# Asymptotic Analysis of a Coupled System of Nonlocal Equations with Oscillatory Coefficients
^{†}^{†}thanks: Support from NSF DMS-1910180 and DMS-1615726 is gratefully acknowledged.

###### Abstract

In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous media that additionally accounts for short-range forces. We consider the vanishing nonlocality limit on the same length scale as the heterogeneity and show that the system’s effective behavior is characterized by a coupled system of local equations that are elliptic in the sense of Legendre-Hadamard. This effective system is characterized by a fourth-order tensor that shares properties with Cauchy elasticity tensors that appear in the classical equilibrium equations for linearized elasticity.

remarkRemark \newsiamremarkhypothesisHypothesis \newsiamthmclaimClaim \headersAsymptotics of a nonlocal systemT. Mengesha and J. Scott

eridynamics, systems of integro-differential equations, periodic homogenization, elliptic systems, asymptotic compatibility, nonlocal-to-local limit

35R09, 74Q05, 35J47

## 1 Introduction and statement of main results

Given and a vector field , we study the asymptotic behavior of solutions to the system of equations given by

(1.1) |

where for each the operator is defined as

(1.2) |

with and the functions and are bounded and nondegenerate. We also assume that is periodic while is periodic in the second variable, both with unit period.

We will show that for a given , the sequence of solutions converge strongly in . Moreover, the limiting vector field solves a strongly coupled system of partial differential equations whose coefficients depend on the effective properties of and .

This work is motivated by the multiscale analysis of the displacement of heterogeneous media in the peridynamic formulation [Silling2000, Silling2007], a non-local continuum theory for deformable media that incorporates long range interactions of material points via a force field. Parametrized operators of the type were first introduced in the work [Alali2012Multiscale] as a means of representing short range forces in the modeling of the deformation of media with heterogeneities. These short range forces are represented at a “microscopic” scale relative to the heterogeneities, hence the dependence of and on . In the absence of the diffusive scaling in the operator , the asymptotic properties of solutions of (1.1) is studied in [Alali2012Multiscale, du2016multiscale, DuMengeshaMultiscale15] for both stationary and dynamic problems. In this special case, using the method of two scale convergence [Allaire-Twoscale, Nguetseng-twoscale], it has been shown that solutions to the nonlocal system of equations that use the operator (1.2) indeed exhibit multiscale behavior and most importantly, their effective or homogenized behavior can be captured by a vector field that is a solution to a (homogenized) system of nonlocal equations. It has also been proved that in the presence of the diffusive scaling but in the absence of oscillatory heterogeneity (i.e. ) that the integral operator converges to the Lamé-Navier differential operator from classical linearized elasticity and solutions to the peridynamic-type nonlocal system of equations (1.1) also converge to the solution to the corresponding equations of linearized elasticity. This, which is usually referred as nonlocal-to-local convergence, has been demonstrated throughout the literature; see for example [Emmrich-Weckner2007, MengeshaDuElasticity, Du-Zhou2010, Silling2008]. This paper makes an effort to marry these two sets of results and thus attempts to provide a more complete picture of asymptotic regimes in peridynamic formulations. To be precise, we study the asymptotic properties of the operator and solutions to (1.1) in the presence of both the diffusive scaling and the oscillatory coefficients corresponding to the same length scale.

Periodic and stochastic homogenization of integro-differential operators is currently developing in a variety of directions; see [schwab2010periodic, bonder2017h] for some examples. The asymptotic analysis we present here follows the argument presented in the recent paper [Piatnitski-Zhizhina] which is focused on the homogenization of nonlocal equations that are based on integral operators with convolution-type kernels. The approach is essentially the classical H–convergence of elliptic (differential) operators [Murat-Tartar, Tartar, jikov2012homogenization] applied to nonlocal problems with integrable kernels. It turns out that, although the operator in (1.2) is vector-valued and the resulting system is strongly coupled, the operator shares important features with the scalar-valued nonlocal operator studied in [Piatnitski-Zhizhina]. In fact, following the approach in [Piatnitski-Zhizhina] we will show that the sequence of operators will “converge” to a second-order system of elliptic differential operators in nondivergence form with variable coefficients. This convergence will be demonstrated via convergence of resolvents, i.e. for large enough the operators converge strongly in to the operator , where

and denotes the identity matrix. The limiting system of partial differential operators closely resembles the equilibrium equation for linearized elasticity in the sense that its fourth-order tensor of coefficients satisfies the Cauchy symmetry relations from linearized elasticity.

To state the precise statement of the main result, let us fix some notations. We denote the space of matrices with real coefficients by . If a vector field has the property that each of its components belongs to a function space , then we denote the associated vector space of vector fields by . For example, Lebesgue spaces of vector fields will be denoted . Function spaces of higher-order tensor fields will be denoted similarly, i.e. the Lebesgue space is written as . We denote the space of Schwarz functions by . We denote the space of bounded linear operators from a Banach space to a Banach space by . We designate the set of infinitesimal rigid displacements as

The kernel , in addition to being in also satisfies

(A1) |

We also assume that is nondegenerate in the sense that there exists and a symmetric cone with vertex at the origin such that

(A2) |

By “symmetric cone with vertex at the origin” we mean that there exists an open subset of the unit sphere with such that the set can be written as

(A3) |

Throughout this work we will identify periodic functions defined on all of with functions defined on the torus . A consequence of the periodicity, such functions satisfy the identity , for every . For any , the space of periodic functions is defined as

For a given positive integer , the Sobolev space consists of functions whose -order tensor of partial derivatives belongs to for all ; precisely,

with norm

It is standard that if and only if and all of its partial derivatives of up to order are all in . Given a vector field , denotes the gradient matrix of given by , and is the third-order tensor of second partial derivatives of given by . Higher order tensors of partial derivatives will also be denoted in a similar fashion. Finally, we use the standard “contraction of indices” convention and Einstein summation notation when denoting actions of tensors unless specified otherwise. We can now state the main result of the paper.

###### Theorem 1

Suppose that and satisfy (A1), (A2), and (A3). Then there exists a constant such that for all , the resolvents converge strongly to as . Precisely, for every , if is a solution to (1.1), then

where solves the equation

(1.3) |

and the operator is a second-order system of linear differential operators whose tensor of coefficients is elastic and is infinitely differentiable.

Some remarks are in order. In the theorem, by elastic tensor we mean a fourth-order tensor satisfying the symmetries

(1.4) |

for every , and that for some positive constants and the inequalities

(1.5) |

hold uniformly in and for all symmetric matrices and . We have used the generic inner product notation . The tensor of coefficients of will be defined in terms of tensor-valued corrector functions that solve a system of auxiliary cell problems. More importantly, for each is of the form

(1.6) |

As a consequence of this and the smoothness assumption (A3) on , is infinitely differentiable.

(In)equalities (1.4) and (1.5) are instrumental in showing that the resolvent is well-defined. In fact, (1.4) and (1.5) imply that satisfies the Legendre-Hadamard condition, namely and

This condition is sufficient to guarantee the existence of the resolvent using a priori estimates for elliptic systems combined with the method of continuity [DoKi09, krylov1996lectures, GiaquintaBook]. In fact, we can apply [DoKi09, Theorem 2.6] to conclude that there exists such that for any and any , there exists a unique vector field with the estimate

proving that the resolvent is a well-defined bounded operator. From the smoothness of and standard regularity theory we have that if , then so is , see [DoKi09, krylov1996lectures, GiaquintaBook].

Finally, if we take and take to be radial, i.e. , then the constant tensor defined in (1.6) is exactly the elasticity tensor associated to the Lamé-Navier system. Specifically, we have

(1.7) |

Above, denotes the Kronecker -function. It is straightforward to check that this constant tensor when used in the definition of gives

(1.8) |

where the Lamé parameter is defined as

This form of the limiting operator is expected; once the heterogeneity is removed from (1.1) the convergence then resembles the nonlocal-to-local limits considered in [Emmrich-Weckner2007, MengeshaDuElasticity, Du-Zhou2010, Silling2008]. The computation of (1.7) is summarized in the appendix.

The paper is organized as follows: In Section 2 we collect tools and general results needed for subsequent sections. In Section 3 we set up the program of asymptotic analysis via correctors and prove Theorem 1, deferring proofs regarding existence of the correctors (such as the solvability of the auxiliary cell problem) to Section 4.

## 2 Tools and preliminaries

In this section we collect tools that we will need throughout the paper. We will also prove some preliminary results related to the main operators of interest. Other general results that we need later in the paper will also be discussed.

### 2.1 Existence and uniform estimates for resolvents

For satisfying the condition (A1) and , define and . We also denote the symmetrized form of by . Then

The function is clearly periodic in both variables. We use this notational convention to introduce the matrix-valued functions

(2.1) |

Notice that for each and any , is positive semi-definite matrix and for any vector , we have . Moreover, by a change of variables, it is clear that . The estimate (A3) on gives for every .

Using these matrix-valued functions, we define the operators and for by

For , we simply write , , and as opposed to , , , and , respectively.

We can then rewrite the main operator introduced in (1.2) as

The operator is a combination of convolution and multiplication operators, and can be rewritten as

where denotes the multiplication operator, i.e. for any bounded function and the convolution is defined in terms of matrix multiplication, i.e.

We will use this formulation of the operator and the following result repeatedly throughout this work. The proof of the following lemma mimics that given in [Piatnitski-Zhizhina] appropriately modified to fit our framework.

###### Lemma 1

###### Proof

The boundedness of follows from a trivial application of Young’s inequality as

and so

To prove the invertibility of we introduce the weighted Lebesgue space with weight . Then

which implies . Thus is a bounded linear operator. Further, is self-adjoint in , and . Indeed, splitting the double integral, interchanging the role of and and using Fubini’s theorem and the fact that ,

Setting in the last line gives . Repeating these steps in reverse order with the roles of and interchanged, one sees that . Thus for we have . This implies by the Cauchy-Schwarz inequality that

and therefore we have

(2.3) |

As a consequence, Range is closed and kernel. It then follows that Rangekernel. Thus is a bounded linear operator. Setting for in (2.3) gives a bound of for the operator norm of over . Notice that the bound is uniform in . Finally, since , with norms comparable with constants independent of the operator is well-defined, linear and bounded, with operator norm bounded independent of .

### 2.2 Some operators on function spaces of periodic functions

As we will see in the next section, the operators and will be applied not only to functions in but also to functions in . We now summarize basic properties of these operators as linear maps on . We begin with the following proposition.

###### Proposition 1

###### Proof

Since the matrix uniformly in , is a bounded linear operator. To see that is invertible, it suffices to show that the symmetric matrix is positive definite, i.e. there exists a constant such that

(2.4) |

To prove this, first note that

(2.5) |

Since , the quantity cannot be identically zero on . Therefore for each we have . Since the mapping is continuous on a compact set, it follows that for all with lower bound independent of by (2.5).

###### Proposition 2

###### Proof

Given , we first show that . This follows from Tonelli’s theorem and Young’s inequality since by periodicity we have

As a consequence, for almost all , . That is, is well-defined for almost all . Linearity of the operator is obvious. A similar argument also shows that for and the function . Now for any , we have after a change of variables, Hölder’s inequality, and Tonelli’s theorem

To show compactness of the operator, we show that is a uniform limit of bounded, compact linear operators. To that end, let and define

where is a matrix with bounded support. We claim that for each , the operator is compact. Assuming this and applying the previous estimate for the operator ,

and thus is compact as it is the limit (in the operator norm) of compact operators .

To see that is compact, we use the periodicity of and to write

Writing , and using that

We will show that consists of – in the appropriate sense – a finite linear combination of convolution and multiplication operators. This fact must be shown carefully, as is periodic and thus does not necessarily belong to a Lebesgue space defined over all of . With this goal in mind, define as