaneously at some F 0 ∈ GL + (2), similar to a saddle point. This makes it very challenging, if not impossible, to find a single weakest rank-one convex candidate for M − . Still, with separate convexity as the most important part of rank-one convexity, it is reasonable to start with the candidate W − magic (F) and check for quasiconvexity. Our first (negative) result in this respect is the following.

Proposition 5.1 The energy
is polyconvex.

Proof
We check for polyconvexity with a Theorem due to Šilhavý (2002), Proposition 4.1 (see also Mielke 2005, Theorem 4.1) for energy functions in terms of the (ordered) singular values. It states that any W : GL + (2) → R of the form is polyconvex if and only if for every γ 1 ≥ γ 2 > 0, there exists such that for all ν 1 ≥ ν 2 > 0, The energy W − magic can be expressed in the form (5.30) with g such that g(x, y) = x y − 2 log(y) for x > y > 0. In particular, for γ 1 ≥ γ 2 > 0, Now, for any given γ 1 ≥ γ 2 > 0, choose In order to establish the proposition, it suffices to show that for any ν 1 ≥ ν 2 > 0, (5.34) Since ν 1 ≥ ν 2 > 0 by assumption, where the final inequality holds since t ≥ 1 + log(t) for all t > 0.

Connection to the Burkholder Functional
Recall that quasiconvexity of W + magic (F) would imply quasiconvexity for all rank-one convex energies with an additive volumetric-isochoric split and a convex isochoric part, while proving the opposite would lead to an example of Morrey's conjecture. Definition 6.1 For an energy function W : GL + (n) → R, we define its Shield transformation (Shield 1967;Ball 1977) as Does an isotropic energy W : GL + (2) → R with additive volumetric-isochoric split exist that is rank-one convex but not quasiconvex?
Note that (classical) convexity of W is not equivalent to the convexity of W # for dimension n ≥ 2; for example, the constant mapping W ≡ 1 is convex, whereas W # (F) = det F is not. Furthermore, in the one-dimensional case, the invariance of the convexity properties can be expressed as the equivalence which is related to the study of so-called reciprocally convex functions (Merkle 2004, Lemma 2.2) and, for f ∈ C 2 (R + ), follows directly from the equality Applying now the Shield transformation, we show a surprising connection between our candidate W + magic (F) and the work of Iwaniec in the field of complex analysis and the so-called Burkholder functional. Using the connection C 2 ∼ = R 2×2 we arrive at a much more general settings of functions W : R 2×2 → R without a volumetricisochoric split defined on the linear space R 2×2 instead GL + (2). 6 To the best of our knowledge, the equivalence between the quasiconvexity of W and W # has only been shown if non-smooth variations ϑ / ∈ C ∞ 0 ( ) are excluded.
is also known as the Burkholder functional 9 (Burkholder 1988;Astala et al. 2012). It is already established that the function B p is rank-one convex and satisfies a quasiconvexity property at the identity F = 1 in the case of smooth non-positive integrands (Astala et al. 2012, Theorem 1.2), namely if ϑ ⊂ C ∞ 0 ( ; R 2 ) and B p (1+∇ϑ(x)) ≤ 0 for all x ∈ , then Lemma 6.4 (Burkholder 1988) The function B p : Proof For the convenience of the reader, we add the proof from Burkholder (1988): We use the connection to complex analysis R 2×2 ∼ = C 2 (cf. "Appendix") where we identify and transform the Burkholder function B p (F) to a function L p : Then B p (F(z, w)) = −L p (z, w) and, for rank-one matrices ξ ⊗ η with ξ, η ∈ R 2 , we find . 9 In the literature, the Burkholder functional is usually defined via B p (F) = p 2 det F + 1 − p 2 |F| 2 |F| p−2 , i.e. with a reversed sign and it is investigated with respect to rank-one concavity and quasiconcavity instead (Astala et al. 2012).
An overview of the relation between our energy candidate W + magic (F) and the Burkholder functional B p (F) is visualized in Fig. 3. Again, while we try to find an example to settle Morrey's question, i.e. show that our energy function W + magic (F) is not quasiconvex, in the topic of martingale study, it is sought to prove that B p (F) is quasiconvex to positively affirm some estimates in function theory, see "Appendix" for more information.
Iwaniec/Burkholder approach: Volumetric-isochoric split approach: the "heart of the matter" [15] Fig. 3 Connection between the Burkholder functional B p (F) and our energy candidate W + magic (F) as a chance to check Morrey's conjecture, where |F| = sup ξ =1 Fξ R 2 = λ max denotes the operator norm (i.e. the largest singular value) of F

Radially Symmetric Deformations
Radial deformations play an important role for planar quasiconvexity. For isotropic materials, it seems reasonable that any global minimizer under radial boundary conditions is radial, but such a symmetry statement may cease to hold for non-convex nonlinear problems 11 (Brock et al. 1996). A deformation is radial if there exists a function v : [0, ∞) → R such that In that case, 3) 11 Let be the unit ball in R n . Then the minimization problem of the non-convex functional R(v) = 1 1 + ∇v(x) 2 dx (7.1) has neither a radial nor a unique solution in the class with the singular values of ∇ϕ(x) given by x . (7.4) Using mappings of this type, it is possible to construct an extensive class of inhomogeneous deformations for which the energy value of W + magic is identical to that of the homogeneous deformation (cf. Lemma 7.3). In particular, this implies that the homogeneous solution to the energy minimization problem is neither unique nor stable.
First, however, it is important to note that radial mappings cannot serve as a counterexample to Morrey's conjecture, since rank-one convexity of an energy is already sufficient for a homogeneous mapping to be energetically optimal among all radial deformations.

Expanding and Contracting Deformations
In general, non-trivial radial deformations do not have the same energy value as the homogeneous deformation ϕ 0 (x) = λx, especially if the energy is strictly separately convex. However, there exist non-trivial examples for the Burkholder energy B p (F) as well as for the energy W + magic (F) for which equality holds. In order to construct such deformations, we first consider the general class of radial functions with v(0) = 0 and v(R) = R which keep the center point and exterior radius constant. More specifically, we consider the subclasses of expanding and contracting functions for which the order of the singular values λ min and λ max remains fixed. 12 The class V R of expanding functions can be described by radial functions for which the gradient of the tangent in a point (r , v(r )) is smaller then the gradient of the secant of the origin with (r , v(r )) for any r ∈ [0, R] as shown in Fig. 4. In contrast, the class V −1 R of contracting functions contains any radial functions v such that the gradient of the tangent in a point (r , v(r )) is larger than the gradient of the secant of the origin with (r , v(r )) for any r ∈ [0, R] as shown in Fig. 4. The connection between V R and V −1 R is given by the inverse function theorem: For an where B R (0) denotes the ball with radius R and center 0.
(7.12) 12 The condition v(r ) r ≥ v (r ) is not trivial and only allows radial functions which inflate all disks with center in zero, i.e. v(r ) ≥ r for all r ∈ [0, R]. Note that if there exists a point r 0 with v(r 0 ) < r 0 , then v (r 0 ) ≤ v(r 0 ) r 0 < 1, which would imply that v(r ) < r for all r ≥ r 0 and thus v(R) < R. The following lemma establishes an analogous result for the function W + magic with for all radial deformations ϕ( Proof We first extend Lemma 7.2 to the function B : GL + (2) → R with for all F ∈ GL + (2), where B 2 (F) = det F; note that B 2 is a null Lagrangian. For expanding deformations defined by V R , we find thus the energy potential of B is indeed constant on the class V R . Next, recall that B and W + magic are related via the Shield transformation, i.e. that B # (F) = 1 2 W + magic (F)− 1 for all F ∈ GL + (2), and that for every expanding radial deformation ϕ, its inverse mapping ϕ −1 is a contracting radial deformation, i.e. that for every v(r ) ∈ V −1 R there exists a unique w(r ) ∈ V R with w = v −1 . We can therefore apply the general formula where F v and F w are the deformation gradients corresponding to v(r ) and w(r ) = v −1 (r ), respectively.
Both the radial deformations ϕ(x) = v( x ) x x defined by v ∈ V R in Lemma 7.2 and v ∈ V −1 R in Lemma 7.3 can be extended by Astala et al. (2012) (i) Choosing any F 0 = λQ instead of 1 for arbitrary λ > 0 and Q ∈ SO(2), (ii) Choosing any point z 0 ∈ instead of the origin, (iii) Combining multiple separate balls B R i (z i ) with i B R i (z i ) = ∅, Fig. 5 Circular packing of several balls with various contracting mappings to achieve an overall nonsymmetric deformation, also called piecewise radial mapping. Note that for the right-hand figure, the contraction in the "outer" circles need to satisfy the conditions stated in (iv) above (iv) Inductively adding more balls B r (z 0 ) which are contained in a previous ball B R (z 0 ) under the assumption that the contracting deformation of the outer ball B R (z 0 ) maintains F 0 = λQ at the boundary of the inner ball B r (z 0 ), to construct an overall non-radially symmetric deformation (cf. Fig. 5) with the same energy level as the homogeneous deformation.

Summary
The class of planar energy functions with an additive volumetric-isochoric split, i.e. functions on GL + (2) of the form where λ 1 , λ 2 > 0 denote the singular values of F and h, f are given real-valued functions, proves to be a rather interesting subject in the context of Morrey's conjecture: On the one hand, if neither h nor f is convex, then W is not rank-one convex; on the other hand, convexity of both f and h already implies that W is quasiconvex. Investigating the remaining cases, we first focused on functions with a convex isochoric part h. In that case, Theorem 4.1 shows that the question whether or not rank-one convexity implies quasiconvexity can be reduced to the question whether the particular (rank-one convex) function W + magic : GL + (2) → R with is quasiconvex. While W + magic is not polyconvex according to Lemma 4.3, determining whether or not W + magic is indeed quasiconvex remains an open problem at this point. The case of volumetric-isochorically split functions with a convex volumetric part f appears to be more involved: In Sect. 5, we identified three "least" rank-one convex functions W − magic (F), W smooth (F), and W 3b (F), none of which is currently known to be quasiconvex. A comprehensive overview for candidates with an additive volumetric-isochoric split is given in Fig. 2.

Symmetry and Stability
The energy potential W + magic also exhibits a number of interesting properties regarding the relation between symmetry and stability: According to Lemma 7.3, W + magic is constant for the non-trivial class of contracting radially symmetric deformations, i.e. W + magic allows for inhomogeneous deformations whose energy level is equal to the homogeneous one. This implies that the homogeneous solution to the energy minimization problem is neither unique nor stable.
As shown in Fig. 5, it is thereby possible to construct a non-symmetrical microstructure for the fully isotropic energy potential W + magic even if the applied boundary conditions are fully symmetrical. This is in stark contrast to the linear convex case, where such symmetrically posed problems generally yield symmetric solutions; in practice, this observation allows for certain simplifications (e.g. by reducing calculations to any quarter of a cylinder) of symmetric linear problems. However, in the nonlinear non-convex case, it is even possible for a symmetrically stated problem (cf. Eq. (7.1)) to not have a stable symmetric solution at all (Brock et al. 1996). This phenomenon, which is exhibited by the energy function W + magic as well, demonstrates a "preference of disorder over order" which can occur in non-(strictly-)quasiconvex problems.
Note that due to Lemma 7.1, radially symmetric deformations cannot be energetically favourable compared to the affine solution even under the weaker assumption of separate convexity. Additionally, in an intriguing article by Kawohl and Sweers (1990), it is shown that the simple assumption ϕ(x, y) = ϕ(−x, y) for the test function ϕ ∈ W 1,∞ 0 ( ) on the domain = (−1, 1) 2 is enough for the transition from rank-one convexity to quasiconvexity. Thus in general, the break of symmetry plays an essential role in stability and, in particular, must be taken into account for the minimization of potentially non-quasiconvex energies.
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