Jacques Sylvain

Lcts

CERAMIX MATRIX MICROCOMPOSITES PREPARED BY P-RCVD WITHIN THE (Ti-Si-B-C) SYSTEM

Nanoscale carbide multilayered interphases were deposited within the (Ti-Si-B-C) system by pressure-Pulsed Reactive Chemical Vapor Deposition (P-RCVD) on single filament Hi-Nicalon fibers and embedded in a SiC matrix sheath. The Reactive method, in which the titanium-containing layer growth involves the consumption of the pre-deposited Si-B-C sub-layer, allowed TiC-and TiB2based films to be obtained with a porous multilayer microstructure as a result of the Kirkendall effect. A first difficulty relied on the protection of the fiber surface which was very sensitive to chemical attack by P-RCVD. This difficulty could be circumvented through a first deposited SiC sub-layer thick enough to protect the surface of the fiber. But, because the porosity volume fraction was still not high enough, the role of mechanical fuse of these pyrocarbon-free interphases could not be evidenced from the microcomposite tensile curves, which remained fully linear up to the failure. Finally, the P-RCVD method was applied to the matrix processing itself. Microcomposites, this time with a pyrocarbon interphase but also with new matrix materials such as Ti3SiC2, were prepared and characterized.

INTRODUCTION

In Ceramic Matrix Composites (CMCs) reinforced by continuous fibers, despite the brittleness of the ceramic, a good toughness can be achieved by adding between the fiber and the matrix a thin film of a compliant material called "interphase". [START_REF] Evans | The physics and mechanics of fibre-reinforced brittle matrix composites[END_REF] Pyrocarbon (PyC) is the material commonly used to play this role, but its poor oxidation resistance is responsible for the loss of strength of these composites when used at high temperature in oxidizing atmosphere. [START_REF] Tressler | Recent developments in fibers and interphases for high temperature ceramic matrix composites[END_REF] The use of interphases only made of carbide and/or boride would allow CMC properties to be maintained in severe conditions. The problem of boro-carbide brittleness and the achievement of a mechanical fuse could be overcome by introducing porosity in the film as proposed by Carpenter et al. [START_REF] Carpenter | Fibers coatings for ceramic matrix composites[END_REF] or by laminating the carbide material itself at the nanometer scale.

A way to process nanoscale multilayered coatings is the use of pressure-Pulsed Chemical Vapor Deposition (P-CVD), a method adapted from CVD. [START_REF] Naslain | The design of the fibre-matrix interfacial zone in ceramic matrix composites[END_REF][START_REF] Naslain | Synthesis of highly tailored ceramic matrix composites by pressure-pulsed CVI[END_REF] Another way is the use of pressure-Pulsed Reactive CVD (P-RCVD) adapted from the RCVD method in which a part of the elements of the deposited coating comes from the substrate. [START_REF] Rapaud | SiC/SiC minicomposites with (PyC/TiC)n interphases processed by pressure-pulsed reactive CVI[END_REF] The first aim of the present contribution was to study new porous and multilayered PyC-free coatings obtained by P-RCVD from a H2/TiCl4 gaseous mixture not reacting with classical initial pure PyC sub-layers but with SiC-PyC, i.e. SiC sub-layers containing PyC, or Si-B-C sub-layers similar to the ones studied by Pallier et al. 7 The second aim was to take advantage of the possibilities offered by the P-RCVD method to prepare and study pure Ti3SiC2 8 as a matrix material that could be an interesting component of CMCs thanks to its damage tolerant properties. [START_REF] Spencer | The reactivity of Ti2AlC and Ti3SiC2 with SiC fibers and powders up to temperatures of 1550°C[END_REF]

EXPERIMENTAL

The experimental study was carried out by using microcomposites. A microcomposite is a model composite consisting of one single fiber coated with the interphase and embedded in the matrix. It represents the elementary cell of the composite. [START_REF] Naslain | Micro/minicomposites: a useful approach to the design and development of non-oxide CMCs[END_REF] The coatings were deposited on Hi-Nicalon single fibers (~14 µm in diameter, from Nippon Carbon, Japan) by P-CVD and P-RCVD at a temperature close to 1373 K in the same CVD device as Jouanny et al. [START_REF] Jouanny | Synthesis of TiC from porous carbon coating on Si-C-O (Nicalon) fibres by reactive chemical vapour deposition in pressure-pulsed mode or at atmospheric pressure[END_REF] When used, the PyC interphase with a thickness of ~100 nm was deposited by P-CVD from C3H8. SiC or Si-B-C sublayers were deposited by P-CVD from a H2/MTS or H2/BCl3/MTS gaseous mixture (MTS: CH3Cl3Si). The TiC porous interphases were obtained by P-RCVD from SiC sub-layers in which some PyC was added with on pulse of propane prior to total RCVD consumption. The number of H2/TiCl4 gas pulses was adjusted so to totally consume the previously deposited SiC-PyC sub-layers. The carbon element of the initial SiC-PyC sub-layers is involved in the TiC growth whereas the silicon is evacuated in the gaseous phase in the form of by-product species. The multilayered coating deposited within the (Ti-Si-B-C) system was obtained from the partial consumption of the Si-B-C sub-layers by the H2/TiCl4 gas pulses. Finally, the Ti3SiC2 coatings were deposited by P-RCVD by total consumption of previously deposited SiC sub-layers and with a higher dilution rate of TiCl4 in H2 than in the case of the TiC growth as explained by Jacques et al. [START_REF] Jacques | Reactive chemical vapor deposition of Ti3SiC2 with and without pressure pulses: Effect on the ternary carbide texture[END_REF] Each kind of microcomposite is referred to as "fiber(f)/interphase/matrix(m)".

For the mechanical characterization, about fifteen microcomposites per batch were tensile tested in a monotonic way according to a single fiber test procedure. [START_REF] Patankar | Weibull distribution as applied to ceramic fibres[END_REF] Before the tensile testing, the diameter of each microcomposite was measured by a laser diffraction technique. [START_REF] Perry | Fibre diameter measurement by laser diffraction[END_REF] The accuracy of the diameter measurement was ± 0.5 µm. The stresses at failure were determined with an accuracy of ± 5 % by dividing the load at failure by the section of the specimen. For each batch of microcomposites, the given mechanical properties were arithmetic means. The microcomposites were observed by scanning electron microscopy (SEM) using backscattered electron (BSE) and secondary electron (SE) detectors. Chemical analyses of coatings were carried out by Auger electron spectroscopy (AES-VG Microlab 310F spectrometer) with simultaneous Ar + etching in order to obtain the concentration depth profiles of boron, carbon, titanium, oxygen and silicon. Coatings were also characterized by X-ray diffraction (XRD, Bruker D8, λCu-Kα1 = 0.154056 nm). In that case, the coating was deposited on a graphite flat substrate placed in the CVD reactor in addition to the fibers.

RESULTS AND DISCUSSION

Microcomposites without PyC interphase TiC porous interphases

Figure 1.a shows a porous TiC interphase deposited directly on the Hi-Nicalon fiber with ten sequences of SiC-PyC-P-CVD/TiC-P-RCVD and then embedded in a 1.5 µm thick SiC matrix. The AES chemical analyses of such coatings (not shown here) only reveal the sole presence of titanium and carbon, silicon being not detected, which therefore confirms the total consumption of the SiC-PyC sub-layers. Although it is made of TiC only, the coating exhibits a multilayer-like microstructure, each TiC sub-layers having a thickness of about 100 nm. Lined up pores can be observed between the dense TiC sub-layers. These pores were produced by the Kirkendall effect based on nonreciprocal mutual solid-state diffusion process through an interface of two materials, i.e. carbon and silicon versus titanium atom diffusion between SiC and TiC. The roughness at the fiber surface shows that the fiber has been slightly attacked by the titanium containing species during the R-CVD process. The damaging of the fiber reinforcement led to an average tensile stress at failure of the SiC(f)/porousTiC/SiC(m) microcomposites of only 800 MPa (table 1) with a brittle behavior (figure 2.d).

In order to avoid the fiber attack, a SiC sub-layer was first deposited by P-CVD at the fiber surface with a relatively large number of H2/MTS pulses. A large part of this SiC protective sublayer, about 150 nm, is still present at the fiber surface. Consequently the fiber was not attacked as evidenced by the smooth surface observed by SEM (figure 1.b). These last SiC(f)/SiC-porousTiC/SiC(m) microcomposites exhibit an average tensile stress at failure of 1070 MPa (table 1). But despite the presence of porosity, no damageable behavior is observed in the tensile curves, which remain fully linear (figure 2.e). Therefore, the volume fraction of the porosity in this carbide interphase is still not high enough to deflect the matrix cracks according to the porous coating concept proposed by Carpenter et al.. (SiBC-TiBC)n interphase Figure 3 shows the fracture surface of microcomposites with (SiBC-TiBC)n interphase after tensile test. On the BSE image (figure 3.a) the Si-B-C sub-layers appear in grey whereas the Ti-B-C sub-layers resulting from the attack of Si-B-C by H2/TiCl4 gas pulses appear in white because of the presence of the heavier titanium element. High magnification observation shows that these ex-P-RCVD sub-layers are dual-layers separated by a row of pores in the middle. As the initial Si-B-C sub-layers were only partially consumed, the Hi-Nicalon fiber was not attacked. The average stress at failure of the SiC(f)/(SiBC-TiBC)n/SiC(m) microcomposites could be as high as 1200 MPa. Some delamination in the multilayered interphase and between the interphase and the matrix can be observed on the fractures surfaces (figure 3.b) but no composite behavior is evidenced on the tensile curves (figure 2.f). The interfaces remain too strong to efficiently deflect the matrix cracks.

Microcomposites with a PyC interphase

As the (SiBC-TiBC)n coating could not fully play the role of an interphase on its own, a similar coating was tested as a part of the matrix and as a complement of a classical PyC interphase (figure 4.a). The XRD pattern reveals the presence of TiB2 and TiC (not shown here). Figure 5 shows the AES concentration depth profile in the microcomposite matrix. From these different results, it appears that the inner side of each Ti-B-C dual-layer contains TiB2 and the outer side contains TiC, both sides being separated by pores. As in the case of the RCVD single layer TiC growth from SiC, silicon from Si-C-B is lost in the gaseous by-products. This time we can not only observe the damageable behavior required for CMCs thanks to the PyC interphase (figure 2.b) but also some stair step-like fracture surfaces in the multilayered coating (figure 4.b). It is worthy of note that, compared to the other batches that have been tested so far, the SiC(f)/PyC/(SiBC-TiBC)n-SiC(m) microcomposites have a lowest Young's modulus (table 1). This result could be due to a matrix microcraking occurring during the cooling after the processing allowed by both the presence of the PyC interphase and the difference of coefficients of thermal expansion between the components: ~8x10 -6 K -1 for TiC and TiB2 versus ~3.5x10 -6 K -1 for the Hi-Nicalon fiber.

Going back to the Ti-Si-C system, a Ti3SiC2 matrix was produced by P-RCVD from totally consumed SiC sub-layers and by increasing the dilution rate of TiCl4 in H2. [START_REF] Jacques | Reactive chemical vapor deposition of Ti3SiC2 with and without pressure pulses: Effect on the ternary carbide texture[END_REF] The very first SiC sublayer deposited on the PyC interphase was, however, thicker than the others in order to protect the interphase and the fiber from the H2/TiCl4 attack. The same matrix coating was deposited simultaneously on a graphite flat substrate and characterized by XRD. The pattern is typical of pure Ti3SiC2, SiC being hardly detectable (figure 6). The fracture surface observation shows that the PyC interphase has allowed extended fiber pull-out (figure 7.a). The thin unconsumed part of the first SiC sub-layer can be observed against the interphase. The main part of the matrix with a thickness of ~2 µm corresponds to pure Ti3SiC2 with a typical plate-like microstructure (figure 7.b). The SiC(f)/PyC/(SiC-)Ti3SiC2(m) microcomposites are non-brittle (figure 2.c) with an average strain at failure up to the one of the Hi-Nicalon fiber (table 1). But the Young's modulus of the microcomposites is well below that of the one of the sole fiber (270 GPa) [START_REF] Bunsell | A review of the development of three generations of small diameter silicon carbide fibres[END_REF] whereas those of the matrix components are above 300 GPa. [START_REF] Finkel | Low temperature dependencies of the elastic properties of Ti4AlN3, Ti3Al1.1C1.8, and Ti3SiC2[END_REF] This very low Young's modulus is likely to be accounted for by a matrix pre-microcracking resulting from the important coefficient of thermal expansion mismatch between the fiber and the Ti3SiC2 matrix (~9x10 -6 K -1 ).

CONCLUSION

PyC-free coating interphases could be deposited on Hi-Nicalon fibers by P-RCVD. A first difficulty relied on the protection of the fiber surface which was very sensitive to chemical attack by titanium containing-species. This difficulty could be circumvented through a first deposited SiC sublayer thick enough to protect the surface of the fiber or with partially consumed Si-B-C sub-layers. However, even if some delamination could be observed, the porous and multilayered (boro-)carbide coatings did not act as mechanical fuses in the SiC(f)/SiC(m) microcomposites. Microcomposites with a PyC interphase and a Ti3SiC2 matrix could be also prepared and tensile tested at room temperature.

FIGURES

  FIGURES

Figure 1 .

 1 Figure 1. SE SEM images of microcomposites with porous TiC interphases deposited on an unprotected fiber (a) and on a fiber protected with a SiC layer (b).

Figure 2 .

 2 Figure 2. Tensile stress-strain curves at room temperature of representative microcomposites of the batches with PyC interphases (a), (b) and (c) and without a PyC interphase (d), (e) and (f). (For clarity curves are offset horizontally.)

Figure 3 .

 3 Figure 3. BSE (a) and SE (b) SEM images of the fracture surface of microcomposites with (SiBC-TiBC) n interphase.

Figure 4 .

 4 Figure 4. Fracture surface SE SEM images of microcomposites with a PyC interphase and a (SiBC-TiBC) n -SiC matrix.

Figure 5 .

 5 Figure 5. AES depth profile analyses of the (SiBC-TiBC) n -SiC matrix.

Figure 6 .

 6 Figure 6. XRD pattern of Ti 3 SiC 2 deposited by P-RCVD on a flat graphite substrate.

Figure 7 .

 7 Figure 7. Fracture surface SE SEM images of microcomposites with a PyC interphase and a Ti 3 SiC 2 matrix.

3 Table 1 .

 31 Tensile mechanical characteristics of the microcomposites. e i and e m are the thicknesses of the interphase and the matrix. V f is the fiber volume fraction. E is the Young's modulus. ε PL and σ PL are the proportional limit strain and stress, ε F and σ F are the strain and stress at failure. Standard deviations are given in brackets.

	batch	microcomposites	ei (µm)	em (µm)	Vf (%)	E (GPa)	εPL (%)	σPL (MPa)	εF (%)	σF (MPa)
	(a)	SiC(f)/PyC/SiC(m)	0.1 3.5 40 (5)	380 (30)	0.28 (0.05)	1060 (260)	0.48 (0.10)	1540 (220)
	(b) SiC(f)/PyC/(SiBC-TiBC)n-SiC(m) 0.1 3.2 47 (3)	260 (20)	0.26 (0.06)	670 (130)	0.56 (0.14)	1160 (190)
	(c)	SiC(f)/PyC/(SiC-)Ti3SiC2(m)	0.1 2.4 55 (3)	190 (40)	0.30 (0.21)	520 (380)	1.13 (0.29)	1650 (380)
	(d)	SiC(f)/porousTiC/SiC(m)	1.0 1.5 52 (2)	340 (70)			0.24 (0.07)	800 (230)
	(e)	SiC(f)/SiC-porousTiC/SiC(m)	1.0 0.6 65 (2)	330 (30)			0.33 (0.06)	1070 (220)
	(f)	SiC(f)/(SiBC-TiBC)n/SiC(m)	1.1 1.5 56 (7)	320 (20)			0.39 (0.11)	1200 (270)

ACKNOWLEDGEMENT

This work was supported by the French national project NaCoMat and by Herakles. The AES analysis was performed by M. Lahaye at the "CeCaMA" from the University of Bordeaux 1. The author is indebted to his young coworkers from LCTS, namely O. Ledain and J. Couvin.


Radu Ignat
email: radu.ignat@math.univ-toulouse.fr

A De Giorgi type conjecture for elliptic systems under the divergence constraint

Keywords: symmetry, De Giorgi conjecture, minimisers, calibrations, Ginzburg-Landau system, divergence constraint, nonlinear Stokes equation

We study the symmetry of transition layers in Ginzburg-Landau type functionals for divergence-free maps in N -dimensions. Namely, we determine a class of nonlinear potentials such that the minimal transition layers in the periodic strip domain Ω = R × T N -1 are one-dimensional where T = R/Z is the 1-torus. In particular, this class includes in dimension N = 2 the nonlinear potentials w 2 with w being an harmonic function or a solution to the wave equation, while in dimension N > 2, this class contains a perturbation of the standard Ginzburg-Landau potential as well as potentials having N + 1 zeros with prescribed transition cost between the zeros. For that, we develop a theory of calibrations for divergence-free maps in dimension N (similar to the theory of entropies for the Aviles-Giga model when N = 2). We also give a necessary condition for finite energy configurations yielding the boundary condition in L 2 and almost everywhere in T N -1 as x 1 → ±∞.

This note represents the summary of the talk of the author given at the Workshop "Calculus of Variations" in Oberwolfach, 2-8 August 2020 and is based on the articles [START_REF] Ignat | A De Giorgi-type conjecture for minimal solutions to a nonlinear Stokes equation[END_REF][START_REF] Ignat | A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips, Special Issue Dedicated to Professor Haim Brezis on the occasion of his 75th birthday[END_REF] written in collaboration with Antonin Monteil. This report will be included in the volume Oberwolfach Reports No. 22/2020 dedicated to that workshop.

Introduction.

We analyse the symmetry of transition layers in some variational models arising in physics where the order parameter is a vector field of vanishing divergence. We develop a theory of calibrations in order to prove that one-dimensional transition layers are the unique global minimisers in these models.

This question is similar to the famous De Giorgi conjecture for minimal surfaces: if

u : R N → R is a C 2 solution to ∆u = dW du (u) in R N with W (u) = 1 4 (1 -u 2 ) 2 such that -1 < u < 1 and ∂ 1 u > 0 in R N , then u is one-dimensional (1D) provided that N ≤ 8. 1
After important contributions of Ghoussoub-Gui, Ambrosio-Cabré etc., Savin proved the conjecture under the additional boundary condition lim

x 1 →±∞ u(x 1 , x ) = ±1 for every x ∈ R N -1 .
(

Finally, Del Pino-Kowalczyk-Wei gave a counter-example satisfying (1) for N ≥ 9. Lately, an intensive research was developed for vector-valued solutions u : R N → R d to the elliptic system ∆u = ∇W (u) in R N for potentials W : R d → R + . The typical potential arising in phase separation models (such as Bose-Einstein condensates with two components, i.e.,

d = 2) is W (u 1 , u 2 ) = 1 2 u 2 1 u 2 2 + Λ(1 -|u| 2 ) 2
for Λ ≥ 0. Under certain boundary conditions, one-dimensional symmetry of solutions has been shown provided monotonicity / growth / stability conditions on solutions in certain dimensions N . Periodic strip. In the following, we focus on the infinite strip domain

Ω = R × T N -1
in x 1 direction where T = R/Z is the flat torus. For d ≥ 1 and nonnegative continuous potentials W : R d → R + , we set the energy functional

E(u) = Ω 1 2 |∇u| 2 + W (u) dx, u : Ω → R d .
In this context, the boundary condition (1) becomes a necessary condition for finite energy configurations:

Lemma 1 ([2]
). Assume that W has a finite number of zeros and

lim inf |u|→∞ W (u) > 0. If E(u) < ∞, then there exist two zeros u ± ∈ R d of W such that lim x 1 →±∞ u(x 1 , •) = u ± in L 2 and a.e. in T N -1 . (2)
Thus, for a.e.

x ∈ T N -1 , x 1 ∈ R → u(x 1 , x ) represents a curve connecting u ± in R d endowed with the singular metric g W = 2W g 0 where g 0 is the Euclidean metric. Let geod W (u -, u + ) = inf 1 -1 2W (γ(t)) | γ| dt : γ ∈ Lip(-1, 1), γ(±1) = u ± . Corollary 2. If a minimal geodesic connecting two zeros u ± of W in (R d , g W ) exists, then any global minimiser of E connecting u ± is 1D, i.e., u = u(x 1 ). Proof. If u : Ω → R d satisfies (2)
, we have:

E(u) = Ω 1 2 |∇ u| 2 + 1 2 |∂ 1 u| 2 + W (u) dx ≥ Ω 1 2 |∇ u| 2 dx + geod W (u -, u + ). (3)
It proves that optimal 1D transition layers connecting u ± are global minimisers; moreover, if u is a global minimiser of E, then ∇ u = 0 a.e. in Ω, so u = u(x 1 ).

Our model. From now on, we assume that N = d and u : Ω → R N is divergencefree. This constraint is natural in certain asymptotic regimes in liquid crystals, elasticity, ferromagnetism etc. In particular, if

ū(x 1 ) = T N-1 u(x 1 , x ) dx
is the x -average of u and E(u) < ∞, then ū is continuous with constant first component, i.e., ū1 = a in R. Moreover, as in Lemma 1, if the set {W (a, •) = 0} is finite and lim inf u 1 →a, |u |→∞ W (u 1 , u ) > 0, then there are two zeros u ± of W (a, •) such that (2) holds (see [START_REF] Ignat | A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips, Special Issue Dedicated to Professor Haim Brezis on the occasion of his 75th birthday[END_REF]Theorem 1.3]). We want to determine W :

R N → R + such that inf E(u) : u : Ω → R N , ∇ • u = 0 with (2) (4)
has only one-dimensional global minimisers. 1 They satisfy the nonlinear Stokes system -∆u + ∇W (u) = ∇p for some pressure p (due to the constraint ∇ • u = 0).

Calibrations.

Inspired by the beautiful paper of Jin-Kohn [START_REF] Jin | Singular perturbation and the energy of folds[END_REF], our strategy is to construct calibrations Φ :

R N → R N such that Ω ∇ • [Φ(u)] dx ≤ E(u) for every u : Ω → R N with ∇ • u = 0 and (2), (5)
and there exists u * satisfying

Ω ∇ • [Φ(u * )] dx = E(u * ), ∇ • u * = 0 and (2). (6)
Note that (5) & (6) yield u * is a global minimiser in (4). The aim is to prove that u * is 1D and that the equality in (5) is achieved only for 1D vector fields u. Roughly speaking, this is related with the equipartition of the energy density for any minimiser u in (4), i.e.,

1 2 |∇u| 2 = W (u) in Ω.
Results for N = 2. The typical example is given by the Aviles-Giga model where W (u) = 1 4 (1 -|u|2 ) 2 ; note that {W (a, •) = 0} is a finite set 2 for every a ∈ R. Jin-Kohn proved that optimal 1D transition layers are global minimizers in (4) (for the reverse implication, see Theorem A below). Moreover, they proved for the potential W δ (u) = 1 4 (1δu 2 2u 2 1 ) 2 with δ > 0 small enough that 1D transition layers are no longer global minimizers in (4). Our main result is the following:

Theorem A ([1]). Assume that W = 1 2 w 2 where w ∈ C 2 (R 2 ) solves the Tricomi equation ∂ 11 w(u) -f (u 1 )∂ 22 w(u) = 0 for every point u = (u 1 , u 2 ) ∈ R 2 with f ∈ C 1 (R) satisfying |f | ≤ 1. If u ± = (a, u ± 2 ) are two zeros of W such that W (a, •) > 0 in the interval (u - 2 , u + 2 ), then any minimizer u ∈ L ∞ in (4) is 1D.
The assumption u ∈ L ∞ can be dropped out provided that |∇w(u)| ≤ Ce β|u| 2 for every u ∈ R 2 for some C, β > 0. If f = 1, then w solves the wave equation (for example, the Aviles-Giga potential). If f = -1, then w is an harmonic function, in particular,

w(u) = u 2 2 -u 2 1 . If f = 1
δ with δ ≥ 1, we recover the potential W δ (in a different regime of δ than in [START_REF] Jin | Singular perturbation and the energy of folds[END_REF]). The proof of Theorem A is based on constructing a calibration Φ such that

∇Φ(u) = α(u) w(u) f (u 1 )w(u) α(u)
for some function α.

Results for general N ≥ 2. We present two strategies to construct calibrations Φ. We denote by Π 0 (resp. Π + ) the projection on traceless N × N matrices (resp. the projection on symmetric matrices). For divergence-free u : Ω → R N , we have:

∇ • [Φ(u)] = ∇Φ(u) : Π 0 ∇u T = Π 0 ∇Φ(u) : ∇u T ≤ 1 2 (|∇u| 2 + |Π 0 ∇Φ(u)| 2 ).
Strategy 1. We impose that |Π 0 ∇Φ(u)| 2 ≤ 2W (u) for every u ∈ R N . In particular, (5) holds. This strategy yields the following result (see [START_REF] Ignat | A De Giorgi-type conjecture for minimal solutions to a nonlinear Stokes equation[END_REF]Theorem 2.11]): Assume that X = {x 0 , . . . , x N } is an affine basis in R N and ρ is a (prescribed) metric on X. Then there exists a potential W such that X = {W = 0}, ρ = geod W on X × X and any minimiser of (4) connecting

u ± ∈ X in a periodic strip in direction ν ⊥ (u + -u -) is 1D. The proof is based on the calibration Φ = ϕν where ϕ ∈ Lip(R N ) satisfies |ϕ(u + ) -ϕ(u -)| = ρ(u + , u -)
for every u ± ∈ X, yielding the potential W = 1 2 |∇ϕ| 2 . Strategy 2. We impose that ∇Φ(u) is symmetric and |Π 0 ∇Φ(u)| 2 ≤ 4W (u) for every u ∈ R N (the constant 4 is crucial here). Then for divergence-free u, [START_REF] Ignat | A De Giorgi-type conjecture for minimal solutions to a nonlinear Stokes equation[END_REF]Proposition 4.12]). This strategy yields a class of potentials W for which (4) has only 1D global minimisers (see [1, Theorem 2.10]). The proof is based on calibrations such that ∇Φ = ∇ 2 Ψ with Ψ solving the wave equation in any directions x i and x j , i.e., ∂ ii Ψ = ∂ jj Ψ in R N ; the potential is then given by W = 1 2 i<j |∂ j Φ i | 2 . In particular, we recover the following extension of the Aviles-Giga potential in dimension N ≥ 3:

∇ • [Φ(u)] = Π 0 ∇Φ(u) : ∇u T = Π 0 ∇Φ(u) : Π + ∇u T ≤ |Π + ∇u| 2 + 1 4 |Π 0 ∇Φ(u)| 2 . It yields (5) since ∇u 2 L 2 (Ω) = 2 Π + ∇u 2 L 2 (Ω) (see
W (u) = 1 4 (1 -|u| 2 ) 2 + |u | 2 (u 2 1 + u 2 2 ) corresponding to Ψ(u) = -u 1 u 2 √ 2 ( u 2 1 +u 2 2 3
+ |u | 2 -1) for every u = (u 1 , u 2 , u ) ∈ R N with u = (u 3 , . . . , u N ).

Perspective. Motivated by micromagnetics, a future problem is to extend this study for divergence-free vector fields u satisfying the nonconvex constraint |u| = 1.

Note that for divergence-free maps u, (3) holds if additionally u1 = a in Ω.

Without the constraint ∇ • u = 0, no global minimisers of E exist when W (u) = 1 4 (1 -|u| 2 ) 2 since two zeros u ± can be connected within the curve {W = 0}, so the infimum of E vanishes.


Arnaud Bourdin

Alberto Papi

Jonathan Corren

J Christian Virchow

Megan Rice

Yamo Deniz

Michel Djandji

Paul Rowe

Ian Pavord

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