Victor A Eremeyev
email: eremeyev.victor@gmail.com

Francesco Dell'isola
email: francescodellisola@uniroma1.itc.boutinclaude.boutin@entpe.fr

Claude Boutin

David Steigmann

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Keywords: Strain gradient elasticity, Linear pantographic sheets, Existence, Uniqueness, Anisotropic Sobolev's space

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g.

Introduction

Mechanical scientists have been recently attracted to the formulation of design and construction criteria of new materials whose behaviour is established apriori. One can say that the aim of this stream of researches is to produce Materials on Demand.Moreprecisely: once fixed the peculiar behaviour of a material which is desirable for optimizing its use in a given application, the aim of aforementioned researches is to find the way for constructing such a material. Materials designed in order to get a specific behaviour are often called metamaterials.

The role of mathematical sciences in the design and constructions of metamaterials recently increased for two reasons: (i) the development of the technology of 3D printing allowed for the transformation of the mathematically conceived structures, geometries and material properties into the reality of precisely built specimens; (ii) the way in which one specifies the set of properties to be realized is specifically mathematical, as this specification exactly consists in choosing the equations which one assumes to be those governing the mechanical response of the conceived metamaterial.

Once more we can say that mathematics is shaping our world, as it is allowing us to design new technological solutions and tools. The present paper deals with a mathematical problem arising in a specific context involving the design of second gradient metamaterials. More precisely: in order to find a class of materials whose deformation energy depends on both first and second gradient of placement field in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF] a microstructured (pantographic) fabric is introduced and its homogenized continuum model (which we call pantographic sheet) is determined. Various aspects of modelling of pantographic lattices are considered in [START_REF] Battista | Frequency shifts induced by large deformations in planar pantographic continua[END_REF][START_REF] Placidi | A review on 2D models for the description of pantographic fabrics[END_REF][START_REF] Placidi | A second gradient formulation for a 2D fabric sheet with inextensible fibres[END_REF][START_REF] Scerrato | Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations[END_REF][START_REF] Turco | Non-standard coupled extensional and bending bias tests for planar pantographic lattices, part I: numerical simulations[END_REF][START_REF] Turco | Non-standard coupled extensional and bending bias tests for planar pantographic lattices, part II: comparison with experimental evidence[END_REF][START_REF] Turco | Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence[END_REF][START_REF] Turco | Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model[END_REF][START_REF] Turco | Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields[END_REF] where discrete and homogenized models are considered. Let us note that one of the sources of generalized continua and models of metamaterials is the homogenization of heterogeneous materials, see, e.g., [START_REF] Del Vescovo | Dynamic problems for metamaterials: review of existing models and ideas for further research[END_REF][START_REF] Reda | Wave propagation in 3D viscoelastic auxetic and textile materials by homogenized continuum micropolar models[END_REF][START_REF] Trinh | Evaluation of generalized continuum substitution models for heterogeneous materials[END_REF] and reference therein. Homogenization may lead also to strain gradient models [START_REF] Challamel | Higher-order gradient elasticity models applied to geometrically nonlinear discrete systems[END_REF][START_REF] Cordero | Second strain gradient elasticity of nano-objects[END_REF]. While the ideas underlying the definition of pantographic microstructures have been exploited up to now only in the context of purely mechanical phenomena, it is expected that when introducing multi-physics effects (as the piezoelectric coupling phenomena exploited as explained in [START_REF] Dell'isola | Piezo-electromechanical (PEM) structures: passive vibration control using distributed piezoelectric transducers[END_REF][START_REF] Giorgio | Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications[END_REF]) the designed meta materials could have even more interesting features. We expect a fortiori that the mathematical tools used in the present context will be of use also in the envisioned more general context.

The linearized equilibrium equations valid in the neighbourhood of a stress free configuration for such pantographic sheets cannot be immediately studied by using the results available in the literature. However we prove in this paper that the standard strategy involving the use of Poincaré inequality, Lax-Milgram Theorem and coercivity of bilinear deformation energy form do apply also in the present more generalized context.

What has to be modified is the Energy space where the solutions, relative to suitable well-posed boundary conditions, are looked for and the Sobolev space which includes this Energy Space.

Indeed the concept of Anisotropic Sobolev Space, whose definition was conceived on purely logical grounds by Sergei M. Nikol'skii, see [START_REF] Nikol'skii | On imbedding, continuation and approximation theorems for differentiable functions of several variables[END_REF], has to be used in order to apply the abstract Hilbertian setting of solution strategy.

We expect that further developments will lead us to study the complete nonlinear problem of deformation of pantographic sheets.

We need to explicitly remark here that we believe it is important to consider the particular case of pantographic lattices whose first gradient energy does not depend on shear deformation. This could be considered either a pathological case or an important exceptional object (in the sense made explicit in [START_REF] Stillwell | Exceptional objects[END_REF] and [START_REF] Turro | Paradigms lost and paradigms found: examples of science extraordinary and science pathological and how to tell the difference[END_REF]). In both cases we believe that the pathology shown by pantographic lattices in absence of first gradient shear energy deserves due attention for what we can eventually understand by studying its ill-posedness or well-posedness. The significance of pathological examples in the development of mathematics has been widely discussed: without listing the many examples of pathological mathematical behaviour which has led to a better understanding of physical and mathematical world (we have found in the entry of Wikipedia (https://en.wikipedia.org/wiki/ Pathological_(mathematics) consulted on 17 April 2017, really interesting) we simply recall the classical example of the Greek discovery of irrational and its consequences (see, e.g., [START_REF] Kline | Mathematical Thought from Ancient to Modern Times[END_REF]). 1 On the other hand if a behavior must be considered as pathological it is indeed subject to the personal intuition and therefore pathology is a concept which should not play a relevant role in scientific theories. What has to be classified as "pathology" depends on many factors including: its context, the training of the group of scientists studying the problem, and their experience: indeed what is regarded as pathological by one researcher may very well be 1 Si parva licet componere magnis (Virgil), i.e., if it be allowable to compare small things with great ones, we would like to cite in the present context a relevant statement by Freeman Dyson "The same pathological structures that the mathematicians invented to break loose from 19-th naturalism turn out to be inherent in familiar objects all around us in nature" in Characterizing Irregularity, Science 200 [1978] (see [START_REF] Dyson | Characterizing irregularity[END_REF] for a deep and further discussion about this point). Moreover we believe to have pointed out in this paper an anomaly in classical elasticity theory (in the technical sense given to this word by Thomas Kuhn, see [START_REF] Kuhn | The Structure of Scientific Revolutions, 3rd edn[END_REF]). Indeed we prove here that the standard use of the hypothesis of ellipticity in the proof of well-posedness needs to be modified if one wants to incorporate linear second gradient elasticity in the body of theory of elasticity. The most conservative readers will be reassured by the fact that in this way it is avoided a crisis (always in the sense of Kuhn) inside this theory.

standard behaviour to another one. The concept of pathologic behaviour seems of therefore of relevance only in the history of scientific thought and in the description of the process of solution finding.

Postulated Deformation Energy for "Long-Fibers" Pantographic Sheets

Pantographic sheets are bidimensional continua whose microstructure is constituted by a lattice of extensible and continuous fibers having bending stiffness and interconnected by (internal) pivots (i.e., pin joints which are not interrupting the material continuity of the pairs of beams to which they are applied). It has to be explicitly remarked that, in general, we are not considering trusses. A truss, by definition, is assumed to comprise a set of independent beams that are connected by means of pin joints connecting only ending points of the beams. This means that if the truss is loaded only with concentrated forces applied to pin joints then each beam (or fiber) can only be either in compression or in extension. We call lattices of beams the most general beams structure involving pin joints (but also possibly clamping devices, or rollers or glyphs).

Roughly speaking, pantographic sheets can be characterized as those lattices of fibers whose microstructure, once pivots are assumed to be ideal and no external constraints are applied, allows for the existence of some homogeneous deformations which do not store deformation energy. These deformations are sometimes called "floppy-modes". In Fig. 1 such a structure is schematically described, while in Fig. 2 a picture of a 3D printed specimen in polyamide is shown. Of course the pivots in the real object can have only an approximate ideal behaviour: they will store some deformation energy and therefore the homogenized corresponding continuum will need to have a non vanishing deformation energy relative to shear deformation. In our opinion, however, the ideal case of vanishing first gradient stiffness deserves a careful attention, as it can suggest to us some features and exotic behaviours which pantographic sheets can exhibit.

The main feature of the considered pantographic structure consists in the presence of "long" continuous fibers constituting two arrays: at each intersection point of one fiber with all fibers of the other array it is present a pin joint (or internal pivot) which is not interrupting the mechanical and geometrical continuity of both interconnecting fibers. This pin joint imposes that the beam sections which it is interconnecting must undergo the same displacement, however it leaves free their relative rotations.

We assume that in the reference configuration the two arrays of fibers are initially orthogonal and we denote D α , α = 1, 2, the unit vectors of their current directions. As a macro model of the system described before we consider a continuum whose reference configuration is given by a (suitably regular) domain ω ⊂ R 2 . By assuming planar motion, the actual configuration of ω is described by the planar macro-placement

χ : ω → R 2 , ( 1
)
whose gradient ∇χ will be denoted by F. For considered "long fibers" pantographic sheets a possible expression for deformation energy is given by (the two possible methods for getting this expression are described in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF] or in [START_REF] Boutin | Linear pantographic sheets: asymptotic micro-macro models identification[END_REF]):

U χ(•) = ω α K α e 2 F D α -1 2 dω + ω α K α b 2 ∇F : D α ⊗ D α • ∇F : D α ⊗ D α F D α 2 - F D α F D α • ∇F : D α ⊗ D α F D α 2 dω + ω K p 2 arccos F D 1 F D 1 • F D 2 F D 2 - π 2 2 dω , (2)
which accounts for stretching (first integral) and bending deformations of fibers (second integral) as well as for the resistance to shear distortion (third integral) related to the variation of the angle between the fibers. Here • is the Euclidean norm in R 2 and : is the double dot product. The twisting deformations in fibers are not considering here. The coefficients K α e > 0 and K α b > 0 are related respectively to the extensional and bending stiffnesses of the fibers at the interpivot scale, while the coefficient K p ≥ 0 models, at macro-level, the shear stiffness of the pantographic sheet and is related to the interaction between the two arrays of fibers via their interconnecting pivots: when these pivots are perfect this interaction is vanishing and K p vanishes. Considering stiffness parameters K α e , K α b , and K p we conclude that pivots are weakest elements in typical pantographic structures. An example of how printed fibers with pivots look like after breakage is given in Fig. 3. Even made of the same Fig. 3 Fibers with pivots material, pivots are thinner than fibers. Thus, for such structures K p is much smaller than the tangent and bending stiffnesses.

In this paper we will start considering small deformations of the sheet in the neighborhood of the reference configuration. Therefore we calculate the second order Taylor expansion for the energy U(χ(•)) in terms of the small parameter η controlling the amplitude of the displacement u starting from the reference configuration. In formulas

χ(X) = X + ηu(X), X ∈ ω. ( 3
)
By introducing the notations H := ∇u, H =: E + W, where E is symmetric and W is skew-symmetric we get formally (where I denotes the identity tensor)

F = I + ηH, ∇F = η∇H.
As a consequence

FD α = D α + ηHD α , FD α • FD α = 1 + 2ηHD α • D α + η 2 HD α • HD α , FD α = 1 + 2ηED α • D α + η 2 HD α • HD α 1 + ηHD α • D α , 1 FD α 1 -ηHD α • D α , 1 FD α 2 1 -2ηHD α • D α , K α e 2 FD α -1 2 K α e 2 (ηHD α • D α ) 2 , F D 1 F D 1 • F D 2 F D 2 (ηD 1 • HD 2 + ηHD 1 • D 2 ), K p 2 arccos F D 1 F D 1 • F D 2 F D 2 - π 2 2 K p 2 (ηD 1 • HD 2 + ηHD 1 • D 2 ) 2 , ∇F : D α ⊗ D α • ∇F : D α ⊗ D α FD α 2 η 2 (∇H : D α ⊗ D α • ∇HD α ⊗ D α ), FD α FD α • ∇FD α ⊗ D α FD α (1 -ηHD α • D α ) 2 (D α + ηHD α ) • η∇HD α ⊗ D α , FD α FD α • ∇FD α ⊗ D α FD α D α • η∇HD α ⊗ D α .
Since D α are unit orthogonal vectors we may replace them by standard basis vectors i α , D α = i α . As a consequence if the displacement u is represented in the basis D α by means of its components

u = u 1 i 1 + u 2 i 2 , u α = u α (x 1 , x 2 ), X = x 1 i 1 + x 2 i 2 ,
then the second order Taylor expansion of the energy ( 2) is given by

U u(•) = ω K 1 e 2 u 2 1,1 + K 2 e 2 u 2 2,2 + K p 2 (u 1,2 + u 2,1 ) 2 + K 1 b 2 u 2 1,22 + K 2 b 2 u 2 2,11 dω, ( 4
)
see also [START_REF] Boutin | Linear pantographic sheets: asymptotic micro-macro models identification[END_REF] for a direct derivation of this energy based on a homogenization procedure exploiting the micro-balance of force and momentum. In [START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model[END_REF] the same energy has been derived with an explicit identification of the constitutive coefficients in terms of the Young's modulus of the fiber materials, of the area and of the moment of inertia of the fiber cross sections and of the distance between the nearest pivots. Here indices after comma denote derivatives, so f ,α is the partial derivative of f with respect to x α , f ,α = ∂ α f ≡ ∂f ∂xα . Since the torsional (shear) stiffness is much smaller than other stiffness parameters used in the model we can study independently this singular limit case. So, in what follows and in the spirit described in the introduction, we mainly consider the mathematically interesting case which is represented by pantographic structures whose shear stiffness is vanishing.

Energy for Pantographic Sheets and Equilibrium Conditions

Let us consider the deformation energy relative to pantographic structures having vanishing shear stiffness. The deformation energy becomes

U u(•) = ω W dω, ( 5
)
where the strain energy density W is given by

W = K 1 e 2 u 2 1,1 + K 2 e 2 u 2 2,2 + K 1 b 2 u 2 1,22 + K 2 b 2 u 2 2,11 .
For derivation of the equilibrium conditions we consider the first variation of U . First we obtain

δU = ω K 1 e u 1,1 δu 1,1 + K 2 e u 2,2 δu 2,2 + K 1 b u 1,22 δu 1,22 + K 2 b u 2,11 δu 2,11 dω.
Then, integrating by parts we transform δU into

δU = ω -K 1 e u 1,11 + K 1 b u 1,2222 δu 1 dω + ω -K 2 e u 2,22 + K 2 b u 2,1111 δu 2 dω + ∂ω n 1 K 1 e u 1,1 -n 2 K 1 b u 1,222 δu 1 + n 2 K 1 b u 1,22 δu 1,2 ds + ∂ω n 2 K 2 e u 2,2 -n 1 K 2 b u 2,111 δu 2 + n 1 K 2 b u 2,11 δu 2,1 ds. ( 6
)
Here n α = i α • n, t α = i α • t, n and t are the unit external normal and unit tangent to the boundary ∂ω vectors, respectively, see Fig. 4. Representing δu 1,2 and δu 2,1 by formulas

δu 1,2 = i 2 • ∇δu 1 , δu 2,1 = i 1 • ∇δu 2
with ∇ defined at the boundary through normal and tangent derivatives

∇ = n ∂ ∂n + t ∂ ∂s ,
where ∂/∂s and ∂/∂n are derivatives with respect to arc length s and normal coordinate, respectively, we obtain that

δu 1,2 = n 2 ∂δu 1 ∂n + t 2 ∂δu 1 ∂s , δu 2,1 = n 1 ∂δu 2 ∂n + t 1 ∂δu 2 ∂s .
Substituting the latter formulae into (6) and again integration by parts with respect to s we obtain that

δU = ω -K 1 e u 1,11 + K 1 b u 1,2222 δu 1 + -K 2 e u 2,22 + K 2 b u 2,1111 δu 2 dω + ∂ω n 1 K 1 e u 1,1 -n 2 K 1 b u 1,222 - ∂ ∂s n 2 t 2 K 1 b u 1,22 δu 1 ds + ∂ω n 2 K 2 e u 2,2 -n 1 K 2 b u 2,111 - ∂ ∂s n 1 t 1 K 2 b u 2,11 δu 2 ds + ∂ω K 1 b u 1,22 n 2 2 ∂ ∂n δu 1 + K 2 b u 2,11 n 2 1 ∂ ∂n δu 2 ds. ( 7
)
Fig. [START_REF] Agranovich | Elliptic boundary problems[END_REF] Deformation of a pantographic sheet

Here for simplicity we assumed that boundary contour ∂ω is a plane curve which is smooth enough, i.e., differentiable and without corner points. In [START_REF] Placidi | Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients[END_REF][START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model[END_REF] the integration by part procedure valid for boundaries with corner points is considered. The form of δU requires that only a class of external loads can be applied: indeed the virtual work of external loads must be consistent with it. So, we must assume that the virtual work of external loads δA is given in the following form:

δA = ω (f 1 δu 1 + f 2 δu 2 ) dω + ∂ω (ϕ 1 δu 1 + ϕ 2 δu 2 ) ds + ∂ω n 2 μ 1 ∂ ∂n δu 1 + n 1 μ 2 ∂ ∂n δu 2 ds. ( 8
)
Here f α are surface loads. Moreover ϕ α and μ α are forces and couples respectively assigned on the part of the boundary ∂ω where u α and/or ∂uα ∂n are not assigned. Therefore we introduce a suitably regular partition of ∂ω into two disjoint subsets ∂ e ω α and ∂ n ω α (or ∂ e ω ⊥ α and ∂ n ω ⊥ α ) on which either displacements (or normal derivatives of displacements) are assigned or their dual quantities are assigned respectively (the index α = 1, 2 refers to the displacement component u α ).

Finally, from the principle of least action δU -δA = 0, and by assuming the following essential boundary conditions,

u 1 = u 0 1 , (x 1 , x 2 ) ∈ ∂ e ω 1 , (9)
u 2 = u 0 2 , (x 1 , x 2 ) ∈ ∂ e ω 2 , (10)
n 2 ∂ n u 1 = ϑ 1 n 2 , (x 1 , x 2 ) ∈ ∂ e ω ⊥ 1 , (11)
n 1 ∂ n u 2 = ϑ 2 n 1 , (x 1 , x 2 ) ∈ ∂ e ω ⊥ 2 , ( 12
)
where u 0 1 , u 0 2 , ϑ 1 , and ϑ 2 are given functions at ∂ω and ∂ n = ∂/∂n, we obtain the equilibrium equations and natural (static) boundary conditions

-K 1 e u 1,11 + K 1 b u 1,2222 -f 1 = 0, (x 1 , x 2 ) ∈ ω, ( 13
) -K 2 e u 2,22 + K 2 b u 2,1111 -f 2 = 0, (x 1 , x 2 ) ∈ ω; ( 14
)
n 1 K 1 e u 1,1 -n 2 K 1 b u 1,222 - ∂ ∂s n 2 t 2 K 1 b u 1,22 = ϕ 1 , (x 1 , x 2 ) ∈ ∂ n ω 1 ; ( 15
)
n 2 K 2 e u 2,2 -n 1 K 2 b u 2,111 - ∂ ∂s n 1 t 1 K 2 b u 2,11 = ϕ 2 , (x 1 , x 2 ) ∈ ∂ n ω 2 ; (16)
K 1 b u 1,22 n 2 2 = n 2 μ 1 , (x 1 , x 2 ) ∈ ∂ n ω ⊥ 1 ; (17)
K 2 b u 2,11 n 2 1 = n 1 μ 2 , (x 1 , x 2 ) ∈ ∂ n ω ⊥ 2 . ( 18
)
It is interesting that ( 13) and ( 14) contain partial derivatives of different orders. For example, (13) contains second derivative with respect to x 1 and fourth derivative with respect to x 2 .

Since the energy has a reduced form (in other words: in it some higher order derivatives are not appearing) we have also reduced boundary conditions. For example, for a fixed boundary and differently from what happens in the case of first gradient elasticity, instead of

∂ n u 1 = 0 we have n 2 ∂ n u 1 = 0 which has a sense if n 2 = 0. Indeed, if n 2 = 0 that corresponds
AB: K 1 b u 1,22 = 0, K 1 b u 1,222 = 0, K 2 e u 2,2 = 0, along BC: K 1 e u 1,1 = 0, K 2 b u 2,111 = 0, K 2 b u 2,11 = 0, along CD: u 1 = 0, u 2 = 0, ∂ 2 u 1 = 0, along DA: u 1 = 0, u 2 = 0, ∂ 1 u 2 = 0.
Clearly, this rectangle gives an example of degenerated boundary conditions, since instead of four conditions as in the general case, here we have only three. This circumstance occurs always when the boundary (or a part of it) is parallel to one of the coordinate axes, i.e., is parallel to one of the fiber directions.

Heuristics

It is evident that the immediate application of the classic methods used for proving existence and uniqueness of the solution of the elastic problem is not possible in the present context [START_REF] Ciarlet | Mathematical Elasticity, vol. I: Three-Dimensional Elasticity[END_REF][START_REF] Eremeyev | Existence of weak solutions in elasticity[END_REF][START_REF] Fichera | Existence theorems in elasticity[END_REF][START_REF] Lebedev | Functional Analysis[END_REF], as coercivity could seem, at first sight, a condition which is not verified (this circumstance is already mentioned in [START_REF] Boutin | Linear pantographic sheets: asymptotic micro-macro models identification[END_REF]). Moreover also the results for second gradient continua proven by Healey et al. [START_REF] Healey | Injective weak solutions in second-gradient nonlinear elasticity[END_REF][START_REF] Mareno | Global continuation in second-gradient nonlinear elasticity[END_REF] are not applicable here, as the energy of pantographic sheets is not coercive with respect to the highest order derivatives. Before framing the problem in the appropriate energy space we present here some heuristic preliminary considerations.

First of all: assume that for a displacement field u * the energy (5) vanishes.

It is trivial to check that as u * 1,1 = 0

u * 1 = f (x 2 ) while, as u * 1,22 = f ,22 = 0 f = a 1 x 2 + b 1
and, finally,

u * 1 = a 1 x 2 + b 1 ,
where a 1 and b 1 are constants. We have that

u * 2 = a 2 x 1 + b 2
with constant a 2 and b 2 independent of a 1 and b 1 .

Note that in the case of plane infinitesimal deformations the rigid body motion is

u r = φ × X + b,
where φ = φi 3 is a constant rotation vector, i 3 = i 1 × i 2 , × is the cross product, and b = b 1 i 1 + b 2 i 2 is a constant vector. So in components the rigid body motion has the form

u 1 = φx 2 + b 1 , u 2 = -φx 1 + b 2 .
It is therefore evident that the "kernel" or "null-space" of the strain energy not only include rigid (infinitesimal) motions (corresponding to a 1 = -a 2 ) but also pure shear that corresponds to elongation/contraction in the directions at an angle ±π/4 with respect to the coordinate axes (when a 1 = a 2 ). The null-space of strain energy density consists of four linear independent modes and all their linear combinations

u * 1 = i 1 , u * 2 = i 2 , u * 3 = i 3 × X, u * 4 = x 2 i 1 .
Instead of u * 4 one can use the equivalent mode x 1 i 2 or the symmetric mode x 2 i 1 + x 1 i 2 . Differently to what happens for first gradient elasticity a fourth mode having vanishing energy appears and it relates to uniform shear in a specific direction.

Clearly well-posedness results must take into account such a property. Secondly: it has to be recalled that boundary conditions producing well-posed problems in the case of second gradient continua are more general than when dealing with first gradient continua (see for more details about generalized boundary conditions, e.g., [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: théorie du second gradient[END_REF][START_REF] Germain | The method of virtual power in continuum mechanics, part 2: microstructure[END_REF][START_REF] Mindlin | On first strain-gradient theories in linear elasticity[END_REF][START_REF] Toupin | Theories of elasticity with couple-stress[END_REF] and the historic works by Piola). The procedure which is used in the aforementioned papers can be summarized as follows (see [START_REF] Auffray | Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids[END_REF][START_REF] Dell'isola | At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola[END_REF]): i) one postulates the principle of virtual work, i.e., the equality between internal and external work expended on virtual displacements; ii) one determines a class of internal work functionals involving second gradients of virtual displacement; iii) one determines, by means of an integration by parts, the class of external work functionals which are compatible with the determined class of internal work functionals.

A consequence (see, e.g., [START_REF] Dell'isola | How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach "á la d'Alember[END_REF][START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: théorie du second gradient[END_REF][START_REF] Mindlin | On first strain-gradient theories in linear elasticity[END_REF][START_REF] Toupin | Theories of elasticity with couple-stress[END_REF]) of the just described procedure is that Neumann problems for considered second gradient deformation energies must include, to be complete, double symmetric and skew-symmetric boundary forces together with forces concentrated on points. To be more precise: the class of so-called natural boundary conditions must include the dual (with respect to work functionals) quantities of normal gradients of virtual displacements. Following Germain, the dual of the tangential part of normal gradient of virtual displacement is a "couple" (i.e., skew-symmetric contact double forces) while the dual of the normal part of normal gradient of virtual displacement is a "double force" (i.e., symmetric contact double forces). For some reasons (initially studied in [START_REF] Dell'isola | Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola's peridynamics and generalized continuum theories[END_REF][START_REF] Dell'isola | Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives[END_REF][START_REF] Eugster | Exegesis of the introduction and Sect. I from "Fundamentals of the Mechanics of Continua" by E. Hellinger[END_REF], but surely further investigations are needed!) this kind of boundary conditions has been considered, sometimes and by some schools of mechanicians, unphysical: the reader is referred to the beautiful paper by Sedov, Leonid Ivanovich, [START_REF] Sedov | Mathematical methods for constructing new models of continuous media[END_REF] for a lucid discussion of this point and its physical, mathematical and epistemological implications.

After having identified the displacements which are in the null space of the deformation energy, a conjecture about mixed boundary conditions which are likely to produce wellposed problems can, consequently, be formulated. Indeed, let us partition the boundary ∂ω of the body ω into two disjoint subsets, i.e., ∂ω e and ∂ω n . We assume that the displacements on ∂ω e are assigned and that the displacements on ∂ω n are free. We call AC the set of C 2 displacements verifying the assigned conditions on ∂ω e . We say that AC is singular if there exist an element u in AC and a displacement field u 0 belonging to the null space for deformation energy (i.e., a displacement having vanishing deformation energy) such that u + u 0 also belongs to AC. We conjecture here (and rigorously prove in the next section) that the considered mixed boundary problem is well-posed if and only if AC is NOT singular. Remark that if the considered elastic energy is a first gradient one and it is positive when regarded as a function of infinitesimal strain tensor then the aforementioned statement reduces to the standard requirement that in well-posed problems the constrained body cannot undergo rigid displacements. The concept of underconstrained system (see [START_REF] Kuznetsov | Underconstrained Structural Systems[END_REF]) has to be modified in order to include the treated case of planar second gradient continua: see, for instance, Figs. 1 and2 for examples of underconstrained pantographic sheets. This point will need further investigations to include the case of pantographic sheets moving in threedimensional space and three-dimensional pantographic bodies.

In the present paper we limit ourselves to consider Dirichlet's and mixed boundary problems in which, on a part of the body boundary, only the displacement is assigned, while the remaining part of the body boundary is left free.

On Ellipticity of Equilibrium Equations

The equilibrium equations ( 13) and ( 14) are very specific. Indeed, unlike classic mechanical models like membranes, plates, or linear elastic solids they are neither elliptic nor strongly elliptic. Let us introduce the differential operators

P 1 (D) = -K 1 e ∂ 2 1 + K 1 b ∂ 4 2 , P 2 (D) = -K 2 e ∂ 2 2 + K 2 b ∂ 4 1 , D = (∂ 1 , ∂ 2 ). ( 19
)
So ( 13) and ( 14) take a symbolic form

P 1 (D)u 1 = f 1 , P 2 (D)u 2 = f 2 . ( 20
)
Then the ellipticity of P 1 (D) and P 2 (D) requires that, see [START_REF] Fichera | Linear Elliptic Differential Systems and Eigenvalue Problems[END_REF][START_REF] Fichera | Existence theorems in elasticity[END_REF],

K 1 b ξ 4 2 = 0, K 2 b ξ 4 1 = 0, ∀ ξ ≡ (ξ 1 , ξ 2 ) = 0,
respectively. Obviously, these conditions are not fulfilled.

Strong ellipticity requires the fulfillment of a stronger condition such as

K 1 b ξ 4 2 ≥ c 1 |ξ | 4 , K 2 b ξ 4 1 ≥ c 2 |ξ | 4 , ∀ ξ ≡ (ξ 1 , ξ 2 ) = 0
with positive constants c 1 and c 2 , see [START_REF] Agranovich | Elliptic boundary problems[END_REF] for the general framework, and [START_REF] Mareno | Global continuation in second-gradient nonlinear elasticity[END_REF] for the strain gradient elasticity. So, it is clear that here the equilibrium equations are not strongly elliptic.

Nevertheless, the considered operators are still elliptic in the sense of Agmon-Douglis-Nirenberg [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF][START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF]. The differential operators P 1 (D) and P 2 (D) belong to the class of quasielliptic or semi-elliptic differential operators. The boundary value problems for quasi-elliptic operators have been intensively studied at the end of the Sixties, see, e.g., [START_REF] Barozzi | Un problema al contorno non omogeneo in un dominio angoloso per equazioni fortemente quasi-ellittiche in due variabili (I)[END_REF][START_REF] Barozzi | Un problema al contorno non omogeneo in un dominio angoloso per equazioni fortemente quasi-ellittiche in due variabili (II)[END_REF][START_REF] Cavallucci | Sulle proprietà differenziali delle soluzioni delle equazioni quasi-ellittiche[END_REF][START_REF] Matsuzawa | On quasi-elliptic boundary problems[END_REF][START_REF] Volevich | Solubility of boundary value problems for general elliptic systems[END_REF]. The quasi-elliptic operators are a particular class of the so-called hypoelliptic operators, see, e.g., [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators, II: Differential Operators with Constant Coefficients[END_REF]. It is easy to prove that P 1 (D) and P 2 (D) are hypoelliptic. Indeed, necessary and sufficient condition for the hypoellipticity of the operator P (D) is the following limiting relation [START_REF] Egorov | Foundations of the Classical Theory of Partial Differential Equations[END_REF][START_REF] Hörmander | The Analysis of Linear Partial Differential Operators, II: Differential Operators with Constant Coefficients[END_REF]

|∇P (ξ )| |P (ξ )| → 0, |ξ | → +∞,
where ∇P = ( ∂P ∂ξ 1 , ∂P ∂ξ 2 ), which is fulfilled for P 1 (D) and P 2 (D). For the theory of elliptic, quasi-elliptic and hypoelliptic operators we refer to [START_REF] Agranovich | Elliptic boundary problems[END_REF][START_REF] Egorov | Foundations of the Classical Theory of Partial Differential Equations[END_REF][START_REF] Hörmander | The Analysis of Linear Partial Differential Operators, II: Differential Operators with Constant Coefficients[END_REF][START_REF] Lions | Non-Homogeneous Boundary Value Problems and Applications[END_REF][START_REF] Palamodov | Systems of linear differential equations[END_REF][START_REF] Rodino | Partial differential equations of multi-quasi-elliptic type[END_REF] and the references therein. In particular, in [43, p. 214] the variational statement of similar problem for an rectangle is discussed.

Existence and Uniqueness of Weak Solutions

Let us now come back to the first variation of the energy functional. For solution we have the principle of virtual work in the form

δU -δA = 0. ( 21
)
For simplicity let us replace δu by a new function v. With v Eq. ( 21) transforms to

B(u, v) ≡ ω K 1 e u 1,1 v 1,1 + K 2 e u 2,2 v 2,2 + K 1 b u 1,22 v 1,22 + K 2 b u 2,11 v 2,11 dω = ω (f 1 v 1 + f 2 v 2 ) dω + ∂ω ϕ 1 v 1 + ϕ 2 v 2 + n 2 μ 1 ∂v 1 ∂n + n 1 μ 2 ∂v 2 ∂n ds. ( 22
)
Here we introduce the bilinear form B(u, v) to denote the quadratic terms in [START_REF] Dell'isola | Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives[END_REF]. Now we introduce the weak solution of the boundary-value problem ( 13)-( 17) as the vector-function u such that the variational equation ( 22) is fulfilled for any test function

v = v 1 i 1 + v 2 i 2 .
The properties of u and v are specialized below.

Without loss of generality, in what follows we use the following dimensionless form of W :

2W = u 2 1,1 + u 2 2,2 + u 2 1,22 + u 2 2,11 . ( 23
)
Now the bilinear form is

B(u, v) = ω (u 1,1 v 1,1 + u 2,2 v 2,2 + u 1,22 v 1,22 + u 2,11 v 2,11 ) dω.
Here we keep the same notations u α and x α for dimensionless displacements and dimensionless coordinates, respectively. So, W has the form of a seminorm in anisotropic Sobolev spaces, see definition in [START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF], and Appendix. More precisely: let u 1 ∈ W (1,2) 2 (ω) and u 2 ∈ W (2,1) 2 (ω), then we have that u ∈ W (1,2) 2 (ω) ⊕ W (2,1) 2 (ω) and

2U(u) = ω u 2 1,1 + u 2 2,2 + u 2 1,22 + u 2 2,11 dω = |u 1 | W (1,2) 2 + |u 2 | W (2,1) 2 ( 24
)
is a seminorm in W (1,2) 2 (ω) ⊕ W (2,1) 2 (ω). Here we introduced the following notations for some auxiliary seminorms

|f | W (1,2) 2 := f ,1 L 2 + f ,22 L 2 , |f | W (2,1) 2 := f ,11 L 2 + f ,2 L 2 ,
whereas it is possible to transform them into norms for instance following the standard choice:

f W (1,2) 2 = f L 2 + |f | W (1,2) 2 , f W (2,1) 2 = f L 2 + |f | W (2,1) 2 . ( 25
)
It clear that the functional space W (1,2)

2 (ω) ⊕ W (2,1)

2

(ω) is constituted exactly by the set of all functions for which ( 24) is finite. We will call energy space E for the considered energy functional any subspace of W (1,2)

2 (ω) ⊕ W (2,1)

2

(ω) which is the completion of one of the previously introduced space AC relative to NONSINGULAR boundary conditions using the norms [START_REF] Dyson | Characterizing irregularity[END_REF]. Remark that when restricted to an energy space the seminorm given by ( 24) becomes a norm. Now the definition of a weak solution for linear pantographic structures can be given as follows.

Definition 1 We call u ∈ E a weak solution of the equilibrium problem (21) if ( 22) is fulfilled for any test function v from a dense set in E.

The bilinear form B(u, v) is continuous and the following inequalities are valid:

B(u, v) ≤ |u 1 | W (1,2) 2 |v 1 | W (1,2) 2 + |u 2 | W (2,1) 2 |v 2 | W (2,1) 2 ≤ u E v E . ( 26
)
For the analysis of existence and uniqueness of weak solutions we start by considering two cases. The simplest case is given by Dirichlet boundary conditions.

Dirichlet's Boundary Conditions

We start by proving the existence and uniqueness for the simplest case when the whole boundary is fixed. Consider the set of equations for the strong formulation of equilibrium problem:

-u 1,11 + u 1,2222 = f 1 , -u 2,22 + u 2,1111 = f 2 , (x 1 , x 2 ) ∈ ω; ( 27
)
u 1 = 0, u 2 = 0, n 2 ∂ n u 1 = 0, n 1 ∂ n u 2 = 0, (x 1 , x 2 ) ∈ ∂ω. ( 28
)
Here the weak solution is defined through the integral equation

B(u, v) = ω (f 1 v 1 + f 2 v 2 ) dω, ∀ v 1 , v 2 ∈ C 2 0 (ω), (29)
which, when assuming that f belongs to L 2 (ω), can be written as

B(u, v) -(f, v) L 2 = 0, ∀ v ∈ C 2 0 (ω).
Using the Poincaré inequalities (Friedrich's inequality, see, e.g., [START_REF] Adams | Sobolev Spaces, 2nd edn[END_REF]) we get

u 1 L 2 ≤ C 1 u 1,1 L 2 , u 2 L 2 ≤ C 1 u 2,2 L 2 (30)
with some constants C 1 and C 2 and, as a consequence, we can establish that , respectively, where with the upper circle we denoted the completion of C 2 0 (ω) (or C ∞ 0 (ω)) with respect to the corresponding norms. So here the energy space E is the anisotropic Sobolev's space

|u 1 | W (1,2) 2 + |u 2 | W (2,1) 2 ≥ C u 1 W (1,2) 2 + u 2 W (2,
• W (1,2) 2 (ω)⊕ • W (2,1) 2 (ω). This means that we have proven that B(u, v) is coercive B(u, u) ≥ C u 2 E .
One can easily prove that (f, v) L 2 is a linear bounded functional in E. Thus, by using Lax-Milgram theorem [START_REF] Evans | Partial Differential Equations[END_REF], the following theorem can be easily proven:

Theorem 1 Let the Cartesian components f 1 and f 2 of f belong to the space L 2 (ω). There exists a weak solution u * ∈ E ≡ • W (1,2) 2 (ω)⊕ • W (2,1) 2
(ω) to the equilibrium problem [START_REF] Egorov | Foundations of the Classical Theory of Partial Differential Equations[END_REF] and [START_REF] Eremeyev | Existence of weak solutions in elasticity[END_REF], which for any v ∈ E satisfies the equation

u * , v E - ω f • v dω = 0.
Furthermore, u * is unique and it is a minimizer of the energy:

F u * = inf u∈E F (u), F (u) ≡ U(u) - ω f • u dω.
Remark 1 Since for the coercivity we need inequalities [START_REF] Evans | Partial Differential Equations[END_REF] which require that only the functions are zero at the boundary, (i.e., u 1 = u 2 = 0 at ∂ω), for uniqueness is suffices to consider only the boundary conditions concerning displacements, without considering the condition on the normal derivatives [START_REF] Eremeyev | Existence of weak solutions in elasticity[END_REF].

Remark 2

We used here L 2 (ω) as a functional space for f. This condition can be weakened using imbedding theorems of E into anisotropic Lebesgue spaces [START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF] and we omit this for simplicity.

For non-homogeneous boundary conditions ( 9)-( 12) we seek a solution in the form u = u * + u 0 , where u 0 is a vector function which satisfies ( 9)-( 12) whereas for u * boundary conditions (28) are assumed. Substituting this representation into ( 27) and ( 9)-( 12) we reduce the non homogeneous boundary-value problem to the previous one, for which we already proved the theorem on existence and uniqueness of weak solutions.

Mixed Boundary Conditions

Somehow more difficult is the case of mixed boundary conditions. In the theory of linear elasticity it is known that for existence and uniqueness it is enough to request that a part of the boundary is fixed: i.e., when rigid displacements are not allowed [START_REF] Ciarlet | Mathematical Elasticity, vol. I: Three-Dimensional Elasticity[END_REF][START_REF] Eremeyev | Existence of weak solutions in elasticity[END_REF][START_REF] Fichera | Existence theorems in elasticity[END_REF][START_REF] Lebedev | Functional Analysis[END_REF]. In our problem it is not enough to require that rigid displacements are ruled out. Indeed as an example we consider two rectangles with one fixed side whereas others are free, see Fig. 6. More precisely, on side AB the displacements are zero: u = 0. The difference in rectangles consists only on their orientation with respect to the coordinate axes, that is to fiber orientations. It is clear that for the first rectangle the solution is not unique since the displacement u = ax 2 i 1 satisfies the equilibrium and boundary conditions for any value of a: the set AC in this circumstance is indeed SINGULAR. For the second rectangle, instead, the set AC is actually NON-SINGULAR and the aforementioned displacement is not a solution of equilibrium problem.

In other words, for rectangle a) we have at least the solutions u = 0 and u = ax 2 i 1 . Obviously we should avoid such situations since even without loading we have an infinity of non-trivial (deformative) solutions. Thus, in what follows we always assume that the boundary conditions are nonsingular.

Let us consider the mixed boundary-value problem formulated by ( 9)- [START_REF] Del Vescovo | Dynamic problems for metamaterials: review of existing models and ideas for further research[END_REF]. Here the energy space E is a subspace of W (1,2) 2 (ω) ⊕ W (2,1) 2 (ω) obtained as the completion of functions C 2 (ω) which verify ( 9)- [START_REF] Boutin | Linear pantographic sheets: asymptotic micro-macro models identification[END_REF].

The weak solution is defined as an element u belonging to E satisfying the equation

B(u, v) = ω (f 1 v 1 + f 2 v 2 ) dω + ∂ω ϕ 1 v 1 + ϕ 2 v 2 + n 2 μ 1 ∂v 1 ∂n + n 1 μ 2 ∂v 2 ∂n ds ( 31
)
for any admissible function v (i.e., a function belonging to a dense subset of E).

Using the same technique we formulate the theorem on existence and uniqueness of the weak solution in E.

Theorem 2 Let the Cartesian components f 1 and f 2 of f belong to the space L 2 (ω), ϕ α ∈ L 2 (∂ n ω α ), μ α ∈ L 2 (∂ n ω ⊥ α ) and assume that the boundary conditions are nonsingular. There exists a weak solution u * ∈ E to the equilibrium problem (12)-( 18) which for any v ∈ E satisfies the equation [START_REF] Fichera | Linear Elliptic Differential Systems and Eigenvalue Problems[END_REF].

Furthermore, u * is unique and it is a minimizer of the functional F (u):

F u * = inf u∈E F (u), F (u) ≡ U(u) - ω f • u dω + ∂ω 1 ϕ 1 u 1 + ϕ 2 u 2 + n 2 μ 1 ∂u 1 ∂n + n 1 μ 2 ∂u 2 ∂n ds.

Existence and Uniqueness Considering Non-zero Shear Stiffness

Using the anisotropic Sobolev spaces we can also prove existence and uniqueness for a more general case by considering energy in the form of (4). In dimensionless form the energy is given by

2U(u) = ω u 2 1,1 + u 2 2,2 + (u 1,2 + u 2,1 ) 2 + u 2 1,22 + u 2 2,11 dω, ( 32
)
and the bilinear form becomes

B(u, v) = ω u 1,1 v 1,1 + u 2,2 v 2,2 + (u 1,2 + u 2,1 )(v 1,2 + v 2,1 ) + u 1,22 v 1,22 + u 2,11 v 2,11 dω. ( 33
)
Differently from the previous case where the energy is exactly a seminorm in W (1,2) 2 (ω) ⊕ W (2,1) 2 (ω), now U is slightly different. Moreover, for the energy [START_REF] Fichera | Existence theorems in elasticity[END_REF] there are only rigid body motions as in the case of the classical plane elasticity. Considering Dirichlet or mixed boundary conditions with u 1 = 0, u 2 = 0 at the part of the boundary we can use the Korn inequality [START_REF] Ciarlet | Mathematical Elasticity, vol. I: Three-Dimensional Elasticity[END_REF][START_REF] Fichera | Existence theorems in elasticity[END_REF][START_REF] Lebedev | Functional Analysis[END_REF] to prove the coercivity condition

B(u, u) ≥ C u 2 E . ( 34
)
Here the energy space E is a subspace of W (1,2) 2 (ω) ⊕ W (2,1) 2 (ω) obtained as the completion of functions C 2 (ω) which verify kinematic boundary conditions in the norm given by [START_REF] Fichera | Existence theorems in elasticity[END_REF]. Using Lax-Milgram theorem we can prove existence and uniqueness of the weak solutions. Let us note that this problem is much easier for analysis than the problem with zero shear stiffness. However the results given in the previous section gives us (i) the logical basis needed for formulating a numerical integration scheme which can be used in the case of small shear stiffness and (ii) a basis for understanding the exotic behaviour of purely second gradient pantographic sheets.

Conclusions

The results presented in this paper allow us to prove existence and uniqueness theorems for the elastic problem in the case of planar pantographic sheets and for a variety of boundary conditions. The main difficulties which we had to confront were: i) the existence of floppy modes, i.e., deformations corresponding to zero deformation energy and ii) the absence in the deformation energies of many higher order derivatives.

Therefore the results by Healey and Chambon [START_REF] Chambon | Uniqueness studies in boundary value problems involving some second gradient models[END_REF][START_REF] Healey | Injective weak solutions in second-gradient nonlinear elasticity[END_REF][START_REF] Mareno | Global continuation in second-gradient nonlinear elasticity[END_REF] could not be applied directly and there was the appearance of a lack of the coercivity of considered energy. Indeed the second gradient deformation energy for pantographic sheets is not coercive if one considers the standard Sobolev Space, whose norm involves all second order derivatives.

However we prove that the standard Hilbertian abstract setting used for solving the elastic problem does not need to be changed. Instead one has to change the definition of the Energy spaces which correspond to the various imposed boundary conditions: they must be regarded as subsets of the Anisotropic Sobolev space whose norm is defined by involving only the derivatives appearing in the considered deformation energy. The abstraction effort due to Nikol'skii (and to Besov and others) which lead him to introduce a wider class of Sobolev spaces was initially motivated only by the need of developing a mathematical theory based on the minimum possible necessary assumptions: Anisotropic Sobolev Spaces include functions which do not posses all higher order weak derivatives.

The abstract tool which he developed allowed us to frame rather naturally the numerical and mathematical problems concerning the equilibrium of linear pantographic sheets. Discussion of the finite element method developed for anisotropic Sobolev's spaces is given in [START_REF] Eastham | The finite element method in anisotropic Sobolev spaces[END_REF].

We are also confident that the same tools will allow us to study non-linear deformations problems. Finally we want to stress once more that the study of apparent pathologies can be a fruitful source of understanding in both mathematics and in the mathematical modelling of physical phenomena. Acknowledgement V.A.E. acknowledges the financial support by the Russian Science Foundation under grant "Methods of microstructural nonlinear analysis, wave dynamics and mechanics of composites for research and design of modern metamaterials and elements of structures made on its base" (No. 15-19-10008).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Appendix: Anisotropic Lebesgue and Sobolev Functional Spaces

In the paper we used the classic and anisotropic Lebesgue and Sobolev functional spaces. For sake of self consistency, we present here some necessary information on this topic. In plane elasticity and other problems of mechanics functional spaces such as Lebesgue space L 2 (ω) and Sobolev's spaces W 1 2 (ω) and W 2 2 (ω) are widely used [START_REF] Ciarlet | Mathematical Elasticity, vol. I: Three-Dimensional Elasticity[END_REF][START_REF] Eremeyev | Existence of weak solutions in elasticity[END_REF][START_REF] Lebedev | Functional Analysis in Mechanics[END_REF][START_REF] Lebedev | Functional Analysis[END_REF]. The norms in these spaces are defined as follows:

f L 2 = ω f 2 dω 1 2 , f W 1 2 = f 2 L 2 + f ,1 2 L 2 + f ,2 2 L 2 1 2 , f W 2 2 = f 2 L 2 + f ,1 2 L 2 + f ,2 2 L 2 + f ,11 2 L 2 + 2 f ,12 2 L 2 + f ,22 2 L 2 1 2 ,
where f = f (x 1 , x 2 ) is a function defined in an open set in the two-dimensional Euclidean space, ω ⊂ R 2 , the boundary of ω is assumed to be smooth enough. Various useful requirements to the boundary of ω are discussed in [START_REF] Adams | Sobolev Spaces, 2nd edn[END_REF][START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF]. The Greek indices take values 1, 2. L 2 (ω), W 1 2 (ω) and W 2 2 (ω) are examples of a separable Hilbert space [START_REF] Adams | Sobolev Spaces, 2nd edn[END_REF].

In this paper we use the various imbedding theorems for Sobolev's spaces, therefore let us recall the general definition of an imbedding. We say that the normed space E is imbedded in the normed space H , and we write E → H to denote this imbedding, if (i) E is a vector subspace of H , and (ii) there exists a constant C such that u H ≤ C u E ∀u ∈ E. For imbedding theorems in Sobolev's spaces we refer to [START_REF] Adams | Sobolev Spaces, 2nd edn[END_REF][START_REF] Lebedev | Functional Analysis in Mechanics[END_REF][START_REF] Lebedev | Functional Analysis[END_REF].

In addition to classic Lebesgue and Sobolev's space we introduce the anisotropic Lebesgue and Sobolev spaces. Here we are restricting ourselves to functions defined on a set of R 2 . Let p = (p 1 , p 2 ) be a multiindex, where 1 < p α < ∞. Then the norm in the anisotropic Lebesgue space L p is defined as

f Lp = f (x 1 , x 2 ) p 1 dx 1 p 2 /p 1 dx 2 1/p 2 .
If p 1 = p 2 = p we use standard notation L p = L p . • Lp is called the mixed norm [START_REF] Adams | Sobolev Spaces, 2nd edn[END_REF]. An anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions, so such functions have generalized derivatives of different order and, in general, different L p in coordinate directions x 1 and x 2 . The theory of the anisotropic Sobolev spaces including imbedding theorems, relations with other Sobolev's spaces and analysis of the coercivity of differential operators is presented in [START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Nikol'skii | On imbedding, continuation and approximation theorems for differentiable functions of several variables[END_REF], see also [START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Triebel | Theory of Function Spaces III[END_REF]. We introduce the multiindex = (l 1 , l 2 ) where l α are natural numbers and the norm

f W p = f Lp + 2 α=1 ∂ lα α f Lp . (35)
So, the set of functions defined on ω and having generalized derivatives such that the introduced norm is finite, is called the anisotropic Sobolev space W p (ω). Obviously, when l 1 = l 2 = l and p 1 = p 2 = p we have the classical Sobolev space W l p . The anisotropic Sobolev's space W p is a separable Banach space whereas W 2 is a Hilbert space.

We also introduce the anisotropic Sobolev space • W p as the closure of C 2 0 (ω) (or C ∞ 0 (ω)) functions in norm [START_REF] Giorgio | Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications[END_REF].

For our purposes we consider two specific anisotropic Sobolev spaces W (1,2) 2 and W (2,1) 2 with norms

f W (1,2) 2 = f L 2 + f ,1 L 2 + f ,22 L 2 , ( 36
)
f W (2,1)

2 = f L 2 + f ,11 L 2 + f ,2 L 2 . ( 37
)
With certain assumptions on the regularity of ω for these spaces, there are the following imbedding theorems [START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF][START_REF] Nikol'skii | On imbedding, continuation and approximation theorems for differentiable functions of several variables[END_REF]:

W 2 (ω) → W 1 2 (ω), W 2 (ω) → C(ω), = (1
, 2), (2, 1) .

Evidently, any function f ∈ W 2 2 (ω) belongs to W 2 (ω) with = {(1, 2), (2, 1)}, but not all elements of W 2 (ω) belong to W 2 2 (ω). For more details on imbeddings in anisotropic Sobolev's spaces and their further generalizations including results on traces of functions we refer to [9-11, 47, 48, 60].

Fig. 1

 1 Fig. 1 Scheme of a pantographic sheet and beams connection trough a pivot

Fig. 2

 2 Fig. 2 3D printed specimen of a pantographic sheet

Fig. 5

 5 Fig. 5 Pantographic rectangle

1 ) 2 with

 12 another constant C. In other words we proved that | • | W (1

Fig. 6

 6 Fig.[START_REF] Barozzi | Un problema al contorno non omogeneo in un dominio angoloso per equazioni fortemente quasi-ellittiche in due variabili (I)[END_REF] Two rectangles with clamped edge with different fiber orientation


Dr Edakkattuparambil Sidharth Shibu

Dr Nadezda Varkentina

Dr Laurent Cognet

Brahim Lounis
email: brahim.lounis@u-bordeaux.fr

Prof Brahim Lounis

Small gold nanorods with tunable absorption for photothermal microscopy in cells

Keywords: photothermal microscopy, gold nanorods, plasmonic materials, cell imaging, density gradient ultrahigh centrifugation

 that practical applications of absorption microscopy were developed in single-molecule research. Photothermal imaging (PhI)

microscopy can indeed reach unprecedented sensitivities for detecting tiny absorbers with absorption cross-section as small as a few 10 -16 cm 2 . [12][13][14] Several types of nano-objects ranging from gold nanoparticles, [15][16][17][18][19] carbon nanotubes [20,21] to quantum dots [22] are detected at the single-particle level leading to refined spectroscopic studies and ultra sensitive imaging applications. PhI displays extremely stable signals and is therefore appealing for biological applications. Furthermore photothermal microscopies are totally insensitive to non-absorbing scatterers, even when large objects with strong refractive index contrasts are present within the surroundings of the imaged nanoparticles, as is often the case in biological samples. [23] Several live cell application of PhI involve the detection of small gold nanospheres in order to perform long term single molecule tracking. [9,24,25] A limitation however arises due to background signal from endogenous cellular components, since these nanoparticles have to be excited at their plasmon resonance, which lies around 530 nm. Although one can also take advantage of this intrinsic absorption to achieve label-free imaging, [26] very small red-shifted absorbing nanoparticles would represent better probes for PhI since near-infrared is a region where the absorption of cell organelles is negligible. The use of gold nanorods as small probes absorbing in the near infra-red, is a promising strategy as they would combine good subcellular accessibility, [9] low contribution from intrinsic cellular signals and perfect photostability.

Gold nanorods are elongated nanoparticles, which display additional red-shifted plasmon resonance as compared to their spherical counterparts. [27] More precisely, transverse or longitudinal plasmon resonances result from the electron cloud oscillations excited by light polarized along their short-axis (diameter) or long-axis (length) respectively. Interestingly, the short axis plasmon resonance is basically independent on nanorods dimensions, while the long-axis plasmon resonance can be tuned over a wide spectral range (from 620 nm to 1100 nm) based on their aspect ratio. Although large diameter nanorods (> 8 nm) have been extensively used in different biological applications, such as bioimaging, delivery, sensing and therapy, [18,28,29] the synthesis of nanorods with smaller diameter is more delicate and are more recent. [30] Here, we report the synthesis, sorting and characterization of mono-disperse gold nanorods. These nanorods display tunable red-shifted plasmon resonance as compared to their spherical counterparts. A dual color PhI microscope was developed to image them down to the single nanorod level and to demonstrate that nanorods are promising basic building blocks for the realization of cellular imaging biomarkers in the near infrared.

Nanorods were synthesized using a modified pH-prompted seedless protocol (see experimental section). [30] The as-prepared solution was first characterized using UV/Vis absorption spectroscopy from 400 nm to 1000 nm (Figure 1A). The absorption spectrum shows the characteristic splitting of the surface Plasmon resonance (SPR) into two modes in case of gold nanorods (transverse at 516 nm and longitudinal ~ 680 nm). [31][32][33] Nanorod morphologies were then characterized by transmission electron microscopy (TEM). All obtained nanorods had lengths smaller than 50 nm and diameters smaller than 7 nm (Figure 1B). More precisely, the distributions of nanorod lengths, diameters and aspect ratios were constructed (Figure 1C-E) revealing that the as-prepared solution was poly-disperse in length, and rather mono-disperse in diameter (6± 1nm).

In order to improve nanorods mono-dispersity with tunable longitudinal plasmon resonance, we performed length sorting by density gradient ultracentrifugation [34,35] (see experimental section). Upon centrifugation of the initial solution, distinct colored bands were visible (Figure 2A). Different fractions were collected and analyzed. Figure 2B displays UV/Vis absorption spectra of the 6 first collected fractions corresponding to the shortest nanorods. A manifold of narrower longitudinal SPR peaks is found as compared to the parent solution while the transverse SPR peaks are almost identical. This indicates that efficient sorting based on nanorod lengths was obtained which results in monodisperse nanorods solutions with tunable plasmon resonance peaks. In the following, we concentrated on the solutions containing the smallest nanorods. Figure 2C andD show TEM images of fractions with peak SPR at 618 nm and 642 nm respectively (image of a fraction containing longer nanorods is shown in Figure S1). From such images we constructed the histograms of nanorod lengths and aspect ratios (Figure 2E-H). We obtained average lengths of 10±1 nm (Figure 2E) and 13±1 (Figure 2G) and aspect ratios of 1.8±0.7 and 2.2±0.7 (Figure 2F and 2H) respectively. The latest are consistent with position of the SPR peaks measured in Figure 2B.

We next characterized by PhI microscopy the nanorod sorted samples at the single-particle level. PhI microscopy is indeed a sensitive imaging modality, which enables detection of nanometer-sized objects solely based on their absorption. It involves the detection of refractive index variations that are induced by photothermal effect in the local environment of an absorbing nanoparticle. [12,[36][37][38] Gold nanoparticles of a few nanometers are efficiently detected by this imaging modality, thanks to their fast relaxation times (ps range) and large absorption sections around their plasmon resonance. In order to excite nanorods either at their transverse or at their longitudinal plasmon resonances, a PhI microscope was built with twocolor excitation beams and a probe beam in the near infrared. The setup uses a low amplitude noise single frequency laser diode at 785 nm acting as probe beam (Innovative Photonic Solutions, TO-56) and a 640 nm laser (Coherent OBIS-FP) to excite nanorods at their longitudinal resonance or a 532 nm laser (Coherent Sapphire) to excite the nanorods at their transverse resonance (Figure 3). The intensities of the absorption beams were modulated at 450 kHz. The three beams were overlaid and focused onto the sample using a high NA objective (60x, NA=1.49). Absorption beams were circularly polarized to ensure that nanorods are equally excited regardless of their orientation in the sample plane. Upon light absorption by a nanorod, temperature elevation induces time-modulated variations of the refraction index in its close environment. The interaction of the probe beam with this index profile produces a scattered field with sidebands at the modulation frequency. The scattered field was then detected through its beat note with the probe field transmitted through the sample, which plays the role of a local oscillator and is extracted by lock-in detection.

Samples were mounted on a piezo-scanner stage that allowed scanning the sample to acquire 2D images of nanorods. All data where acquired with integration times of 5 ms/pixel using a resolution of 100 nm/pixel. The setup also included white light trans-illumination and a CCD camera for bright-field imaging of the biological samples.

Samples were first prepared by spin coating a mixture of (1:1 ratio) CTAB stabilized nanorods and polyvinyl alcohol (PVA, 1.5%) on the surface of plasma cleaned glass slides. A drop of silicon oil was then added on the sample to ensure homogenous heat diffusion. PhI images of nanorod fraction with a SPR peak at 642 nm were acquired using laser excitation at 532 nm (Figure 3B) and 640 nm (Figure 3C) with the same intensities at the imaging plane.

One to one correlation is found in the two images indicating that identical objects are detected at the two excitations wavelengths. The signal intensities originating from ~100 single nanorods excited at the two wavelengths were compared as shown in the histograms of Figure 3D-E. Several observations indicate that that single nanorods are detected at the two wavelengths. First, both distributions are narrow which also indicates that the dispersions of nanorod dimensions (both diameter and lengths) are very small within such length-sorted fractions. Second, signal dispersion at 640nm excitation also includes a tail towards the small signals attributed to a contribution from the out-of polarization-plane orientation of the rods. Indeed, for individual nanorods, absorption at 532 nm under circularly polarized excitation is weakly dependent on nanorod orientations [39] since the transverse mode is always excited (Figure 3D) while absorption at 640 nm circularly polarized excitation is dependent on nanorod orientation since the longitudinal mode of nanorods with long axis normal to the polarization plane are not excited. To further support that single nanorods are detected in We also note that a three fold higher intensity is observed under 640 nm excitation compared to 532 nm excitation at the peak of the histograms. This was expected for nanorods where the longitudinal SPR is more intense than the transverse one. Noteworthy in ensemble spectra presented in Figure 2B, the longitudinal SPR absorption peak is less than three times the transverse one due to orientation averaging of nanorods in solutions. PhI images and corresponding intensity histograms of other fractions are given in the supporting information (Figure S2). All of these fractions show similar trend in their intensity histograms, confirming that the synthesis protocol and DGU sorting leads to narrow size dispersion of nanorod solutions.

For bioimaging applications requiring very small nano-labels to access restricted cellular areas or complex tissue organizations, the signal to cellular background ratio at which small nanolabels can be detected is commonly the limiting factor. The smaller the nano-labels, the more difficult is the task due to the increasing weakness of detectable signal originating from the labels. In the case of PhI, intrinsic signal due to residual absorption of green light by cell mitochondria, lead to background signals that can reach that PhI signals of 5 nm gold nanoparticles excited at their plasmon resonance, i.e. around 532 nm. nanorods which we can detect with high signal to noise ratio at 640 nm while maintaining a small size, should thus represent a promising strategy in the context of cellular imaging with photothermal microscopy. In the following, we imaged nanorods in cellular environments. Fixed COS-7 cells were chosen because these cells were previously shown to display intense intrinsic PhI signals originating from mitochondrial light absorption. After fixation COS-7 cells were incubated with 5 nM solution of nanorods in PBS containing 3% bovine serum albumin followed by extensive rinsing steps to remove non-immobilized nanorods. In conclusion, we developed a novel strategy for photothermal imaging based on gold nanorods, which present strong optical absorption tunable from the red to the near IR. For biological applications, the use of these nanorods minimizes background signal from the cell organelles. We anticipate that they will constitute the next generation photothermal probe to study complex molecular dynamics in biological systems owing to their small size, tunable NIR-absorption, absolute photostability and chemical suitability for surface functionalization and bioconjugation.

Experimental Section

Nanorod synthesis: Briefly, HAuCl 4 .3H 2 O (90 mL, 1 mM) was added to 270 mL cetyltrimethyl ammonium bromide (CTAB; 0.2 M) followed by AgNO 3 (9 mL, 4 mM). The solution was gently shaken. HCl (405 µL, 37%) was then added to adjust the pH to ~1. During this step, the color of the solution evolves from dark yellow to orange. Subsequently, 1260 µL of L-ascorbic acid (AA; 78.8 mM) was injected and gently shaken until colorless.

Immediately, ice-cold NaBH 4 (540 µL, 10 mM) was slowly added and the solution was kept overnight at constant temperature (~30 °C). Appearance of dark blue color indicates the formation of nanorods. Excess CTAB was removed by centrifuging the parent solution at 20k rpm for 1.5h. The pellet was collected and re-dispersed in 10 mM CTAB (4 mL) solution.

TEM imaging: Monolayers of positively charged nanorods (due to CTAB encapsulating layer) were deposited and dried under ambient conditions on carbon coated Cu grids (200 mesh) that were negatively charged using a glow discharge technique (K950X Turbo evaporator with a 350X glow discharge head; Emitech, France).

Nanorods length sorting by density gradient ultrahigh-centrifugation (DGU):

The densitygradient consisted of 4 layers of decreasing concentration of ethylene glycol (EG) from bottom to top (volume ratios, 80%, 70%, 60%, 50%) in 10mM CTAB, cooled at 4 °C before use. A 4 mL volume of nanorod solution was drop-casted on the top of cooled density gradient tube and centrifuged at a speed of 10k rpm for 3.5 h at 15 °C (Beckman Coulter ultrahigh centrifuge).

Cell culture and fixation: COS7 cells were cultured plated on #1 glass slides up to 60% confluence in DMEM medium supplemented with streptomycin (100 µg/ml), penicillin (100 U/ml), and 10 % bovine serum in a humidified atmosphere (95 %) at 5 % CO 2 and 37 o C. Cells were used for 12-14 passages and were transferred every 4 days. Before imaging, the cells were fixed in methanol at -30 ºC, washed with phosphate buffered saline (PBS) and stored in PBS at 4ºC until incubated with nanorods.

Figure

  Figure 3B-C, we also collected images with at 640 nm excitation beam having four different

Figure 4

 4 displays white light images of the cells and the corresponding PhI images recorded using 532 nm and 640 nm excitation. Accordingly to Figure 3, punctual signals originating from individual immobile nanorods can be identified on the cells in the two images. Importantly cellular (mitochondrial) structures are clearly visible under 532 nm excitation (Figure S3), which complicate the identification of nanorods around the mitochondria (arrow) at this excitation wavelength. In contrast, background signals originating from mitochondria are notably reduced under 640 nm excitation (Figure S4). In addition, individual nanorods display notably higher PhI signals 640 nm excitation as compared to 532 nm excitation, facilitating their detection of in cellular environments. The combination of these two effects thus participated to a clear enhancement of signal to cellular background ratio of detection of nanorods in cells.

Figure 1 .

 1 Figure 1. UV/Vis absorption spectrum (A), TEM image (B) and corresponding length (C),

Figure 2 .

 2 Figure 2. (A) Photographs of DGU tube before and after centrifugation. (B) UV/Vis

Figure 3 .

 3 Figure 3. (A) Schematics of the two-color PhI microscope with excitations at 532 nm or 640

Acknowledgements

This work was supported by CNRS, Agence Nationale de la Recherche (ANR-14-CE09-0018-01), Conseil Régional d'Aquitaine, the France-BioImaging national infrastructure (ANR-10-INBS-04-01) and IdEx Bordeaux (ANR-10-IDEX-03-02, LAPHIA) and SIRIC BRIO. E.S.S. acknowledges Marie Curie Individual Fellowship (655890 funding).

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

The table of contents entry

We report the synthesis, sorting and characterization of mono-disperse gold nanorods with overall dimension around 10 nm in length and less than 6 nm in diameter. They display strong and tunable red-shifted plasmon resonance, in a region where cellular absorption is reduced. A dual color photothermal microscope is developed to demonstrate that these nanorods are promising single molecule probes for cellular imaging in the near infrared. Keyword: photothermal microscopy, gold nanorods, plasmonic materials, cell imaging, density gradient ultrahigh centrifugation Edakkattuparambil Sidharth Shibu 1,2 , Nadezda Varkentina 1,2 , Laurent Cognet


Jorge Martinez-Gil
email: jorgemar@unex.es

An Overview of Textual Semantic Similarity Measures Based on Web Intelligence

Keywords: Similarity measures, Web Intelligence, Web Search Engines, Information Integration

Computing the textual similarity between terms (or short text expressions) that have the same meaning but which are not lexicographically similar is a key challenge in many computer related fields. The problem is that traditional approaches to semantic similarity measurement are not suitable for all situations, for example, many of them often fail to deal with terms not covered by synonym dictionaries or are not able to cope with acronyms, abbreviations, buzzwords, brand names, proper nouns, and so on. In this paper, we present and evaluate a collection of emerging techniques developed to avoid this problem. These techniques use some kinds of web intelligence to determine the degree of similarity between text expressions. These techniques implement a variety of paradigms including the study of co-occurrence, text snippet comparison, frequent pattern finding, or search log analysis. The goal is to substitute the traditional techniques where necessary.

Introduction

Textual semantic similarity measurement consists of computing the similarity between terms, statements or texts, which have the same meaning, but which are not lexicographically similar [START_REF] Li | An approach for Measuring Semantic Similarity between Words Using Multiple Information Sources[END_REF]. This is an important problem in a lot of computer related fields, for instance, in data mining, information retrieval, or even, natural language processing. The traditional approach for solving this problem has been to use manually compiled dictionaries such as WordNet [START_REF] Budanitsky | Evaluating WordNet-based Measures of Lexical Semantic Relatedness[END_REF].

The question is that a lot of (sets of) terms (acronyms, abbreviations, buzzwords, brand names, and so on) are not covered by these kinds of dictionaries; therefore, similarity measures that are based on this kind of resource cannot be used directly in these cases.

On the other hand, Collective Intelligence (CI) is a field of research that explores the potential that collaborative work has to solve a number of problems. It assumes that when a group of individuals collaborate with each other, intelligence that otherwise did not exist suddenly emerges. We use the name Web Intelligence (WI) when these users use the Web as a means of collaboration. We want to profit from the fact that web users provide rich sets of information that can be converted into knowledge reusable for solving problems related to semantic similarity measurement. To perform our experiments, we are going to use Google [START_REF] Brin | The Anatomy of a Large-Scale Hypertextual Web Search Engine[END_REF] which is a web search engine owned by Google Inc. and is currently the most popular search engine on the Web according to Alexa Ranking. However, we see no problem in using any other similar search engine.

So, in this paper we review and evaluate the most promising methods to determine the degree of semantic similarity between (sets of) terms using some kind of web intelligence. We are especially interested in those methods that are able to measure the similarity between emerging terms or expressions which are not frequently covered in dictionaries, including a new branch of methods designed by us which consists of using the historical search patterns from web search engines.

The rest of this paper is organized as follows: Section 2 describes related approaches that are proposed in the literature currently available. Section 3 describes the methods for semantic similarity measurement including the study of co-occurrence, text snippet comparison, frequent pattern finding, or trend analysis. Section 4 presents a statistical evaluation of the presented methods, and finally, we draw conclusions and put forward future lines of research.

Related Work

Much work has been developed over the last few years proposing different ways to measure semantic similarity. According to the specific knowledge sources exploited and the way in which they are used, different families of methods can be identified. These families are:

-Edge Counting Measures: taking into account the position of the terms in a given dictionary or taxonomy. -Information Content Measures: measuring the difference of the information content of the two terms as a function of their probability of occurrence in a corpus. -Feature based Measures: measuring the similarity between terms as a function of their properties or based on their relationships to other similar terms. -Hybrid Measures: combining all of the above. Now we propose creating a new category, called WI measures, for trying to determine the semantic similarity between terms using content generated by web users. The rest of this paper explains, evaluates, and discusses the semantic similarity measurement of terms using the Google search engine, but it is applicable to the rest of existing web search engines.

Google-Based Techniques

The problem which we are addressing consists of trying to measure the semantic similarity between two given (sets of) terms a and b. Semantic similarity is a concept that extends beyond synonymy and is often called semantic relatedness in the literature. According to Bollegala et al.; a certain degree of semantic similarity can be observed not only between synonyms (lift and elevator), but also between meronyms (car and wheel) or hyponyms (leopard and cat) [START_REF] Bollegala | Measuring semantic similarity between words using web search engines[END_REF].

In this paper, we use the expression semantic similarity in order to express that we are comparing the meaning of terms instead of comparing their associated lexicography. For example, the terms house and mouse are quite similar from a lexicographical point of view but do not share the same meaning at all. We are only interested in the real world concept that they represent, considering that a similarity score of 0 stands for complete inequality and 1 for equality of the concepts being compared.

From our point of view, the methods for measuring semantic similarity using web search engines can be categorized as follows:

-Co-occurrence methods, which consist of measuring the probability of co-occurrence of the terms on the Web. -Frequent patterns finding methods, which consist of finding similarity patterns in the content indexed by the web search engine. -Text snippet comparison methods, which consist of determining the similarity of the text snippets from the search engines for each term pair. -Trend analysis methods, which consist of comparing the time series representing the historical searches for the terms.

Co-ocurrence methods

On the Web, probabilities of term co-occurrence can be expressed by hits. In fact, these formulas are measures for the probability of co-occurrence of the terms a and b [START_REF] Cilibrasi | The Google Similarity Distance[END_REF]. The probability of a specific term is given by the number of hits returned when a given search engine is presented with this search term divided by the overall number of web pages possible returned. The joint probability p(a, b) is the number of hits returned by a web search engine, containing both search term a and search term b divided by the overall number of web pages returned.

One of the most outstanding works in this field is the definition of the Normalized Google Distance (NGD) [START_REF] Cilibrasi | The Google Similarity Distance[END_REF]. This distance is a measure of semantic similarity derived from the number of hits returned by the Google search engine for a given (set of) keyword(s).

N GD(a, b) = mx{log hit(a), log hit(b)} -log hit(a, b) log M -mn{log hit(a), log hit(b)} (1)
Other measures of this kind are: Pointwise Mutual Information (PMI), Dice, Overlap Coefficient, or Jaccard, all of which are explained by Bollegala et al. [START_REF] Bollegala | Measuring semantic similarity between words using web search engines[END_REF]. When these measures are used on the Web, it is necessary to add the prefix Web-; WebPMI, WeDice, and so on. All of them are considered probabilistic because given a web page containing one of the terms, these measures try to compute the probability of that web page also containing the other term. These are their corresponding formulas:

W ebP M I(a, b) = log p(a, b) p(a) • p(b) (2)
W ebDice(a, b) = 2 • p(a, b) p(a) + (b) (3) W ebOverlap(a, b) = p(a, b) min(p(a), p(b)) (4) W ebJaccard(a, b) = p(a, b) p(a) + p(b) -p(a, b) (5)
Despite its simplicity, the idea behind these measures is that terms with similar meanings tend to be close to each other because it seems to be empirically supported that synonyms often appear together in web pages [START_REF] Cilibrasi | The Google Similarity Distance[END_REF], while terms with dissimilar meanings tend to be farther apart, and therefore, present low similarity values.

Frequent patterns finding

This group of techniques belongs to the field of machine learning, and consists of looking for similarity patterns in the websites that are indexed by the web search engines. One of the most popular techniques was proposed by Bollegala et al. [START_REF] Bollegala | Measuring semantic similarity between words using web search engines[END_REF] and proposes looking for such regular expressions as "a also known as b", "a is a b", "a is an example of b", and so on. This is because this kind of expression indicates semantic similarity between the two (set of) terms.

A high number of occurrences of these kinds of patterns provides us with evidence for the similarity between the two terms, but it is necessary to perform some preliminary studies about what is 'a high number' according to the problem that we wish to address. This can be done, for example, by studying the number of results offered by the web search engines for perfect synonyms. Moreover, it is necessary to take into account that these expressions should be tested in two ways, because the similarity between a and b is by definition equal to the similarity between b and a.

In our study, we are going to use a method for measuring the occurrences of such expressions as "a is a b" OR "b is an a". The maximum will be obtained after training the algorithm with some perfect synonyms. For example, try to imagine these perfect synonyms appear, on average, 1 million times together in the same regular expression. Then, 1 million occurrences will be the maximum and 0 occurrences the minimum. A pattern which appears 210,000 times on the web search engine results will present a similarity score of 0.21.

Text snippet comparison

This kind of technique comprises of capturing the text snippets which are generated by the web search engines when offering the results, just after searching for these terms. These text snippets can be processed in order to be compared using well-known algorithms for determining the similarity between short texts. In this way, we can determine the similarity between two terms based on their associated text snippets.

Moreover, one of the best algorithms for comparing the text snippets is Latent Semantic Analysis (LSA) which is a kind of statistical technique for representing the similarity of terms by analyzing a large text corpus. This technique uses a singular value decomposition approach, namely, a general form of factor analysis for condensing a very large matrix of text content into a smaller matrix [START_REF] Scott | Indexing by Latent Semantic Analysis[END_REF].

In our study, we are going to use the first text snippet for each term and the LSA algorithm, this LSA algorithm has been borrowed from [START_REF] Wolfe | Use of Latent Semantic Analysis for Predicting Psychological Phenomena: Two Issues and Proposed Solutions[END_REF]. More elaborated techniques can be applied. For example, it is possible to capture the n first snippets and try to look for similarities one by one, in this way the uncertainty of dealing with an appropriate/relevant text snippet can be avoided.

Trend analysis

People may search things on the Web in order to find information related to a given topic. We want to take advantage of this in order to detect similarities between terms and short text expressions. To do this, we are going to work with time series, i.e. collections of observations of well-defined data items obtained through repeated measurements, because the web search engines often store the user queries in this way in order to offer or exploit this information in an efficient manner in the future.

The similarity problem in time series implies that by using two sequences of real numbers representing the measurements of a variable at equal time intervals; the similarity can be defined and computed. Maybe the most intuitive solution could consist of viewing each sequence as a point in n-dimensional Euclidean space, and defining similarity between sequences as Lp(X, Y ), this solution would be easy to compute but there is a problem because there are no actual scales used in data from the web search engines due to the results being normalized and, therefore it is not clear what the exact numbers are.

In order to avoid this kind of problem, we propose using four different ways to compute the semantic similarity: Co-occurrence of Terms in Search Patterns, Computing the Relationships between Search Patterns, Outlier Coincidence on Search Patterns, and Forecasting comparisons. The great advantage of our proposal is that any of the proposed methods take into account the scale of the results, but other kinds of characteristics.

Co-occurrence of Terms in Search Patterns

The first algorithmic method that we propose consists of measuring how often two terms appear in the same query. Co-occurrence of terms in a given corpus is usually used as an indicator of semantic similarity in the literature. We propose adapting this paradigm to our purposes. To do this, we are going to compute the joint probability p(a, b) so that a user query may contain the search terms over time. For example, if we look for the co-occurrence of the terms lift and elevator over time, we can see that these two terms appear frequently, so we have evidence of their semantic similarity.

The method that we propose for measuring the similarity using the notion of co-occurrence means using the following formula: n. years terms co -occur n. years registered in the log

We think that the proposed formula is appropriate because it computes a score according to the fact that the terms never appear together or appear together every year. In this way a similarity score of 0 stands for complete inequality and 1 for equality of the input terms.

Correlation between Search Patterns

The correlation between two variables is the degree to which there is a relationship between them. Correlation is usually expressed as a coefficient which measures the strength of a relationship between the variables. We propose using two measures of correlation: Pearson and Spearman.

The first measure of correlation that we propose, i.e. Pearson's correlation coefficient, is closely related to the Euclidean distance over a normalized vector space. Using this measure means that we are interested in the shape of the time series instead of their quantitative values. The philosophy behind this technique is that similar concepts may present almost exactly the same shape in their associated time series and, therefore, semantic similarity between them is presumed to be very high. This similarity can be computed as follows (where the terms a and b are substituted by their corresponding time series):

Sim(a, b) = cov(a, b) σ a σ b = E[(a -µ a )(b -µ b )] σ a σ b (7)
The second measure of correlation that we propose using is the Spearman coefficient which assesses how well the relationship between two variables can be described using a monotonic function. If there are no repeated values, a perfect Spearman correlation occurs when each of the variables is a perfect monotone function of the other. This is the formula used to compute it:

Sim(a, b) = 1 - 6 d 2 i n(n 2 -1) (8)

Outlier Coincidence on Search Patterns

There is no rigid mathematical definition of what constitutes an outlier. Grubbs said that "An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs" [START_REF] Grubbs | Procedures for Detecting Outlying Observations in Samples[END_REF]. So our proposal suggests looking for elements of a time series that distinctly stand out from the rest of the series. Outliers can have many causes. Once we have discarded a Google malfunction, we have to assume that outliers in search patterns occur due to historical events, and that users search for information related to this historical event at the same time but maybe using different lexicographies.

Various indicators are used to identify outliers. We are going to use the proposal of Rousseeuw and Leroy who affirm that an outlier is an observation which has a value that is more than 2.5 standard deviations from the mean [START_REF] Rousseeuw | Robust Regression and Outlier Detection[END_REF].

Forecasting Comparison

Our forecasting comparison method compares the prediction of the (sets of) terms for the coming months. There are many methods for time series forecasting, but the problem is that people's behavior cannot be predicted, or at least, can be notably influenced by complex or random causes. For example, it is possible to predict searches related to holidays every summer, but it is not possible to predict searches related to balls. Anyway, we wish to obtain a quantitative result for the quality of this method in order to compare it with the others.

To do that, we propose training a neural network in order to predict the results of the searches. We can establish the similarity between two terms on the basis of the similarity between these predictions. We have chosen a forecasting based on neural networks and discarded such techniques as moving average or exponential smoothing. Moving average uses past observations weighted equally, while exponential smoothing assigns exponentially decreasing weights as the observation gets older. The reason for our choice is that neural networks have been widely used successfully as time series forecasters for real situations [START_REF] Patuwo | Forecasting with artificial neural networks: The state of the art[END_REF].

Evaluation

We have created a new dataset which has been rated by a group of 20 people who come from several countries, indicating a value of 0 for non similar terms and 1 for totally similar terms. This dataset is specially designed to evaluate terms that are not frequently included in dictionaries but which are used by people daily. In this way, we will be able to determine the most appropriate method for comparing the semantic similarity of emerging terms. This could be useful in very dynamic domains such as medicine, finance, sms language, social networks, technology, and so on. Table 1 shows the term pairs and the mean for the values obtained after asking the people to comment on their similarity. The comparison between this dataset and our results is made using the Pearson's Correlation Coefficient, which is a statistical measure for the comparison of two matrices of numeric values. Therefore the results can be in the interval [-1, 1], where -1 represents the worst case and 1 represents the best case. This coefficient allows us to measure the strength of the relation between human ratings of similarity and computational values. However, Pirro stated that it is also necessary to evaluate the significance of this relation [START_REF] Pirro | A semantic similarity metric combining features and intrinsic information content[END_REF]. To do 2 Ranking for the algorithms tested using the benchmark dataset which contains terms that do not appear in dictionaries very frequently that, we have used the p-value technique, which shows how unlikely a given correlation coefficient will occur given no relation in the population. We have obtained that, for our sample, all values above 0.3 are statistically significant.

A larger dataset would be necessary to confirm the significance of the rest of the tests. On the other hand, in order to compare the emerging methods with the existing ones; we consider techniques which are based on dictionaries. We have chosen the Path Length algorithm which is a simple node counting approach. The similarity score is inversely proportional to the number of nodes along the shortest path between the definitions. The shortest path occurs when the two definitions are the same [START_REF] Pedersen | WordNet::Similarity -Measuring the Relatedness of Concepts[END_REF]. A dictionary-based approach proposed by Lesk which consists of finding overlaps in the definitions of the two terms. The relatedness score is the sum of the squares of the overlap lengths [START_REF] Lesk | Information in Data: Using the Oxford English Dictionary on a Computer[END_REF]. An ontology-based technique from Leacock and Chodorow which takes into account the depth of the taxonomy in which the definitions are found [START_REF] Leacock | Using Corpus Statistics and WordNet Relations for Sense Identification[END_REF]. An information-based technique proposed by Resnik, which computes common information between concepts represented by their common ancestor subsuming both concepts found in the taxonomy to which they belong [START_REF] Resnik | Using Information Content to Evaluate Semantic Similarity in a Taxonomy[END_REF]. Finally, the Vector Pairs technique which works by comparing the co-occurrence vectors from the WordNet definitions of concepts [START_REF] Banerjee | Extended Gloss Overlaps as a Measure of Semantic Relatedness[END_REF].

Table 2 shows the results for the benchmark dataset. As can be seen, the emerging methods are much better than those based on dictionaries. The reason is that by using Google, it is possible to have access to fresher and up to date content. On the other hand, we can see that the best methods are those based on co-occurrence, pattern finding and in part for trend analysis. Text snippet comparison seems to less effective, but these results may be influenced by the fact that our implemented method is simple. More complex methods based on this paradigm could be better, at least when solving specific scenarios. Finally, we have found that the classic methods (Vector pairs, Lesk, Path Length and Resnik), thus, those based on dictionaries are much worse than the majority of the emerging ones, thus, our initial hypothesis is confirmed. Moreover, it is necessary to take into account that most of the methods explained here are apt for optimization, although this step is beyond the scope of this work.

Conclusions

In this paper, we have presented and evaluated a set of novel techniques for determining the semantic similarity between (sets of) terms which of using knowledge from web search engines. All methods reviewed have been evaluated using a benchmark dataset for terms which are not often included in dictionaries, taxonomies or thesaurus. As a result, we have shown experimentally that some of the methods based on Web Intelligence significantly outperform existing methods when evaluating this kind of dataset.

For future work, we want to avoid the cognitive bias associated with the fact that people rate our term pairs in many different ways according to their cultural background. There are terms that are perfect synonyms to us, but people from other cultures do not agree (and vice versa), so in the future it will be necessary to reach a common agreement on the data used to evaluate the different approaches. Moreover, we are going to keep working towards applying novel time series comparison algorithms, because we think that is an area little explored and can lead to success if the appropriate time series algorithms are used. The final goal is to determine which the best approaches for solving this problem are, and implement them in real information systems where the automatic computation of semantic similarity between terms may be necessary.

Table 1

 1 Benchmark dataset containing the similarity scores for a set of terms and expressions which are not frequently covered by dictionaries

	Term	Term	Score
	peak oil	apocalypse	0.056
	bobo	bohemian	0.185
	windmills	offshore	0.278
	copyleft	copyright	0.283
	tweet	snippet	0.314
	subprime	risky business	0.336
	imo	in my opinion	0.376
	buzzword	neologism	0.383
	quantitave easing	money flood	0.410
	glamping	luxury camping	0.463
	slumdog	underprivileged	0.482
	i18n	internationalization	0.518
	vuvuzela	soccer horn	0.523
	pda	computer	0.526
	sustainable	renewable	0.536
	sudoku	number place	0.538
	terabyte	gigabyte	0.573
	ceo	chief executive officer	0.603
	tanorexia	tanning addiction	0.608
	the big apple	New York	0.641
	asap	as soon as possible	0.661
	qwerty	keyboard	0.676
	thx	thanks	0.784
	vlog	video blog	0.788
	wifi	wireless network	0.900
	hi-tech	high technology	0.903
	app	application	0.915