# On the sum of two powered numbers

J. Brüdern and O. Robert

00footnotetext: Keywords : Powered numbers, sumsets00footnotetext:
MSC(2020):11B13

Many of the historic diophantine problems concern additive properties of
powers of natural numbers. Fermat’s Last Theorem, the Catalan conjecture and
problems of Waring’s type are among the most familiar representatives in this
class of questions. In his beautiful survey article Mazur [2] suggests to
consider rounded versions of these problems. Rather than insisting to work
with perfect powers, one is then led to analyse the additive properties of
natural numbers $m$ whose kernel $k(m)$, defined as the largest squarefree
divisor of $m$, is small in terms of $m$. To make this precise, fix a real
number $\theta\in(0,1]$ and define the set

$\mathscr{A}(\theta)=\\{m\in{\mathbb{N}}\colon k(m)\leq m^{\theta}\\}.$

For $\theta=1/l$ with $l\in\mathbb{N}$ this set constitutes the powered
numbers, in Mazur’s terminology. The motivation is that $i(m)=\log m/\log
k(m)$ measures multiplicities in the prime factorisation of $m$, and for all
$l\in\mathbb{N}$ one has $i(m^{l})\geq l$ so that the $l$-th powers form a
subset of $\mathscr{A}(1/l)$, and elements of $\mathscr{A}(1/l)$ should then
substitute for $l$-th powers in Mazur’s rounded diophantine problems. As
demonstrated in [1], the number $S_{\theta}(x)$ of elements in
$\mathscr{A}(\theta)$ that do not exceed $x$ satisfies the inequalities

(1) $x^{\theta}\ll S_{\theta}(x)\ll x^{\theta+\varepsilon}$

whenever $\varepsilon$ is a given positive real number and $x$ is large in
terms of $\varepsilon$. Hence, if one replaces $l$-th powers by members of
$\mathscr{A}(1/l)$, then the new set is roughly of the same density.

In this short communication, we discuss the binary linear equation $u+v=n$
with the variables $u,v$ restricted to $\mathscr{A}(\theta)$. Thus, we are
interested in the sumset

$\mathscr{A}(\theta)+\mathscr{A}(\theta):=\\{u+v\colon
u,v\in\mathscr{A}(\theta)\\},$

and in particular, we wish to determine the values of $\theta$ where this
sumset contains all but finitely many natural numbers. Our main result comes
close to the rounded version of representations by sums of two squares.

######  Theorem.

Let $C=2\sqrt[4]{27}$. Then, for all natural numbers $n$ with $n\geq 4$ there
are natural numbers $m_{j}$ with $m_{j}\geq 2$ $(j=1,2)$ and

(2) $n=m_{1}+m_{2},\quad k(m_{j})\leq C\sqrt{m_{j}}\quad(j=1,2).$

This conclusion has an immediate consequence for our initial question.

######  Corollary.

Let $\theta>\frac{1}{2}$. Then all large natural numbers are the sum of two
numbers in $\mathscr{A}(\theta)$.

Since there are certainly no more than $S_{\theta}(x)^{2}$ natural numbers
below $x$ in the sumset $\mathscr{A}(\theta)+\mathscr{A}(\theta)$, it follows
from (1) that for all $\theta<\frac{1}{2}$, the sumset
$\mathscr{A}(\theta)+\mathscr{A}(\theta)$ has density $0$ in the natural
numbers. Therefore, the conclusion in the corollary is certainly false for
$\theta<\frac{1}{2}$.

Our theorem fails to settle the case $\theta=\frac{1}{2}$. However, the set
$\mathscr{A}(\frac{1}{2})$ is somewhat denser than the set of squares, and it
is very likely that the condition on $\theta$ in the corollary can be relaxed
to $\theta\geq\frac{1}{2}$. It is worthwhile to consider the vincinity of
$\frac{1}{2}$ on a finer scale. In our companion paper [1] we established
asymptotic formulae for $S_{\theta}(x)$ and related counting functions. As a
consequence of [1, (1.15)], whenever $\varepsilon>0$ and $x$ is large in terms
of $\varepsilon$, one has

$S_{\frac{1}{2}}(x)\gg\sqrt{x}\exp\big{(}(2-\varepsilon)\sqrt{\log x/\log\log
x}\big{)}.$

Further, as a special case of [1, Theorem 3], for any real number $\gamma$,
one finds

(3) $\\#\\{m\leq x\colon k(m)\leq\sqrt{m}(\log m)^{\gamma}\\}=(\log
x)^{\gamma}S_{\frac{1}{2}}(x)\Big{(}1+O\Big{(}\sqrt{\frac{\log\log x}{\log
x}}\Big{)}\Big{)}.$

In light of these estimates, it seems safe to conjecture that for any given
real number $\gamma$, a sufficiently large natural number $n$ has a
representation $n=m_{1}+m_{2}$ with

(4) $k(m_{j})\leq\sqrt{m_{j}}(\log m_{j})^{\gamma}\quad(j=1,2)$

We expect the expansion (3) to persist if the role of the logarithm in the
factors $\log m$ and $\log x$ is played by a function that grows somewhat
faster, and one may then hope to be able to tighten the condition (4) to

$k(m_{j})\leq\sqrt{m_{j}}\exp\big{(}-\sqrt{\log m_{j}/\log\log m_{j}})$

or thereabout, and still conclude that $n=m_{1}+m_{2}$ has solutions with this
condition in place, at least when $n$ is large.

The proof of the theorem is entirely elementary. We write
$A=\frac{1}{2}\sqrt[4]{27}$. Let $a$ and $b$ be the integers defined by

(5) $2^{a}<\sqrt{n}/A\leq 2^{a+1},\quad 3^{b}<A\sqrt{n}\leq 3^{b+1}.$

Then $a\geq 1$ and $b\geq 1$ for $n\geq 7$. Next, define $U$ and $V$ by

(6) $U=[n/2^{a}]-1,\quad n=2^{a}U+V.$

Then $2^{a}U=n-\xi 2^{a}$ for some real number $\xi\in[1,2)$, and so
$2^{a}\leq V<2^{a+1}.$ Since $2$ and $3$ are coprime, there are integers $W$,
$w$ with

(7) $V=-2^{a}W+3^{b}w.$

The theory of linear diophantine equations shows that in this last equation it
can be arranged that $1\leq w\leq 2^{a}$. With this choice for $w$ we put

(8) $u=2^{a}(U-W),\quad v=3^{b}w.$

By (6) and (7), we have $n=u+v$. Further, the constraints on $w$ and the
inequalities (5) ensure that

(9) $3^{b}\leq v\leq 2^{a}3^{b}<n.$

By (9), we first see that $u\geq 1$, and then by (8) that $u\geq 2^{a}$. It
only remains to estimate the kernels of $u$ and $v$. By construction, we have

$k(v)\leq 3k(w)\leq 3w=\sqrt{v}\,\frac{\sqrt{w}}{3^{b/2-1}}.$

By (5) and the range for $w$, we see

$3^{2-b}{w}<\frac{2^{a}}{3^{b-2}}<\frac{\sqrt{n}/A}{A\sqrt{n}/27}\leq\frac{27}{A^{2}}.$

This yields

$k(v)<\frac{\sqrt{27}}{A}\sqrt{v}.$

The treatment of $u$ is similar. We use $u=2^{a}(U-W)=n-v$. Then

$k(u)\leq 2k(U-W)\leq 2(U-W),$

and hence

$k(u)\leq 2^{1-a}u=\sqrt{u}\,\sqrt{n-v}\,2^{1-a}.$

But $2^{1-a}\leq 4A/\sqrt{n}$, and then

$k(u)\leq 4A\sqrt{u}.$

Now choose $m_{1}=u$ and $m_{2}=v$ to confirm (2) whenever $n\geq 7$. For
$4\leq n\leq 6$ one may use $m_{1}=2$ and $m_{2}=n-2$.

## References

  * [1] J. Brüdern and O. Robert. On the distribution of powered numbers. Forthcoming.
  * [2] B. Mazur. Questions about powers of numbers. Notices Amer. Math. Soc. 47 (2000), no. 2, 195–202.

Jörg Brüdern
Universität Göttingen
Mathematisches Institut
Bunsenstrasse 3–5
D 37073 Göttingen
Germany
<EMAIL_ADDRESS>
Olivier Robert
Université de Lyon, Université de Saint-Étienne
Institut Camille Jordan CNRS UMR 5208
25, rue Dr Rémy Annino
F-42000, Saint-Étienne
France
<EMAIL_ADDRESS>


# A dual variational principle for nonlinear dislocation dynamics

Amit Acharya Department of Civil & Environmental Engineering, and Center for
Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA 15213, email:
<EMAIL_ADDRESS>
###### Abstract

A dual variational principle is defined for the nonlinear system of PDE
describing the dynamics of dislocations in elastic solids. The dual
variational principle accounting for a specified set of initial and boundary
conditions for a general class of PDE is also developed.

This paper is dedicated to Roger Fosdick on the occasion of his 85th birthday

##  1 Introduction

Unlike the physics of the microscopic structure of sub-atomic particles (e.g.
‘core’ of an electron), much more is physically known about the microscopic
structure of dislocations and their mutual interactions, as well as their
interactions with applied loads, within a (nonlinear) elastic crystal, both
through direct experimental observation and through lattice statics/molecular
dynamics/density functional theory calculations. Due to this knowledge,
physically well-justified and transparent mathematical models can be posited
for the phenomena, with the possibility of systematic refinement to include
more detail when deemed necessary after mathematical study and comparison with
experiment. There is a long and distinguished history of the study of
dislocations in elasticity in the classical setting, see, e.g., [1, 2, 3, 4,
5], the continuously distributed setting, e.g., [6, 7, 8], [9, including
second-order effects] and [10], and the connections of some of the kinematic
aspects of dislocations to non-Riemannian Geometry [11, 7, 12]. As well,
techniques for developing well-set, classical thermomechanical theories of the
mechanics of continuous media comprising different types of materials
exhibiting strongly nonlinear behavior and satisfying the relevant invariances
and material symmetries are available [13, 14, 15, 16] and [10]. These ideas
and techniques have been synthesized and extended to produce the theory/model
of dislocation mechanics stated in [17], as reviewed in [18]. The theory
admits the minimal specification of an energy density function $\psi(W)$,
where $W$ is the inverse elastic distortion field (not necessarily a
gradient), and that of a dislocation velocity field, the function
$V_{s}(\alpha,W,\rho)$ in (3), which, when guided by the requirements of being
proportional to its derived thermodynamic driving force, is a specified
function of the thermodynamically derived Cauchy stress tensor $T_{ij}=-\rho
W_{ki}\psi^{\prime}_{kj}$ and the dislocation density tensor $\alpha_{ij}$,
admitting a scalar or matrix of material constants representing dislocation
mobility. Here, $\psi^{\prime}_{ij}=\partial_{W_{ij}}\psi$, and it suffices to
use a rectangular Cartesian coordinate system and tensor components w.r.t its
basis in this Section. The time variable is represented by the symbol $t$ and
not used as an index.

For prescribed static dislocation fields the framework is shown to be able to
compute the stress and energy fields of such distributions in bodies of
arbitrary geometry and general elastic symmetries [19, 20]. Similarly for
prescribed dislocation velocity field, the setup is shown to be able to
compute the evolution of the dislocation field [20]. And the evolution in the
fully coupled case also has been shown to work well to predict nonsingular
dislocation cores, dislocation annihilation, dissociation and stress-mediated
interaction when restricted to dislocation motion within a planar layer in a
3–d body [21] within a ‘small deformation’ ansatz.

The phenomenon of macroscopic plasticity of crystalline materials corresponds
to the collective dynamical behavior of a very large number of dislocation
curves in an elastic body under generally time-dependent loads. While
experimental observations and real practical applications of plasticity
abound, it is fair to say that there does not exist a fundamental theory that
arises as a coarse-graining of nonlinear dislocation dynamics as described
above (or by any other model). The phenomenon of plasticity shows fascinating
dynamical changes as a function of initial conditions and tamely evolving
driving loads - e.g., yielding, Stage I, II, III, IV behaviors as a function
of applied load temperature and initial crystal orientation, intricate
patterned dislocation microstructure formation such as cells and sub-grain
boundaries to name only a few - with no established fundamental theory for
understanding them (the phenomenon is even richer, with rapidly driven
situations also being of theoretical and practical interest). It is in this
context that we would like to use a path integral implementation of the
dynamics represented by (3) to evaluate how much of the reality of macroscopic
plasticity can be understood by the combination of the model and the
technique. The rough expectation is to be able to interpret drastic changes of
overall behavior observed in reality as statistical phase transitions as
understood in Effective/Statistical Field Theory.

A first step in this program is to define an action functional for the system
(3) which, in the first instance, does not emanate from a variational
principle; it is this objective that is tackled in Secs. 2, 3, and 4, refining
the work in [18] following the ideas in [22]. Section 5 develops the
variational principle accounting for a specified set of initial and boundary
conditions for a nonlinear system of second-order PDE expressed as a first-
order system.

Variational principles for ‘small deformation,’ static dislocation mechanics
and internal stress problems by the method of ‘eigenstrains’ is presented in
[23, 24], and there is a modern literature involving rigorous analysis
reviewed, in detail in [19]. Our work involves finite-deformation, nonlinear
dislocation dynamics including inertia.

##  2 The essential idea: An optimization problem for an algebraic system of
equations111I thank Vladimir Sverak for insisting on the ‘simplest,’
transparent explanation of the ideas in [18, 22]. This brief Section is a
result of that effort.

Consider a generally nonlinear system of algebraic equations in the variables
$x\in\mathbb{R}^{n}$ given by

$A_{\alpha}(x)=0,$ (1)

where $A:\mathbb{R}^{n}\to\mathbb{R}^{N}$ is a given function (a simple
example would be $A_{\alpha}(x)=\bar{A}_{\alpha i}\,x^{i}-b_{\alpha}$,
$\alpha=1\ldots N,i=1\ldots n$, where $\bar{A}$ is a constant matrix, not
necessarily symmetric (when $n=N$), and $b$ is a constant vector). We allow
for all possibilities $0<n\lesseqqgtr N>0$.

The goal is to construct an objective function whose critical points solve the
system (1) (when a solution exists) by defining an appropriate
$x^{*}\in\mathbb{R}^{n}$ satisfying $A_{\alpha}(x^{*})=0$.

For this, consider first the auxiliary function

$\widehat{S}_{H}(x,z)=z^{\alpha}A_{\alpha}(x)+H(x)$

(where $H$ belongs to a class of scalar-valued function to be defined shortly)
and define

$S_{H}(z)=z^{\alpha}A_{\alpha}(x_{H}(z))+H(x_{H}(z))$

with the requirement that the system of equations

$z^{\alpha}\frac{\partial A_{\alpha}}{\partial x^{i}}(x)+\frac{\partial
H}{\partial x^{i}}(x)=0$ (2)

be solvable for the function $x=x_{H}(z)$ through the choice of $H$, and any
function $H$ that facilitates such a solution qualifies for the proposed
scheme.

In other words, given a specific $H$, it should be possible to define a
function $x_{H}(z)$ that satisfies

$z^{\alpha}\partial_{x^{i}}A_{\alpha}(x_{H}(z))+\partial_{x^{i}}H(x_{H}(z))=0\quad\forall
z\in\mathbb{R}^{N}$

(the domain of the function $x_{H}$ may accommodate more intricacies, but for
now we stick to the simplest possibility). Note that (2) is a set of $n$
equations in $n$ unknowns regardless of $N$ ($z$ for this argument is a
parameter).

Assuming this is possible, we have

$\frac{\partial S_{H}}{\partial
z^{\beta}}(z)=A_{\beta}(x_{H}(z))+\left(z^{\alpha}\frac{\partial
A_{\alpha}}{\partial x^{i}}(x_{H}(z))+\frac{\partial H}{\partial
x^{i}}(x_{H}(z))\right)\frac{\partial x^{i}_{H}}{\partial
z^{\beta}}(z)=A_{\beta}(x_{H}(z)),$

using (2). Thus,

  * •

if $z_{0}$ is a critical point of the objective function $S_{H}$ satisfying
$\partial_{z^{\beta}}S_{H}(z_{0})=0$, then the system $A_{\alpha}(x)=0$ has a
solution defined by $x_{H}(z_{0})$;

  * •

if the system $A_{\alpha}(x)=0$ has a unique solution, say $y$, and if
$z^{H}_{0}$ is any critical point of $S_{H}$, then
$x_{H}\left(z^{H}_{0}\right)=y$, for all admissible $H$.

  * •

If $A_{\alpha}(x)=0$ has non-unique solutions, but
$\partial_{z^{\beta}}S(z)=0$ ($N$ equations in $N$ unknowns) has a unique
solution for a specific choice of the function $z\mapsto x_{H}(z)$ related to
a choice of $H$, then such a choice of $H$ may be considered a selection
criterion for imparting uniqueness to the problem $A_{\alpha}(x)=0$.

  * •

Finally, to see the difference of this approach with the Least-Squares (LS)
Method, for a linear system $\bar{A}x=b$, the LS governing equations are given
by

$\bar{A}^{T}\bar{A}z=\bar{A}^{T}b,$

with LS solution defined as $z$ even when the original problem $\bar{A}x=b$
does not have a solution (i.e., when $b$ is not in the column space of
$\bar{A}$). The LS problem always has a solution, of course. In contrast, in
the present duality-based approach with quadratic $H(x)=\frac{1}{2}x^{T}x$ the
governing equation is

$\bar{A}\bar{A}^{T}z=b$

with solution to $\bar{A}x=b$ given by $x=\bar{A}^{T}z$, and the problem has a
solution only when $\bar{A}x=b$ has a solution, since the column spaces of the
matrices $\bar{A}$ and $\bar{A}\bar{A}^{T}$ are identical.

As a practical matter, the latter approach appears to have, at least in
principle, advantages for solving large, consistent, underdetermined systems
as the size of the matrix $\bar{A}\bar{A}^{T}$ is much smaller than that of
$\bar{A}^{T}\bar{A}$ in this situation, with due consideration paid to
conditioning-related robustness issues (cf. [25, 26], [27, pp. 299-300]).

##  3 A class of variational principles for nonlinear dislocation mechanics

We implement the idea of Sec. 2 to define an action(s) for the nonlinear
partial differential equations of dislocation mechanics given by

$\displaystyle 0$ $\displaystyle=e_{jrs}\partial_{r}W_{is}+\alpha_{ij}$ (3)
$\displaystyle 0$
$\displaystyle=\partial_{t}W_{ij}+\partial_{j}(W_{ik}v_{k})-v_{k}e_{rkj}\alpha_{ir}-e_{jrs}\alpha_{ir}V_{s}(\alpha,W,\rho)=\partial_{t}W_{ij}+v_{k}\partial_{k}W_{ij}+W_{ik}\partial_{j}v_{k}-e_{jrs}\alpha_{ir}V_{s}(\alpha,W,\rho)$
$\displaystyle 0$ $\displaystyle=\partial_{t}\rho+\partial_{k}(\rho v_{k})$
$\displaystyle 0$ $\displaystyle=\partial_{t}(\rho v_{i})+\partial_{j}(\rho
v_{i}v_{j})+\partial_{j}(\rho W_{ki}\psi^{\prime}_{kj}).$

The physical basis of this system of equations is explained in [18]. Briefly,
the first equation is the equation of elastic incompatibility. The second
reflects the compatibility between the velocity gradient, the rate of the
change of the elastic distortion and the rate of permanent deformation
produced by the motion of dislocations. The third equation is balance of mass,
and the fourth, the balance of linear momentum. Setting $\alpha=0$ in the
system above gives the equations of nonlinear elasticity written in an
Eulerian setting.

First define the functional

$\displaystyle\widehat{S}_{H}[A,W,\theta,\rho,\lambda,v,B,\alpha]=\int_{[0,T]\times\Omega}\,dtd^{3}x$
$\displaystyle\,-W_{ij}\partial_{t}A_{ij}-W_{ik}v_{k}\partial_{j}A_{ij}-A_{ij}v_{k}e_{rkj}\alpha_{ir}-A_{ij}e_{jrs}\alpha_{ir}V_{s}(\alpha,W,\rho)$
$\displaystyle-\rho\partial_{t}\theta-\rho v_{k}\partial_{k}\theta$
$\displaystyle-\rho v_{i}\partial_{t}\lambda_{i}-\rho
v_{i}v_{j}\partial_{j}\lambda_{i}-\rho
W_{ki}\psi^{\prime}_{kj}\partial_{j}\lambda_{i}$ $\displaystyle-
e_{jrs}W_{is}\partial_{r}B_{ij}+B_{ij}\alpha_{ij}$
$\displaystyle+H(W,\rho,v,\alpha),$

which is obtained by converting (3) to scalar form by taking inner products
with the ‘dual’ fields

$D=(A,\theta,\lambda,B),$

integrating by parts on the space-time domain assuming the dual fields vanish
on the boundary of the domain, and adding the potential $H$. Now define

$U:=(W,\rho,v,\alpha)\qquad\mbox{and}\qquad\mathcal{D}:=(\partial_{t}A,\nabla
A,A,\partial_{t}\theta,\nabla\theta,\partial_{t}\lambda,\nabla\lambda,\nabla
B,B)$

(note ‘$\mathcal{D}\neq D$’) and require that there exists a function

$U_{H}(\mathcal{D})=(W_{H}(\mathcal{D}),\rho_{H}(\mathcal{D}),v_{H}(\mathcal{D}),\alpha_{H}(\mathcal{D}))$
(4)

such that for the functional $S_{H}[A,\theta,\lambda,B]$ of the dual fields
defined as

$\int_{[0,T]\times\Omega}\,dtd^{3}x\
\mathcal{L}_{H}(\mathcal{D},U_{H}(\mathcal{D}))=S_{H}[A,\theta,\lambda,B]:=\widehat{S}_{H}[A,W_{H}(\mathcal{D}),\theta,\rho_{H}(\mathcal{D}),\lambda,v_{H}(\mathcal{D}),B,\alpha_{H}(\mathcal{D})],$
(5)

the first variation is given by (we suppress the subscript H on the elements
of $U_{H}$ for notational simplicity)

$\displaystyle\delta S_{H}=\int_{[0,T]\times\Omega}\,dtd^{3}x$
$\displaystyle\,-W_{ij}(\mathcal{D})\partial_{t}\delta
A_{ij}-W_{ik}(\mathcal{D})v_{k}(\mathcal{D})\partial_{j}\delta A_{ij}-\delta
A_{ij}v_{k}(\mathcal{D})r_{rkj}\alpha_{ir}(\mathcal{D})$ (6)
$\displaystyle\qquad\qquad\qquad\qquad-\delta
A_{ij}e_{jrs}\alpha_{ir}(\mathcal{D})V_{s}(\alpha(\mathcal{D}),W(\mathcal{D}),\rho(\mathcal{D}))$
$\displaystyle-\rho(\mathcal{D})\partial_{t}\delta\theta-\rho(\mathcal{D})v_{k}(\mathcal{D})\partial_{k}\delta\theta$
$\displaystyle-\rho(\mathcal{D})v_{i}(\mathcal{D})\partial_{t}\delta\lambda_{i}-\rho(\mathcal{D})v_{i}(\mathcal{D})v_{j}(\mathcal{D})\partial_{j}\delta\lambda_{i}-\rho(\mathcal{D})W_{ki}(\mathcal{D})\psi^{\prime}_{kj}(W(\mathcal{D}))\partial_{j}\delta\lambda_{i}$
$\displaystyle-e_{jrs}W_{is}(\mathcal{D})\partial_{r}\delta B_{ij}+\delta
B_{ij}\alpha_{ij}(\mathcal{D}),$

a condition that is satisfied if the system

$\displaystyle\frac{\partial\mathcal{L}_{H}}{\partial
W_{lp}}=-\partial_{t}A_{lp}-v_{p}\partial_{j}A_{lj}-A_{ij}e_{jrs}\alpha_{ir}\frac{\partial
V_{s}}{\partial W_{lp}}(\alpha,W,\rho)-e_{jrp}\partial_{r}B_{lj}$ (7)
$\displaystyle-\rho\left(\psi^{\prime}_{lj}(W)\partial_{j}\lambda_{p}\right.\left.+\
W_{ki}\psi^{\prime\prime}_{kjlp}\partial_{j}\lambda_{i}\right)$
$\displaystyle+\frac{\partial H}{\partial W_{lp}}(W,\rho,v,\alpha)=0$
$\displaystyle\frac{\partial\mathcal{L}_{H}}{\partial\rho}=-A_{ij}e_{jrs}\alpha_{ir}\frac{\partial
V_{s}}{\partial\rho}(\alpha,W,\rho)-\partial_{t}\theta-
v_{k}\partial_{k}\theta-
v_{i}\partial_{t}\lambda_{i}-v_{i}v_{j}\partial_{j}\lambda_{i}$
$\displaystyle-W_{ki}\psi^{\prime}_{kj}\partial_{j}\lambda_{i}$
$\displaystyle+\frac{\partial H}{\partial\rho}(W,\rho,v,\alpha)=0$
$\displaystyle\frac{\partial\mathcal{L}_{H}}{\partial
v_{p}}=-W_{ip}\partial_{j}A_{ij}-A_{ij}e_{rpj}\alpha_{ir}-\rho\partial_{p}\theta-\rho\partial_{t}\lambda_{p}-\rho
v_{j}\partial_{j}\lambda_{p}$ $\displaystyle-\rho
v_{i}\partial_{p}\lambda_{i}$ $\displaystyle+\frac{\partial H}{\partial
v_{p}}(W,\rho,v,\alpha)=0$
$\displaystyle\frac{\partial\mathcal{L}_{H}}{\partial\alpha_{lp}}=-A_{lj}v_{k}e_{pkj}-A_{lj}e_{jps}V_{s}(\alpha,W,\rho)-A_{ij}e_{jrs}\alpha_{ir}\frac{\partial
V_{s}}{\partial\alpha_{lp}}(\alpha,W,\rho)$ $\displaystyle+B_{lp}$
$\displaystyle+\frac{\partial H}{\partial\alpha_{lp}}(W,\rho,v,\alpha)=0,$

can be solved in the form of

$(W,\rho,v,\alpha)=U_{H}(\mathcal{D}).$

This is so, since solving (7) defines $U_{H}(\mathcal{D})$ that ensures
$\frac{\partial\mathcal{L}_{H}}{\partial U}(\mathcal{D},U_{H}(\mathcal{D}))=0$
which then implies

$\frac{\partial\mathcal{L}_{H}}{\partial
U}(\mathcal{D},U_{H}(\mathcal{D}))\cdot\frac{\partial
U_{H}}{\partial\mathcal{D}}(\mathcal{D})\cdot\delta\mathcal{D}=0\qquad\mbox{for
all}\ \mathcal{D}.$

Note that (6) then is simply

$\delta S_{H}=\int_{[0,T]\times\Omega}\,dtd^{3}x\
\dfrac{\partial\mathcal{L}_{H}}{\partial\mathcal{D}}\cdot\delta\mathcal{D},$

and requiring

$\delta S_{H}=0\ \mbox{for all variations}\ \delta D\ \mbox{that vanish on the
boundary of}\ \Omega\times[0,T]$

shows that the Euler-Lagrange equations of the functional $S_{H}$ defined in
(5) are the equations of (3) with the substitution

$(W,\rho,v,\alpha)=U_{H}(\mathcal{D}).$

This is so because $\mathcal{L}_{H}$ is necessarily linear in its first
argument, see (5).

To summarize, the primal equations (3) of dislocation mechanics are the Euler-
Lagrange equations of any of the dual functionals, written in terms of
particular specific combinations (mappings) of the dual fields for each choice
of the function $H$, each specific mapping defining the primal fields. Thus
one may think of the primal fields as “gauge invariant” observable
combinations of the dual fields (“gauge fields”) satisfying one specific set
of equations (the primal system). While this is not how gauge fields appear in
traditional gauge theories of physics, it is interesting that a completely
different starting point and approach raise somewhat similar invariance
structures that may be interpreted as symmetries.

As for the plausibility of being able to solve the algebraic system (7) given
a specific $\mathcal{D}$, consider $H$ to be separately quadratic in each of
its arguments, say $U_{A}$, with large in magnitude coefficient, so, e.g.,
$H=\frac{1}{2}\alpha_{W}W_{ij}W_{ij}+\cdots$, with $1\ll|\alpha_{W}|$. Then,
assuming the solution of the Euler-Lagrange equations are bounded in some
appropriate sense, (7) can indeed be solved to define $U_{H}(\mathcal{D})$,
and it has to be made sure that the solutions of the Euler-Lagrange equations
(using this function) indeed satisfy the assumed bounds. To ensure that this
latter condition is satisfied one has a large class of $H$ functions to
operate with but, at any rate, this is a delicate question of analysis,
including how the requirement can be relaxed if required, in the context of
solving the dual variational problem (and not necessarily its Euler-Lagrange
equations).

We end this section with the following remarks:

  * •

Our system (3) does not involve multi-valued fields or non-simply connected
domains for defining dislocation dynamics, but is fully capable of
representing the topological charge of dislocation lines with its ingredients.

  * •

Based on the explorations of stress-coupled dislocation motion presented in
[28, 21], the ‘primal’ system requires a ‘core-energy’ in the form of the
dependence of the energy function $\psi$ on the dislocation density $\alpha$
as well. This results in the dislocation velocity depending on the
$\mbox{curl}\,\alpha$. Such a dependence is accommodated within our ‘action-
generating’ scheme by adding an extra variable and equation to the system (3)
of the form $e_{jrs}\partial_{r}\alpha_{is}=\beta_{ij}$ and writing the
dislocation velocity as $V_{s}=V_{s}(\alpha,W,\rho,\beta)$. This would have
the effect of increasing the number of fields in the dual problem as well.

It is an interesting question whether the precise definition of a formally
‘small’ core energy contribution with a small parameter representing
microscopic physics can make a difference in the development of an accurate
model for the prediction of macroscopic behavior, and whether such a device
should be allowed in the class of models admitted. Physically, in the context
of the physics of dislocation dynamics, there appears to be no reason to
exclude the possibility of the importance of such effects and, in fact, allows
more precise physics to be incorporated in the description of gross
macroscopic behavior (which is, admittedly, a double-edged sword in the
context of coarse-graining). Some evidence to support such an expectation is
also provided by the mathematically rigorous study of the inviscid Burgers
equation, ‘regularized’ by a small viscous effect in one case and by
dispersion in another [29, 30, 31].

Based on the above observation, one advantage of the ‘dual’ formalism proposed
herein may be that when the microscopic physics to be added is not even
qualitatively understood with certainty, working with a regularization on the
dual side, may be guided solely by the aim of producing a ‘good’ dual
extremal, i.e., with guaranteed existence in an appropriate function space.
Doing so appears to require no modification to the physics of the primal
problem, and then the limit of dual solutions, as the regularization parameter
vanishes, may be studied.

  * •

In the context of an action functional that simply has as its Euler-Lagrange
equation the given system of PDEs, the proposed scheme delivers, at least
formally and under the stated requirements, what is needed. However, if the
action functional is to be used in a path integral, dual fields $D$ other than
extremals matter as well. In this sense it is reasonable to demand that the
added potential $H$ in $\mathcal{L}_{H}$ be subject to further requirements of
invariance that may obstruct the inversion process required to define the
function $U_{H}(\mathcal{D})$. In case such a restriction is so severe as not
to allow the definition of even a single ‘change of variables’
($U_{H}(\mathcal{D})$), through the choice of some $H$), one can retain both
the fields $W,A$ and still obtain a relevant action functional, as shown in
[18].

##  4 Linear dislocation mechanics

We illustrate the proposed technique with a very closely related one (using a
Legendre transform, cf. [32, 22]) in the simplified setting of linear
dislocation mechanics with a prescribed dislocation velocity field $V$ in
space-time along with the ansatz

$\displaystyle U_{ij}$ $\displaystyle:=\delta_{ij}-W_{ij}$ $\displaystyle
T_{ij}$ $\displaystyle:=C_{ijkl}U_{kl},$

ignoring all nonlinearities in (3) and assuming the mass density field $\rho$
to a be specified field. The ansatz is justified for small elastic distortions
$(U)$ about the ground state (cf., [18]). We note that $C_{ijkl}$ is
necessarily symmetric in $(k,l)$ and $(i,j)$ so that it is not invertible on
the space of all second order tensors (and hence the stress only depends on
the elastic strain, the symmmetric part of $U$). With these assumptions, the
system (3) may be expressed as

$\displaystyle 0$ $\displaystyle=\partial_{t}(\rho
v_{i})-\partial_{j}(C_{ijkl}U_{kl})$ (8) $\displaystyle 0$
$\displaystyle=\partial_{j}v_{i}-\partial_{t}U_{ij}-e_{jrs}\alpha_{ir}V_{s}$
$\displaystyle 0$ $\displaystyle=e_{jrs}\partial_{r}U_{is}-\alpha_{ij}.$

Taking inner products of these equations with the dual fields
$D=(\lambda,A,B)$ that vanish on the boundary and utilizing an arbitrary
function $M$ convex in the list of arguments $M(v,U,\alpha)$ we define the
functional

$\displaystyle\widehat{S}[A,U,B,\alpha,\lambda,v]=\int_{[0,T]\times\Omega}\,dtd^{3}x$
$\displaystyle\ v_{i}(-\partial_{j}A_{ij}-\rho\partial_{t}\lambda_{i})$
$\displaystyle\
+U_{ij}(\partial_{t}A_{ij}-e_{sjr}\partial_{r}B_{is}+C_{ijkl}\partial_{l}\lambda_{k})$
$\displaystyle\ +\alpha_{ir}(-A_{ij}e_{jrs}V_{s}-B_{ir})-M(U,\alpha,v)$

Defining

$p:=(-\partial_{j}A_{ij}-\rho\partial_{t}\lambda_{i},\partial_{t}A_{ij}-e_{sjr}\partial_{r}B_{is}+C_{ijkl}\partial_{l}\lambda_{k},-A_{ij}e_{jrs}V_{s}-B_{ir});\qquad\qquad
Q:=(v,U,\alpha)$

and $M^{*}(p)$ the Legendre transform of $M(Q)$ given by

$\displaystyle(v_{M}(p),U_{M}(p),\alpha_{M}(p))=:Q_{M}(p)$
$\displaystyle=(\partial_{Q}M)^{-1}(p)$ (9) $\displaystyle M^{*}(p)$
$\displaystyle=Q_{M}(p)\cdot p-M(Q_{M}(p))$
$\displaystyle\partial_{p}M^{*}(p)$ $\displaystyle=Q_{M}(p)$

(well-defined because of the convexity of $M(Q)$), we define the dual action,
$S_{M}[D]$,

$\widehat{S}[A,U_{M}(p),B,\alpha_{M}(p),\lambda,v_{M}(p)]=:S_{M}[D]=\int_{[0,T]\times\Omega}\,dtd^{3}x\
M^{*}(p)$

whose first variation is given by (after an integration by parts)

$\displaystyle\delta S_{M}$ $\displaystyle=\int_{[0,T]\times\Omega}dtd^{3}x\
Q_{M}(p)\,\delta p$ $\displaystyle=\int_{[0,T]\times\Omega}dtd^{3}x\
\delta\lambda_{i}\left(\partial_{t}(\rho
v_{i}(p))-\partial_{j}(C_{ijkl}U_{kl}(p))\right)$
$\displaystyle\qquad\qquad\qquad\,+\ \delta
A_{ij}\left(\partial_{j}(v_{i}(p))-\partial_{t}(U_{ij}(p))-e_{jrs}\alpha_{ir}(p)V_{s}\right)$
$\displaystyle\qquad\qquad\qquad\,+\ \delta
B_{is}(e_{srj}\partial_{r}(U_{ij}(p))-\alpha_{is}(p)),$

(where we have dropped the subscript M on the dual-to-primal mapping fields
for notational convenience). Thus, the dual Euler-Lagrange equations are the
system (8) expressed in terms of the dual fields through the mapping codified
in (9)2, regardless of the convex potential $M$ chosen to define the dual
functional $S_{M}$.

This exercise exposes an interesting fact in a simple setting. Clearly, for
$M$ to be convex in $U$ it cannot be invariant as it has to depend on the
skew-symmetric part of the latter - and rotational invariance/invariance under
superposed rigid deformations in the linear setting precludes such a
dependence. However, the use of such a potential in the dual theory does not
in any way obstruct the definition of correct physics as embodied in the
Euler-Lagrange equations solved.

##  5 Dual variational principle for a primal problem with initial and
boundary conditions

Consider the system of PDE:

$\displaystyle\partial_{t}u_{I}$
$\displaystyle=\mathbb{A}_{IJ}\,u_{J}+\mathbb{B}_{IJk}\,\partial_{k}u_{J}+\mathbb{C}_{IJkl}\,\partial_{l}B_{Jk}+\mathfrak{f}_{I}(u,B,C)+\partial_{k}(\mathfrak{A}_{Ik}(u,B,C))$
(10) $\displaystyle\partial_{i}u_{I}$ $\displaystyle=B_{Ii}$
$\displaystyle\partial_{j}B_{Ii}$ $\displaystyle=C_{Iij},$

where $\mathbb{A},\mathbb{B},\mathbb{C}$ are arrays of real constants,
$\mathfrak{f}_{I}$ and $\mathfrak{A}_{Ij}$ are, for each $I,j$, given, real-
valued, smooth functions of the arguments shown, uppercase Latin indices span
$1$ to $n$, and lowercase Latin indices representing space-dimensions span $1$
to $1\leq d\leq 3$, and $t$ is time.

The functions $\mathfrak{f},\mathfrak{A}$ do not contain any terms linear in
the array $(u,B,C)$.

Let the initial and boundary conditions for (10) be

$\displaystyle u_{I}(x,0)=\overline{u}^{(i)}_{I}(x),\qquad x\in\Omega$ (11)
$\displaystyle\mathfrak{A}_{Ik}(u(x,t),B(x,t),C(x,t))\,n_{k}(x)$
$\displaystyle\qquad+\left(\mathbb{B}_{IJk}u_{J}(x,t)+\mathbb{C}_{IJik}B_{Ji}(x,t)\right)n_{k}(x)=\overline{\tau}_{I}(x,t),\qquad(x,t)\in\partial\Omega_{\tau}(x)\times(0,T]$
$\displaystyle
u_{I}(x,t)=\overline{u}^{(b)}_{I}(x,t),\qquad(x,t)\in\partial\Omega_{u}(x)\times(0,T]$
$\displaystyle\partial_{i}u_{I}(x,t)=B_{Ii}(x,t)=\overline{B}_{Ii}(x,t),\qquad(x,t)\in\partial\Omega_{\nabla
u}(x)\times(0,T]$

where $n$ represent the outward unit normal to the boundary of the domain
$\partial\Omega$, the functions with overhead bars are prescribed, and the
subsets $\partial\Omega_{\tau},\partial\Omega_{u},\partial\Omega_{\nabla u}$
of the spatial boundary $\partial\Omega$ can very well be empty for a specific
problem.

The initial and boundary conditions (11) for the primal system (10) is simply
a set of conditions that encompass the commonly encountered ones for up to
second-order systems of partial differential equations; the present work does
not deal with the question of well-posedness of the system (10) with this
specified set of initial and boundary conditions. For instance, it may very
well be that in a specific problem, well-possedness requires
$\partial_{i}u_{I}=B_{Ii}$ not to be specified on any part of the boundary of
$\Omega$. In that case $\partial\Omega_{\nabla}u$ can be chosen to be the
empty set, and then, as suggested by (12) below, the dual field $\rho$ needs
to be prescribed on the entirety of $\partial\Omega$ for all times.

Our scheme [22] then suggests defining

$\displaystyle\widehat{S}[u,B,C,\lambda,\gamma,\rho]=\int_{\Omega\times(0,T)}d^{3}xdt$
$\displaystyle-\partial_{t}\lambda_{I}u_{I}-\lambda_{I}\mathfrak{f}_{I}(u,B,C)+\mathfrak{A}_{Ik}(u,B,C)\partial_{k}\lambda_{I}$
$\displaystyle-
u_{I}\mathbb{A}_{JI}\lambda_{J}+u_{I}\,\mathbb{B}_{JIk}\,\partial_{k}\lambda_{J}+B_{Ii}\,\mathbb{C}_{JIil}\,\partial_{l}\lambda_{J}$
$\displaystyle-u_{I}\partial_{i}\gamma_{Ii}-\gamma_{Ii}B_{Ii}$ $\displaystyle-
B_{Ii}\partial_{j}\rho_{Iij}-\rho_{Iij}C_{Iij}+H(u,B,C,x,t),$

_where we have allowed for $H$ to depend on $(x,t)$ as well. This generality
is practically useful, especially when $H$ plays the role of a selection
criterion for non-unique solutions of the primal problem_.

Next define the arrays $U,P,L,F$ and the function $M$:

$\displaystyle U:=(u_{I},B_{Ii},C_{Iij}),\qquad L:=(\lambda_{I},\
-\partial_{k}\lambda_{I})$ $\displaystyle
P:=(\partial_{t}\lambda_{I}+\partial_{i}\gamma_{Ii}+\mathbb{A}_{JI}\lambda_{J}-\mathbb{B}_{JIk}\partial_{k}\lambda_{J},\quad\gamma_{Ii}+\partial_{j}\rho_{Iij}-\mathbb{C}_{JIil}\,\partial_{l}\lambda_{J},\quad\
\rho_{Iij}),\qquad F:=(\mathfrak{f}_{I},\mathfrak{A}_{Ij})$ $\displaystyle
M(U,L,x,t):=H(U,x,t)-L\cdot F(U)$

(noting that $F$ is indeed a function of only $U$).

Following [22], we now ask that $M$ (enabled by the choice of $H$) be such
that

$\exists\
U_{H}(P,L,x,t)\quad\mbox{satisfying}\quad\partial_{U}M(U_{H}(P,L,x,t),L,x,t)=P,\
\forall P$

and in terms of this define

$M^{*}(P,L,x,t):=U_{H}(P,L,x,t)\cdot P-M(U_{H}(P,L,x,t),L,x,t).$

We then replace $U$ in $\widehat{S}$ by $U_{H}(P,L,x,t)$ to define the dual
functional

$\displaystyle S[\lambda,\gamma,\rho]$
$\displaystyle:=\int_{\Omega\times(0,T)}d^{3}xdt\ -P\cdot
U_{H}(P,L,x,t)+M(U_{H}(P,L,x,t),L,x,t)$
$\displaystyle=-\int_{\Omega\times(0,T)}d^{3}xdt\ M^{*}(P,L,x,t)$

whose first variation is given by

$\delta S=\int_{\Omega\times(0,T)}d^{3}xdt\,-\partial_{P}M^{*}\cdot\delta
P-\partial_{L}M^{*}\cdot\delta L.$

Now, a calculation detailed in [22, Sec. 6], shows that

$\partial_{P}M^{*}(P,L,x,t)=U_{H}(P,L,x,t),\qquad\partial_{L}M^{*}(P,L,x,t)=F(U_{H}(P,L,x,t)).$

Then, using the notation $U_{H}(P,L,x,t)=\left(u^{H}_{I}(P,L,x,t),\
B^{H}_{Ii}(P,L,x,t),\ C^{H}_{Iij}(P,L,x,t)\right),I=1\ \mbox{to}\ n;\ i,j=1\
\mbox{to}\ d$,

$\displaystyle\delta S$
$\displaystyle=\int_{\Omega\times(0,T)}d^{3}xdt\,-U_{H}(P,L,x,t)\cdot\delta
P-F(U_{H}(P,L,x,t))\cdot\delta L$
$\displaystyle=\int_{\Omega\times(0,T)}d^{3}xdt\,-u^{H}_{I}(P,L,x,t)(\partial_{t}\delta\lambda_{I}+\partial_{i}\delta\gamma_{Ii}+\mathbb{A}_{JI}\delta\lambda_{J}-\mathbb{B}_{JIk}\partial_{k}\delta\lambda_{J})$
$\displaystyle\qquad\qquad\qquad\qquad-B^{H}_{Ii}(P,L,x,t)(\delta\gamma_{Ii}+\partial_{j}\delta\rho_{Iij}-\mathbb{C}_{JIil}\,\partial_{l}\delta\lambda_{J})-C_{Iij}^{H}(P,L,x,t)\delta\rho_{Iij}$
$\displaystyle\qquad\qquad\qquad\qquad-\mathfrak{f}_{I}(U_{H}(P,L,x,t))\delta\lambda_{I}+\mathfrak{A}_{Ik}(U_{H}(P,L,x,t))\partial_{k}\delta\lambda_{I},$

and collecting terms,

$\displaystyle\delta S$
$\displaystyle=\int_{\Omega}d^{3}x\,\delta\lambda_{I}(x,0)u_{I}^{H}(P(x,0),L(x,0),x,0)-\int_{\Omega}d^{3}x\,\delta\lambda_{I}(x,T)u_{I}^{H}(P(x,T),L(x,T),x,T)$
$\displaystyle\quad+\int_{\Omega\times(0,T)}d^{3}xdt\
\delta\lambda_{I}\left(\partial_{t}u^{H}_{I}(P,L,x,t)-\mathfrak{f}_{I}(U_{H}(P,L,x,t))-\partial_{k}\left(\mathfrak{A}_{Ik}(U_{H}(P,L,x,t))\right)\right)$
$\displaystyle\quad+\int_{\Omega\times(0,T)}d^{3}xdt\
\delta\lambda_{I}\left(-\mathbb{A}_{IJ}u^{H}_{J}(P,L,x,t)-\mathbb{B}_{IJk}\partial_{k}\left(u^{H}_{J}(P,L,x,t)\right)-\mathbb{C}_{IJil}\partial_{l}\left(B^{H}_{Ji}(P,L,x,t)\right)\right)$
$\displaystyle\quad+\int_{0}^{T}dt\int_{\partial\Omega}da\
\delta\lambda_{I}\left(\mathbb{M}_{jk}\mathfrak{A}_{Ij}(U_{H}(P,L,x,t))+u^{H}_{J}(P,L,x,t)\mathbb{B}_{IJk}+\mathbb{C}_{IJik}B^{H}_{JI}(P,L,x,t)\right)\,n_{k}$
$\displaystyle\quad-\int_{0}^{T}dt\int_{\partial\Omega}da\
\delta\gamma_{Ii}\,u_{I}^{H}(P,L,x,t)\,n_{i}$
$\displaystyle\quad+\int_{\Omega\times(0,T)}d^{3}xdt\,\delta\gamma_{Ii}\left(\partial_{i}u_{I}^{H}(P,L,x,t)-B_{Ii}^{H}(P,L,x,t)\right)$
$\displaystyle\quad-\int_{0}^{T}dt\int_{\partial\Omega}da\
\delta\rho_{Iij}B^{H}_{Ii}(P,L,x,t)\,n_{j}$
$\displaystyle\qquad+\int_{\Omega\times(0,T)}d^{3}xdt\,\delta\rho_{Iij}\left(\partial_{j}B^{H}_{Ii}(P,L,x,t)-C^{H}_{Iij}(P,L,x,t)\right).$

From the above calculation we read off the modification required to $S$ to
account for the specified initial and boundary conditions:

$\displaystyle S_{ibvp}[\lambda,\gamma,\rho]$
$\displaystyle:=-\int_{\Omega\times(0,T)}d^{3}xdt\,M^{*}(P(x,t),L(x,t),x,t)$
(12)
$\displaystyle\quad-\int_{\Omega}d^{3}x\,\lambda_{I}(x,0)\,\overline{u}^{(i)}_{I}(x)-\int_{0}^{T}dt\int_{\partial\Omega_{\tau}}da\,\lambda_{I}(x,t)\,\overline{\tau}_{I}(x,t)$
$\displaystyle\quad+\int_{0}^{T}dt\int_{\partial\Omega_{u}}da\,\gamma_{Ii}(x,t)\,\overline{u}^{(b)}_{I}(x,t)\,n_{i}(x)+\int_{0}^{T}dt\int_{\partial\Omega_{\nabla
u}}da\,\rho_{Iij}(x,t)\overline{B}_{Ii}(x,t)\,n_{j}(x)$ with
$\displaystyle\quad\lambda_{I}(x,T)=\mbox{`arbitrarily' prescribed and
}\delta\lambda_{I}(x,T)=0,x\in\Omega$
$\displaystyle\quad\lambda_{I}(x,t)=\mbox{`arbitrarily' prescribed and
}\delta\lambda_{I}(x,t)=0,(x,t)\in\partial\Omega\backslash\partial\Omega_{\tau}\times(0,T]$
$\displaystyle\quad\gamma_{Ii}(x,t)=\mbox{`arbitrarily' prescribed and
}\delta\gamma_{Ii}(x,t)=0,(x,t)\in\partial\Omega\backslash\partial\Omega_{u}\times(0,T]$
$\displaystyle\quad\rho_{Iij}(x,t)=\mbox{`arbitrarily' prescribed and
}\delta\rho_{Iij}(x,t)=0,(x,t)\in\partial\Omega\backslash\partial\Omega_{\nabla
u}\times(0,T],$

(with prescriptions chosen to avoid discontinuities at ‘space-time corners’ of
boundary of the space-time domain $\Omega\times(0,T)$).

We end by noting that in ongoing work these ideas have been used to
successfully formulate and compute approximate solutions (with minimal error)
of the heat equation and the first-order wave equation in bounded domains, in
one space dimension and time. These parabolic and hyperbolic equations are
solved by a common methodology based on computing weak solutions to degenerate
elliptic boundary value problems in a space-time domain, involving oblique
natural boundary conditions. Uniqueness of solutions to the corresponding dual
problems with the initial-boundary conditions developed following the above
ideas is also shown.

As already noted in the Introduction, the primary application envisioned for
this variational principle is in enabling a Feynman Path Integral based
statistical analysis of dislocation dynamics. A second major application is in
designing an efficient and robust numerical scheme for the system (3), which
is a system of Hamilton-Jacobi equations. It also has the potential of
providing a pathway to the rigorous analysis of the system in the hands of
bona-fide experts in PDE and variational calculus.

## References

  * [1] G. Weingarten, “Sulle superficie di discontinuità nella teoria della elasticità dei corpi solidi,” _Rend. Reale Accad. dei Lincei, classe di sci., fis., mat., e nat., ser. 5_ , vol. 10.1, pp. 57–60, 1901.
  * [2] V. Volterra, “Sur l’équilibre des corps élastiques multiplement connexes,” in _Annales scientifiques de l’École normale supérieure_ , vol. 24, 1907, pp. 401–517.
  * [3] R. Peierls, “The size of a dislocation,” _Proceedings of the Physical Society_ , vol. 52, no. 1, p. 34, 1940.
  * [4] F. Nabarro, “Dislocations in a simple cubic lattice,” _Proceedings of the Physical Society (1926-1948)_ , vol. 59, no. 2, p. 256, 1947.
  * [5] J. P. Hirth and J. Lothe, _Theory of dislocations_. Krieger, 1982.
  * [6] J. F. Nye, “Some geometrical relations in dislocated crystals,” _Acta Metallurgica_ , vol. 1, no. 2, pp. 153–162, 1953.
  * [7] E. Kröner, “Continuum theory of defects,” in _Physics of Defects, Les Houches Summer School Proceedings_ , R. Balian, M. Kléman, and J.-P. Poirier, Eds., vol. 35. North-Holland, Amsterdam, 1981, pp. 217–315.
  * [8] T. Mura, “Continuous distribution of moving dislocations,” _Philosophical Magazine_ , vol. 8, no. 89, pp. 843–857, 1963.
  * [9] J. R. Willis, “Second-order effects of dislocations in anisotropic crystals,” _International Journal of Engineering Science_ , vol. 5, no. 2, pp. 171–190, 1967.
  * [10] N. Fox, “A continuum theory of dislocations for single crystals,” _IMA Journal of Applied Mathematics_ , vol. 2, no. 4, pp. 285–298, 1966.
  * [11] K. Kondo, “Non-Riemannian geometry of imperfect crystals from a macroscopic viewpoint,” _RAAG Memoirs of the unifying study of the basic problems in engineering science by means of geometry_ , vol. 1, pp. 6–17, 1955.
  * [12] B. A. Bilby, R. Bullough, and E. Smith, “Continuous distributions of dislocations: a new application of the methods of non-riemannian geometry,” _Proceedings of the Royal Society of London, A_ , vol. 231, no. 1185, pp. 263–273, 1955.
  * [13] C. Truesdell and R. Toupin, “The Classical Field Theories,” in _Principles of classical mechanics and field theory/Prinzipien der Klassischen Mechanik und Feldtheorie_. Springer, 1960, pp. 226–858.
  * [14] C. Truesdell and W. Noll, “The Non-linear Field Theories of Mechanics,” in _The non-linear field theories of mechanics_. Springer, 2004, pp. 1–579.
  * [15] B. D. Coleman and W. Noll, “The thermodynamics of elastic materials with heat conduction and viscosity,” in _The foundations of mechanics and thermodynamics_. Springer, 1974, pp. 145–156.
  * [16] B. D. Coleman and M. E. Gurtin, “Thermodynamics with internal state variables,” _The journal of chemical physics_ , vol. 47, no. 2, pp. 597–613, 1967.
  * [17] A. Acharya, “Constitutive analysis of finite deformation field dislocation mechanics,” _Journal of the Mechanics and Physics of Solids_ , vol. 52, no. 2, pp. 301–316, 2004.
  * [18] ——, “An action for nonlinear dislocation dynamics,” _Journal of the Mechanics and Physics of Solids_ , vol. 161, p. 104811, 2022.
  * [19] R. Arora and A. Acharya, “A unification of finite deformation ${J}_{2}$ von-Mises plasticity and quantitative dislocation mechanics,” _Journal of the Mechanics and Physics of Solids_ , vol. 143, p. 104050, 2020\.
  * [20] R. Arora, X. Zhang, and A. Acharya, “Finite element approximation of finite deformation dislocation mechanics,” _Computer Methods in Applied Mechanics and Engineering_ , vol. 367, p. 113076, 2020.
  * [21] X. Zhang, A. Acharya, N. J. Walkington, and J. Bielak, “A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations,” _Journal of the Mechanics and Physics of Solids_ , vol. 84, pp. 145–195, 2015.
  * [22] A. Acharya, “Variational principles for nonlinear PDE systems via duality,” Quarterly of Applied Mathematics, https://doi.org/10.1090/qam/1631, Article electronically published on September 26, 2022, https://arxiv.org/abs/2108.08902.
  * [23] E. Kröner, _Kontinuumstheorie der versetzungen und eigenspannungen_. Springer, 1958, vol. 5.
  * [24] V. L. Berdichevsky, _Variational Principles of Continuum Mechanics_. Springer, 2009.
  * [25] A. George, M. T. Heath, and E. Ng, “Solution of sparse underdetermined systems of linear equations,” _SIAM journal on scientific and statistical computing_ , vol. 5, no. 4, pp. 988–997, 1984.
  * [26] M. T. Heath, _Scientific computing: an introductory survey, revised second edition_. SIAM, 2018.
  * [27] G. H. Golub and C. F. Van Loan, _Matrix computations_. JHU press, 2013.
  * [28] A. Acharya and L. Tartar, “On an equation from the theory of field dislocation mechanics,” _Bollettino dell’Unione Matematica Italiana_ , vol. (9) IV, pp. 409–444, 2011.
  * [29] P. D. Lax, _Hyperbolic systems of conservation laws and the mathematical theory of shock waves_. SIAM, 1973.
  * [30] ——, “On dispersive difference schemes,” _Physica D: Nonlinear Phenomena_ , vol. 18, no. 1-3, pp. 250–254, 1986.
  * [31] P. D. Lax and C. D. Levermore, “The zero dispersion limit for the Korteweg-deVries KdV equation,” _Proceedings of the National Academy of Sciences_ , vol. 76, no. 8, pp. 3602–3606, 1979.
  * [32] J. Zaanen, F. Balm, and A. J. Beekman, “Crystal gravity,” _SciPost Physics_ , vol. 13, p. 039, 2022.