Fractional Piola identity and polyconvexity in fractional spaces

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Abstract

In this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption of polyconvexity of the energy density, and, as a consequence, the existence of solutions to the associated Euler–Lagrange system of nonlinear fractional PDE. The main ingredient is the fractional Piola identity, which establishes that the fractional divergence of the cofactor matrix of the fractional gradient vanishes. This identity implies the weak convergence of the determinant of the fractional gradient, and, in turn, the existence of minimizers of the nonlocal energy. Contrary to local problems in nonlinear elasticity, this existence result is compatible with solutions presenting discontinuities at points and along hypersurfaces.

Introduction

In the last years there has been a renewed interest in variational problems involving the so-called Riesz s-fractional gradient, which for a function uCc(Rn) is defined asDsu(x)=cn,spvxRnu(x)u(y)|xy|n+sxy|xy|dy,xRn, where pvx stands for the principal value centred at x, and cn,s is a suitable constant. In the recent references [32], [33], variational principles for functionals depending on this fractional gradient are addressed, as well as the fractional PDE derived from those as equilibrium equations. The authors consider typical calculus of variations problems, with standard growth conditions in which the classical (local) gradient is substituted by Dsu. This naturally leads to the consideration of the spaceHs,p(Rn)={uLp(Rn):DsuLp(Rn,Rn)}. It is also of interest the affine subspace of functions verifying a complement value condition; to be precise, given gHs,p(Rn) and a bounded domain ΩRn, we considerHgs,p(Ω)={uHs,p(Rn):u=g in Ωc}, where Ωc stands for the complement of Ω in Rn. References [32], [33] deal with integral functionals of the formI(u)=RnW(x,u(x),Dsu(x))dx with p-growth conditions, and prove the existence of minimizers in Hgs,p under the fundamental hypothesis of convexity of W in the last variable, as well as natural coercivity conditions. Taking advantage of the fractional framework and the properties of Riesz potentials and Fourier transform, they also show some remarkable results on the functional spaces Hs,p, including a fractional Sobolev-type inequality, the compact embedding into Lp and the equivalence with Bessel spaces (see [1], [38], [28]). The book [26, Ch. 15] also pays attention to the s-fractional gradient, providing a proof not based on Fourier transform of a fractional fundamental theorem of calculus, also proved in [32], which is used for showing a Sobolev type inequality from Hs,p to Lp. Another reference also dealing with the fractional gradient in the case p=1 is [29].

Even more recent references are [8], [35]. In [35], the s-fractional gradient together with the s-fractional divergence are studied in a systematic manner. Several important properties are given, such as the uniqueness up to a multiplicative constant of the fractional gradient under natural requirements (invariance under translations and rotations, homogeneity under dilations and some continuity properties in an appropriate functional space), and some fractional calculus rules. The results in [35] established, both from a mathematical and physical perspective, what was pointed out earlier in [32], [33], [26], namely, that the s-fractional gradient is the natural definition for a fractional differential object. We agree with the previous references on the claim that the s-fractional gradient deserves more attention in the literature, and likely there will be both more theoretical studies and applications in different contexts. In [8], it is addressed the space of functions u whose total fractional variation is finite, naturally leading to the definition of the space BVs(Rn) and to an s-fractional Caccioppoli perimeter concept. Several interesting results are shown, including a continuous embedding of fractional Sobolev spaces into BVs, a Sobolev-type inequality, a coarea formula, an s-fractional isoperimetric inequality and a natural s-fractional analogue of De Giorgi's notion of reduced boundary.

The definition of the fractional gradient (1) has an important drawback when thinking about certain applications, for instance, in nonlocal solid mechanics, since it requires an integration in the whole space for the computation of the gradient. This is somehow meaningless in those contexts and, in addition, makes it difficult the extension to more realistic boundary conditions. In [24], a definition of a nonlocal gradient in bounded or unbounded domains, for which (1) is a particular case, is introduced. Moreover, the localization of this nonlocal gradient when the nonlocality parameter (horizon) δ goes to zero is analyzed, showing the convergence to the local gradient in different topologies. For a domain ΩRn, the nonlocal gradient is defined asDΩ,δu(x)=pvxΩu(x)u(y)|xy|xy|xy|ρδ(xy)dy, where the function ρδ, typically radial, is an integral kernel reflecting the forces of interaction exerted by particles separated by a positive distance smaller than δ. Typically ρδ vanishes outside the ball of radius δ centred at the origin. The oldest references we are aware of regarding nonlocal gradients of this kind used for models in continuum mechanics are [13], [12], where a nonlocal model of elasticity was proposed. More recently, a nonlocal alternative theory in solid mechanics, named peridynamics, has been proposed in [37], [36]. The development of peridynamics in the last years has been impressive; however, most of the work until now concerns linear elastic models. In [22], [23], a linear elasticity model in the context of peridynamics is rigorously studied, proving the existence and uniqueness of solutions and their convergence to the local Lamé system of linear elasticity as the horizon goes to zero. In these references a projected version of nonlocal gradient (3) is used in the context of the so-called state based peridynamic model. Another interesting approach going from a nonlocal peridynamic framework with finite horizon to a nonlocal fractional situation with infinite horizon in the linear case, appears in the recent references [19], [30], [31]. It is also worth mentioning [14], where both the ill- and well-posedness of the classical Eringen model of nonlocal elasticity are addressed.

In this investigation we deepen in the existence issue for vector variational problems involving the s-fractional gradient, as well as the PDE derived from those as equilibrium conditions. Thus, we consider the more difficult vectorial case under conditions weaker than convexity. To be precise, we establish the existence of minimizers in Hs,p under the polyconvexity assumption of the integrand. Polyconvexity is a central notion in calculus of variations, with essential implications on the existence and stability of solutions in solid mechanics, and particularly in elasticity [2], [9]. In order to obtain our results, we follow the usual steps as for classical polyconvex variational problems, namely, we show that the determinant (or any minor) of the fractional gradient matrix Dsu is continuous with respect to weak convergence in Hs,p. Similarly to the classical case, we need a fractional version of the Piola identity, highlighting this as the key ingredient and the most remarkable contribution of this work. In this new situation we follow [24] to define a fractional divergence and establish an integration by parts formula (see also [26], [35]). We adapt the techniques developed there for some spaces of nonlocal type to our Hs,p spaces. More concretely, as in (1), the Riesz s-fractional gradient of a vector field u:RnRn isDsu(x)=cn,spvxRnu(x)u(y)|xy|n+sxy|xy|dy,xRn, where ⊗ denotes the tensor product, and the Riesz s-fractional divergence of a vector field ψ:RnRn is defined asdivsψ(x)=cn,spvxRnψ(x)+ψ(y)|xy|n+sxy|xy|dy,xRn. As mentioned above, we prove the fractional Piola identity DivscofDsu=0 (where Divs means the s-divergence by rows), then we show the weak continuity of detDsu, the weak lower semicontinuity of polyconvex functionals in Hs,p, and finally we settle the existence of minimizers for (2). We believe that the fractional Piola identity is a result of interest in itself. On the one hand, it may serve to show analogous versions in the fractional or nonlocal situations of classical results in whose proof the Piola identity is invoked, as for instance, the change of variables formula for surface integrals. On the other hand, it may also be useful in other fractional or nonlocal models in different contexts, such as fluid mechanics [11]. Furthermore, an extension to a nonlocal Piola identity for nonlocal gradients defined on bounded domains (see (3)) is easy from the proof we provide here in the fractional framework.

In the last decade there has been a great deal of work on fractional PDE of elliptic type involving the fractional Laplacian in some way. Our results here enlarge this theory by giving an existence result of minimizers of nonlinear fractional vector variational problems based on polyconvexity, which implies, in turn, an existence result of solutions to nonlinear fractional PDE systems. The amount of references on nonlocal equations and fractional Laplacian is overwhelming, so for situations related to this work we just cite the survey [27], the paper [33] and the references therein. On the other hand, we would like to point out the relationship of our study with nonlocal elasticity and peridynamics. As mentioned above, the variational principle considered in this paper is not an appropriate model in solid mechanics, but a version in bounded domains of the functional (2) involving the nonlocal gradient (3), satisfying additional requirements in order to be physically consistent, fits into the peridynamics state-based model for large deformations [36]. Whereas in Hs,p(Rn), the structural functional analysis facts necessary to prove an existence theory for functionals like (2) are known (continuous and compact embeddings into Lp), those are still unknown for an analogous version of this space in bounded domains. In this sense, and since the proof provided for the fractional Piola identity may be adapted in a more or less straightforward way to bounded domains, we think this study may be seen as a first step towards a rigorous mathematical theory of nonlocal hyperelasticity. Furthermore, one primary interest for us is that Hs,p is larger than W1,p, and functions in Hs,p may exhibit singularities prohibited in W1,p, as we point out in Section 2. We would like to emphasize that, contrary to classical elasticity, both singularities along points (cavitation) and hypersurfaces are compatible with the existence of solutions in Hs,p (see Theorem 6.1). This seems to indicate that the Lp norm of Dsu not only contributes to the elastic energy, but also to a kind of surface energy, since the latter is necessary in the modelling of such singularities (see, e.g., [4], [25], [10], [18]).

The outline of the paper is the following. Section 2 introduces the functional space of fractional type Hs,p(Rn) and its main properties. We also include examples of functions exhibiting singularities belonging to these spaces but not to Sobolev spaces. Section 3 contains some calculus facts in Hs,p, such as the formulas for the fractional derivative of a product and the fractional integration by parts. In Section 4 we prove the fractional Piola identity. Section 5 shows the weak continuity of minors in Hs,p. Finally, in Section 6 we prove the existence of minimizers of (2) for polyconvex integrands, and obtain the associated Euler–Lagrange system of fractional PDE.

Section snippets

Functional analysis framework

This section introduces general properties of the functional space Hs,p. We start by setting the definition of principal value. Given a function f:RnR and xRn such that fL1(B(x,r)c) for every r>0, we define the principal value centred at x of Rnf, denoted bypvxRnforpvxf, aslimr0B(x,r)cf, whenever this limit exists. We have denoted by B(x,r) the open ball centred at x of radius r, and by B(x,r)c its complement. As most integrals in this work are over Rn, we will use the symbol ∫ as a

Calculus in Hs,p

In this section we present some calculus rules of nonlocal functionals related to Hs,p, notably, an integration by parts formula.

We start with a sufficient condition for the s-fractional gradient to be defined everywhere.

Lemma 3.1

Let 0<α<s<1 and φC0,α(Rn)C0,1(Rn). ThensupxRn|φ(x)φ(y)||xy|n+sdy<. If, in addition, φ has compact support then DsφLr(Rn), for every r[1,].

Proof

Let L and C be, respectively, the Lipschitz and α-Hölder constants of φ. Then, for every xRn,|φ(x)φ(y)||xy|n+sdyB(x,1)L|xy|

Fractional Piola Identity

In this section we introduce a fractional version of the Piola Identity. This is the main step in order to prove the existence of solutions for our fractional energy, since it will allow us to prove the weak continuity in Hs,p of the determinant of the s-fractional gradient. Recall that the classical Piola identity asserts that, for smooth enough functions u:ΩRnRn one has DivcofDu=0. Of course, cof denotes the cofactor matrix, which satisfies cofAAT=(detA)I for every ARn×n.

Contrary to the

Weak continuity of the determinant

In this section we prove that any minor (determinant of a submatrix) of Dsu is a weakly continuous mapping in Hs,p. We start by expressing a nonlocal integration by parts formula for the minors of Dsu that involves the operator Kφ of Lemma 3.2. Recall that for any FRn×n and 1in we denote by Fi the i-th row of F.

Lemma 5.1

Let kN be with 1kn. Consider indices 1i1<<ikn and 1j1<<jkn and the functionsM=Mi1,,ik;j1,,jk,M¯=M¯i1,,ik;j1,,jk,N˜=N˜i1,,ik of Definition 4.1. Let pk1, qpp1 and 0<s<1

Existence of minimizers and equilibrium conditions

In this section we prove the existence of minimizers in Hs,p of functionals of the formI(u):=W(x,u(x),Dsu(x))dx, under natural coercivity and polyconvexity assumptions. We also derive the associated Euler–Lagrange equation, which is a fractional partial differential system of equations.

We recall the concept of polyconvexity (see, e.g., [2], [9]). Let τ be the number of submatrices of an n×n matrix. We fix a function μ:Rn×nRτ such that μ(F) is the collection of all minors of an FRn×n in a

Declaration of Competing Interest

Authors declare that they do not have competing interest.

Acknowledgements

This work has been supported by the Spanish Ministerio de Ciencia, Innovación y Universidades through projects MTM2017-83740-P (J.C.B. and J.C.), and MTM2017-85934-C3-2-P (C.M.-C.).

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