Characterizations of Symmetric Polyconvexity
Abstract
Symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory. Due to the complexity of this notion, a common approach is to retreat to necessary and sufficient conditions that are easier to handle. This article focuses on symmetric polyconvexity, which is a sufficient condition. We prove a new characterization of symmetric polyconvex functions in the two and threedimensional setting, and use it to investigate relevant subclasses like symmetric polyaffine functions and symmetric polyconvex quadratic forms. In particular, we provide an example of a symmetric rankone convex quadratic form in 3d that is not symmetric polyconvex. The construction takes the famous work by Serre from 1983 on the classical situation without symmetry as inspiration. Beyond their theoretical interest, these findings may turn out useful for computational relaxation and homogenization.
1 Introduction
1.1 Summary of Results

(i) \(\varepsilon \mapsto \det \varepsilon \) is not symmetric polyconvex in \(d=2,3\);

(ii) \(\varepsilon \mapsto  \det \varepsilon \) is symmetric polyconvex in \(d=2\), but not in \(d=3\);

(iii) \(\varepsilon \mapsto (\mathrm{cof\,}\varepsilon )_{ii}\), \(i=1,2,3\) is symmetric polyconvex in \(d=3\), while \(\varepsilon \mapsto \mathrm{cof\,}\varepsilon \) is not.
We show that symmetric polyconvexity in 2d and 3d can be characterized as follows: any symmetric polyconvex function in 2d corresponds exactly to a convex function of all minors that is nonincreasing with respect to the determinant (see Theorem 4.1). In particular, this includes (ii) above and elucidates why (i) is not symmetric polyconvex. For 3d, we prove in Theorem 5.1 that any symmetric polyconvex function can be represented as a convex function of first and second order minors (hence, no dependence on the determinant) whose subdifferential with respect to its cofactor variable is negative semidefinite. Notice that this result is in correspondence with example (iii) above. The difficulty here lies in identifying these characterizations. The proofs build on monotonicity properties of convex functions expressed in terms of their (partial) subdifferentials (cf. Lemmas 3.3 and 3.4), as well as on some standard tools from convex analysis, properties of semiconvex functions in the classical setting, and on basic algebraic relations for minors.
1.2 Organization of the Article
The paper is structured as follow: in Section 2, we recall the common generalized notions of convexity in the symmetric setting and review selected results from the literature regarding their characterization, properties and relations. The section is concluded with the above mentioned simple motivating examples. After some notational remarks and preliminaries in Section 3, we address characterizations of symmetric polyconvexity in 2d in Section 4 and in 3d in Section 5. Moreover, we consider symmetric polyconvex quadratic forms in Sections 4.2 and 5.2, where we also present the proof of Theorem 5.7. Finally, symmetric polyaffine functions are the topic of Sections 4.3 and 5.3. In the remainder of the introduction, we comment further on the relevance of the notion of (symmetric) polyconvexity in the calculus of variations and elasticity theory.
1.3 Connections with (Symmetric) Quasiconvexity and Relaxation
Polyconvexity is a sufficient condition for quasiconvexity, which has been the subject of intensive investigation since its introduction by Morrey [39] in 1952. It constitutes the central concept for the existence theory of vectorial integral problems in the calculus of variations, generalizing the notion of convexity to multidimensional variational problems. If we consider the integral functional E in (1.1) with suitable assumptions on the integrand W and the space of deformations, quasiconvexity of W is necessary and sufficient for weak lower semicontinuity of E. Along with coercivity, quasiconvexity is thus a key ingredient for proving existence of minimizers of E (e.g. among all deformations with given boundary values) via the direct method in the calculus of variations, see e.g. [2, 5, 12, 16]. As for weak lower semicontinuity of the functional \({\mathcal {E}}\) in the geometrically linear theory of elasticity, cf. (1.2), symmetric quasiconvexity of the elastic energy density f takes over the role of quasiconvexity of W [4, 22, 26]. This means that f satisfies Jensen’s inequality for all symmetrized gradient test fields, or equivalently that the function is quasiconvex when composed with the linear projection of \({\mathbb {R}}^{d\times d}\) to symmetric matrices, cf. Proposition 2.2. For a further discussion on the connections between linearized (smallstrain) and nonlinear (finitestrain) elasticity theory we refer the reader e.g. to [3, 7, 20, 31].
Despite its importance, the notion of quasiconvexity is still not fully understood in all its facets due to its complexity. A common approach is therefore to retreat to the weaker and stronger conditions like rankone convexity and polyconvexity, as they are easier to deal with and yet allow to make useful conclusions about quasiconvex functions [16, 41]. The study of generalized notions of convexity for functions with specific properties has helped to gain valuable new insight and to advance the field. Relevant classes include onehomogeneous functions [17, 40], functions obtained by compositions with transposition [34, 42], or functions with different types of invariances, such as isotropic functions [16, 18, 37, 38, 49, 58, 59], quadratic forms with linearly elastic cubic, cyclic and axisreflection symmetry [29], and functions of linear strains [9, 10, 32, 44, 60, 61]. In this spirit, the characterizations of symmetric polyconvexity proved in this article contribute to a deeper understanding of quasiconvex functions that are invariant under symmetrization.
1.4 Applications to the Translation Method
Our characterization of symmetric polyconvex (and thus rankone convex) quadratic forms in 2d (cf. (1.3) or Proposition 4.5) yields an explanation of why the translator \(\varepsilon \mapsto \det \varepsilon \) is often a good choice. It was used for example in the 2d setting of [9] in the derivation of a relaxation formula for twowell energies with possibly unequal moduli. Indeed, if we rewrite the right hand side of (1.8) as \((fq)^{\mathrm{c}}  h^{\mathrm{c}} + h + q \ge (f h  q)^{\mathrm{c}} + h + q\) with a convex quadratic function \(h:{\mathcal {S}}^{2\times 2}\rightarrow {\mathbb {R}}\), we see in view of the first equation in (1.3) that working with just \(\varepsilon \mapsto \alpha \det \varepsilon \) for \(\alpha >0\) as a translator is equivalent to using all symmetric rankone convex quadratic forms. The second equation in (1.3) indicates that the analogous observation is true in three dimensions for \(\varepsilon \mapsto A:\mathrm{cof\,}\varepsilon \) with \(A\in {\mathcal {S}}^{3\times 3}\) positive semidefinite. Translators of this type play a key role in the derivation of the bounds in [9, 44]. Our characterization result hence provides structural insight into the choice of translators in the abovementioned literature.
In the classical setting, Firoozye [25] showed that a translation bound optimized over rankone convex quadratic forms and NullLagrangians is at least equally good as polyconvexification, and even strictly better for some threedimensional functions. His proof of this latter statement is based on Serre’s example in (1.4). Our example (1.6) in the 3d symmetric setting clearly implies that, in contrast to 2d, considering symmetric rankone convex quadratic forms as translators will in general give better bounds than using just symmetric polyconvex ones. Whether there are situations when combining symmetric rankone convex quadratic forms with other symmetric polyconvex functions leads to improved results remains an open question for future research; notice that we do not have any nontrivial NullLagrangians at hand in the symmetric setting, cf. Proposition 5.10.
2 Different Notions of Symmetric SemiConvexity
Here, we are interested in functions that are independent of the skewsymmetric part of its variables \(F\in {\mathbb {R}}^{d\times d}\), that is, functions that depend only on the symmetric part of F as motivated by geometrically linear elasticity theory, cf. Section 1. As documented there, also here the semiconvexity notions are of interest, i.e., symmetric quasi, poly and rankone convex functions. The special class of semiconvex functions that are independent of skewsymmetric parts motivates the concept of symmetric semiconvexity. According to the following definition, we call a function defined on the space of \({\mathcal {S}}^{d\times d}\) symmetric semiconvex, if its natural extension to all matrices in \({\mathbb {R}}^{d\times d}\) is semiconvex in the conventional sense, cf. work by Zhang [60, 62], where symmetric semiconvexity is called semiconvexity on linear strains:
Definition 2.1
Note that in particular, \({{\tilde{f}}}(F) = f(F^s) = {{\tilde{f}}}(F^s) = {{\tilde{f}}}(F^T)\) for any \(F\in {\mathbb {R}}^{d\times d}\), i.e., \({{\tilde{f}}}\) is invariant under symmetrization. As an aside, we mention that a corresponding definition of symmetric convex functions is possible. By linearity of \(\pi _d\), a function is symmetric convex if and only if it is convex.
Next we will collect and review some classical, as well as more recent, results in the context of symmetric semiconvex functions. The following characterizations of symmetric quasi and rankone convexity for general dimensions d are straightforward to show and appear to be wellknown, see e.g. [22, 60].
Proposition 2.2
 (i)f is symmetric quasiconvex if and only if for every \(\varepsilon \in {\mathcal {S}}^{d\times d}\),$$\begin{aligned} f(\varepsilon )\le \inf _{\varphi \in C^{\infty }_c((0,1)^d;{\mathbb {R}}^d)}\int _{(0,1)^d} f(\varepsilon +(\nabla \varphi )^s)\;\mathrm {d}{x}; \end{aligned}$$(2.1)
 (ii)f is symmetric rankone convex if and only iffor all \(\lambda \in (0,1)\) and \(\varepsilon , \eta \in {\mathcal {S}}^{d\times d}\) compatible, i.e. \(\varepsilon \eta = a\odot b := \tfrac{1}{2}(a\otimes b + b\otimes a)\) for some \(a, b \in {\mathbb {R}}^{d}\).$$\begin{aligned} f(\lambda \varepsilon +(1\lambda )\eta )\le \lambda f(\varepsilon )+(1\lambda )f(\eta ) \end{aligned}$$(2.2)
Equivalently, \(t\mapsto f(\varepsilon + t a\odot b)\) is convex for any \(\varepsilon \in {\mathcal {S}}^{d\times d}\) and any \(a,b \in {\mathbb {R}}^d\).
Remark 2.3
 (a)
Notice that many works involving semiconvex functions defined on linear strains, such as [10, 21, 22, 46], take the characterizations of Proposition 2.2 as a starting point and definition.
 (b)In [52], a function \(f:{\mathcal {S}}^{d\times d}\rightarrow {\mathbb {R}}\) is called quasiconvex, if for every \(\varepsilon \in {\mathcal {S}}^{d\times d}\),where \(D^2\psi \) denotes the Hessian matrix of \(\psi \), cf. also [23]. This notion is strictly weaker than the symmetric quasiconvexity in the sense of Definition 2.1. Since for every \(\psi \in C_c^{\infty }((0,1)^d)\) the gradient \(\nabla \psi \) is an admissible test field in (2.1), the asserted implication is immediate. To see that it is strict, we consider in 2d the function$$\begin{aligned} f(\varepsilon )\le \inf _{\psi \in C_c^{\infty }((0,1)^d)} \int _{(0,1)^d} f(\varepsilon + D^2\psi )\;\mathrm {d}{x}, \end{aligned}$$which Šverák in [52] proved to be quasiconvex. However, \(f_0\) is not symmetric rankone convex, and therefore not symmetric quasiconvex (see (2.4) below), since the following map, which is the composition of a compatible line with \(f_0\), is not convex:$$\begin{aligned} f_0(\varepsilon ) = {\left\{ \begin{array}{ll} \det \varepsilon &{} \text {if }\varepsilon \text { is positive definite},\\ 0 &{} \text {otherwise,} \end{array}\right. } \qquad \varepsilon \in {\mathcal {S}}^{2\times 2}, \end{aligned}$$where \({\mathbb {I}}\) is the identity matrix, and \(e_1, e_2\) are the standard unit vectors in \({\mathbb {R}}^2\).$$\begin{aligned}&{\mathbb {R}}\rightarrow {\mathbb {R}}, \quad t\mapsto f_0({\mathbb {I}}+ 2te_1\odot e_2)\\&\quad = {\left\{ \begin{array}{ll} \det ({\mathbb {I}}+ 2te_1\odot e_2) = 1t^2 &{} \text {for }t\in (1,1)\\ 0 &{}\text {otherwise,} \end{array}\right. } \end{aligned}$$Similarly, we show in the 3d setting that the following quasiconvex functions from [52]with \(l=1,2\), are not symmetric rankone convex. For \(l=1\) and \(l=2\), we use the compatible lines \(t\mapsto \mathrm{diag}(1,1,1) + 2te_1\odot e_2\) and \(t\mapsto \mathrm{diag}(1,1, 1) + 2 t e_1\odot e_2\), respectively.$$\begin{aligned} f_l(\varepsilon ) = {\left\{ \begin{array}{ll} \det \varepsilon  &{} \text {if }\varepsilon \text { has exactly }l\text { negative eigenvalues},\\ 0 &{} \text {otherwise,} \end{array}\right. } \qquad \varepsilon \in {\mathcal {S}}^{3\times 3}, \end{aligned}$$
 (c)
Contrary to expectations that may arise in the light of Proposition 2.2, symmetric polyconvexity of a function \(f:{\mathcal {S}}^{d\times d}\rightarrow {\mathbb {R}}\) according to Definition 2.1 is not the same as f being a convex function of symmetric quasiaffine maps (or NullLagrangians). Indeed, since there are no nontrivial NullLagrangians in the symmetrized context (cf. Section 4.3 for \(d=2\) and Proposition 5.10 for \(d=3\)), the latter property equals convexity of f, and is strictly stronger than symmetric polyconvexity, cf. Example 2.4.
 (d)Linearized elasticity can be viewed within the general \({\mathcal {A}}\)free framework [26, 43, 54]. With the secondorder constantrank operator \({\mathcal {A}}\) defined for \(V\in C^\infty ((0,1)^d;{\mathbb {R}}^{d\times d})\) by$$\begin{aligned} ({\mathcal {A}}V)_{jk} = \sum _{i=1}^d \partial _{ik}^2 V_{ji} + \partial _{ij}^2 V_{ki}  \partial ^2_{jk}V_{ii}  \partial _{ii}^2 V_{jk}, \quad j,k=1, \ldots , d, \end{aligned}$$(2.3)
We point out that, even though the notions of symmetric quasiconvexity and \({\mathcal {A}}\)quasiconvexity with \({\mathcal {A}}\) as in (2.3) fall together, there is a conceptual difference between symmetric polyconvexity and \({\mathcal {A}}\)polyconvexity due to the lack of nontrivial NullLagrangians in the symmetric setting (see c) above). Recall that a function is called \({\mathcal {A}}\)polyconvex if it can be represented as the composition of a convex function with an \({\mathcal {A}}\)quasiaffine one, cf. [45, Definition 2.5]. Whereas any \({\mathcal {A}}\)polyconvex function has to be convex in the symmetric setting, there are symmetric polyconvex functions that are nonconvex (cf. Theorems 4.1 and 5.1).
The following basic example served us as a motivation for the characterization results of symmetric polyconvex functions in Theorems 4.1 and 5.1:
Example 2.4
Let \(d=2,3\). The determinant map \({\mathcal {S}}^{d\times d}\rightarrow {\mathbb {R}}\), \(\varepsilon \mapsto \det \varepsilon \) is not symmetric rankone convex, and therefore neither symmetric quasi nor polyconvex.
In the 3d case, both \(\varepsilon \mapsto \det \varepsilon \) and \(\varepsilon \mapsto  \det \varepsilon \) fail to be symmetric rankone convex. However, by taking the diagonal \(2\times 2\) minors, simple examples of symmetric rankone convex functions can be constructed. A direct adaptation of the 2d argument above shows that the maps \(\varepsilon \mapsto (\mathrm{cof\,}\varepsilon )_{ii}\) for \(\varepsilon \in {\mathcal {S}}^{3\times 3}\) with \(i=1,2,3\) are symmetric polyconvex, while \(\varepsilon \mapsto (\mathrm{cof\,}\varepsilon )_{ii}\) are not.
3 Preliminaries
Before proving the results announced in the introduction, we use this section to collect further relevant notation and auxiliary results.
3.1 Notation
This work focuses on the space dimensions \(d=2,3\). We write \(a\cdot b\) with \(a, b\in {\mathbb {R}}^d\) for the standard inner product on \({\mathbb {R}}^d\), and use the scalar product \(A:B=\sum _{i, j=1}^d A_{ij} B_{ij}\) for \(d\times d\) matrices A and B. The latter induces the Frobenius norm \(A^2:=A:A\) on \({\mathbb {R}}^{d\times d}\). Moreover, \(e_i\) with \(i=1, \ldots , d\) are the standard unit vectors in \({\mathbb {R}}^d\), \((a\otimes b)_{ij} = a_ib_j\) with \(i, j=1, \ldots , d\) for \(a, b\in {\mathbb {R}}^d\) is the tensor product of a and b, and \(a\odot b=\frac{1}{2}(a\otimes b + b\otimes a)\) with \(a, b\in {\mathbb {R}}^d\). Further, \(\mathrm{diag}(\lambda _1, \ldots , \lambda _d)\) with \(\lambda _i\in {\mathbb {R}}\) is our notation for diagonal \(d\times d\) matrices.
Let us denote by \({\mathcal {S}}^{d\times d}\) the set of symmetric matrices in \({\mathbb {R}}^{d\times d}\). Any \(F\in {\mathbb {R}}^{d\times d}\) can be decomposed into its symmetric and antisymmetric part, i.e. \(F=F^s+F^a\) with \(F^s=\frac{1}{2}(F+F^T)\in {\mathcal {S}}^{d\times d}\) and \(F^a=\frac{1}{2}(FF^T)\). For the subsets of positive and negative semidefinite matrices in \({\mathcal {S}}^{d\times d}\) we use the notations \({\mathcal {S}}^{d\times d}_+\) and \({\mathcal {S}}^{d\times d}_\), respectively.
3.2 Properties of Symmetric Matrices and Their Minors
Lemma 3.1
 (i)
\(q_A\) is convex;
 (ii)
\(q_A\) is rankone convex;
 (iii)
A is positive semidefinite.
Proof
3.3 Convex Functions, Subdifferentials and Monotonicity
For vectors \(y, {\bar{y}}\in {\mathbb {R}}^n\) the order relation \(y \le {\bar{y}}\) is to be understood componentwise, that is as \(y_i \le {\bar{y}}_i\) for \(i=1, \ldots , n\), and analogously for \(y\ge {\bar{y}}\). Monotonicity of a function \(h:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is also defined componentwise, that is, h is called nonincreasing (nondecreasing) if \(h(y)\ge h({\bar{y}})\) for all \(y, {\bar{y}}\in {\mathbb {R}}^n\) with \(y \le {\bar{y}}\) (\(y\ge {\bar{y}}\)).
Lemma 3.2
The next lemma generalizes the elementary observation that every bounded and convex function \({\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is constant to the situation where one has only partial bounds on the growth behavior of the function in selected variables.
Lemma 3.3
 (i)
The function g is nonincreasing (nondecreasing) in the second variable if and only if \(\partial _2 g(x, y)\subset (\infty , 0]^n\) (\(\partial _2 g(x,y)\subset [0, \infty )^n\)) for every \((x,y) \in {\mathbb {R}}^m\times {\mathbb {R}}^n\);
 (ii)
If \(\partial g(x, y)\subset {\mathbb {R}}^m\times (\infty , 0]^n\) (\(\partial g(x, y)\subset {\mathbb {R}}^m\times [0, \infty )^n\)) for all \((x, y) \in {\mathbb {R}}^m\times {\mathbb {R}}^n\), then \(\partial _2 g(x, y)\subset (\infty , 0]^n\) (\(\partial _2 g(x, y)\subset [0, \infty )^n\)) for every \((x, y)\in {\mathbb {R}}^m\times {\mathbb {R}}^n\);
 (iii)
If there exists \({\hat{x}}\in {\mathbb {R}}^m\) such that \(\partial _2 g({\hat{x}}, y)\subset (\infty , 0]^n\)\((\partial _2 g({\hat{x}}, y)\subset [0, \infty )^n)\) for all \(y\in {\mathbb {R}}^n\), then \(\partial _2 g(x, y)\subset (\infty , 0]^n\) (\(\partial _2 g(x, y)\subset [0, \infty )^n\)) for all \((x, y)\in {\mathbb {R}}^m\times {\mathbb {R}}^n\);
 (iv)If there exists \(({\hat{x}}, {\hat{y}})\in {\mathbb {R}}^{m}\times {\mathbb {R}}^n\) and a constant \(C>0\) such thatthen \(\partial _2 g(x, y)\subset (\infty , 0]^n\) (\(\partial _2 g(x, y)\subset [0, \infty )^n\)) for all \((x, y)\in {\mathbb {R}}^m\times {\mathbb {R}}^n\).$$\begin{aligned} g({\hat{x}}, y)\le C \qquad \text {for all }y\ge {\hat{y}} (y\le {{\hat{y}}}), \end{aligned}$$(3.14)
Proof
We will only prove the primary statements, since those in brackets follow analogously. Part (i) results directly from the singlevariable case mentioned above.
After suitable identifications, the above results also apply to functions defined on Cartesian products between \({\mathbb {R}}^{d\times d}\), \({\mathcal {S}}^{d \times d}\), and \({\mathbb {R}}^n\). Moreover, we have the following lemma, which we will use in the proof of Theorem 5.1:
Lemma 3.4
 (i)
If \(\partial g(\varepsilon , \eta )\subset {\mathcal {S}}^{d\times d}\times {\mathcal {S}}^{d\times d}_\) for all \(\varepsilon , \eta \in {\mathcal {S}}^{d\times d}\), then \(\partial _2 g(\varepsilon , \eta )\subset {\mathcal {S}}^{d\times d}_\) for every \(\varepsilon , \eta \in {\mathcal {S}}^{d\times d}\);
 (ii)If there exists \({\hat{\varepsilon }}\in {\mathcal {S}}^{d\times d}\) and a constant \(C>0\) such thatthen \(\partial _2 g(\varepsilon , \eta )\subset {\mathcal {S}}^{d\times d}_\) for all \(\varepsilon , \eta \in {\mathcal {S}}^{d\times d}\).$$\begin{aligned} g({\hat{\varepsilon }}, x\otimes x)\le C \qquad \text {for all }x\in {\mathbb {R}}^d, \end{aligned}$$
Proof
4 Symmetric Polyconvexity in 2d
In this section, we provide a characterization of symmetric polyconvex functions and study symmetric polyaffine functions as well as symmetric polyconvex quadratic forms in 2d.
4.1 Characterization of Symmetric Polyconvexity in 2d
The next theorem gives a necessary and sufficient condition for a real function on \({\mathcal {S}}^{2\times 2}\) to be symmetric polyconvex. While a classical polyconvex function can be expressed as a convex function of minors, symmetric polyconvex functions in the 2d case can be represented as convex functions of minors satisfying an additional monotonicity condition in the variable of the determinant.
Theorem 4.1
Proof
Remark 4.2
 a)
Due to Lemma 3.3 i), an equivalent way of phrasing the necessary and sufficient condition is that the convex function g satisfies \(\partial _2 g(\varepsilon , t)\subset (\infty , 0]\) for all \(\varepsilon \in {\mathcal {S}}^{2\times 2}\) and \(t\in {\mathbb {R}}\).
 b)We point out that the representation of a function \(f:{\mathcal {S}}^{2\times 2}\rightarrow {\mathbb {R}}\) in terms of a convex function of \(\varepsilon \) and \(\det \varepsilon \) is in general not unique. For instance, letfor \(\varepsilon \in {\mathcal {S}}^{2\times 2}\) and \(t\in {\mathbb {R}}\), and consider$$\begin{aligned} g(\varepsilon , t) = (\mathrm{tr\,}\varepsilon )^2  t\quad \text { and }\quad h(\varepsilon , t) = \varepsilon ^2 + t \end{aligned}$$$$\begin{aligned} f(\varepsilon )=g(\varepsilon , \det \varepsilon ) = (\mathrm{tr\,} \varepsilon )^2\det \varepsilon , \quad \varepsilon \in {\mathcal {S}}^{2\times 2}. \end{aligned}$$(4.2)
Generally speaking, it depends on the way a function \(f:{\mathcal {S}}^{2\times 2}\rightarrow {\mathbb {R}}\) is given, whether deciding about symmetric polyconvexity of f is immediate or not directly obvious. If, however, f can be expressed as a convex function depending on \(\det \varepsilon \) only, then Corollary 4.3 below gives a simple criterion.
In Example 2.4, we convinced ourselves that \(\varepsilon \mapsto  \det \varepsilon \) is symmetric polyconvex, whereas \(\varepsilon \mapsto \det \varepsilon \) is not. As a consequence of Theorem 4.1, the following more general result can be obtained:
Corollary 4.3
Indeed, in light of the next lemma the proof follows immediately.
Lemma 4.4
Proof
To prove the second part of the statement, observe that for every \(t\in {\mathbb {R}}\) one can find a symmetric matrix whose determinant equals t. \(\quad \square \)
At the end of the section, we turn in more detail to two classes of polyconvex functions.
4.2 Symmetric Polyconvex Quadratic Forms
In two dimensions, the three classes of symmetric polyconvex, quasiconvex, and rankone convex quadratic forms are identical. In view of Definition 2.1, this is a consequence of the corresponding wellknown property of classical semiconvex quadratic forms [16, Theorem 5.25]. The following characterization constitutes a refinement for the symmetric case of a result by Marcellini [36, Equation (11)], see also [16, Lemma 5.27].
Proposition 4.5
 (i)
f is symmetric polyconvex;
 (ii)there exist \(\alpha \ge 0\) and a convex quadratic form \(h:{\mathcal {S}}^{2\times 2}\rightarrow {\mathbb {R}}\) such that$$\begin{aligned} f(\varepsilon )=h(\varepsilon )\alpha \det \varepsilon \quad \text {for all }\varepsilon \in {\mathcal {S}}^{2\times 2}; \end{aligned}$$
 (iii)there exists \(\alpha \ge 0\) such that$$\begin{aligned} f(\varepsilon ) +\alpha \det \varepsilon \ge 0 \quad \text {for all }\varepsilon \in {\mathcal {S}}^{2\times 2}. \end{aligned}$$
Proof
Since a quadratic form is convex if and only if it is nonnegative, (ii) and (iii) are equivalent. The implication “\((ii)\Rightarrow i)\)” is an immediate consequence of Theorem 4.1. To see that (i) implies (iii), we use that by [16, Lemma 5.27, p.192] there exists \(\alpha \in {\mathbb {R}}\) such that \({{\tilde{f}}}(F) + \alpha \det F\ge 0\), \(F\in {\mathbb {R}}^{2\times 2}\). Hence, \({\tilde{h}}:{\mathbb {R}}^{2\times 2}\rightarrow {\mathbb {R}}\) defined by \({{\tilde{h}}}(F) = \tilde{f}(F) + \alpha \det F\) is quadratic and thus convex. Note that in particular, \(f(\varepsilon ) + \alpha \det \varepsilon \ge 0\) for all \(\varepsilon \in {\mathcal {S}}^{2\times 2}\).
4.3 Symmetric Polyaffine Functions
Summing up, we observe that f is symmetric polyaffine if and only if it is affine.
5 Symmetric Polyconvexity in 3d
This section is devoted to a discussion of symmetric polyconvex functions in 3d. After providing a characterization of symmetric polyconvexity, we subsequently discuss a few examples and two important subclasses of symmetric polyconvex functions in three dimensions, these are symmetric polyconvex quadratic forms and symmetric polyaffine functions.
5.1 Characterization of Symmetric Polyconvexity in 3d
The following theorem constitutes the main result of this paper:
Theorem 5.1
Proof
We subdivide the proof into the natural two steps, proving first that (5.1) is necessary for symmetric polyconvexity, and then that it is also sufficient.
Next, we show that g is constant in the determinant. Then, in Step 1b, the asserted negative semidefiniteness of matrices in the subdifferential with respect to the second variable of g is established.
 (i)
There exists a convex function \(k:{\mathbb {R}}^{3\times 3}\times {\mathbb {R}}^{3\times 3}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) such that \({\tilde{f}}(F)\ge k(F, \mathrm{cof\,}F, \det F)\) for all \(F\in {\mathbb {R}}^{3\times 3}\);
 (ii)Let \(F,F_i \in {\mathbb {R}}^{3\times 3}\) and \(\lambda _i\in [0,1]\) with \(i=1,2,\dots , n=20\) such thatThen, \({{\tilde{f}}}(F)\le \sum _{i=1}^n \lambda _i {{\tilde{f}}}(F_i)\).$$\begin{aligned}&\sum _{i=1}^{n} \lambda _i=1,\ \ F=\sum _{i=1}^{n}\lambda _iF_i,\ \ \mathrm{cof\,}F\nonumber \\&\quad =\sum _{i=1}^{n}\lambda _i\mathrm{cof\,}F_i, \quad \text {and} \ \ \det F=\sum _{i=1}^{n}\lambda _i\det F_i. \end{aligned}$$(5.5)
Remark 5.2
 (a)As a consequence of Theorem 5.1, any symmetric polyconvex function f satisfies a monotonicity assumption with regard to the diagonal minors. Precisely, there exists a convex function \(g:{\mathcal {S}}^{3\times 3}\times {\mathbb {R}}^3\times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}\) that is nonincreasing in its second argument such thatcf. Section 3.1 for notations. In view of Lemma 3.3 i) and ii), this follows from the fact that diagonal entries of negative semidefinite matrices are always nonpositive.$$\begin{aligned} f(\varepsilon ) = g(\varepsilon , \mathrm{cof\,}_{\mathrm{diag}}\, \varepsilon , \mathrm{cof\,}_{\mathrm{off}}\,\varepsilon ), \quad \varepsilon \in {\mathcal {S}}^{3\times 3}, \end{aligned}$$
 (b)Note that a function \(f:{\mathcal {S}}^{3\times 3}\rightarrow {\mathbb {R}}\) given as in (5.1) with a convex function g whose partial subdifferential regarding the second variable is not negative semidefinite may still be symmetric polyconvex. Indeed,which depends increasingly on the diagonal cofactors, admits an alternative representation that is in accordance with Theorem 5.1, namely$$\begin{aligned} f(\varepsilon ) = \varepsilon _{11}^2+ \varepsilon _{22}^2 + 2\varepsilon _{12}^2 + (\mathrm{cof\,}\varepsilon )_{33}, \quad \varepsilon \in {\mathcal {S}}^{3\times 3}, \end{aligned}$$Thus, f is in fact symmetric polyconvex.$$\begin{aligned} f(\varepsilon ) = (\varepsilon _{11} +\varepsilon _{22})^2  (\mathrm{cof\,}\varepsilon )_{33}, \quad \varepsilon \in {\mathcal {S}}^{3\times 3}. \end{aligned}$$(5.6)
 (c)We emphasize that the representation of a symmetric polyconvex function according to Theorem 5.1 is not unique. To see this, take for example f as in (5.6) and observe that it can equivalently be expressed as$$\begin{aligned} f(\varepsilon ) = \tfrac{1}{2}(\varepsilon _{11} + \varepsilon _{22})^2 + \tfrac{1}{2} \varepsilon _{11}^2 + \tfrac{1}{2}\varepsilon _{22}^2 +\varepsilon _{12}^2,\quad \varepsilon \in {\mathcal {S}}^{3\times 3}. \end{aligned}$$
The 3d analogon of Corollary 4.3 is based on an auxiliary result extending Lemma 4.4.
Lemma 5.3
 (i)If there exists a function \(h: {\mathbb {R}}\rightarrow {\mathbb {R}}\) such thatthen h is constant;$$\begin{aligned} g(\varepsilon , \mathrm{cof\,}\varepsilon ) = h(\det \varepsilon )\quad \text {for all }\varepsilon \in {\mathcal {S}}^{3\times 3}, \end{aligned}$$
 (ii)If there exists a continuous function \(h:{\mathcal {S}}^{3\times 3}\rightarrow {\mathbb {R}}\) satisfyingthen \(h=g(0, \cdot )\).$$\begin{aligned} g(\varepsilon , \mathrm{cof\,}\varepsilon ) = h(\mathrm{cof\,}\varepsilon )\quad \text {for all }\varepsilon \in {\mathcal {S}}^{3\times 3}, \end{aligned}$$
Proof
First, we argue that, both in i) and ii), the function g is independent of the first variable. This follows as in the proof of Lemma 4.4, just replacing (3.8) with (3.9). Notice that every matrix with vanishing cofactor has in particular zero determinant.
In the following, let \(g_0 = g(0, \cdot ):{\mathcal {S}}^{3\times 3}\rightarrow {\mathbb {R}}\). As a consequence of the separate convexity of g, the auxiliary function \(g_0\) is convex, and we have that \(g_0(\mathrm{cof\,}\varepsilon ) = g(0, \mathrm{cof\,}\varepsilon ) = g(\varepsilon , \mathrm{cof\,}\varepsilon )\) for any \(\varepsilon \in {\mathcal {S}}^{3\times 3}\).
As for i), we use Lemma 3.3 iii) to infer from the observation that \(g_0(te_i\otimes e_i) = g_0(\mathrm{cof\,}(te_j\otimes e_j + e_k\otimes e_k)) = h(0)\) for all \(t\in {\mathbb {R}}\) and any circular permuation (ijk) of (123) that \(g_0\) is constant in the variables corresponding to the diagonal entries. Thus, plugging diagonal matrices into the identity \(g_0(\mathrm{cof\,}\varepsilon ) = h(\det \varepsilon )\) for \(\eta \in {\mathcal {S}}^{3\times 3}\) shows that h is constant.
Remark 5.4
 a)
The statement in Lemma 5.3ii) clearly fails, if we do not require h to be continuous. To see this, we first recall that, in the 3d case, the rank of any cofactor matrix is not equal to 2. Indeed, if \({{\,\mathrm{rank}\,}}\varepsilon =0\) or \({{\,\mathrm{rank}\,}}\varepsilon =1\), then \(\mathrm{cof\,}\varepsilon =0\), so its rank is 0. If \({{\,\mathrm{rank}\,}}\varepsilon =2\), we have by Cramer’s rule (3.3) and Sylvester’s rank formula that \({{\,\mathrm{rank}\,}}\mathrm{cof\,}\varepsilon + {{\,\mathrm{rank}\,}}\varepsilon \le 3\) and hence, \({{\,\mathrm{rank}\,}}\mathrm{cof\,}\varepsilon \le 1\). Finally, if \(\varepsilon \) is invertible, so is \(\mathrm{cof\,}\varepsilon \) and \({{\,\mathrm{rank}\,}}\mathrm{cof\,}\varepsilon =3\). Now, let \(g:{\mathcal {S}}^{3\times 3}\times {\mathcal {S}}^{3\times 3}\rightarrow {\mathbb {R}}\) be the zero function, and choose \(h:{\mathcal {S}}^{3\times 3}\rightarrow {\mathbb {R}}\) such that \(h(\varepsilon ) = 1\) for all \(\varepsilon \in {\mathcal {S}}^{3\times 3}\) with \({{\,\mathrm{rank}\,}}\varepsilon = 2\), and \(h(\varepsilon )=0\) otherwise. Then, since \({{\,\mathrm{rank}\,}}\mathrm{cof\,}\varepsilon \ne 2\), \(g(\varepsilon , \mathrm{cof\,}\varepsilon ) = h(\mathrm{cof\,}\varepsilon )\) for any \(\varepsilon \in {\mathcal {S}}^{3\times 3}\), but \(h\ne 0 = g(0,\cdot )\).
 b)Let \(g:{\mathcal {S}}^{3\times 3}\times {\mathcal {S}}^{3\times 3}\rightarrow {\mathbb {R}}\) be convex. If there is a continuous function \(h:{\mathcal {S}}^{3\times 3}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) such thatthen results analogous to Lemma 5.3i) and ii) cannot be expected. That is, h may be nonconstant in the last variable and \(h(\cdot , 0) \ne g(0,\cdot )\). Due to \(\det (\mathrm{cof\,}\varepsilon ) = (\det \varepsilon )^2\) for all \(\varepsilon \in {\mathcal {S}}^{3\times 3}\), considering the functions$$\begin{aligned} g(\varepsilon , \mathrm{cof\,}\varepsilon ) = h(\mathrm{cof\,}\varepsilon ,\det \varepsilon )\quad \text {for all } \varepsilon \in {\mathcal {S}}^{3\times 3}, \end{aligned}$$$$\begin{aligned} g(\varepsilon , \eta )= 0\quad \text { and }\quad h(\eta , t) = t^2\det \eta , \quad \varepsilon , \eta \in {\mathcal {S}}^{3\times 3}, t\in {\mathbb {R}}, \end{aligned}$$
The following statement is an immediate consequence of Theorem 5.1 and Lemma 5.3, and it allows us to decide in special situations, if a function is symmetric polyconvex or not, without having to find a function g as required in Theorem 5.1:
Corollary 5.5
In contrast to the 2d setting, Corollary 5.5i) shows that there exists no symmetric polyconvex function in 3d that is given as a nonconstant function of the determinant only.
5.2 Symmetric Polyconvex Quadratic Forms
Finally, we turn our attention to quadratic forms in 3d. As a consequence of Theorem 5.1, we obtain a characterization of symmetric polyconvexity for this class of functions, which reminds of the characterization of polyconvex quadratic forms in [16, p. 192].
Proposition 5.6
 (i)
f is symmetric polyconvex;
 (ii)there exist a convex quadratic form \(h:{\mathcal {S}}^{3\times 3}\rightarrow {\mathbb {R}}\) and a matrix \(A\in {\mathcal {S}}^{3\times 3}_+\) such that$$\begin{aligned} f(\varepsilon ) = h(\varepsilon )A:\mathrm{cof\,}\varepsilon \quad \text {for all }\varepsilon \in {\mathcal {S}}^{3\times 3}; \end{aligned}$$(5.7)
 (iii)there is \(A\in {\mathcal {S}}^{3\times 3}_+\) such that$$\begin{aligned} f(\varepsilon ) + A:\mathrm{cof\,}\varepsilon \ge 0\quad \text {for all }\varepsilon \in {\mathcal {S}}^{3\times 3}. \end{aligned}$$
Proof
Next we present an example of a quadratic form which is symmetric rankone convex but not symmetric polyconvex. As already mentioned in the introduction, this is motivated by a corresponding result in the classical setting by Serre [50].
Theorem 5.7
Proof
The remainder of the proof is subdivided into the natural two steps.
Step 1:fis symmetric rankone convex. Due to (5.9), one has that \(f(a\odot b) = f_0(a\odot b)  \eta a\odot b^2 \ge 0\) for all \(a, b\in {\mathbb {R}}^3\). Hence, f is symmetric rankone convex by (2.5).
Step 2:fis not symmetric polyconvex. Let \(A=(A_{ij})_{ij}\in {\mathcal {S}}^{3\times 3}_+\) be fixed but arbitrary. In view of Proposition 5.6iii), it is enough to find one \(\varepsilon _A\in {\mathcal {S}}^{3\times 3}\) such that \(f(\varepsilon _A)+A:\mathrm{cof\,}{\varepsilon _A}<0\).
In the second case, \(a\odot b\) has rank two. Thus \(\mathrm{cof\,}(a\odot b) \ne 0\). Hence, by (3.6), we can assume without loss of generality that \(a_2b_1a_1b_2\ne 0\). If \(a_3\ne 0\) and \(b_3\ne 0\), then \(a_3 (a_2 b_1a_1 b_2) b_3\ne 0\). If \(a_3=0\), then, by Lemma 5.8, the coefficients \(a_1,a_2\) and \(b_3\) cannot vanish. If in addition \(b_1\ne 0\), then \(a_1(a_2b_3a_3b_2)b_1 =a_1a_2b_1b_3\ne 0\). If \(b_1=0\), then \(b_2\ne 0\) by Lemma 5.8, and we conclude that \(a_2(a_1b_3a_3b_1)b_2= a_1a_2b_2b_3\ne 0\). Hence, at least one of the expressions \(a_3 (a_2 b_1a_1 b_2) b_3\), \(a_1(a_2 b_3a_3 b_2)b_1\) and \(a_2(a_1b_3a_3b_1)b_2\) is nonzero.
The proof of Theorem 5.7 makes use of the following auxiliary result on the structure of minimizers of \(f_0\) in \({\mathcal {M}}\), cf. (5.9):
Lemma 5.8
 (i)
\((a_i,b_i)\ne (0,0)\) for all \(i\in \{1,2,3\}\), and
 (ii)
\((a_i,a_j)\ne (0,0)\) and \((b_i,b_j)\ne (0,0)\) for all \( i, j\in \{1,2,3\}\), \(i\ne j\).
Proof
Finally, we observe that for those quadratic forms that are representable as linear combinations of \(2\times 2\) minors, the notions of symmetric polyconvexity and symmetric rankone convexity are in fact identical.
Corollary 5.9
 (i)
f is symmetric polyconvex;
 (ii)
f is symmetric rankone convex;
 (iii)
A is positive semidefinite.
Proof
In particular, it follows from Corollary 5.9 that any symmetric rankone convex quadratic form given in terms of a linear combination of offdiagonal cofactor entries can only be trivial, since there is no positive semidefinite symmetric matrix with vanishing diagonal entries.
5.3 Symmetric Polyaffine Functions
In the classical setting, it is wellknown that for realvalued functions defined on \({\mathbb {R}}^{3\times 3}\) the properties of being polyaffine, quasiaffine (or NullLagrangian) and rankone affine are equivalent, see e.g. [16, Theorem 5.20]. Consequently, by Definition 2.1 the corresponding statement is true in the symmetric setting.
Next we show that a function f in 3d is symmetric polyaffine (or symmetric quasiaffine or symmetric rankone affine) if and only if f is affine. The proof follows from the characterization of symmetric polyconvex functions from Theorem 5.1 in conjunction with Lemma 5.3.
Proposition 5.10
Proof
If the function f is of the form (5.13), then, clearly, it is symmetric polyaffine.
Assume, now, that f is symmetric polyaffine.
Notes
Acknowledgements
CK gratefully acknowledges the support by a Westerdijk Fellowship from Utrecht University. This work was initiated during research visits of CK and OB to the University of Würzburg. The latter were funded by a DAAD travel grant.
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