Regularity for a geometrically nonlinear flat Cosserat micropolar membrane shell with curvature

We consider the rigorously derived thin shell membrane $\Gamma$-limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation $m:\omega\subset{\mathbb R}^2\to{\mathbb R}^3$ and the orthogonal microrotation tensor field $R:\omega\subset{\mathbb R}^2\to SO(3)$. The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet type energy term $|DR|^2$. We use Rivi\`ere's regularity techniques of harmonic map type systems for our system which couples harmonic maps to $SO(3)$ with a linear equation for $m$. The additional coupling term in the harmonic map equation is of critical integrability and can only be handled because of its special structure.


Regularity background and setting of the problem
This paper is a contribution to the wide field of regularity theory of harmonic map type equations.Driven by the application to geometrically nonlinear flat Cosserat shell models, we extend known regularity results to a system that couples a harmonic map equation with another uniformly elliptic equation.The system we consider is of the form Div S(Dm, R) = 0, (1.1) 2) The unknown functions here are the midsurface deformation m ∈ W 1,2 (ω, R 3 ) and the microrotation R ∈ W 1,2 (ω, SO(3)), while ω ⊂ R 2 is a smooth domain.Moreover, there are functions Ω R ∈ L 2 (ω, (R 2 ) * ⊗ so(3)) and the force stress tensor S(Dm, R) ∈ L 2 (ω, R 3×2 ) involved, and • is some bilinear product explained later.
The function Ω R is the same that makes the harmonic map equation for harmonic mappings to SO(3) ⊂ R 3×3 .The theory of harmonic map equations of 2-dimensional domains (to any sufficiently smooth compact target manifold, here SO(3)) has a long history.It has been proven in 1948 by Morrey [44] that minimizing weakly harmonic maps are smooth.In 1981, Grüter [30] generalized that to conformal weakly harmonic maps, and then in 1984 Schoen [69] to stationary ones.The regularity proof for general weakly harmonic maps was then found in 1990 by Hélein [32] [33].(Note that in our case, the target mainfold SO(3) is a Lie group, and in this case the harmonic map problem has a lot of interesting extra stucture, many aspects of which are covered in Helein's book [34].)Later, in 2007, Rivière [63] revisited harmonic map type equations and asked for which Ω R all weak solutions of (1.3) on a two-dimensional domain are smooth.It turned out that Ω R need not come from the harmonic map equation (in which case it can be seen as the anti-symmetrized tensor derived from the second fundamental form of the target manifold), but for the regularity result only the skew-symmetry of Ω R is needed.This gave deeper insight in the structures necessary to have regularity results, and it is Rivière's philosophy that we rely upon.
It should be pointed out that, with Ω R and DR being in L 2 , the nonlinear term Ω R • DR in (1.3) is only in L 1 , and if it would not have any further structure, it would be difficult to start with any regularity theory, due to the lack of an L p -theory working for p = 1.But it turns out that the product Ω R • DR, after a suitable gauge transformation, is the sum of products of divergence-free vector fields and gradients in L 2 , which is known to be in the Hardy space H 1 rather than L 1 .This little bit of extra regularity is enough to perform regularity theory.Now let us have a look at our equation (1.2).Compared with (1.3), it has an extra term skew(Dm • S(Dm, R))R, and again, with DR ∈ L 2 , S(Dm, R) ∈ L 2 , and R ∈ L ∞ , this has only L 1 -integrability.But once more, DR is a gradient, and S(Dm, R) is divergence free due to equation (1.1).This time, we have the product of a gradient Dm, a divergence free vector field S(Dm, R), and a bounded function R. Based on a crucial estimate by Coifman, Lion, Meyer and Semmes [14], Rivière and Struwe [64] were able to handle such products in their work on partial regularity in dimensions ≥ 3.They encountered such products in the course of their proof for the equation (1.3) without any extra terms, and we can modify their arguments to handle our extra term from the coupling.The handling of the first equation, which is linear in m with some right-hand side, is easier, in principle.But we have to do the iteration procedure for both equations simultaneously in the proof of Hölder continuity, resulting in some technicalities.Once we have that, classical Schauder theory helps with the higher regularity of m, while for the second equation controlling the smoothness of R, we still need some machinery.

Engineering background and application
The Cosserat model is one of the best known generalized continuum models [13].It assumes that material points can undergo translation, described by the standard deformation map ϕ : U → R 3 and independent micro rotations described by the orthogonal tensor field R : U → SO (3), where U ⊂ R 3 describes the smooth reference configuration of the material.Therefore, the geometrically nonlinear Cosserat model induces immediately the Lie-group structure on the configuration space R 3 × SO (3).
Both fields are coupled in the assumed elastic energy W = W (Dϕ, R, DR) and the static Cosserat model appears as a two-field minimization problem which is automatically geometrically nonlinear due to the presence of the non-abelian rotation group SO (3).Material frame-indifference (objectivity) dictates left-invariance of the Lagrangian W under the action of SO(3) and material symmetry (here isotropy) implies right-invariance under action of SO (3).
In the early 20th century the Cosserat brothers E. and F. Cosserat introduced this model in its full geometrically nonlinear splendor [17] in a bold attempt to unify field theories embracing mechanics, optics and electrodynamics through a common principal of least action.They used the invariance of the energy under Euclidean transformations [15,4] to deduce the correct form of the energy and to derive the equations of balance of forces (variations w.r.t the deformation ϕ, the force-stress tensor may loose symmetry [56]) and balance of angular momentum (variations w.r.t.rotations R).The Cosserat brothers did not provide, however, any specific constitutive form of the energy since they were not interested in applications.
While the appearance of an additional rotational field R for describing the elastic response of bulk material is requiring getting used to, such an appearance is most natural in the case of shell-theory.There, the Frenet-Darboux trièdre [16] (trièdre caché in the terminology of the Cosserats, trihedron) naturally plays a role and it is no big step to assume that this orthogonal field is supposed to be kinematically independent of the former (trièdre mobile).Hence the Cosserat approach [16]; the independent rotation field R describes the rotations of the cross-sections of the shell (including in-plane drill rotations about the normal n m to the midsurface m) and these cross-sections are all allowed to shear with respect to the normal of the midsurface (Re 3 = n m ).
On this basis, very efficient ad-hoc Cosserat shell-models have been introduced, see e.g.[2,3].A special case of these shell models is the family of Reissner-Mindlin shells in which the in-plane rotations are discarded (no drill energy) [37] and one is left with a one director theory [39] 1 .Upon identifying/constraining the trièdre mobile with the trièdre caché (microrotation equals continuum rotation, Cosserat couple modulus µ c → ∞), canonical shell models of Kirchhoff-Love type emerge [27].However, engineers would often prefer the Cosserat shell models since these yield nonlinear balance equations of second order [67,72,73,35,62,7].
The precise derivation of Cosserat shell models may proceed in several different ways: integration of equilibrium equations through the thickness [18,62], direct modeling as a two-dimensional directed surface [2,3,29], or the derivation approach, which starts from a three-dimensional variational problem and introduces certain assumptions for the deformation behavior through the thickness.The second author has introduced this derivation procedure based on the geometrically nonlinear Cosserat model in his habilitation thesis [50,
In this method, one needs to choose an energy scaling regime and obtains typically either membrane or bending like theories [21,39,40,41] when starting from classical finite strain elasticity [21,22,23].However, the Γ-limit membrane model [39,40] has a serious shortcoming which is connected to the necessary relaxation step: it does not predict any resistance against compression and averages out the expected fine scale wrinkling response.The situation is strikingly different when starting from a three-dimensional Cosserat model, as done in [51].This is true since the bulk-Cosserat model already features a curvature term (derivatives of R) which "survives" the membrane scaling.The Cosserat membrane Γ-limit with remaining curvature effects can be used as an effective surrogate model to describe ultra thin graphen mono-layers.Graphen is the name given to a single atomic layer of carbon atoms tightly packed into a two-dimensional honeycomb lattice (see Figure 2).It can be wrapped up to form fullerenes, rolled into nanotubes [74] or stacked into graphite.It's stiffness properties are extreme.Such a graphen layer has resistance against in-plane stretch and curvature changes but it's thickness is so small, that a classical membrane-bending model (where the bending terms scale with h 3 while the membrane terms with h) is clearly insufficient.It is simply impossible to speak about the "thickness" of graphen in a classical continuum framework.Researchers then usually resort to introducing an "effective bending rigidity" in order to apply concepts from classical shell theory.This can be completely avoided in the Cosserat membrane model.
In this paper we will consider, for the first time, the challenging regularity questions for the flat shell Cosserat membrane Γ-limit.To the best knowledge of the authors, such a regularity investigation for the flat Cosserat membrane shell has never been undertaken.Two recent previous contributions consider the regularity issue for the geometrically isotropic nonlinear Cosserat bulk equations [24,42], both times restricting attention to the uni-constant Dirichlet curvature energy |DR| 2 , leading to a ∆R-term in the Euler-Lagrange equations and allowing the sophisticated techniques for harmonic map type systems to be used.
This paper is now structured as follows.After this introduction and the introduction of our notation, in Section 3 we will introduce the three-dimensional isotropic Cosserat model, together with a short discussion of suitable representations for the curvature term.Following, in Section 4, we briefly describe the dimensional descent towards a membrane shell, juxtaposing the result of the Γ-limit procedure and a formal engineering approach.In Section 5 we introduce the final two-dimensional Cosserat membrane shell model together with some pertinent notations and simplifications.The remainder of the paper is devoted to showing the interior Hölder regularity of these weak solutions.In the appendix we gather further useful calculations like the three-dimensional Euler-Lagrange equations in dislocation tensor format, we present a more engineering oriented derivation of the two-dimensional Euler-Lagrange equations and give a glimpse on a related Reissner-Figure 2: Deformed graphen mono atomic-layer resisting against in-plane stretches (membrane effects) and curvature.Classical continuum models are not any more suitable, since there is no tangible thickness, c.f. [74].Graphen is thought to be the strongest among all known materials.Nevertheless it is soft in the sense that it can be easily bent due to its one atom thin nature.
Mindlin model.Finally, we show some numerical experiments of the flat Cosserat membrane shell model in compression.

Notation
Let a, b ∈ R 3 .We denote the scalar product on R 3 with a, b R 3 and the associated vector norm with The set of real-valued 3×3 second order tensors is denoted by R 3×3 .The standard Euclidean scalar product on R 3×3 is given by X, Y R 3×3 = tr(XY T ), and the associated norm is |X| 2 = X, X R 3×3 .If 1 3 denotes the identity matrix in R 3×3 , we have tr(X) = X, 1 3 .For an arbitrary matrix X ∈ R 3×3 we define sym(X) = 1 2 (X + X T ) and skew(X) = 1 2 (X − X T ) as the symmetric and skew-symmetric parts, respectively and the trace free deviatoric part is defined as dev X = X − 1 n tr(X)1 n , for all X ∈ R n×n .We let Sym(n) and Sym + (n) denote the symmetric and positive definite symmetric tensors, respectively.The Lie-algebra of skewsymmetric matrices is denoted by so(3) := {X ∈ R 3×3 | X T = X} and the Lie-algebra of traceless tensors is defined by sl(3) := {X ∈ R 3×3 | tr(X) = 0}.We consider the orthogonal decomposition X = dev sym X + skew X + 1  3 tr(X) • 1 3 = sym X + skew X.The canonical identification of so(3) and R 3 is given by axl : so(3) → R 3 and its inverse Anti : R 3 → so(3).We note the following properties and

.2)
A matrix having the three column vectors R 1 , R 2 , R 3 will be written sometimes as The matrix Curl and matrix Div are defined row-wise as For ϕ ∈ C 1 (U, R 3 ) and for every vector (x, y, z) ∈ R 3 , we write The mapping m : ω ⊂ R 2 → R 3 will always denote the deformation of the midsurface ω and we write Moreover, we will use the notations where A 1 , A 2 may be numbers-, vector-, or matrix-valued functions on ω of the same type.Note that it is also customary to write Curl instead of Div ⊥ , but the latter underscores the symmetry of (D, Div) with (D ⊥ , Div ⊥ ), hence we reserve Curl for three-dimensional domains.We assume that h > 0 with h 1.The three-dimensional flat thin domain U h ⊂ R 3 is introduced as We also need to define the projection operator on the first two columns and the operator 3 Three-dimensional geometrically nonlinear isotropic Cosserat model The underlying three-dimensional isotropic Cosserat model can be described in terms of the standard deformation mapping ϕ : U ⊂ R 3 → R 3 and an additional orthogonal microrotation tensor R : The goal is to find a minimiser of the following isotropic energy The problem will be supplemented by Dirichlet boundary conditions for the deformation ϕ but the microrotations R can be left free.Here, µ > 0 is the standard elastic shear modulus, κ 3D = 3λ+2µ 3 > 0 is the three dimensional elastic bulk modulus (with λ the second elastic Lamé parameter) and µ c ≥ 0 is the so-called Cosserat couple modulus, a 1 , a 2 , a 3 are non-dimensional non-negative weights and L c > 0 is a characteristic length.The energy (3.1) is the most general isotropic quadratic representation for the Cosserat model in terms of the nonsymmetric Biot type stretch tensor U = R T Dϕ (first Cosserat deformation tensor [17]) and the curvature measure R T Curl R (physically linear, small strain, but geometrically nonlinear).We call the second order dislocation density tensor [10].Due to the orthogonality of dev sym, skew and tr(.)1 3 , the curvature energy provides a complete control of For example, we can express the uni-constant isotropic curvature term where we have used (4.4) and shows that (3.1) controls DR in L 2 (U, R 3×3×3 ).
In this setting, the minimization problem is strictly convex in the strain and curvature measures (U , α) but highly non-convex w.r.t (ϕ, R).Existence of minimizers for (3.1) with µ c > 0 has been shown first in [49], see also [19,43,53,49,38,10,51].The partial regularity of minimizers/statonary solutions is investigated in [24,42] under additional assumptions.Note also that in [24], the first author gives an example of a solution that exhibits a point singularity.
The Cosserat couple modulus µ c controls the deviation of the microrotation R from the continuum rotation polar(Dϕ) in the polar decomposition of Dϕ = polar(Dϕ) • Dϕ T Dϕ, cf.[59].
For µ c → ∞ the constraint R = polar(Dϕ) is generated and the model would turn into a Toupin couple stress model.

Connections to the Oseen-Frank energy in nematic liquid crystals
In nematic liquid crystals one considers the unit-director field n : U ⊂ R 3 → S 2 , minimizing the threeparameter frame-indifferent "curvature energy" [71] The uni-constant approximation 2 For this, we note the identity (see, [5] eq (2.5) and [1] eq (2.6)) valid for all sufficiently smooth vector fields v : The corresponding Euler-Lagrange equations for the uni-constant case are (see e.g.[1]) see equation (A.56) for a self-contained derivation.Since (3.8) and (3.9) are just the energy and Euler-Lagrange equation for harmonic maps to spheres, all regularity theorems for harmonic maps apply.In the three dimensional case, minimizers are smooth up to a discrete set of singularities.Stationary solutions have a co-dimension 1 singular set.In the two dimensional case, all weak solutions of (3.9) are smooth, see Section (1.1) for the literature on this.For K 1 , K 2 , K 3 positive and different, any minimizer to (3.6) is smooth except for a closed set of Hausdorff dimension strictly less than 1, cf. [31].Ball and Bedford [6] consider the sublinear regime |Dn| q , 1 < q < 2.
4 Dimensional descent towards a membrane model

Membrane Γ-limit
We are interested in a situation, where the reference configuration is flat with uniform shell thickness h > 0, i.e. the reference configuration is taken to be of the form (see Figure 3 ) The goal is to derive a limit two-dimensional problem, posed over the referential midsurface ω ⊂ R 2 , as Figure 3: Process of dimensional reduction.Flat reference configuration with height h and deformed configuration.
h → 0. This has been achieved in [58] based on Γ-convergence arguments and using the nonlinear membrane scaling.We say that the dimensionally reduced model is a membrane, since no dedicated bending terms appear in the problem.However, since the Cosserat model already includes curvature terms (those depending on space derivatives DR), these curvature terms "survive" in the Γ-limit procedure and scale with h, while canonical bending terms scale with h 3 .This sets the Cosserat membrane model apart from more canonical membrane models [54].
For the Γ-limit procedure it is useful to re-express the curvature energy from (3.1) in terms of the so-called second order wryness tensor [60,18] (second Cosserat deformation tensor [17]) 3 is skew-symmetric, we have the following relation [25,26,61] By using these formulas we note Now using (4.2), we obtain where a 1 = a 1 , a 2 = a 2 and a 3 = 4a 3 .Altogether we get with a 1 = b 1 > 0, a 2 = b 2 > 0 and b 3 = a1 3 + a 3 > 0. Thus, the variational problem (3.1) can be equivalently expressed as Applying the nonlinear scaling [20], allows to rewrite the problem on a domain ] with unit thickness in terms of properly scaled variables ϕ , R in (thickness) z-direction The descaled Γ-limit of E 3D h as h → 0 is then given by [51] where m : ω ⊂ R 2 → R 3 describes the deformation of the midsurface, R : ω ⊂ R 2 → SO(3) and where the matrix is in the form (see [27]) . Thus we can write the Γ-limit minimization problem as 3 If we assume that in the underlying Cosserat bulk curvature energy we have the uni-constant expression then the homogenized curvature energy is given by [10,20] 3 Note the four fold appearance of the harmonic mean H, i.e.

Alternative engineering ad-hoc dimensional descent
In [48] the three-dimensional Cosserat model has been reduced to a flat shell problem by proposing an engineering ansatz for the deformation ϕ and the microrotation R over the shell thickness.We let again m : ω ⊂ R 2 → R 3 denote the midsurface deformation, U := R T (Dm|R 3 ) the non-symmetric membrane stretch tensor and R : Since we are only interested in the membrane like response, we will neglect terms related to bending effects right away while keeping the curvature change 4 scaling with h.
The dimensionally reduced energy reads then [48, (4.5)] transverse shear energy Letting µ c → ∞ in the reduced membrane model implies on the on hand that R 3 = n m is normal to the midsurface m and on the other hand skew(R T (Dm|n m )) = 0 implies R = polar(Dm|n m ) (trièdre cacheé).
In contrast to the representation of the energy in (4.18) the rigorously derived Γ-limit membrane model [54] has the energy (see equation (4.14)) . Thus, the engineering formulation in (4.18) coincides with the membrane Γ-limit if and only if and 4 The missing Cosserat bending terms scaling with h 3 are of the type [48, (4.5)] and the uni-constant case would appear for µ = µc, λ = 0.
One can show that the latter implies for the engineering Poisson number ν := λ 2(µ + λ) the bound ν > − 1 2 (instead of ν > −1 for three-dimensional linear elasticity). 6  5 The two-dimensional Euler-Lagrange equations Henceforth, we skip all unnecessary material parameters in (4.18) in order to arrive at a compact representation.Again, we consider the midsurface deformation m : ω ⊂ R 2 → R 3 and the orthogonal microrotation tensor R : ω ⊂ R 2 → SO(3).We set h = 1 and assume the normalization µ Moreover, we set κ = 3κ hom 2 .Thus, the corresponding energy function describing the two-dimensional membrane shell problem is We assume µ, µ c , κ to be positive.Remember that we have defined a linear operator P : R 3×3 → R 3×3 by Using the mutual orthogonality of dev sym X, skew X, and (tr X)1 3 , we can write down the functional in a simplified form, it reads Now we are going to calculate the Euler-Lagrange equations for the dimensionally reduced problem based on E. The first variation of E in the direction of (ϑ, 0) : and the first variation in the direction of (0, Q) : (5.5) 5 In linear elasticity theory for the displacement u : U ⊂ R 3 → R 3 , the common bulk modulus κ appears in the form µ | dev sym Du| 2 + κ 2 tr(Du) 2 and not as µ | dev sym Du| 2 + κ 3 tr(Du) 2 , which would be more natural from the perspective of orthogonality of dev sym Du and tr(Du) • 1 3 .
6 2λ + µ > 0 and µ > 0 Using P * = P and P(X T ) = P(X) T such that P * P = P 2 , and observing π 12 (v 1 |v 2 |v 3 ) = (v 1 |v 2 ), we rewrite these as The pair of Euler-Lagrange equations then consists of and Note that it is not true that X T P 2 (X) = X T P * PX is symmetric for all matrices X; this is because P is not a matrix.Therefore, (Dm|0)P 2 (Dm|0) T R is not automatically orthogonal to T R SO(3).And this term, being formally only in L 1 due to Dm being in L 2 , makes the structure of the equation interesting, as explained in Section 1.1.
For readability, we introduce a product which shares aspects of scalar products and matrix multiplication.We define we rewrite the second term of (5.8) as Noting that the projection of any matrix This means that the pair of Euler-Lagrange equations (5.7)-(5.8)can be rewritten as Div S(Dm, R) = 0, (5.12) The latter is a relation rather than an equation, but we can rewrite it as an equation.In geometric analysis, this is usually done using the second fundamental form of SO( 3), but we present the calculation in a more elementary way.Our aim is to calculate the tangential part (∆R) of ∆R.
Differentiating R R T ≡ 1 3 gives Differentiating R T R ≡ 1 3 twice and summing over i, we find implying (5.16) For any fixed matrix R ∈ SO(3), we have T R SO(3) = R so(3), where so(3) is the space of skew-symmetric matrices in R 3×3 .The projections of any X ∈ R 3×3 to T R SO(3) or its orthogonal complement [T R SO(3)] T therefore are (5.17) Therefore, we can calculate the orthogonal component of ∆R as We have used (5.16) in the second "=", and (5.14) in the third.We now abbreviate and hence have Combining with the result of (5.13), we have calculated the tangential part of the left-hand side of (5.8) as and thus have derived the Euler-Lagrange equations in their final form.We summarize Div S(Dm, R) = 0, (5.20) where here Remark 5.1.In engineering language, (5.20) is the balance of forces, while 5.21 is the balance of angular momentum equation.The tensor is the non-symmetric Biot-type stress tensor (symmetric if µ c = 0), while is the first Piola-Kirchhoff type force-stress tensor.Note the analogy to the corresponding tensors in the 3D-Cosserat model presented in (A.7) and (A.8).

Regularity
The objective of this section is to prove our main theorem.

Hölder regularity
We observe that the last term in (5.21) is, up to "skew" and the harmless factor R, the product of a "gradient" Dm with a divergence-free quantity S(Dm, R), with both factors in L 2 .As we know from [14], such a product is in the Hardy space H 1 rather than just in L 1 , and we will use arguments from [64] that tell us how to handle the additional R factor.A standard source for the Hardy space H 1 is in Chapter III of Stein's book [70].Note that [64](see also [24]) is about harmonic maps in ≥ 3 dimensions, and it is Rivière's paper [63] about two-dimensional harmonic maps that is mostly the basis of what we are doing here.Schikorra [68] found some simplification to the arguments of [63] and [64], and the most accessible account of all these arguments to date is the textbook [28] which allows us to handle the Euler-Lagrange equation (5.21) quite flexibly.Note that our equation (5.21) is more general than the equations of the form ∆R − Ω • DR = 0 studied in those papers and the book [28], since we have the extra term −R skew(Dm • S(Dm, R)) of order 0 in R. We are lucky that we have the additional structure coming from S(Dm, R) being divergence-free, again implying that up to a bounded factor the extra term is in H 1 .Without that additional information, we would not know how to incorporate that into the existing regularity theory.
It will be crucial to use Morrey norms, at least locally.We say that u ∈ L p (U ) is in the Morrey space M p,s (U ) if Having this, we define the Morrey norm by u M p,s (U ) := [u] M p,s (U ) + u L p (U ) .We need the following lemmas.The first one is a special case of Lemma A.1 in [68], in the spirit of similar estimates from [14].This is where Hardy-BMO duality comes in as a hidden ingredient of our proof.Lemma 6.3.There is a constant C such that for all choices of x 0 ∈ R 2 , r > 0, and functions a ∈ with Div Γ = 0 in the weak sense on B r (x 0 ), we have Another one, due to Rivière [63] and Schikorra [68], can be found as a special case of Theorem 10.57 in [28].
We also need a version of the Hodge decomposition theorem.This one is a special case of [36, Corollary 10.5.1], adapted from the differential forms version to 2-dimensional vector calculus as in [28,Corollary 10.70] Lemma 6.5.Let p ∈ (1, ∞).On B r (x 0 ) ⊂ R 2 , every 1-form V ∈ L p (B r (x 0 ), (R 2 ) * ) can be decomposed uniquely as where α ∈ W 1,p (B r (x 0 )), β ∈ W 1,p 0 (B r (x 0 )), and h ∈ C ∞ (B r (x 0 ), (R 2 ) * ) is harmonic.Moreover, there is a constant C depending only on p, such that (6.4) We now start our regularity proof.Our first step is local Hölder continuity.
Proof.We write B ρ for any ball B ρ (x 0 ) ⊂ ω.We assume r to be small enough such that B 2r (x 0 ) ⊂ ω.We will collect more smallness conditions on r during the proof.
We choose G according to Lemma 6.4 and find, abbreviating Now we Hodge-decompose G −1 DR according to Lemma 6.5.We find almost everywhere in B r .Using the well-known relations Div D = Div ⊥ D ⊥ = ∆ and Div D ⊥ = Div ⊥ D = 0, we calculate and for any constant R 0 ∈ R 3×3 (not necessarily a rotation).Both terms on the right-hand side, multiplied with some ϕ ∈ W 1,3 0 (B r , R 3×3 ), can be estimated using Lemma 6.3.Choosing a
We let Combining the duality of L 3/2 and L 3 and (6.7) with (6.9) and (6.10), we find, using ϕ = 0 on ∂B r , Here, in the second "≤", we have used Lemma 6.3.And in the fourth "≤", we have used Using (6.8), we can also estimate the L 3/2 -norm of D ⊥ g.We find This time, we have used DG L 2 (Br) ≤ 3 Ω R L 2 (Br) ≤ 3 ε, and the Sobolev embedding W 1,2/3 → L 6 for R.
For h, being harmonic, we have the standard estimate for any 0 < ρ < r.From (6.6), and then (6.11) and (6.12), we hence infer Now we are going to derive a similar estimate for Dm L 3/2 .Hodge-decompose S(Dm, R), i.e.
with α ∈ W 1,2 0 (B r , R 3×2 ), and χ ∈ W 1,2 (B r , L(R 2 , R 3×2 )) harmonic.This time, there is no term of the form Dζ, since Div of the left-hand side is 0. This would imply that ζ is harmonic, and so would be Dζ, which hence can be absorbed into χ.We have, abbreviating P R for the linear mapping ξ Using the same ideas as before, and defining Proceeding exactly as above, we find In order to do so, we have used We divide (6.14) and (6.18) by ρ 1/3 and combine them into We now assume r ≤ ε, where ε > 0 is yet to be determined.For formal reasons, we also add ρ on both sides, which gives for some suitable constant C 0 .Now we fix ρ := r 12C0 and ε := (12C 0 ) −4/3 , making C 0 ( ρ r + (ε + r)( r ρ ) 1/3 ) = 1 6 .Abbreviating θ := 1 12C0 , we thus have This holds for all B θr (x 0 ) and B 2r (x 0 ) ⊂ ω which share the same center x 0 .But clearly, we can replace B 2r (x 0 ) with any ball B s (y 0 ) ⊃ B 2r (x 0 ) which is still in ω.All smallness assumptions made so far for B r (X 0 ) will now also be assumed for s, that is s ≤ ε, Ω R L 2 (Bs(y0)) ≤ ε, and Dm L 2 (Bs(y0)) ≤ ε.We then have which is valid for all r, s, x 0 , y 0 such that B 2r (x 0 ) ⊂ B s (y 0 ) ⊂ ω.Then the B θρ (x 0 ) cover all of B θs/2 (y 0 ).Hence, on the left-hand-side, we can take the infimum over all feasible r and x 0 , and find  With p = 3 2 , n = 2, the last estimate (6.24) and Lemma 6.7 imply R, m ∈ C 0,β loc (ω), which is the Hölder regularity asserted in Proposition 6.6.Remark 6.8.It is essential that we are working in the critical dimension n = 2 here, even though this may not be too obvious in the preceding proof which uses methods developed for supercritical dimensions.But the arithmetic of the exponents crucially uses n = 2.In particuar, Lemma 6.3 for n > 2 is only available with exponents adding up to n instead of ( 3 2 , 1 2 ).But we would not succeed in finding similarly good estimates in the corresponding Morrey spaces.

Higher regularity
In this subsection, we are going to complete the proof of Theorem 6.1.
Proof.Remember we have the equations Div S(Dm, R) = 0, (6.25) where for ξ ∈ R 3×2 we have defined and Abbreviating L R (ξ) := π 12 (2 R P 2 (R T (ξ, 0))), we rewrite the first equation (6.25) as For every R ∈ SO(3), L R : R 3×2 → R 3×2 is a linear mapping satisfying the Legendre condition (uniform positivity) because of where here λ := min{µ, µ c , κ} is independent of R, hence we have a uniformly elliptic operator m → Div L R (Dm).For this operator, classical Schauder theory applies once it depends Hölder continuously on x through R(x).And it does, because we already know R ∈ C 0,β loc for some β > 0. We use the following version of Schauder theory.The proof is well known, a good reference is [28,Theorem 5.19] which reads as follows.Lemma 6.9.Let u ∈ W 1,2 loc (U, R m ) be a solution to with A satisfying the Legendre-Hadamard condition and having its components A αβ ij in C 0,σ loc (U) for some σ ∈ (0, 1).If F α i ∈ C 0,σ loc (U), then also Du is of class C 0,σ loc (U).From what was proven in the last section, we know that both L R and the right-hand side of (6.27) are in C 0,β loc locally, hence Lemma 6.9 implies that m ∈ C 1,β loc for some β > 0. This simplifies the discussion of the regularity of R, because the Dm-terms in the equation for ∆R are now locally bounded.We can therefore rewrite it as ∆R + a(x, DR) = 0 , ( where the function a depends on R and Dm additionally, but those are locally bounded.The function satisfies Since DR ∈ L 2 , this means that ∆R is in L 1 , but L 1 is just not enough to perform regularity theory for R.However, the structure of the equation almost allows to apply the higher regularity theory for harmonic maps, where we could deal with C|DR| 2 instead of C(|DR| 2 + 1) on the right-hand side.A simple formal trick will care for that condition.Let with values in SO(3) × R.Then, letting ã(x, Du) := (a(x, Du 0 ), 0), we have where here Now we can follow the regularity theory for harmonic maps for a while.Note that in [45], Lemma 3.7 and Proposition 3.2 assume u to be a harmonic map, but the proof uses only |∆u| ≤ C|Du| 2 instead of the full harmonic map equation.We therefore can apply Lemmas 3.6 and 3.7, Proposition 3.2 in [45] to our u and find that Du ∈ L ∞ loc .This means that the second term in (6.29) is in L p loc for all p > 1, and standard L p -theory gives us u ∈ W 2,p loc for all p > 1.The Sobolev embedding W 1,p → C 0,1−2/p for p > 2 then gives us Du ∈ C 0,β loc with β > 0. Together with the result for m, we now have Once we have this, we can iterate the Schauder estimates, i.e. differentiate the equations and apply Lemma 6.9 to partial derivatives of m, R instead of m and R alone.Thus we find that (m, R) ∈ C k,β loc for our β > 0 and all k ∈ N, which means we have proven that m and R are smooth on the interior of the domain.

Body forces
It is physically reasonable to consider the equations with an additional external body force term in the first equation of balance of forces, Div S(Dm, R) = f, (6.34) . By integrating in one direction and setting we can always assume f = Div F. Note that F depends on the first component (x 0 ) 1 of the center of the ball B r (x 0 ) on which we are momentarily working.We have F ∈ W 1,2 (ω, R 3×2 ), implying F ∈ L p (ω, R 3×2 ) for all p ∈ [1, ∞).Now we may rewrite the first equation as Div(S(Dm, R) − F) = 0. (6.36) We will need to estimate DF, which we calculate via Since we can always assume r ≤ 1, we have proven The regularity theory for the more general equation including forces goes pretty much along the lines of the F = 0 case presented in Section 6.1.We only indicate the necessary modifications.We rewrite (6.7) as In (6.11), we replace S(Dm, R) L 2 (Br) by S(Dm, R) − F L 2 (Br) .Choosing the radius of B r sufficiently small, we can also assume that F L 2 (Br) ≤ ε, hence we can estimate S(Dm, R) − F L 2 (Br) by C(ε + r) just as we did for S(Dm, R) L 2 (Br) in (6.11).But we also have an additional term on the right-hand side of that estimate.Using the boundedness of G −1 and R, it is estimated as follows, assuming also F L 3 (Br) ≤ ε.We have which can be absorbed in the right-hand side of (6.11).Hence the conclusion of (6.11) continues to hold also in the F = 0 case.
The second modification we have to make is that we now Hodge-decompose S(Dm, R) − F, which means The additional term involving F on the right-hand side of (6.16) can be written as −Div ⊥ (F −F Br ).In (6.17), (F − F Br ) can be processed exactly like (R − R B R ), resulting in an additional Cr Dψ L 3 (Br) DF L 3/2 (Br) , which can be estimated using (6.38) and ψ ∈ U as follows, making the additional smallness assumption This additional term in (6.17) now contributes to the right-hand side of (6.18), but here enlarges only the r 4/3 and ρ 4/3 terms that are there, anyway.By the same argument, taking F into account also contributes only to more of s 4/3 terms in which updates (6.19).Hence the contributions of the modified versions of both (6.17) and (6.19) do not change the conclusion of (6.18).Now that we have adapted (6.11) and (6.18) to nonvanishing body forces, we can conclude Hölder continuity just as in the end of Section 6.1, under the weak assumption of f being in W 1,2 .If we assume f ∈ C ∞ instead, both f and F are bounded, and the higher regularity proof from Section 6.2 goes through with hardly any modification.Note, for example, that (6.30) continues to hold.

Remarks on a special case
Our system simplifies considerably when µ = µ c = κ, which makes P the identity8 .Even though this assumption is not too natural from the point of applications, we would like to comment briefly on that case.
The simplified variational functional reads now which has the Euler-Lagrange equations (c.f.(5.7)) and The point here is that the last term in the second equation now depends on Dm only linearly, making it an L 2 -term instead of L 1 (the L 1 -part is cancelled by the skew-operator).But harmonic map type equations with a right-hand side in L 2 have been studied by Moser in quite some generality, see the book [45] for an excellent exposition of the methods.
In particular, Moser has two theorems that help us here.Here, N ⊂ R n is a compact manifold, U ⊂ R d a domain, and II is the second findamental form of the target manifold, which corresponds to our term quadratic in DR, i.e. loc ∩ W 1,4 loc (U \ Σ, N ).While those theorems are highly nontrivial, it is standard to deduce regularity of the solutions to our model in the special case considered here.Theorem 6.12 (interior regularity for µ = µ c = κ).Any solution (m, R) ∈ W 1,2 (ω, R 3 × SO(3)) of the simplified problem (6.45)-(6.46) is smooth on the interior of the domain ω.

Conclusion and open problems
We have deduced interior Hölder regularity for a Dirichlet type geometrically nonlinear Cosserat flat membrane shell.The model is objective and isotropic but highly nonconvex.Therefore, our regularity result is astonishing and shows again the great versatility of the Cosserat approach compared to other more classical models.At present, we are limited to treating the uni-constant curvature case |DR| 2 since only then can sophisticated methods for harmonic functions with values in SO(3) be employed.This calls for more effort of researchers to generalize the foregoing.Progress in this direction would also allow to consider the full Cosserat membrane-bending flat shell [48,49,50,9].Another case warrants further attention: taking the Cosserat couple modulus µ c = 0 in the model (in-plane drill allowed, but no energy connected to it) may still allow for regular minimizers.However, even the existence of minimizers remains unclear at present since it hinges on some sort of a priori regularity for the rotation field R (the non-quadratic curvature term |DR| 2+ε , ε > 0, together with zero Cosserat couple modulus µ c = 0 allows for minimizers [52,47]).Finally, it is interesting to understand regularity properties of Cosserat shell models with curved initial geometry [25,26,27].
We expect some boundary regularity to hold, too.On the geometric analysis side, an adaptation of Rivière's boundary methods to problems with continuous Dirichlet boundary data has been performed in [46], which one could try to use.But with a view on applications, partially free boundary problems would probably be more interesting.

A.4 A glimpse on a Reissner-Mindlin type flat membrane shell model
It is interesting to compare our Cosserat flat membrane shell model (allowing for existence of minimizers and their full regularity) with one that would appear closer to classical approaches.For this sake we consider a Reissner-Mindlin flat membrane shell model next.
In case of the one-director geometrically nonlinear, physically linear Reissner-Mindlin flat membrane shell model without independent drilling rotations, the problem can be described as a two-field minimization for the midsurface m : ω ⊂ R 2 → R 3 and the unit-director field d : ω ⊂ R 2 → S 2 of the elastic energy  (A.39) Here, the membrane energy part is not rank-one elliptic due to the presence of the membrane strain Dm T Dm − 1 2 .The uniconstant curvature energy could be generalized to the Oseen-Frank form, cf.subsection 3.1.We note that d , ∂xm 2 + d , ∂ym 2 = |Dm T d| 2 R 2 , and look for simplicity at the energy

Figure 1 :
Figure 1: The mapping m : ω ⊂ R 2 → R 3 describes the deformation of the flat midsurface ω ⊂ R 2 .The Frenet-Darboux frame (in blue, trièdre caché) is tangent to the midsurface.The independent frame mapped by R ∈ SO(3) is the trièdre mobile (in red, not necessary tangent to the midsurface).Both fields m and R are coupled in the variational problem.
i II(u)(∂ i u, ∂ i u) = Ω u • Du in our case.Theorem 6.10.[45, Theorem 4.1] Suppose u ∈ W 1,2 (U, N ) is a stationary solution of ∆u − i II(u)(∂ i u, ∂ i u) = f , in U, for a function f ∈ L p (U, R n ),where p > d 2 and p ≥ 2. Then there exists a relatively closed set Σ ⊂ U of vanishing (d − 2)-dimensional Hausdorff measure, such that u ∈ C 0,α loc (U \ Σ, N ) for a number α > 0 that depends only on m, N , and p. Theorem 6.11.[45, Theorem 4.2] Under the assumptions of the previous theorem, if n ≤ 4 and p = 2, we also have u ∈ W 2,2