Existence and rigidity of the vectorial Peierls-Nabarro model for dislocations in high dimensions

Denote the unit torus $\mathbb{R}/\mathbb{Z}$ as $\mathbb{T}$. Instead of minimizing the total energy (1.1) on $\mathbb{R}^{3}$, we consider the model on $\mathbb{R}^{2}\times\mathbb{T}$. Correspondingly, the slip plane $\Gamma$ is replaced by $\Gamma^{\prime}$ which is defined as

A standard calculation of the first variation of the total energy (1.1) derives the following Euler–Lagrange equation satisfied by minimizers of (1.1) in the sense of Definition 1. The proof of this lemma can be found in Appendix B or our previous work [19].

For the special case in (1.7), where $\gamma$ only depends on the shear displacement $u_{1}^{+}$, we can simplify and decouple the system (1.13) into two independent equations and finally drive a reduced scalar equation of $u_{1}^{+}$, i.e. (1.15). In details, one can employ the Dirichlet to Neumann map and the elastic extension introduced in [19] to reduce the problem from $\mathbb{R}^{2}\times\mathbb{T}$ onto $\Gamma^{\prime}$, i.e., to equations of $(u_{1}^{+},u_{3}^{+})$ on the split plane $\Gamma^{\prime}$. Second, if further employing (1.7), one can derive a linear representation formula between $u_{1}^{+}$ and $u_{3}^{+}$ on the Fourier side, i.e.,

Here $\hat{u}_{i}^{+}(k_{1},k_{2}),i=1,3$ are the Fourier transform of $u_{i}^{+},i=1,3$ with frequency vector $\bm{k}=(k_{1},k_{2}),k_{1}\in\mathbb{R},k_{2}\in 2\pi\mathbb{Z}$. Substituting (1.14) into system (1.13), an independent equation of $u_{1}^{+}$ is derived, which contains a pseudo-differential operator $\mathcal{L}$ defined on $H^{1}(\mathbb{R}\times\mathbb{T})$:

The derivation of (1.15) is standard and can be found in Section 2. Therefore, as long as we can solve the non-local semi-linear equation of $u_{1}^{+}$, i.e. the first equation in (1.15), we can also find $u_{3}^{+}$ by (1.14), and then derive the solution of the original system (1.13). For brevity, we will omit the superscript ’$+$’ in the following sections. We call (1.15) the reduced scalar equation.

To solve (1.15), a meaningful observation is that we can write down an explicit solution to it for certain double-well potential $\gamma$’s. Highly compatible with dislocations in Halite, the cosine potential $\gamma_{0}=\dfrac{1}{\pi^{2}}(\cos(\pi u)+1)$ in the PN model implements an explicit solution [6, 17] to (1.15) and (1.14):

In particular, $u_{1}$ in (1.16) is a layer solution (see Definition 3) since it is strictly monotonic in $x$ direction and satisfies assumption (1.9). In fact, (1.16) is a good candidate for minimizers of total energy (1.1) in the sense of Definition 1. We will prove that this solution is the unique minimizer up to translations which concludes the question on existence and rigidity. We remark here that this is just a concrete example of our general result: for general double-well type potentials, we prove that minimizers of the total energy $F$ (see (1.37)) in function set (1.35) exist and they are layer solutions (see Definition 3). Moreover, they are unique up to translations.

The existence and rigidity of the vectorial PN model are important in understanding dislocations. Previous literature on the vectorial PN model mainly focused on numerical simulations [40, 42, 28, 36] and physical experiments [42, 39], while only few rigorous mathematical results [19] were derived. In this section, we aim at mathematically formulating those important but unsolved problems into a framework and embedding our result into this macroscopic framework.

Consider the Euler-Lagrange equation of the vectorial model. We first observed that the second and fourth equations of (1.13) can be rewritten as

where $\mathcal{A}$ is a pseudo-differential operator with Fourier symbol $\hat{\mathcal{A}}$. Equation (1.17) is a nonlocal reduced elliptic equations on slip plane $\Gamma$, which is an open system. Meanwhile, the misfit potential here may depend on both $u_{1}$ and $u_{3}$: $\gamma=\gamma(u_{1},u_{3})$. As far as we know, neither existence nor rigidity of (1.17) was studied by previous literatures.

In fact, if the misfit potential is carefully selected [42, 28, 36, 6], solutions of (1.17) determine the underlying structure of dislocations in crystals such as Cu ($\nu=0.36$) and Al ($\nu=0.33$), no matter straight ones or curved ones. Considering symmetry of the crystal lattice, authors of [40] adopted a truncated Fourier expansion for the generalized stacking fault energy (see eq.(14) in [40]) as the misfit energy $\gamma(u_{1},u_{3})$, with different coefficients for Cu and Al.

Numerical simulations in [40] indicated that straight edge dislocations in both Cu (figure 4 in [40]) and Al (figure 5 in [40]) possess the following structure: the displacement in the shear direction (i.e. $u_{1}$) is a layer solution (see Definition 2) and the displacement in the transverse direction (i.e. $u_{3}$) is a solitary wave. As claimed by authors in [40], this numerical result agrees with data from experiments on real materials [39].

To understand this consistency between the numerical result and the experimental data, we consider (1.17) where the Poisson ratio $\nu=0$, which is exactly the case for cork. This special case is less obscure since equations for $u_{1}$ and $u_{3}$ are decoupled now. Furthermore, we assume that the misfit potential consists of two parts which depend merely on $u_{1}$ and $u_{3}$ respectively:

Suppose that $\gamma_{1}$ is a double-well potential (see (1.8)) and $\gamma_{3}$ is the nonlinear potential in the Benjamin-Ono equation [7], i.e.

If $\bm{u}$ is a minimizer of (1.1) with boundary conditions

then separately, $u_{1}$ and $u_{3}$ satisfy

For $i=1$, because $\gamma_{1}$ is a double-well potential, [9] proved that (1.21) admits layer solutions $u_{1}$ (unique up to translations) in the sense of Definition 2. For $i=3$, (1.21) is the traveling wave form of the Benjamin-Ono equation which admits

as a solitary solution [7]. These solutions also satisfy the boundary condition (1.20). These special solutions partially explain the structure of minimizers observed in [40, 39], i.e., it is a layer solution in the shear direction while it is a solitary wave in the transverse direction.

As far as we know, a complete answer on the rigidity of minimizers is still unknown even for the case $\nu=0$. More explicitly, does the De Giorgi conjecture hold in this case? Does (1.17) (or (1.21)) admit any other solution? Existing evidence indicates negative results. If $(-\Delta)^{1/2}$ is replaced by $-\Delta$ in (1.21) and we take $i=3$, the author of [11] proved that there exist solutions being a soliton in the $x$ direction while being periodic (but non-constant) in the $z$ direction. Thus, the one-dimensional symmetry (or the De Giorgi conjecture) fails in this case. This evidence strongly indicates the existence of high-dimensional solutions of $u_{3}$ in (1.21). High dimensionality physically indicates the existence of curved dislocations but the construction of a counterexample for the De Giorgi conjecture in the nonlocal case is still open.

For general cases where the Poisson ratio $\nu$ is non-zero, neither existence nor rigidity result is proved to our best knowledge. In particular, no matter $u_{3}$ has a one-dimensional profile or not, no conclusion can be drawn on the rigidity of $u_{1}$.

Another question regarding the De Giorgi conjecture is also of great interest: for what misfit potential $\gamma$, there exist one-dimensional solutions for (1.17)? For what misfit potential $\gamma$, the De Giorgi conjecture holds, i.e. all solutions of (1.17) are one dimensional? No previous study has ever considered these problems as far as we know.

In summary, the rigidity and existence of the vectorial PN model is an important problem that is central to studies of dislocations, both straight and curved dislocations. Our contribution to this macroscopic framework is that, under the assumption $\gamma=\gamma(u_{1})$, i.e. $\gamma$ only depends on $u_{1}$, a complete answer to existence and rigidity is justified even for high dimensions. See the following sections in the introduction.

We remind our audience here that we will focus on the case where $\gamma$ only depends on $u_{1}$ and $\gamma$ is a double-well type potential (see (1.8)) in the following sections.

We extend the discussion to any dimension $d\geq 1$ and clarify the set up. Denote $\Omega_{d}:=\mathbb{R}\times\mathbb{T}^{d-1}$, consider the high-dimensional reduced scalar equation in $\Omega_{d}$:

The potential function $\gamma\in C^{2}(\mathbb{R})$ is a double-well potential that satisfies (1.8). The linear operator $\mathcal{L}$ is a convolution-type singular integral operator [38] which is defined as

whose convolution kernel $K(\bm{w})$ can be written as

where $\bm{w}=(x,\bm{y}),\ x\in\mathbb{R},\ \bm{y}\in\mathbb{T}^{d-1}$.

We impose several assumptions on the operator $\mathcal{L}$ and its kernel $H$. To clarify these assumptions, we first introduce the Fourier transform on $\Omega_{d}$ and the Sobolev spaces $H^{s}(\Omega_{d})$.

Denote $\Omega_{d}^{\prime}=\mathbb{R}\times(2\pi\mathbb{Z})^{d-1}$. The Fourier transform on $\Omega_{d}$ is understood as a composition in two directions: the Fourier transform on $\mathbb{R}$ in the $x$ direction and the Fourier series expansion on $\mathbb{T}^{d-1}$ in the $\bm{y}$ direction. Denote $\bm{\nu}=(\xi,\bm{k})$ where $\xi\in\mathbb{R}$ and $\bm{k}\in(2\pi\mathbb{Z})^{d-1}$, then the Fourier transform of $u(\bm{w})$, denoted as $\hat{u}(\bm{\nu})$, is defined as

Thus the Fourier transform on $\Omega_{d}$ maps functions defined on $\Omega_{d}$ into functions defined on $\Omega_{d}^{\prime}$. For any $s>0$, we define Sobolev spaces $H^{s}(\Omega_{d})$ in the classical way on the Fourier side:

$H^{s}(\Omega_{d}),\ s>0$ are Hilbert spaces with inner product

Denote the norm induced by this inner product as $\|\cdot\|_{H^{s}(\Omega_{d})}$. We also define the homogeneous norm $\|\cdot\|_{\dot{H}^{s}(\Omega_{d})}$ as

Now we are ready to impose assumptions of $\mathcal{L}$ in (1.24) and introduce several important properties of it. We assume that:

1. (A)

   (symbol of order 1) The Fourier symbol of $\mathcal{L}$ is positive with same order as $|\bm{\nu}|$, i.e. for any $\bm{\nu}\in\Omega_{d}^{\prime}=\mathbb{R}\times(2\pi\mathbb{Z})^{d-1}$, there exist positive constants $c$ and $C$ such that

   (1.29)

   $\displaystyle\widehat{\mathcal{L}u}(\bm{\nu})=\sigma_{\mathcal{L}}(\bm{\nu})\hat{u}(\bm{\nu}),\ c|\bm{\nu}|\leq\sigma_{\mathcal{L}}(\bm{\nu})\leq C|\bm{\nu}|$

2. (B)

   (positivity and continuity) $H(\bm{z}):\mathbb{R}^{d}\to\mathbb{R}$ is positive and continuous on $\mathbb{R}^{d}\setminus\{\bm{0}\}$.

3. (C)

   (homogeneity) For any $\bm{z}\neq\bm{0}$ and $a>0$,

   (1.30)

   $\displaystyle\ H(a\bm{z})$

   $\displaystyle=a^{-d-1}H(\bm{z}).$

4. (D)

   (symmetry) For any $\bm{z}\in\mathbb{R}^{d}$, $H(\bm{z})=H(-\bm{z})$.

The assumptions we impose on $\mathcal{L}$ and its kernel $K$ include two important cases. First, in dimension $d=2$, the non-local operator in equation (1.15) derived from the PN model is included if the Poisson ratio $\nu\in(-1/2,1/3)$. In this case, the operator $\mathcal{L}$ has Fourier symbol $|\bm{k}|^{3}/((1-\nu)k_{1}^{2}+k_{2}^{2})$ (see (1.15)) which is of the same order as $|k|$, so Assumption (A) is satisfied. Moreover, the authors of [14] proved that $K$ satisfies assumption (B), (C) and (D) if and only if $\nu\in(-1/2,1/3)$. So equation (1.15) is included in this context if $\nu\in(-1/2,1/3)$. Second, in arbitrary dimensions, if we take $\nu=0$, then $\mathcal{L}=(-\Delta)^{1/2}$ defined on $\Omega_{d}$ is also included. In this case, the Fourier symbol is exactly $|k|$ and according to [26], there exists a constant $C_{d}>0$ such that

So all assumptions are satisfied.

We remark here that we adopt two different but equivalent definitions for $\mathcal{L}$: one is as a Fourier multiplier and the other is as a singular convolution. The result that these two definitions for fractional Laplacian $(-\Delta)^{\alpha},\alpha\in(0,1)$ are equivalent is thoroughly investigated in [26]. For equation (1.15), the equivalence of these two definitions is also well-studied in [14]. So in later sections, we will switch between these two definitions for the sake of convenience.

Before presenting the main results, we introduce the fractional Allen-Cahn equation [5]:

Here the double-well type potential $\gamma\in C^{2}(\mathbb{R})$ satisfying (1.8) is exactly the misfit potential in the PN model. Taking $\nu=0$ in (1.15), we see that (1.32) is a special case of (1.23) and (1.15). (1.32) has already been thoroughly investigated in the literatures [9, 32, 37, 34, 35, 15]. In particular, the well-posedness result of (1.32) is completely developed. A long standing conjecture named after De Giorgi [12, 21] (which originally discussed the local case, i.e., one replaces $(-\Delta)^{1/2}$ by $-\Delta$ in (1.32), but then generalized to the non-local case (1.32)) is proved for dimension $d\leq 8$. The De Giorgi conjecture claims that any layer solution (see below) to (1.32) is a simple 1D profile for dimensions $d\leq 8$. In the classical Allen-Cahn equation, this conjecture is optimal in the sense that a counterexample in dimension $d=9$ is constructed [13]. The layer solution in the De Giorgi conjecture is defined as:

The layer solution is also of main interest in the PN model since it models a dislocation profile that monotonically converges to two stable states at far field in the shear direction.

Now we are ready to articulate our existence and rigidity result on the PN model and (1.23). Although the PN model and (1.23) share the common double-well nonlinearity and non-localness with the fractional Landau-Ginzburg equation (see (1.32)), the main difference between our setting and previous work on (1.32) is that we work on a partially periodic domain $\Omega_{d}$ and Hilbert spaces $H^{s}(\Omega_{d})$ while previous work focused on the whole domain $\mathbb{R}^{d}$ and Banach spaces $C^{k,\alpha}(\mathbb{R}^{d})$. This discrepancy in the setting urges us to develop more appropriate methods while referring to some valuable techniques introduced in previous work.

For the existence problem, although we know that (1.16) is a solution to (1.15) and (1.14), we are still not aware of whether a minimizer of the total energy (1.1) in function set (1.35) exists. In [9], authors worked on Hölder spaces $C^{k,\alpha}(\mathbb{R}^{d}),\ k=0,1,2,$ and derived some Schauder’s estimates of the weak solution to (1.32). Based on these estimates, they proved the existence of the classical solutions to (1.32) for $d=1$ by considering the harmonic extension of (1.32) on the upper half-plane. In [32], the authors adopted the direct method in calculus of variations minimizing the total energy on a subset of $L^{1}_{\mathrm{loc}}(\mathbb{R})$, i.e.

They proved the existence and uniqueness of the minimizer in one dimension and the existence result is generalized to any dimension $d$.

For the high-dimensional equation (1.23), we will follow the idea of [32] by using the direct method in the calculus of variations to prove that the minimizer of a functional $F(u)$ exists. However, instead of requiring the far field assumption (1.9), we only consider $H^{1/2}$ perturbation of a given 1D profile $\eta$ who satisfies (1.9):

Here $\eta(x,\bm{y})$ is a smooth 1D profile, i.e. $\eta(x,\bm{y})=\eta(x)$ for any $(x,\bm{y})\in\Omega_{d}$, satisfying

We will abuse the notation $\eta$ to represent the profile defined on either $\Omega_{d}$ or $\mathbb{R}$. We remark here that the weak $H^{1/2}$ regularity does not ensure any far field limit behavior in any dimension, even in dimension $d=1$.

The functional $F$ that we aim to minimize is the perturbed version of the total energy (1.1):

Here $\gamma\in C^{2}(\mathbb{R})$ is the double-well type potential considered in (1.23) that satisfies (1.8).

Starting from functions only with $H^{1/2}$ regularity and even not necessarily satisfying the far field limit condition, we construct a minimizer with $H^{2}$ regularity (in fact smooth) that also satisfies the desired rigidity result that we aim to prove: it is a layer solution with 1D symmetry.

The critical technique in the proof is the energy decreasing rearrangement method in [32]. This method relies on the rearrangement inequality (see Lemma 3.1) whose proof is quite elementary. However, driven by this basic inequality, the energy decreasing method is powerful in proving monotonicity and 1D symmetry. We will introduce this method in Section 3.1.

For the rigidity problem, it worth mentioning the De Giorgi conjecture on the fractional Ginzburg-Landau equation (1.32). It claims that at least for $d\leq 8$, layer solutions to (1.32) are in fact just 1D profiles. Here a layer solution is defined in Definition 3. This conjecture was proved for dimension $d=2$ in [9] and finally completely proved by Savin in his series of work [34, 35, 32]. We also mention the asymptotic analysis for the sharp interface limit of the fractional diffusion-reaction equation with isotropic/anisotropic nonlocal kernel of order $\frac{1}{r^{d+2s}},s\in(0,1)$ in [4, 20, 23, 30] .

The common approach to prove the De Giorgi conjecture is to develop a Liouville-type theorem and then apply the theorem on ratios of partial derivatives in different directions $u_{x_{i}}/u_{x_{1}},i=1,2,...,d-1$. Then one can conclude that there exist constants $c_{i},i=1,2,...,d-1$ such that

So $u$ in fact only depends on $x_{1}$ variable and hence is a 1D profile. For (1.23), we can prove the following theorem which is true for any dimension $d\geq 1$, not only for dimension $d\leq 8$:

We emphasize here that the main reason of Theorem 2 being true in any dimension $d\geq 1$ instead of only dimensions less than eight is that we fully employed the compactness of the torus $\mathbb{T}^{d-1}$. The compactness ensures convergence of a sequence which is a key step in our proof (see the proof of Theorem 2). As we explained in Section 1.2, without periodicity, the De Giorgi conjecture may fail in the classical case [11]. Therefore, domain $\Omega^{d}=\mathbb{R}\times\mathbb{T}^{d-1}$ is critical to our result which is also physically meaningful since it incorporates the periodicity in materials.

Instead of using Liouville type theorems, we prove Theorem 2 by analyzing the spectrum of a linear operator. This method sufficiently respects the maximal property of operator $\mathcal{L}$ (see Lemma 2.1) which is realized by the positivity assumption (Assumption (B)). Utilized in our previous work [18], this spectral analysis method is straightforward and appropriate in our setting since the working spaces are selected as Hilbert spaces $H^{s}(\Omega_{d})$ instead of Banach spaces $C^{k,\alpha}(\mathbb{R}^{d})$ in [9]. Under this setting, we can use the perturbation theory of self-adjoint operators on Hilbert spaces [25].

Specifically speaking, suppose that $u$ is a given solution to (1.23) that satisfies conditions in Theorem 2. Differentiating on both sides of equation (1.23), we see that $u_{x}$ and $u_{y_{i}},i=1,2,...,d-1$ are solutions to the following non-local linear elliptic equation of $\phi$ on $\Omega_{d}$ which is given by:

Equivalently, $u_{x}$ and $u_{y_{i}},i=1,2,...,d-1$ are eigenfunctions of eigenvalue 0 for the linear operator linearized along profile $u$:

Therefore, as long as we can prove that 0 is a simple eigenvalue of $L$, i.e. the eigenspace of 0 is only 1 dimension, then we prove that $u_{x}$ and $u_{y_{i}},i=1,2,...,d-1$ are linearly dependent, which indicates 1D symmetry. This is the main idea and approach we will utilize to prove Theorem 2.

As a direct corollary of Theorem 2, we can prove that both layer solutions and minimizers of $F$ on $\mathcal{A}$ are unique up to translations. Define $\mathcal{A}_{\ell}$ and $\mathcal{A}_{m}$ as

i.e. $\mathcal{A}_{\ell}$ is the set of layer solutions to (1.23) with $H^{1}$ regularity and $\mathcal{A}_{m}$ is the set of minimizers of $F$ on set $\mathcal{A}$. Then we can prove the following theorem:

Theorem 3 provides a compatible physical interpretation of the PN model in three dimensions (with periodicity in the transverse direction): if we assume exclusive dependence of the misfit potential on the shear displacement, then the equilibrium dislocation on the slip plane only admits shear displacements. Furthermore, this uniquely (up to translations) determined shear displacement is a strictly monotonic 1D profile connecting two stable states. This reduces the vectorial PN model to the two-dimensional PN model which was investigated in our previous work [19]. In summary, we have the following theorem:

These four theorems (Theorem 1, 2, 3 and 4) are the main results for this work which completely close the problem of existence and rigidity in a general setting including the original PN model with Poisson ratio $\nu\in(-1/2,1/3)$. Following this logic, we will first conduct a preliminary analysis in Section 2 to assist readers to bridge some gaps in understanding the derivation of (1.23) and be aware of some important properties of the linear operator $\mathcal{L}$. Then we prove Theorem 1 in Section 3 and Theorem 2, 3 and 4 in Section 4. Finally, the spectral analysis of operator $L$ is established in Section 5 which proves that 0 is simple and the principle eigenvalue of $L$. For facts in functional analysis and details in the spectral analysis, readers may refer to Appendix A; for proofs of some lemmas in the proof of the theorems, readers may refer to Appendix B.

Denote the Fourier transform of $u_{i}^{+}(x,z),i=1,2,3$ as $\hat{u}_{i}^{+}(\bm{k}),i=1,2,3$ where $\bm{k}=(k_{1},k_{2}),k_{1}\in\mathbb{R},k_{2}\in 2\pi\mathbb{Z}$ is the frequency vector. Given $\bm{u}$ that satisfies equation (1.13), one can rewrite $(\sigma_{12}^{+},\sigma_{32}^{+})$ on $\Gamma^{\prime}$ as a linear transform of $(u_{1}^{+}(\bm{k}),u_{3}^{+}(\bm{k}))$ on the Fourier side: