Setting the Free Material Design problem through the methods of optimal mass distribution

The paper deals with the Free Material Design (FMD)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\text {FMD})}$$\end{document} problem aimed at constructing the least compliant structures from an elastic material, the constitutive field of which plays the role of design variable in the form of a tensor valued measure λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} supported in the design domain. Point-wise the constitutive tensor is referred to a given anisotropy class H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {H}$$\end{document}, while the integral of a cost c(λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(\lambda )$$\end{document} is bounded from above. The convex p-homogeneous elastic potential j is parameterized by the constitutive tensor. The work puts forward the existence result and shows that the original problem can be reduced to the Linear Constrained Problem (LCP)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathrm {LCP})$$\end{document} known from the theory of optimal mass distribution by G. Bouchitté and G. Buttazzo. A theorem linking solutions of (FMD)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\text {FMD})}$$\end{document} and (LCP)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathrm {LCP})$$\end{document} allows to effectively solve the original problem. The developed theory encompasses several optimal anisotropy design problems known in the literature as well as it unlocks new ones. By employing the derived optimality conditions we give several analytical examples of optimal designs.

given static load F: (1.1) where, for a 4th order elasticity tensor H ∈ H and a symmetric 2nd order strain tensor ξ ∈ S d×d , function j returns the elastic energy density j(H , ξ). The elasticity or the Hooke tensor H plays the role of a parameter for the Hooke's law of elasticity: j(H , · ) is the elastic potential inducing the constitutive law σ ∈ ∂ j(H , ξ), where σ ∈ S d×d is the stress tensor, and the subdifferential is computed with respect to the second argument. The strain field e(u) = 1 2 ∇u + (∇u) is the symmetric part of the gradient of a vectorial function u representing the displacement field. The load is modelled by a vector valued measure F ∈ M( ; R d ). Since the body is assumed not to be kinematically fixed, i.e. there is no Dirichlet condition imposed on u, an extra condition on F must be enforced: is said to be balanced if the resultant force and the resultant moment are zero, namely: For a chosen cost function c : H → R + by Free Material Design (FMD) we shall call the problem of finding the constitutive field being a tensor valued measure λ ∈ M( ; H ) that minimizes the compliance: In the context of the linear theory of elasticity, i.e. for quadratic potential j(H , ξ) = 1 2 H ξ, ξ , and with the trace cost c(H ) = Tr H , the (FMD) problem has been for the first time considered in [31], where the set H consisted of all positive semi-definite Hooke tensors. Soon then, in [4,5] the (FMD) was reformulated as a problem in which only one scalar variable is involved: μ := Tr λ. One of the first existence results may be found in [36] where the Hooke field was assumed to be a function, i.e. λ = C L d , and, to gain compactness in L ∞ , the author imposed an additional local constraint Tr C (x) ≤ c 0 . Similar assumption was kept throughout the papers [25,38] or [26]. The present work puts forward a far more general framework where (FMD) is in fact a family of optimal design problems parameterized by: the elastic potential j, the cost function c, and the set of admissible Hooke tensors H . The set H may be chosen as any closed convex cone contained in the set of positive semi-definite Hooke tensors. The cost function c is any norm on the space of Hooke tensors restricted to H . Finally, for a chosen p ∈ (1, ∞), the potential j : H × S d×d → R + will be subject to the following assumptions: • for each H ∈ H there hold: (H1) the function j(H , · ) is real-valued, non-negative and convex on S d×d ; (H2) the function j(H , · ) is positively p-homogeneous on S d×d ; • whilst for each ξ ∈ S d×d there hold: (H3) the function j( · , ξ) is concave and upper semi-continuous on H ; (H4) the function j( · , ξ) is one-homogeneous on H ; (H5) if ξ = 0 then there exists H ∈ H such that j(H , ξ) > 0.
Based on assumptions (H1)-(H5), in Sect. 2 for a fixed function u the properties of the elastic energy functional M( ; H ) λ → j λ, e(u) are established, including its upper semi-continuity in the weak-* topology proved in Proposition 2.2. This renders C = C(λ) convex and weak-* lower semi-continuous, which shall in turn furnish the existence result: Theorem 1.1 Assuming that j satisfies (H1)-(H5) the Free Material Design problem (FMD) admits a solutionλ once F is balanced.
In the case when j is quadratic, c = Tr, and H is the set of all anisotropic Hooke tensors -which shall be referred to as the Anisotropic Material Design (AMD) setting -in works [18,19] the (FMD) problem has been reformulated on the formal level to a pair of mutually dual problems: where, in the AMD setting only, ρ = ρ 0 = | · | is the Euclidean norm on the space of symmetric matrices. The matrix-valued measure τ ∈ M ; S d×d plays a role of the stress field that equilibrates the load F or, adopting the language of the Optimal Transport Problem (cf. [35]), transports certain parts of F to its other parts. The equilibrium equation −div τ = F is intended in the sense of distributions on the whole R d . The equality sup (P) = min (P * ) is a result of a standard duality argument, which also furnishes existence in (P * ).
In this work the general (FMD) problem is proved to be equivalent to the pair of problems (P), (P * ), starting with the following variational equality:

Theorem 1.2 The minimum value of compliance in (FMD) equals
where Z = sup P = inf P * and p = p/( p − 1) is the Hölder conjugate exponent.
The equality (1.2) depends on ρ, ρ 0 : S d×d → R + being mutually polar real closed gauge functions constructed through finite dimensional programs: where H 1 is the convex compact subset of Hooke tensors H ∈ H satisfying the unit cost condition c(H ) ≤ 1; the convex conjugate function j * is intended with respect to the second argument. Functions ρ, ρ 0 involve all the parameters H , j, c; hence, the pair of gauges encodes in (P), (P * ) the actual setting of the (FMD) problem. In parallel to research on the Free Material Design problem the so-called Mass Optimization Problem (MOP) was put forth in [8]. In (MOP) we seek an elastic material or "mass" distribution μ ∈ M + ( ) that minimizes the compliance. In [8] one finds that (MOP) is equivalent to the pair of problems (P) and (P * ) as well; the authors gave a rigorous proof of the passage. One of the paramount differences between the two design problems, however, is the following: for (MOP) the functions ρ, ρ 0 are data that fully constitute the law of elasticity, whereas here the gauges are to be constructed in accordance with (1.3). The present work essentially adapts and generalizes the methods of (MOP) to provide a rigorous mathematical framework for the family of (FMD) problems. We shall directly build upon the work [10] where the (MOP) theory is already further developed.
After [10], the two displacement-based maximization problems: the one in (1.1) and the problem (P) require relaxation of the differentiability condition. Based on the compactness result established in [8], solutionû of a relaxed problem (P) is attained on a set of continuous a.e. differentiable functions with e(û) ∈ L ∞ ( ; S d×d ). A simple adaptation of the result from [10] proves that the same may be inferred about solutionǔ of (1.1) provided that λ =λ is optimal for the (FMD) problem. Such a solutionǔ proves to be the displacement field in the optimally designed structure.
In works [8,10] the solution of (MOP) was recast based on the solution of the pair (P), (P * ) or, as it was there referred to, of the Linear Constrained Problem (LCP). One of the main results in the present work generalizes those ideas towards the (FMD) problem. If bŷ τ we denote a solution of (P * ), we may decompose it toτ =σμ whereσ ∈ L ∞ μ ( ; S d×d ) satisfies ρ 0 (σ ) = 1μ-a.e. In (LCP) the information on the Hooke tensor function C =Ĉ is implicit and may be point-wise recovered based on the finite dimensional minimization problem in (1.3): for any σ ∈ S d×d byH 1 (σ ) we shall understand the set of minimizers H ∈ H 1 of j * (H , σ ). The definition of quadruple solving (LCP) may be given: (i)û solves the relaxed problem (P); (ii)τ =σμ ∈ M ; S d×d solves (P * ) with ρ 0 (σ ) = 1μ-a.e.; (iii)Ĉ is anyμ-measurable selection of the multifunction x →H 1 σ (x) .
The proof of Theorem 1.3 is long yet not very technical as it mostly uses variational inequalities and equalities, including the one in Theorem 1.2. The proposed method of solving (FMD), however, cannot be deemed fully universal as long as existence of solution of (LCP) in the sense of Definition 1.3 is not assured. Existence of the optimal tripleû,μ,σ was already proved in [8,10]; here the novel question is as follows: does there always exist aμ-measurable functionĈ that is point-wise optimal in the sense that In some particular settings of (FMD) this is straightforward, e.g. for the AMD setting there exists an explicit formula: H ∈H 1 (σ ) ⇔ H = σ ⊗ σ . The general statement is herein provided by the more technical Lemma 3.1 where the multifunction S d×d σ →H 1 σ ∈ 2 H 1 \∅ is proved to be upper semicontinuous, cf. [15].
The present paper is organized as follows. In Sect. 2 we study weak-* upper semicontinuity of the energy integral functional M( ; H ) λ → j(λ, ) ∈ R + to guarantee existence of solution of (FMD). Section 3 is devoted to equivalence of the (FMD) problem and the abstract (LCP); Theorems 1.2 and 1.3 are proved, which delivers the main method of this work. Section 4 again generalizes the ideas of the work [10]: we give necessary and sufficient conditions for the quadruple (u, μ, σ, C ) to solve the (FMD) problem; due to lack of differentiability of u in general we employ the tools of μ-tangential calculus introduced in [12] and developed in [9,10]. Section 5.1 presents a number of examples of settings of (FMD): e.g. by choosing H to be the set of isotropic Hooke tensors we recover the Isotropic Material Design (IMD) problem already considered in [17,21]; cf. also the work on the Young Modulus Design in [20]. Moreover, by utilizing the general framework of the herein developed theory, new settings are proposed, including the Fibrous Material Design (FibMD) problem that is proved to be equivalent to the renown Michell problem, cf. [28] or [13]. Finally, by employing the newly derived optimality conditions, analytical solutions of a simple design problem are given in Sect. 5.2 for the settings: AMD, IMD, FibMD.

Formulation of the design problem
Within a d-dimensional ambient space R d we shall consider an elastic body: a plate in case of d = 2 or a solid for d = 3. Let us begin by discussing the convention for the finite dimensional spaces used in this text. By S d×d we shall understand the space of symmetric d × d matrices. Since S d×d is isomorphic to the space of symmetric 2nd-order tensors we shall sometimes refer to its elements as: strain tensors ξ ∈ S d×d or stress tensors σ ∈ S d×d . We endow S d×d with a scalar product ξ, σ := d i, j=1 ξ i j σ i j and the corresponding Euclidean norm |ξ | := ( ξ, ξ ) 1/2 . The eigenvalues of ξ ∈ S d×d shall be denoted by λ i (ξ ), while the trace will read Index of notation General notation (V is any finite dimensional vector space): the i-th eigenvalue of a matrix ξ ∈ S d×d or of an operator H ∈ L (S d×d ) Tr σ , Tr H the trace being the sum of the respective eigenvalues the symmetric part of the gradient, i.e. 1 a closed gauge ρ = ρ(ξ) -a non-negative convex positively one-homogeneous lower semi-continuous function -and its polar ρ 0 = ρ 0 (σ ) . By a tensor product x ⊗ y of vectors x, y ∈ R d we will understand a (possibly asymmetric) d × d matrix A whose components are A i j = x i y j ; it is the unique matrix that satisfies Az = y, z x for each z ∈ R d . Clearly x ⊗ x ∈ S d×d for any x ∈ R d , while x s ⊗ y ∈ S d×d will stand for the symmetrization 1 2 (x ⊗ y + y ⊗ x). The identity matrix in S d×d will be denoted by I.
The symbol L (S d×d ) will stand for the space of symmetric linear operators from S d×d to S d×d . We shall work with an abstract form of operators H ∈ L (S d×d ), i.e. without invoking their matrix representation. The notion of an eigenvalue of an operator  1/2 . For σ ∈ S d×d by a tensor product H = σ ⊗ σ we understand the unique element of L (S d×d ) satisfying: H ξ = ξ, σ σ for any ξ ∈ S d×d . By Id ∈ L (S d×d ) we will denote the identity operator, namely Id ξ = ξ for all ξ ∈ S d×d .
In the classical elasticity the anisotropy of the body is point-wise characterized by a Hooke tensor: a 4-th order tensor that enjoys certain symmetries and is positive semi-definite. In fact, the set of Hooke tensors is isomorphic to the closed convex cone (2.1) We thus agree that henceforward by Hooke tensors we shall mean the operators H ∈ L + (S d×d ) precisely (slightly abusing the terminology).
In the sequel we will restrict the admissible class of anisotropy by admitting Hooke tensors in a chosen subcone of L + (S d×d ), more accurately: H is an aribtrary non-trivial closed convex cone contained in L + (S d×d ).
We now display some cases of cones H that will be of interest to us:

Example 2.1
The subcone H may be chosen so that the condition H ∈ H implies a certain type of anisotropy symmetry, for instance H = H iso will be the set of isotropic Hooke tensors; we have the characterization where the non-negative numbers K , G are the so-called bulk and shear moduli, respectively. In the case of plane elasticity, i.e. d = 2, for later purposes we give the relation between the moduli and the pair: Young modulus E, Poisson ratio ν It must be stressed that some symmetry classes generate cones that are non-convex. This is the case with classes that distinguish directions, e.g. orthotropy, cubic symmetry.

Example 2.2
Let us denote by H axial the set of uni-axial Hooke tensors, i.e.
where by S d−1 we mean the unit sphere in R d . Above A = η ⊗ η is an element of S d×d whilst A ⊗ A is an operator. Further, we shall abuse notation by writing η ⊗ η ⊗ η ⊗ η := (η ⊗ η) ⊗ (η ⊗ η). The set H axial is clearly a cone yet it is non-convex for d > 1, and, thus, a natural step is to consider the smallest closed convex cone containing H axial , i.e. its closed convex hull: where we used the fact that in a finite dimensional space the convex hull of a closed cone is closed. This family of Hooke tensors relates to materials that are made of 1D fibres.
The constitutive law, i.e. the point-wise relation between the stress tensor σ ∈ S d×d and the strain tensor ξ ∈ S d×d , will be parameterized by the Hooke tensor H ∈ H ; therefore, the elastic energy will depend on two arguments: Throughout the present section we shall assume that, for a chosen exponent p ∈ (1, ∞), the function j satisfies assumption (H1)-(H4) given in the introduction; let us note that, for the time being, we shall not need the ellipticity assumption (H5). We henceforward agree that the subdifferential ∂ j(H , ξ) will be intended with respect to the second variable, and, similarly, we shall understand the convex conjugate j * . This way the constitutive law σ ∈ ∂ j(H , ξ) may be rewritten as the equality ξ, σ = j(H , ξ) + j * (H , σ ).

Example 2.3
The simplest case of a function j for p = 2 is the one from linear elasticity: It is easy to see that the assumptions (H1)-(H4) are satisfied by the function above. The assumption (H5) is virtually put on the set H as it has to contain "enough" Hooke tensors.
In the optimization problem herein considered the Hooke field, being a tensor valued measure λ ∈ M ; H , plays a role of the design variable. A natural constraint to impose on λ is to bound its total cost by a constant C 0 ; therefore, we must choose a cost integrand c : H → R + that satisfies essential properties: convexity, positive homogeneity, lower semicontinuity on H , and c(H ) = 0 ⇔ H = 0. Since H is a closed convex cone consisting of positively semi-definite tensors, for every non-zero H ∈ H we have −H / ∈ H . Then, it is easily seen that every such function c extends to a norm on the whole space L (S d×d ). It is thus suitable that we choose the cost function c as restriction of any norm on L (S d×d ) to H .

Example 2.4
In the pioneering work on the Free Material Design problem [31] the cost function c was proposed as the trace function, i.e.
This is an exceptional example of a cost function c for it is linear on H .
By ⊂ R d we shall understand a bounded connected open set. According to Radon-Nikodym theorem any measure λ ∈ M( ; H ) can be decomposed as follows: that is μ can be computed as the variation measure c(λ), while C is the Radon-Nikodym derivative dλ/(d c(λ)). The condition that C (x) ∈ H for μ-a.e. x virtually defines M( ; H ) as a subset of the Banach space M ; L (S d×d ) ; since H is a cone, this definition is meaningful as it does not depend on the norm c. In (2.5) the information on the Hooke tensor field λ has been split into two: information on the distribution of elastic material μ and information on the anisotropy C . The measure μ, whose support may be a proper subset of , determines the topology and shape of the elastic body. Additional geometric features of the design may be point-wise identified via the space tangent to measure T μ (see e.g. [12]) -measure λ may account for lower dimensional structural elements such as membrane shells, ribs, or bars.
As already announced in the introduction, we state the Free Material Design problem (FMD) of designing in a feasible domain an elastic anisotropic body of minimum compliance under a load being vector valued measure F ∈ M ; R d : where compliance C(λ) is defined variationally in (1.1).

Elastic energy as integral functional on the space of tensor valued measures. The existence result
The compliance functional C : It is well established (see e.g. [22]) that to show convexity and weak-* lower semi-continuity of C it is enough to show this property for each functional in such family or, in this particular case, we have to prove that for any continuous strain field ∈ C ; S d×d the functional is concave and weakly-* upper semi-continuous (above we have utilized the decomposition λ = C μ from (2.5) and one-homogeneity of j with respect to the first argument). Concavity follows directly from assumption (H3), and the present subsection is devoted to proving the upper semi-continuity result. We shall start by proving that the integrand j itself is u.s.c. jointly in both arguments. Beforehand we make an observation:

Remark 2.1
In convex analysis a convex function restricted to a convex subset of a linear space can be equivalently treated as a function defined on the whole space if extended by +∞. Since the real function j : H × S d×d → R + is concave with respect to the first variable H , we can by analogy speak of an extended real function j : such that j(H , ξ) = −∞ for any ξ ∈ S d×d and any H ∈ L (S d×d )\H . This way, the functional (2.6) may be naturally extended to the whole Banach space of Radon measures M ; L (S d×d ) . Then, the condition on the energy j λ, = j C , dμ > −∞ enforces that C ∈ H μ-a.e. and, therefore, λ must lie in the set M( ; H ). This is convenient since in the (FMD) problem we may only require that λ ∈ M( ; L (S d×d )), thus avoiding analysis of the weak-* closedness of M( ; H ).

Proposition 2.1
The function j is upper semi-continuous on the product L (S d×d ) × S d×d , i.e. jointly in variables H and ξ .
Proof We fix a pair (H ,ξ) ∈ H × S d×d . Let us take any ball U ⊂ L (S d×d ) centred at H and introduce a compact set K = U ∩ H . We observe that for any fixed ξ ∈ S d×d the set { j(H , ξ) : H ∈ K } is bounded in R. The zero lower bound follows from non-negativity of j| H , whereas, since j( · , ξ) is real-valued concave and upper semi-continuous on H , it achieves its finite maximum on K . According to [32,Theorem 10.6], the shown point-wise boundedness combined with convexity of every j(H , · ) imply that the family of functions { j(H , · ) : H ∈ K } is equi-continuous on any bounded subset of S d×d . Upon fixing ε > 0 we may thus choose δ 1 > 0 such that where it must be stressed that K does not depend on ε. Due to the upper semi-continuity of j( · ,ξ), we can also choose δ 2 > 0 for which B(H , δ 2 ) ⊂ U and Instead of proving upper semi-continuity of the concave functional (2.6), we may focus on lower semi-continuity of the convex functional λ → j − (λ, ) where simply j − := − j. Then, the idea is to use the classical Reshetnyak's theorem; however, non-positivity of the integrand j − requires an additional trick involving minorization of j − : Proposition 2.2 For any ∈ C ; S d×d the functional λ → j λ, is concave and weakly-* upper semi-continuous on the space M ; L (S d×d ) .

Proof
The idea is to show there exists a continuous function G : Since G is continuous and j − is lower semi-continuous jointly on L (S d×d ) × S d×d by Proposition 2.1, we see by uniform continuity of that the function g is lower semi-continuous jointly on × L (S d×d ). Then, owing to non-negativity of g and its convexity together with positive one-homogeneity with respect to the second variable, the functional λ → g x, λ(dx) is weakly-* lower semi-continuous on M ; L (S d×d ) due to Reshetnyak's theorem, cf. [2, Theorem 2.38]. We observe that for any ∈ C( ; S d×d ) hence, the functional λ → j − λ, is a sum of a continuous linear functional (the function G • : → L (S d×d ) is uniformly continuous) and weakly-* lower semi-continuous functional on M ; L (S d×d ) , which through equality j λ, = − j − λ, furnishes the thesis.
To conclude the proof we must therefore show existence of the function G. First we show that for every ξ ∈ S d×d the proper convex l.s.c. function j − ( · , ξ) : L (S d×d ) → R is subdifferentiable at the origin, i.e. ∂ 1 j − (0, ξ) = ∅ where in this proof by ∂ 1 j − we shall understand the subdifferential with respect to the first argument. By [32,Theorem 23.3], the scenario ∂ 1 j − (0, ξ) = ∅ can occur only if there exists a direction H ∈ L (S d×d ) such that the directional derivative with respect to the first argument j − H (0, ξ) equals −∞. Since j − ( · , ξ) is positively one-homogeneous, our argument for subdifferentiability at the origin amounts to verifying that j − (H , ξ) > −∞ for every H in a unit sphere in L (S d×d ). But this is trivial since, by definition, the function j − does not take the value −∞.
We have thus arrived at a multifunction : \∅ that is convex-valued and closed-valued. According to [29, Theorem 3.2"], in order to show that there exists a continuous selection G of it suffices to show that is l.s.c. (in the sense of theory of multifunctions). This boils down to proving that the function , and all amounts to showing lower semi-continuity of j − = − j on L (S d×d ) × S d×d , which is guaranteed by Proposition 2.1 herein.
The first existence result may readily be proved:

Proposition 2.3 Assuming that j satisfies (H1)-(H4) the Free Material Design problem (FMD) admits a solution whenever C min < ∞.
Proof Compliance functional C is a point-wise supremum of functionals λ → u, F − j λ, e(u) that are convex and weakly-* lower semi-continuous due to Proposition 2.2. Therefore C is itself convex and weakly-* l.s.c. Since c is a norm, minimization in (FMD) runs over a compact subset of M ; L (S d×d ) (see Remark 2.1), and the thesis follows by the Direct Method of Calculus of Variations.
We note that Proposition 2.3 does not employ the ellipticity condition (H5). In Sect. 3 we shall show that, under this additional assumption, finiteness of C min is equivalent to load F being balanced, which will deliver Theorem 1.1 announced in the introduction.

The linear constrained problem and the variational equality
We start the passage to the Linear Constrained Problem by showing the variational equality constituting Theorem 1.2 announced in the introduction: where Z is the supremum in the primal problem With definition (1.1) of C(λ) plugged into (FMD) problem, we arrive at a min-max problem: We shall see that inf and sup above can be swapped, which will allow to formulate a variant of [10, Theorem 1], but first we introduce some additional notions. The functionj : S d×d → R + shall represent the strain energy that is maximal with respect to the admissible anisotropy represented by Hooke tensor H ∈ H of a unit c-cost: As a point-wise supremum of a family of convex functions { j(H , · ) : H ∈ H 1 } the functionj is convex as well. Furthermore, since each j(H , · ) is positively p-homogeneous by assumption (H2), the functionj inherits this property. Next, due to concavity and upper semi-continuity of j( · , ξ) together with compactness of H 1 , we see thatj , and, in particular,j is finite on S d×d , which, in conjunction with its convexity, renders continuity ofj. It is natural to definē being a non-empty, convex, and compact subset of H 1 for every ξ ∈ S d×d . The short over-bar · will be consistently used in the sequel to mark maximization with respect to Hooke tensor and should not be confused with long over-bar · denoting e.g. the closure of a set.
We have just showed thatj is a convex, continuous, and positively p-homogeneous function, and it is well-known (see [32,Corollary 15.3.1]) that it can be written as where ρ : S d×d → R + is a closed gauge -a non-negative, convex, lower semi-continuous, and positively one-homogeneous function. Since ρ is finite valued it is in fact continuous.
We are ready to prove the first link between problems (FMD) and (P) being the equality in Theorem 1.2: Proof of Theorem 1.2 In (3.2) the set over which the infimum is taken is weakly-* compact. Therefore, by acknowledging Proposition 2.2, we easily verify the assumptions of Ky Fan's min-max theorem (cf. [37, Theorem 2.10.2]), which allows to interchange inf and sup: where for any continuous stress field ∈ C( ; S d×d ) we definē where we decomposed λ to C μ with c(C ) = 1 μ-a.e. (the symbolJ is not to be confused with l.s.c. regularization of a functional J ). Further we fix a strain field . For any pair C , μ admissible above we easily find an estimate We shall show that the right hand side of this inequality is attainable for a certain candidateλ . Due to the continuity ofj and of , the functionj ( · ) is continuous on the compact set as well, and, thus, there existsx ∈ such that j ( ) ∞ =j (x) . We putλ = C 0H (x) δx whereH (x) is any Hooke tensor from the non-empty setH 1 (x) , and δx is the Dirac delta measure atx. It is trivial to check that c(λ ) = C 0 , whilst Next we use a technique that was e.g. applied in [24]: by substitution u = t u 1 we obtain where, under the assumption that u 1 , F is non-negative, in the last step we have computed the maximum with respect to t which was attained fort = u 1 , F /C 0 p −1 . Since the function ( · ) p is increasing for non-negative arguments, the thesis follows.
The remainder of this subsection focuses on the analysis of (P) and its dual (P * ). This pair of problems has been already studied in [8,10,13]. In what follows we shall rely on the arguments known from those papers, and the details shall be skipped for brevity.
The closed gauge function ρ 0 : S d×d → R + will stand for the polar to ρ; namely, for a stress tensor σ ∈ S d×d where we recall that ξ, σ := d i, j=1 ξ i j σ i j . Then, to any measure τ ∈ M ; S d×d we may assign a non-negative Borel measure ρ 0 (τ ) that for any Borel set B ⊂ can be defined via the integral formula ρ 0 (τ )(B) : is any measure with respect to which τ is absolutely continuous. Owing to [33,Theorem 6], we obtain the integral representation of the support function of a set in C( ; S d×d ): The function ρ is a continuous gauge; hence, there exists a constant C 2 > 0 such that ρ( ) ∞ ≤ C 2 ∞ for each ∈ C( ; S d×d ). As a result, the set ∈ C( ; S d×d ) : ρ( ) ≤ 1 in has a non-empty interior in C( ; S d×d ) (with respect to the topology of uniform convergence). Then, by duality argument that employs a standard algorithm from [22, Chapter III], we arrive at the dual problem with the zero duality gap: where the equilibrium equation is intended in the sense of distributions on the whole space R d , more precisely: The tensor valued measure τ ∈ M ; S d×d models the stress field. With the trial functions ϕ treated as virtual displacement functions, the right hand side above is known as the virtual work principle. Note that ϕ above may not vanish on ∂ , and, therefore, a Neumann boundary condition is accounted for in −div τ = F, possibly a non-homogeneous one if F charges ∂ . Solutionτ of (P * ) exists as long as Z is finite, which is a part of the duality result. According to [13, Proposition 2.1], for existence of a stress field τ that equilibrates a load F it is necessary and sufficient that the latter is balanced in the sense of Definition 1.1 (we shall independently recover this result). Let us define the space U 0 of rigid body displacement functions: At this point of the work the ellipticity assumption (H5) enters. It may seem weak as it allows degenerate tensors H ∈ H for which there exist non-zero strains ξ such that j(H , ξ) = 0. Nevertheless, (H5) guarantees thatj(ξ ) = 0 if and only if ξ = 0, and, therefore, the same must hold for ρ. It is thus straightforward that: Proposition 3.1 Under assumptions (H1)-(H5) the mutually polar functions ρ, ρ 0 : S d×d → R + are finite, continuous, convex positively one-homogeneous, and they satisfy for some positive constants C 1 , C 2 : In other words, ρ, ρ 0 are mutually dual norms on S d×d if and only if they are symmetric.
Since in general the problem (P) does not attain a solution, it must be relaxed, which shall consist in taking the closure of the set of admissible smooth displacement functions in the topology of uniform convergence on and showing suitable compactness. Clearly, as a subset of C( ; R d ), the set U 1 is unbounded for it contains the linear subspace U 0 . The "rigid body part" of a function u can be eliminated by means of a linear projection operator It may be defined such that Thanks to coerciveness of ρ stated in Proposition 3.1, the compactness result can be obtained identically as in [8]: it requires using Korn's inequality in L q twice for some q > d and then exploiting the Morrey's embedding theorem.

Proposition 3.2 Let be a bounded domain with Lipschitz boundary. Then, under assump-
With the compactness result at our disposal, we may readily propose the relaxation of the problem (P): Recall that F is balanced if and only if u, F = 0 ∀ u ∈ U 0 . Then, once F is balanced, it is straightforward to show that any maximizing sequence u n ∈ U 1 can be modified to another maximizing sequence u n − P U 0 u n . As a result we obtain:

Corollary 3.1 Under the prerequisites of Proposition 3.2 the number Z is finite if and only if F is balanced. In that case
In particular, problems (P) and (P * ) attain their solutions.
Thanks to Proposition 3.1 and equality Z = inf P * , from Corollary 3.1 we recover the result that the equilibrium equation −div τ = F has a solution τ if and only if F is balanced. Finally, by combining Corollary 3.1 and Proposition 2.3, we establish the existence result for the (FMD) problem stated in Theorem 1.1 in the introduction.
In the sequel of this work we assume that ∂ is Lipschitz regular and that (H1)-(H5) hold for the function j (unless clearly stressed otherwise).

The displacement and stress solutions of the elasticity problem
Classically, we say that a vector displacement function u and a tensor stress function σ solve the elasticity problem for a load F whenever: (i) σ equilibrates the load F; (ii) σ and the strain e(u) point-wise satisfy the constitutive law of elasticity. With the Hooke tensor field given by a measure λ = C μ ∈ M ; H the two conditions may be written as (i) −div(σ μ) = F and (ii) σ ∈ ∂ j C , e(u) μ-a.e. The constitutive law written in this fashion is meaningful for any measure λ only if u is of C 1 class (see also Corollary 4.2).
In general, both solutions u and σ can be found by solving independent variational problems. In case of the displacement function, it is well established that one must solve the maximization problem that defines the compliance; we recall the formula: To establish existence of solution, the maximization problem must be relaxed. Once j(C , · ) satisfies a suitable ellipticity condition, the relaxed solution may be found in a Sobolev space with respect to measure μ denoted by W 1, p μ which was proposed in [12] and then developed in e.g. [10]; see also [6] on the application of higher order weighted Sobolev space in elasticity of beams with degenerate width distribution. In the present paper it is crucial that the energy potential may be degenerate, e.g. in the sense that j C , may vanish for some non-zero ∈ L p μ ( ; S d×d ) on a set of non-zero measure μ. Therefore, the theory put forward in [12] cannot be directly applied to every pair of measures F and λ. One would have to design another functional space that depends on the structure of j(C , · ) and, in general, that may contain functions u of low regularity, including those of jump-type discontinuities.
The situation is better if λ is optimally chosen for F, i.e. it solves (FMD). Then, by a straightforward generalization of [10, Proposition 2], we could show that amongst the sequences u n ∈ (D(R d )) d that are maximizing for (3.8) we can always choose one satisfying sup n ρ e(u n ) ∞ < ∞. More precisely, it is justified to introduce the following notion: Definition 3.1 For a balanced load F assume that λ solves the (FMD) problem. Then, by a relaxed solution of (3.8) we shall understand a function u ∈ C( ; R d ) such that there exists a sequence u n ∈ (D(R d )) d that is maximizing for (3.8) and satisfies: u n → u uniformly in , and ρ e(u n ) ≤ (Z /C 0 ) p / p in for every n.
The problem whose solution occurs to be the stress σ is the one that is dual to problem (3.8). Similarly as in the previous section, the duality argument will be standard. Beforehand, however, for a fixed λ = C μ we must examine the functional j(C , · ) dμ : L p μ ( ; S d×d ) → R. As a first step we shall make sure that conjugation and integration operations commute, i.e. that j(C , · ) dμ * : L p μ ; S d×d → R is equal to j * (C , · ) dμ, where j * : H × S d×d → R is the convex conjugate with respect to the second variable. A simple scenario when such commutation holds is when the integrand j C ( · ), · : ×S d×d → R + is a Carathéodory function, cf. [22]:

Proposition 3.3 For a given Radon measure
(i) for μ-a.e. x the function j C (x), · is continuous; (ii) for every ξ ∈ S d×d the function j C ( · ), ξ is μ-measurable.
Proof The statement (i) follows easily from the assumption (H1) since every convex function that is finite on the whole finite dimensional space is automatically continuous. For any ξ ∈ S d×d the function j C ( · ), ξ is a composition of an upper semi-continuous function j( · , ξ) (cf. assumption (H3)) and a μ-measurable function C , hence the claim (ii).
Next, by employing assumptions (H1)-(H4), it is straightforward to infer that for C ∈ L ∞ μ ( ; C ) the functional j(C , · ) dμ is convex and continuous on the Lebesgue space L p μ ( ; S d×d ). Combining this fact with Proposition 3.3 unlocks, once again, the standard algorithm from [22, Chapter III], and we readily arrive at the problem dual to (3.8): where again λ = C μ. As a part of the duality result we infer that the minimizer of (3.9) exists whenever C(λ) < ∞; therefore, contrarily to the displacement-based problem (3.8), the problem (3.9) does not require relaxation. The infimum in (3.9) yields an alternative, dual definition of compliance C(λ); it allows to give upper bounds for C(λ), which will be well utilized while proving Theorem 1.3.

Remark 3.1
Should it exist, solution σ of the problem (3.9) depends on the particular choice of μ and C that gives λ = C μ. For instance, if for given μ we put μ 1 = αμ, for some α > 0, solution σ 1 for this pair would satisfy the scaling property: σ 1 = 1 α σ . To put it differently, it is the field τ = σ μ that is invariant of the chosen representation λ = C μ. Accordingly, τ may be considered an absolute stress field whilst σ = dτ dμ (the Radon-Nikodym derivative) can be interpreted as the relative stress field, i.e. relative with respect to the elastic material's distribution μ. Since in this work we always enforce that c(C ) = 1 μ-a.e., speaking of solution σ of (3.9) (see (iii) in Definition 1.2) should not cause any confusion.

Designing the anisotropy at a point-the underlying finite dimensional program
The functionj and, therefore, also the function ρ are expressed via finite dimensional program (3.3) where function j enters. In the present subsection it will appear that a "mirror" relation may be established between the polar ρ 0 and the conjugate function j * , which will be fundamental for connecting two of minimization problems in this work: (P * ) and (3.9). By definition of polar ρ 0 , for any pair (ξ, σ ) ∈ S d×d × S d×d there always holds ξ, σ ≤ ρ(ξ ) ρ 0 (σ ). We shall say that a pair (ξ, σ ) satisfies the extremality condition for ρ and its polar whenever we have an equality ξ, σ = ρ(ξ ) ρ 0 (σ ). Another result of this subsection will link this extremality condition to satisfying the constitutive law σ ∈ ∂ j(H , ξ) forH ∈ H 1 optimally chosen for σ , namely forH belonging to the set This relation will be utilized whilst formulating the optimality conditions for the (FMD) problem in Sect. 4.
We first investigate the properties of the convex conjugate j * ; by its definition, for a fixed H ∈ H we get a function j * (H , · ) : S d×d → R expressed by the formula (3.10) Convexity and l.s.c. of j * (H , · ) follow by the well established properties of convex conjugates. We look at the subdifferential ∂ j(H , · ) : S d×d → 2 S d×d . Almost by definition for while the opposite inequality on the right hand side, known as Fenchel's inequality, holds always. By recalling positive p-homogeneity of j(H , · ), it is well known that the following repartition of energy holds (see e.g. [32])

Remark 3.2
For a given H ∈ H the function j * (H , · ) may admit infinite values: take for instance H ∈ H that is a singular operator and j(H , ξ) = 1 2 H ξ, ξ ; then, j * (H , σ ) = ∞ for any σ lying outside the range of H . At this point it is not clear whether the function j * ( · , σ ) is proper for arbitrary σ ∈ S d×d , i.e. we question the strength of the assumption (H5). A positive answer to this question shall be a part of the theorem that we state below. Theorem 3.1 Let ρ : S d×d → R + be the real closed gauge function defined by (3.4), and by ρ 0 denote its polar. Then, the following statements hold: (i) For every stress σ ∈ S d×d the setH 1 (σ ) is a non-empty compact convex set, and where the continuous functionj * : S d×d → R + is the convex conjugate ofj. Moreover, (ii) For a triple (ξ, σ,H ) ∈ S d×d × S d×d × H satisfying: ρ(ξ ) ≤ 1, σ = 0,H ∈ H 1 the following conditions are equivalent: (1) there hold the extremality conditions: (2) the constitutive law is satisfied:

Moreover, for each of the conditions (1), (2) to be true, it is necessary that ρ(ξ ) = 1. (iii) The following implication is true for every non-zero
while, in general,H 1 (σ ) may be a proper subset ofH 1 (ξ ).
Proof For the proof of the statement (i) we fix a matrix σ ∈ S d×d ; then, directly by definition of the convex conjugate (3.10) we obtain a min-max problem: Above, to interchange the order of inf and sup we again used Ky Fan's theorem. The first equality in (3.14) is proved, while, considering (3.4), the second one is well established, see [33,Corollary 15.3.1]. Consequently,j * (σ ) < ∞ by Proposition 3.1; therefore, owing to convexity and lower semi-continuity of H → j * (H , σ ) (cf. Proposition 3.4) and compactness of H 1 , we infer non-emptiness, convexity, and compactness ofH 1 (σ ). Since ρ 0 (σ ) > 0 for a non-zero σ (cf. Proposition 3.1), the moreover part of the claim (i) follows by (3.13).
For the implication (2) ⇒ (1) we assume that for a triple ξ, σ,H with ρ(ξ ) ≤ 1, ρ 0 (σ ) = 1, H ∈ H 1 the constitutive law (3.15) is satisfied. Then, by (3.12) we have the repartition of energy: ξ, σ = p j(H , ξ) = p j * (H , σ ). The following chain of estimates holds: and, therefore, all the inequalities above are in fact equalities; in particular we have which proves implication (2) ⇒ (1) and the "moreover part" of the assertion (ii). We demonstrate some of the statements of Theorem 3.1 for the AMD setting of the Free Material Design problem:
Next, for a non-zero σ the point (iii) of Theorem 3.1 furnishes: where we used the fact thatH 1 (ξ σ ) is a singleton. Therefore, we have obtained The latter results were obtained in [18] by solving the problem min H ∈H 1 j * (H , σ ) directly.

Remark 3.4
It is worth noting that, in general, neither of the setsH 1 (ξ ) or evenH 1 (σ ) is a singleton, see Examples 5.1 and 5.2 in Sect. 5.1.

Recasting a solution of the Free Material Design problem from a solution of the Linear Constrained Problem
We are finally moving to the result that is central for this work: we shall prove Theorem 1.3 that allows to move from a quadruple (û,μ,σ ,Ĉ ) that solves the Linear Constrained Problem in the sense of Definition 1.3 to a quadruple (ǔ,μ,σ ,Č ) solving the Free Material Design problem in the sense of Definition 1.2, and vice versa. As noted in the introduction, all the fields constituting the two solutions are already proved to exist except for the Hooke functionĈ being anyμ-measurable selection of the multivalued map x →H 1 σ (x) . The following result settles this issue: Lemma 3.1 For a given Radon measure μ ∈ M + ( ) let γ : → S d×d be a μ-measurable function. We consider the multifunction γ : → 2 H 1 \∅ that is closed-valued and convexvalued: Then, there exists a μ-measurable selection C γ : → H 1 of the multifunction γ , namely Proof It suffices to prove that the multifunction σ →H 1 (σ ) is upper semi-continuous on S d×d . Then, it is also a measurable multifunction, and, thus, there exists a Borel measurable selectionH : S d×d → H 1 , i.e.H (σ ) ∈H 1 (σ ) for every σ ∈ S d×d , see [15,Corollary III.3 and Theorem III.6]. Then, C γ :=H • γ : → H 1 is μ-measurable as a composition of Borel measurable and μ-measurable functions.
We now show upper semi-continuity of the multifunction σ →H 1 (σ ). Since H 1 is compact, it is enough to show that its graph G is closed, see [27]. Thanks to equality (3.14) one can write The functionj * = 1 p (ρ 0 ( · )) p is real-valued and continuous, see Proposition 3.1. The function j * : H 1 ×S d×d → R is jointly lower semi-continuous by Proposition 3.4; therefore, so is the mapping (H , σ ) → j * (H , σ ) −j * (σ ). As a result, the graph G is closed in S d×d × H 1 , which finishes the proof. This result renders the statement of Theorem 1.3 an effective method of solving the Free Material Design problem. We prove the theorem now: Proof of Theorem 1. 3 Let us first assume that the quadruple (û,μ,Ĉ ,σ ) is a solution of (LCP). By definitionτ =σμ is a solution of the problem (P * ) and ρ 0 (τ ) = ρ 0 (σ )μ =μ. Since ρ 0 (σ ) = 1μ-a.e., owing to the "moreover part" of the claim (i) in Theorem 3.1, we deduce that c(Ĉ ) = 1μ-a.e. as well, and the same concernsČ . We verify thatλ =Čμ is a feasible Hooke tensor field by computing: sinceτ is a minimizer for (P * ). In order to prove thatλ is a solution for C min , it suffices to show that C(λ) ≤ C min where C min = Z p / p C 0 p −1 by Theorem 1.2.
We observe thatμ-a.e. ρ 0 (σ ) = Z C 0 ρ 0 (σ ) = Z C 0 . Since there holdsσμ =σμ =τ , the equilibrium equation −div(σμ) = F is satisfied. Due to the p -homogeneity of j * (H , · ) (see Proposition 3.4), the fieldČ =Ĉ is both a measurable selection for x →H 1 σ (x) and x →H 1 σ (x) . Then, by the dual stress-based version of the elasticity problem (3.9) where in the first equality we have used the fact thatČ (x) ∈H 1 σ (x) forμ-a.e. x, the second one is by the assertion (i) in Theorem 3.1, whilst the last equality is due to the fact that dμ = C 0 . This proves optimality ofλ, and we have only equalities in the chain above. The first equality implies thatσ solves the dual elasticity problem (3.9) for λ =λ =Čμ. Next, we must show thatǔ is a relaxed solution for (1.1), see Definition 3.1. Sinceû solves (P), there exists a sequenceû n ∈ U 1 such that û n −û ∞ → 0 and û n , F → Z . By definition of U 1 , we have ρ e(û n ) ≤ 1, and maximizing sequence for (1.1) may be found asǔ n := (Z /C 0 ) p / pû n . Indeed, by (3.3), (3.4) we see that where we have used the fact that lim n→∞ ǔ n , F = (Z /C 0 ) p / p lim n→∞ û n , F = (Z /C 0 ) p −1 Z and that Z p / p C 0 p −1 = C min = C(λ) by optimality ofλ. Therefore,ǔ n is a maximizing sequence for (1.1), thus concluding the proof of the first implication.
Conversely, we assume that the quadruple (ǔ,μ,Č ,σ ) is a solution of the (FMD) problem (by definition we have c(Č ) = 1μ-a.e.). The Hölder inequality furnishes (3.19) and the equalities hold only if ρ 0 (σ ) isμ-a.e. constant and only if dμ = C 0 (excluding the case whenσ is a zero function, which is justified by the prerequisite C min > 0). The following chain of inequalities holds true: where: -the first and the second equality acknowledge thatλ =Čμ solves (FMD) and, respectively, thatσ is a minimizer in (3.9); -the remaining inequality and equality in the first line are by Theorem 3.1, claim (i); -to pass to the second line we use (3.19); -the last inequality is by admissibility ofσμ for (P * ), which is due to −div(σμ) = F coming from admissibility ofσ for (3.9); -the last equality is the statement of Theorem 1.2.
To finish the proof we have to show thatû = C 0 Z p / pǔ is a solution for (P). It is straightforward to show thatû ∈ U 1 based on Definition 3.1 of the relaxed solution for (1.1), and, thus, we only have to verify whether û, F = Z . One can easily show that forǔ being a relaxed solution for C(λ) there holds the repartition of energy ǔ, F = p C(λ) (see [10,Proposition 3]).

Corollary 3.2
The relative stressσ , that due to the load F occurs in the structure of the optimal Hooke tensor distributionλ =Čμ, is uniform in the sense thať

Optimality conditions for the Free Material Design problem
In order to efficiently verify whether a given quadruple (u, μ, σ, C ) is optimal for (FMD) problem, we shall derive the system of optimality conditions. Due to the much simpler structure of the problem (LCP) and the link between the two problems in Theorem 1.3, it is more natural to pose the optimality conditions for (LCP). Since the form of the latter problem is similar to the one from the paper [10] (see also the earlier work [8]), we will build upon the results given therein: in addition we must involve the Hooke tensor function C .
In [10] one of the optimality conditions is local for it relates the stress σ (x) to the strain (x) at μ-a.e. x. Since solutions u ∈ U 1 of the problem (P) may be non-differentiable, the notion e(u) is not, in general, well defined μ-a.e. in the classical or the weak sense. To fix this issue the authors of [10] employ the apparatus of μ-tangential calculus, introduced for the first time in [12], see also [9]. It allows to define a μ-tangential gradient ∇ μ u that (for a scalar function u) is an element of L 1 μ ( ; R d ).
Below we quickly review the main aspects of the μ-tangential calculus. Its theory was generalized in [10,Section 4], making it possible to work with a wide range of linear differential operators A : (D(R d )) m → C( ; V ) for any natural m and any finite dimensional space V . In what follows we specify the general framework from [10] by choosing m = d, V = S d×d and A = e to arrive at the μ-tangential strain operator e μ = e μ (u).
Let μ ∈ M + ( ) be any Radon measure. For the graph of the operator e with the space of smooth functions as its domain, i.e. for G := u, e(u) : u ∈ (D(R d )) d , we take the following closure: where u ∞ := sup x∈ |u(x)|, and * stands for the weak-* convergence. In general G is no longer a graph of any operator, namely the weakly-* closed space is non-trivial. According to [10,Proposition 4], the space N is decomposable in the following sense: there exists a μ-measurable multifunction x is a linear subspace of S d×d , it is therefore closed. By construction (and the terminology borrowed from the case of A = ∇, see [9,12]) point-wise N μ (x) receives an interpretation of the space of matrices normal to μ at x. The next step consists in defining the space of μ-tangential matrices by taking the orthogonal complement of N μ (x). This line of reasoning leads to the definition: where is any element of L ∞ μ ( ; S d×d ) such that (u, ) ∈ G.

Remark 4.1
The choice of in the definition above may be ambiguous: in general, for u ∈ dom(e μ ) we have different pairs (u, 1 ), (u, 2 ) ∈ G where ζ := 1 − 2 = 0 is an element of N by the very definition of the latter. Nonetheless, thanks to characterization (4.2), we have μ (x) ζ(x) = 0 for μ-a.e. x; therefore, the operator e μ is well defined after all.

Remark 4.2
It is straightforward to show that for any smooth function u ∈ (D(R d )) d there holds e μ (u) = μ e(u). This property can be easily extended to functions u of C 1 class.

Remark 4.4 (Characterization of M μ for the multi-junction measures μ)
In order to effectively characterize the space M μ , a fairly wide class of geometric measures μ can be considered. By a multi-junction measure we shall understand a measure of the form μ = N i=1 μ i where for each i: μ i = m i H k i S i with m i being a positive constant, k i being an integer in {1, . . . , d}, and S i being a k i -dimensional manifold of C 2 class. In addition, we assume that μ i (S i ) = 0 whenever i = i . If for each x ∈ S i by T S i (x) one denotes the classical tangent space, then: and, as a result, for any matrix ξ ∈ S d×d where P S i (x) is the matrix of orthogonal projection onto T S i (x). For details see [9,10,12].
The μ-tangential objects defined above can be analysed from the "dual perspective" that was, in fact, a point of departure in the pioneering work [12]. Let us consider a stress field τ ∈ M( ; S d×d ) and its decomposition to τ = σ μ for μ ∈ M + ( ) and σ ∈ L 1 μ ( ; S d×d ). If F is any balanced load, the condition on the relative stress σ point-wisely lying in M μ turns out to be necessary for the equilibrium. Indeed, by virtue of [10, Lemma 2] we have: for μ-a.e. x, e μ (u), σ dμ = u, F ∀ u ∈ dom(e μ ).
Below we prove that any admissible function u ∈ U 1 is an element of the domain of e μ . On top of that, we show how the condition ρ e(u) ≤ 1 in holding for smooth u ∈ U 1 can be translated to U 1 for μ-a.e. x. The result below is almost identical to [10, Lemma 1], but we display the proof for the reader's convenience:
Proof For a fixed u ∈ U 1 let {u n } ⊂ U 1 ⊂ (D(R d )) d be a sequence such that u n → u uniformly on . Due to the coerciveness of ρ guaranteed by Proposition 3.1, from the fact that ρ e(u n ) ≤ 1 in we infer that sup n e(u n ) L ∞ μ < ∞. Therefore, (up to choosing a subsequence) e(u n ) * in L ∞ μ ( ; S d×d ) for some function , and, thus, u ∈ dom(e μ ) by Definition 4.1 and, moreover, = e μ (u) + ζ where ζ ∈ N . For any Borel set B ⊂ the convex functional → B ρ( ) dμ is lower semi-continuous for the weak-* topology on L ∞ μ ( ; S d×d ), and, thus, starting by acknowledging (4.2) and formula (4.7): The "moreover part" readily follows due to arbitrariness of the Borel set B.
The optimality conditions for the Linear Constrained Problem may readily be given:

The quadruple solves (LCP) if and only if the following optimality conditions are met:
(4.11) Moreover, the pair of conditions (iii), (iv) may be equivalently put as a constitutive law of elasticity: (4.12) Proof Considering Defintion 1.3 that explicitly stipulates the condition (iv), the first part of the claim will follow once we prove that: u and τ = σ μ solve problems (P) and, respectively, (P * ) if and only if conditions (i), (ii), (iii) hold true. Since (i) and (ii) are the admissibility criteria for (P * ) and (P), they may be further assumed to be true. As a result the following chain may be written down where: -the first equality is the integration by parts formula in (4.4) that holds since −div(σ μ) = F and u ∈ U 1 ⊂ dom(e μ ); -the inequality acknowledges that max P = min P * , cf. Corollary 3.1; -the last equality exploits the fact that σ (x) ∈ M μ (x) for μ-a.e. x (see (4.4) again) and the formula (4.8): we have ρ 0 (σ ) = ρ 0 μ (x, σ ) μ-a.e. as a result. Using the zero-gap result max P = min P * again, we deduce that u and τ = σ μ are solutions if and only if in (4.13) we have equalities only. By the "moreover part" of Proposition 4.2 and by the very definition of polarity, we have e μ (u), σ ≤ ρ 0 μ (σ ) μ-a.e. Consequently, equalities in (4.13) hold if and only if e μ (u), σ = ρ 0 μ (x, σ ) μ-a.e., which is exactly (iii) since ρ 0 μ (x, σ ) = ρ 0 (σ ) = 1. The first part of the claim is proved. The "moreover part" of the assertion follows directly by Corollary 4.1 and the fact that ρ 0 (σ ) = 1 μ-a.e. Indeed, by acknowledging Proposition 4.2 again, for μ-a.e. x the triple ξ = e μ (u)(x), σ (x),H = C (x) satisfies the prerequisites of the corollary.
Owing to the "moreover part" of the theorem above, we can recover the equations of elasticity for the optimal body undergoing a non-smooth displacementǔ (cf. − div(σμ) = F,σ ∈ ∂ j μ x,Č , e μ (ǔ) μ-a.e. (4.14) Proof The stated pair of conditions holds true for any solution (û,μ,Ĉ ,σ ) of (LCP) by virtue of Theorem 4.1. Considering the characterization (3.12) written for j μ , one easily checks that this pair of conditions is preserved under the transformation to (ǔ,μ,Č ,σ ) in accordance with Theorem 1.3.

Case study and examples of optimal structures
In Example 3.1 we have computed: ρ, ρ 0 together with the extremality conditions for ξ, σ and the sets of optimal Hooke tensorsH 1 (ξ ),H 1 (σ ) in the Anisotropic Material Design setting (AMD) which assumed that H = L + (S d×d ) (all Hooke tensors are admissible) and j(H , ξ) = 1 2 H ξ, ξ (linearly elastic material). The computed functions and sets virtually define the problem (LCP) (in the AMD setting) which, via Theorem 1.3, paves the way to solution of the original (FMD) problem.
In Sect. 5.1 below we will compute ρ, ρ 0 andH 1 (ξ ),H 1 (σ ) for other settings of the (FMD) problem. They will vary in the choice of both H and j. Afterwards, in Sect. 5.2, a simple (FMD) problem will be solved analytically in all of the proposed settings.

Case b) the determinant of σ is positive
We shall focus on the case when λ 1 (σ ), λ 2 (σ ) > 0. Once again, there is unique ξ σ with ρ(ξ σ ) = 1 and satisfying (5.12): necessarily ξ σ = I. Therefore, any unit vector η is an eigenvector of ξ σ (but not necessarily of σ ) with eigenvalue equal to one and, thus, By plugging this form of H into (5.13) we obtain the characterization for σ ∈ S d×d which is not a singleton. The setH 1 (σ ) is in fact very rich: it may be showed that non-zero fibres may be laid out in any direction η i ∈ S d−1 .
Case c) σ is of rank one It is not restrictive to assume that λ 1 (σ ) = 0, λ 2 (σ ) > 0 and so σ = λ 2 (σ ) v 2 (σ )⊗v 2 (σ ). In this case there are infinitely many ξ σ such that ρ(ξ σ ) = 1 and (5.12) holds. We can, however, test (5.13) with only one: , which is necessarily equal toH 1 (σ ) due to point (iii) of Theorem 3.1. Eventually, for a rank-one stress σ the set of optimal Hooke tensors may be written as a singletonH where we used the fact that ρ 0 (σ ) = |σ | in this case. By comparing to Example 3.1, we learn that AMD and FibMD share the optimal Hooke tensor at points where σ is rank-one.
For the next step, we wish to consider an energy function different than j(H , ξ) = 1 2 H ξ, ξ . For isotropic tensors H ∈ H iso the quadratic function can be easily generlized to exponents p ∈ (1, ∞) different than 2, thus arriving at the so called power-laws, see e.g. [16]. Instead, in order to make the fairly general assumptions (H1)-(H5) more worthwhile, we shall construct an important energy integrand that is non-linear with respect to H : According to Remark 4.7, the functions j + and j − satisfy conditions (H1)-(H4). They are proposals of elastic potentials of materials that are incapable of withstanding compressive and, respectively, tensile stresses. In fact, once the baseline energy function is chosen as j(H , ξ) = 1 2 H ξ, ξ for isotropic Hooke tensors H = H (K , G) (see Example 2.1), then the potential j − (H , · ) can be employed to recast the 2D masonry material model that was put forth in [23] in a more elementary way. Indeed, if ε 1 ≤ ε 2 stand for the two eigenvalues of ξ ∈ S 2×2 , then one may compute that: for K > 0, while j(H , ξ) = 0 otherwise. The formula above coincides with [23,Eq. (3.19)]. Let us note that j − (H , ξ) varies in H (i.e. in K , G) non-linearly provided that we move within the second regime. Although the starting function j is assumed to satisfy the ellipticity condition (H5), it is lost for either of the functions j + , j − . More precisely, owing to first formulas in (5.17), (5.18), we infer that j + (H , ξ) = 0 (resp. j − (H , ξ) = 0) for every H ∈ H if and only if ξ ∈ S d×d − (resp. ξ ∈ S d×d + ). In order to restore the condition (H5), we define a function j ± : H ×S d×d → R + that shall model a composite material that is dissymmetric for tension and compresion: where κ + , κ − are positive reals and p is the homogeneity exponent of j(H , · ). Effectively j ± (H , ξ) = 0 for each H ∈ H 1 if and only if ξ ∈ S d×d + ∩ S d×d − = {0}. In summary, the function j ± : H × S d×d → R + satisfies all the conditions (H1)-(H5), and, thus, the Free Material Design problem is well posed for the material that j ± models. At the same time, based on the analysis of the formula (5.19) we infer that we can easily obtain j ± that is non-linear with respect to H , which justifies the need for a more general assumption (H3) kept throughout this work.
The newly proposed energy function j ± can be now exploited to modify each of the foregoing settings of (FMD). We shall focus on the FibMD setting only: 2 H ξ, ξ as in Example 5.3. In contrast to Example 5.2, we have no linearity of j ± with respect to H , and, therefore, for given ξ ∈ S d×d we must test j ± (H , ξ) with tensors H in the whole co(H axial ) instead of just H axial .
Nevertheless, computations similar to those from Example 5.2 show thatj ± (ξ ) = 1 2 ρ ± (ξ ) 2 with the ρ ± being a non-symmetric generalization of the spectral norm: For σ ∈ S d×d the polar ρ 0 ± reads where ρ 0 is the polar to the spectral norm, see (5.11); it is worth to note that Tr σ enters the formula with a sign. The formula for ρ 0 ± was already reported in [28,Section 3.5]. The extremality conditions between ξ and σ for ρ ± and ρ 0 ± are very similar to those displayed for the FibMD setting (see (5.12)); thus, we shall neglect to write them down. The same goes for characterizations of the setsH 1 (ξ ) andH 1 (σ ); we merely show a formula for being a universal (but in general non-unique) element of the setH 1 (σ ).

Remark 5.1
The pair of variational problems (P) and (P * ) with ρ and ρ 0 specified in Example 5.2 are well known to constitute the Michell problem which is the one of finding the least-weight truss-resembling structure for the permissible stresses equal in tension and compression:s + =s − ∈ R + , cf. [34] and [13]. An extensive coverage of the Michell structures may be found in [28]. Herein, the Michell problem is recovered as the FibMD setting of the Free Material Design problem. Another works, where a link between the Michell problem and optimal design of elastic body was made, are [1,3,7,14,30], where the Michell problem was recast through an asymptotic analysis of optimal shape obtained by homogenization method in the high-porosity regime. To the knowledge of the present authors, however, until now no elastic design problem discussed in the literature was showed to be equivalent to the Michell problem for uneven permissible stresses in tension and compressions + =s − , whereas here it naturally emerges through the pair of problems (P) and (P * ) in the FibMD± setting discussed in Example 5.4 provided one chooses κ + /κ − =s + /s − .

Examples of solutions of the Free Material Design problem in settings: AMD, IMD, FibMD and FibMD±
For a load F that simulates the uni-axial tension test, we are to solve the Free Material Design problem in the several settings listed in this paper. Thanks to Theorem 1.3, we may solve the corresponding (LCP) problem instead, for which we have at our disposal the optimality conditions from Theorem 4.1. Our strategy will be to first propose a competitor (u, μ, σ, C ) for which we shall validate the optimality conditions. The displacement solutions u are non-unique as they can be modified by any rigid body displacement function u 0 ∈ U 0 .  We see that ρ 0 (σ ) = |σ | = 1 and Tr C = 1, which are the initial prerequisites in Theorem 4. 1.

Case b) the Isotropic Material Design setting
For the IMD setting, the norms ρ and ρ 0 are given in (5.2) and (5.3), respectively. We put forward a quadruple that shall be checked for optimality: x 1 e 1 − 2 − √ 2 2 x 2 e 2 , μ = 2 + √ 2 2 First we check that Tr C = 2K + 2 · 2G = 1, thus C ∈ H 1 as assumed in Theorem 4.1. Since the stress field τ = σ μ is identical to the one from Case a), the optimality condition (i) in Theorem 4.1 clearly holds. The function u is again smooth, so we compute e(u) = 2+ and therefore u ∈ U 1 , which validates the optimality condition (ii); moreover ρ 0 (σ ) = 1 as required in Theorem 4.1. We move on to check the remaining optimality conditions in the version (iii,iv). Since μ above coincides with the one from Case a) (up to a multiplicative constant), the formulas for M μ and μ given therein are also correct here. By acknowledging Remarks 4.2, 4.3, we have μ-a.e. σ, e μ (u) = σ, μ e(u) = σ, e(u) , where we used the fact that σ ∈ M μ μ-a.e. We easily check that σ, e(u) = 1, and condition (iii) follows: σ, e μ (u) = 1 μ-a.e. Then, one may find that the moduli K , G agree with the characterization of the setH 1 (σ ) in (5.5), hence the condition (iv), namely C ∈H 1 (σ ) μ-a.e., which proves that quadruple (u, μ, σ, C ) is optimal for (LCP) in the IMD setting. In order to be complete, we will show that the optimality condition (iii,iv)' holds as well. It is clear that for L 2 -a.e. x ∈ R, where M μ (x) = S 2×2 , we have j μ x, C (x), · = j C (x), · . Meanwhile, for H 1 -a.e. x ∈ [A 0 , B 0 ] the tensors ζ ∈ N μ (x) are exactly those of the form ζ = e 2 s ⊗ η, where η ∈ R 2 and s ⊗ is the symmetrized tensor product. Hence, for H 1 -a.e.
x ∈ [A 0 , B 0 ], after performing the minimization (being non-trivial here) we obtain j μ x, C (x), ξ = inf The equations above are verified after elementary computations; in particular, using formulas (5.22) for optimal K , G gives the Young modulus and the Poisson ratio: The computed value of Young modulus E turns out to be maximal among all pairs K , G ≥ 0 satisfying Tr C = 2K + 2 · 2G ≤ 1.

Case c) the Fibrous Material Design setting
In the case of the Fibrous Material Design setting it is enough to shortly note that the quadruple (u, μ, σ, C ) proposed in Case a) is also optimal in the FibMD setting: indeed, both e(u) and σ are of rank one; thus, spectral norm ρ e(u) and its polar ρ 0 (σ ) (see (5.11)) coincide with |e(u)| and |σ |, respectively. Moreover, again for a rank-one field σ , the sets H 1 (σ ) are identical for AMD and FibMD, see (5.16) and the comment below.
Further, the same solution (5.20), (5.21) of (LCP) will be shared by the FibMD± setting provided that one assumes κ + = 1. This is a consequence of σ being positive definite μ-a.e.