$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^\infty$ via the singular value problem

For $\mathrm{H} \in C^2(\mathbb{R}^{N \times n})$ and $u : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}^N$, consider the system \[ \label{1}\mathrm{A}\_\infty u\, :=\,\Big(\mathrm{H}\_P \otimes \mathrm{H}\_P + \mathrm{H}[\mathrm{H}\_P]^\bot \mathrm{H}\_{PP}\Big)(\mathrm{D} u): \mathrm{D}^2 u\, =\,0. \tag{1}\]We construct $\mathcal{D}$-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our $\mathcal{D}$-solutions are $W^{1,\infty}$-submersions and are obtained without any convexity hypotheses for $\mathrm{H}$, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions $n\neq N$.


Introduction
Let H ∈ C 2 (R N ×n ) be a given function and Ω ⊆ R n a given open set, n, N ∈ N. In this paper we are interested in the problem of existence of appropriately defined generalised solutions with given Dirichlet boundary conditions to the second order PDE system (1.1) A ∞ u := H P ⊗ H P + H[H P ] ⊥ H P P (Du) : D 2 u = 0.
In the above, the subscript P denotes the derivative of H with respect to its matrix variable, while Du(x) = D i u α (x) α=1,...,N i=1,...,n ∈ R N ×n , D 2 u(x) = D 2 ij u α (x) α=1,...,N i,j=1,...,n ∈ R N ×n 2 s , denote respectively the gradient matrix and the hessian tensor of (smooth) maps u : Ω ⊆ R n −→ R N . The notation "[H P ] ⊥ " symbolises the orthogonal projection on the orthogonal complement of the range of the linear map H P (P ) : R n −→ R N : (1.2) [H P (P )] ⊥ := Proj R(H P (P )) ⊥ . ). Calculus of Variations in L ∞ , as the field is known today, was initiated by G. Aronsson in the 1960s who studied the scalar case N = 1 quite systematically ([A1]- [A7]). Since then the area has been developed marvellously due to both the intrinsic mathematical interest and the importance for applications. In particular, the theory of Viscosity Solutions of Crandall-Ishii-Lions played a fundamental role in the study of the generally singular solutions to the scalar version of (1.1). When N = 1, the respective single equation simplifies to (1.4) A ∞ u = H P (Du) ⊗ H P (Du) : D 2 u = 0 and is known as the "Aronsson equation". For a pedagogical introduction to the scalar case with numerous references see e.g. [K7, C] (see also [ACJ, BEJ, SS, CR] for some relevant work and [Pi] for a comparison between the Viscosity Solutions approach and the Baire Category one).
On the other hand, except perhaps the deep vectorial contributions in [BJW1,BJW2] which however were restricted to the study of the functional only, until the early 2010s the case N ≥ 2 remained terra incognita. The foundations of the general vector case have been laid in a series of recent papers of one of the authors ([K1]- [K6], [K8]- [K12]) and also in collaboration with Abugirda, Manfredi and Pryer ([AK, KP, KM]). An important special case of (1.1) to which the results of this paper apply is the archetypal model of the so-called ∞-Laplace system which arises by taking as H the square of the Euclidean norm of R N ×n : In index form, (1.5) becomes N β=1 n i,j=1 An additional difficulty in the study of the non-divergence system (1.1) which is not present in the scalar case of (1.4) is that the operator A ∞ has discontinuous coefficients, even when it is applied to C ∞ maps. Actually, even in the model case of (1.5), the example u(x 1 , x 2 ) = e ix1 − e ix2 is a 2 × 2 analytic ∞-Harmonic map near the origin of R 2 , but the rank of the gradient jumps from 2 off the diagonal {x 1 = x 2 } to 1 on it and [Du] ⊥ is discontinuous. The emergence of interfaces whereon the coefficients are discontinuous is a general phenomenon studied in some detail in [K1, K3, K4]. A manifestation of this fact is that in the genuine vectorial case of rank rk(Du) ≥ 2, the appropriate minimality notion connecting (1.1) to (1.3) is not the obvious extension of Aronsson's scalar notion of Absolute Minimisers (see [K2, K10, K11]). Perhaps the greatest difficulty associated to the study of (1.1) is that it is quasilinear, non-divergence and non-monotone and all standard approaches in order to define generalised solutions based on integration-by-parts or on the maximum principle seem to fail. Motivated partly by the systems arising in Calculus of Variations in L ∞ , the second author has recently proposed in [K9, K8] a new efficient theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable mappings to be admitted as solutions. This approach of the so-called D-solutions is duality-free and is based on the probabilistic representation of derivatives which do not exist classically. The tools is the weak* compactness of the difference quotients considered in the Young measures valued into a compactification of the "space of jets".
For the special case at hand of 2nd order systems like (1.1)-(1.5), the definition can be motivated as follows: let u be a strong a.e. W 2,∞ loc (Ω, R N ) solution to (1.6) F Du, D 2 u = 0, a.e. on Ω.
We are looking for a meaningly way to interpret the hessian rigorously for just W 1,∞ loc (Ω, R N ) because this is the natural regularity class for (1.1) arising from the L ∞ variational problem for (1.3). For reasons to become apparent in a moment, we restate (1.6) as: for any compactly supported Φ ∈ C c R N ×n 2 s , we have That is, we view the classical hessian map D 2 u as a probability-valued map Ω ⊆ R n −→ P R N ×n 2 s given by the Dirac measure at itself x → δ Du(x) . Further, we restate that D 2 u is the limit in measure of the difference quotients of the gradient D 1,h Du as h → 0 by writing The weak* convergence of (1.8) is in the Young measures valued into R N ×n 2 s , that is the set of weakly* measurable probability-valued maps Ω ⊆ R n −→ P R N ×n 2 s (for details see Section 2 that follows and [CFV, FG, V, P, FL]). The hope arising from (1.7)-(1.8) is that it might be possible for general probability-valued "diffuse hessians" to arise for just W 1,∞ loc maps which will no longer be the concentration masses δ D 2 u . This is indeed possible if we embed R N ×n 2 s into its 1-point Alexandroff compactification By considering (δ D 1,h Du ) h =0 as Young measures valued into R N ×n 2 s , subsequential weak* limits always do exist in the set of Young measures Ω ⊆ R n −→ P R N ×n 2 s and we may define generalised hessians as follows: Definition 1 (Diffuse Hessians, cf. [K8]). For any u ∈ W 1,∞ loc (Ω, R N ), we define the diffuse hessians D 2 u as the infinitesimal subsequential limits of difference quotients of Du in the Young measures valued into the sphere R N ×n 2 s : Obviously, where {e 1 , ..., e n } is the usual basis of R n . By the weak* compactness of the set Y Ω, R N ×n 2 s , every map possesses at least one diffuse hessian (and actually exactly one if it is twice differentiable in measureà la Ambrosio-Malý [AM], see [K8]).
Definition 2 (Lipschitzian D-solutions of 2nd order PDE systems, cf. [K8]). Let The notion of generalised solution given by Definitions 1 & 2 will be our central notion of solution for (1.1). By the motivation of the notion and the results of Section 2 it follows that it is compatible with strong and classical solutions (for more see [K8]- [K12]). More precisely, if a D-solution is twice differentiable a.e. on Ω, then (1.8) says that all diffuse derivatives coincide a.e. on Ω and are given by the Dirac measure δ D 2 u at the hessian. Hence, the integral formula (1.7) reduces to the definition of strong a.e. solutions.
The principal result of this paper is the existence of D-solutions to the Dirichlet problem for (1.1) when N ≤ n. These solutions have extra geometric properties, being W 1,∞ -submersions and solving a certain system of vectorial Hamilton-Jacobi equations associated to (1.1)-(1.3). Our only assumptions on H are that • H is non-negative, C 1 on R N ×n and C 2 on {P ∈ R N ×n : rk(P ) = N }, • H(P ) depends on P via P P ∈ R N ×N s , • rk(H P (P )) = N when rk(P ) = N .
In particular, we do not assume any kind of convexity of "BJW-quasiconvexity" or level-convexity for H, not even C 2 smoothness on the whole of R N ×n but merely on the open set of matrices of full rank. The notion of "BJW-quasiconvexity" introduced and studied in [BJW1,BJW2] is the correct notion for weak* lower semi-continuity of L ∞ functionals. Examples of H to which our results apply are given by H 1 (P ) = |AP | α for α > 1 and A ∈ R N ×N ≥0 and by H 2 (P ) = (|P | 2 + 1) −1 . The example H 2 does not even have rank-one convex sublevel sets. The solutions we construct are obtained by the celebrated Baire Category method of  and [D, DP]) for a certain 1st order differential inclusion relevant to (1.1) which we expound after the statement: Theorem 3 (Existence of D-solutions to the Dirichlet problem). Let N ≤ n and H : • the derivative h X (X) is symmetric and det(h X (X)) = 0 for X > 0.
Then, for any open Ω ⊆ R n and g ∈ W 1,∞ (Ω, R N ), the Dirichlet problem for (1.1) has an infinite set of D-solutions in the class of Lipschitz submersions Namely, there is an infinite set of u ∈ A such that for any Proposition 4 (Geometric properties of D-solutions to (1.10)). In the setting of Theorem 3, we also have the following extra properties for our D-solutions: for any there is an infinite set of solutions A c ⊆ A such that each u ∈ A c solves the coupled system of Hamilton-Jacobi equations Theorem 3 and Proposition 4 generalise one of the results of [K8] in which the the conclusions above were established in the special case n = N and H(P ) = |P | 2 , corresponding to the ∞-Laplacian (1.5).
Informally, the idea of the proofs of Theorem 3 and Proposition 4 is as follows. An inspection of (1.1) shows that it can be contracted as Hence, if we could prove for some C ≥ 0 existence of solutions to the differential inclusion we would obtain a solution to (1.12) (i.e. to (1.1)) because then D H(Du) ≡ 0 and also [H P (Du)] ⊥ since if H P (Du) has full rank the orthogonal complement of its range trivialises. However, the preceding arguments make sense only for classical or strong solutions. The proof of Theorem 3 has two main parts. We first use the Baire Category method to establish the existence of W 1,∞ strong a.e. solutions to system (1.11) of Corollary 4 and then we utilise the machinery of D-solutions to make the above heuristics rigorous.
Our main ingredient for the solvability of (1.11) is a result of independent interest about the solvability of the following fully non-linear system, usually referred to as the prescribed singular value problem: (1.13) λ i (Du) = 1, a.e. on Ω, i = 1, ..., n ∧ N, u = g, on ∂Ω.
In (1.13), {λ 1 (Du), ..., λ n∧N (Du)} denotes the set of singular values of the matrix Du, namely the eigenvalues of the matrix (Du Du) 1/2 in increasing order and we symbolise n ∧ N = min{n, N }. Systems of PDEs involving singular values, mostly related to non-convex problems in Calculus of Variations, have been considered by several authors (cf. for instance [Cr, BCR, DR, DPR]). In particular the problem of finding sufficient conditions on the boundary datum g in order to get existence of solutions to problem (1.13) has been addressed, in the special case n = N , in [DM1,DM2,DT]. To the best of our knowledge no results are known for the case n = N . Accordingly, we establish the following result.
The proof of the previous theorem can be obtained as an application of the general existence theory for differential inclusion via the Baire Category method (cf. [DP]), in the same spirit as in the N = n case. It relies on the characterisation of the rank-one convex envelope of the set of matrices as proved in Theorem 11.
In order to address the question of the existence of solutions to the problem (1.11) with g ∈ W 1,∞ (Ω, R N ), some comments on the admissible regularity of the boundary datum in Theorem 5 are in order. Indeed, the piecewise affinity of the datum g can be weakened to Lipschitz continuity if we restrict slightly the bound on the norm λ n∧N (Dg). This result, precisely stated in Corollary 6 that follows, is a simple consequence of the convexity of the rank-one convex hull of E (cf. Theorem 11) and of the approximation result proved in [DM2, Corollary 10.21].
Corollary 6. Let Ω ⊆ R n be an open set. Assume that g ∈ W 1,∞ (Ω, R N ) is such that for some δ > 0 we have λ n∧N (Dg) ≤ 1 − δ, a.e. on Ω. Then there exists a infinite set of solutions u ∈ W 1,∞ g (Ω, R N ) to the system (1.13).
The rest of the paper is organised as follows. In the next section we recall some known results about Young measures valued into spheres. In Section 3 we provide the proof of the existence of solutions for the prescribed singular value problem and in the last section we prove existence and geometric properties of D-solutions to the problem (1.10).

Young measures valued into spheres
Here we collect some basic material taken from [K8] which can be found in different guises and in greater generality e.g. in [CFV, FG]. Let E ⊆ R n be measurable and consider the L 1 space of strongly measurable maps valued in the continuous functions over the sphere R N ×n 2 s (for details on these spaces see e.g. [Ed, FL]): The Banach space C R N ×n 2 s will be endowed with the standard supremum norm. The above L 1 space consists of Carathéodory functions Φ : The dual of this (separable) Banach space is The dual space above consists of measure-valued maps x → ϑ(x) which are weakly* measurable, that is, for any fixed Ψ ∈ C R N ×n 2 s , the function where " · " symbolises the total variation on the real (signed) Radon measures. The closed unit ball of L ∞ w * is sequentially weakly* compact and the duality pairing Definition (Young Measures). The subset of the unit sphere of L ∞ w * consisting of probability-valued maps is the set of Young measures from E into the compactifica- We finally note the following well known facts (for their proofs see e.g. [FG]): (a) The set of Young measures above is convex and sequentially weakly* compact.

The prescribed singular value problem
In this section we prove Theorem 5 and Corollary 6. To this aim we start by recalling for the convenience of the reader some well known results about generalised convex hulls of sets of matrices (for further details we refer to the books (1) f is said to be polyconvex if there exists a convex function g : for every λ ∈ [0, 1] and every Q, R ∈ R N ×n with rk(R − Q) = 1.
It is well known that if a function is polyconvex, then it is rank one convex. Next we recall the corresponding notions of convexity for sets.
Definition 8. Let E be a subset of R N ×n .
(1) We say that E is polyconvex if there exists a convex set K ⊆ R τ (N,n) such that Q ∈ R N ×n : T (Q) ∈ K = E.
Definition 9. The polyconvex and rank one convex hulls of a set E ⊆ R N ×n are, respectively, the smallest polyconvex and rank one convex sets containing E and are, respectively, denoted by Pco E and Rco E.
Obviously one has the following inclusions: E ⊆ Rco E ⊆ Pco E ⊆ co E, where co E denotes the convex hull of E. Let us also recall the next characterisation of the rank one convex hull: where R 0 coE = E and R i+1 co E = Q ∈ R N ×n : It is well known that, for E ⊆ R N ×n , (cfr. [D, Proposition 2.36]) Analogous representations to (3.2) can be obtained in the polyconvex and rank one convex cases: For a matrix A ∈ R N ×n , we denote by λ i (A), with 0 ≤ λ i (A) ≤ λ i+1 (A), i = 1, . . . , N ∧ n, the singular values of A, that is, the eigenvalues of the matrix (A A) 1 2 ∈ R n×n . The following singular value decomposition theorem can be deduced from [HJ, Theorem 2.6.3].
Theorem 10. Let A ∈ R N ×n . Then there exist U ∈ R N ×N , V ∈ R n×n unitary matrices and D ∈ R N ×n rectangular diagonal matrix such that A = U DV t .
Using the singular values we can define the unitarily invariant norms on R N ×n (known as Ky Fan k-norms (cf. [HJ,Section 7.4

.8]) as
Choosing k = 1 we obtain the useful property of the maximum singular value λ n∧N (A) of being a norm in R N ×n .
As done in [DM1,Theorem 7.16] for the case n = N , in the next theorem we characterise the rank-one convex hull of the set It turns out that, in this particular case where the singular values are all equal, Rco E is indeed a convex set, as co E = Rco E = Pco E.
Theorem 11. Let E be defined by (3.3). Then co E = Pco E = Rco E, Proof. Let X := {Q ∈ R N ×n : λ n (Q) ≤ 1}. The claim will follow from the inclusions co E ⊆ X and X ⊆ Rco E. The first one is a direct consequence of the convexity of the function A → λ n (A) (that is convex being a norm) and of the characterisation of the convex envelope of a set given by (3.2). To prove the second inclusion, by the decomposition Theorem 10, it is enough to prove that a general rectangular diagonal matrix D ∈ X is in the rank-one convex envelope of the subset of rectangular diagonal matrices in E. The proof then is identical to that of [D,Theorem 7.17] where we have to consider the notation diag(a 1 , . . . , a n ) as a rectangular diagonal matrix made by a square diagonal block with entries (a 1 , . . . , a n ) and a (N − n) × n or (n − N ) × N zero block accordingly with the case N > n or n > N . The characterisation of the interior follows from the continuity of A → λ n (A) (λ n is a norm) and from the fact that if Q ∈ int Rco E then λ n (Q) = 1. The last claim can be easily proved by contradiction, assuming without loss of generality that Q is diagonal and perturbing it by considering Q + εZ where Z ∈ R N ×n is such that (Z) n∧N,n∧N = 1 and the other entries of Z are null. Then Q + εZ would not belong to Rco E, which is a contradiction.
To prove Theorem 5 we can apply the general existence theorem of [DP] (note that similar existence theorems have been obtaind by Kirchheim in [Ki1,Ki2] and by Müller andŠverák in [MS1, MS2]). To this aim we need the notion of Approximation Property defined below.
Definition 12 (Approximation Property). Let E ⊆ K (E) ⊆ R N ×n . The sets E and K (E) are said to have the approximation property if there exists a family of closed sets E δ and K (E δ (2) for every > 0 there exists δ 0 = δ 0 ( ) > 0 such that dist(Q, E) ≤ for every Q ∈ E δ and δ ∈ [0, δ 0 ]; (3) if Q ∈ int K (E) then Q ∈ K (E δ ) for every δ > 0 sufficiently small.
We therefore have the following result (cf. [DP, Theorem 7 and 9] and also [D, Section 10.2.1]).
Theorem 13. Let E ⊆ R N ×n be compact and suppose that there exist E δ and K (E δ ) = Rco E δ satisfying the approximation property with K = int Rco E. Let Ω ⊆ R n be open and g ∈ Aff piec Ω, R N be such that Then there exists (a dense set of ) u ∈ W 1,∞ g Ω, R N such that Du (x) ∈ E, a.e. x ∈ Ω. Dg (x) ∈ C, a.e. x ∈ Ω, where C ⊆ K is compact (cf. Corollary 10.21 in [DM1]).
Indeed it is easy to check that To check the second condition we observe that for any fixed Q ∈ E δ , then up to a multiplication by unitary matrices (see Theorem 10), we can assume that Q = (1 − δ)Ĩ, whereĨ is the rectangular identity matrix belonging to E. Therefore Q−Ĩ ≤ cδ. The third condition can be easily verified arguing as in the last part of the proof of Theorem 11. We can therefore prove Theorem 5 by applying Theorem 13. The claim of Corollary 6 can be proved by arguing as before and appealing to Remark 14.

D-solutions to the PDE system arising in vectorial Calculus of Variations in L ∞
In this section we establish the proofs of Theorem 3 and of Proposition 4 by utilising Corollary 6 which was proved in Section 3.
Proof of Proposition 4. We begin by showing the next consequence of the solvability of the singular value problem.
Next, by the Spectral Theorem, for each such u there is a measurable map with values in the orthogonal matrices One of the referees of this paper brought to our attention that the tensor Du Du is the so called Left-Cauchy-Green Tensor which appears in nonlinear elasticity.
a.e. on Ω. The claim thus ensues.
Now we complete the proof of the proposition. By our assumptions on H (see Theorem 3), for any u as above we have H(Du) = h Du Du = h c 2 I and by splitting the identity as where α, β = 1, ..., N and i, j = 1, ..., n. In order to show that u is a D-solution of (1.1), we must show that for any diffuse hessian of u as m → ∞, we have (4.6) for a.e. x ∈ Ω and any fixed Φ ∈ C c R N ×n 2 s . By (1.11), we have H Du(x) = const for a.e. x ∈ Ω. Fix such an x ∈ Ω, 0 < |h| < dist(x, ∂Ω) and i ∈ {1, ..., n}. Then, by Taylor's theorem, we have

This implies the identity
(4.7) Let now h m → 0 as m → ∞ be an infinitesimal sequence giving rise to a diffuse hessian as in (4.5). We define the "error" tensor  for a.e. x ∈ Ω, where α, β = 1, ..., N , i, j = 1, ..., n and m ∈ N. Then, in view of (4.8) and of the following consequence of ( a.e. on Ω and for all α = 1, ..., N . In view of definition (4.4), we rewrite (4.9) compactly as for a.e. x ∈ Ω. By multiplying by Φ(D 1,hm Du), this gives (4.10) for a.e. x ∈ Ω. Since Du ∈ L ∞ (Ω, R N ×n ), by the continuity of the translations in L 1 we have Du(· + he i ) −→ Du, i = 1, ..., n, as h → 0, in L 1 loc (Ω, R N ×n ) and hence along, if necessary, a further subsequence (m k ) ∞ 1 we have Du x + h m k e i −→ Du(x), for a.e. x ∈ Ω, as k → ∞. Since H ∈ C 1 (R N ×n ) and Du ∈ L ∞ (Ω, R N ×n ), the Dominated Convergence Theorem and (4.8) imply that (4.11) |E m k | −→ 0, in L 1 loc (Ω), as k → ∞. Let E ⊆ Ω be a compact set. Then, (4.10) gives (4.12) We define the Carathéodory maps which are elements of the Banach space L 1 E, C R N ×n 2 s by the compactness of the support of Φ and of the set E. We also have that which yields that the right hand side vanishes as k → ∞ as a consequence of(4.11). The weak*-strong continuity of the duality pairing (2.1), (4.13) and (4.5) allow us to pass to the limit as k → ∞ in (4.12) and deduce E R N ×n 2 s Φ(X) A ∞ Du(x) : X d D 2 u(x) (X) dx = 0.
Since E ⊆ Ω is an arbitrary compact set, the theorem ensues.

Remarks and open questions
In this final section we make some further comments regarding the results stated in Theorem 3, pointing also towards certain relevant open questions which we have not answered in this work.
• An interesting question which we not attempt to answer herein regards the relation of D-and viscosity solutions in the scalar case of N = 1, in particular in relation to the first order vectorial equation H(Du) = c. The content of Theorem 3 is that all W 1,∞ strong solutions to the first order system (1.11) are D-solutions to (1.1). This is a kind of generalisation of the scalar implication that "differentiable solutions to the Eikonal equation |Du| = c are ∞-Harmonic". However, we refrained from mentioning this explicitly earlier since, when restricted to the scalar case, the notion of D-solutions is generally weaker than that of viscosity solutions. In the recent paper [K10] some further insights regarding the behaviour of C 1 D-solutions and their relation to the variational problem are presented, even though they do not answer this particular question.
• A further interesting open question regards whether C 1 -regular D-solutions to the system are in some sense "critical points" to the functional. Clearly, C 1 is a very optimistic regularity expectation for putative minima or extrema of any kind and one might hope that this prominent class would have the most favourable properties. At present we do not have a definitive answer, primarily because there is no clear way of how to define an adequate notion of "critical points" for supremal functionals.
• We close by noting that one might be tempted to call the solutions we construct "critical points" of the functional, in the sense that they solve the equilibrium equations whilst not minimising the functional. However, the reader should note that our solutions are not critical points in the classical sense of Calculus of Variations for integral functionals. The situation for supremal functionals, even in the scalar case, is indeed far from being well understood. In particular, for supremal functionals, even global minimisers may not solve the equations! To the best of our knowledge, nowhere in the literature any kind of L ∞ -notion of critical points has been defined. The only existing variational notions are those of absolute minimisers and their vectorial generalisations (∞-minimal maps in [K2] and tight maps in [SS]).