A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,\xi)$ is quadratic and convex in $\xi$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,\xi)$, depending smoothly on $\xi$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$.

Assume that g is the trace of a map u 0 ∈ W 1,2 (B, R 2 ) that satisfies det ∇u 0 = 1 a.e. in B, so that the class is, in particular, nonempty.The constrained minimization problem that we study in this paper is then to find (5) min u∈A g E(u) in A g .Concrete instances of g for which A g is nonempty include: (a) g(x) := Ax, x ∈ ∂B, where A is any constant matrix in SL + (2, R), and (b) g(θ) := 1 Note that the latter is the trace of the so-called N −covering map expressed in plane polar coordinates, and where we employ the notation e R (θ) = (cos θ, sin θ).Definition 1.1.(Stationary point) We say that u is a stationary point of E(•) if there exists a function λ, which we shall henceforth refer to as a pressure, belonging to W 1,1 (B) and such that div (∇ ξ f (x, ∇u) + 2λ(x) cof ∇u) = 0 in D ′ (B).(7) The first main result we obtain shows that if u is a stationary point of the energy E whose corresponding pressure λ satisfies, in addition, the assumption that then u is a global minimizer of E. We think of the estimate (8) as characterizing 'smallness' of the pressure; concrete examples (such as can be found in [3,Proposition 3.4] or [5], for instance) show that such an estimate need not hold in general.In the following we assume that g has been chosen and fixed so that A g is non-empty.
Theorem 1.2 (Uniqueness under small pressure).Let the energy functional E(u) be given by (1), and let f (x, ξ) be given by (2), where M ∈ L ∞ (B, R 16 ) is symmetric and satisfies (3) for some ν > 0. Let u be a stationary point of E in the sense of (7) and assume that the corresponding pressure λ satisfies Then u is a global minimizer of E in A g .Moreover, if the inequality is strict, i.e. |∇λ(x)R| < In our second main result we provide an explicit integrand f (x, ξ) of the form (2) whose corresponding energy functional E is minimized in A g by the N −covering map u N .Here, g is the trace of u N as defined in (6).For its construction we make use of Theorem 1.2.A novelty of our approach is that, in order to apply Theorem 1.2, we develop a method to compute the corresponding pressure explicitly.

Theorem 1.3 (Counterexample to regularity). Let g be the trace of the
, where ν > 0. Then the following statements are true: 2 Here the norm is defined via and satisfies M (x) ≥ νId for any x ∈ B \ {0} and any N ∈ N \ {1}.
(ii) The maps x → M (x) and x → f (x, ξ), for any ξ ∈ R 2×2 \ {0}, are discontinuous at 0. (iii) The maps x → M (x) and x → f (x, ξ), for any ξ ∈ R 2×2 \ {0}, belong to W 1,q \ W 1,2 for any 1 ≤ q < 2 with the spaces (B, R 16 ) and (B) respectively.(iv) The map is a stationary point of E, as defined in (7), and the corresponding pressure λ exists and satisfies λ ∈ W 1,q (B) for any 1 ≤ q < 2. (v) Moreover, the map u N is the unique global minimizer of E in the class A g .(vi) The minimum energy is given by The problem of studying a functional of the form E(u) is of interest not least because the regularity and/or uniqueness of minimizers of such incompressible problems cannot necessarily be determined a priori.Concerning uniqueness in the compressible setting, works including but not limited to John [7], Knops and Stuart [10], Sivaloganathan [16], Zhang [22], and Sivaloganathan and Spector [17] provide conditions under which the uniqueness of a global minimizer can be expected.By contrast, a striking example given by Spadaro [18] clearly demonstrates that global minimizers need not be unique, even under full displacement boundary conditions.When the domain of integration, or reference configuration, is an annulus, a number of papers, including those of John [7], Post and Sivaloganathan [12], Taheri [20], and Morris and Taheri [11,21], address uniqueness.With the topology of the annulus at their disposal, a multiplicity of solutions/equilibria can be generated by working with certain homotopy classes.For example, Morris and Taheri [11,21] consider functionals of the form W (x, s, ξ) = F (|x| 2 , |s| 2 )|ξ| 2 /2, with F ∈ C 2 , on the annulus A and admissible maps A = W 1,2 id (A, R 2 ), and show that there are countably many solutions, with exactly one for each homotopy class.
In the homogeneous, incompressible elasticity setting, Knops and Stuart [10, Section 6] (see also [15]) show that the equilibirum solutions they consider are, when subject to affine displacement boundary conditions, global minimizers of the associated energy.Recent results [4] show that there are polyconvex energies with inhomogeneous integrands that, under pure displacement boundary conditions, possess countably many pairs of equal-energy stationary 3 Here the multiplication is understood through its action on ξ ∈ R 2×2 which is given by

points. It is an open question whether the lowest-energy pair of such stationary points represent global minimizers.
The regularity of equilibrium solutions or mininimzers in incompressible variational problems is, like its compressible counterpart, a delicate matter.Ball [1, §2.6] points out that any minimizer of the Dirichlet energy in the class W 1,2 (B; R 2 ), subject to the pointwise (incompressibility) constraint det ∇u = 1 a.e and boundary condition u(1, θ) = 1 √ 2 (cos(2θ), sin(2θ)), must fail to be C 1 .There is evidence to suggest that the double-covering map u 2 (see ( 6)) itself may be the global minimizer in that particular problem [3,5].A partial regularity result for Lipschitz minimizers that are subject to a type of monotonicity condition was established in [6], and Karakhanyan [8,9] proves that, in the case of the Dirichlet energy, bounds on the so-called dual pressure lead, by a sophisticated argument, to the conclusion that suitably defined equilibrium solutions must belong to the Hölder class C 1 2 loc .It is perhaps significant that Karakhanyan's results, like ours, also depend on 'pressure bounds', but more significant still that the maps he deals with are measure-preserving.The double-covering map u 2 mentioned above, and indeed the N −covering maps which form the basis of the counterexample to regularity in Section 3 of this paper, do not preserve L 2 −measure in the sense of [19, Eq.(24)], for example, and so are less relevant to physically realistic models of elasticity.
It seems that pressure regimes can be used to divide the sorts of incompressible problems we consider into two classes.The double-covering problem introduced by Ball appears to lie in the 'high pressure' regime 4 , whereas the problem we focus on falls, by design, into the 'low pressure' regime, where we can say a bit more.
Let v, u ∈ A g and suppose that u is a stationary point of E in the sense of (7).To compare E(v) and E(u) we set η = v − u and expand E(v) = E(u + η) as Our problem, as expressed in (5), is made more tractable by the observation made in [3] that the stationarity condition (7) allows us, at the expense of incorporating a pressure term, to rewrite the final, affine-in-∇η term in the expansion above as a term that is quadratic in ∇η, namely In particular, For the details, see (12) and the foregoing discussion.
Plan of the paper: The main purpose of Section 2 will be to prove the uniqueness result, Theorem 1.2.We begin by giving two technical lemmas, the first of which enables us to decompose certain expressions in terms of Fourier modes.Section 2 concludes with the proof of Theorem 1.2, together with an argument which shows that the prefactor 3 4 By which we mean that the pressure λ 2 , say, appearing in the equilibrium equations associated with u 2 obeys ||R ∇λ 2 ||∞ = 3ν, when adapted to the notation we use in this paper.The prefactor of ν in the latter exceeds the prefactor 3 2 appearing in the condition (9) of Theorem 1.2, which is why we refer to this as the 'high pressure' regime.
appearing in (9) can be replaced by 1 when λ depends on just one of the variables R, θ.See Corollary 2.3.The focus of Section 3 is Theorem 1.3.In order to obtain this result we first develop a method to compute the pressure explicitly: this is done for a quite general situation in Lemma 3.1, and then more concretely in Lemmata 3.2-3.3.
1.1.Notation.For a 2 × 2−matrix A the cofactor is given by (11) cof A = a 22 −a 21 −a 12 a 11 , and we define the matrix J via When ϕ is suitably differentiable, we recall that det ∇ϕ = Jϕ R • ϕ τ , where ϕ R and ϕ τ = 1 R ϕ θ are, respectively, the radial and angular derivatives of ϕ.We use L 2 to denote two-dimensional Lebesgue measure.For any k ∈ N\{0} and f : .
All other notation is either standard or is defined when it is used.

Uniqueness in the small pressure regime
To prove Theorem 1.2 we need two technical lemmas.The first contains basic properties of functions in the class W 1,1 (B) that satisfy R∇λ L ∞ (B,R 2 ) < ∞, and it relies on a standard Fourier decomposition which, when applied to η ∈ C ∞ (B, R 2 ), is given by: and, for any j ≥ 1, where For later use, we set η := η − η (0) .Lemma 2.1.Let λ ∈ W 1,1 (B) and assume that R∇λ L ∞ (B,R 2 ) < ∞.Then the following statements are true: iii) It holds Proof.i)-iv) For a proof of these points, see [3, Lem 3.2 and Prop 3.2].The argument given there still applies if λ depends on x instead of R. v) Assuming ϕ ∈ C ∞ c (B), a computation shows: The result follows by bringing the rightmost term to the left-hand side and dividing by two.Note, as a last step, that one needs to upgrade the above equation to hold not just for ϕ ∈ C ∞ c (B) but instead for all ϕ ∈ W 1,2 0 (B).This is slightly delicate because of the weak spaces involved: for a proof, see [3,Lem 3.2

.(iv)].
vi) This is a version of (v) in which we emphasise the way that the above expression depends on ϕ (0) .Again, we assume ϕ ∈ C ∞ c (B), and we start by noting ϕ, θ = φ, θ , hence, then the rightmost term is just together with the above we get The uniqueness condition will be of the form ∇λ(x)R L ∞ (B,R 2 ) ≤ C, for some constant C > 0 and where λ will be the corresponding pressure to some stationary point.A priori, the condition only guarantees the existence of ∇λ(x)R in a suitable space.In the next lemma we show that λ and ∇λ themselves exist in a suitable space, which, in particular, allows one to make use of the technical lemma above.
Lemma 2.2.Let µ : B → R be a function satisfying Proof.The proof is straightforward.Indeed, it holds that where the latter integrand is integrable for all 1 ≤ p < 2.
We are now in a position to prove the main statement of this section.

Proof of Theorem 1.2:
Let u ∈ A g be a stationary point with pressure λ, let v ∈ A g be arbitrary and set η := v − u ∈ W 1,2 0 (B, R 2 ).We start expanding the energy via Expanding the Jacobian of η and exploiting the fact that both u and v satisfy det ∇u = det ∇v = 1 a.e.yields det ∇η = −cof ∇u • ∇η a.e.By the latter identity and the fact that (u, λ) satisfies ( 7), H can be written as (12) H(u, η) = 2 Now by noting that the 0−mode is only a function of R, we get 2,R η1 ).Instead of just λ, θ on the right hand side of the latter equation we would like to have the full gradient of λ.This can be achieved by using the basic relations e θ • e θ = 1 and e R • e θ = 0 to obtain Arguing similarly for (II), and a short computation shows By Hölder's inequality we get 5 2 and a weighted Cauchy-Schwarz Inequality, we see Next we recall an elementary Fourier estimate (see, for instance, [3, Proof of Proposition 3.3]), which states that for any φ ∈ C ∞ (B) it holds (14 Applying the Cauchy-Schwarz inequality and ( 14), and then combining some of the norms yields L 2 (dx) 5 Since we are not only interested in a qualitative but rather a quantitative estimate, we need to specify which norm we pick on R 2 .For this the above Hölder estimate is given more carefully by This is the reason why we defined the norm of Making use of η, θ = η, θ , which is true since the zero-mode does not depend on θ, and η, Choosing a = √ 3 √ 2 and again combining norms gives where D(η) := ∇η 2 L 2 (dx) denotes the Dirichlet energy.This yields which, since M ξ • ξ ≥ ν|ξ| 2 for all ξ ∈ R 2 , completes the proof.The prefactor 9) is the best we have for general λ at the moment.If, however, circumstances are such that λ depends on only one of R, θ throughout B, then condition ( 9) can be replaced by the weaker assumption Corollary 2.3.Let the conditions of Theorem 1.2 be in force, but with (15) replacing (8), and assume that either λ(x) = λ(R) or λ(x) = λ(θ) for all x ∈ B. Then the conclusions of Theorem 1.2 hold.
Proof.(i) (λ(x) = λ(R).)This case is significantly simpler and one can argue more along the lines of the proof of [3,Prop.3.3].The reason is that in this case it still holds that where η = η − η (0) eliminating the 0−mode.Then applying of [3, Lemma 3.2.(iv)]yields Using Hölder's inequality, λ ′ (R)R L ∞ (0,1) ≤ ν, and Fourier estimate (14) we get (ii) extend g to a one-homogeneous function u(R, θ) := Rg(θ), and compute, in Lemma 3.1, a PDE which must be satisfied by both u and an associated λ in order that u is a stationary point of E in the sense of ( 7); (iii) fix g = tr (u N ) and construct, in Lemma 3.2, a suitable f (x, ξ) such that the PDE in step (ii) can be solved for λ, and (iv) verify that the small pressure condition stipulated in Corollary 2.3 is satisfied by λ, and hence that u N is the unique global minimizer of the associated energy E. We first examine conditions on M necessary for a to be a stationary point of E in the sense of (7).Notation: Recall the notation for 2d polar coordinates {e R , e θ } := {(cos θ, sin θ), (− sin θ, cos θ)}.
Additionally, we will use {e N R , e N θ } := {(cos(N θ), sin(N θ)), (− sin(N θ), cos(N θ))} for any N ∈ N.Moreover, we will use the notation M ijgk = (M (e i ⊗ e j )) • (g ⊗ e k ) for any combination of i, j, k ∈ {R, θ} and any map g ∈ R 2 .Especially, if g = e N l for some l ∈ {R, θ} we will use M ij(N l)k for short.
where g obeys Jg • g ′ = 1 a.e. in [0, 2π), and let u = Rg(θ) ∈ A g be a stationary point of the energy E as defined in (7).
Then there exists a corresponding pressure λ ∈ W k,p (B, R) and it satisfies the following system of equations a.e. in B : Proof.Let u = Rg(θ) ∈ A g be a stationary point.If there exists a corresponding pressure λ ∈ W 1,p then u is a solution of (17 For now, let us assume that λ ∈ W 1,p (B, R).In order to derive the system of equations above, we enter the explicit form of u and the representation η = (η • e R )e R + (η • e θ )e θ into the stationarity condition.By some further calculations, which are mainly integrations by parts, we obtain (16).In the last step of the proof we discuss the existence of λ ∈ W 1,p .
Step 3: Existence of the pressure λ ∈ W 1,p (B, R) : The equations above can be rewritten as where h = (h 1 , h 2 ).We know that div (λcof ∇u) ∈ L p (dx) iff h(M, g) ∈ L p ( dx R ), with obvious notation.Now consider h 1 (M, g) (the argument being similar for h 2 ) and define Then for h 11 ∈ L p ( dx R ) we need M ∈ L ∞ (dx) and g, g ′′ ∈ L p , which is true by assumption.Now, by Sobolev imbedding, we have W 2,p ֒→ W 1,∞ ([0, 2π), R 2 ), and hence, in order for h 12 ∈ L p ( dx R ), and bearing in mind that g, g ′ ∈ L ∞ , it is enough to require that ∇M ∈ L p (dx).This is exactly how we chose the classes for M and g.This guarantees the existence of div (λcof ∇u) = (cof ∇u)∇λ ∈ L p (dx).By further noting that g, g ′ ∈ L ∞ , it is immediate that ∇u ∈ L ∞ (dx), and since det ∇u = 1 a.e. in B, we may write ∇λ = (cof ∇u) T h(M, g) R ∈ L p (dx).
In particular, when M and g are specified, ∇λ is specified and it belongs to the class L p (B), reverting to the traditional notation.One can argue similarly for the higher integrability.
We now specify g = tr (u N ) and compute the pressure under the assumptions that M depends only on θ and is diagonal with respect to the basis of polar coordinates.with ν > 0 and α, β, γ, δ ≥ ν for any θ ∈ [0, 2π).Furthermore, suppose u = Rg(θ) ∈ A g is a stationary point of the energy E, as defined in (1).Then there exists a corresponding pressure λ ∈ W k,p (B) and it satisfies the following system of equations a.e. in B :