Abstract
Lattice defects in crystalline materials create long-range elastic fields which can be modelled on the atomistic scale using an infinite system of discrete nonlinear force balance equations. Starting with these equations, this work rigorously derives a novel far-field expansion of these fields. The expansion is computable and is expressed as a sum of continuum correctors and discrete multipole terms which decay with increasing algebraic rate as the order of the expansion increases. Truncating the expansion leaves a remainder describing the defect core structure, which is localised in the sense that it decays with an algebraic rate corresponding to the order at which the truncation occurred.
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1 Introduction
The field of continuum solid mechanics has been highly successful in providing robust predictions of material behaviour at a wide range of length-scales. In crystalline materials in particular, it is recognised that the predictions made by the equations of linear elasticity are valid with tolerable errors even when resolving features such as defects whose characteristic size is close to that of the interatomic spacing. This said, when considering processes which involve the genuine thermodynamic, electronic and chemical properties of such defects, the fundamental discreteness of matter becomes crucial and no single continuum model can be sufficient to completely capture the fine detail of a material’s behaviour at this scale. Moreover, it is exactly these fine details which determine phenomena such as a material’s yield strength and behaviour under cyclic loading, both of which are crucial to understand for engineering applications.
As a result, a range of theoretical techniques have been developed over the last 60 years which seek to predict and compute defect behaviour in crystals, connecting discrete and continuum models of these materials. Broadly, these approaches can be divided into two categories, namely concurrent and sequential modelling strategies. Models in the former class combine discrete and continuum models into a single system which can then be numerically solved simultaneously, while those in the latter class generally involve iteration over separate models acting at different scales. In particular, the last 25 years has seen a great deal of research activity focused on concurrent strategies, with one of the most significant developments in this area being the quasicontinuum method [1] and many variants thereof [2]. In contrast, the present work revisits sequential strategies for accurately modelling defects, but with a new perspective that recent progress in the study of multi-scale models has enabled, and our results on the structure of crystalline defects have direct consequences for the design and evaluation of more general models, including both purely atomistic and concurrently coupled models.
Starting from a discrete energy for a material defined on an infinite domain [3], we develop a hierarchy of linear continuum PDE systems which can (in principle) be efficiently derived and solved numerically. The solutions to these PDE systems form a sequence of smooth predictors which describe the far-field behaviour of the lattice strain around a defect to within arbitrary accuracy, and thus provide increasingly accurate boundary conditions to be used on the discrete model when confined to a finite domain. The key idea behind our approach is to exploit the knowledge that the variation in the strain field generally decays smoothly away from the core of any localised defect, and hence the far-field behaviour is more and more accurately predicted by the solutions of the continuum PDE models we define. These far-field approximate solutions are coupled with the properties of the discrete defect core, encapsulated by the spatial moments of the acting forces and expressed as a multipole expansion. The relative simplicity of these moments provides an elegant, computable way to transfer information from the nonlinear discrete problem to the continuum hierarchy. The coupling moreover is “weak” in the sense that a term in the continuum hierarchy only requires information on the multipole terms of strictly lower order. Indeed all terms are defined and are computable in sequence without concurrent coupling.
Our approach has connections with classical approaches to modelling defects in continuum linear elasticity using the defect dipole tensor [4, 5] (also known as the elastic dipole tensor or the double force tensor) and to Sinclair’s work on atomistic models of fracture [6, 7]. More recent related work includes that of Trinkle and coworkers, where lattice Green’s functions have been used to improve the accuracy of defect computations [8, 9], along with mathematical developments in our understanding of the regularity of discrete strain fields advanced by the authors and coworkers [3, 10,11,12]. In particular, [12] explores some of the initial ideas of our mathematical strategy in a simplified setting. The approach presented here unifies many of the ideas involved in these previous works into a single framework and expands them systematically to higher orders.
More generally, the structural results we present here serve as a useful tool in any discussion of crystalline defects where high accuracy is required. While the present work focuses primarily on the analytical background, in Section 3.3 we will briefly sketch out how our results might be used as a foundation for a rigorous numerical analysis of defect algorithms and also provide a path to systematically improve their accuracy.
1.1 Methodology
Our starting point is to consider a total energy \({\mathcal {E}}(u)\) for displacements u of an infinite lattice \(\Lambda \) of atoms. We will make the mild assumption that the total energy of a displacement is expressed as a sum of site energies, that is contributions to the total energy arising from the environment of each atom. Under the assumption of frame indifference, this energy (and the site energies which make up the total energy) must depend only upon the relative displacements between atoms.
An equilibrium displacement \({\bar{u}}\) of the energy \({\mathcal {E}}\) satisfies the force balance equation
and it is this infinite system of discrete equations which we study. Since equilibrium displacements \({\bar{u}}\) exhibit decay properties away from the core of point defects and dislocations [3], we can develop these equations around the state of zero displacement to derive approximate equations for the equilibrium displacement \({\bar{u}}\) in the far field. This comes in two steps:
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As a first step, we can expand the site potential around zero to obtain linear and nonlinear lattice operators, which still depend on finite differences (atomic bonds); and
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As a second step, we can use an expansion for the displacement itself to replace finite differences with a gradient and higher derivatives, obtaining continuum PDE approximations to the discrete lattice equations.
Note already, that the latter Taylor expansion requires sufficiently smooth continuum displacements and cannot be applied to the discrete \({\bar{u}}\) itself, so some care needs to be taken when applying this strategy. The PDE approximations come in the form of a hierarchy of corrector equations. Crucially, each one of these corrector equations has the same form: the continuum linear elasticity (CLE) equation must be solved but with different right-hand sides (forces) that depend on previous terms in the expansion. Such hierarchies of corrector equations have been explored by other authors before, for example, see [13] for a formal hierarchy of similar corrector equations in the defect-free setting. In the present work, we take great care to define all terms rigorously, ensure sufficient regularity, and provide sharp estimates for all resulting error terms.
We then combine these continuum correctors with a multipole series which can be obtained from the moments of linearised residual forces. Overall, this enables us to obtain far-field approximations to arbitrarily high order for \({\bar{u}}\), which is characterised by a discrete remainder whose locality (decay) is precisely controlled.
While our general approach and many of our theoretical results are generic, some significant technical and conceptual challenges come into play when applying the results to specific defects. Specifically, questions regarding the geometry and the relation of a defect state to a reference lattice, as well as the precise properties of the lattice Green’s function after adjusting for the geometry. These challenges can sometimes be overcome adhoc in leading order (as in [3] for edge dislocations) but require a more detailed understanding for higher orders. Our main focus here is the general methodology. We will therefore restrict ourselves to two defect types that are geometrically relatively simple: general point defects and (straight) screw dislocations.
1.2 Outline
The paper is organised as follows: In Section 2, we develop our general theory independent of specific defects. We present our main theoretical results concerning the decay of displacement fields and their approximation by multipole expansions for linearised lattice models in Section 2.2 and then outline the derivation of the continuum corrector PDEs in Section 2.3.
In Section 3, we apply our methodology to our nonlinear atomistic models of point defects in Theorem 5 and screw dislocations in Theorem 7, and explain the implications for the convergence of numerical methods which exploit the result. Section 4 presents our conclusions, and discusses the outlook for extending and applying these results in the future.
The proofs of our decay estimates for the linearised models are then provided in Section 5, a full discussion of the lattice Green’s function in Section 6, and the proofs of Theorems 5 and 7 and in Section 7.
2 General Results
2.1 Models and notation
\({\textbf {The Atomistic Energy and General Notation.}}\) Our results are concerned with modelling of crystalline defects, that is local regions of non-uniform atom arrangements embedded in a homogeneous host crystal. In this section we will start with the homogeneous setting itself. In Section 3 we will then look at the description of defect configurations and discuss how the homogeneous results can be applied there.
The homogeneous crystal is described by a Bravais lattice \(\Lambda := A \mathbb {Z}^d\) as the reference, where \(d \geqq 2\) and \(A \in \mathbb {R}^{d \times d}\) is non-singular, and displacements \(u :\Lambda \rightarrow \mathbb {R}^N\), where we allow \(N \ne d\) in order to model a range of scenarios (for example, in some models for pure screw dislocations, we have \(d = 2\), \(N = 3\), while in anti-plane shear \(d = 2\), \(N = 1\)).
We denote discrete differences by \(D_\rho u (\ell ) := u(\ell + \rho ) - u(\ell )\) for \(\ell , \rho \in \Lambda \). Later on we will also look at higher discrete differences which we denote by \(D_{\varvec{\rho }} = D_{\rho _1} ... D_{\rho _j}\) for a \(\varvec{\rho } = (\rho _1, ..., \rho _j) \in \Lambda ^j\).
Next we look at an interaction neighbourhood \({\mathcal {R}}\subset \Lambda / \{ 0\}\). We assume throughout that \({\mathcal {R}}\) is finite, \({\mathcal {R}}= -{\mathcal {R}}\), and that it spans the lattice \({{\,\mathrm{span}\,}}_\mathbb {Z}{\mathcal {R}}= \Lambda \). Based on \({\mathcal {R}}\) we define the discrete difference stencil \(Du(\ell ) := D_{{\mathcal {R}}} u(\ell ) := (D_\rho u (\ell ))_{\rho \in {\mathcal {R}}}\). Again, later we will consider higher discrete differences and in particular apply D k times for which we use the simple notation \(D^k u = D \dots D u\). With the discrete difference stencil we can formally write down the energy of u as
where \(V :\mathbb {R}^{N \times {\mathcal {R}}} \rightarrow \mathbb {R}\) is the site energy. We will assume throughout that \(V \in C^{K}(\mathbb {R}^{N \times {\mathcal {R}}})\) for some K and satisfies the natural and very mild symmetry assumption
Note that this is only a formal definition of the energy as the sum might not converge. All these quantities might also need to be adjusted to inhomogeneous generalisations \({\mathcal {E}}^{\mathrm{def}}\), \(\Lambda ^{\mathrm{def}}\), \({\mathcal {R}}_\ell \), and \(V_\ell \) to allow for a desired defect structure. Both of these aspects will be discussed in detail in Section 3. In particular, we will make \({\mathcal {E}}\) precise and establish differentiability properties for any u in the discrete energy space \({\mathcal {H}}^{1}\) which is defined by
For future reference we also define the dense subspace \({\mathcal {H}}^{c}\subset {\mathcal {H}}^{1}\) of displacements with compact support,
Variations of \({\mathcal {E}}\) can then be written as
More generally, we use the notation \(T[a_1, \dots , a_k]\) for a multi-linear operator T. For symmetric operators we will shorten the notation further and write \(T[a]^k:= T[a,\dots ,a]\) if \(a_1=\dots = a_k = a\) or \(T[a]^m[b]^{(k-m)}\) if m inputs are a and \(k-m\) are b. We denote the Hessian at zero by
It will be convenient to interpret these objects as linear functionals, belonging to \(({\mathcal {H}}^{1})^*\), acting on the last test function. We will often use a point wise representation based on the \(\ell ^2\) scalar product. For example,
and specifically
where \({{\,\mathrm{Div}\,}}A = -\sum _{\rho \in {\mathcal {R}}} D_{-\rho } A_{\cdot \rho }\) is the discrete divergence for a matrix field \(A :\Lambda \rightarrow \mathbb {R}^{N\times {\mathcal {R}}}\).
In this notation we can also write down the force equilibrium equations \(\delta {\mathcal {E}}(u)=0\) in the pointwise form
\({\textbf {Lattice Stability and The Green's Function}}\) We assume throughout that the Hamiltonian \(H=\delta ^2 {\mathcal {E}}(0)\) is stable and will equivalently call the lattice stable (see [10]), which by definition is the case if and only if there exists a \(c_0 > 0\) such that
For stable operators H there exists a lattice Green’s function \({\mathcal {G}}: \Lambda \rightarrow \mathbb {R}^{N \times N}\) such that \(H[{\mathcal {G}}e_k](\ell ) = e_k \delta _{\ell ,0}\) for all \(1 \leqq k \leqq N\) and such that
see [3]. Here we used the notation \({|{\ell } |}_{0} := |\ell |+ 2\) to write decay rates in the discrete setting in a more compact form which we will do throughout. We will also often just write \({\mathcal {G}}_k := {\mathcal {G}}e_k\).
\({\textbf {The Cauchy--Born Continuum Model}}\) Our atomistic lattice model naturally gives rise to corresponding continuum model based on the Cauchy–Born rule. In the continuum setting, one has an energy of the form
for a displacement \(u :\mathbb {R}^d \rightarrow \mathbb {R}^N\). The energy density W is given by the Cauchy–Born rule
for any \(M \in \mathbb {R}^{N \times d}\), where \(c_{\mathrm{vol}} = |\det A |> 0\) is the volume of a lattice cell.
We will later see in more detail the usefulness and limitations of the nonlinear continuum model for defect problems. We will also make heavy use of the linearised continuum problem for our corrector equations. These are given through the continuum Hamiltonian \(H^{\mathrm{C}}= \delta ^2 {\mathcal {E}}^{\mathrm{C}}(0)\). In our pointwise notation the equilibrium equation is
or
where \(\mathbb {C} = \nabla ^2 W(0)\). These are the standard continuum linear elasticity (CLE) equations.
The lattice stability (4) in particular also implies the Legendre-Hadamard stability of \(\mathbb {C}\) (see [10, 11]), so that (7) is elliptic and allows for a continuum Green’s function or fundamental solution \(G^{\mathrm{C}} :\mathbb {R}^d / \{0\} \rightarrow \mathbb {R}^{N \times N}\) that solves
in the distributional sense for \(1 \leqq k \leqq N\).
\({\textbf {Notation for Tensors}}\) To work with various higher order tensor products throughout, we establish a precise but compact notation: Given a k-tuple of vectors in \(\mathbb {R}^d\), \(\varvec{\sigma }=(\sigma ^{(1)}, \dots , \sigma ^{(k)} ) \in (\mathbb {R}^d)^k\) we denote their k-fold tensor product
The vector space spanned by these tensor products is denoted \((\mathbb {R}^d)^{\otimes k}\), and it is easy to see this space is isomorphic to \(\mathbb {R}^{d^k}\) with the natural basis
We also write
when considering the k-fold tensor product of a single vector \(v \in \mathbb {R}^d\).
Let \(S_k\) denote the usual symmetric group of all permutations which act on the integers \(\{1,\ldots ,k\}\). This action can be extended to k-tuples and tensor products by defining
for any \(\pi \in S_k\) and \(\varvec{\sigma } \in (\mathbb {R}^d)^k\). For any \(\varvec{\sigma }\in (\mathbb {R}^d)^k\), we define the symmetric tensor product by
The space spanned by these symmetric tensors is then denoted by \((\mathbb {R}^d)^{\odot k}\), and is a vector subspace of \((\mathbb {R}^d)^{\otimes k}\).
The natural scalar product on \((\mathbb {R}^d)^{\otimes k}\) and \((\mathbb {R}^d)^{\odot k}\) is denoted by A : B for \(A,B \in (\mathbb {R}^d)^{\otimes k}\) and, as usual, is defined to be the linear extension of
In the atomistic setting, we also have to work with the finite set \({\mathcal {R}} \subset \Lambda \) as a reference set instead of the usual \(\{1,...,d\}\). This results in the tensor spaces \((\mathbb {R}^{{\mathcal {R}}})^{\otimes k}\) and \((\mathbb {R}^{{\mathcal {R}}})^{\odot k}\). As \({\mathcal {R}}\) is still a finite reference set, nothing changes except for the labeling.
The following special case will come up regularly: for \(u :\Lambda \rightarrow \mathbb {R}^N\) we have \(D^k u(\ell ) \in \mathbb {R}^N \otimes (\mathbb {R}^{{\mathcal {R}}})^{\otimes k}\). For a tensor \(\mathbf{C}\in (\mathbb {R}^{{\mathcal {R}}})^{\otimes k}\), given component-wise as the sum \(\mathbf{C}= \sum _{\varvec{\rho } \in {\mathcal {R}}^{k}} \mathbf{C}_{\varvec{\rho }} E_{\varvec{\rho }}\), we can then use the scalar product for each \(D^k u_i\), \(1 \leqq i \leqq N\) and can thus extend the notation to write
2.2 General results for the linearised equation
At the most fundamental level, our results concern the characterisation of the far-field behaviour of lattice displacements \(u : \Lambda \rightarrow \mathbb {R}^N\) that are close to equilibrium in the far-field. More precisely, given a stable Hamiltonian H as introduced in the previous section we will characterise the decay of a general lattice displacement u provided that the (linearised) residual forces \(f(\ell ) := H[u](\ell )\) decay sufficiently rapidly as \(|\ell |\rightarrow \infty \). In Section 2.3 we will then show how to use these results for the linearised operator to obtain characterisations of the far-field behaviour of equilibrium displacements in our full nonlinear interaction model.
If \(\ell \mapsto H[u](\ell ) \otimes \ell ^{\otimes j} \in \ell ^1(\Lambda )\) we define the j-th moment
With this definition we have the following result:
Theorem 1
Assume \(|H [u](\ell ) |\lesssim {|{\ell } |}_{0}^{-d-p} \log ^\alpha {|{\ell } |}_{0}\) with \(p, \alpha \in \mathbb {N}_0\), as well as \({\mathcal {I}}_i=0\) for \(i=0, \dots , p-1\). Then, for \(j = 1, 2\),
If \(j \geqq 3\), then Equation (9) is still true under the additional assumption that \(|D^m H [u](\ell ) |\lesssim {|{\ell } |}_{0}^{-d-p-m} \log ^\alpha {|{\ell } |}_{0}\) for \(m=1,...,j-2\).
Remark 1
If H[u] satisfies the assumptions for some \(\alpha < -1\) instead of \(\alpha \in \mathbb {N}_0\), then there is no logarithmic factor needed in the result, that is,
The same holds true for the Theorems 2 and 4 below, if \(\alpha < -1\).
Remark 2
When initially reading both this theorem and the theorems in Sections 2 and 3, we suggest to ignore the logarithmic terms and focus only on the algebraic rates. However, the treatment of the logarithmic terms is an important aspect of both our theorems and proofs since they appear to be intrinsic to the expansion, and not due to suboptimal estimates.
Suppose now that we have a general elastic field u with non-vanishing moments (8) but still fast decay of the residual forces. Then, we can decompose u into a truncated multipole expansion — higher order derivatives of the lattice Green’s function defined in Section 2.1 — corresponding to the non-vanishing moments and a far-field remainder that exhibits the improved decay established in Theorem 1. This idea is made precise in the next result.
Theorem 2
Assume \(|H [u] |\lesssim {|{\ell } |}_{0}^{-d-p} \log ^\alpha {|{\ell } |}_{0}\) with \(p, \alpha \in \mathbb {N}_0\). Furthermore, let \({\mathcal {S}} \subset \Lambda \) be linear independent with \({{\,\mathrm{span}\,}}_\mathbb {Z}{\mathcal {S}}=\Lambda \). Then there are coefficients \(b^{(i,k)} \in (\mathbb {R}^{{\mathcal {S}}})^{\odot i}\), \(i=0, \dots , p-1\), \(k=1,\dots , N\), such that
and the remainder decays as
for \(j=1,2\).
If \(j \geqq 3\), then Equation (12) is still true under the additional assumption that \(|D^m H [u] |\lesssim {|{\ell } |}_{0}^{-d-p-m} \log ^\alpha {|{\ell } |}_{0}\) for all \(m=1,...,j-2\).
Remark 3
In Lemma 14 we will in fact see that the \(b^{(i,k)}\) can be obtained through a linear relationship from the force moments (8).
It is convenient to have a continuum reformulation of this multipole expansion, to avoid having to work with the discrete Green’s function and its discrete derivatives. Towards that end we exploit the connections between continuum and discrete Green’s functions and derive higher order continuum approximations of the discrete Green’s function. Specifically, in Section 6, we will construct a sequence of continuum kernels with the following properties:
Theorem 3
There are unique kernels \(G_n \in C^{\infty }(\mathbb {R}^d/\{0\}; \mathbb {R}^{N \times N})\) such that
for all \(\ell \in \Lambda \) and \(j,p \in \mathbb {N}_0\) and such that the \(G_n\) are positively homogeneous of degree \((2-2n-d)\) if \(n \geqq 1\) or \(d \geqq 3\), while in the case \(n=0, d=2\) we have \(G_0(\ell ) = A \log |\ell |+ \varphi (\ell )\), where \(A \in \mathbb {R}^{N \times N}\) and \(\varphi \) is 0-homogeneous. Furthermore, we find \(G_0 = G^{\mathrm{C}}\).
The \(G_n\), \(n>0\), are higher order corrections resolving the atomistic-continuum error. We will give precise definitions of all the \(G_n\) in Section 6. In addition we want to point out that the \(G_n\) are practically computable via Fourier methods. A formula for that is also given in Section 6.
Returning to the multipole expansion, if \(p=1,2\) in Theorem 2, then one can replace the lattice Green’s function \({\mathcal {G}}\) with the continuum Green’s function \(G^{\mathrm{C}}\). However, for a higher order continuum description, higher order \(G_n\) need to be used. If we also use Taylor expansions to get actual derivatives we get a pure continuum expansion.
Theorem 4
Assume \(|H [u] |\lesssim {|{\ell } |}_{0}^{-d-p} \log ^\alpha {|{\ell } |}_{0}\) with \(p, \alpha \in \mathbb {N}_0\). Then there are \(a^{(i,n,k)} \in (\mathbb {R}^{d})^{\odot i}\) such that
and the remainder decays as
for \(j=1,2\).
If \(j \geqq 3\), then Equation (13) is still true under the additional assumption that \(|D^m H [u] |\lesssim {|{\ell } |}_{0}^{-d-p-m} \log ^\alpha {|{\ell } |}_{0}\) for all \(m=1,...,j-2\).
2.3 The full far-field expansion
Theorem 2 lays out a path on how to construct good far-field approximations for a solution \({\bar{u}}\) of the atomistic equations \(\delta {\mathcal {E}} ({\bar{u}})=0\). Instead of looking at the solutions directly it suffices to construct an approximate solution of the equations \({\hat{u}}\) such that the remainder r defined by \({\bar{u}}=: {\hat{u}} + r\) has small linearized forces H[r] in the far field. For this approach to be useful, it is desirable that \({\hat{u}}\) is both easy to understand analytically and practically computable.
Our goal is to construct smooth continuum approximations through the addition of successive corrector terms \(u_i^{\mathrm{C}}\) up to the desired order. We write \({\bar{u}}={\hat{u}}_p+r_p= u_0^{\mathrm{C}} + u_1^{\mathrm{C}} + \dots + u_{p}^{\mathrm{C}} + r_p\) and aim to achieve \(|H[r_p] |\lesssim {|{\ell } |}_{0}^{-d-p-1}\), so that with Theorem 2 we have that
with a remainder \(w_p\) satisfying \(|w_p |\lesssim {|{\ell } |}_{0}^{1-d-j-p}\); that is, the remainder is highly localised around the defect core.
The precise statements for both point defects and screw dislocations are given in Section 3. The full rigorous construction of the \(u_i^{\mathrm{C}}\) is given in the proofs in Section 7. We do however want to formally outline this construction here.
The first step is a Taylor expansion of the energy around the lattice or, more precisely, the potential V around zero, giving that
Separating out the linear terms and inserting the ansatz \({\bar{u}}= {\hat{u}}_p + r_p = u_0 + u_1 + \dots + u_{p} + r_p\) gives
The next step then is to Taylor expand the discrete differences in \(H[u_i]\) and \(\delta ^{k+1} {\mathcal {E}}(0)[{\hat{u}}_p]^{k}\) leading to continuum differential operators. In particular, the leading order term for \(H[u_i]\) is \(c_{\mathrm{vol}} H^{\mathrm{C}}[u_i]\) followed by higher order differential operators.
Formally, \(u_i\) is of the order \({|{\ell } |}_{0}^{2-d-i}\) with each derivative or discrete difference adding one order of decay. This allows us to group all the resulting terms into operators \({\mathcal {S}}_i\), based on their order of decay. We thus obtain
We can therefore use the PDE
to define \(u_i^{\mathrm{C}}\). While the just outlined procedure does in fact work as a definition of the operators \({\mathcal {S}}_i\), we will actually give a fully explicit definition later, specifically Equation (78) in Section 7.
When we look at all the details of this construction in Section 7, we will also see that \(u_i\) as used on the right hand side of (15) has to include the multipole terms of that order, which we can write as
In particular, it is then important to know that with the help of Theorem 4 we can use the their smooth continuum variant instead of the discrete one.
The \({\mathcal {S}}_i\) in (15) are in general non-linear and higher order differential operators. Crucially though, \({\mathcal {S}}_i\) only depends on the terms \(u_0, \dots , u_{i-1}\) and not \(u_i\) itself. That means that the equation defining \(u_i\) is always the same second order, elliptic continuum linear elasticity equation \(H^{\mathrm{C}}[u_i^{\mathrm{C}}] = f_i\) for some residual forces \(f_i\).
With \(u_i^{\mathrm{C}}\) defined this way, most of the terms on the right hand side of (14) cancel out and a precise estimate of the higher order errors gives the desired estimate for \(H[r_{p-1}]\).
In Section 7, we give a precise definition of the \({\mathcal {S}}_i\) in the corrector equation (15). The grouping of the terms into different \({\mathcal {S}}_i\) depends on the dimension d. As an example, for \(d=2\), the first three \(S_i\) are given by
Recall from (6) that \(W(A) = \frac{1}{c_{\mathrm{vol}}} V((A\rho )_{\rho \in {\mathcal {R}}})\) is the Cauchy–Born energy density for \(A\in \mathbb {R}^{N \times d}\) and \(H_{\mathrm{SG}}\) is a linear differential operator describing a strain-gradient term in linear elasticity. It is defined by
If, on the other hand, \(d=3\), then
3 Far Field Expansion for Crystalline Defects
We now demonstrate how our general structural results can be directly applied to obtain precise characterisations of the discrete elastic far-fields surrounding crystalline defects. Our results apply directly to points defects and screw dislocations; edge and mixed mode dislocations require additional ideas due to their more non-trivial lattice topology and will therefore not be discussed here. In addition, we briefly outline how these characterisations give rise to novel algorithms for simulating such defects.
3.1 Point defects
We consider point defects first. We briefly review the setting of [3] to motivate the formulation of our main result in this context. First, we assume that the point defect has a reference configuration \(\Lambda ^{\mathrm{def}}\subset \mathbb {R}^d, d \geqq 2\) which is locally finite and homogeneous outside some defect radius \(R^{\mathrm{def}}\), meaning \(\Lambda ^{\mathrm{def}}{\setminus } B_{R^{\mathrm{def}}} = \Lambda {\setminus } B_{R^{\mathrm{def}}}\).
Let \({\mathcal {R}}_\ell \) denote a finite interaction range for each site \(x \in \Lambda ^{\mathrm{def}}\) and assume that there is a family of site energies \(V_\ell \in C^K(\mathbb {R}^{d {\mathcal {R}}_\ell }), \ell \in \Lambda ^{\mathrm{def}}\). Moreover, assume \(N=d\) and that there is a homogeneous interaction range \({\mathcal {R}}\) and site energy \(V \in C^K(\mathbb {R}^{d {\mathcal {R}}})\) for all sites of \(\Lambda \), and that \({\mathcal {R}}_\ell = {\mathcal {R}}, V_\ell = V\) for all \(\ell \in \Lambda ^{\mathrm{def}}{\setminus } B_{R^{\mathrm{def}}}\). The potential energy under a displacements \(u : \Lambda ^{\mathrm{def}}\rightarrow \mathbb {R}^d\) and \(u : \Lambda \rightarrow \mathbb {R}^d\) are then, respectively, given by
With these definitions \({\mathcal {E}}\) is then well defined on \({\mathcal {H}}^{c}\) and has a unique continuous extension to \({\mathcal {H}}^{1}\). The same is true with \({\mathcal {E}}^{\mathrm{def}}\) for analogously defined \({\mathcal {H}}^{c}(\Lambda ^{\mathrm{def}})\) and \({\mathcal {H}}^{1}(\Lambda ^{\mathrm{def}})\). Furthermore with \(V \in C^K\) we also find \({\mathcal {E}}, {\mathcal {E}}^{\mathrm{def}} \in C^K\), all of which is shown in [3, Lemma 1].
Allowing \(\Lambda ^{\mathrm{def}}\ne \Lambda \) in \(B_{R^{\mathrm{def}}}\) admits defects such as vacancies and interstitials, while allowing inhomogeneity of \(V_{\ell }\) admits impurities and foreign interstitials.
A point defect can be thought of as a finite-energy equilibrium of \({\mathcal {E}}\), that is, equilibrium displacements \({\bar{u}}^{\mathrm{def}} \in {\mathcal {H}}^{1}(\Lambda ^{\mathrm{def}})\) such that
A notationally convenient approach is to simply project \({\bar{u}}^{\mathrm{def}}\) to the homogeneous lattice. That is, we define \({\bar{u}} : \Lambda \rightarrow \mathbb {R}^d\) by
This is of course only one of many possible projections, which we have made only for the sake of notational convenience. Our subsequent results are essentially independent of how this projection is performed. Most importantly, because we have \({\bar{u}} = {\bar{u}}^{\mathrm{def}}\) outside the defect core we obtain that
for \(|\ell |\) large enough. Here \(\delta _\ell (\ell ') := \delta _{\ell \ell '}\). With a small amount of additional work one can in fact show that
where \(g : \Lambda \rightarrow \mathbb {R}^{d \times {\mathcal {R}}}\) with \(\mathrm{supp}(g) \subset B_{R^{\mathrm{def}}}\). This motivates the setup for our next result.
Theorem 5
Choose \(p \geqq 0, J \geqq 0\) and suppose that \(V \in C^{K}(\mathbb {R}^{d \times {\mathcal {R}}})\), such that \(K \geqq J+2+ \max \{0,\lfloor \frac{p-1}{d} \rfloor \}\). Let \(g : \Lambda \rightarrow \mathbb {R}^{d \times {\mathcal {R}}}\) with compact support, and let \({\bar{u}} \in {\mathcal {H}}^{1}(\Lambda )\) such that
Then, there exist \(u_i^{\mathrm{C}} \in C^\infty \) and coefficients \(b^{(i,k)}\) such that
and such that the \(u_i^C\) satisfy the PDEs in Equation (59) with \(u_i^C=0\) for \(0 \leqq i \leqq d\). Furthermore, the remainder \(r_{p+1}\) satisfies the estimate
for \(j= 1, \dots , J\).
Remark 4
Note that the \(u_i^C\) are not unique, as it is possible to add terms that do not change the equation and have a decay that falls under the remainder estimate. This feature should not be surprising, as the expansion provided is correct only up to a certain order of decay of the remainder. In fact, we will see in the proof that, beyond satisfying the PDE (59), the only additional constraint on \(u_i^C\) in our construction is a certain decay estimate. Proposition 15 will discuss this constraint in detail.
Remark 5
In particular, up to order \(p=d\), we have the pure multipole expansion
Effects from the nonlinearity and higher order derivatives are only noticeable in terms beyond that.
Remark 6
In the point defect case it is likely possible to reduce the number of logarithms in the estimate somewhat for all orders. We do not want to explore this in detail but want to point out the log-free estimate for low orders which directly follows from (23) and estimates on the lattice Green’s function based on Theorem 3. To be precise, for \(p<d\) we have that
3.2 Screw dislocations
Now let us consider screw dislocations. Again, our modelling follows the setup in [3] and [12]. We consider a straight screw dislocation with periodic behaviour along the dislocation line so that we can project to the lattice to a two-dimensional lattice on the normal plane to describe the behaviour. Hence we have \(d=2\) and \(N=3\) meaning \(u :\Lambda \subset \mathbb {R}^2 \rightarrow \mathbb {R}^3\), though N is left arbitrary in the following to include for example the case in [12] where \(N=1\).
Again we have a finite interaction range \({\mathcal {R}}\) and a site energy \(V \in C^K(\mathbb {R}^{d {\mathcal {R}}})\). The potential energy is then formally given by
However that sum will usually not converge, so we follow [3] and consider the energy differences instead.
Here \(u_{\mathrm{CLE}}\) is the continuum linear elasticity solution.
More precisely, \(u_{\mathrm{CLE}}\) solves
where \(b \in \mathbb {R}^3\), \(b \parallel e_3\), is the Burgers vector of the screw dislocation, \({\hat{x}} \in \mathbb {R}^2\) is the reference position of the dislocation core and \(\Gamma := \{ x \in \mathbb {R}^2 : x_2 = {\hat{x}}_2, x_1 \geqq {\hat{x}}_1 \}\) a branch-cut chosen such that \(\Gamma \cap \Lambda = \emptyset \). We want to point out that the precise positioning of \({\hat{x}}\) is not crucial and does not have physical meaning as the difference between two shifted solutions is in the energy space \({\mathcal {H}}^{1}\).
Equation (29) was missed in [3] but is in fact crucial for the results there to be true. It encodes the assumption that the system has zero net force and thus avoids spurious solutions of the type \(g(x)= u_{\mathrm{CLE}}(x) + G^C(x-{\hat{x}})\). However, the standard construction of a solution \(u_{\mathrm{CLE}}\), which can be found, for example, in the book by Hirth and Lothe [14], already takes it into account. Therefore, the results of [3] and later works that build on it remain correct provided such a solution \(u_{\mathrm{CLE}}\) is employed.
The following observation links it to the atomistic setting:
Proposition 6
Let \(u \in C^3(\mathbb {R}^2 / \Gamma ), \nabla u \in C^2(\mathbb {R}^2 {\setminus } \{{\hat{x}}\})\) solve (25), (26), and (27) with \(|\nabla ^j u |\lesssim |\ell -{\hat{x}} |^{-j}\) for \(j=1,2,3\). Then
Therefore (29) is equivalent to either of these sums vanishing.
Indeed, the property \(\sum \delta {\mathcal {E}}(u)(\ell ) =0\) is heavily used in [3] and we will use their results here.
With the definition (24) the energy \({\mathcal {E}}\) is then well defined on \(u_{\mathrm{CLE}} + {\mathcal {H}}^{c}\) and has a unique continuous extension to \(u_{\mathrm{CLE}} + {\mathcal {H}}^{1}\). WM \(V \in C^K\) we also find that \({\mathcal {E}} \in C^K\); see [3, Lemma 3].
Theorem 7
Suppose that \(V \in C^{K}(\mathbb {R}^{N \times {\mathcal {R}}})\), \(K \geqq J+2+ p\) with \(p \geqq 0\) and \(J \geqq 2\). Let \({\bar{u}} \in {\mathcal {H}}^{1}(\Lambda )\) solve
Then, there exist \(u_i^{\mathrm{C}} \in C^\infty \) and coefficients \(b^{(i,k)}\) such that
and such that the \(u_i^C\) satisfy the PDEs (59) and \(u_0^{\mathrm{C}}=u_{\mathrm{CLE}}\). Furthermore, the remainder \(r_{p+1}\) satisfies the estimate
for \(j= 1, \dots , J\).
Remark 7
Contrary to the point defect none of these terms are expected to vanish in general, except for a few special cases which are explored in [12]. In particular, the regularity assumption cannot be weakened as in the point defect case. Indeed, our general theory without looking at any special cases requires \(K \geqq J+2+ \lfloor \frac{p}{d-1} \rfloor \). As far as our proof goes the number of logarithms is optimal for \(d=2\), though probably not for higher dimension. We also expect this to be generic for the theorem itself as indeed the \(u_i^C\) will (in general) contain higher and higher logarithmic terms. However, in special cases these logarithmic terms in the \(u_i^C\) do not necessarily always appear, as explored in [12].
3.3 Accelerated convergence of cell problems
An immediate application of the defect expansions of Theorems 5 and 7 is that they suggest a novel family of numerical schemes that exploit these expansions to accelerate the simulation of crystalline defects. In this section, we motivate this by briefly sketching out one such scheme. However, a practical implementation requires multiple challenging additional steps (see Remark 8), hence we postpone this to future work.
Consider the equilibration of a point defect or a screw dislocation near the origin as in Theorems 5 and 7 . We define a family of restricted displacement spaces
where atoms are clamped in their reference configurations outside a ball with radius R. Then we can approximate (22), (30) by the Galerkin projection
where \({\bar{u}}_R \in {\mathcal {U}}_R\).
Under suitable stability conditions it is then shown in [3] that
for R sufficiently large, where \(p \in \{0, 1\}\). This convergence is an almost immediate corollary of the decay estimate \(|Dr_1(\ell ) |\lesssim {|{\ell } |}_{0}^{-d} \log ^p {|{\ell } |}_{0}\). (For energy minima, [3] can be applied directly while for saddle points the analysis of [15] can be readily adapted.) Our aim now is to accelerate this relatively slow convergence by providing an improved far-field boundary condition.
The overarching principle is to
-
1.
replace the naive far-field predictor
$$\begin{aligned} {\hat{u}}_0 := {\left\{ \begin{array}{ll} 0, &{} \text {point defects}, \\ u_{\mathrm{CLE}}, &{} \text {dislocations} \end{array}\right. } \end{aligned}$$with the higher-order predictor
$$\begin{aligned} {\hat{u}}_p := \sum _{i = 0}^p u_i^{\mathrm{C}} \end{aligned}$$ -
2.
and to enlarge the admissible corrector space with the multipole moments
$$\begin{aligned} {\mathcal {U}}_R^{(p)} := \big \{ v : \Lambda \rightarrow \mathbb {R}^N \mid&\,\, v = \sum _{i = 1}^p \sum _{k = 1}^N b^{(i,k)} : D^i_{{\mathcal {S}}} {\mathcal {G}}_k + w, \\&\hbox {for free coefficients}\ b^{(i,k)}\hbox { and }\mathrm{supp}~ w \subset \Lambda \cap B_R \big \} \end{aligned}$$That is, the corrector displacement is now parametrised by its values in the computational domain \(B_R \cap \Lambda \) and by the coefficients of the multipole terms.
We can then consider the pure Galerkin approximation scheme: find \({\bar{v}}_R \in {\mathcal {U}}^{(p)}_R\), \({\bar{u}}_{p, R} = {\hat{u}}_p +{\bar{v}}_R\), such that
The arguments of [3] leading to (34) are generic Galerkin approximation arguments, leveraging the strong stability condition; they can be followed verbatim up to the intermediate result (Céa’s Lemma)
The existence of \({\bar{v}}_R\) is implicitly guaranteed through an application of the inverse function theorem, due to the fact that the right-hand side in this estimate approaches zero as \(R \rightarrow \infty \). To estimate the right-hand side we can insert the exact tensors \(b^{i,k}\) from the solution representation of Theorems 5, 7 into \(v_R\), in order to obtain \(D{\bar{u}} - D{\hat{u}}_p - Dv_R = Dr_{p+1} - Dw_R\), where \(r_{p+1}\) is the core remainder term, and hence,
We can now define \(w_R\) to be a suitable truncation of \(r_{p+1}\) to the computational domain \(B_R\). The details are given in [3, Thm. 2] and immediately yield
Theorem 8
Suppose that \({\bar{u}}\) is a strongly stable solution of (22) or (30); that is, there exists a stability constant \(c_0 > 0\) such that
then, for R sufficiently large, there also exists a solution \({\bar{u}}_R \in {\mathcal {U}}_R^{(p)}\) to the Galerkin scheme (35) such that
Remark 8
The scheme (35) cannot be implemented as is since the energy difference functional cannot be evaluated for a displacement with infinite range. However, this highly idealised scheme is of immense theoretical value in that it highlight what could potentially be achieved if this challenge can be overcome. Any practical scheme will necessarily have to engage in the approximate evaluation of the multipole tensors \(b^{(i)}\), for which there are several promising possibilities that we will explore in separate works.
A second challenge for practical implementations is the fast and accurate evaluation of the higher-order far-field predictor \({\hat{u}}_p\). All of these approximations require suitable controlled approximation to \({\mathcal {E}}\), somewhat analogous to quadrature rules or other kinds of variational crimes in the classical numerical analysis context.
4 Conclusions and Outlook
The main result of the present paper is the fact that the elastic field surrounding a defect in a crystalline solid may be represented to within arbitrary accuracy with three “low-dimensional” ingredients:
-
1.
a series of continuum fields specified through PDEs;
-
2.
a series of multipole moments; and
-
3.
a highly localised discrete core correction.
More specifically, we have shown that by increasing the accuracy of components (1) and (2), the core correction (3) becomes increasingly local. While there is a certain amount of interaction between the components (1) and (2), there is no coupled problem that needs to be solved at any point. Indeed, both series are obtained sequentially order by order and the PDE defining the term of a given order in (1) only depends on lower-order terms of the multipole expansion, but not on the multipole term of the same order.
There is a clear path to exploit our results in the analysis and design of improved numerical schemes to simulate crystalline defects (cf. Sec. 3.3). Such schemes could either explicitly or implicity rely on our results, but in either case they require substantial additional work and will be pursued separately.
Our presentation here is restricted to simple lattices and a limited class of defects. Generalisations do require additional technical difficulties to be overcome, but there appears to be no fundamental limitation to extend the method and the results to multi-lattices and a range of other defects in some form. To conclude, we briefly discuss some of these possibilities and limitations.
-
Edge and mixed dislocations: Edge and mixed dislocations are technically more challenging as they create a mismatch that affects the two-dimensional reference lattice. To leading order it suffices to correct the CLE solution with an ad-hoc transformation \(u_0 = u_{\mathrm{CLE}} \circ \xi ^{-1}\), see [3], though the analysis also becomes a bit more technical still. For higher orders however more care needs to be taken, not just in the choice of \(\xi \) but also in the effect such transformations have on the PDEs. Furthermore, many arguments have additional technical complications due to the need of slip operators to describe the elastic strain.
-
Cracks: The full extension of our results to crack geometries appears to be considerably more challenging, as the homogeneous lattice is no longer a particularly good global reference. Thus already the discussion of Green’s function is significantly more involved [16]. Formally, we still expect our overall strategy to apply and it is interesting to note that due to different orders of decay the first higher order corrector is already needed to even define the model in the first place rather than “just” improve on it.
-
Energy differences for defect transitions: The precise characterisation of the far-field strain in terms of defect continuum fields and a multipole expansion suggests that some level of cancellation in energy differences, for example between a saddle point and energy minimum as observed in [15], could be precisely tracked and characterised. Moreover, such results may then also explain improved convergence rates of numerical schemes for energy differences that are often seen in practice.
-
Convergence of numerical schemes: A consequence of our analysis, with direct practical value is the construction of improved approximate cell problems that leverage the explicit low-dimensional structure of defect fields that we identified. We have given a hint at how this might be achieved in Theorem 8 but much additional work is needed to formulate practical schemes along these lines.
The same line of work can also lead to robust new numerical schemes and analysis of existing schemes for the defect dipole tensor specifically (also called the elastic dipole tensor). Such schemes are of important and ongoing interest in defect physics (for example, [17, 18]). In particular, our approach to these terms developed here naturally includes the anisotropic case as well as extensions to higher multipole tensors.
-
Higher-order dislocation dynamics models: A further consequence is that we hope that the expansion of the far-field strain we have obtained here allows us to go beyond traditional dislocation dynamics approaches, which rely upon the leading-order CLE description of these defects. By using the structure of our expansion to studying the effect of applied stress fields on defect cores, we can provide more detailed atomistic input into such models. This suggests a route to better connect dislocation dynamics and atomistic approaches, bridging the scale and language gaps between these two simulation methodologies.
-
Dynamics and statistical mechanics: Statistical mechanics models, such as free energies or transition rates could in principle benefit from an analysis within our new framework. For example, in the harmonic approximation, the analysis of [15] could be taken as a starting point. It is far less clear whether more nonlinear models could also benefit, and it appears certain that finding similar coarse-grained descriptions of full dynamics of a crystalline far-field would require very different ideas.
5 Proofs - Decay Estimates
In this section we want to prove Theorems 1 and 2. First, however, let us cite the following lemma:
Lemma 9
(Conversion to divergence form) Let \(\alpha \in \mathbb {R}\), \(q > d\), and \(f\,:\,\Lambda \rightarrow \mathbb {R}^N\) such that \(|f(\ell ) |\leqq C_f {|{\ell } |}_{0}^{-q} \log ^{\alpha } {|{\ell } |}_{0}\) for all \(\ell \in \Lambda \) and \(\sum _{\ell \in \Lambda }f(\ell ) = 0\). Then there exists \(g\,:\,\Lambda \rightarrow \mathbb {R}^{N} \otimes \mathbb {R}^{{\mathcal {S}}}\) and a constant C independent of f and \(\ell \) such that \(f = -\mathrm{Div}_{{\mathcal {S}}} g\) and \( |g(\ell ) |\leqq CC_f{|{\ell } |}_{0}^{-q+1} \log ^{\alpha } {|{\ell } |}_{0}\) for all \(\ell \in \Lambda \).
Proof
For \(\alpha =0\) this is [3, Corollary 1]. However the addition of logarithmic terms is trivial. Indeed, one can construct g in the exact same way and can easily carry the logarithmic term through by including them in the weighted norms used in the proof. \(\quad \square \)
The decay estimate for the lowest order found in [3] is indeed based on Lemma 9 which then allows for a partial summation in the Green’s function representation sum of the remainder given in Equation (37). The key idea for the higher order decay estimates is that Lemma 9 can actually be extended to higher orders based on vanishing higher order moments
as long as one only tries to write symmetric parts in divergence form; see Proposition 12 below. We will then use this higher order divergence form with more precise higher order partial summation in specific parts of the lattice in the Green’s function representation sum of the remainder given in Equation (37) to arrive at the new decay estimates. We begin by establishing two further auxiliary results.
In that follows, we will use \(\varvec{\rho } = (\rho _1, ..., \rho _j) \in {\mathcal {R}}^j\subset (\mathbb {R}^d)^j\), \(j \geqq 1\), and analogously for \(\varvec{\sigma }\). Also, recall that \(D_{\varvec{\rho }} = D_{\rho _1} ... D_{\rho _j}\).
Lemma 10
Given a linearly independent set of vectors \({\mathcal {S}} \subset \mathbb {R}^d\) (which must necessarily have \(\#{\mathcal {S}}=k\leqq d\)), the set of tensors \(\{ \varvec{\sigma }^\odot :\varvec{\sigma } \in {\mathcal {S}}^k\}\) is also linearly independent in \((\mathbb {R}^d)^{\otimes k}\). Furthermore, \(\varvec{\sigma }^\odot = \varvec{\rho }^\odot \) with \(\varvec{\sigma }, \varvec{\rho } \in {\mathcal {S}}^k\), if and only if \(\varvec{\rho } = \pi (\varvec{\sigma })\) for some permutation \(\pi \in S_k\).
Proof
Although the proof is straightforward, and likely well-known, we present it for the sake of convenience: Define a scalar product \((\cdot , \cdot )_{{\mathcal {S}}}\) on \(\mathbb {R}^d\) for which \({\mathcal {S}}\) forms part of an orthonormal basis. This induces a scalar product on the space of tensors by multi-linear extension of \((\sigma ^\otimes , \rho ^\otimes )_{{\mathcal {S}}} := \prod _{j=1}^k (\sigma ^{(j)}, \rho ^{(j)})_{{\mathcal {S}}}\).
Consider the scalar product
If \(\varvec{\rho }\ne \pi (\varvec{\sigma })\) for all \(\pi \in S_k\), then for each \(\pi \in S_k\), there exists an index i such that \(\sigma ^{(i)}\ne \rho ^{(\pi (i))}\), and consequently \((\sigma ^{(i)},\rho ^{(\pi (i))})_{{\mathcal {S}}}=0\). This entails that each of the products summed in the expression above is zero. Since \(\varvec{\rho }^\odot \) and \(\varvec{\sigma }^\odot \) are both non-zero and orthogonal in this inner product, they cannot be equal.
The same argument entails that if \(\varvec{\sigma }^\odot = \varvec{\rho }^\odot \), there must exist \(\pi \in S_k\) such that \(\varvec{\rho }= \pi (\varvec{\sigma })\). On the other hand, if \(\varvec{\rho }= \pi (\varvec{\sigma })\), we see that
We deduce that \(\varvec{\sigma }^\odot \) and \(\varvec{\rho }^\odot \) are either identical (if and only if \(\varvec{\sigma }\) is a permutation of \(\varvec{\rho }\)) or mutually orthogonal in the \({\mathcal {S}}\)-inner product, which implies the stated result. \(\quad \square \)
Lemma 11
Let \(\varvec{\sigma } \in \Lambda ^p\), then
Proof
This identity follows from two observations: First,
for a polynomial r in \(\ell \) of degree at most \(j-2\). Second,
for any such polynomial. A simple induction then shows the result. \(\quad \square \)
We can now turn towards to a crucial result converting a discrete force field into higher order divergence form.
We will slightly abuse notation in the following and let the symmetric part \({{\,\mathrm{sym}\,}}A\) of a tensor \(A \in \mathbb {R}^{N} \otimes (\mathbb {R}^{\mathcal {S}})^{\otimes k}\) denote the symmetrical part in the later indices only, namely \({{\,\mathrm{sym}\,}}(A) := \frac{1}{k!}\sum _{\pi \in S_k} A_{\cdot , \pi (\varvec{\sigma })}\). For higher order tensor fields we will follow the usual convention from the continuum that \({{\,\mathrm{Div}\,}}\) always applies to the last component.
Proposition 12
(Conversion to higher order divergence form) Let \({\mathcal {S}} \subset \Lambda \) be linearly independent with \({{\,\mathrm{span}\,}}_\mathbb {Z}{\mathcal {S}}=\Lambda \). Also let \(p, q \in \mathbb {N}\), \(q+1-p > d\), \(\alpha \in \mathbb {R}\), and \(f^{(0)}\,:\,\Lambda \rightarrow \mathbb {R}^N\) such that \(|f^{(0)}(\ell ) |\leqq C_f{|{\ell } |}_{0}^{-q} \log ^{\alpha } {|{\ell } |}_{0}\) for all \(\ell \in \Lambda \). Also assume p moments vanish, that is, \({\mathcal {I}}_j=0\) for \(0 \leqq j \leqq p-1\). Then there exist \(f^{(k)}\,:\,\Lambda \rightarrow \mathbb {R}^{N} \otimes (\mathbb {R}^{\mathcal {S}})^{\otimes k}\), \(1 \leqq k \leqq p\) and a constant C such that
and \( |f^{(k)}(\ell ) |\leqq CC_f {|{\ell } |}_{0}^{-q+k} \log ^{\alpha } {|{\ell } |}_{0}\) for all \(\ell \in \Lambda \) and all \(1 \leqq k \leqq p\).
Proof
\(p=1\) is covered by Lemma 9. By induction, assume the statement is true for a \(p \in \mathbb {N}\) and we now have \(q-p>d\) as well as \(p+1\) vanishing moments. In particular, we have already constructed the desired \(f^{(0)}, \dots , f^{(p)}\). Now \(|{{\,\mathrm{sym}\,}}f^{(p)}(\ell )|\lesssim {|{\ell } |}_{0}^{-(q-p)}\) and \(q-p>d\).
We claim that
for all \(j=0, \dots , p\). This is clear for \(j=0\). By induction, if this is true for \(j<p\), then
Note that the decay of \(f^{(j+1)}\) is needed not just for the sums to exist but also for the partial summation to be true. This establishes (36).
Applying Lemma 11, with \(j = p\), and using that \({\mathcal {I}}_p=0\) we obtain
According to Lemma 10, the set of the tensors \({{\,\mathrm{sym}\,}}\varvec{\sigma }^\otimes \) is linearly independent. Additionally, \(\varvec{\sigma }^\odot = \varvec{\rho }^\odot \) implies \({{\,\mathrm{sym}\,}}f^{(p)}_{\cdot , \varvec{\sigma }} = {{\,\mathrm{sym}\,}}f^{(p)}_{\cdot , \varvec{\rho }}\) as \(\varvec{\rho } = \pi (\varvec{\sigma })\) for some permutation \(\pi \). Therefore, \(\sum _\ell {{\,\mathrm{sym}\,}}f^{(p)}(\ell ) =0\) and we can apply Lemma 9 to find \(f^{(p+1)}\) with the desired properties. \(\quad \square \)
To prepare for the Proof of Theorem 1, we fix some \(u \in {\mathcal {H}}^{1}\) and write \(f^{(0)} := H[u]\). We extend the lattice Green’s function approach developed in [3] to estimate u. The Green’s function satisfies
for all \(u \in {\mathcal {H}}^{c}\). As the right hand side is not invariant under adding a constant to u, this cannot directly translate to general \(u \in {\mathcal {H}}^{1}\), but the situation looks better for derivatives.
Lemma 13
Let \(u \in {\mathcal {H}}^{1}\) and assume that \(|f^{(0)}(\ell )|= |H[u]|\lesssim |\ell |^{-\gamma }_0\) for some \(\gamma >1\). Then, for all \(\varvec{\rho } = (\rho _1, ..., \rho _j) \in {\mathcal {R}}^j\), \(j \geqq 1\), we have that
Proof
Due to the decay assumption on \(f^{(0)}\) the sum converges absolutely. Furthermore, for \(u \in {\mathcal {H}}^{c}\) the statement is clearly true. The right hand side is a well-defined, continuous, linear functional on \({\mathcal {H}}^{1}\). The result is straightforward when \(d \geqq 3\) or \(j \geqq 2\) as in this case the left hand side is also a continuous, linear functional because \(|D_{\varvec{\rho }}{\mathcal {G}}_k |\in \ell ^2(\Lambda )\) if and only if \(4 < d+2j\).
To include the case where \(d=2\) and \(j=1\) we have to be a bit more careful. Note that, for \(u \in {\mathcal {H}}^{c}\),
As the right hand side is now a well-defined, continuous, linear functional on \({\mathcal {H}}^{1}\), we find for, all \(u \in {\mathcal {H}}^{1}\), that
If we now have a \(u \in {\mathcal {H}}^{1}\) where additionally \(|f^{(0)} |\lesssim {|{\ell } |}_{0}^{-\gamma }\), then the sum \(\sum _{z\in \Lambda }f^{(0)}(z)D_{\varvec{\rho }}{\mathcal {G}}_k(\ell -z)\) converges (absolutely). Consider a smooth cutoff function \(\eta _R\), such that \(\eta _R :\mathbb {R}^d \rightarrow [0,1]\) satisfies \(\eta _R(z)=1\) for \(|z |\leqq R\) and that \(\eta _R(z)=0\) for \(|z |\geqq 2R\), as well as \(|\nabla ^k \eta _R |\lesssim R^{-k}\) for \(k=1, ..., j\). With that, we see that
as the remaining term can be estimated by
That is, (37) holds for all \(u \in {\mathcal {H}}^{1}\) with \(|f^{(0)}(\ell ) |\lesssim {|{\ell } |}_{0}^{-\gamma }\). \(\quad \square \)
Proof of Theorem 1
We now use the Green’s function representation (37) to estimate \( D_{\varvec{\rho }} u(\ell )\). Let us consider \(|f^{(0)} |\lesssim {|{\ell } |}_{0}^{-d-p} \log ^{\alpha } {|{\ell } |}_{0}\) where we include both the case \(\alpha \in \mathbb {N}_0\) and \(\alpha < -1\) to include Remark 1. Then let us assume there are p vanishing moments.
Although our argument is related to the lower order equivalent in [3], it is technically more complex as both the higher derivatives and the higher order divergence form lead to partial summations which, crucially, have to be performed on only specific and separate parts of the lattice. We will therefore provide full details.
This can be done in a very clean way by splitting the sum (37) into four regions; see Figure 1 for a visualisation: Region 1 is the far field, where \(|z |\) and \(|\ell -z|\) are comparable. Region 2 is the intermediary area where \(|z |\), \(|\ell |\), and \(|\ell -z|\) are all comparable, and region 3 and 4 are the areas around \(z=0\) and \(z=\ell \), where either \(|z |\) can be small but \(|\ell |\) and \(|\ell -z|\) are comparable or \(|z-\ell |\) can be small but \(|\ell |\) and \(|z|\) are comparable. Inserting estimates for the residual \(f^{(0)}\) and the lattice Green’s function \({\mathcal {G}}\) and using this split of the sum indeed gives sharp estimates in absolute value. However, as we will show below, this is only a good estimate if both \(p=0\) and \(j=1,2\). If either \(p \geqq 1\) or \(j \geqq 3\) then the sum in (37) exhibits significant large scale cancellation effects. To get sharp estimates in these cases, we will remove these cancelling terms via separate partial summations in region 3 and 4. The required discrete derivatives are directly available in the case \(j \geqq 3\) or are obtained with the help of Proposition 12. To avoid discrete boundary terms, we will not split the four regions sharply but use smooth cutoff functions. The boundary terms in the partial summation are then spread out and can be treated like terms in region 2.
Splitting \(\mathbb {R}^d\) into four regions as used in the proof of Theorem 1
So, let us take a smooth cutoff \(\eta _\ell :\mathbb {R}^d \rightarrow [0,1]\) satisfying \(\eta _\ell (z)=1\) for \(|z |\leqq |\ell |/4\) and \(\eta _\ell (z)=0\) for \(|z |\geqq |\ell |/2\), as well as \(|\nabla ^k \eta _\ell |\lesssim |\ell |^{-k}\) for \(k=1, ..., j\). As discussed, we now split the sum in (37) according to the four regions:
We first estimate the far-field term
where the logarithmic term is estimated trivially by \(\log ^{\alpha } {|{z} |}_{0} \leqq \log ^{\alpha } {|{\ell } |}_{0}\) for negative \(\alpha \), while for \(\alpha \in \mathbb {N}_0\) the estimate instead follows from partial integrations of the resulting one-dimensional radial integral \(\int _{ |\ell |}^\infty r^{1-j-d-p} \log ^{\alpha } r \,\mathrm{d}r\).
The intermediary area is even more direct as we can just estimate the functions uniformly and multiply by the number of lattice points in the area
Next, \(T_3\) can be estimated by
Hence, we have \(|T_3 |\lesssim {|{\ell } |}_{0}^{2-d-j}\) if either \(p>0\) or \(\alpha <-1\). If, on the other hand, \(p=0\) and \(\alpha \in \mathbb {N}_0\), then we obtain \(|T_3 |\lesssim {|{\ell } |}_{0}^{2-d-j} \log ^{\alpha +1}|\ell |\).
Finally, for \(T_4\) the estimate is
Putting all four estimates together, this completes the proof in the special case where both \(p=0\) and \(j =1,2\).
For \(p \geqq 1\), we need a better estimate on \(T_3\). We choose \(f^{(m)}\) according to Proposition 12. We claim that, for \(0 \leqq m \leqq p\),
Let us prove (40) by induction over m. Clearly, it is true for \(m=0\) since the second term on the right-hand side is identical to the left-hand side. Given its validity for a m, with \(m+1 \leqq p\), we now employ Proposition 12 and summation by parts to obtain
The last term is concentrated in the annulus where \(D_\tau \eta \) is non-zero and can be estimated by
which completes the proof of (40).
Using (40) for \(k=p\), we find that
Therefore, \(|T_3 |\lesssim {|{\ell } |}_{0}^{2-j-d-p}\) for \(\alpha < -1\) and \(|T_3 |\lesssim {|{\ell } |}_{0}^{2-j-d-p} \log ^{\alpha +1} {|{\ell } |}_{0}\) for \(\alpha \in \mathbb {N}_0\). This finishes the proof for \(j=1,2\) and arbitrary p.
Finally, for \(j \geqq 3\), we need a better estimate for \(T_4\). In this case, let us split the difference directions as \(\varvec{\rho }=(\varvec{\sigma },\varvec{\tau })\), with \(\varvec{\sigma } \in {\mathcal {R}}^2\). Then,
Note that if at least one discrete derivative falls on \(\eta _\ell \) the estimate is similar to \(T_2\) as \(|z |\), \(|\ell |\), \(|\ell - z |\) are then comparable for all non-zero terms. We thus get
Note that we freely used the estimates of \(|D_{{\mathcal {S}}}^{j-2}f^{(0)} |\) for \( |D_{-{\mathcal {S}}}^{j-2}f^{(0)} |\), as \({\mathcal {S}}\) spans the lattice and thus these estimates are equivalent. \(\quad \square \)
Theorem 2 is almost an immediate consequence of Theorem 1. The only additional ingredient we need for the proof is the following lemma:
Lemma 14
In the setting of Theorem 2 define a function \(v_b\) parameterised by \(b = (b^{(i,k)})_{i,k}\) by
Then there are unique \(b^{(i,k)} \in (\mathbb {R}^{{\mathcal {S}}})^{\odot i}\), \(i=1, \dots , p-1\), \(k=1,\dots , N\), such that \({\mathcal {I}}_j(u)={\mathcal {I}}_j(v_b)\) for all \(j=0, \dots , p-1\).
Proof
For any b one calculates that
Note that \(H [D_{\varvec{\rho }} {\mathcal {G}}_k] = D_{\varvec{\rho }} \delta _0 e_k\). In particular, there are no summability problems as this expression has compact support. With that in mind we can use Lemma 11 to calculate that
Therefore, to determine \(b^{(j,k)}\) uniquely by induction over j, it is sufficient to show that the linear map
given by
is a bijection. By dimensionality (recall that \({\mathcal {S}}\) is chosen to be a basis), this is equivalent to T being one–to–one which follows from Lemma 10. \(\quad \square \)
Proof of Theorem 2
Let \(v_b=\sum _{i=0}^{p-1} \sum _{k=1}^n b^{(i,k)} : D_{\mathrm{{\mathcal {S}}}}^i {\mathcal {G}}_k\), and choose b according to Lemma 14. As the moments are linear, the result then follows directly from Theorem 1. \(\quad \square \)
Note that even though the choice of b in Lemma 14 is unique, that does not necessarily mean that the choice of b for Theorem 2 is unique. Indeed, it can happen that for certain b the sum \( \sum _{k=1}^N b^{(i,k)} : D_{\mathrm{{\mathcal {S}}}}^i {\mathcal {G}}_k\) exhibits sufficient cancellation to be part of the error estimate, for example if the coefficients b correspond to a discrete approximation of the CLE equation.
\({\textbf {Decay estimates for the far-field continuum equations}}\) We also want to have decay estimates for the equations (15) introduced in Section 2.3. These can be obtained along similar lines to the discrete case. However, we are purely interested in the far-field which simplifies things a bit and avoids a few additional complications specific to the continuum case.
Proposition 15
Let \(p, \alpha \in \mathbb {N}_0\). For any \(f \in C^{\infty }(\mathbb {R}^d / B_{R_0/2}(0);\mathbb {R}^N)\) with
for all \(j\in \mathbb {N}_0\), there exists a \(u \in C^{\infty }(\mathbb {R}^d / B_{R_0}(0);\mathbb {R}^N)\) and \({\tilde{C}}_j\) with
for all \(j\in \mathbb {N}\), such that
Remark 9
It is important to point out that the additional logarithm in the result is a generic aspect of the problem and not just a limitation of our proof. To see that consider \(\alpha =0\) and \(p=1\). Specifically, let us take a f with \(f = {{\,\mathrm{div}\,}}g\), \(|g(\ell ) |\lesssim |\ell |^{-d}\), and \(|f(\ell ) |\lesssim |\ell |^{-d-1}\). Then our proof below actually shows that
Even for something as simple as \(g(\ell ) : = E_{11} |\ell |^{-d}\) one directly gets a logarithm from the integral. That means the logarithm term naturally comes from a summing of the stresses (for \(p=1\)) around the defect core.
Of course there are special cases where there is no logarithmic term appearing. For \(p=1, \alpha =0\) that would be any setting with enough cancellation such that
For example, the first corrector equation in [12] satisfies \(\int _{\partial B_R(0)} g(z) \,\mathrm{d}z =0\) and indeed there was no logarithmic term.
Proof
(Sketch of the Proof) As the equation only needs to be satisfied outside of \(B_{R_0}\), we can assume without loss of generality that \(f \in C^{\infty }(\mathbb {R}^d;\mathbb {R}^N)\), by multiplying f with an appropriate cutoff function. Our ultimate goal will be to define \(u := f *G^{\mathrm{C}}\) and follow the same estimates as in the discrete case. The main problem in adapting the discrete proof to the continuum is the need to replicate Lemma 9 and Lemma 12 which allow conversion to appropriate divergence form for \(p \geqq 1\), and we sketch out the route to the analogues of these results here.
We follow the general idea of the proof in [3], defining \(f_0:=f\), \(f_{n+1}(\ell ):=3^d f_n(3\ell )\), and
By direct computation, it can be checked that
and using the bound on f assumed in the statement and \(p \geqq 1\), one concludes that \(f_n \rightarrow 0\) uniformly in any \(\mathbb {R}^d / B_{\delta }(0)\). Similarly, we may show that the sequence \(g_n\) is uniformly summable over the same sets, and we define \(g:= \sum _{n=0}^{\infty } g_n\). Using the relation (41), we find \(f = -{{\,\mathrm{div}\,}}g\) in \(\mathbb {R}^d / \{0\}\). It also easy to check that \(g \in C^\infty (\mathbb {R}^d / \{0\})\) and that g satisfies the decay \(|g(\ell ) |\lesssim |\ell |^{-d-p+1} \log ^{\alpha } |\ell |\) away from zero. In the continuum setting, a possible consequence of this construction is that g may have a singularity at \(\ell =0\). However, since we are only interested in the far-field behaviour, we can simply remove any singularity by redefining g on the interior of \(B_{R_0}(0)\) such that \(g \in C^\infty (\mathbb {R}^d; \mathbb {R}^{N \times d})\), and then redefine \(f:= {{\,\mathrm{div}\,}}g\). This operation does not affect the behaviour outside \(B_{R_0}(0)\), and we have therefore recovered the equivalent of Lemma 9.
A similar construction and inductive argument to that used in the proof of Lemma 12 allows us to generate an extension of f to \(f=f^{(0)} \in C^\infty (\mathbb {R}^d; \mathbb {R}^N)\) such that there are tensor fields \(f^{(k)} \in C^\infty (\mathbb {R}^d; \mathbb {R}^N \otimes (\mathbb {R}^d)^{\otimes k})\) for \(1 \leqq k \leqq p\), such that \({{\,\mathrm{sym}\,}}f^{(k-1)}= -{{\,\mathrm{div}\,}}f^{(k)}\) for \(1 \leqq k \leqq p\) satisfying the decay estimate \(|f^{(k)}(\ell ) |\lesssim |\ell |^{-d-p+k} \log ^{\alpha } |\ell |\) for \(\ell \) bounded away from zero. To avoid generating a singularity at \(\ell =0\) in this case, again we redefine the functions in \(B_{R_0}(0)\) starting at the highest order \(f^{(p)}\).
With this ability to transform f to (higher-order) divergence form, one can now follow the proof of Theorem 1 almost verbatim to estimate the derivatives of \(u=f *G^{\mathrm{C}}\) using the split into the four different contributions defined in (39). Note that the singularity of \(G^{\mathrm{C}}\) is not a problem as \(G^{\mathrm{C}}\) and its first derivative are locally integrable everywhere, and we only need to take further derivatives of the Green’s function in the estimate of \(T_3\), which is the part away from that singularity where \(G^{\mathrm{C}}\) is smooth. Indeed, in the estimate of \(T_4\), we can use the fact that we have assumed estimates for derivatives of f of arbitrary order and only put one derivative on the Green’s function instead of two. \(\quad \square \)
6 The Lattice Green’s Function and Series Expansion
In this section, we define the lattice Green’s function, and construct a series expansion in terms of continuum kernels proving Theorem 3. The general strategy is to expand the inverse of the discrete Fourier multiplier corresponding to the lattice operator H as a series of homogeneous terms about \(k=0\), and then use tools developed by Morrey in [19] to provide integral representations of these terms in real space which allow us to explicitly characterise the decay and even homogeneity of each term. In the first parts we will also follow some ideas that were developed in [20] for the simpler scalar case where \(N=1\).
6.1 Fourier transforms
We begin by providing definitions of the Fourier transforms we deploy here. For \(f:\Lambda \rightarrow \mathbb {R}^n\), we define the Fourier transform \({\mathcal {F}}_a[f]:{\mathcal {B}}\rightarrow \mathbb {R}^n\), where \({\mathcal {B}}\) is the Brillouin zone of the lattice, via
The inverse is given by
Note that \(|{\mathcal {B}} |= \frac{(2 \pi )^d}{ c_{\mathrm{vol}}}\), where \(c_{\mathrm{vol}}\) is the volume of a fundamental cell of \(\Lambda \).
We note the following properties:
-
If \(\varvec{\rho } = (\rho _1,\ldots ,\rho _j)\in {\mathcal {R}}^j\), then
$$\begin{aligned}{\mathcal {F}}_a[D_{\varvec{\rho }}f](k) = \prod _{n=1}^j (\mathrm e^{i k\cdot \rho _n}-1) {\mathcal {F}}_a[f](k). \end{aligned}$$ -
For each \(m\in \mathbb {N}\), we have that
$$\begin{aligned} \sum _{\ell \in \Lambda } f(\ell )\otimes \ell ^{\otimes m} = i^m \nabla ^m {\mathcal {F}}_a[f](0). \end{aligned}$$if f decays fast enough such that the sum converges absolutely.
We also use the standard continuum Fourier transform
defined for \(f \in L^1(\mathbb {R}^d; \mathbb {R}^N)\) and extended to the space of tempered distributions \({\mathcal {S}}'(\mathbb {R}^d; \mathbb {R}^N)\) by duality. As usual the inverse is given by
6.2 Fourier multiplier series manipulation
Next, we formally carry out a series development of the inverse of the discrete lattice operator Fourier multiplier, which we will subsequently analyse in detail.
As before consider the Hamiltonian \(H=\delta ^2 {\mathcal {E}}(0)\) given pointwise by \(H [u]= -{{\,\mathrm{Div}\,}}\big (\sum _{\rho \in {\mathcal {R}}} \mathbf{C}_{\sigma \rho } D_\rho u\big )_{\sigma \in {\mathcal {R}}}\), where \(\mathbf{C}= \nabla ^2 V(0) \in \mathbb {R}^{({\mathcal {R}}\times N) \times ({\mathcal {R}}\times N)}\) and we write \(\mathbf{C}_{\sigma \rho } = (\mathbf{C}_{\sigma i\rho j})_{ij}\in \mathbb {R}^{N \times N}\). \(\mathbf{C}\) satisfies the symmetry \(\mathbf{C}_{i \rho k \sigma } = \mathbf{C}_{k \sigma i \rho }\) as a second derivative and \(\mathbf{C}_{i \rho k \sigma } = \mathbf{C}_{i (-\rho ) k (-\sigma )}\) due to Equation (2). Then (see [10, 11]) H has a representation as a Fourier multiplier
where the matrices \(A_\rho \) are implicitly defined through the second identity.
As before we assume lattice stability (4). In reciprocal space this entails that there exists \(c_0>0\) such that
where \({{\,\mathrm{Id}\,}}\) is the \(N\times N\) identity matrix. We note that \({\widehat{H}}\in C^\infty _{\text {per}}({\mathcal {B}})\), and by virtue of lattice ellipticity, \({\widehat{H}}\) is strictly positive definite except when \(k=0\). This entails that the matrix inverse \({\widehat{H}}^{-1}(k)\) exists everywhere in \({\mathcal {B}}\) except at \(k=0\), and moreover,
We define
which is 2n homogeneous in k. Indeed, as \(4\sin ^2(\frac{1}{2}x)=2(1-\cos (x))\) we have the globally convergent power series
A standard procedure can not be used to invert this series. Defining \(S = {\widehat{H}}-{\widehat{H}}_2\), it is straightforward to check that \(|{\widehat{S}}(k) |\lesssim |k |^4\) for k in a neighbourhood of 0. Also, since \(c_0 |k |^2{{\,\mathrm{Id}\,}}\leqq {\widehat{H}}_2(k)\) for any k, we see that \(|{\widehat{H}}_2^{-1/2}(k) |\lesssim |k |^{-1}\).
This allows us to write
for all \(k \ne 0\) sufficiently small as we have
In a neighbourhood of 0, we thus have the absolutely convergent series representation
Now, by inserting
and re-summing, we may express \(H^{-1}\) as a series of terms with increasing homogeneity:
Explicitly, \({\mathcal {A}}_{2n}(k)\) is defined by the finite sum
This means \({\mathcal {A}}_{2n}\) is 2n-homogeneous.
In cases where the first term in the expansion of \(H_2\) is a multiple of the identity, that is \({\widehat{H}}_2(k)=h_2(k){{\,\mathrm{Id}\,}}\), the series simplifies to
and yet further simplification can be made if H is a multiple of the identity matrix, for example, in the scalar setting \(N=1\), which is the case considered in [20].
6.3 The lattice Green’s function and its development at infinity
With the previous series development complete, we may introduce and analyse the lattice Green’s function, which acts as an inverse to the lattice operator H.
If \(d\geqq 3\), we define the lattice Green’s function to be
and if \(d=2\), we set
where \(\gamma \) is the Euler–Mascheroni constant. To demonstrate that the limit in the latter definition exists, we take \(0< \delta < \varepsilon \) and calculate
In the above argument, we have used that \({\hat{H}}^{-1}-{\mathcal {A}}_{-2}\) is bounded in a neighbourhood of \(k=0\), and the fact that \(|1-e^{i\sigma \ell r} |=O(|\ell |r)\) for all suitably small r. Rearranging, we have established a uniform bound independent of \(\delta \), and hence the limit exists.
We now use the series expansion (42) to define a series of functions \(G_{n}\) which approximate \({\mathcal {G}}\). Note that \({\mathcal {A}}_{2n-2}\) is locally integrable and defines a tempered distribution unless both \(n=0\) and \(d=2\). Except in this special case, we therefore define
In the special case where \(n=1\) and \(d=2\) we define
in the distributional sense, that is
for any \(\varphi \in {\mathcal {S}}(\mathbb {R}^d; \mathbb {R}^N)\).
With this definition a simple direct calculation also shows that
and indeed \(G^{\mathrm{C}}=G_0\).
6.4 Alternative representation of \(G_n\)
We now connect the definitions made above to alternative representations considered by Morrey in [19, chap. 6.2]. These alternative representations will provide us with the ability to directly deduce regularity, decay, and even homogeneity. Furthermore, this representation is promising for computational uses as only a finite surface integral in Fourier space is required for its evaluation.
We begin by setting \(P = \lceil \frac{d+2n-1}{2}\rceil \), and \(h= 2P+2-2n-d\), which we note satisfies \(h\geqq 1\). Then, we define
for \(\ell \ne 0\), where \(\Delta \) is the Laplacian, and
We remark that these functions satisfy \(J_l'(w)=iJ_{l+1}(w)\). We also use the following definition from [19, Def. 6.1.4]:
Definition 1
A function \(\varphi :\mathbb {R}^d \rightarrow \mathbb {R}^{N \times N}\) is called essentially homogeneous of order s if either
-
\(s<0\) and the components of \(\varphi \) are all positively homogeneous of order s, or
-
\(s \in \mathbb {N}_0\) and \(\varphi (\ell ) = \varphi _1(\ell ) + \varphi _2(\ell ) \log |\ell |\), where the components of \(\varphi _1\) are positively homogeneous of order s, and the components of \(\varphi _2\) are homogeneous polynomials in the components of \(\ell \) of degree s.
We directly want to point out though, that for the purpose of our results here we are only interested in the cases \(s<0\) and \(s=0\). So either \(\varphi \) is positively homogeneous of a negative degree, or \(\varphi (\ell ) = \varphi _1(\ell ) + A \log |\ell |\), where A is a constant and \(\varphi _1\) is 0-homogeneous. With this definition in place, we state the following result:
Lemma 16
\(G_{n}^{{\mathcal {M}}}\) is well-defined, smooth for \(\ell \ne 0\), and \(G_{n}^{{\mathcal {M}}}\) is essentially \((2-2n-d)\)-homogeneous. Furthermore, \(G_{n}^{{\mathcal {M}}}\) is positively \((2-2n-d)\)-homogeneous if either \(2-2n-d<0\) or d is odd.
Proof
This is part of [19, Thm. 6.2.1]. Note that \({\mathcal {A}}_{2n}\) being matrix-valued does not change the argument. \(\quad \square \)
In our setting, the only case that might fail to be positively homogeneous is \(G_{0}^{{\mathcal {M}}}\) for \(d=2\). As pointed out above, in that case essentially homogeneous means that \(G_{0}^{{\mathcal {M}}} (\ell ) = \varphi (\ell ) + C_1 \log |\ell |\), where \(\varphi \) is positively 0-homogeneous and \(C_1 \in \mathbb {R}^{N \times N}\).
We will now show that \(G_{n}\) and \(G_{n}^{{\mathcal {M}}}\) are in fact identical. Let us first give a short motivation for \(G_{n}^{{\mathcal {M}}}\). The basic idea of the definition (44) is to use the homogeneity of \({\mathcal {A}}_{2n-2}\) to isolate the radial component in the Fourier integral and then integrate it out. Naïvely, this approach leads to a integral of the form \(\int _0^\infty r^l e^{irw} \,\mathrm{d}r\) which does not exist for any l and real w. The idea behind the P Laplacians is to ensure that the resulting l is non-negative. If one then adds a small imaginary part to w, the integral is indeed finite and can be computed. This leads to the \(J_l\).
Lemma 17
On \(\mathbb {R}^d / \{0\}\) the distribution \(G_{n}\) is represented by a function which we will also call \(G_{n}\) and we have \(G_{n}= G_{n}^{{\mathcal {M}}}\) on \(\mathbb {R}^d / \{0\}\).
Proof
As discussed in the motivation directly before the Lemma, we first let \(\Im (w)>0\) and \(l \geqq 0\). Then
Take any \(\delta >0\), and let [i, w] be the line segment in the complex plane, for which we note that \(z\in [i,w]\) implies \(\Im (z)>0\).
Define
which is easily seen to satisfy \((z\varphi _0(z))' = \frac{e^z-1}{z}\). Using the definition of \(J_0\) and the definition and property of \(\varphi _0\) just noted, we then find
where we define
In (45), the first term is the desired integral for \(l=-1\), the second term is a renormalization of the singular part for r close to 0, and the third term is of lesser importance and will go to zero at the end.
We can simplify \(p^\delta _0(w)\) further, as the exponential integral \(E_1\) satisfies
for \(z \in \mathbb {C}/ \mathbb {R}_{\leqq 0}\), where \(\gamma \) is the Euler–Mascheroni constant, and \([z,\infty ]\) is any contour in \(\mathbb {C}/ \mathbb {R}_{\leqq 0}\) connecting z to positive real infinity. It follows that
Now, defining
and
for \(h \geqq 0\) we may calculate inductively that
This calculation was restricted to \(\Im (w)>0\) to ensure that the integrals converge. But for \(h \geqq 1\) that is true even for real w. So we now can take the limit \(\Im (w) \rightarrow 0\) and see that for all \(w \in \mathbb {R}\), \(\delta >0\), and \(h \geqq 1\), we have that
Using the definition in (44), we have obtained the following representation for \(G_{n}^{{\mathcal {M}}}\):
We now inspect the three integrals in turn. Clearly, for the first,
in the sense of tempered distributions. For the second integral, as \(p^\delta _h\) is a polynomial of degree \(h = 2P +2 -2n -d\), we find \( (-\Delta )^P p^\delta _h ( \ell \sigma )=0\) except when \(d=2\) and \(n=0\), in which case \( (-\Delta )^P p^\delta _h ( \ell \sigma )= \gamma + \log \delta \). The integrand in the final integral contains the factor \((-\Delta )^P \delta (i\ell \sigma )^{h+1} \varphi _{h}(\delta i \ell \sigma )\), which goes to 0 uniformly on compact sets as \(\delta \rightarrow 0\). Since \(\delta \) was arbitrary, we may pass to this limit, and we do indeed find \(G_m= G_m^{{\mathcal {M}}}\) on \(\mathbb {R}^d / \{0\}\). \(\quad \square \)
After having constructed the kernels \(G_n\) and having established their properties, we can now prove the expansion of the lattice Green’s function.
Proof of Theorem 3
We have already established the regularity and homogeneity properties of the \(G_n\). In the following, fix \(j \geqq 0\) and \(\varvec{\rho }=(\rho _1, \dots , \rho _j) \in {\mathcal {R}}^j\). We then still need to show that
for all \(\ell \), p. We claim it is in fact sufficient to show the result holds for any p but where the error decays with two orders less, that is
for all \(\ell \). If this estimate holds, then using Equation (47) with \(p+1\) and the triangle inequality entails that
but since we have that \(D_{\varvec{\rho }} G_{p+1}(\ell ) = O(|\ell |^{-2p-d-j})\) due to the homogeneity of \(G_{p+1}\), this establishes (46).
To show (47), fix a smooth cutoff \(\eta :\mathbb {R}^d \rightarrow [0,1]\) with \(\eta (k)=1\) in a neighbourhood of 0 and \({{\,\mathrm{supp}\,}}\eta \subset \subset {\mathcal {B}}\). Then set \(G_{n}^\eta := c_{\mathrm{vol}}{\mathcal {F}}^{-1}_c [{\mathcal {A}}_{2n-2} \eta ]\) and \(G_{n}^{1-\eta } := c_{\mathrm{vol}}{\mathcal {F}}^{-1}_c [{\mathcal {A}}_{2n-2} (1-\eta )]\). If \(n=0\) and \(d=2\), set
instead. If \(2n-2 > -d \), note that \({\mathcal {A}}_{2n} \eta \) is integrable and has support in \({\mathcal {B}}\), therefore
as well as
In all cases we find
with
as long as m is small enough to ensure that \(r_m\) is integrable. The smoothness of \(\eta \), and the properties of \({\hat{H}}^{-1}\) and \({\mathcal {A}}_{2n}\) discussed in Section 6.2 ensure that \(r_m\) is smooth away from \(k=0\) and is bounded by \(C |k |^{2p-2m + j}\) in a neighbourhood of \(k=0\). It follows that \(r_m \in L^1\) if \(2m \leqq 2p + j + (d-1)\) and therefore (48) implies that
For \(G_{n}^{1-\eta }\), we observe that \(D^{\alpha } \big ( {\mathcal {A}}_{2n-2} (1-\eta )\big )\) is integrable for all sufficiently large \(|\alpha |\), and therefore \(G_{n}^{1-\eta }\) has super-algebraic decay at infinity. The same therefore also holds true for any \(D_{\varvec{\rho }} G_{n}^{1-\eta }\).
The only remaining claim now is the uniqueness. Assume we have two such series of kernels \((G_n)_n\) and \(({\tilde{G}}_n)_n\). By induction, for \(p\in \mathbb {N}_0\) let us assume we already know that \(G_n = {\tilde{G}}_n\) for all \(n<p\), we need to show that \(G_p = {\tilde{G}}_p\).
First exclude the case where both \(p=0\) and \(d=2\). Then we know that \(G_p\) and \({\tilde{G}}_p\) are positively homogeneous of order \((2-2p-d)\). Using both estimates of order p without any derivatives, we obtain
for all \(\ell \in \Lambda \). Combined with the homogeneity, we thus have that
Given any \(\varepsilon >0\) we can therefore find an \(R>0\) such that
for all \(x \in \{ \frac{\ell }{|\ell |} :\ell \in \Lambda , |\ell |> R \}\). As this set is dense in \(\partial B_1(0)\) and both functions are continuous, (49) holds for all \(x \in \partial B_1(0)\). As \(\varepsilon >0\) was arbitrary we thus have \(G_p = {\tilde{G}}_p\) on \(\partial B_1(0)\) and by homogeneity on all of \(\mathbb {R}^d / \{0\}\).
In the case that \(p=0\) and \(d=2\) the argument is only slightly more complex. We write \(G_0 = A \log |\ell |+ \varphi \) and \({\tilde{G}}_0 = {\tilde{A}} \log |\ell |+ {\tilde{\varphi }}\) with \(A,{\tilde{A}} \in \mathbb {R}^{N \times N}\) and \(\varphi , {\tilde{\varphi }}\) both 0-homogeneous. Then
for all \(\ell \in \Lambda \). As \(\varphi \) and \({\tilde{\varphi }}\) are bounded, we can divide by \(\log |\ell |\) and send \(|\ell |\rightarrow \infty \) to see that \(A={\tilde{A}}\). The identity \(\varphi ={\tilde{\varphi }}\) then follows the same way as before. \(\quad \square \)
We can now rewrite the truncated multipole expansion of Theorem 2 in continuum terms.
Lemma 18
where
for all \(j \in \mathbb {N}_0\).
Proof
This is an immediate consequence of the expansion of the lattice Green’s function from Theorem 3 and a Taylor expansion of the discrete differences in \(D_{\varvec{\rho }}G_{n}\). \(\quad \square \)
As a consequence we obtain Theorem 4.
Proof of Theorem 4
The result follows by combining Theorem 2 and Lemma 18. \(\square \)
7 Proofs - Far Field Expansions for Crystalline Defects
In this section we prove the results of Section 3. Before we get to the main results, let us start with a preliminary proof.
Proof of Proposition 6
\(\delta {\mathcal {E}}(u)(\ell )\) is summable according to [3, Lemma 10]. The same then holds true for \(H[u](\ell )\). The idea of the proof is to use a cutoff \(\eta _R\) and sum by parts so that only the far away annulus \(B_{2R} {\setminus } B_R\) contributes. Here, the linearisation and discretisation errors are small and we can go from the nonlinear forces to the linearised forces and all the way to the continuum forces. So, consider a smooth cutoff function \(\eta _R\), such that \(\eta _R :\mathbb {R}^d \rightarrow [0,1]\) satisfies \(\eta _R(z)=1\) for \(|z |\leqq R\) and \(\eta _R(z)=0\) for \(|z |\geqq 2R\), as well as \(|\nabla \eta _R |\lesssim R^{-1}\). We then have that
as the linearisation error sums to \(O(R^{-1})\). Furthermore,
as the discretisation error is also \(O(R^{-1})\). \(\quad \square \)
Now let us come to the main topic of this section, the proofs of Theorem 5 and Theorem 7.
At the outset, we recall from the setting discussed in Sections 2 and 3 that we assume the site potentials (and hence the potential energy functional) are of class \(C^K\), the number of derivatives we want to estimate is \(J \geqq 2\) and the dimension of the problem is d. We now prove the expansion and the error estimates by induction over \(p \geqq 0\) as long as
in the point defect case and
in the case of the screw dislocation, where we always have that \(d=2\). We also note that all results only need to be proven for \(|\ell |\geqq R\) for some sufficiently large R.
7.1 The case \(p=0\)
The case \(p=0\) is a consequence of results in [3] for both the point defect case and the case of screw dislocation. To be precise, it follows from the assumptions of both Theorem 5 and Theorem 7 that \(J \leqq K-2\). In this setting, we may apply Theorem 1 in [3], which states that the solution \({\bar{u}}\) in the point defect cases satisfies
In the screw dislocation case, we may similarly apply Theorem 5 in [3], implying that
In order to combine and summarise these results, we define
while \(\nabla u_0 \in C^\infty (\mathbb {R}^d/\{0\};\mathbb {R}^N)\) with
7.2 The case \(p>0\)
We now proceed inductively from the case \(p=0\).
\({\textbf {Taylor expansion.}}\) We follow the construction that we formally motivated in Section 2.3 and expand the solution \({\bar{u}}\) by performing a Taylor expansion of the equilibrium equation for the forces
for all \(\ell \) sufficiently large, where \(g=0\) in the screw dislocations case and g has compact support for point defects.
Recall from Section 3 that \({\mathcal {E}}\) inherits the same regularity as the homogeneous site potential V, and is therefore \(C^K\). For such a variation of the energy we use the pointwise notation for the corresponding \(\ell ^2\) representative. Specifically, that means
A Taylor expansion around the homogeneous lattice \(u=0\) up to order \({\tilde{K}}\leqq K-2\) gives
We will determine the precise order \({\tilde{K}}\) for this expansion as part of our arguments later. We note that we already have explicit expressions for the first two terms in the sum: A homogeneous (Bravais) lattice is always in equilibrium, \(\delta {\mathcal {E}}(0)=0\), and the second term is \(\delta ^2 {\mathcal {E}}(0)[{\bar{u}}] = H[{\bar{u}}]\). For convenience, we will define \({\mathcal {T}}_{p+1}^{(1)}\) to be the integral error term and also include the compactly supported \({{\,\mathrm{Div}\,}}g\), that is
so that we may rewrite (55) as
\({\textbf {Expansion of solution.}}\) Next, we make the inductive assumption that we have already obtained \(u_i\) for \(0 \leqq i \leqq p-1\), and that we can further decompose
which are respectively continuum- and lattice-valued functions, so that we may write
where \(r_p\) is some remainder. These functions will be assumed inductively to satisfy certain properties which we now make precise:
-
We will assume that the former terms, \(u_i^{\mathrm{C}}\in C^{\infty }(\mathbb {R}^d/ B_{R_0}(0))\), \(i \leqq p-1\), have been constructed to solve a sequence of linear elliptic PDEs,
$$\begin{aligned} H^{\mathrm{C}}[u_i^{\mathrm{C}}]= {\mathcal {S}}_i(u_0^{\mathrm{C}}, u_0^{\mathrm{CMP}}, \dots , u_{i-1}^{\mathrm{C}}, u_{i-1}^{\mathrm{CMP}}) \end{aligned}$$(59)and satisfy decay estimates
$$\begin{aligned} |\nabla ^j u_i^{\mathrm{C}} (\ell )|\leqq C_j |\ell |^{2-d-j-i} \log ^{i} |\ell |\quad \text {for all }j\in \mathbb {N}_0. \end{aligned}$$(60)We note that \({\mathcal {S}}_i\) are forcing terms which we determine as part of our argument.
-
We assume that the latter terms in (57) have been constructed to take the form
$$\begin{aligned} u_i^{\mathrm{MP}} = \sum _{k=1}^N b^{(i,k)} : D_{\mathrm{{\mathcal {S}}}}^i {\mathcal {G}}_k \end{aligned}$$(61)for some constant tensors \(b^{(i,k)}\). In particular, \(H[u_i^{\mathrm{MP}}](\ell ) =0\) for \(|\ell |\) large enough, and we assume \(u_0^{\mathrm{MP}}=0\).
-
We assume that the remainder estimate
$$\begin{aligned} |D^jr_{p}(\ell ) |\lesssim {|{\ell } |}_{0}^{2-d-p-j} \log ^{p} {|{\ell } |}_{0} \end{aligned}$$(62)holds for \(j=1,\dots ,J\).
In light of (61), whenever useful, we may also rewrite the multipole contributions in terms of smooth continuum kernels by applying Lemma 18. Indeed, by combining the terms of equal homogeneity, Lemma 18 implies that we may write
where \(u_i^{\mathrm{CMP}} \in C^\infty (\mathbb {R}^d/ \{0\};\mathbb {R}^N)\) is positively homogeneous of degree \((2-d-i)\) which implies
for all \(j \in \mathbb {N}_0\), and we have the additional error estimate
for all \(j \in \mathbb {N}_0\). Using the functions \(u^{\mathrm{CMP}}\), we therefore have the representation
Using the inductive assumptions made above, we may substitute (58) into the left-hand side of (56), and write
The goal is now to split the remainder \(r_p\) into the next continuum contribution \(u_p^{\mathrm{C}}\) and a new intermediary term \(s_p\), and find the correct equation for \(u_p^{\mathrm{C}}\) so the residual becomes even smaller in the far-field. Then we can apply Theorem 2 to \(s_p\) to split this term yet further into multipole terms up to order p and a new remainder \(r_{p+1}\) with the desired improved decay.
\({\textbf {Taylor expansion error.}}\) With the assumptions above, we turn back to the Taylor expansion (55), and establish the choice of order, \({\tilde{K}}\). Our aim is to fully resolve all orders of decay up to and including \({|{\ell } |}_{0}^{-d-p}\), and we have the decay estimates \(|D^j {\bar{u}} |\lesssim {|{\ell } |}_{0}^{2-d-j}\) and the stronger \(|D^j {\bar{u}} |\lesssim {|{\ell } |}_{0}^{1-d-j}\) in the case of a point defect, as \(u_0=0\). Formally, this means that we expect that \(|\delta ^{k+1} {\mathcal {E}}(0)[{\bar{u}}^{\otimes k}] (\ell ) |\lesssim {|{\ell } |}_{0}^{(1-d)k-1}\) or \({|{\ell } |}_{0}^{-dk-1}\) in the case of a point defect. These orders of decay come from estimating a k-fold tensor product of \(D{\bar{u}}\) with the decay mentioned, and one further additional power arising from the discrete divergence. In particular, to ensure that the order of decay matches or exceeds that of \(|\ell |_0^{-d-p}\), we require \({\tilde{K}} \geqq \frac{p}{d-1}+1\), or in the case of a point defect \({\tilde{K}} \geqq \frac{p-1}{d}+1\). Indeed, this is satisfied by assumption on K, if we set \({\tilde{K}} := K-J-1\). We also note that the expression of \({\mathcal {T}}_{p+1}^{(1)}\) involves \({\tilde{K}}+2=K-J+1\) derivatives of \({\mathcal {E}}\) and thus V, as well as one divergence, so \(J-2\) further derivatives are available. Let us use these considerations and the choice \({\tilde{K}} := K-J-1\) to rigorously estimate \({\mathcal {T}}_{p+1}^{(1)}\).
Lemma 19
With \({\tilde{K}}=K-J-1\), the remainder term in the Taylor expansion (55) satisfies
for all \(j=0, \dots , J-2\).
Proof
For \(\ell \) large enough, \(-{{\,\mathrm{Div}\,}}g(\ell ) =0\), hence we have
In the case of a screw dislocation we can use the estimates (53) and (54) for \(u_0\) and \(r_1\) to see that \(|D^j {\bar{u}} |\lesssim {|{\ell } |}_{0}^{-j}\) for \(1 \leqq j \leqq J\). By distributing the outer discrete derivatives over the tensor product and using this estimate, we deduce that
This proves the result in this case as \({\tilde{K}}=K-J-1 \geqq p+1\) and \(d=2\).
For a point defect, we have \(u_0=0\). More specifically, according to (51) we have \(|D^j {\bar{u}} |\lesssim {|{\ell } |}_{0}^{1-d-j}\) for \(1 \leqq j \leqq J\). Using this estimate and distributing the outer discrete derivative as in the previous case, it follows that
And since \({\tilde{K}} = K-J-1 \geqq \lfloor \frac{p-1}{d}\rfloor +1\) we can estimate the exponent by
\(\square \)
\({\textbf {Error terms due to discrete far-field residuals.}}\) With \({\mathcal {T}}_{p+1}^{(1)}\) controlled, we now go back to work on (66). As the next step we want to replace \({\bar{u}}\) on the right hand side by a smooth continuum approximation which naturally gives us another error term from the remaining discrete far-field terms. Recalling (65), we have
We choose to decompose \(r_p = u^{\mathrm{C}}_p+s_p\), where we will specify \(u^{\mathrm{C}}_p\) as the solution to a PDE later in the construction. For now, take any function \(u^{\mathrm{C}}_p \in C^\infty (\mathbb {R}^d / B_{R}(0);\mathbb {R}^N)\) that satisfies
for all \(j \in \mathbb {N}\), some radius \(R>0\), and some constants \(C_j>0\).
Then define \({\bar{u}}_p := \sum _{i=0}^p u^{\mathrm{C}}_i +\sum _{i=0}^{p-1}u^{\mathrm{CMP}}_i\), so that
Based on this, we can substitute (68) into the series on the right-hand side of (66), giving
where
collects all the terms that contain contribution from \(s_p\) or \(w_p^{\mathrm{MP}}\). To estimate derivatives of \(w_p^{\mathrm{MP}}\), we can use (64). For \(s_p\) we can combine (62) and (67) to obtain
for \(j=1,\dots ,J\).
This allows us to estimate the second error term.
Lemma 20
The estimate
holds for all \(j=0, \dots , J-2\).
Proof
We note that \({\mathcal {T}}_{p+1}^{(2)}\) can be written as a sum of differences,
For the \(k=2\) term, we use (68) and apply the discrete product rule to obtain
We note that the shifts \(\rho \) appearing in the first estimate which arise from the discrete product rule remain below some finite radius \(R=R(J)\). Using the estimates for \(w_p^{\mathrm{MP}}\) and \(s_p\) gives the desired overall estimate.
For \(k \geqq 3\) a similar argument provides the same estimate, as any additional factor just improves the estimate by at least \({|{\ell } |}_{0}^{-1} \log {|{\ell } |}_{0} \lesssim 1\). \(\quad \square \)
\({\textbf {Continuum approximation.}}\) We want to construct a sequence of PDEs to resolve (69) to sufficient accuracy and thus describe the far-field behaviour of \({\bar{u}}\). To that end, we note that a finite difference of any smooth function v can be Taylor expanded to give
for some \(\theta \in [0,1]\). By iterating this process of Taylor expansion, a similar series approximation can be constructed for higher-order finite differences. In particular, for a second-order finite difference, we obtain
where \(\theta _l\) lies in the convex hull of the set of points \(\{0,\rho ',\rho ,\rho '+\rho \}\) for \(l=1,\ldots ,M-1\).
To start, we may use this expansion to approximate the action of H on a smooth function. We note that H was defined by (3) as
The symmetry assumption (2) implies \(\nabla ^2 V(0)_{\rho \rho '} = \nabla ^2 V(0)_{(-\rho ) (- \rho ')}\). Therefore all the odd orders cancel and using (71) we get that there exist tensors \(\mathbb {C}^{2n}\in \mathbb {R}^{N^2\times d^{2n}}\) for \(n\geqq 1\) such that, for any smooth function v,
where \({\mathcal {T}}^{(3,1)}\) collects the Taylor error terms.
In coordinates, these tensors are
for \(a,b=1,\ldots ,N\), \(j_l=1,\ldots ,d\).
In particular, in leading order, we find that \(H[v] \approx c_{\mathrm{vol}} H^{\mathrm{C}}[v]\), as
where we used that
based on the definition of the Cauchy–Born energy density W in (6).
Construction of \(\varvec{u^{\mathrm{C}}_p.}\) We recall that so far, we have Taylor expanded the site potential and substituted a truncated series \({\bar{u}}_p\) on the right-hand side of (55) to obtain (69), that is
We note that the remaining series on the left and right-hand sides of this equation involve only the action of discrete difference operators on smooth continuum functions, and so we may now perform a discrete-to-continuum approximation to handle these terms.
On the left, we insert (73) from the discussion of \(H^{\mathrm{C}}\) as the leading order into the full expansion (72) to see that there exist tensors \(\mathbb {C}^{n}\in \mathbb {R}^{N^2\times d^n}\) for \(n\geqq 3\) such that for a smooth function v,
We can perform a similar construction for the higher-order terms in the Taylor expansion on the right-hand side of (69), so that in general, there exist tensors \(\mathbb {C}^{{\varvec{j}}}\) in the natural tensor space which is isomorph to \(\mathbb {R}^{N^{k+1}\times d^n}\) for each \({\varvec{j}} = (j_1, ..., j_k)\) that satisfies \(1 \leqq j_m \leqq M_k-1\) and \(\sum _{m=1}^k j_m = n\) for some \(n \geqq k+1\). This allows us to approximate \(\delta ^{k+1} {\mathcal {E}}(0)[v^{\otimes k}](\ell )\) as
Each of the tensors \(\mathbb {C}^{{\varvec{j}}}\) is a sum of tensor products formed of components of \(\nabla ^{k+1}V(0)\) and a n-fold tensor products of vectors from \({\mathcal {R}}\). The final term
collects all the terms that include at least one Taylor error for v. And overall, after inserting the specific v, we can combine these errors to
Our aim is now to use the decay estimates of the terms \(u^{\mathrm{C}}_i\) and \(u^{\mathrm{CMP}}_i\) to identify and collect all terms in (76) with the same rate of decay into a forcing term on the right-hand side of (59), thereby defining the terms \({\mathcal {S}}_q\) which satisfy the estimate \(|\nabla ^j{\mathcal {S}}_q |\lesssim {|{\ell } |}_{0}^{-d-j-q} \log ^{\gamma _q} {|{\ell } |}_{0}\) for all j, some \(\gamma _q\), and all \(|\ell |\) large enough. Recall from (60) and (63) that for \(|\ell |\) large enough
and for notational convenience in the following argument, define \({\tilde{u}}_i := u^{\mathrm{C}}_i+u^{\mathrm{CMP}}_i\) for \(i <p\) and \({\tilde{u}}_p:= u^{\mathrm{C}}_p\), which satisfies that
for all \(j \in \mathbb {N}_0\) while overall \(\sum _{i=0}^p {\tilde{u}}_i = {\bar{u}}_p\).
Let us consider multi-indices \({\varvec{i}}=(i_1, \dots , k)\) for choosing multiple \(u_i\) in a nonlinear term and \({\varvec{j}}=(j_1, \dots , j_k)\) for the number of derivatives on each term. We write \( |{\varvec{i}} |_1 = \sum _m i_m\) for the sum of the components.
By virtue of estimate (77), and the properties of tensor products, we have that
We can therefore identify terms with the decay rate \({|{\ell } |}_{0}^{-d-q}\) in (76) by looking only at the indices in
Now that we can precisely identify all these terms we define
We only divide by \(c_{\mathrm{vol}}\) as we want to factor it out from the resulting equation.
We collect all the remaining orders in
Note that as long as we choose the \(M_k\) large enough, the definition of the \(S_q\), \(q \leqq p\), in (78) is independent of their precise choice. Any additional terms from choosing \(M_k\) even larger only appear in \({\mathcal {T}}^{(4)}_{p+1}\).
Lemma 21
The following estimates holds:
hold for all \(j\in \mathbb {N}_0\).
Proof
The order of decay in (79) is clear by construction, we only need to check the number of logarithmic terms. Indeed, the number of logarithmic terms needed is the maximum of all possible \(\sum _m i_m = |{\varvec{i}} |_1\). For \(k \geqq 2\) and \(n \geqq k+1\) derivatives we have \(|{\varvec{i}} |_1 = d+ q + (2-d)k-|{\varvec{j}} |_1\) which is maximal if \(|{\varvec{j}} |_1=k+1\) and \(k=2\), therefore \(|{\varvec{i}} |_1 \leqq 1-d+q \leqq q-1\) for all terms with \(k \geqq 2\). For the linear terms we directly see that we get at most \(q-1\) logarithmic terms from \(u_{q-1}\).
(80) is trivial as we can choose \(M_k\) as large as we want.
And (81) is just the same estimate as (79), where now we look at various \(q \geqq p+1\). The dominant part being the estimate for \(q= p+1\). Having discrete differences instead of gradients does not change anything. \(\square \)
We note the separation between terms which derive from third- and higher-order derivatives of \({\mathcal {E}}\) for \(k>1\) on the first line of (78), and those which derive from H on the second line. A distinction arises partly due to the functions involved, and because we retain the leading order approximation \(H\approx c_{\mathrm{vol}} H^{\mathrm{C}}\) on the left-hand side of (59).
For given q, it can be checked that under the constraints on the indices, the maximal index i of any \({\tilde{u}}_i\) involved in one of the terms in the first sum is \(i = q-(d-1)\leqq q-1\), and in the second, when \(i=q-1\). It follows that
as we assumed in (59). Furthermore, as \({\tilde{K}}\) and the \(M_k\) are chosen sufficiently large, the definition of \(S_q\), \(1 \leqq q \leqq p\), given here does not depend on the induction step p and thus remains the same over the iteration. So given that we assume \(H^{\mathrm{C}}[u^{\mathrm{C}}_q] = {\mathcal {S}}_q\), these terms are fixed iteratively. For \(q=p\), we can now specify \(u^{\mathrm{C}}_p\) to be a solution to
for \(\ell \) sufficiently large, so that these terms cancel on both sides of the equation as well, and we generate a new term in our series expansion of \({\bar{u}}\).
Indeed, as \({\mathcal {S}}_p \in C^\infty (\mathbb {R}^d / B_R(0); \mathbb {R}^N)\) for some radius \(R>0\) and satisfies
there according to (79), we can use Proposition 15 to see that there exists a \(u^{\mathrm{C}}_p \in C^\infty (\mathbb {R}^d / B_{2R}(0); \mathbb {R}^N)\) that solves \(H^{\mathrm{C}}[u^{\mathrm{C}}_p] = {\mathcal {S}}_p\) and satisfies the estimate
for \(|\ell |\) large enough. \(u^{\mathrm{C}}_p\) can of course be extended arbitrarily for small \(|\ell |\).
In summary, we may use the definition of \({\mathcal {S}}_q\) and the remainder terms \({\mathcal {T}}_{p+1}^{(1)}\), \({\mathcal {T}}_{p+1}^{(2)}\), \({\mathcal {T}}_{p+1}^{(3)}\), and \({\mathcal {T}}_{p+1}^{(4)}\) to rewrite (69) as
In view of the definition of \(u^{\mathrm{C}}_i\) as a solution to (59), it follows that this equation reduces to the remainder equation
which we study in the next section.
\({\textbf {Series remainder estimate.}}\) Note that (82) is not a linear equation for \(s_p\), as \(s_p\) also appears on the right hand side in \({\mathcal {T}}_{p+1}^{(2)}\). In the leading order estimates in [3] this was resolved through techniques coming from regularity theory which we rely on for \(p=0\). For the higher orders, that is \(p>0\), we can use the (suboptimal) estimates that are available from the previous order. That turns out to be sufficient as the appearance of \(s_p\) in \({\mathcal {T}}_{p+1}^{(2)}\) only occurs in nonlinear terms which give additional decay, see Equation (70) and Lemma 20.
Combining the error estimates provided by in Lemma 19, Lemma 20 and Lemma 21, we have that
for all \(j=0, \dots , J-2\). This means we can apply Theorem 2 with \(p+1\) instead of p to get that
and
for \(j=1, \dots , J\), as desired.
\({\textbf {Conclusion.}}\) To conclude the induction step we still have to look at that the possible addition of lower order terms from (84) as they change the multipole terms from one iteration to the next. Instead of discussing under what conditions these terms vanish, we allow for them to be non-zero and consider the more detailed consequences. Indeed, this is fine as long as \(u_0^{\mathrm{MP}}=0\) remains true as this is part of the results and as long as \(u_i^{\mathrm{CMP}}\), \(i=1,\dots ,p-1\), remain unchanged which is important since they are part of the continuum equations (59). The first just follows from the fact that \({\mathcal {I}}_0[v]=0\) due to the divergence form of \({\mathcal {T}}_{p+1}\) combined with sufficient decay. For the second part, let us use the continuum expansion of the multipole terms to write
We know that \(v_i^{\mathrm{CMP}}\) is \((2-d-i)\)-homogeneous. Combining the decay estimate (70) for \(s_p\) with the decay estimates for \(r_{p+1}\) and \({\tilde{w}}^{\mathrm{MP}}_{p+1}\), we find that
for \(|\ell |\) large enough. This is only compatible with the homogeneity if \(v_i^{\mathrm{CMP}}=0\) for \(i=1,\dots ,p-1\). That ends the induction step and thus the proof of Theorem 5 and 7.
As a last step, we see that for a point defect, for any d, we have \(u_0^{\mathrm{C}}=0\). We then also find that \(u_1^{\mathrm{C}}=\dots = u_{d}^{\mathrm{C}}=0\) as \({\mathcal {S}}_{i}=0\) for \(1 \leqq i \leqq d\). The first non-trivial terms arises when \(u_1^{\mathrm{CMP}}\) appears in the nonlinearity, namely
This proves Remark 5.
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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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The authors Julian Braun and Christoph Ortner were supported by EPSRC Grant EP/R043612/1.
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Braun, J., Hudson, T. & Ortner, C. Asymptotic Expansion of the Elastic Far-Field of a Crystalline Defect. Arch Rational Mech Anal 245, 1437–1490 (2022). https://doi.org/10.1007/s00205-022-01810-3
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DOI: https://doi.org/10.1007/s00205-022-01810-3