# On the $\Gamma-$convergence of the Allen-Cahn functional with boundary
conditions

Dimitrios Gazoulis  Department of Mathematics and Applied Mathematics,
University of Crete, 70013 Heraklion, Greece, and Institute of Applied and
Computational Mathematics, FORTH, 700 13 Heraklion, Crete, Greece
<EMAIL_ADDRESS>
###### Abstract.

We study minimizers of the Allen-Cahn system. We consider the
$\varepsilon-$energy functional with Dirichlet values and we establish the
$\Gamma$-limit. The minimizers of the limiting functional are closely related
to minimizing partitions of the domain. Finally, utilizing that the triod and
the straight line are the only minimal cones in the plane together with
regularity results for minimal curves, we determine the precise structure of
the minimizers of the limiting functional, and thus the limit of minimizers of
the $\varepsilon$-energy functional as $\varepsilon\rightarrow 0$.

##  1\. Introduction

In this work we are concerned with the study of vector minimizers of the
Allen-Cahn $\varepsilon$-functional,

(1.1)
$\begin{gathered}J_{\varepsilon}(u,\Omega)=\int_{\Omega}(\frac{\varepsilon}{2}|\nabla
u|^{2}+\frac{1}{\varepsilon}W(u))dx\\\
u:\Omega\rightarrow\mathbb{R}^{m}\end{gathered}$

where $\Omega\subset{\mathbb{R}}^{n}$ is an open set and $W$ is a $N$-well
potential with $N$ global minima.

Let

(1.2) $u_{\varepsilon}:=\min_{v\in
W^{1,2}(\Omega;{\mathbb{R}}^{m})}\\{J_{\varepsilon}(v,\Omega):v|_{\partial\Omega}=g_{\varepsilon}|_{\partial\Omega}\\}\;,\;\textrm{where}\;\;g_{\varepsilon}\in
W^{1,2}(\Omega;{\mathbb{R}}^{m})$

Thus $u_{\varepsilon}$ is a weak solution of the system

(1.3) $\begin{cases}\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}W_{u}(u_{\varepsilon})=0\;\>,\;\textrm{in}\;\>\Omega\\\
u_{\varepsilon}=g_{\varepsilon}\;\>,\;\textrm{on}\;\>\partial\Omega\end{cases}$

We study the asymptotic behavior of $u_{\varepsilon}$ within the framework of
$\Gamma$-convergence. Moreover, we analyze the relationship between minimizers
of the Allen-Cahn system and minimizing partitions subject to Dirichlet
boundary conditions. For some particular choices of boundary conditions, we
will determine the structure of the minimizers of the limiting functional.

We now briefly introduce some of the well known results in the scalar case.
The notion of $\Gamma$-convergence was introduced by E. De Giorgi and relates
phase transition type problems with the theory of minimal surfaces. One
application of $\Gamma$-convergence is the proof of existence of minimizers of
a limiting functional, say $F_{0}$, by utilizing an appropriate sequence of
functionals $F_{\varepsilon}$ that we know they admit a minimizer and the
$\Gamma$-limit of $F_{\varepsilon}$ is $F_{0}$. And also vice versa ([11]). We
can think of this notion as a generalization of the Direct Method in the
Calculus of Variations i.e. if $F_{0}$ is lower semicontinuous and coercive we
can take $F_{\varepsilon}=F_{0}$ and then $\Gamma-$lim
$F_{\varepsilon}=F_{0}$.

There are many other ways of thinking of this notion, such as a proper tool in
studying the asymptotic behavior of minimizers of functionals.

Let $X$ be the space of the measurable functions endowed with the $L^{1}$ norm
and

$\displaystyle
F_{\varepsilon}(u,\Omega)=\begin{cases}\int_{\Omega}\frac{\varepsilon}{2}|\nabla
u|^{2}+\frac{1}{\varepsilon}W(u)dx\;\;,\;u\in W^{1,2}(\Omega;\mathbb{R})\cap
X\\\
+\infty\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\;\textrm{elsewhere
in}\;\>X\end{cases}$ $\displaystyle
F_{0}(u,\Omega)=\begin{cases}\sigma\mathcal{H}^{n-1}(Su)\;\;,\;u\in
BV(\Omega;\\{-1,1\\})\cap X\\\
+\infty\;\;\;\;\;\;\;\;\;\;\;\;\;,\;\textrm{elsewhere
in}\;\>X\end{cases}\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\textrm{where}\;\>W:\mathbb{R}\rightarrow[0,+\infty)\;,\;\\{W=0\\}=\\{-1,1\\}\;,\;\sigma=\int_{-1}^{1}\sqrt{2W(u)}du$
$\displaystyle\textrm{and}\;\>Su\;\>\textrm{is the interface that separates
the phases}\;-1\;\>\textrm{and}\;+1.$

Let now $u_{\varepsilon}$ be a minimizer of $F_{\varepsilon}$ subject to a
mass constraint, that is, $\int_{\Omega}u=V\in(0,|\Omega|)$. The asymptotic
behavior of $u_{\varepsilon}$ was first studied by Modica and Mortola (see
[12] and [13]). Later Sternberg [17] generalized these results for minimizers
with volume constraint. Furthermore, Owen, Rubinstein and Sternberg in [16]
and Ansini, Braides and Piat in [3], among others, studied the asymptotic
behavior of the minimizers subject to Dirichlet values for the scalar case.

As mentioned previously, one of the most important outcomes of
$\Gamma$-convergence in the scalar phase transition type problems is the
relationship with minimal surfaces. More precisely, the well known theorem of
Modica and Mortola states that the $\varepsilon$-energy functional of the
Allen-Cahn equation $\Gamma$-converges to the perimeter functional that
measures the perimeter of the interface between the phases (i.e.
$\Gamma-\textrm{lim}\>F_{\varepsilon}=F_{0}$). So the interfaces of the
limiting problem will be minimal surfaces.

This relationship is deeper as indicated in the De Giorgi conjecture (see [8])
which states that the level sets of global entire solutions of the scalar
Allen-Cahn equation that are bounded and strictly monotone with respect to
$x_{n}$, are hyperplanes if $n\leq 8$. The relationship with the Bernstein
problem for minimal graphs is the reason why $n\leq 8$ appears in the
conjecture. The $\Gamma$-limit of the $\varepsilon$-energy functional of the
Allen-Cahn equation is a possible motivation behind the conjecture.

In addition, Baldo in [4] and Fonseca and Tartar in [10] extended the
$\Gamma$-convergence analysis for the phase transition type problems to the
vector case subject to a mass constraint and the limiting functional measures
the perimeter of the interfaces separating the phases, and thus there is a
relationship with the problem of minimizing partitions. In section 4 we
analyze this in the set up of Dirichlet boundary conditions.

Acknowledgements: I wish to thank my advisor Professor Nicholas Alikakos for
his guidance and for suggesting this topic as a part of my thesis for the
Department of Mathematics and Applied Mathematics at the University of Crete.
Also, I would like to thank Professor P. Sternberg and Professor F. Morgan for
their valuable comments on a previous version of this paper, which let to
various improvements.

##  2\. Hypotheses and Basic Lemmas

Hypothesis on $W$: (H1) $W\in
C^{2}({\mathbb{R}}^{m};[0,+\infty))\;,\;\\{W=0\\}=\\{a_{1},a_{2},...,a_{N}\\}\;,\>N\in\mathbb{N}\;\>,a_{i}$
are the global minima of $W$ and such that

$\displaystyle\xi^{T}W_{uu}(a_{i})\xi\geq
2c^{2}|\xi|^{2}\;\;\;,\;i=1,2,...,N.$

Assume also that $\;W_{u}(u)\cdot u>0$ and $W(u)\geq c_{1}|u|^{2}\>$, if
$|u|>M$.

Hypothesis on the Dirichlet Data: (H2) (i) $g_{\varepsilon}\in
C^{1,\alpha}(\overline{\Omega})\;,\>|g_{\varepsilon}|\leq M$ (uniformly in
$\varepsilon)\>,\;\>|g_{\varepsilon}|_{1,\alpha}\leq\dfrac{M}{\varepsilon^{1+\alpha}}$
and $g_{\varepsilon}\rightarrow g_{0}$ in $L^{1}(\Omega).\\\ $

(ii) $J_{\varepsilon}(g_{\varepsilon},\Omega_{2}\setminus\Omega)\leq
C_{0}\;\>(\textrm{independent of}\>\varepsilon>0)$, where
$\Omega_{\rho}:=\\{\rho x:x\in\Omega\\}\;,\>\rho>0$ and for simplicity suppose
$\Omega$ is a convex set that contains the origin in
$\mathbb{R}^{n}.\;g_{\varepsilon}$ is trivially extended in each
$\Omega_{\rho}$, for $\rho>1$, being constant in each line segment that
connects $\partial\Omega$ with $\partial\Omega_{\rho}$ and the extension of
this line segment passes through the origin.

For $i\neq j\;\>,\;i,j\in\\{1,2,...,N\\}$, let $U\in
W^{1,2}(\mathbb{R};\mathbb{R}^{m})$ be the 1D minimizer of the action

(2.1)
$\begin{gathered}\sigma_{ij}:=\min\int_{-\infty}^{+\infty}(\frac{1}{2}|U^{\prime}|^{2}+W(U))dt<+\infty\;\;\;,\\\
\lim_{t\rightarrow-\infty}U(t)=a_{i}\;\;,\;\;\lim_{t\rightarrow+\infty}U(t)=a_{j}\;\>,\;U(\mathbb{R})\in\mathbb{R}^{m}\setminus\\{W=0\\}\end{gathered}$

where $U$ is the connection that connects $a_{i}$ to
$a_{j}\;,\>i,j\in\\{1,2,...,N\\}$ and suppose that all of which have equal
energy, i.e. $\sigma_{ij}=\sigma>0\;,\>i,j\in\\{1,2,3\\}$. If we have a
symmetric potential $W$ for example, then (2.1) $\sigma_{ij}=\sigma$ holds.
111This assumption that all actions are equal is not necessary for the
$\Gamma$-limit. It is only used to determine the precise structure of the
minimizers in the last section.

Note: We note that the convexity assumption of the domain in (H2)(ii) could be
relaxed by considering the convex hull of a set in general as we see in the
example in Figure 1 below. However, we assume convexity for convenience.

$g_{0}=a_{1}$$g_{0}=a_{2}$$g_{0}=a_{3}$$g_{0}=a_{2}$$\Omega$$\tilde{\Omega}\
=\ convex\ hull\ of\ \Omega$Figure 1.(a)(b)$g_{0}=a_{2}$

######  Lemma 2.1.

For every critical point $u_{\varepsilon}\in W^{1,2}(\Omega;\mathbb{R}^{m})$,
satisfying (1.3) weakly together with the assumptions (H1) and (H2)(i) it
holds

$\displaystyle||u_{\varepsilon}||_{L^{\infty}}<M\;\;\;\;\textrm{and}\;\;\;\;||\nabla
u_{\varepsilon}||_{L^{\infty}}<\frac{\tilde{C}}{\varepsilon}$

###### Proof.

By linear elliptic theory, we have that $u_{\varepsilon}\in
C^{2}(\Omega;{\mathbb{R}}^{m})$. Set
$v_{\varepsilon}(x)=|u_{\varepsilon}(x)|^{2}$, then

$\displaystyle\Delta v_{\varepsilon}=2W_{u}(u_{\varepsilon})\cdot
u_{\varepsilon}+2|\nabla
u_{\varepsilon}|^{2}>0\;\;\;\;for\;\;|u_{\varepsilon}|>M$

Hence $\max|u_{\varepsilon}|^{2}\leq M^{2}$ if $v_{\varepsilon}$ attains max
in the interior of $B_{1}$.

On the other hand (from (H2)), $\max_{\partial B_{1}}|u_{\varepsilon}|\leq M$.
Thus $\max_{\overline{B}_{1}}|u_{\varepsilon}|\leq M.\\\ $ For the gradient
bound, consider the rescaled problem $y=\frac{x}{\varepsilon}$, denote by
$\tilde{u}\>,\>\tilde{g}$ the rescaled $u_{\varepsilon}\>,\>g_{\varepsilon}$,
so by elliptic regularity

$\displaystyle|\tilde{u}|_{1,\alpha}\leq
C(||\tilde{u}||_{L^{\infty}}+|\tilde{g}|_{1,\alpha})\leq 2CM$
$\displaystyle\Rightarrow||\nabla\tilde{u}||_{L^{\infty}}\leq
2CM\Rightarrow|\nabla u_{\varepsilon}|\leq\frac{\tilde{C}}{\varepsilon}$

∎

######  Lemma 2.2.

Let $u_{\varepsilon}$ defined in (1.2), then
$J_{\varepsilon}(u_{\varepsilon})=\int_{\Omega}(\frac{\varepsilon}{2}|\nabla
u_{\varepsilon}|^{2}+\frac{1}{\varepsilon}W(u_{\varepsilon}))dx\leq C\;\>,\;C$
independent of $\varepsilon>0$, if $\Omega$ is bounded.

###### Proof.

Without loss of generality we will prove Lemma 2.2 for $\Omega=B_{1}$ (or else
we can cover $\Omega$ with finite number of unit balls and the outside part is
bounded by (H2)(ii)).

Substituting $y=\dfrac{x}{\varepsilon}$,

$\displaystyle
J_{\varepsilon}(u_{\varepsilon})=\int_{B_{\frac{1}{\varepsilon}}}(\frac{\varepsilon}{2}|\nabla_{y}\tilde{u}_{\varepsilon}|^{2}\>\frac{1}{\varepsilon^{2}}+\frac{1}{\varepsilon}W(\tilde{u}_{\varepsilon}))\varepsilon^{n}dy$

where $\tilde{u}_{\varepsilon}=u_{\varepsilon}(\varepsilon y)$ and for
$\varepsilon=\dfrac{1}{R}$,

$\displaystyle\Rightarrow
J_{\varepsilon}(u_{\varepsilon})=\varepsilon^{n-1}\int_{B_{\frac{1}{\varepsilon}}}(\frac{1}{2}|\nabla_{y}\tilde{u}_{\varepsilon}|^{2}+W(\tilde{u}_{\varepsilon}))dy=\frac{1}{R^{n-1}}\int_{B_{R}}(\frac{1}{2}|\nabla_{y}\tilde{u}_{R}|^{2}+W(\tilde{u}_{R}))dy=\frac{1}{R^{n-1}}\tilde{J}_{R}(\tilde{u}_{R})$

So, $\tilde{u}_{R}$ is minimizer of
$\tilde{J}_{R}(v)=\int_{B_{R}}(\frac{1}{2}|\nabla v|^{2}+W(v))dx$ and from
Lemma 2.1, it holds $|\tilde{u}_{R}|+|\nabla\tilde{u}_{R}|\leq M$ and via the
comparison function (see [1] p.135), for $r>1$

$\displaystyle
v(x)=\begin{cases}a_{1}\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{for}\;\>|x|\leq
r-1\\\
(r-|x|)a_{1}+(|x|-r+1)\tilde{u}_{R}(x)\;,\;\textrm{for}\;\>|x|\in(r-1,r]\\\
\tilde{u}_{R}(x)\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{for}\;\>|x|>r\end{cases}$

we have

$\displaystyle\tilde{J}_{R}(\tilde{u}_{R})\leq J(v)\leq
CR^{n-1}\;\;,\;C\;\>\textrm{independent of}\;\>R.$

Thus

$\displaystyle
J_{\varepsilon}(u_{\varepsilon})=\frac{1}{R^{n-1}}\tilde{J}_{R}(\tilde{u}_{R})\leq
C\;\;(C\;\>\textrm{independent of}\;\>\varepsilon>0)$

∎

######  Lemma 2.3.

Let $u_{\varepsilon}$ defined in (1.2), then $u_{\varepsilon}\rightarrow
u_{0}$ in $L^{1}$ along subsequences and $u_{0}\in BV(\Omega;\mathbb{R}^{m})$.
In addition,
$u_{0}=\sum_{i=1}^{N}a_{i}\chi_{\Omega_{i}}\;\;,\;\mathcal{H}^{n-1}(\partial\Omega_{i})<\infty$
and $|\Omega\setminus\cup_{i=1}^{N}\Omega_{i}|=0$.

###### Proof.

Claim: $||u_{\varepsilon}||_{BV(\Omega;\mathbb{R}^{m})}<C$. This claim
together with the $L^{1}$ convergence of $u_{\varepsilon}$ holds by
Proposition 4.1 in [4].222In [4] there is also a technical assumption for the
potential (see (1.2) in p.70), that is utilized in order to obtain
equiboundedness of $u_{\varepsilon}$. However, we obtain this fact in Lemma
2.1 from the assumption (H1) and (H2)(i) instead. Also $u_{0}\in
BV(\Omega;\mathbb{R}^{m}).\\\ $

From Lemma 2.2, we have

$\displaystyle\frac{1}{\varepsilon}\int_{\Omega}W(u_{\varepsilon}(x))dx\leq
C\;\>\;(C\;\>\textrm{independent of}\;\>\varepsilon>0)$

Since $|u_{\varepsilon}|\leq M$ and $W$ is continuous in
$\overline{B}_{M}\subset{\mathbb{R}}^{m}\>\Rightarrow
W(u_{\varepsilon})\leq\tilde{M}$, therefore by the dominated convergence
theorem we obtain

$\displaystyle\int_{\Omega}W(u_{0}(x))dx=0\Rightarrow
u_{0}\in\\{W=0\\}\;\>a.e.\;\>\Rightarrow
u_{0}=\sum_{i=1}^{N}a_{i}\chi_{\Omega_{i}}$

where $\chi_{\Omega_{i}}$ have finite perimeter since $u_{0}\in
BV(\Omega;\mathbb{R}^{m})$ (see [9]).

The proof of Lemma 2.3 is complete. ∎

######  Proposition 2.4.

It holds that

(2.2) $\begin{gathered}\int_{\Omega^{\prime}}|D(\phi_{k}\circ
u_{0})|=\sum_{i=1,i\neq
k}^{N}\sigma\mathcal{H}^{n-1}(\partial^{*}\Omega_{k}\cap\partial^{*}\Omega_{i}\cap\Omega^{\prime})\\\
k=1,2,..,N\;\>,\textrm{for every
open}\;\>\Omega^{\prime}\subset\Omega\end{gathered}$

where $\partial^{*}\Omega_{k}$ is the reduced boundary of $\Omega_{k}$ and
$\phi_{k}(z)=d(z,a_{k})\;,\>k=1,2,...,N,$ where $a_{k}$ are the zeros of $W$
and $d$ is the riemannian metric derived from $W^{1/2}$, that is

(2.3) $\displaystyle
d(z_{1},z_{2})=\inf\\{\int_{0}^{1}\sqrt{2}W^{1/2}(\gamma(t))|\gamma^{\prime}(t)|dt:\gamma\in
C^{1}([0,1];{\mathbb{R}}^{2}),\gamma(0)=z_{1},\>\gamma(1)=z_{2}\\}$

###### Proof.

By Lemma 2.3, $u_{0}=\sum_{i=1}^{N}a_{i}\chi_{\Omega_{i}}$ ,
$\Omega_{i}\cap\Omega_{j}=\emptyset\;\>,\>i\neq j$ and
$|\Omega\setminus\cup_{i=1}^{N}\Omega_{i}|=0.\\\ $ Arguing as in the proof of
Proposition 2.2 in [4], we utilize the coarea formula for $BV$ functions (see
[9] Theorem 5.9)

(2.4) $\displaystyle\int_{\Omega^{\prime}}|D(\phi_{k}\circ
u_{0})|=\int_{-\infty}^{+\infty}\mathcal{P}_{\Omega^{\prime}}(\\{x\in\Omega^{\prime}:\phi_{k}(u_{0}(x))\leq
t\\})dt$

for every open $\Omega^{\prime}\subset\Omega$ where
$\mathcal{P}_{\Omega^{\prime}}(V)=\mathcal{H}^{n-1}(\partial^{*}V\cap\Omega^{\prime}$).

We now observe that $\phi_{i}(a_{j})=d(a_{j},a_{i})=\sigma>0$ (see (2.1)), for
every $i\neq j\;,\>i,j\in\\{1,...,N\\}$ (and $\phi_{i}(a_{i})=0$). This can be
seen by the equipartition of the energy for 1D minimizers (see Theorem 2.1 in
[1]).

It holds that

(2.5) $\\{x\in\Omega^{\prime}:\phi_{k}(u_{0}(x))\leq
t\\}=\begin{cases}\Omega^{\prime}\;\;\;,\;\textrm{for}\;\>t\geq\sigma\\\
\Omega_{k}\cap\Omega^{\prime}\;\;\;,\;\textrm{for}\;\>t\in[0,\sigma)\\\
\emptyset\;\;\;,\;\textrm{for}\;\>t<0\end{cases}$

So, we have

(2.8) $\displaystyle\begin{gathered}\int_{\Omega^{\prime}}|D(\phi_{k}\circ
u_{0})|=\int_{0}^{\sigma}\mathcal{P}_{\Omega^{\prime}}(\\{x\in\Omega^{\prime}:\phi_{k}(u_{0}(x))\leq
t\\})dt=\sigma\mathcal{H}^{n-1}(\partial^{*}\Omega_{k}\cap\Omega^{\prime})\\\
=\sigma\sum_{i=1,i\neq
k}^{N}\mathcal{H}^{n-1}(\partial^{*}\Omega_{k}\cap\partial^{*}\Omega_{i}\cap\Omega^{\prime})\end{gathered}$

and for the last equality we utilized the fact that
$\partial^{*}\Omega_{k}=\cup_{i=1,i\neq
k}^{N}(\partial^{*}\Omega_{k}\cap\partial^{*}\Omega_{i})\cup N$ where
$\mathcal{H}^{n-1}(N)=0$ (see Proposition 2.2 in [4]).

The proof of Proposition 2.4 is complete. ∎

Let $\mu$ and $\nu$ be two regular Borel measures on $\Omega$ we denote by
$\mu\bigvee\nu$ the smallest regular positive measure which is greater than or
equal to $\mu$ and $\nu$ on all borel subsets of $\Omega$, for $\mu\;,\>\nu$
being two regular positive Borel measures on $\Omega.$ We have

$\displaystyle(\mu\bigvee\nu)(\Omega)=\sup\\{\mu(A)+\nu(B):A\cap
B=\emptyset,\;A\cup B\subset\Omega,\;A\;\>\textrm{and}\;\>B\;\>\textrm{are
open sets in}\;\>\Omega\\}$

Now let

$\displaystyle\bigvee_{k=1}^{N}\int_{\Omega}|D(\phi_{k}\circ
u_{0})|:=\sup\\{\sum_{k=1}^{N}\int_{A_{k}}|D(\phi_{k}\circ
u_{0})|:\cup_{k=1}^{N}A_{k}\subset\Omega,$ $\displaystyle A_{i}\cap
A_{j}=\emptyset\>,\>i\neq j,\;A_{i}\;\>\textrm{open sets in}\;\>\Omega\\}$

Furthermore, reasoning again as in the proof of Proposition 2.2 in [4] we
have,

(2.9) $\bigvee_{k=1}^{N}\int_{\Omega}|D(\phi_{k}\circ
u_{0})|=\frac{\sigma}{2}\sum_{i,j=1\;,\>i\neq
j}^{N}\mathcal{H}^{1}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}\cap\Omega)$

##  3\. The $\Gamma$-limit with boundary conditions

Let $J_{\varepsilon}$ defined in (1.1), we define for $\rho>0$,

(3.1)
$\tilde{J}_{\varepsilon}(u,\Omega_{\rho})=\begin{cases}J_{\varepsilon}(u,\Omega_{\rho})\;\>,\;\textrm{if}\;\>u=g_{\varepsilon}\;\>\textrm{on}\;\>\mathbb{R}^{n}\setminus\Omega_{\rho}\;\>,\;u\in
H^{1}_{loc}(\mathbb{R}^{n};\mathbb{R}^{m})\\\
+\infty\;\;\;\;\;\;\;\;\;,\;\;\textrm{otherwise}\end{cases}$

where $\Omega_{\rho}$ as in (H2)(ii), $\>g_{\varepsilon}\rightarrow
g_{0}\;\>\textrm{in}\;\>L^{1}(\Omega_{\rho})\;,\>\textrm{and}\>\;g_{\varepsilon}$
as in (H2)(i),(ii) and $g_{0}$ takes values on $\\{W=0\\}$ arguing as in Lemma
2.3. Also, $g_{0}$ is initially defined in the trace sense for BV functions
and then trivially extended in $\mathbb{R}^{n}\setminus\Omega_{\rho}$ being
constant in each line that that passes through $\partial\Omega_{\rho}$ and the
origin of $\mathbb{R}^{n}$ and belongs in
$\mathbb{R}^{n}\setminus\Omega_{\rho}$. $\Omega$ is convex, so the
intersection of such line with $\partial\Omega_{\rho}$ is a single point
(since the origin belongs in $\Omega$, we have that
$\Omega_{\rho}\subset\Omega$ for $\rho<1$ and $\Omega\subset\Omega_{\rho}$ for
$\rho>1).$

Let

(3.2)
$J_{0}(u,\Omega_{\rho})=\begin{cases}\bigvee_{k=1}^{N}\int_{\Omega_{\rho}}|D(\phi_{k}\circ
u)|\;\>,\;\textrm{if}\;\>u\in BV(\Omega_{\rho};\\{W=0\\})\\\
+\infty\;\;\;\;\;\;\;\;\;,\;\;\textrm{otherwise}\end{cases}$

and we obtain that, if $J_{0}(u,\Omega_{\rho})<+\infty$, then

(3.3) $\displaystyle J_{0}(u,\Omega_{\rho})=\sigma\sum_{1\leq i<j\leq
N}\mathcal{H}^{n-1}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}\cap\Omega_{\rho})=\sigma\mathcal{H}^{n-1}(S(u)\cap\Omega_{\rho})$

where $S(u)$ is the interface of $u$ separating the phases.

Finally we define

(3.4)
$\tilde{J}_{0}(u,\Omega_{\rho})=\begin{cases}J_{0}(u,\Omega_{\rho})\;\>,\;\textrm{if}\;\>u\in
BV(\Omega_{\rho};\\{W=0\\})\;\>\textrm{and}\;u=g_{0}\;\>\textrm{on}\;\>\mathbb{R}^{n}\setminus\Omega_{\rho}\\\
+\infty\;\;\;\;\;\;\;\;\;,\;\;\textrm{otherwise}\end{cases}$

We can write
$J_{\varepsilon},J_{0},\tilde{J}_{\varepsilon},\tilde{J}_{0}:L^{1}(\Omega_{\rho};\mathbb{R}^{n})\rightarrow\overline{\mathbb{R}}$,
where $\overline{\mathbb{R}}=\mathbb{R}\cup\\{\infty\\}$ and the
$\Gamma$-convergence will be in respect to the $L^{1}$ topology.

In [4] it has been proved that $J_{\varepsilon}\;\>\Gamma-$converges to
$J_{0}$ with mass constraint, but it also holds without mass constraint. We
will point out this more clearly in the proof of Theorem 3.1 below, in which
we are going to prove that
$\tilde{J}_{\varepsilon}(u,\Omega)\;\>\Gamma-$converges to
$\tilde{J}_{0}(u,\overline{\Omega})$. Also, in [3, Theorem 3.7] there is a
$\Gamma$-convergence result with boundary conditions in the scalar case which
we utilize for proving Theorem 3.1. In other words, in Theorem 3.1 below, we
prove that provided $J_{\varepsilon}\;\>\Gamma$-converges to $J_{0}$, then we
establish the $\Gamma$-limit of $\tilde{J}_{\varepsilon}$, that is, the
functional $J_{\varepsilon}$ with the constraint of Dirichlet values.

######  Theorem 3.1.

Let $J_{\varepsilon}$ be defined by (1.1) and
$\tilde{J}_{\varepsilon}\;,\>\tilde{J}_{0}$ defined in (3.1) and (3.4)
respectively. Then

(3.5) $\Gamma-\lim_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u,\Omega)=\tilde{J}_{0}(u,\overline{\Omega})$

where we extend $u$ by setting $u=g_{0}$ on $\mathbb{R}^{2}\setminus\Omega.\\\
$

######  Remark 3.2.

Note that the domain of $\tilde{J}_{0}$ is the closure of $\Omega$, which
means that there is a boundary term (see also (2.9) in [16] for the analog in
the scalar case). More precisely, by Proposition 2.4 above and Theorem 5.8 in
[9] we can write

(3.6)
$\begin{gathered}\tilde{J}_{0}(u,\overline{\Omega})=\frac{1}{2}\sum_{i=1}^{N}\int_{\overline{\Omega}}|D(\phi_{i}\circ
u)|\\\ =\frac{1}{2}\sum_{i=1}^{N}\int_{\Omega}|D(\phi_{i}\circ
u)|+\frac{1}{2}\sum_{i=1}^{N}\int_{\partial\Omega}|T(\phi_{i}\circ
u)-T(\phi_{i}\circ g_{0})|\>d\mathcal{H}^{n-1}\\\
\textrm{where}\;\>T\;\>\textrm{is the trace operator
for}\;\>BV\;\>\textrm{functions}.\end{gathered}$

###### Proof.

We begin by proving the $\Gamma-\liminf$ inequality.

Let $u_{\varepsilon}\in L^{1}(\Omega;\mathbb{R}^{m})$ such that
$u_{\varepsilon}\rightarrow u$ in $L^{1}(\Omega;\mathbb{R}^{m})$. If
$u_{\varepsilon}\notin H^{1}_{loc}$ or $u_{\varepsilon}\neq g_{\varepsilon}$
on $\mathbb{R}^{m}\setminus\Omega$, then
$\tilde{J}_{\varepsilon}(u_{\varepsilon},\Omega)=+\infty$ and the liminf
inequality holds trivially. So, let $u_{\varepsilon}\in
H^{1}_{loc}(\Omega;\mathbb{R}^{m})$ such that $u_{\varepsilon}\rightarrow u$
in $L^{1}$ and $u_{\varepsilon}=g_{\varepsilon}$ on
$\mathbb{R}^{m}\setminus\Omega$.

Let $\rho>1$, we have

(3.7)
$\tilde{J}_{\varepsilon}(u_{\varepsilon},\Omega)=J_{\varepsilon}(u_{\varepsilon},\Omega_{\rho})-J_{\varepsilon}(g_{\varepsilon},\Omega_{\rho}\setminus\Omega)$

and

(3.8) $J_{\varepsilon}(g_{\varepsilon},\Omega_{\rho}\setminus\Omega)\leq
c(\rho^{n-1}-1)\int_{-\infty}^{+\infty}\frac{1}{2}|U^{\prime}|^{2}+W(U)=O(\rho^{n-1}-1)$

(by (H2)(ii) and (2.1))

Hence, by (3.7), for every $u_{\varepsilon}$ converging to $u$ in $L^{1}$ such
that $u_{\varepsilon}=g_{\varepsilon}$ on $\mathbb{R}^{m}\setminus\Omega$ and
$\liminf_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u_{\varepsilon},\Omega)<+\infty$, we have that

(3.9) $\liminf_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u_{\varepsilon},\Omega)\geq\liminf_{\varepsilon\rightarrow
0}J_{\varepsilon}(u_{\varepsilon},\Omega_{\rho})-O(\rho^{n-1}-1)$

Also, by the liminf inequality for $J_{\varepsilon}$ (see Theorem 2.5 in [4]),
we can obtain

(3.10) $\liminf_{\varepsilon\rightarrow
0}J_{\varepsilon}(u_{\varepsilon},\Omega_{\rho})\geq\bigvee_{k=1}^{N}\int_{\Omega_{\rho}}|D(\phi_{k}\circ
u)|=J_{0}(u,\Omega_{\rho})$

Thus, by (3.9) and (3.10), passing the limit as $\rho$ tends to $1$ we have
the liminf inequality

(3.11) $\liminf_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u_{\varepsilon},\Omega)\geq
J_{0}(u,\overline{\Omega})$

We now prove the $\Gamma-$limsup inequality. Let $u\in
BV(\Omega;\\{a_{1},a_{2},...,a_{N}\\})$ be such that $u=g_{0}$ on
$\mathbb{R}^{m}\setminus\Omega.\\\ $ a) We first assume that $u=g_{0}$ on
$\mathbb{R}^{m}\setminus\Omega_{\rho_{1}}$ with $\rho_{1}<1$.

As we observe in the proof of Theorem 2.5 in [4] (see in particular the proof
of Lemma 3.1), the $\Gamma$-limsup inequality for $J_{\varepsilon}$ also holds
without the mass constraint. Also, since the $\Gamma$-liminf inequality holds,
the $\Gamma$-limsup inequality is equivalent with

(3.12) $J_{0}(u,\Omega)=\lim_{\varepsilon\rightarrow
0}J_{\varepsilon}(u_{\varepsilon},\Omega)$

for some sequence $u_{\varepsilon}$ converging to $u$ in
$L^{1}(\Omega_{\rho_{1}};\mathbb{R}^{m})$. So let $u_{\varepsilon}$ be a
sequence converging to $u$ in $L^{1}(\Omega_{\rho_{1}};\mathbb{R}^{m})$ such
that (3.12) is satisfied. In particular $u_{\varepsilon}$ converges to $g_{0}$
on $\mathbb{R}^{m}\setminus\Omega_{\rho_{1}}$.

Now, utilizing the sequence $u_{\varepsilon}$ obtained from (3.12), we will
modify it by a cut-off function so that the boundary condition is satisfied.
Let $\phi_{\varepsilon}$ be a cut-off function between
$U=\Omega_{\frac{1+\rho_{1}}{2}}$ and $U^{\prime}=\Omega$ and let
$V=\Omega\setminus\overline{\Omega}_{\rho_{1}}$. By Lemma 3.2 in [3] 333Lemma
3.2 in [3] can be extended in the vector case with minor modifications., we
have

(3.13) $\displaystyle
J_{\varepsilon}(u_{\varepsilon}\phi_{\varepsilon}+(1-\phi_{\varepsilon})g_{\varepsilon},\Omega)\leq
J_{\varepsilon}(u_{\varepsilon},\Omega)+J_{\varepsilon}(g_{\varepsilon},V)+\delta_{\varepsilon}(u_{\varepsilon},g_{\varepsilon},U,U^{\prime},V)$

(where $g_{\varepsilon}$ is extended in $V$ trivially).

By the assumptions on $u_{\varepsilon}$ and (H2)(ii) we also have

$\displaystyle u_{\varepsilon}\rightarrow
g_{0}\;,\;\;\;\;\;g_{\varepsilon}\rightarrow
g_{0}\;\;\;\textrm{in}\;\>L^{1}(V).$

Hence we get (again by Lemma 3.2 in [3])444The condition
$\sup_{\varepsilon>0}(J_{\varepsilon}(u_{\varepsilon},U^{\prime})+J_{\varepsilon}(g_{\varepsilon},V))<+\infty$
in Lemma 3.2 in [3] holds from Lemma 2.2 and (H2).

$\displaystyle\lim_{\varepsilon\rightarrow
0}\delta_{\varepsilon}(u_{\varepsilon},g_{\varepsilon},U,U^{\prime},V)=0$

and by (3.7), (3.8) and (3.13)

$\displaystyle\Gamma-\limsup_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(\tilde{u}_{\varepsilon},\Omega)\leq\tilde{J}_{0}(u,\Omega)$

where
$\tilde{u}_{\varepsilon}=u_{\varepsilon}\phi_{\varepsilon}+(1-\phi_{\varepsilon})g_{\varepsilon}$
and $\tilde{u}_{\varepsilon}=g_{\varepsilon}$ in
$\mathbb{R}^{n}\setminus\Omega.\\\ \\\ $ b) In the general case we consider
$\rho_{1}<1$ and we define $u_{\rho_{1}}(x)=u(\frac{1}{\rho_{1}}x)$. By the
previous case (a) and (3.3)

(3.17) $\displaystyle\begin{gathered}\Gamma-\limsup_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u_{\rho_{1}},\Omega)\leq\tilde{J}_{0}(u_{\rho_{1}},\Omega)=\sigma\mathcal{H}^{n-1}(S(u_{\rho_{1}})\cap\Omega)\\\
\leq\sigma\mathcal{H}^{n-1}(S(u)\cap\overline{\Omega})+O(1-\rho_{1}^{n-1})\\\
=\tilde{J}_{0}(u,\overline{\Omega})+O(1-\rho_{1}^{n-1})\end{gathered}$

Since $u_{\rho_{1}}$ converges to $u$ as $\rho_{1}$ tends to $1$, if we denote

$\displaystyle
J^{\prime}(u_{\rho_{1}},\Omega):=\Gamma-\limsup_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u_{\rho_{1}},\Omega)$

then by the lower semicontinuity of the $\Gamma-$upper limit (see e.g.
Proposition 1.28 in [5]) and (3.17)

(3.18) $\Gamma-\limsup_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u_{\rho_{1}},\Omega)\leq\liminf_{\rho_{1}\rightarrow
1}J^{\prime}(u_{\rho_{1}},\Omega)\leq\tilde{J}_{0}(u,\overline{\Omega})$

Hence by (3.11) and (3.18) we get the required equality (3.5).

∎

##  4\. Minimizing partitions and the structure of the minimizer

In this section we begin with the basic definitions of minimizing partitions.
Then we underline the relationship of minimizing partitions in
$\mathbb{R}^{2}$ with the minimizers of the functional $\tilde{J}_{0}$ and by
imposing the appropriate Dirichlet conditions, we analyze the structure of the
minimizer of $\tilde{J}_{0}$ that we obtain from the $\Gamma$-limit. Utilizing
a Bernstein type theorem for minimizing partitions we can explicitly compute
the energy of the minimizer in Proposition 4.4 and by regularity results in
[14] we can determine the precise structure of a minimizer subject to the
particulary boundary conditions in Theorem 4.6. In the last subsection we note
that we can extend these results to the mass constraint case. Finally, in
subsection 4.2 we make some comments for the limiting minimizers in dimension
three.

Let $\Omega\subset\mathbb{R}^{n}$ open, occupied by $N$ phases. Associated to
each pair of phases $i$ and $j$ there is a surface energy density
$\sigma_{ij}$, with $\sigma_{ij}>0$ for $i\neq j$ and
$\sigma_{ij}=\sigma_{ji}$, with $\sigma_{ii}=0$. Hence, if $A_{i}$ denoted the
subset of $\Omega$ occupied by phase $i$, then $\Omega$ is the disjoint union

$\displaystyle\Omega=A_{1}\cup A_{2}\cup...\cup A_{N}$

and the energy of the partition $A=\\{A_{i}\\}_{i=1}^{N}$ is

(4.1) $\displaystyle E(A)=\sum_{1\leq i<j\leq
N}\sigma_{ij}\mathcal{H}^{n-1}(\partial A_{i}\cap\partial A_{j})$

where $\mathcal{H}^{n-1}$ is the $(n-1)$-Hausdorff measure in
$\mathbb{R}^{n}$. If $\Omega$ is unbounded, for example
$\Omega=\mathbb{R}^{n}$ (we say then that $A$ is complete), the quantity above
in general will be infinity. Thus, for each $W$ open, with
$W\subset\subset\Omega$, we consider the energy

(4.2) $E(A;W)=\sum_{0<i<j\leq N}\sigma_{ij}\mathcal{H}^{n-1}(\partial
A_{i}\cap\partial A_{j}\cap W)$

######  Definition 4.1.

The partition $A$ is a minimizing $N$-partition if given any
$W\subset\subset\Omega$ and any $N$-partition $A^{\prime}$ of $\Omega$ with

(4.3) $\bigcup_{i=1}^{N}(A_{i}\triangle A_{i}^{\prime})\subset\subset W$

we have

$\displaystyle E(A;W)\leq E(A^{\prime};W)$

the symmetric difference $A_{i}\triangle A_{i}^{\prime}$ is defined as their
union minus their intersection, that is, $A_{i}\triangle
A_{i}^{\prime}=(A_{i}\cup A_{i}^{\prime})\setminus(A_{i}\cap
A_{i}^{\prime}).\\\ $

To formulate the Dirichlet problem, given a partition $C$ of $\partial\Omega$
up to a set of $\mathcal{H}^{n-1}$-measure zero, we may prescribe the boundary
data for $A$:

$\displaystyle(\partial_{\Omega}A)_{i}=\partial
A_{i}\cap\partial\Omega=C_{i}\;,\;\;\;\;i=1,...,N.$

Now the energy is minimized subject to such a prescribed boundary.

######  Remark 4.2.

Note that the minimization of the functional $\tilde{J}_{0}(u,\Omega)$ is
equivalent to minimizing the energy $E(A;\Omega)$ under the appropriate
Dirichlet conditions.

$1$$2$$3$$\theta_{1}$$\theta_{2}$$\theta_{3}$Figure
2.$\hat{\theta}_{1}$$\hat{\theta}_{2}$$\hat{\theta}_{3}$$\sigma_{1}{}_{2}$$\sigma_{1}{}_{3}$$\sigma_{2}{}_{3}$

We now state a well known Bernstein-type theorem in $\mathbb{R}^{2}.\\\ \\\ $
Theorem: Let $A$ be a complete minimizing partition in $\mathbb{R}^{2}$ with
$N=3$ (three phases), with surface tension coefficients satisfying

(4.4) $\sigma_{ik}<\sigma_{ij}+\sigma_{jk}\;\;\;,\;\textrm{for}\;\>j\neq
i,k\;\>\textrm{with}\;\>i,j,k\in\\{1,2,3\\}.$

Then $\partial A$ is a triod.

For a proof and related material we refer to [20] and the expository [2].

In Figure 2 we show a triod with angles $\theta_{1},\theta_{2},\theta_{3}$,
and the corresponding triangle with their supplementary angles
$\hat{\theta}_{i}=\pi-\theta_{i}$ . For these angles Young’s law holds, that
is,

(4.5)
$\frac{\textrm{sin}\hat{\theta}_{1}}{\sigma_{23}}=\frac{\textrm{sin}\hat{\theta}_{2}}{\sigma_{13}}=\frac{\textrm{sin}\hat{\theta}_{3}}{\sigma_{12}}$

In our case we have $\sigma_{ij}=\sigma>0$ for $i\neq j$, therefore we have by
Young’s law $\theta_{i}=\dfrac{2\pi}{3}\;\>,\;i=1,2,3$. As a result of Theorem
2 above, we expect that, by imposing the appropriate boundary conditions, the
minimizer $u_{0}$ of
$\tilde{J}_{0}(u,\overline{B}_{1})\;,\;B_{1}\subset\mathbb{R}^{2}$ which we
obtain from the $\Gamma$-limit will be a triod with angles $\dfrac{2\pi}{3}$
restricted in $B_{1}$ and centered at a point $x\in B_{1}.\\\ $

We now recall Steiner’s problem that gives us some geometric intuition about
this fact.

Let us take three points $A,\>B$ and $C$, arranged in any way in the plane.
The problem is to find a fourth point $P$ such that the sum of distances from
$P$ to the other three points is a minimum; that is we require $AP+BP+CP$ to
be a minimum length.

If the triangle $ABC$ possesses internal angles which are all less than
$120^{o}$, then $P$ is the point such that each side of the triangle, i.e.
$AB,\>BC$ and $CA$, subtends an angle of $120^{o}$ at $P$. However, if one
angle, say $A\hat{C}B$, is greater than $120^{o}$, then $P$ must coincide with
$C.$

The Steiner’s problem is a special case of the Geometric median problem and
has a unique solution whenever the points are not collinear.

###  4.1. The structure of the minimizer in the disk

In order to obtain precise information about the minimizer of the limiting
functional $\tilde{J}_{0}(u,\overline{B}_{1})\>,\;B_{1}\subset\mathbb{R}^{2}$,
we impose some particular boundary conditions that “fit well” with the
geometric intuition that we have from the minimizing partitions. We will also
prove that the minimizer is unique.

So, we assume the following, (H2) (iii)
$g^{\varepsilon}(1,\theta)=a_{1}\;,\;\>\theta\in(C\varepsilon,\frac{2\pi}{3}-C\varepsilon)\;,\;g^{\varepsilon}(1,\theta)=a_{2}\;,\>\theta\in(\frac{2\pi}{3}+C\varepsilon,\frac{4\pi}{3}-C\varepsilon)$
and
$g^{\varepsilon}(1,\theta)=a_{3}\;,\;\theta\in(\frac{4\pi}{3}+C\varepsilon,2\pi-C\varepsilon)$
(in polar coordinates) and connecting $a_{1}$ with $a_{2}$ in
$(\frac{2\pi}{3}-C\varepsilon,\frac{2\pi}{3}+C\varepsilon)$ and similarly in
the remaining intervals.

We see that $g_{\varepsilon}\rightarrow g_{0}$ in $L_{1}$, where
$g_{0}(\theta)=a_{1}\chi_{(0,\frac{2\pi}{3})}+a_{2}\chi_{(\frac{2\pi}{3},\frac{4\pi}{3})}+a_{3}\chi_{(\frac{4\pi}{3},2\pi)}\;,\;\theta\in(0,2\pi).\\\
$

######  Remark 4.3.

For the mass constraint case, by classical results of Almgren’s improved and
simplified in White [19] for minimizing partitions with surface tension
coefficients $\sigma_{ij}$ satisfying the strict triangle inequality (see
(4.4)), $\Omega_{j}$ can be taken open with $\partial\Omega_{j}$ real analytic
except possibly for a singular part with Hausdorff dimension at most $n-2$.
Therefore
$\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}=\partial\Omega_{i}\cap\partial\Omega_{j}\;,\>\mathcal{H}^{n-1}$-a.e.,
where $u_{0}=\sum_{i=1}^{N}a_{i}\chi_{\Omega_{i}}$ is the minimizer of $J_{0}$
with a mass constraint. Also, Morgan in [15] has proved regularity of
minimizing partitions in the plane subject to mass constraint. However, we
deal with the problem with boundary conditions, so we cannot apply these
regularity results directly.

The problem of minimizing partitions subject to boundary conditions, in
contrast to the mass constraint case, might not always admit a minimum, we
provide two examples if Figure 3 below.

$A$$B$$C$$D$$a_{1}$$a_{2}$$g_{0}=a_{1}$$a_{1}$$\Omega$(a)$(b)$Figure
3.$A$$B$$\Omega^{\prime}$$g_{0}=a_{1}$$a_{2}$

However a minimizer will exist for the minimization problem $\min_{u\in
BV(\Omega;\\{W=0\\})}\tilde{J}_{0}(u,\overline{\Omega})$, for instance the one
we obtain from the $\Gamma$-limit, which will form a “boundary layer” in the
boundary of the domain instead of internal layer (i.e. the interface
separating the phases). Particularly, in both (a) and (b) in Figure 3 above,
$u_{0}=a_{1}$, a.e. will be a minimizer of $\tilde{J}_{0}$ and

$\displaystyle\tilde{J}_{0}(u_{0},\overline{\Omega})=\frac{1}{2}\sum_{i=1}^{3}\int_{\partial\Omega}|T(\phi_{i}\circ
u_{0})-T(\phi_{i}\circ
g_{0})|d\mathcal{H}^{1}=\sigma|AB|=\tilde{J}_{0}(u_{0},\overline{\Omega^{\prime}})$

When there are no line segments in the boundary of the domain or when $g_{0}$
does not admit jumps nearby such line segments, then we expect that there are
no boundary layers and the boundary term in the energy of $\tilde{J}_{0}$
vanishes (see Remark 3.2), otherwise we could find a minimizer with strictly
less energy. In the cases where the boundary term vanishes we can write
$\tilde{J}_{0}(u_{0},\overline{\Omega})=\tilde{J}_{0}(u_{0},\Omega)$. This can
be proved rigorously in the case where $\Omega=B_{1}$ and assuming (H2)(iii),
utilizing also Proposition 3.2 in [14] as we will see in Theorem 4.6.

######  Proposition 4.4.

Let $(u_{\varepsilon})$ be a minimizing sequence of
$\tilde{J}_{\varepsilon}(u,B_{1})$. Then $u_{\varepsilon}\rightarrow u_{0}$ in
$L^{1}$ along subsequence with $u_{0}\in BV(B_{1};\\{a_{1},a_{2},a_{3}\\})$
and $u_{0}$ is a minimizer of $\tilde{J}_{0}(u,\overline{B}_{1})$ subject to
the limiting Dirichlet values (H2)(iii), where we extend $u$ by setting
$u=g_{0}$ on $\mathbb{R}^{2}\setminus B_{1}$.

In addition, we have

(4.6) $\displaystyle\sum_{1\leq i<j\leq
3}\mathcal{H}^{1}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}\cap\overline{B}_{1})=3$

where
$u_{0}=a_{1}\chi_{\Omega_{1}}+a_{2}\chi_{\Omega_{2}}+a_{3}\chi_{\Omega_{3}}.$

###### Proof.

From Lemma 2.2, 2.3 it holds that if $u_{\varepsilon}$ is a minimizing
sequence for $\tilde{J}_{\varepsilon}(u,B_{1})$, then
$\tilde{J}_{\varepsilon}(u_{\varepsilon},B_{1})\leq C$ and thus
$u_{\varepsilon}\rightarrow u_{0}$ in $L^{1}$ along subsequence. The fact that
$u_{0}$ is a minimizer of $\tilde{J}_{0}$ is a standard fact from the theory
of $\Gamma-$convergence. It can be seen as follows.

Let $w\in BV(\overline{B_{1}},\\{a_{1},a_{2},a_{3}\\})$ such that $w=g_{0}$ on
$\mathbb{R}^{2}\setminus B_{1}$, then from the limsup inequality in Theorem
3.1, we have that there exists $w_{\varepsilon}\in
H^{1}_{loc}(\mathbb{R}^{2};\mathbb{R}^{m})\;\>,\;w_{\varepsilon}=g_{\varepsilon}$
on $\mathbb{R}^{2}\setminus B_{1}$ such that $w_{\varepsilon}\rightarrow w$ in
$L^{1}$ and $\limsup_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(w_{\varepsilon},B_{1})\leq\tilde{J}_{0}(w,\overline{B}_{1})$.
Now since $u_{\varepsilon}$ is a minimizing sequence for
$\tilde{J}_{\varepsilon}(u,B_{1})$ and from the liminf inequality in Theorem
3.1, we have

(4.7)
$\begin{gathered}\tilde{J}_{0}(u_{0},\overline{B}_{1})\leq\liminf_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(u_{\varepsilon},B_{1})\leq\liminf_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(w_{\varepsilon},B_{1})\\\
\leq\limsup_{\varepsilon\rightarrow
0}\tilde{J}_{\varepsilon}(w_{\varepsilon},B_{1})\leq\tilde{J}_{0}(w,\overline{B}_{1})\end{gathered}$

For proving (4.6), we utilize Theorem 2 above (i.e. Theorem 2 in [2]). Since
the triod is a minimizing 3-partition in $\mathbb{R}^{2}$ we have that for any
$W\subset\subset\mathbb{R}^{2}$ and any partition it holds that $E(A,W)\leq
E(V,W)$, where suppose that $A=\\{A_{1},A_{2},A_{3}\\}$ is the partition of
the triod and $V=\\{V_{1},V_{2},V_{3}\\}$ is a 3-partition in
$\mathbb{R}^{2}.$

We have
$u_{0}=a_{1}\chi_{\Omega_{1}}+a_{2}\chi_{\Omega_{2}}+a_{3}\chi_{\Omega_{3}}$
such that $u_{0}=g_{0}$ on $\partial B_{1}$ and extend $u_{0}$ in
$\mathbb{R}^{2}$, being the triod with $\theta_{i}=\dfrac{2\pi}{3}$ in
$\mathbb{R}^{2}\setminus B_{1}$. This defines a 3-partition in
$\mathbb{R}^{2}$, noted as
$\tilde{\Omega}=\\{\tilde{\Omega}_{i}\\}_{i=1}^{3}$. Since the triod is a
minimizing 3-partition in the plane, we take any
$W\subset\subset\mathbb{R}^{2}$ such that $B_{2}\subset\subset W$ and
$\bigcup_{i=1}^{3}(A_{i}\triangle\tilde{\Omega}_{i})\subset\subset W$, so we
have

(4.8) $E(A,W)=E(A,\overline{B}_{1})+E(A,W\setminus\overline{B}_{1})\leq
E(\tilde{\Omega},W)=E(\tilde{\Omega},\overline{B}_{1})+E(\tilde{\Omega},W\setminus\overline{B}_{1})$

where $A$ is the partition of the triod.

Now since
$E(A,W\setminus\overline{B}_{1})=E(\tilde{\Omega},W\setminus\overline{B}_{1})$
(by the way we extended $u_{0}$ in $\mathbb{R}^{2}$) and $\\\
E(A,\overline{B}_{1})=\sigma\sum_{1\leq i<j\leq 3}\mathcal{H}^{1}(\partial
A_{i}\cap\partial A_{j}\cap\overline{B}_{1})=3\sigma$ (since $\partial
A_{i}\cap\partial A_{j}\cap\overline{B}_{1}$ are radi of $B_{1}$), we conclude

(4.9) $\begin{gathered}3\sigma\leq
E(\tilde{\Omega},\overline{B}_{1})=\tilde{J}_{0}(u_{0},\overline{B}_{1})\\\
\Leftrightarrow 3\leq\sum_{1\leq i<j\leq
3}\mathcal{H}^{1}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}\cap\overline{B}_{1})\end{gathered}$

The inequality $\sum_{1\leq i<j\leq
3}\mathcal{H}^{1}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}\cap\overline{B}_{1})\leq
3$ can be obtained by the minimality of $u_{0}$ in comparison with the
partition of the triod in $B_{1}$ (in particular, consider as a test function
$\tilde{u}=a_{1}\chi_{A_{1}}+a_{2}\chi_{A_{2}}+a_{3}\chi_{A_{3}}).\\\ $ ∎

######  Remark 4.5.

Arguing similarly as in Proposition 4.4 above, we can obtain that for every
ball $B_{R}$, the energy of the limiting minimizer will be
$\tilde{J}_{0}(u_{0},\overline{B}_{R})=3\sigma R$, for every $R>0$, thus we
can obtain an entire minimizer in the plane (by a diagonal argument) and the
partition that it defines will be a minimal cone, since
$\dfrac{\mathcal{H}^{1}(\partial\Omega_{i}\cap\partial\Omega_{j}\cap
B_{R})}{\omega_{1}R}=C_{0}$ (see [20]).

Finally, we will prove that the minimizer of $\tilde{J}_{0}$ in
$\overline{B}_{1}$ is unique, that is, the only minimizer is the triod
restricted to $B_{1}$ centered at the origin. For this result, we will need
some regularity results from [14].

######  Theorem 4.6.

Let
$u_{0}=a_{1}\chi_{\Omega_{1}}+a_{2}\chi_{\Omega_{2}}+a_{3}\chi_{\Omega_{3}}$
be a minimizer of $\tilde{J}_{0}(u,\overline{B}_{1})$ subject to the limiting
Dirichlet values (H2)(iii). Then $\partial\Omega_{i}\cap\partial\Omega_{j}$
are radi of $B_{1}\;,\>|\Omega_{i}|=\frac{1}{3}|B_{1}|\;\>(i\neq j)$ and the
minimizer is unique (as in Figure 4 below).

$A$$B$$C$$\Omega_{1}$$\Omega_{2}$$\Omega_{3}$$g_{0}=a_{1}$$g_{0}=a_{2}$$g_{0}=a_{3}$Figure
4.$\frac{2\pi}{3}$

###### Proof.

Firstly, we show that the minimizing partition of $B_{1}$ with respect to the
boundary conditions defined from $g_{0}$, is a $(M,0,\delta)$-minimal for
$\delta>0$ and therefore $(M,cr^{\alpha},\delta)$-minimal (see Definition 2.1
in [14]). If not, let $S$ be the partition defined from $u_{0}$, we can find a
Lipchitz function $\phi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ such that

$\displaystyle\mathcal{H}^{1}(S\cap W)>\mathcal{H}^{1}(\phi(S\cap W))$

with $W=\mathbb{R}^{2}\cap\\{x:\phi(x)\neq x\\}$, diam$(W\cup\phi(W))<\delta$
and dist$(W\cup\phi(W),\mathbb{R}^{2}\setminus B_{1})>0$.

So if we consider the partition

$\displaystyle\tilde{S}=\begin{cases}S\;\;,\;S\cap W=\emptyset\\\ \phi(S\cap
W)\;\;,\;S\cap W\neq\emptyset\end{cases}$

then the boundary of the partition defined by $\tilde{S}$ will satisfy the
boundary conditions (since dist$(W\cup\phi(W),\mathbb{R}^{2}\setminus
B_{1})>0$) and also $\mathcal{H}^{1}(\tilde{S})<\mathcal{H}^{1}(S)$ which
contradicts the minimality of $S.$

Thus, by Proposition 3.2 in [14], we have that the unique smallest
$(M,cr^{\alpha},\delta)$-minimal set consists of three line segments from the
three vertices defined from $g_{0}$ (i.e. the jump points in $\partial B_{1}$)
meeting at $\frac{2\pi}{3}$. The meeting point is unique and it is the origin
of $B_{1}$. Thus, the line segments
$\partial\Omega_{i}\cap\partial\Omega_{j}=\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}$
(since the boundary of the partition is piecewise smooth) are radi of $B_{1}$
and $|\Omega_{i}|=\frac{1}{3}|B_{1}|$. ∎

######  Remark 4.7.

If we modify the hypothesis (H2)(iii) in the Dirichlet conditions such that
the limit of $g_{\varepsilon}$ becomes
$g_{0}(\theta)=\sum_{i=1}^{3}a_{i}\>\chi_{I_{i}}(\theta)\;,\;\theta\in[0,2\pi)\;,\;I_{i}\subset[0,2\pi)$
connected and $\cup_{i=1}^{3}I_{i}=[0,2\pi)$, then we have an analogous
structure and uniqueness of the minimizer as illustrated in the Figure 5
below. The proof of Proposition 4.4 and Theorem 4.6 will be similar, the only
difference is that the center of the triod will be a different point in
$B_{1}$ than the origin such that the angle will remain $\dfrac{2\pi}{3}$, if
the largest angle of the triangle defined by the points $\partial
I_{i}\cap\partial I_{j}$ is less than $\dfrac{2\pi}{3}$ (otherwise see (a) in
Proposition 3.2 in [14]). Also, the sum of the interfaces in (4.6) will be
different than three, depending on the particular points that appear in the
limiting boundary conditions (i.e. $\partial I_{i}\cap\partial I_{j}$).

$A$$B$$C$$\frac{2\pi}{3}$$\frac{2\pi}{3}$$g_{0}=a_{1}$$g_{0}=a_{2}$$g_{0}=a_{3}$Figure
5.

###  4.2. Minimizers in dimension three

In this subsection we will briefly make some comments for the structure of
minimizers in $\mathbb{R}^{3}$. If we impose the appropriate boundary
conditions in $B_{R}\subset\mathbb{R}^{3}$ and
$\\{W=0\\}=\\{a_{1},a_{2},a_{3}\\}\;,\>g_{\varepsilon}\rightarrow
g_{0}\;\>\textrm{in}\;\>L^{1}(B_{R};\mathbb{R}^{3})$ such that the partition
in $\partial B_{R}$ defined by $g_{0}$ is equal to the partition of
$(C_{tr}\times\mathbb{R})\cap\partial B_{R}$, where $C_{tr}$ is the triod as
in Figure 2 (with equal angles), then by Theorem 3 in [2], arguing as in
Proposition 4.4 (see also Remark 4.5), we can obtain

$\displaystyle\tilde{J}_{0}(u,B_{R})=\frac{3}{2}\sigma\pi R^{2}$

which gives

$\displaystyle\frac{\mathcal{H}^{2}(\partial\Omega_{i}\cap\partial\Omega_{j}\cap
B_{R})}{\omega_{2}R^{2}}=\frac{3}{2}$

where $\omega_{2}$ is the volume of the 2-dimensional unit ball (see [20]).
That is, the partition that the minimizer defines can be extended to a minimal
cone in $\mathbb{R}^{3}$. Now since the only minimizing minimal cones are the
triod and the tetrahedral cone (see [18]), then the minimizer of
$\tilde{J}_{0}$ is such that $u_{0}=\sum_{i=1}^{3}a_{i}\chi_{\Omega_{i}}$,
where $\Omega=\\{\Omega_{i}\\}_{i=1}^{3}$ is the partition of
$(C_{tr}\times\mathbb{R})\cap B_{R}$.

Similarly, if $\\{W=0\\}=\\{a_{1},a_{2},a_{3},a_{4}\\}$ and we impose the
Dirichlet conditions such that $g_{0}$ defines the partition of the
tetrahedral cone intersection with $\partial B_{R}$, then again
$u_{0}=\sum_{i=1}^{4}a_{i}\chi_{\Omega_{i}}$, where
$\Omega=\\{\Omega_{i}\\}_{i=1}^{4}$ is the partition of the tetrahedral cone
restricted in $B_{R}$.

###  4.3. Minimizers in the disc for the mass constraint case

In this last subsection, we note that the result in Theorem 4.6 above can be
extended also to the mass constraint case (see [4]). However, in this case the
uniqueness will be up to rigid motions of the disc (see Theorem 3.6 and
Theorem 4.1 in [6]).

Let $u_{0}$ be a minimizer of $J_{0}(u,B_{1})\;,\>B_{1}\subset\mathbb{R}^{2}$
defined in (3.3) subject to the mass constraint $\int_{B_{1}}udx=m$ (or
consider the minimizer $u_{0}$ of Theorem p.70 in [4]) and
$\\{W=0\\}=\\{a_{1},a_{2},a_{3}\\}$. Then
$u_{0}=\sum_{i=1}^{3}a_{i}\chi_{\Omega_{i}}$, where
$\Omega_{1},\Omega_{2},\Omega_{3}$ is a partition of $B_{1}$ which minimizes
the quantity

$\displaystyle\sum_{1\leq i<j\leq
3}\sigma\mathcal{H}^{1}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j})$

among all other partitions of $B_{1}$ such that
$\sum_{i=1}^{3}|\Omega_{i}|a_{i}=m.$

If $a_{i}\;,\>a_{j}$ are linear independent for $i\neq
j\;,\>i,j\in\\{1,2,3\\}$ and $m=\frac{1}{3}|B_{1}|\sum_{i=1}^{3}a_{i}$, then
we obtain that $|\Omega_{i}|=\frac{1}{3}|B_{1}|$ (also by the fact that
$\sum_{i=1}^{3}|\Omega_{i}|=|B_{1}|$). Now by Theorem 4.1 in [6] we conclude
that the minimizer is a standard graph (that is
$\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}=\partial\Omega_{i}\cap\partial\Omega_{j}$
are line segments in this case) and unique up to a rigid motion of the disc.

This result holds more generally for some choice of $m>0$ (i.e.
$m_{1},m_{2}>0$), by Theorem 4.1 in [6] and the boundary of the partition is
consisted of three circular arcs or line segments meeting at an interior
vertex at $120$ degrees angles, reaching orthogonally $\partial B_{1}$ and so
that the sum of geodesic curvature is zero.

## References

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11institutetext:  University of Illinois, Chiacgo, USA 22institutetext:
University of Glasgow, Glasgow, UK
22email<EMAIL_ADDRESS>22email<EMAIL_ADDRESS>22email:
<EMAIL_ADDRESS>
#  Evaluating the Explainability of Neural Rankers

Saran Pandian The work was conducted at Dhirubhai Ambani Institute of
Information and Communication Technology, the author’s previous affiliation.11
Debasis Ganguly 22 Sean MacAvaney 22

###### Abstract

Information retrieval models have witnessed a paradigm shift from unsupervised
statistical approaches to feature-based supervised approaches to completely
data-driven ones that make use of the pre-training of large language models.
While the increasing complexity of the search models have been able to
demonstrate improvements in effectiveness (measured in terms of relevance of
top-retrieved results), a question worthy of a thorough inspection is - “how
explainable are these models?”, which is what this paper aims to evaluate. In
particular, we propose a common evaluation platform to systematically evaluate
the explainability of any ranking model (the explanation algorithm being
identical for all the models that are to be evaluated). In our proposed
framework, each model, in addition to returning a ranked list of documents,
also requires to return a list of explanation units or rationales for each
document. This meta-information from each document is then used to measure how
locally consistent these rationales are as an intrinsic measure of
interpretability - one that does not require manual relevance assessments.
Additionally, as an extrinsic measure, we compute how relevant these
rationales are by leveraging sub-document level relevance assessments. Our
findings show a number of interesting observations, such as sentence-level
rationales are more consistent, an increase in complexity mostly leads to less
consistent explanations, and that interpretability measures offer a
complementary dimension of evaluation of IR systems because consistency is not
well-correlated with nDCG at top ranks.

###### Keywords:

IR Evaluation, Neural Ranking Models, Explainability

##  1 Introduction

A neural ranking model (NRM) involves learning a data-driven parameterised
similarity function between queries and documents [8, 19, 26, 5, 16]. Despite
achieving state-of-the-art effectiveness (measured in terms of relevance of
search results), NRMs suffer from poor interpretability of their underlying
working mechanism due to two main reasons. First, they lack a closed form
expression involving the human interpretable fundamental components of an IR
similarity function (i.e., term frequency, inverse document frequency and
document length) [22]. Second, due to the fact that the similarity function of
an NRM operates at the level of embedded representations of documents and
queries, it is difficult to determine which terms present within a document
are mainly responsible for contributing to its retrieval status value [2]. To
the best of our knowledge, there is no work in IR research that has evaluated
the quality of explanations of NRMs in an objective manner, e.g., in terms of
consistency and correctness akin to some of the work done for the broader
class of predictive models [17, 1, 27].

#### Our Contributions.

First, we introduce a framework for an offline evaluation of the
explainability of a ranking model. The evaluation protocol requires
participating IR systems to report a list of explanation units or rationales
comprised of text snippets of arbitrary lengths in addition to a ranked list
of retrieved documents. Second, we show how these rationales can be evaluated
both in an intrinsic and extrinsic manner - the later requiring sub-document
level relevance. We conduct a range of experiments to compare the explanation
qualities across a range of different NRMs using a common explanation
methodology - that of occlusion commonly used in the literature [9, 20, 11].
An important finding of our experiments is that we demonstrate that the IR
systems that produce the most relevant results are not necessarily the most
explainable.

##  2 Related Work

In IR, there has been little effort in investigating the explanation
effectiveness of neural models. The article [25] proposes a LIME-based local
explanation methodology for IR models and also proposes a metric that measures
the overlap of explanation units with the relevant terms (bag-of-words
representation of the set of relevant documents for a query). However, the
metric proposed in this paper is different from that of [25] in that our
metric factors in intrinsic consistency of the explanations, and our proposed
extrinsic measure makes use of sub-document level relevance.

Another thread of work towards a quantitative evaluation of IR model
explanations is in the form of how well a simple linear model either comprised
of the fundamental functional components - term frequency, idf, document
length etc. fits a complex black-box model [22], or that how well it conforms
to the axioms of IR [12] yielding a notion of fidelity. An yet another
approach of IR trustworthiness evaluation involves the notion of measuring how
consistently does an IR model conform to user expectations of the similarity
of the top-retrieved documents based on information need change across query
reformulations [23].

Our evaluation framework for IR explanation effectiveness is different from
those of [22, 12] in the sense that ours does not involve evaluating how well
a simpler model fits a more complex one, and it also differs from [23] in that
our evaluation directly concerns model explanations in the form of rationales
instead of analysing a model’s behaviour across pairs of queries.

##  3 Evaluation Framework

Figure 1:  The proposed workflow for measuring effectiveness of explainable IR
models via intrinsic and extrinsic explanation effectiveness measures in
addition to relevance-based ones. The meta-information comprised of the
explanation units or rationales (shown in blue) is the additional output
either obtained from a ranking model itself or with the help of a common
explanation methodology (as is the case in our experiments).

In our proposed evaluation workflow, each IR model $M^{\theta}$, in addition
to outputting a ranked list of $k$ documents $M^{\theta}_{k}(Q)$ for a query
$Q$, also outputs a list of the most important text segments as rationales for
each retrieved document. Our evaluation is oblivious of the exact method by
which these explanations are obtained for each document - possible options
being applying a local explanation algorithm for post-hoc per-document
explanations via LIME [20, 25], SHAP [11] etc., or by making this process an
integral part of an IR model by application of feature occlusion based methods
[9, 28] to estimate the importance at sub-document level.

#### Intrinsic Explainability Measure.

Given a list of $m$ rationales for each top-ranked document $D_{i}$ ($D_{i}\in
M^{\theta}_{k}(Q)$, i.e., the top-$k$ set for a query $Q$), we obtain its
representation $D^{(i)}$ by concatenating the text from each rationale (each
being a segment of $D_{i}$ itself), i.e.,

$\phi_{m}(D)=D^{(i)}=d^{(i)}_{1}\oplus\ldots d^{(i)}_{m}.$ (1)

Next, to measure how consistent are these rationales, we use this rationale-
based representation of each document to recompute its score with the black-
box neural model. Subsequently, we rerank the top-$k$ with these modified
scores and compute the agreement of this re-ranked top-$k$ with the original
one, i.e., the ranking agreement between the explanation-based representation
of documents and their original ones. Formally speaking, we define the
consistency metric as Mean Rank Correlation (MRC) as

$\text{MRC}(\theta)=\frac{1}{|\mathcal{Q}|}\sum_{Q\in\mathcal{Q}}\sigma(M^{\theta}_{k}(Q),\hat{M}^{\theta}_{k}(Q)),$
(2)

where $\mathcal{Q}$ is a set of queries, $M^{\theta}_{k}(Q)$ denotes the
top-$k$ set obtained with the original document content,
$\hat{M}^{\theta}_{k}(Q)$ denotes the re-ranked set of documents scored with
the meta-information of explanation rationales obtained with $\phi$ (i.e., the
representation $D^{(i)}$ of Equation 1), and $\sigma$ denotes a rank
correlation measure, such as Kendall’s $\tau$.

The reason the metric MRC (Equation 2) addresses the consistency of an
explanation model can be attributed to the following argument. Consider two
neural models $\theta_{1}$ and $\theta_{2}$ both with the same underlying
explanation mechanism $\phi$, which is either a part of the model or a stand-
alone local explanation methodology such as LIME for ranking [25]. A higher
value of $\text{MRC}(\theta_{1})$ indicates that the model $\theta_{1}$
provides more faithful explanations for its observed top list of documents as
compared to $\theta_{2}$, because the model $\theta_{1}$ when presented with
only the rationales for the top documents still scores them in a relatively
similar manner thus ensuring a higher correlation with the original list. The
value being lower for $\theta_{2}$, on the other hand, indicates that the
rationales themselves do not reflect the true reason behind the relative score
computation of the model $\theta_{2}$.

#### Extrinsic Explainability Measure.

The evaluation metric MRC of Equation 2 is intrinsic in nature because it does
not rely on the availability of relevance assessments of the rationales
themselves. Consequently, while MRC is useful to find if a model is more
consistent in its explanations than another, it cannot explicitly answer if a
model’s rationales align well with a human’s perception of relevance. Assuming
that each document $D_{i}\in M^{\theta}_{k}(Q)$ is comprised of a list of
$n_{i}$ relevant passages (may be sentences or paragraphs) of the form
$R(D_{i})=\\{r^{(i)}_{1},\ldots,r^{(i)}_{n_{i}}\\}$111 If $D_{i}$ is non-
relevant, $n_{i}=0$ and $R(D_{i})=\emptyset$., the main idea now is to check
whether the rationales overlap with the relevant passages. Since the
explanation rationales are arbitrary segments of text without being restricted
to passage boundaries, a simple and effective way of measuring the degree of
overlap between the explanation units and the relevant passages is to
aggregate the matches in their content. More precisely, we compute MER (Mean
Explanation Relevance) as

$\text{MER}(\theta)=\frac{1}{|\mathcal{Q}|mk}\sum_{Q\in\mathcal{Q}}\sum_{i=1}^{k}\sum_{j=1}^{m}\max_{j^{\prime}=1}^{n_{i}}\omega(d^{(i)}_{j},r^{(i)}_{j^{\prime}}),$
(3)

where $\omega$ is a similarity function. We use the cosine-similarity as a
concrete realisation of $\omega$. Similar to MRC, a high value of the MRE
metric is preferable because it indicates that the rationales demonstrate a
substantial overlap with relevant content222Implementation of the proposed
explainability evaluation measures are available at
https://github.com/saranpandian/XAIR-evaluation-metric.

Note that in our evaluation setup, we use the same evaluation algorithm for
all IR systems, which means that our metrics do not report how effective is an
explanation model itself (akin to, e.g., the prior work of comparing the
fidelity scores of LIME with SHAP [11]); rather these metrics in our setup
capture how effectively can an IR model be explained with a specific
explanation algorithm.

##  4 Experiment Setup

###  4.1 Research Questions, Datasets and IR Models

We investigate the following research questions in our experiments.

  * •

RQ-1: What are the relative variations in the explanation consistency
(intrinsic evaluation measure - MRC) across different IR models, i.e., are
some models more _explainable_ than others?

  * •

RQ-2: Does MRC provide an aspect of system evaluation that is complementary to
that of relevance?

  * •

RQ-3: What are the relative variations in the relevance-based explanation
consistency (i.e, the extrinsic evaluation measure - MER) across different IR
models?

  * •

RQ-4: Does the extrinsic explanation evaluation measure (MER of Equation 3)
also induce a different relative ordering of IR systems as compared to
evaluating them only by relevance?

For investigating the first two research questions, we conducted experiments
on the passage and document collections of MS-MARCO [14]. As topic sets for
our experiments, we use the TREC DL’20 topic set, comprising 54 queries. The
reason we employed the two different collections - one where the retrievable
units are short passages (average $3.4$ sentences for the MS-MARCO passages),
and the other where they are much larger (average of $55.7$ sentences for MS-
MARCO documents) is to investigate the effect of document length on the
intrinsic and the extrinsic explainability measures.

Recall that computing the extrinsic explainability measure of Equation 3
requires the availability of sub-document level information. Although the MS-
MARCO document collection is constituted of the text units of the MS-MARCO
passage collection, they use different identifiers, as a result of which it is
not possible to directly use the relevance assessment data of the passage
collection as the desired sub-document level relevance information of the
document collection. To construct an approximate sub-document level
assessments of the MS-MARCO document collection, we make use of passage-level
relevance in the document corpus using a document-passage mapping. We built
this mapping by matching through the “QnA” version of the MS-MARCO
dataset333https://github.com/microsoft/MSMARCO-Question-Answering, which
provides the URLs of each passage. Documents were matched to the MS-MARCO
document ranking corpus IDs via URL, and passages were matched to the MS-MARCO
passage ranking corpus IDs through exact text matching (after correcting for
character encoding errors present in the passage corpus but not in QnA).

We assess the explainability of a variety of neural ranking models (NRMs) with
a diverse set of architectures and training regimes in this work.

  * •

BM25 [21] is a classic lexical retrieval model with a closed form functional
expression involving term frequency, inverse document frequency and document
length. We include BM25 in our experiments mainly as a point of comparison for
the explanation effectiveness results of the NRMs.

  * •

ColBERT [8] is a multi-representation, ‘late interation’ model that computes
relevance based on the sum of the maximum query token similarities in a given
document. It represents a strong late interaction model.

  * •

TCT-ColBERT [10] is a single-representation dense retrieval model that was
trained from ColBERT using distillation444We use the castorini/tct_colbert-
msmarco model checkpoint.. It represents a strong bi-encoder dense retrieval
model.

  * •

MonoT5 [15] is a cross-encoder model that is trained to predict “true” or
“false” given a prompt that includes the query and document text. We use the
‘castorini/monot5-base-msmarco’ model checkpoint.

  * •

MonoElectra [18] is a cross-encoder model based on the ELECTRA foundation
trained with hard negatives [3]555We use the crystina-z/monoELECTRA_LCE_nneg31
model checkpoint..

All the NRMs in our experiments operate with the re-ranking based setting,
i.e., they employ a sparse index (specifically BM25 to retrieve the top-1000
results) which are then re-ranked by the NRM. Following the reranking step, we
generate the meta-information in the form of rationales for the top $k=10$ and
$k=50$ top documents (more details in Section 4.2).

To obtain the top-$k$ for the document ranking task, we first segmented each
document into non-overlapping chunks of $3$ sentences, which ensures that the
content fits within the $512$ token limit of the underlying transformer models
of the NRMs. Next, the score of a document is obtained by taking an
aggregation over the scores of the individual chunk. Specifically, we used
‘max’ as an aggregation operator to compute the score of a document as
reported in [29].

###  4.2 Explanation Model Settings

As the explanation model for generating the rationales in our proposed
evaluation framework (the meta-information corresponding to each document
retrieved in the top-list), we use an occlusion-based approach [9]. The main
advantage of an occlusion-based approach (commonly as counter-factual
explanations in recommender models [24]) is its run-time efficiency in
comparison to approaches such as LIME for ranking (LIRME [25]) which fit a
linear regressor to the data of relative score changes with occlusion. We note
that the occlusion-based explanations are not tied to a specific model
architecture; they can be applied to any relevance model that operates on the
text of the query and document.

#### Explanations for Passage Retrieval.

We employ two different granularities for generating the rationales - the
first at the level of sentences, and the second at the level of word windows
of length $w$ ($w$ being a parameter). Accordingly, we sample $n$ segments
(i.e., $n$ number of sentence or word windows as per the chosen granularity,
$n$ being a parameter) from a document $D$, and then mask out the selected
segments to construct a pseudo-document $D^{\prime}$. Similar to [25], we then
compute the relative score change induced by this masking process, which as
per the local explanation principle yields the relative importance of the
masked segment. The importance weights across the $n$ segments are then
distributed uniformly. The occlusion-based weights are then accumulated for
each segment after each sampling step. Formally,

$\phi(d_{i})=\frac{1}{n}\frac{|\theta(Q,D)-\theta(Q,D-\bigcup_{j=1}^{n}d_{j})|}{\theta(Q,D)}$
(4)

where $d_{i}$ a segment of $D$ ($i=1,\ldots,n$). Finally, we output the
top-$m$ segments with the highest weights (as per the $\phi$ values computed
via Equation 4) as the rationales for retrieving $D$ for query $Q$.

#### Explanations for Document Retrieval.

The process is largely similar to that of passage retrieval with a small
number of differences as follows. Firstly, due to the large size of the
documents ($55.7$ sentences on an average), it is not possible to encode an
entire document’s representation via a transformer model during the training
or the inference phase. Due to this reason, we partition each document into
fixed length chunks, and then obtain the overall score of the document by
computing the maximum over the individual scores for each chunk [29]. As the
number of sentences defining a chunk, we use the value of $3$ (this was chosen
so that each chunk in the document collection is approximately of the same
size as the retrievable units of the MS-MARCO passage collection, the average
length being $3.4$ sentences).

Secondly, due to the large length of the documents we restrict the minimum
granularity of rationales to individual sentences. Similar to explaining
passage retrieval with $m$ word windows, for document retrieval experiments
also we report $m$ top explanation units with the highest fidelity scores
(similar to Equation 4), the explanation units being sentences for document
retrieval. We use the same parameter name - $m$ to indicate the number of
explanations units; whether this applies for a word or a sentence is to be
understood from the context. We iterate through each sentence in a chunk and
compute the relative score change induced by its occlusion, i.e.,

$\phi(d_{i})=\frac{\theta(Q,D)-\theta(Q,D-d_{i})}{\theta(Q,D)}.$ (5)

We then employ a greedy approach and select the one with the highest
$\phi(d_{i})$ value and remove it from the document $D$. We then the repeat
this step $m-1$ more times to eventually yield a total of $m$ sentences as the
rationales for an IR model $\theta$.

##  5 Explainability Evaluation

Table 1:  A comparison of the relevance and the explanation consistency of
different IR models on the MS-MARCO passage collection for top-10 and top-50
search results with both sentences and word windows employed as explanation
units. For these experiments, we set the size of the word windows to $5$ and
the number of rationales to $6$ for the word windows, and $1$ for the
sentences. The best results in terms of relevance and the explanation
effectiveness of the NRMs are bold-faced.

Explanation Effectiveness Relevance (nDCG) MRC (Sentences) MRC (Word windows)
Model top-10 top-50 top-10 top-50 top-10 top-50 BM25 0.4910 0.4889 0.4000
0.1503 0.3029 0.1232 ColBERT 0.6900 0.6400 0.4502 0.1923 0.2790 0.1058 TCT-
ColBERT 0.6900 0.6328 0.2938 0.1480 0.1926 0.0974 MonoT5 0.7133 0.6660 0.3481
0.2230 0.1844 0.1102 MonoElectra 0.7460 0.6958 0.4165 0.2262 0.2644 0.1303

#### Passage Ranking Results.

Table 1 presents the results of evaluating different IR systems on the MS-
MARCO passage collection in terms of relevance and our proposed intrinsic
explanation evaluation measure MRC. The extrinsic explanation measure - MER
cannot be computed for the MS-MARCO passage collection as there is no sub-
document level information available for these retrievable units. We now
present the following key observations from Table 1.

Relevance-based observations: As expected, all the NRMs significantly
outperform BM25 in terms of relevance-only based evaluation, which is
consistent with the findings of existing research [4].

Relevance vs. Explainability: Interestingly, the model with the best retrieval
effectiveness is not necessarily the most well-explainable model. This can be
seen from the fact that the MRC@10 values for the best performing model in
terms of relevance, i.e., MonoElectra, are not the best ones (ColBERT MRC@10
values are better). This answers RQ-2 in affirmative in the sense that the
explanation quality potentially provides an alternate way towards IR model
evaluation different from that of relevance.

NRMs with the best explanation consistencies: BM25 mostly yields more
consistent explanations in comparison to NRMs. This can be seen by comparing
the gray values under the two column groups with the corresponding non-gray
ones across the same column, e.g., BM25 achieves the highest MRC@10 of
$0.3029$ with word-window based rationales, which is higher than the best
explanation effectiveness obtained by an NRM ($0.2790$ with ColBERT).

(a) $m=1$

(b) $m=3$

(c) $m=5$

Figure 2:  Effects of varying the size of the explanation units on the
intrinsic consistency of the explanations (MRC of Equation 2) on the top-10
search results obtained with several NRMs. A comparison across the plots
reveals the the effect of the variations in the number of rationales provided
as explanations ($m$). A general observation is that a higher number of
explanations coupled with larger explanation units tend to provide more
consistent explanations.

Sentence vs. word-window rationales: Table 1 shows that the MRC values of the
sentence level rationales are higher than that of the word window based ones,
which indicates that sentence-level rationales are more consistent than word-
window based ones. This can likely be attributed to the fact that, firstly,
sentences retain more informative content of a document which is useful for
providing consistent explanations, and that secondly, transformer-based
approaches due to their underlying language models usually favour well-formed
sentences [13]. However, short word segments are more preferable from a user’s
perspective because reading them potentially requires less effort, as a result
of which existing work on text-based explanations have commonly used short
word windows as rationales [20, 11, 25, 6]. It is also observed that the
sentence rationales of NRMs are more consistent in comparison to the BM25
ones. This is due to the fact that the linguistic coherence of word sequences
in sentence-level explanations work well for the NRMs, BM25 being oblivious of
word ordering.

Explaining top-10 vs. top-50 results: From Table 1, we see that the MRC@50
values are lower than those of MRC@10, which indicates that it is easier to
explain a smaller number of search results. Again, this is expected because
documents towards the very top ranks would actually contain text segments that
are potentially relevant to the information need thus amenable to more
consistent explanations.

Table 2:  A comparison of the relevance and explanation consistency (both
intrinsic and extrinsic) of different IR models on the MS-MARCO document
collection for top-10 and top-50 search results with sentence-level
rationales. Similar to Table 1, BM25 results (shown in gray) are included as a
point for comparison with the NRM explanation effectiveness. The best results
in terms of relevance and the explanation effectiveness of the NRMs are bold-
faced. The number of sentences used as rationales ($m$) was set to $1$ for
these results.

Explanation Effectiveness Relevance (nDCG) Intrinsic (MRC) Extrinsic (MER)
Model top-10 top-50 top-10 top-50 top-10 top-50 BM25 0.5213 0.5325 0.1660
0.1798 0.2024 0.1823 ColBERT 0.5675 0.5686 0.2064 0.2801 0.1846 0.1647 TCT-
ColBERT 0.5787 0.5654 0.2420 0.2721 0.2069 0.1701 MonoT5 0.6045 0.6048 0.2133
0.3265 0.2039 0.1703 MonoElectra 0.6185 0.6051 0.2493 0.2863 0.1959 0.1739

#### Document Ranking Results.

Similar to Table 1, in Table 2 we show the results on the MS-MARCO document
ranking task. For this task, since documents are much larger than MS-MARCO
passages, we restrict the granularity of explanation units to sentences only.
In addition to the intrinsic explainability measure, Table 2 also reports
values for the extrinsic one obtained with sub-document level relevance.
Following are the key observations from Table 2.

Intrinsic explanation consistency: The results are similar to that of the
passage task (Table 1) in the sense that we note considerable variations in
the reported MRC values across IR models. Also, similar to the passage task,
we observe that the intrinsic explanation consistency of the NRMs is better
than that of BM25’s.

Extrinsic explanation consistency: It is observed from the MER columns of
Table 2 that the extrinsic evaluation effectiveness is not necessarily
correlated with the intrinsic one, e.g., the best model in terms of intrinsic
consistency (MRC) of explanations of top-10 documents is MonoElectra (similar
to the passage ranking task), whereas the best model with the extrinsic
explanation consistency is TCT-ColBERT. This indicates that the parameterised
ranking function of TCT-ColBERT puts more attention, on an average, to the
relevant pieces of text in comparison to the other NRMs.

In relation to RQ-3, we can thus comment that similar to the MRC variations,
we do observe a noticeable variations in the extrinsic explainability measure
as well indicating that some NRMs are substantially better explainable than
others in terms of the relevance of the rationales.

Explaining top-10 vs. top-50 results: Different from Table 1, in Table 2 we
observe that the intrinsic explanations for the top-50 results are better than
the top-10 ones (compare MRC@10 vs. MRC@50 across the different IR models).
This can be attributed to the fact that due to the large length of the
documents, it is still likely to find some partially relevant sentences even
in documents that are not at the very first search result page (e.g., within
top-10). An NRM by putting attention to these partially relevant sentences can
yield consistent explanations even within the top-50 set of documents.

However, the trend is reversed for the extrinsic measure MER, where we observe
that the top-10 results are better than top-50 ones, the reason being it is
more likely for the rationales of the top-10 documents to overlap with the
relevant passages.

#### Further Analysis.

For the passage ranking task, we now investigate the effects of the variations
in i) the number of rationales used to explain the search results, i.e., $m$,
and ii) the size of the explanations in terms of the number of words or
sentences depending on the granularity - word windows, or sentences, i.e.,
$w$. Figure 2 shows the effect of variations in the length of the rationales
(in terms of the number of words constituting the explanation units) for $3$
different values of the number of explanations provided.

We observe that, first, the explanation consistency increases with an increase
in the number of rationales provided (compare the MRC values across the
different plots). This is expected because with a higher number of rationales,
each document’s attention-focused representation (Equation 1) tends towards
its original representation, i.e., the explanations themselves tend to cover
the entire document. While this shows up as increased values of MRC, as
pragmatic reasons, too large a value of $m$ is not desirable. Moreover, we
also observe that for most NRMs, increasing the rationale length leads to more
consistent explanations. Too large text windows can lead to inclusion of
potentially non-relevant text in the explanations, which due to the semantic
incoherence with the queries can lead to decreased MRC values (as can be seen
from the drops in MRC from $w=9$ to $w=11$ in most cases).

(a) Intrinsic Consistency

(b) Extrinsic Consistency

Figure 3:  Effect of the number of rationales on the explanation consistency
metrics across different NRMs for the MS-MARCO document ranking task.

(a) BM25

(b) ColBERT

(c) TCT-ColBERT

(d) MonoT5

(e) MonoElectra

Figure 4:  Per-query comparisons of the relevance and intrinsic explanation
consistency measures (MRC) for different IR models on the MS-MARCO passage
ranking task.

We now conduct a similar parameter variation analysis for the document ranking
task. Since the explanations are at sentence-level, this task does not involve
the parameter $w$ (the size of explanations). Figures 3(a) and 3(b) show the
effects of variations of the number of rationales on the intrinsic and the
extrinsic explanation consistencies, respectively. While with an increase in
$m$, the intrinsic measure mostly increases, a reverse trend is observed for
the sub-document relevance-based extrinsic approach. The argument on why MRC
increases with an increase in the number of rationales is the same as that of
passages, i.e., the attention-focused representation of a document tends to
better represent its core topic. On the other hand, a likely reason for the
decrease in the MER values is that the sentences with lower explanation
weights (but still included as a part of the explanations due to relatively
higher values of $m$) represent segments of a document that are potentially
not relevant to the query. This means that the proportion of sub-document
level relevant information in the explanations (Equation 3) progressively
decreases.

(a) BM25

(b) ColBERT

(c) TCT-ColBERT

(d) MonoT5

(e) MonoElectra

Figure 5:  Per-query comparisons of the relevance and intrinsic explanation
consistency measures (MRC) for different IR models on the MS-MARCO document
ranking task.

(a) BM25

(b) ColBERT

(c) TCT-ColBERT

(d) MonoT5

(e) MonoElectra

Figure 6:  Per-query comparisons of the relevance and extrinsic explanation
consistency measures (MER) for different IR models on the MS-MARCO document
ranking task.

#### Explanation Effectiveness vs. Relevance.

In relation to research questions RQ-2 and RQ-4, we now investigate the
correlation between a relevance measure (specifically, nDCG@10 as used in our
experiments) and the explanation consistency measures. Tables 1 and 2 already
demonstrated that the best performing NRM in terms of relevance is not
necessarily the one yielding the most consistent explanations. This
observation is reinforced by Figure 4 which shows a scatter-plot of the per-
query nDCG@10 vs. the MRC values. An interesting observation from Figure 4 is
that all the models (even BM25) registers only a small correlation between the
relevance and explanation consistency. MonoT5 even yields a negative
correlation of MRC with nDCG@10. This suggests that explanation consistency
can potentially serve as an evaluation dimension complementary to that of
relevance.

Similar observations can also be made from Figures 5 and 6. MER is slightly
more correlated with relevance because it involves measuring an overlap of the
rationales with the relevant segments of a document. However, the insights
gained from the MER metric is still different from both relevance and the
intrinsic consistency, which, in turn, suggests that MER like MRC can
potentially be used as another dimension to evaluate NRMs.

Table 3: The top-2 documents of the best performing query in terms of the
intrinsic consistency measure for MonoElectra on the MS-MARCO passage ranking
task. The top-3 rationales are shown with 3 different color shades (a deeper
shade indicating a higher attention weight).

Query Rationales of the top-$3$ documents What is the UN FAO? United Nations
Food and Agriculture Organization (FAO) The Food and Agriculture Organization
ofFood and Agriculture Organization of the United Nations is an agency that
leads international effortsthat leads international efforts to defeat hunger.
The Food and Agriculture Organization ofFood and Agriculture Organization of
the United Nations is an agency of the United Nations that leads international
efforts to defeat hunger. (MRC@10 = 0.9112) Role of the FAO. The Food and
Agriculture OrganizationThe Food and Agriculture Organization of the United
Nations isof the United Nations is an agency of the Unitedan agency of the
United Nations that leads international efforts to defeat hunger. Serving both
developed and developing countries, FAO acts as a neutral forum where all
nations meet as equals to negotiate agreements and debate policy.

#### Sample Explanations.

Table 3 shows an example query with the best intrinsic consistency measure on
the MS-MARCO passage ranking task. It can be seen that rationales provided for
this query does also qualitatively indicate pertinent explanations. This can
be seen from the fact that the NRM (MonoElectra) has been able to bridge the
semantic gap between the query term ‘FAO’, and the ones in the top-ranked
documents such as ‘Food’, ‘Agricultural’ and ‘Organization’. The same argument
also applies for the query term ‘UN’, and the document terms ‘United’ and
‘Nations’.

#### Concluding Remarks.

We introduced a evaluation framework wherein rationales – represented as text
snippets from the document – are available alongside search results. These
rationales can then be evaluated alongside traditional relevance measures to
assess the interpretability of the models. Using a basic occlusion-based
technique for producing rationales, we find that the systems that produce the
most relevant results may not necessarily be the most explainable – both in
terms of intrinsic and extrinsic measures of explainability.

As a future work, tools could be developed to enable easier qualitative
evaluation of rationales [7], and user studies could be conducted to test how
helpful rationales are to end-users.

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