#  Validity and failure of the integral representation
of $\Gamma$-limits of convex non-local functionals

Andrea Braides and Gianni Dal Maso
SISSA, via Bonomea 265, Trieste, Italy

###### Abstract

We prove an integral-representation result for limits of non-local quadratic
forms on $H^{1}_{0}(\Omega)$, with $\Omega$ a bounded open subset of
$\mathbb{R}^{d}$, extending the representation on $C^{\infty}_{c}(\Omega)$
given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a
counterexample showing that a corresponding representation may not hold if we
consider analogous functionals in $W^{1,p}_{0}(\Omega)$, with $p\neq 2$ and
$1<p\leq d$.

MSC codes: 49J45, 31C25, 46E35, 31B15.

Keywords: $\Gamma$-convergence, Dirichlet forms, non-local functionals,
integral representation.

Preprint SISSA 06/2023/MATE

##  1 Introduction

In this paper we continue our investigation on functionals defined on Sobolev
spaces in which a non-local part, in the form of a double integral, is present
beside a usual local part depending on the gradient. In general this question
can be formulated as the characterization of limits of functionals of the form

$F_{k}(u)=\int_{\Omega\times\Omega}f_{k}(u(x)-u(y))d\mu_{k}(x,y)+\int_{\Omega}g_{k}(x,\nabla
u(x))\,dx,$

defined in some Sobolev space $W^{1,p}_{0}(\Omega)$ with $p>1$. Different
types of stability of such a class can be studied: in [2] we have given a
notion of convergence on measures $\mu_{k}$ that guarantees the separate
stability of the integrals on $\Omega\times\Omega$ and on $\Omega$, while in
[3] we have explored conditions under which a limit of a sequence of such
functionals may still be of this form, but the integrands of the limit are
determined by the interaction between the local and non-local terms. The
theory of Dirichlet forms [6] gives the stability of such a class under the
only condition that $f_{k}$ and $g_{k}$ be quadratic, with very mild
conditions on the measures $\mu_{k}$ (see [9]).

In this paper we analyze the properties of the $\Gamma$-limits of these
quadratic functionals, and show that, rather surprisingly, the same stability
property does not hold if we consider only a slight variation of quadratic
forms; namely, when all integrands are $p$-th powers with $p\neq 2$. Previous
examples of lack of stability were known in the case of relaxation results for
non-convex functionals $F$, where the local term is dropped (or, equivalently,
if and $f_{k}=f$ is not convex and $g_{k}\equiv 0$). In that case, the lower-
semicontinuous envelope in the weak $L^{p}$ topology of a double integral can
be non-representable in the same form [1, 7, 8]. Our counterexample shows that
a similar issue arises even for sequences of convex and equicoercive
functionals in $W^{1,p}_{0}(\Omega)$.

In this paper, we first show a result connected to the theory of Dirichlet
forms. If we consider quadratic forms defined in $H^{1}_{0}(\Omega)$ of the
type

$\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}a_{k}(x,y)\,dxdy+\int_{\Omega}|\nabla
u(x)|^{2}\,dx,$ (1.1)

with $\Omega$ a bounded open set in $\mathbb{R}^{d}$ and $a_{k}$ positive
functions equibounded in $L^{1}(\Omega\times\Omega)$ and not concentrating on
the boundary, we prove that the corresponding $\Gamma$-limit, in the weak
topology of $H^{1}_{0}(\Omega)$, can be written on functions $u\in
C^{\infty}_{c}(\Omega)$ in the form

$\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}d\mu(x,y)+\int_{\Omega}|\nabla
u(x)|^{2}\,dx,$

where $\mu$ is a positive bounded Radon measure (see Theorems 2.2 and 2.4, and
Proposition 2.7). The main effort is spent in proving the boundedness of such
a measure, which does not seem to follow directly from the representation
obtained from the Beurling-Deny formula [6, Theorem 4.5.2].

We will show that if $p\neq 2$ (and $1<p\leq d$) this does not hold. More
precisely, a counterexample can be obtained as follows. Given $x_{0}\in\Omega$
and a sequence $\varepsilon_{k}$ of positive numbers converging to $0$, we set

$a_{k}(x,y)=\begin{cases}|B_{\varepsilon_{k}}(x_{0})|^{-1}&\hbox{ if }y\in
B_{\varepsilon_{k}}(x_{0})\\\ 0&\hbox{otherwise,}\end{cases}$

where $B_{r}(x)$ denotes the ball of centre $x$ and radius $r$. In this case
the functionals

$F_{k}(u)=\int_{\Omega\times\Omega}|u(x)-u(y)|^{p}a_{k}(x,y)\,dxdy+\int_{\Omega}|\nabla
u(x)|^{p}\,dx,$ (1.2)

defined for $u\in W^{1,p}_{0}(\Omega)$, have a $\Gamma$-limit, in the weak
topology of $W^{1,p}_{0}(\Omega)$, that can be directly expressed on that
space as

$F(u)=\int_{\Omega}|u(x)-m_{p}(u)|^{p}\,dx+\int_{\Omega}|\nabla
u(x)|^{p}\,dx,$ (1.3)

where $m_{p}(u)$ is the unique minimizer of
$t\mapsto\int_{\Omega}|u(x)-t|^{p}\,dx$. In the proof of this result the
inequality $p\leq d$ is crucial, implying that a single point has zero
$p$-capacity; this also explains the fact that the $\Gamma$-limit is
independent of $x_{0}$.

If $p\neq 2$ we will show that there exist no continuous function
$f\colon\mathbb{R}^{2}\to\mathbb{R}$ and no non-negative bounded Radon measure
$\mu$ on $\Omega\times\Omega$ such that

$\int_{\Omega}|u(x)-m_{p}(u)|^{p}\,dx=\int_{\Omega\times\Omega}f(u(x),u(y))d\mu(x,y)$
(1.4)

for all $u\in C^{\infty}_{c}(\Omega)$ (see Corollary 3.10).

Note that the representation of $F$ as above is not in contrast with the
representability as a double integral when $p=2$. Indeed, in that case

$m_{2}(u)={1\over|\Omega|}\int_{\Omega}u(y)\,dy,$

so that

$\displaystyle\int_{\Omega}|u(x)-m_{2}(u)|^{2}\,dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}|u(x)|^{2}\,dx-{1\over|\Omega|}\Bigl{(}\int_{\Omega}u(y)\,dy\Bigr{)}^{2}$
(1.5) $\displaystyle=$ $\displaystyle{1\over
2|\Omega|}\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}\,dxdy,$

and $\mu$ is just a multiple of the Lebesgue measure on $\Omega\times\Omega$.

The same observations lead to an example of failure of integral
representability in the theory of relaxation. This can be obtained by
considering the functional defined on $C^{1}_{c}(\Omega)$ by

$F_{x_{0}}(u)=\int_{\Omega}|u(x)-u(x_{0})|^{p}\,dx+\int_{\Omega}|\nabla
u(x)|^{p}\,dx,$

where $x_{0}$ is a given point in $\Omega$. Then the lower-semicontinuous
envelope with respect to the weak topology of $W^{1,p}_{0}(\Omega)$ is given
by the same $F$ as in (1.3), so that it cannot be represented in an integral
form. Note that the first term in $F_{x_{0}}(u)$ can be interpreted as an
integral on $\Omega\times\Omega$ with respect to the $d$-dimensional Hausdorff
measure restricted to $\Omega\times\\{x_{0}\\}$, which is the weak limit of
the measures $\mu_{k}=a_{k}\,dxdy$ defined above.

The plan of the paper is as follows. Section 2 is dedicated to the quadratic
case. We first apply the Beurling-Deny formula to obtain a representation on
$C^{\infty}_{c}(\Omega)$ of the $\Gamma$-limit $F$ of the functionals $F_{k}$
in (1.1) involving two measures $\mu$ and $\nu$ on $\Omega\times\Omega$ and
$\Omega$, respectively (Theorem 2.2). We then analyze some properties of such
measures deriving from the estimates satisfied by $F$, proving that both
measures are capacitary and finite (Theorems 2.4 and 2.5). Using some
additional lower-semicontinuity and truncation properties, satisfied by the
$\Gamma$-limit, we then extend the integral representation to the whole of
$H^{1}_{0}(\Omega)$ (Corollary 2.8). Section 3 is devoted to the
counterexample described above. We show that the $\Gamma$-limit of functionals
(1.2) is given by (1.3). We then extend (1.4) to characteristic functions
$u=1_{A}$ and show that, in this case, the right-hand and left-hand sides of
this equality depend only on the measure of $A$. A careful inspection of the
form of this dependence shows that they must be different if $p\neq 2$,
concluding the counterexample (Corollary 3.10).

##  2 The case $p=2$

Throughout the paper $\Omega$ is a connected bounded open subset of
$\mathbb{R}^{d}$, with $d\geq 1$, even though some limit arguments become
trivial if $d=1$. Let $a_{k}\in L^{1}(\Omega\times\Omega)$ be non-negative
functions such that

$\|a_{k}\|_{L^{1}(\Omega\times\Omega)}\leq M\hbox{ for all }k$ (2.1)

for some $M>0$, and consider the functionals

$F_{k}(u):=\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}a_{k}(x,y)\,dxdy+\int_{\Omega}|\nabla
u(x)|^{2}\,dx$ (2.2)

defined for $u\in H^{1}_{0}(\Omega)$. Since they are equicoercive in the weak
topology of $H^{1}_{0}(\Omega)$, we can use the the sequential
characterization of $\Gamma$-limits in the weak topology given in [4,
Proposition 8.10].

Note that each $F_{k}$ satisfies the following truncation property:
$F_{k}(\Psi(u))\leq F_{k}(u)$ for every $1$-Lipschitz function
$\Psi\colon\mathbb{R}\to\mathbb{R}$ with $\Psi(0)=0$ and for every $u\in
H^{1}_{0}(\Omega)$. Moreover, (2.1) implies

$F_{k}(u)\leq M\,({\rm osc}_{\Omega}u)^{2}+\int_{\Omega}|\nabla
u(x)|^{2}\,dx,$ (2.3)

where ${\rm osc}_{\Omega}u:={\rm ess\,sup}_{\Omega}u-{\rm ess\,inf}_{\Omega}u$
denotes the oscillation of $u$ on $\Omega$.

######  Proposition 2.1.

Assume that $F_{k}$ $\Gamma$-converges in the weak topology of
$H^{1}_{0}(\Omega)$ to a functional $F$. Then $F$ satisfies the following
properties:

(a) the domain of $F$, $D(F):=\\{u\in H^{1}_{0}(\Omega):F(u)<+\infty\\}$, is a
linear space containing $H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$;

(b) $F$ is a quadratic form; that is, there exists a bilinear form $B\colon
D(F)\times D(F)\to\mathbb{R}$ such that $F(u)=B(u,u)$ for every $u\in D(F)$;

(c) the space $D(F)$ endowed with the norm $\|\cdot\|_{F}$ defined as

$\|u\|_{F}=\bigl{(}\|u\|^{2}_{L^{2}(\Omega)}+F(u)\bigr{)}^{1/2}$ (2.4)

is a Hilbert space;

(d) we have $F(\Psi(u))\leq F(u)$ for every $1$-Lipschitz function
$\Psi\colon\mathbb{R}\to\mathbb{R}$ with $\Psi(0)=0$ and for every $u\in
H^{1}_{0}(\Omega)$;

(e) the space $C^{\infty}_{c}(\Omega)$ is dense in $D(F)$ with respect to the
norm $\|\cdot\|_{F}$.

###### Proof.

(a) If $u\in H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$ then (2.3) gives

$F(u)\leq\liminf_{k\to+\infty}F_{k}(u)\leq\bigl{(}2\|u\|_{L^{\infty}(\Omega)}\bigr{)}^{2}M+\|\nabla
u\|_{L^{2}(\Omega;\mathbb{R}^{d})}^{2};$ (2.5)

(b) follows from general properties of $\Gamma$-convergence (see [4, Theorem
11.10]); (c) following from the lower semicontinuity of $F$ by a standard
argument; (d) can be obtained from the truncation property of $F_{k}$ using
the definition of $\Gamma$-limit.

As for (e), we first prove that $H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$ is
contained in the closure of $C^{\infty}_{c}(\Omega)$ with respect to the norm
$\|\cdot\|_{F}$. Indeed, for every $u\in H^{1}_{0}(\Omega)\cap
L^{\infty}(\Omega)$ there exist $u_{k}\in C^{\infty}_{c}(\Omega)$ converging
to $u$ in $H^{1}_{0}(\Omega)$ and with
$\|u_{k}\|_{L^{\infty}(\Omega)}\leq\|u\|_{L^{\infty}(\Omega)}$. From this and
(2.5) we deduce that $\|u_{k}\|_{F}$ is equibounded, which implies that
$u_{k}\rightharpoonup u$ weakly in the Hilbert space $D(F)$, showing the
desired inclusion. It remain to prove that $H^{1}_{0}(\Omega)\cap
L^{\infty}(\Omega)$ is dense in $D(F)$. To that end, if $u\in D(F)$ we can
consider $u_{m}=\Psi_{m}(u)$, where $\Psi_{m}(t)=(m\wedge t)\vee(-m)$ is a
truncation operator. Then $u_{m}\in H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$,
$u_{m}\to u$ in $H^{1}_{0}(\Omega)$, and $F(u_{m})\leq F(u)$ by (d). This
again implies that $\|u_{m}\|_{F}$ is equibounded, so that $u_{m}$ weakly
converges to $u$ in $D(F)$, concluding the proof. ∎

######  Theorem 2.2.

Assume that $F_{k}$ $\Gamma$-converges in the weak topology of
$H^{1}_{0}(\Omega)$ to a functional $F$. Then there exist two positive Radon
measures $\mu$ and $\nu$ on $\Omega\times\Omega$ and $\Omega$, respectively,
such that

(a) for $u\in C^{\infty}_{c}(\Omega)$

$F(u)=\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}d\mu(x,y)+\int_{\Omega}|u(x)|^{2}d\nu(x)+\int_{\Omega}|\nabla
u(x)|^{2}\,dx;$ (2.6)

(b) $\mu$ is symmetric; i.e., $\mu(A\times B)=\mu(B\times A)$ for every pair
of Borel sets $A$ and $B$ contained in $\Omega$;

(c) setting $\Delta:=\\{(x,x):x\in\mathbb{R}^{d}\\}$, we have

$\mu((\Omega\times\Omega)\cap\Delta)=0.$ (2.7)

###### Proof.

From the previous proposition it follows that the bilinear form $B$ defined
therein is a Dirichlet form (see [6]), and that $C^{\infty}_{c}(\Omega)$ is a
core for $B$. Consequently by the Beurling-Deny Theorem ([6, Theorem 2.2.2]),
we have the decomposition

$F(u)=F^{n}(u)+F^{\ell}(u)+F^{c}(u)\hbox{ for every $u\in
C^{\infty}_{c}(\Omega)$},$

where $F^{\ell}$ is a local term, $F^{n}$ is a non-local term, and $F^{c}$ is
a local term depending only on the derivatives. More precisely, there exist a
symmetric matrix of Radon measures $\mu_{ij}$, and two positive Radon measures
$\mu$ and $\nu$ such that

$\displaystyle
F^{n}(u)=\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}d\mu(x,y),\quad
F^{\ell}(u)=\int_{\Omega}|u(x)|^{2}d\nu(x),$ (2.8) $\displaystyle
F^{c}(u)=\sum_{i,j=1}^{d}\int_{\Omega}{\partial u(x)\over\partial
x_{i}}{\partial u(x)\over\partial x_{j}}d\mu_{ij}(x)$ (2.9)

for every $u\in C^{\infty}_{c}(\Omega)$. Note that it is not restrictive to
suppose that $\mu$ is symmetric and (2.7) holds since $|u(x)-u(y)|^{2}=0$ on
$\Delta$.

We now show that in our case $\displaystyle F^{c}(u)=\int_{\Omega}|\nabla
u(x)|^{2}\,dx$. Note that

$\displaystyle\int_{\Omega}|\nabla u(x)|^{2}\,dx\leq F(u)\leq M\,({\rm
osc}_{\Omega}u)^{2}+\int_{\Omega}|\nabla u(x)|^{2}\,dx$ (2.10)

for every $u\in C^{\infty}_{c}(\Omega)$, using the lower semicontinuity of the
first integral for the first inequality and (2.3) for the second one.

Given $\omega\in C^{\infty}_{c}(\Omega)$ and $\xi\in\mathbb{R}^{d}$, let
$\varphi$ and $\psi$ be defined by

$\varphi(x)=\omega(x)\cos(x\cdot\xi)\ \hbox{ and }\
\psi(x)=\omega(x)\sin(x\cdot\xi).$

By a direct computation we have

$\displaystyle
F^{c}(\varphi)+F^{c}(\psi)=F^{c}(\omega)+\sum_{i,j=1}^{d}\int_{\Omega}\omega^{2}(x)\xi_{i}\xi_{j}d\mu_{ij}(x),$
$\displaystyle\int_{\Omega}|\nabla\varphi(x)|^{2}\,dx+\int_{\Omega}|\nabla\psi(x)|^{2}\,dx=\int_{\Omega}|\nabla\omega(x)|^{2}\,dx+\int_{\Omega}\omega^{2}(x)|\xi|^{2}dx,$

so that

$\displaystyle\Big{|}\sum_{i,j=1}^{d}\int_{\Omega}\omega^{2}(x)\xi_{i}\xi_{j}d\mu_{ij}(x)-\int_{\Omega}\omega^{2}(x)|\xi|^{2}dx\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}F(\varphi)-\int_{\Omega}|\nabla\varphi(x)|^{2}\,dx\Big{|}+\Big{|}F(\psi)-\int_{\Omega}|\nabla\psi(x)|^{2}\,dx\Big{|}+\Big{|}F(\omega)-\int_{\Omega}|\nabla\omega(x)|^{2}\,dx\Big{|}$
$\displaystyle+F^{\ell}(\varphi)+F^{\ell}(\psi)+F^{\ell}(\omega)+F^{n}(\varphi)+F^{n}(\psi)+F^{n}(\omega)$

Using (2.10) we get

$\displaystyle\Big{|}\sum_{i,j=1}^{d}\int_{\Omega}\omega^{2}(x)\xi_{i}\xi_{j}d\mu_{ij}(x)-\int_{\Omega}\omega^{2}(x)|\xi|^{2}dx\Big{|}$
$\displaystyle\leq
12M\|\omega\|^{2}_{L^{\infty}(\Omega)}+3F^{\ell}(\omega)+F^{n}(\varphi)+F^{n}(\psi)+F^{n}(\omega).$
(2.11)

Let $K={\rm supp}\,\omega$. We observe that

$\displaystyle F^{n}(\varphi)$ $\displaystyle=$ $\displaystyle\int_{K\times
K}|\omega(x)\cos(x\cdot\xi)-\omega(y)\cos(y\cdot\xi)|^{2}d\mu(x,y)$
$\displaystyle+\int_{K\times(\Omega\setminus
K)}|\omega(x)\cos(x\cdot\xi)|^{2}d\mu(x,y)+\int_{(\Omega\setminus K)\times
K}|\omega(y)\cos(y\cdot\xi)|^{2}d\mu(x,y)$ $\displaystyle\leq$ $\displaystyle
4\|\omega\|^{2}_{L^{\infty}(\Omega)}\mu(K\times
K)+\int_{K\times(\Omega\setminus
K)}|\omega(x)|^{2}d\mu(x,y)+\int_{(\Omega\setminus K)\times
K}|\omega(y)|^{2}d\mu(x,y)$ $\displaystyle\leq$ $\displaystyle
4\|\omega\|^{2}_{L^{\infty}(\Omega)}\mu(K\times
K)+\int_{\Omega\times\Omega}|\omega(x)-\omega(y)|^{2}d\mu(x,y)$
$\displaystyle\leq$ $\displaystyle
4\|\omega\|^{2}_{L^{\infty}(\Omega)}\big{(}\mu(K\times
K)+M\big{)}+\int_{\Omega}|\nabla\omega(x)|^{2}\,dx,$

where in the last inequality we again use (2.10). Using the analogous estimate
for $F^{n}(\psi)$, from (2.10) and (2) we get

$\displaystyle\hskip-28.45274pt\bigg{|}\sum_{i,j=1}^{d}\int_{\Omega}\omega^{2}(x)\xi_{i}\xi_{j}d\mu_{ij}(x)-\int_{\Omega}\omega^{2}(x)|\xi|^{2}dx\bigg{|}$
$\displaystyle\leq\|\omega\|^{2}_{L^{\infty}(\Omega)}(8\mu(K\times
K)+20M)+5\int_{\Omega}|\nabla\omega(x)|^{2}\,dx.$

Applying this estimate with $\xi$ replaced by $\lambda\xi$ we obtain

$\displaystyle\hskip-56.9055pt\bigg{|}\sum_{i,j=1}^{d}\int_{\Omega}\omega^{2}(x)\xi_{i}\xi_{j}d\mu_{ij}(x)-\int_{\Omega}\omega^{2}(x)|\xi|^{2}dx\bigg{|}$
$\displaystyle\leq$
$\displaystyle{1\over\lambda^{2}}\Big{(}\|\omega\|^{2}_{L^{\infty}(\Omega)}(8\mu(K\times
K)+20M)+5\int_{\Omega}|\nabla\omega(x)|^{2}\,dx\Big{)}.$

Letting $\lambda\to+\infty$ we then have

$\sum_{i,j=1}^{d}\int_{\Omega}\omega^{2}(x)\xi_{i}\xi_{j}d\mu_{ij}(x)=\int_{\Omega}\omega^{2}(x)|\xi|^{2}\,dx.$

By polarization we have

$\sum_{i,j=1}^{d}\int_{\Omega}\omega^{2}(x)\xi_{i}\eta_{j}d\mu_{ij}(x)=\int_{\Omega}\omega^{2}(x)\xi\cdot\eta\,dx\qquad\hbox{
for all }\xi,\eta\in\mathbb{R}^{d}.$

Taking $\xi,\eta\in\\{e_{1},\ldots,e_{d}\\}$ and using the arbitrariness of
$\omega$, we obtain that $\mu_{ij}=0$ for $i\neq j$ and
$\mu_{ii}=\mathcal{L}^{d}$, so that $F^{c}(u)=\int_{\Omega}|\nabla
u(x)|^{2}\,dx$. Using this we also have

$F^{n}(u)+F^{\ell}(u)\leq M\,({\rm osc}_{\Omega}(u))^{2}$ (2.12)

for all $u\in C^{\infty}_{c}(\Omega)$. ∎

The following example shows that a non-trivial measure measure $\nu$ may
indeed appear in the limit.

######  Example 2.3.

Let $d=2$ and $\Omega=(0,1)\times(0,1)$, with
$a_{k}(x,y)=\alpha_{k}(x)+\alpha_{k}(y)$, where

$\alpha_{k}(x)=\begin{cases}k&\hbox{ if }x\in R_{k}:=(0,1)\times(0,{1\over
k}),\\\ 0&\hbox{ otherwise.}\end{cases}$

Then (2.1) is satisfied with $M=2$. We now show that the corresponding $F_{k}$
converge to the functional given by

$F(u)=2\int_{\Omega}|u(x)|^{2}\,dx+\int_{\Omega}|\nabla u(x)|^{2}\,dx,$

which corresponds to $\mu=0$ and $\nu=2{\cal L}^{d}$.

In order to prove the liminf inequality we fix $u_{k}$ converging weakly to
$u$ in $H^{1}_{0}(\Omega)$. We have

$\displaystyle\int_{\Omega\times\Omega}|u_{k}(x)-u_{k}(y)|^{2}a_{k}(x,y)\,dxdy$
$\displaystyle=2k\int_{R_{k}}|u_{k}(x)|^{2}\,dx-4k\int_{R_{k}}u_{k}(x)\,dx\int_{\Omega}u_{k}(y)\,dy+2\int_{\Omega}|u_{k}(y)|^{2}\,dy.$
(2.13)

Moreover, using a Poincaré-inequality argument in $R_{k}$, we obtain

$k\int_{R_{k}}|u_{k}(x)|^{2}\,dx\leq{1\over k}\int_{R_{k}}|\nabla
u_{k}(x)|^{2}\,dx.$ (2.14)

This gives

$\lim_{k\to+\infty}k\int_{R_{k}}|u_{k}(x)|^{2}\,dx=0$ (2.15)

since the right-hand side in (2.14) converges to $0$. By Hölder’s inequality
we also obtain

$\lim_{k\to+\infty}k\int_{R_{k}}|u_{k}(x)|\,dx=0.$

These limits imply that, by (2.3),

$\lim_{k\to+\infty}\int_{\Omega\times\Omega}|u_{k}(x)-u_{k}(y)|^{2}a_{k}(x,y)\,dxdy=2\int_{\Omega}|u(x)|^{2}\,dx.$

By the lower semicontinuity of the gradient term this shows that
$\liminf\limits_{k\to+\infty}F_{k}(u_{k})\geq F(u)$. On the other hand, (2.3)
with $u_{k}=u$ shows that $\lim\limits_{k\to+\infty}F_{k}(u)=F(u)$, completing
the proof of the $\Gamma$-convergence of $F_{k}$ to $F$.

We now analyze the properties of the measures $\mu$ and $\nu$ given in Theorem
2.2 in order to extend the representation result to the whole
$H^{1}_{0}(\Omega)$.

######  Theorem 2.4.

Let $F\colon C^{\infty}_{c}(\Omega)\to[0,+\infty)$ be such that there exist
exist two positive Radon measures $\mu$ and $\nu$ on $\Omega\times\Omega$ and
$\Omega$, respectively, that satisfy (a), (b), and (c) of Theorem 2.2. Suppose
in addition that there exists $M>0$ such that

$F(u)\leq M\,({\rm osc}_{\Omega}u)^{2}+\int_{\Omega}|\nabla u(x)|^{2}\,dx$
(2.16)

for all $u\in C^{\infty}_{c}(\Omega)$. Then

(a) the measures $\mu$ and $\nu$ are uniquely determined;

(b) $\mu(\Omega\times\Omega)<+\infty$ and $\nu(\Omega)<+\infty$.

###### Proof.

We begin by proving that $\nu$ is a finite measure. From (2.16) and (2.6) we
first obtain

$\int_{\Omega}|u(x)|^{2}d\nu(x)\leq M\,({\rm osc}_{\Omega}u)^{2}\hbox{ for all
$u\in C^{\infty}_{c}(\Omega)$. }$

Approximating the constant $1$ by an increasing sequence of non-negative
functions $u_{k}\in C^{\infty}_{c}(\Omega)$ we obtain that $\nu(\Omega)\leq
M$.

We now complete the proof of claim (b), showing that the measure $\mu$ is
finite. We preliminarily note that from (2.16) and (2.6) we also obtain

$\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}d\mu(x,y)\leq M\,({\rm
osc}_{\Omega}u)^{2}\hbox{ for all $u\in C^{\infty}_{c}(\Omega)$. }$ (2.17)

Since the proof is rather complex we first consider the case $d=1$, hoping it
may clarify the arguments used. For given $\eta>0$ we let

$\Delta_{\eta}:=\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}:|x-y|\leq\eta\\},$
(2.18)

and cover $(\Omega\times\Omega)\setminus\Delta_{\eta}$ ‘in the average’ by a
family of ‘checkerboard-type’ sets depending on two parameters $\alpha$ and
$\beta$, showing that the covering has some average properties independent of
$\eta$.

With given $\alpha,\beta\in\mathbb{R}$, with $\beta>0$, we define

$A_{\alpha,\beta}=\bigcup_{h\\!\ \rm even}[\alpha+h\beta,\alpha+(h+1)\beta),$
(2.19)

and

$E_{\alpha,\beta}=\bigcup_{h+k\\!\ \rm
odd}([\alpha+h\beta,\alpha+(h+1)\beta)\times[\alpha+k\beta,\alpha+(k+1)\beta)).$
(2.20)

Note that $E_{\alpha+k\beta,\beta}=E_{\alpha,\beta}$ for all $k\in\mathbb{Z}$
and

$E_{\alpha,\beta}=(\alpha,\alpha)+E_{0,\beta};$ (2.21)

moreover, we have

$(\Omega\times\Omega)\cap E_{\alpha,\beta}=((\Omega\cap
A_{\alpha,\beta})\times(\Omega\setminus
A_{\alpha,\beta}))\cup((\Omega\setminus A_{\alpha,\beta})\times(\Omega\cap
A_{\alpha,\beta})).$ (2.22)

We claim that

$\mu((\Omega\times\Omega)\cap E_{\alpha,\beta})\leq M,$ (2.23)

where $M$ is defined in (2.16). To prove the claim we take a sequence
$u_{k}\in C^{\infty}_{c}(\Omega)$ such that $0\leq u_{k}\leq 1$ and
$u_{k}(x)\to 1_{A_{\alpha,\beta}}(x)$ for all $x\in\Omega$. We then have

$\lim_{k\to+\infty}\int_{(\Omega\cap A_{\alpha,\beta})\times(\Omega\setminus
A_{\alpha,\beta})}|u_{k}(x)-u_{k}(y)|^{2}d\mu(x,y)=\mu((\Omega\cap
A_{\alpha,\beta})\times(\Omega\setminus A_{\alpha,\beta})).$ (2.24)

If this latter measure is finite this limit is obtained by using the Dominated
Convergence Theorem; otherwise, it follows by applying Fatou’s Lemma. From
(2.17), (2.24), and the analogous limit for $\mu((\Omega\setminus
A_{\alpha,\beta})\times(\Omega\cap A_{\alpha,\beta}))$, we obtain the claim
thanks to (2.22).

The next argument is, given $\eta>0$, to determine $\varepsilon>0$ such that,
setting $D_{\varepsilon}=\\{(\alpha,\beta):0\leq\alpha<\beta,\
\varepsilon\leq\beta<2\varepsilon\\}$, we have

$\int_{D_{\varepsilon}}1_{E_{\alpha,\beta}}(x,y)d\alpha
d\beta\geq\Big{(}{1\over 2}-\eta\Big{)}|D_{\varepsilon}|$ (2.25)

for every $(x,y)\in(\Omega\times\Omega)\setminus\Delta_{\eta}$, where
$\Delta_{\eta}$ us defined in (2.18). Once (2.25) is proved we obtain

$\displaystyle\Big{(}{1\over
2}-\eta\Big{)}|D_{\varepsilon}|\mu((\Omega\times\Omega)\setminus\Delta_{\eta})$
$\displaystyle\leq$
$\displaystyle\int_{\Omega\times\Omega}\int_{D_{\varepsilon}}1_{E_{\alpha,\beta}}(x,y)d\alpha
d\beta d\mu(x,y)$ $\displaystyle=$
$\displaystyle\int_{D_{\varepsilon}}\int_{\Omega\times\Omega}1_{E_{\alpha,\beta}}(x,y)d\mu(x,y)d\alpha
d\beta$ $\displaystyle\leq$
$\displaystyle|{D_{\varepsilon}}|\mu(E_{\alpha,\beta})\leq|{D_{\varepsilon}}|M$

by (2.23). Dividing by $|{D_{\varepsilon}}|$ we obtain that

$\displaystyle\Big{(}{1\over
2}-\eta\Big{)}\,\mu((\Omega\times\Omega)\setminus\Delta_{\eta})\leq M.$

Taking into account (2.7) we obtain

$\mu(\Omega\times\Omega)\leq 2M$ (2.26)

by the arbitrariness of $\eta$, concluding the proof of the boundeness of
$\mu$.

It remains to prove (2.25). By Fubini’s Theorem we have

$\int_{D_{\varepsilon}}1_{E_{\alpha,\beta}}(x,y)d\alpha
d\beta=\int_{\varepsilon}^{2\varepsilon}\mathcal{L}^{1}(\\{\alpha\in[0,\beta):(x,y)\in
E_{\alpha,\beta}\\})d\beta.$

We now claim that $\mathcal{L}^{1}(\\{\alpha\in[0,\beta):(x,y)\in
E_{\alpha,\beta}\\})$ depends on $z:=y-x$ and that, setting

$\gamma_{z}(\beta):=\mathcal{L}^{1}(\\{\alpha\in[0,\beta):(x,y)\in
E_{\alpha,\beta}\\}),$

we have

$\gamma_{z}(\beta)=|z-2m_{\beta}(z)\beta|,$

where $m_{\beta}(z)\in\mathbb{Z}$ is the unique integer such that

$(2m_{\beta}(z)-1)\beta\leq z<(2m_{\beta}(z)+1)\beta.$

We first observe that, by the periodicity of $E_{\alpha,\beta}$, the set
$\\{\alpha\in\mathbb{R}:(x,y)\in E_{\alpha,\beta}\\}$ is a periodic subset of
$\mathbb{R}$ of period $\beta$. Moreover, using (2.21), we have

$\\{\alpha\in\mathbb{R}:(x,y)\in
E_{\alpha,\beta}\\}=\\{\alpha\in\mathbb{R}:(0,z)\in
E_{\alpha,\beta}\\}+\alpha_{\beta}(x),$

with $\alpha_{\beta}(x):=x-\beta\lfloor{x\over\beta}\rfloor$, where
$\lfloor\cdot\rfloor$ denotes the integer part; hence,

$\mathcal{L}^{1}(\\{\alpha\in[0,\beta):(x,y)\in
E_{\alpha,\beta}\\})=\mathcal{L}^{1}(\\{\alpha\in[0,\beta):(0,z)\in
E_{\alpha,\beta}\\})$

by the periodicity of the set, which proves the fact that $\gamma_{z}$ indeed
depends only on $z$.

Figure 1: geometrical interpretation of $\gamma_{z}$; the grey zone represents
$E_{0,\beta}$.

We now observe that, using the periodicity of $E_{0,\beta}$, we can write

$\gamma_{z}(\beta)=\mathcal{L}^{1}(\\{\alpha\in[0,\beta):(0,z)\in(\alpha,\alpha)+E_{0,\beta}\\})=\mathcal{L}^{1}(\\{\alpha\in[0,\beta):(0,z)+(\alpha,\alpha)\in
E_{0,\beta}\\}),$

so that we have $\gamma_{z}(\beta)=0$ when $|z|=m\beta$ for $m$ even and
$\gamma_{z}(\beta)=\beta$ for $|z|=\beta m$ for $m$ odd. Otherwise, the
function $\gamma_{z}$ is the piecewise-affine interpolation determined by
these conditions on $\beta\mathbb{Z}$. In Fig. 1 we give a pictorial
representation of the value $\gamma_{z}(\beta)$, given by the length of the
projection on the $x$-coordinate axis of the intersection of the line
$\\{(0,z)+(\alpha,\alpha):\alpha\in\mathbb{R}\\}$ with
$E_{0,\beta}\cap([0,\beta)\times\mathbb{R})$. In the figure we have pictured
the cases of a generic $z$ and of the two possibilities, $z$ even or odd, when
$z\in\mathbb{Z}$ (dashed lines).

In terms of $\gamma_{z}$, (2.25) is equivalent to

$\int_{\varepsilon}^{2\varepsilon}\gamma_{z}(\beta)d\beta\geq\Big{(}{1\over
2}-\eta\Big{)}|D_{\varepsilon}|.$ (2.27)

Note that $|D_{\varepsilon}|={3\over 2}\varepsilon^{2}$, but in most of the
computations we do not need this explicit value.

Figure 2: a part of the graph of $\gamma_{z}$.

In order to prove (2.27) note that

$\int_{|z|\over 2m}^{|z|\over 2m-2}\gamma_{z}(\beta)d\beta\geq\int_{|z|\over
2m+1}^{|z|\over 2m-1}(\beta-\gamma_{z}(\beta))d\beta$ (2.28)

for every $m\in\mathbb{N}$ with $m\geq 2$ (this is just a comparison between
the areas of the two triangles in Fig. 2). We define $h=h(|z|,\varepsilon)$
and $k=k(|z|,\varepsilon)$ setting

$h=\max\Big{\\{}m\in\mathbb{N}:\varepsilon\leq{|z|\over 2m}\Big{\\}},\qquad
k=\min\Big{\\{}m\in\mathbb{N}:{|z|\over 2m}<2\varepsilon\Big{\\}},$

and observe that $k\leq h$. From (2.28), summing on the set of integer $m$
with $k<m\leq h$ we deduce that

$\displaystyle\int_{\varepsilon}^{2\varepsilon}\gamma_{z}(\beta)d\beta$
$\displaystyle\geq$ $\displaystyle\int_{|z|\over 2h}^{|z|\over
2k}\gamma_{z}(\beta)d\beta$ $\displaystyle\geq$ $\displaystyle\int_{|z|\over
2h+1}^{|z|\over
2k+1}(\beta-\gamma_{z}(\beta))d\beta\geq\int_{\varepsilon}^{2\varepsilon}(\beta-\gamma_{z}(\beta))d\beta-|z|{2\varepsilon\over
k^{2}},$

so that

$\int_{\varepsilon}^{2\varepsilon}\gamma_{z}(\beta)d\beta\geq{1\over
2}|D_{\varepsilon}|-|z|{\varepsilon\over
k^{2}}=|D_{\varepsilon}|\Big{(}{1\over 2}-{2|z|\over
3k^{2}\varepsilon}\Big{)}.$ (2.29)

Note that ${1\over k}<{4\varepsilon\over|z|}$, so that ${2|z|\over
3k^{2}\varepsilon}\leq{32\varepsilon\over 3|z|}$. By choosing
$\varepsilon<{3\over 32}\eta^{2}$ we then have ${2|z|\over
3k^{2}\varepsilon}\leq\eta$ for all $z$ with $|z|\geq\eta$, and estimate
(2.27) holds.

If $d>1$, for all $i\in\\{1,\ldots,d\\}$ and $\alpha,\beta\in\mathbb{R}$, with
$\beta>0$, we consider the set

$E^{i}_{\alpha,\beta}=\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}:(x_{i},y_{i})\in
E_{\alpha,\beta}\\},$

where $E_{\alpha,\beta}$ is defined in (2.20). Correspondingly, we define

$A^{i}_{\alpha,\beta}=\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}:(x_{i},y_{i})\in
A_{\alpha,\beta}\\},$

where $A_{\alpha,\beta}$ is defined in (2.19). As in (2.22) we get

$(\Omega\times\Omega)\cap E^{i}_{\alpha,\beta}=((\Omega\cap
A^{i}_{\alpha,\beta})\times(\Omega\setminus
A^{i}_{\alpha,\beta}))\cup((\Omega\setminus
A^{i}_{\alpha,\beta})\times(\Omega\cap A^{i}_{\alpha,\beta})).$

Repeating the steps in the proof in the case $d=1$ we obtain, as in (2.25),

$\int_{D_{\varepsilon}}1_{E^{i}_{\alpha,\beta}}(x,y)d\alpha
d\beta\geq\Big{(}{1\over 2}-\eta\Big{)}|D_{\varepsilon}|\quad\hbox{for all
$i\in\\{1,\ldots,d\\}$,}$

and hence that $\mu((\Omega\times\Omega)\setminus\Delta^{i})\leq 2M$, where
$\Delta^{i}=\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}:x_{i}=y_{i}\\}$.
Since $\Delta=\bigcap_{i=1}^{d}\Delta^{i}$, and
$\mu((\Omega\times\Omega)\cap\Delta)=0$, we deduce that

$\displaystyle\mu(\Omega\times\Omega)\leq 2dM,$

which concludes the proof of (b).

In order to prove (a), we first note that for all disjoint open subsets $A$
and $B$ of $\Omega$, thanks to (2.6) and the symmetry of $\mu$ we have

$\displaystyle 2\mu(A\times(\Omega\setminus
A))+\nu(A)=\lim_{k\to+\infty}\Big{(}F(u_{k})-\int_{\Omega}|\nabla
u_{k}(x)|^{2}\,dx\Big{)},$ $\displaystyle 2\mu(B\times(\Omega\setminus
B))+\nu(B)=\lim_{k\to+\infty}\Big{(}F(v_{k})-\int_{\Omega}|\nabla
v_{k}(x)|^{2}\,dx\Big{)},$ $\displaystyle 2\mu((A\cup
B)\times(\Omega\setminus(A\cup B)))+\nu(A\cup B)$ $\displaystyle\hskip
113.81102pt=\lim_{k\to+\infty}\Big{(}F(u_{k}+v_{k})-\int_{\Omega}|\nabla
u_{k}(x)+\nabla v_{k}(x)|^{2}\,dx\Big{)},$

where $u_{k},v_{k}$ are sequences in $C^{\infty}_{c}(\Omega)$ with $0\leq
u_{k}\leq 1_{A}$ and $0\leq v_{k}\leq 1_{B}$, such that $u_{k}(x)\to 1_{A}(x)$
and $v_{k}(x)\to 1_{B}(x)$ for all $x\in\Omega$. Summing up the first two
equations above and subtracting the third one we have

$\displaystyle\mu(A\times B)$ $\displaystyle=$ $\displaystyle{1\over
2}\Big{(}\mu(A\times(\Omega\setminus A))+\mu(B\times(\Omega\setminus
B))-\mu((A\cup B)\times(\Omega\setminus(A\cup B)))\Big{)}$ $\displaystyle=$
$\displaystyle\Phi_{F}(A,B),$

where

$\Phi_{F}(A,B):={1\over
4}\lim_{k\to+\infty}\big{(}F(u_{k})+F(v_{k})-F(u_{k}+v_{k})\big{)}.$

This shows that if $\mu_{1}$ and $\mu_{2}$ are symmetric bounded Borel
measures and satisfy (2.6) then

$\mu_{1}(A\times B)=\mu_{2}(A\times B)$ (2.30)

for all disjoint open subsets $A$ and $B$ of $\Omega$. This property can be
extended first to disjoint compact subsets of $\Omega$ and then to disjoint
Borel subsets of $\Omega$.

To extend this equality to arbitrary Borel subsets of $\Omega$ we fix $\eta>0$
and write

$A=\bigcup_{i=1}^{m_{\eta}}A_{i},\qquad B=\bigcup_{j=1}^{n_{\eta}}B_{j},$

where $A_{i}$ and $B_{j}$ are Borel partitions of $A$ and $B$, respectively,
with diam$(A_{i})<{\eta\over 2}$ and diam$(B_{j})<{\eta\over 2}$ for all
$i,j$. Setting $\mathcal{D}_{\eta}=\\{(i,j):A_{i}\cap B_{j}\neq\emptyset\\}$
and observing that $A\times B=\bigcup_{(i,j)}(A_{i}\times B_{j})$ and that
$\bigcup_{(i,j)\in\mathcal{D}_{\eta}}(A_{i}\times B_{j})\subset\Delta_{\eta}$,
by (2.30) we obtain that

$\displaystyle|\mu_{1}(A\times B)-\mu_{2}(A\times B)|$ $\displaystyle=$
$\displaystyle\Big{|}\mu_{1}\Big{(}\bigcup_{(i,j)\in\mathcal{D}_{\eta}}(A_{i}\times
B_{j})\Big{)}-\mu_{2}\Big{(}\bigcup_{(i,j)\in\mathcal{D}_{\eta}}(A_{i}\times
B_{j})\Big{)}\Big{|}$ $\displaystyle\leq$
$\displaystyle\mu_{1}((\Omega\times\Omega)\cap\Delta_{\eta})+\mu_{2}((\Omega\times\Omega)\cap\Delta_{\eta}).$

Therefore, if $\mu_{1}$ and $\mu_{2}$ also satisfy (2.7), then, by letting
$\eta\to 0$ we obtain that $\mu_{1}(A\times B)=\mu_{2}(A\times B)$ for all
pairs of Borel sets, and hence that $\mu_{1}=\mu_{2}$. Finally, by (2.6) we
deduce that for every $u\in H^{1}_{0}(\Omega)$ the integral
$\int_{\Omega}|u(x)|^{2}d\nu(x)$ is uniquely determined by $F$, which gives
the uniqueness of such a $\nu$, and concludes the proof of (a). ∎

In the following theorem we use the classical notion of capacity, and
$\widetilde{u}(x)$ denotes the precise representative of a function $u\in
H^{1}_{0}(\Omega)$, which is defined up to sets of zero capacity (see [5,
Sections 4.7 and 4.8]). A similar result can be proved using the intrinsic
capacity of the Dirichlet form $F$ and the corresponding precise
representatives (see [6, Theorem 4.5.2]).

######  Theorem 2.5.

Let $F\colon H^{1}_{0}(\Omega)\to[0,+\infty)$ and let $\mu$ and $\nu$ be two
bounded positive Radon measures on $\Omega\times\Omega$ and $\Omega$,
respectively, that satisfy (a), (b), and (c) of Theorem 2.2. Suppose also that
$F$ be lower semicontinuous in the weak topology of $H^{1}_{0}(\Omega)$ and
that there exists $M>0$ such that (2.16) holds for all $u\in
C^{\infty}_{c}(\Omega)$. Then

(a) if $B\subset\Omega$ is a Borel set with zero capacity, then
$\mu(B\times\Omega)=\nu(B)=0$;

(b) for every $u\in H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$

$F(u)=\int_{\Omega\times\Omega}|\widetilde{u}(x)-\widetilde{u}(y)|^{2}d\mu(x,y)+\int_{\Omega}|\widetilde{u}(x)|^{2}d\nu(x)+\int_{\Omega}|\nabla
u(x)|^{2}\,dx;$ (2.31)

(c) if, in addition, $F(\Psi_{m}(u))\leq F(u)$ for all $u\in
H^{1}_{0}(\Omega)$ and $m\in\mathbb{N}$, where $\Psi_{m}(t)=(m\wedge
t)\vee(-m)$, then (2.31) holds for every $u\in H^{1}_{0}(\Omega)$.

###### Proof.

We observe that, by the strong continuity of $u\mapsto\int_{\Omega}|\nabla
u(x)|^{2}\,dx$,

$u\mapsto\int_{\Omega\times\Omega}|u(x)-u(y)|^{2}d\mu(x,y)+\int_{\Omega}|u(x)|^{2}d\nu(x)\
\hbox{ is lower semicontinuous}$ (2.32)

on $C^{\infty}_{c}(\Omega)$ with respect to the strong topology of
$H^{1}_{0}(\Omega)$.

Let $K$ be a compact subset of $\Omega$ with zero capacity. We now prove that

$\displaystyle\mu(K\times(\Omega\setminus K))=0\quad\hbox{ and
}\quad\nu(K)=0.$ (2.33)

Given $\eta>0$ let $U$ be an open set such that $K\subset U\subset\Omega$ and

$\displaystyle\mu((U\setminus K)\times\Omega)\leq\eta\quad\hbox{ and
}\quad\nu(U\setminus K)\leq\eta,$ (2.34)

and let $w\in C^{\infty}_{c}(\Omega)$ be such that $0\leq w\leq 1$ on the
whole $\Omega$, $w=1$ in a neighbourhood of $K$, and $w\leq{1\over 4}$ on
$\Omega\setminus U$. Since $K$ has zero capacity there exist a sequence
$u_{k}\in C^{\infty}_{c}(\Omega)$ converging to $w$ strongly in
$H^{1}_{0}(\Omega)$, such that $0\leq u_{k}\leq 1$ on the whole $\Omega$,
$u_{k}=0$ on $K$ and $u_{k}=w$ on $\Omega\setminus U$. Then we have

$\displaystyle\hskip-14.22636pt\int_{\Omega\times\Omega}|u_{k}(x)-u_{k}(y)|^{2}d\mu(x,y)+\int_{\Omega}|u_{k}(x)|^{2}d\nu(x)$
$\displaystyle\leq$ $\displaystyle 2\int_{K\times(\Omega\setminus
U)}|w(y)|^{2}d\mu(x,y)+\int_{(\Omega\setminus U)\times(\Omega\setminus
U)}|w(x)-w(y)|^{2}d\mu(x,y)$ $\displaystyle+\mu((U\setminus
K)\times(U\setminus K))+2\mu((U\setminus K)\times(\Omega\setminus
U))+\int_{\Omega\setminus U}|w(x)|^{2}d\nu(x)+\nu(U\setminus K)$
$\displaystyle\leq$ $\displaystyle{1\over 8}\mu(K\times(\Omega\setminus
U))+\int_{(\Omega\setminus U)\times(\Omega\setminus
U)}|w(x)-w(y)|^{2}d\mu(x,y)$ $\displaystyle+\int_{\Omega\setminus
U}|w(x)|^{2}d\nu(x)+3\eta,$

while

$\displaystyle\hskip-14.22636pt\int_{\Omega\times\Omega}|w(x)-w(y)|^{2}d\mu(x,y)+\int_{\Omega}|w(x)|^{2}d\nu(x)$
$\displaystyle\geq$ $\displaystyle 2\int_{K\times(\Omega\setminus
U)}|1-w(y)|^{2}d\mu(x,y)+\int_{(\Omega\setminus U)\times(\Omega\setminus
U)}|w(x)-w(y)|^{2}d\mu(x,y)$ $\displaystyle+\nu(K)+\int_{\Omega\setminus
U}|w(x)|^{2}d\nu(x).$

Hence, noting that $1-w(y)\geq{3\over 4}$ if $y\in\Omega\setminus U$, from the
convergence of $u_{k}$ to $u$ and (2.32) we obtain

$\displaystyle{9\over 8}\mu(K\times(\Omega\setminus U))+\nu(K)\leq{1\over
8}\mu(K\times(\Omega\setminus U))+3\eta.$

Taking into account that $\mu(K\times(U\setminus K))\leq\eta$ thanks to (2.34)
and the symmetry of $\mu$ we obtain

$\displaystyle\mu(K\times(\Omega\setminus K))+\nu(K)\leq 4\eta,$

and (2.33) is proved by the arbitrariness of $\eta$.

We now claim that

$\displaystyle\mu(K\times K)=0,$ (2.35)

Given $\eta>0$ we can find a finite number of compact sets $K_{i}$ such that
$K=\bigcup_{i}K_{i}$ and diam$K_{i}\leq{\eta\over 2}$. Since

$\displaystyle K\times K\subset\bigcup_{i,j}(K_{i}\times
K_{j})=\bigcup_{(i,j)\in\mathcal{D}}(K_{i}\times
K_{j})\cup\bigcup_{(i,j)\not\in\mathcal{D}}(K_{i}\times K_{j}),$

where $\mathcal{D}=\\{(i,j):K_{i}\cap K_{j}\neq\emptyset\\}$, we have

$\displaystyle\mu(K\times
K)\leq\mu\Big{(}\bigcup_{(i,j)\in\mathcal{D}}(K_{i}\times
K_{j})\Big{)}+\sum_{(i,j)\not\in\mathcal{D}}\mu(K_{i}\times K_{j}).$

Since $K_{j}\subset\Omega\setminus K_{i}$ if $(i,j)\not\in\mathcal{D}$, by
(2.33) applied to $K_{i}$ the terms in the last sum are all zero. On the other
hand $\bigcup_{(i,j)\in\mathcal{D}}(K_{i}\times K_{j})\subset\Delta_{\eta}$,
where $\Delta_{\eta}$ is defined in (2.18), so that $\mu(K\times
K)\leq\mu((\Omega\times\Omega)\cap\Delta_{\eta})$. Since
$\mu((\Omega\times\Omega)\cap\Delta)=0$ and $\mu$ is finite we obtain (2.35)
by letting $\eta\to 0$.

Finally, (2.33) and (2.35) give that $\mu(K\times\Omega)=\nu(K)=0$ for any $K$
compact set with zero capacity. Claim (a) is then obtain by approximation of
$B$ with compact sets contained in $B$.

In order to prove claim (b), by proceeding as in the proof of Proposition
2.1(e) for all $u\in H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$ we have a
sequence $u_{k}\in C^{\infty}_{c}(\Omega)$ converging strongly to $u$ in
$H^{1}_{0}(\Omega)$, such that
$\|u_{k}\|_{L^{\infty}(\Omega)}\leq\|u\|_{L^{\infty}(\Omega)}$ and $u_{k}$
converge weakly to $u$ with respect to the Hilbert structure induced by the
norm defined in (2.4). By Mazur’s theorem we obtain a new sequence $v_{k}\in
C^{\infty}_{c}(\Omega)$ converging strongly to $u$ in $H^{1}_{0}(\Omega)$,
such that $\|v_{k}\|_{L^{\infty}(\Omega)}\leq\|u\|_{L^{\infty}(\Omega)}$ and
$v_{k}$ converge strongly to $u$ both in $H^{1}_{0}(\Omega)$ and with respect
to the Hilbert structure induced by the norm defined in (2.4). In particular
$F(v_{k})\to F(u)$ and, upon passing to a subsequence, $v_{k}\to\widetilde{u}$
quasi-everywhere (in the sense of capacity). Together with the uniform bound,
this implies that

$\lim_{k\to+\infty}\int_{\Omega\times\Omega}|v_{k}(x)-v_{k}(y)|^{2}d\mu(x,y)=\int_{\Omega\times\Omega}|\widetilde{u}(x)-\widetilde{u}(y)|^{2}d\mu(x,y)$
$\lim_{k\to+\infty}\int_{\Omega}|v_{k}(x)|^{2}d\nu(x)=\int_{\Omega}|\widetilde{u}(x)|^{2}d\nu(x),$

and claim (b) follows.

In order to prove (c), let $G\colon H^{1}_{0}(\Omega)\to[0,+\infty]$ be
defined by the right-hand side in (2.31), and let $u\in H^{1}_{0}(\Omega)$.
Let $u_{m}=\Phi_{m}(u)$ and note that $\lim\limits_{m\to+\infty}G(u_{m})=G(u)$
by the Monotone Convergence . Since $F(u_{m})=G(u_{m})$ to conclude the proof
it is enough to note that $\lim\limits_{m\to+\infty}F(u_{m})=F(u)$, which
follows from the hypothesis on truncations and the lower semicontinuity of
$F$. ∎

######  Remark 2.6.

Note that equality (2.31) may not hold on the whole $H^{1}_{0}(\Omega)$ if the
additional assumption in (c) is dropped. For instance, if $G$ is defined by
the right-hand side in (2.31), and $F$ is defined as equal to $G$ except on a
single $u_{0}\in H^{1}_{0}(\Omega)\setminus L^{\infty}(\Omega)$, where we set
$F(u_{0})=0$, then $F$ is lower semicontinuous, $F$ and $G$ are equal on
$H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$, but equality does not hold in the
whole $H^{1}_{0}(\Omega)$.

######  Proposition 2.7.

In addition to the hypotheses of Theorem 2.2, suppose that $a_{k}$ satisfies
the following condition: for every $\varepsilon>0$ there exists a compact set
$K_{\varepsilon}\subset\Omega$ such that

$\int_{(\Omega\times\Omega)\setminus(K_{\varepsilon}\times
K_{\varepsilon})}a_{k}(x,y)\,dxdy\leq\varepsilon\hbox{ for every
}k\in\mathbb{N}.$ (2.36)

Then the measure $\nu$ in Theorem 2.2 is the null measure.

###### Proof.

Let $u\in C^{\infty}_{c}(\Omega)$ such that $0\leq u\leq 1$ and $u=1$ on
$K_{\varepsilon}$. Then, by using $u$ as test function in the $\Gamma$-limit
we have

$\displaystyle F(u)$ $\displaystyle\leq$
$\displaystyle\liminf_{k\to+\infty}\int_{(\Omega\times\Omega)\setminus(K_{\varepsilon}\times
K_{\varepsilon})}|u(x)-u(y)|^{2}\,dxdy+\int_{\Omega}|\nabla u(x)|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle\varepsilon+\int_{\Omega}|\nabla
u(x)|^{2}\,dx.$

Since $F(u)=F^{n}(u)+F^{\ell}(u)+\int_{\Omega}|\nabla u(x)|^{2}\,dx$ we
conclude that $F^{\ell}(u)\leq\varepsilon$ for all such $u$.

We now fix a compact $K$ in $\Omega$ and for each $\varepsilon>0$ take
$u_{\varepsilon}\in C^{\infty}_{c}(\Omega)$ with $0\leq u_{\varepsilon}\leq 1$
and $u_{\varepsilon}=1$ on $K\cup K_{\varepsilon}$. By the estimate above we
have

$\nu(K)\leq\int_{\Omega}|u_{\varepsilon}(x)|^{2}d\nu(x)=F^{\ell}(u_{\varepsilon})\leq\varepsilon.$

By the arbitrariness of $\varepsilon$ we obtain that $\nu(K)=0$ for all $K$
compact of $\Omega$, which proves the claim. ∎

The following corollary improves the conclusions of Theorem 2.2 in light of
Theorems 2.4 and 2.5 and of Proposition 2.7.

######  Corollary 2.8.

Let $F_{k}$ be given by (2.2), with $a_{k}$ satisfying (2.1). Suppose that
$F_{k}$ $\Gamma$-converges to $F$ with respect to the weak topology in
$H^{1}_{0}(\Omega)$. Then there exist two positive finite Radon measures $\mu$
and $\nu$ on $\Omega\times\Omega$ and $\Omega$, respectively, such that

(a) $\mu$ is symmetric and $\mu((\Omega\times\Omega)\setminus\Delta)=0$;

(b) $\mu(B\times\Omega)=\nu(B)=0$ for all Borel sets $B\subset\Omega$with zero
capacity;

(c) for every $u\in H^{1}_{0}(\Omega)$

$F(u)=\int_{\Omega\times\Omega}|\widetilde{u}(x)-\widetilde{u}(y)|^{2}d\mu(x,y)+\int_{\Omega}|\widetilde{u}(x)|^{2}d\nu(x)+\int_{\Omega}|\nabla
u(x)|^{2}\,dx;$ (2.37)

If, in addition, $a_{k}$ satisfies property (2.36), then $\nu=0$.

###### Proof.

This corollary is an immediate consequence of Theorems 2.2, 2.4, and 2.5,
noting that the hypothesis of Theorem 2.5(c) is satisfied thanks Proposition
2.1(d). The last statement follows from Proposition 2.7. ∎

######  Remark 2.9 (extension to general double integrals).

The conclusions of Corollary 2.8 remain valid if we consider the functionals
defined by

$F_{k}(u)=\int_{\Omega\times\Omega}|\widetilde{u}(x)-\widetilde{u}(y)|^{2}d\mu_{k}(x,y)+\int_{\Omega}|\nabla
u(x)|^{2}\,dx$ (2.38)

for every $u\in H^{1}_{0}(\Omega)$, with condition (2.1) substituted by

$\mu_{k}(\Omega\times\Omega)\leq M\hbox{ for every }k\in\mathbb{N},$ (2.39)

and (2.36) substituted by

$\mu_{k}((\Omega\times\Omega)\setminus(K_{\varepsilon}\times
K_{\varepsilon}))\leq\varepsilon\hbox{ for every }k\in\mathbb{N}.$ (2.40)

##  3 The counterexample

We fix $p\in(1,+\infty)$. For simplicity of notation we suppose $0\in\Omega$,
and let $B_{r}$ be the ball of centre $0$ and radius $r$. We fix a sequence of
positive numbers $\varepsilon_{k}$ converging to $0$, and define the
functionals $F_{k}\colon W^{1,p}_{0}(\Omega)\to\mathbb{R}$ by setting

$F_{k}(u)={1\over|B_{\varepsilon_{k}}|}\int_{\Omega}\int_{B_{\varepsilon_{k}}}|u(x)-u(y)|^{p}\,dy\,dx+\int_{\Omega}|\nabla
u(x)|^{p}\,dx$ (3.1)

for every $u\in W^{1,p}_{0}(\Omega)$.

######  Definition 3.1.

For all $u\in L^{p}(\Omega)$ we define $m_{p}(u)$ as the unique minimum point
of

$t\mapsto\int_{\Omega}|u(x)-t|^{p}\,dx.$

######  Lemma 3.2.

The map $m_{p}\colon L^{p}(\Omega)\to\mathbb{R}$ is continuous.

###### Proof.

Let $u_{k}\to u$ in $L^{p}(\Omega)$. Then the sequence $m_{p}(u_{k})$ is
bounded since

$|m_{p}(u_{k})|\leq\|u_{k}-m_{p}(u_{k})\|_{L^{p}(\Omega)}+\|u_{k}\|_{L^{p}(\Omega)}\leq
2\|u_{k}\|_{L^{p}(\Omega)}.$

We can suppose, upon subsequences, that $m_{p}(u_{k})\to t_{0}$. With fixed
$t\in\mathbb{R}$, we can pass to the limit in the inequalities

$\int_{\Omega}|u_{k}(x)-m_{p}(u_{k})|^{p}\,dx\leq\int_{\Omega}|u_{k}(x)-t|^{p}\,dx,$

and obtain that

$\int_{\Omega}|u(x)-t_{0}|^{p}\,dx\leq\int_{\Omega}|u(x)-t|^{p}\,dx,$

which concludes the proof. ∎

######  Theorem 3.3.

If $p\in(1,d]$ then the $\Gamma$-limit of $F_{k}$ with respect to the weak
$W^{1,p}_{0}$-convergence is the functional $F$ defined by

$F(u)=\int_{\Omega}|u(x)-m_{p}(u)|^{p}\,dx+\int_{\Omega}|\nabla u(x)|^{p}\,dx$
(3.2)

for every $u\in W^{1,p}_{0}(\Omega)$.

###### Proof.

Let $u_{k}\rightharpoonup u$ weakly in $W^{1,p}_{0}(\Omega)$. Then also
$u_{k}\to u$ strongly in $L^{p}(\Omega)$. Hence, using Jensen’s inequality,
the minimality of $m_{p}(u_{k})$, and applying Lemma 3.2, we get

$\displaystyle\hskip-56.9055pt\liminf_{k\to+\infty}{1\over|B_{\varepsilon_{k}}|}\int_{\Omega}\int_{B_{\varepsilon_{k}}}|u_{k}(x)-u_{k}(y)|^{p}\,dy\,dx$
$\displaystyle\geq$
$\displaystyle\liminf_{k\to+\infty}\int_{\Omega}\bigg{|}u_{k}(x)-{1\over|B_{\varepsilon_{k}}|}\int_{B_{\varepsilon_{k}}}u_{k}(y)\,dy\bigg{|}^{p}dx$
$\displaystyle\geq$
$\displaystyle\liminf_{k\to+\infty}\int_{\Omega}|u_{k}(x)-m_{p}(u_{k})|^{p}dx\geq\int_{\Omega}|u(x)-m_{p}(u)|^{p}dx.$

Since the term $\int_{\Omega}|\nabla u(x)|^{p}\,dx$ is lower semicontinuous,
this proves the liminf inequality.

To prove the upper bound, we first construct a recovery sequence if $u=0$ in a
neighbourhood of $0$. In this case, let $v_{k}$ be the $p$-capacitary
potential of $B_{\varepsilon_{k}}$ with respect to $\Omega$; that is, the
minimizer of

$\min\Big{\\{}\int_{\Omega}|\nabla v(x)|^{p}\,dx:v\in
W^{1,p}_{0}(\Omega),v=1\hbox{ on }B_{\varepsilon_{k}}\Big{\\}}.$

Since $p\leq d$ it is known that $v_{k}\to 0$ in $W^{1,p}_{0}(\Omega)$ (see
e.g. [5, Section 4.7]). We then set $u_{k}=u+m_{p}(u)v_{k}$, and obtain

$\limsup_{k\to+\infty}F_{k}(u_{k})=\limsup_{k\to+\infty}\Bigl{(}\int_{\Omega}|u_{k}(x)-m_{p}(u)|^{p}\,dx+\int_{\Omega}|\nabla
u_{k}(x)|^{p}\,dx\Bigr{)}=F(u)$

Since $F$ is continuous in $W^{1,p}_{0}(\Omega)$ and the set of function
$C^{\infty}_{c}(\Omega)$ which are $0$ in a neighbourhood of $0$ is dense in
$W^{1,p}_{0}(\Omega)$, the claim follows. ∎

######  Remark 3.4 ($\Gamma$-limit in $W^{1,p}(\Omega)$).

If $\Omega$ is a bounded open set with Lipschitz boundary, then the
functionals defined by (3.1) for $u\in W^{1,p}(\Omega)$ $\Gamma$-converge with
respect to the $L^{p}(\Omega)$ convergence to the functional defined by (3.2)
for $u\in W^{1,p}(\Omega)$. Indeed, in the proof we only use the boundary
condition to deduce the equi-coerciveness of the functionals, a property that
is also assured by the regularity of $\partial\Omega$.

We now want to show that $F$ cannot be represented in the form

$F(u)=\int_{\Omega\times\Omega}f(u(x),u(y))d\mu(x,y)+\int_{\Omega}g(u(x))d\nu(x)+\int_{\Omega}|\nabla
u(x)|^{p}\,dx$

for $u\in C^{\infty}_{c}(\Omega)$, where $f\colon\mathbb{R}^{2}\to\mathbb{R}$
and $g\colon\mathbb{R}\to\mathbb{R}$ are continuous functions, while $\mu$ and
$\nu$ are two positive bounded Radon measures on $\Omega\times\Omega$ and
$\Omega$, respectively. To that end we examine the two integrals with respect
to $\mu$ and $\nu$ separately from the third one.

######  Proposition 3.5.

Let $f\colon\mathbb{R}^{2}\to\mathbb{R}$ and $g\colon\mathbb{R}\to\mathbb{R}$
be continuous functions, and let $\mu$ and $\nu$ be two positive bounded Radon
measures on $\Omega\times\Omega$ and $\Omega$, respectively. Suppose that

$\int_{\Omega}|u(x)-m_{p}(u)|^{p}\,dx=\int_{\Omega\times\Omega}f(u(x),u(y))d\mu(x,y)+\int_{\Omega}g(u(x))d\nu(x)$
(3.3)

holds for $u\in C^{\infty}_{c}(\Omega)$, then the same equality holds also for
$u=1_{A}$, for all $A$ open of $\Omega$; that is,

$\int_{\Omega}|1_{A}(x)-m_{p}(1_{A})|^{p}\,dx=\int_{\Omega\times\Omega}f(1_{A}(x),1_{A}(y))d\mu(x,y)+\int_{\Omega}g(1_{A}(x))d\nu(x).$
(3.4)

###### Proof.

Preliminarily, note that, taking $u=0$ in (3.3), we obtain

$0=f(0,0)\,\mu(\Omega\times\Omega)+g(0)\,\nu(\Omega).$

It is then not restrictive to assume that

$f(0,0)=g(0)=0,$ (3.5)

up to substituting $f(s,t)$ with $f(s,t)-f(0,0)$ and $g(s)$ with $g(s)-g(0)$.

Let now $A$ be an open set relatively compact in $\Omega$, and let $u_{k}$ be
a sequence in $C^{\infty}_{c}(\Omega)$ converging pointwise to $1_{A}$ and
such that $0\leq u_{k}(x)\leq 1_{A}(x)$. By the convergence of $u_{k}$ to
$1_{A}$ in $L^{p}(\Omega)$ and Lemma 3.2 we have the convergence of the left-
hand term in (3.3) to the corresponding term in (3.4). As for the right-hand
side of (3.3), it suffices to apply the Dominated Convergence Theorem. ∎

######  Remark 3.6 (computation of $m_{p}(u)$ for characteristic functions).

For any measurable set $A$ the constant $m_{p}(1_{A})$ is obtained by
minimizing

$\int_{\Omega}|1_{A}(x)-t|^{p}\,dx=|A||1-t|^{p}+(|\Omega|-|A|)|t|^{p}.$

The minimal $t\in[0,1]$ is determined by
$(|\Omega|-|A|)t^{p-1}=|A|(1-t)^{p-1}$; that is, we have

${1-t\over t}=\Bigl{(}{|\Omega|-|A|\over|A|}\Bigr{)}^{1/(p-1)}\quad\hbox{and
}\quad
m_{p}(1_{A})={|A|^{1/(p-1)}\over(|\Omega|-|A|)^{1/(p-1)}+|A|^{1/(p-1)}}.$

######  Remark 3.7.

From the previous remark we have that

$\int\limits_{\Omega}|1_{A}(x)-m_{p}(1_{A})|^{p}\,dx=\Phi_{p}(|A|),$ (3.6)

where

$\Phi_{p}(s):={s(|\Omega|-s)^{p/(p-1)}+(|\Omega|-s)s^{p/(p-1)}\over((|\Omega|-s)^{1/(p-1)}+s^{1/(p-1)})^{p}}={s(|\Omega|-s)\over((|\Omega|-s)^{1/(p-1)}+s^{1/(p-1)})^{p-1}}.$
(3.7)

The following proposition relates the function $\Phi_{p}$ defined in (3.7) and
the measure $\mu$.

######  Proposition 3.8.

Under the assumptions of Proposition 3.5 for all $A,B$ open sets in $\Omega$
with $A\cap B=\emptyset$ we have

$\Phi_{p}(|A|)+\Phi_{p}(|B|)-\Phi_{p}(|A|+|B|)=C_{f}(\mu(A\times
B)+\mu(B\times A)),$ (3.8)

where $C_{f}=f(1,0)+f(0,1)-f(1,1)$.

###### Proof.

From (3.4) and (3.6) we have

$\displaystyle\Phi_{p}(|A|)=f(1,1)\mu(A\times
A)+f(1,0)\mu(A\times(\Omega\setminus A))$ $\displaystyle\hskip
128.0374pt+f(0,1)\mu((\Omega\setminus A)\times A)+g(1)\nu(A),$ (3.9)

and analogous formulas for $\Phi_{p}(|B|)$ and
$\Phi_{p}(|A|+|B|)=\Phi_{p}(|A\cup B|)$, from which the claim follows. ∎

######  Proposition 3.9.

If there exists a bounded Radon measure $\mu$ on $\Omega\times\Omega$ such
that (3.8) holds, then $p=2$.

###### Proof.

Take $A_{1}$, $A_{2}$, and $B$ disjoint open subsets of $\Omega$, and let
$s_{1}=|A_{1}|$, $s_{2}=|A_{2}|$, and $t=|B|$. From (3.8) we then have

$\Phi_{p}(s_{1}+s_{2})+\Phi_{p}(t)-\Phi_{p}(s_{1}+s_{2}+t)=C_{f}(\mu((A_{1}\cup
A_{2})\times B)+\mu(B\times(A_{1}\cup A_{2})))$ $=C_{f}(\mu(A_{1}\times
B)+\mu(B\times A_{1})+\mu(A_{2}\times B)+\mu(B\times A_{2}))$
$=\Phi_{p}(s_{1})+\Phi_{p}(t)-\Phi_{p}(s_{1}+t)+\Phi_{p}(s_{2})+\Phi_{p}(t)-\Phi_{p}(s_{2}+t),$

or, equivalently, that for every fixed $t\in(0,|\Omega|)$ the function

$g(s)=\Phi_{p}(s)+\Phi_{p}(t)-\Phi_{p}(s+t)$

is additive on $(0,|\Omega|-t)$, which implies that there exists a constant
$c_{t}$ such that $g(s)=c_{t}s$. In particular, taking into account the
differentiability of $\Phi_{p}$, we have

$\Phi^{\prime\prime}_{p}(s)-\Phi^{\prime\prime}_{p}(s+t)=g^{\prime\prime}(s)=0\hbox{
for all }s,t\in(0,|\Omega|)\hbox{ such that }s+t<|\Omega|.$

This implies that $\Phi^{\prime\prime}_{p}$ is constant, so that it equals a
second-order polynomial $P$.

It is now convenient to write $\Phi_{p}(s)=s(|\Omega|-s)(h_{p}(s))^{1-p}$,
where

$h_{p}(s)=s^{1/(p-1)}+(|\Omega|-s)^{1/(p-1)}.$

Since $h_{p}(s)\neq 0$ we have $P(s)=0$ if and only if $s=0$ or $s=|\Omega|$,
so that $P(s)=\kappa s(|\Omega|-s)$ for some constant $\kappa$. This implies
that $h_{p}(s)=\kappa$ for every $s\in(0,|\Omega|)$ and then also for $s=0$
and $s=|\Omega|$ by continuity. In particular, this gives

$|\Omega|^{1/(p-1)}=h_{p}(0)=h_{p}\Bigl{(}{|\Omega|\over
2}\Bigr{)}=2\Bigl{(}{|\Omega|\over 2}\Bigr{)}^{1/(p-1)},$

which holds only if $p=2$. ∎

Combining the previous results, we are now in a position to prove that $F$
cannot be represented in an integral form when $1<p\leq d$ and $p\neq 2$.

######  Corollary 3.10.

Let $1<p\leq d$ and let $F$ be the $\Gamma$-limit, with respect to the weak
$W^{1,p}_{0}$-convergence, of the sequence $F_{k}$ defined by (3.1). Suppose
that there exist two real valued continuous functions $f$ and $g$, defined on
$\mathbb{R}^{2}$ and $\mathbb{R}$, and two positive bounded Radon measures on
$\Omega\times\Omega$ and $\Omega$, respectively, such that

$F(u)=\int_{\Omega\times\Omega}f(u(x),u(y))d\mu(x,y)+\int_{\Omega}g(u(x))d\nu(x)+\int_{\Omega}|\nabla
u(x)|^{p}\,dx$ (3.10)

for every $u\in C^{\infty}_{c}(\Omega)$. Then $p=2$ and in this case we have
$\mu={1\over 2|\Omega|}\mathcal{L}^{2d}$ and $\nu=0$, while $f(s,t)=|s-t|^{2}$
for every $s,t\in\mathbb{R}$.

###### Proof.

By Theorem 3.3 the functional $F$ is given by (3.2). By (3.10) this implies
that the assumptions of Proposition 3.5 are satisfied, and by Proposition 3.8
we can apply Proposition 3.9, which gives $p=2$. The explicit form of of $f$,
$\mu$, and $\nu$ follows from (1.5) and (3.2). ∎

Acknowledgements. This paper is based on work supported by the National
Research Project (PRIN 2017BTM7SN) ”Variational Methods for Stationary and
Evolution Problems with Singularities and Interfaces”, funded by the Italian
Ministry of University and Research. The authors are members of the Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

## References

  * [1] A. Braides. A simplified counterexample to the integral representation of the relaxation of double integrals. ArXiv article https://arxiv.org/abs/2212.11200
  * [2] A. Braides and G. Dal Maso. Continuity of some non-local functionals with respect to a convergence of the underlying measures. J. Math. Pures Appl 170 (2023), 136–149 .
  * [3] A. Braides and G. Dal Maso. Compactness for a class of integral functionals with interacting local and non-local terms. Calc. Var. Partial Diff. Equations, to appear.
  * [4] G. Dal Maso. An Introduction to $\Gamma$-convergence. Birkhäuser, Basel, 1994.
  * [5] L.C. Evans and R.F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, 1992.
  * [6] M. Fukushima. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam, 1980.
  * [7] C. Kreisbeck and E. Zappale. Loss of double-integral character during relaxation. SIAM J. Math. Anal. 53 (2021), 351–385.
  * [8] C. Mora-Corral and A. Tellini. Relaxation of a scalar nonlocal variational problem with a double-well potential. Calc. Var. Partial Differential Equations 59 (2020), 67.
  * [9] U. Mosco. Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 (1994), 368–421.


# Deflection of light by magnetars in the generalized Born-infeld
electrodynamics

Jin Young Kim 111E-mail address<EMAIL_ADDRESS>Department of Physics,
Kunsan National University, Kunsan 54150, Korea

###### Abstract

We study the deflection of light by a magnetic dipole field in the generalized
Born-Infeld electrodynamics. Using the effective index of refraction and the
trajectory equation based on geometric optics, we compute the weak bending
angle of light passing on the equator of the magnetic dipole. In the limit
where the classical Born-Infeld parameter is infinite, the bending angle agree
with the one computed from the Euler-Heisenberg Lagrangian. We also compute
the bending angle using the geodesic equation of the effective metric induced
by a massive object with magnetic dipole. In the massless limit the bending
angle agrees with the computation using the trajectory equation. We apply the
result to magnetars to estimate the order of magnitude for the bending angle.

deflection of light, Born-Infeld electrodynamics, magnetar

##  I Introduction

The classical Born-Infeld action was introduced to solve the infinite self-
energy of a point charge in Maxwell’s electrodynamics [1, 2]. Regardless of
the existence of classical Born-Infeld electrodynamics, in the strong field
regime, the nonlinear electrodynamic effects emerge from the vacuum
polarization of quantum electrodynamics. The one-loop effective action is
known as the Euler-Heisenberg Lagrangian [3, 4]. In the nonlinear
electromagnetic theory, the speed of light in the presence of strong electric
or magnetic field background is not constant and depends on the field
strengths. In the classical Born-Infeld theory, the electric field strength
that the nonlinear effect becomes significant is estimated of the order
$10^{20}{\rm V/m}$ [5] corresponding to magnetic induction of the order
$10^{11}{\rm T}$. In quantum electrodynamics, the relevant field strength is
$B_{c}=m_{e}^{2}c^{2}/e\hbar\simeq 4.4\times 10^{9}{\rm T}$ [4].

There have been many ground laboratory experiments to observe the quantum
electrodynamic nonlinearity [6, 7, 8]. However, at strong field strengths
unaccessible on Earth, it seems that charged or magnetized astronomical
objects might give a deeper understanding of nonlinear electrodynamics.
Various nonlinear electrodynamic models are adopted to many topics of
astrophysics and cosmology [9, 10, 11, 12]. Interests in astrophysical compact
objects with extreme electromagnetic field are increasing, especially, in
charged black holes and magnetars.

In geometric optics a light ray passing regions with nonuniform index of
refraction bends towards higher index of refraction. The light ray passing the
massive astronomical objects also bends by the general relativistic effect.
The path of light is described by the geodesic equation. The light bending by
the gradient in the gravitational field of a massive object known as the
gravitational lensing is a very useful tool in astronomy and astrophysics.
When the light ray is passing not close to the compact objects, the deflection
is small and one can compute the bending angle in weak field approximation
[13]. If the compact objects have strong electric or magnetic field, the path
of light is also affected by the nonlinear interaction between the light and
the strong background field. One can compute the trajectory from the geodesic
equation including both mass and charge (or magnetic dipole). If one focuses
on the bending by purely nonlinear electromagnetic effect, one can compute the
bending angle in the geometric optics formalism [14, 15, 16].

In the previous work of the author, the deflection angle of light by Einstein-
Born-Infeld black hole was computed in the weak bending limit [17]. The key
point is that, due to nonlinear electrodynamic effects, photons do not
propagate along the null geodesics of the background spacetime. They propagate
along the so-called effective geodesics [18, 19, 20]. Magnetic fields are more
useful than electric fields in many areas of high energy astrophysics, for
example, magnetized neutron stars. The purpose of this paper to compute the
bending angles of light induced by strong magnetic field of magnetars in the
generalized Born-Infeld electrodynamics.

The organization of the paper is as follows. In Sec. II, we begin by briefly
reviewing the generalized Born-Infeld action and the equations of motion. We
consider the propagation of light in the magnetic field background induced by
a magnetic dipole. Using the trajectory equation based on geometric optics, we
compute the weak bending angle when the light is passing on the equatorial
plane of the magnetic dipole. In the limit where the classical Born-Infeld
parameter is infinite we compare the bending angle with the result obtained
from the Euler-Heisenberg Lagrangian. In Sec. III, we consider a massive
object possessing a magnetic dipole moment to see the general relativistic
effects. From the static axially symmetric solution for the metric and the
electromagnetic four-potential, we compute the bending angle using the
effctive geodesics on the equatorial plane. In the massless limit, we show
that the result agrees with the bending angle computed from the trajectory
equation. In Sec. IV, as a possible application of the result to astrophysics,
we estimate the order of magnitude of the bending angle for magnetars. Finally
in Sec. V, we summarize and discuss our results.

##  II Bending angle using geometric optics

The classical Born-Infeld action is described by the Lagrangian

${\cal
L}_{BI}=\beta_{0}^{2}\left(1-\sqrt{1+\frac{2S}{\beta_{0}^{2}}-\frac{P^{2}}{\beta_{0}^{4}}}\right),$
(1)

where $\beta_{0}$ is the classical Born-Infeld parameter characterizing the
possible maximum value of the field strength, $S$ and $P$ are Lorentz-
invariants defined by

$S=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}({\bf B}^{2}-{\bf
E}^{2}),~{}~{}~{}P=\frac{1}{4}F_{\mu\nu}{\tilde{F}}^{\mu\nu}={\bf E}\cdot{\bf
B},$ (2)

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
tensor, and
${\tilde{F}}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$
is its dual tensor. In the limit $\beta_{0}\to\infty$ the above action reduces
to the Maxwell action. Here we use the unit system with
$1/4\pi\epsilon_{0}=\mu_{0}/4\pi=\hbar=c=1$.

In quantum electrodynamics the one-loop correction of the vacuum polarization
also induces non-linear terms, the Euler-Heisenberg Lagrangian,

${\cal
L}_{EH}=\frac{8}{45}\frac{\alpha^{2}}{m_{e}^{4}}S^{2}+\frac{14}{45}\frac{\alpha^{2}}{m_{e}^{4}}P^{2}.$
(3)

As mentioned before the classical Born-Infeld parameter $\beta_{0}$ is
estimated much larger than $E_{c}$. Because the classical Born-Infeld
Lagrangian is the same as the Maxwell Lagrangian for large $\beta_{0}$, the
quantum correction of the classical Born-Infeld electrodynamics to the leading
order would be the same as the Euler-Heisenberg Lagrangian. Thus, the
effective action of the generalized (classical + quantum) Born-Infeld
electrodynamics can be written as [21]

${\cal
L}=\beta^{2}\left(1-\sqrt{1+\frac{2S}{\beta^{2}}-\frac{P^{2}}{\beta^{2}\gamma^{2}}}\right),$
(4)

where $\beta$ and $\gamma$ are defined as

$\frac{1}{\beta^{2}}\equiv\frac{1}{\beta_{0}^{2}}+\frac{16}{45}\frac{\alpha^{2}}{m_{e}^{4}},~{}~{}~{}\frac{1}{\gamma^{2}}\equiv\frac{1}{\beta_{0}^{2}}+\frac{28}{45}\frac{\alpha^{2}}{m_{e}^{4}}.$
(5)

The equations of motion, obtained from the Euler-Lagrange equation and the
Bianchi identity, are

$\partial_{\mu}\left[\frac{1}{\cal
R}\left(F^{\mu\nu}-\frac{P}{\gamma^{2}}{\tilde{F}}^{\mu\nu}\right)\right]=0,$
(6) $\partial_{\mu}{\tilde{F}}^{\mu\nu}=0,$ (7)

where

${\cal R}=\sqrt{1+\frac{2S}{\beta^{2}}-\frac{P^{2}}{\beta^{2}\gamma^{2}}}.$
(8)

The propagation of light in a static uniform magnetic field, assuming that the
background magnetic field ($\bar{\bf B}$) is much stronger than the photon’s
field and is perpendicular to the direction of photon, can be described by the
effective index of refraction given by [21]

$n_{\bot}=\left(1+\frac{{\bar{\bf
B}}^{2}}{\beta^{2}}\right)^{\frac{1}{2}},~{}~{}~{}n_{\parallel}=\left(1+\frac{{\bar{\bf
B}}^{2}}{\gamma^{2}}\right)^{\frac{1}{2}}.$ (9)

where $n_{\bot}$ ($n_{\parallel}$ ) is for the light mode polarized
perpendicular (parallel) to the background magnetic field. Assuming that the
photon is traveling in the $x$-direction and $\bar{\bf B}$ is in the
$z$-direction, $n_{\bot}=n_{y}$ and $n_{\parallel}=n_{z}$. Note that in the
classical Born-Infeld theory where $\beta=\gamma$ the two indices of
refraction are the same. However, they are different including the quantum
correction so that vacuum birefringence effect is relevant.

Because the effective index of refraction depends on the background field, the
light ray can be bent continuously when the background field is non-uniform.
We consider one of the simple cases to compute the bending angle within the
geometric optics formalism. In the weak bending approximation where the impact
parameter is large compared with the radius of the lensing object, the
trajectory equation based on Snell’s law can be written as [15, 16]

$\frac{d{\bf u}}{ds}=\frac{1}{n}({\bf u}\times{\nabla}n)\times{\bf u},$ (10)

where $s$ is the differential distance $ds=|d\vec{\bf
r}|=\sqrt{dx^{2}+dy^{2}+dz^{2}}$ and ${\bf u}={d{\bf r}}/{ds}$ is the unit
vector in the direction of light. It has been confirmed that the bending angle
computed from this trajectory equation exactly agrees with the one obtained
from the eikonal equation [14, 16].

As far as the index of refraction is close to one, Eq. (10) can be written as

$\frac{d{\bf u}}{ds}=\frac{1}{n}({\bf u}_{0}\times{\nabla}n)\times{\bf
u}_{0},$ (11)

where ${\bf u}_{0}$ is the direction of the incoming light. When the light ray
is coming from $x=-\infty$ and moving to $x=+\infty$ with impact parameter
$b$, ${\bf u}_{0}=(1,0,0)$, the trajectory equation in the leading order can
be written as

$\frac{d^{2}x}{ds^{2}}=0,~{}~{}~{}\frac{d^{2}y}{ds^{2}}=\frac{1}{n}\frac{\partial
n}{\partial y},~{}~{}~{}\frac{d^{2}z}{ds^{2}}=\frac{1}{n}\frac{\partial
n}{\partial z}.$ (12)

The first equation gives $ds=dx$ and we have

$\frac{d^{2}y}{dx^{2}}=\frac{1}{n}\frac{\partial n}{\partial
y},~{}~{}~{}\frac{d^{2}z}{dx^{2}}=\frac{1}{n}\frac{\partial n}{\partial z}.$
(13)

We will use Eq. (13) to compute the bending angles of light for the effective
indices of refraction given by Eq. (9). If we take $\varphi$ as the angle
between $\bf u$ and the $x$ axis, the slope of the trajectory is
$y^{\prime}=dy/dx=\tan\varphi\simeq\varphi$. Then the bending angle can be
obtained by integration

$\Delta\varphi=y^{\prime}(\infty)-y^{\prime}(-\infty).$ (14)

Now we consider the bending of light by a magnetic dipole. The magnetic
induction induced by a magnetic dipole $\bf m$, located at the origin, is

${\bar{\bf B}}=\frac{3({\bf m}\cdot{\bf r}){\bf r}}{r^{5}}-\frac{\bf
m}{r^{3}}.$ (15)

Contrary to the electric field by a Coulomb charge which is isotropic, the
magnetic induction by a magnetic dipole is anisotropic. So the bending angle
depends on the orientation of magnetic dipole relative to the incoming light
ray. For simplicity of computation, we consider the simple case when the light
ray is passing in the equatorial plane of the magnetic dipole (Fig. 1).

Figure 1: Schematic of light bending when a light ray passes on the equatorial
plane of a magnetic dipole

Taking the direction of magnetic dipole as the $z$-axis ${\bf m}=\mu\hat{z}$,
the magnetic induction on the equator is given by

${\bar{\bf B}}=-\frac{\mu}{r^{3}}{\hat{z}},$ (16)

where $r=\sqrt{x^{2}+y^{2}}$. The indices of refraction can be written as

$n_{\bot}=\left(1+\frac{\mu^{2}}{\beta^{2}r^{6}}\right)^{\frac{1}{2}},~{}~{}~{}n_{\parallel}=\left(1+\frac{\mu^{2}}{\gamma^{2}r^{6}}\right)^{\frac{1}{2}}.$
(17)

Because $\bar{\bf B}={\bar{\bf B}}(x,y)$ on the equator, the index refraction
does not depend on the coordinate $z$. Thus, there is no bending in the
$z$-direction from Eq. (13). The bending angle in the $y$-direction for the
perpendicular mode can be computed from

$y^{\prime\prime}=-3\frac{\mu^{2}}{\beta^{2}}\frac{y}{r^{8}}\frac{1}{1+\frac{\mu^{2}}{\beta^{2}}\frac{1}{r^{6}}}.$
(18)

Because the slope $dy/dx=y^{\prime}$ of the trajectory $y(x)$ is the direction
of light, the boundary conditions that the incoming ray with impact parameter
$b$ is in the direction of $+x$-direction can be written as

$y(-\infty)=b,~{}~{}~{}~{}y^{\prime}(-\infty)=0.$ (19)

Integrating Eq. (18) with the boundary conditions, the bending angle obtained
from Eq. (14) is

$\Delta\varphi_{\bot}=\int_{-\infty}^{\infty}y^{\prime\prime}dx=-3\pi\left[1-\frac{1}{3}\frac{1}{\sqrt{1+a^{2}}}-\frac{\sqrt{2}}{3}\frac{1}{(1-a^{2}+a^{4})^{1/4}}\left(1+\frac{1-a^{2}/2}{\sqrt{1-a^{2}+a^{4}}}\right)^{1/2}\right],$
(20)

where $a$ is defined as

$a\equiv\left(\frac{\mu}{\beta b^{3}}\right)^{1/3}.$ (21)

The bending angle for the parallel mode can be obtained by replacing $\beta$
with $\gamma$ in Eq. (18).

Let us check whether Eq. (20) reduces to the bending angle computed from the
Euler-Heisenberg Lagrangian when the classical Born-Infeld parameter becomes
infinite. By series expansion for small $a$, we obtain

$\Delta\varphi_{\bot}=-\frac{15\pi}{16}\frac{\mu^{2}}{\beta^{2}b^{6}}+\cdots.$
(22)

The negative sign means that the bending occurs toward the magnetic dipole. In
the limit $\beta_{0}\to\infty$, replacing $1/\beta^{2}$ with
$16\alpha^{2}/45m_{e}^{4}$ from Eq. (5), the leading order term becomes

$\Delta\varphi_{\bot}=-\frac{\pi}{3}\frac{\alpha^{2}}{m_{e}^{4}}\frac{\mu^{2}}{b^{6}}.$
(23)

For the parallel mode we obtain
$\Delta\varphi_{\parallel}=(7/4)\Delta\varphi_{\bot}$. This result exactly
matches the bending angle obtained from the Euler-Heisenberg Lagrangian [14,
16].

##  III Bending angle using geodesic equation

In the previous section, we computed the bending angle by a magnetic dipole
using the trajectory equation, which is based on geometric optics. If we are
interested in the light bending by a compact astronomical object with strong
magnetic field we have to consider the general relativistic effect. The
bending of light by an electrically charged Born-Infeld black hole was
computed using the geodesic equation [17]. In the limit where the mass of
black hole is going to zero, it was confirmed that the bending angle agree
with the one computed from the geometric optics formalism. Here we compute the
bending angle of light by a compact object with mass and magnetic dipole
moment.

For simplicity and comparision with the result in the previous section, we
consider the light bending in the equator of the static magnetic dipole. We
start from the Einstein-Born-Infeld action

$S=\int d^{4}x\sqrt{-g}\left(\frac{R}{16\pi}+\frac{1}{4\pi}{\cal L}\right),$
(24)

where ${\cal L}$ is given by Eq. (4). We use the unit system with
$1/4\pi\epsilon_{0}=\mu_{0}/4\pi=G=c=1$. The equations of motion can be
obtained by varying the action with respect to $g_{\mu\nu}$ and $A_{\mu}$ with
the Bianchi identity. Since we consider only the background magnetic component
of $F_{\mu\nu}$ is nonzero (${\bar{\bf B}}\neq 0,~{}{\bar{\bf E}}=0$), there
is no contribution from the invariant $P$. In this case the equations of
motion are given by

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=8\pi
T_{\mu\nu},$ (25)
$\displaystyle\nabla\left(\frac{F_{\mu\nu}}{\sqrt{1+\frac{2S}{\beta^{2}}}}\right)$
$\displaystyle=0,$ (26)

where $T_{\mu\nu}$ is the energy momentum tensor given by

$T_{\mu\nu}=\frac{1}{4\pi}\left[\beta^{2}\left(1-\sqrt{1+\frac{2S}{\beta^{2}}}\right)g_{\mu\nu}+\frac{F_{\mu\alpha}F^{\alpha}_{~{}~{}\nu}}{\sqrt{1+\frac{2S}{\beta^{2}}}}\right].$
(27)

These field equations have axially symmetric solution for the magnetic dipole
moment ${\vec{\mu}}=\mu\hat{z}$ at the center. Because of the axial symmetry,
it is convenient to use the cylindrical coordinates, $x^{\mu}=(t,r,z,\phi)$,
with the following ansatzs for the metric and the electromagnetic four-
potential

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}dx^{\mu}dx^{\nu}=f(r,z)dt^{2}-g(r,z)(dr^{2}+dz^{2})-r^{2}h(r,z)d\phi^{2},$
(28) $\displaystyle A_{\mu}$ $\displaystyle=$ $\displaystyle(0,0,0,-\psi).$
(29)

The power series solutions in powers of gravitational constant was obtained by
Martin and Pritchett [22]. Up to the first order in the gravitational
constant, the solutions are given by

$\displaystyle f(r,z)$ $\displaystyle=$ $\displaystyle
1-\frac{2GM}{X}+\frac{G\mu^{2}z^{2}}{X^{6}},$ (30) $\displaystyle g(r,z)$
$\displaystyle=$ $\displaystyle
1+\frac{2GM}{X}-\frac{G\mu^{2}(r^{4}-6r^{2}z^{2}+2z^{4})}{2X^{8}},$ (31)
$\displaystyle h(r,z)$ $\displaystyle=$ $\displaystyle
1+\frac{2GM}{X}-\frac{G\mu^{2}z^{2}}{X^{6}},$ (32) $\displaystyle\psi(r,z)$
$\displaystyle=$ $\displaystyle\frac{\mu
r^{2}}{X^{3}}\left(1+\frac{GM}{2X}\right),$ (33)

where $X$ is the spherical distance $X=\sqrt{r^{2}+z^{2}}$, $M$ is the mass
and we recover $G$ to show the order of powers of the gravitational constant.
The quantum electrodynamic correction to first order for the magnetic
potential, in a flat space-time background, has been computed in spherical
polar coordinates [23]. Similarly the higher order corrections for the
magnetic four-potential $\psi(r,z)$, in curved spacetime background, can also
be considered in cylindrical coordinates.

In the linear Maxwell electrodynamics coupled to gravity, photons and
gravitons follow the same null geodesic made by mass and magnetic dipole.
However, in the nonlinear electrodynamics coupled to gravity, the null
geodesic of the electromagnetic wave is different from the null geodesic of
the gravitational wave due to the nonlinear coupling of the electromagnetic
wave to the background electromagnetic field. The modification of light cone
condition can be represented by [18, 24, 25]

$\left(g^{\mu\nu}+4\frac{{\cal L}_{SS}}{{\cal
L}_{S}}F^{\mu\alpha}F^{\nu}_{~{}~{}\alpha}\right)k_{\mu}k_{\nu}=0,$ (34)

where $k_{\mu}$ is the 4-vector of the propagating photon and ${\cal L}_{S}$
denotes the derivative of ${\cal L}$ with respect to $S$. The effective metric
that makes $k_{\mu}$ a null vector is

${\tilde{g}}^{\mu\nu}=g^{\mu\nu}+4\frac{{\cal L}_{SS}}{{\cal
L}_{S}}F^{\mu\alpha}F^{\nu}_{~{}~{}\alpha}.$ (35)

Substituting Eqs. (28) and (29) into Eq. (35), up to the quadratic order in
Born-Infeld parameter, we obtain

${\tilde{g}}^{\mu\nu}=\pmatrix{g^{00}&0&0&0\cr
0&g^{11}(1-\psi_{r}^{2}/\beta^{2}r^{2})&\psi_{r}\psi_{z}/\beta^{2}r^{2}&0\cr
0&\psi_{r}\psi_{z}/\beta^{2}r^{2}&g^{22}(1-\psi_{z}^{2}/\beta^{2}r^{2})&0\cr
0&0&0&-g^{33}[1-(\psi_{r}^{2}+\psi_{z}^{2})/\beta^{2}r^{2}]},$ (36)

where $\psi_{r}={\partial\psi}/\partial r$ and
$\psi_{z}={\partial\psi}/\partial z$.

Figure 2: Bending angle $\Delta\varphi$ of light ray on the equatorial plane
of the magnetic dipole. The distance of the closest approach $r_{0}$ is the
minimum of the radial coordinate $r$, $b$ is the impact parameter and $r_{s}$
is the radius of the compact object with magnetic dipole.

The trajectory of weak bending on the equatorial plane ($z=0$) can be computed
from the following effective metric, up to the first order in $G$ and second
order in $\mu$,

$ds^{2}_{eq}=B(r)dt^{2}-A(r)dr^{2}-C(r)d\phi^{2},$ (37)

where

$\displaystyle B(r)$ $\displaystyle=$ $\displaystyle 1-\frac{2GM}{r},$ (38)
$\displaystyle A(r)$ $\displaystyle=$ $\displaystyle
1+\frac{2GM}{r}-\frac{G\mu^{2}}{2r^{4}}+\frac{\mu^{2}}{\beta^{2}r^{6}},$ (39)
$\displaystyle C(r)$ $\displaystyle=$ $\displaystyle
1+\frac{2GM}{r}+\frac{\mu^{2}}{\beta^{2}r^{6}}.$ (40)

Following the notation and procedure in [13], the bending angle of light
coming from infinity can be obtained from (Fig. 2)

$\Delta\varphi=2|\varphi(r_{0})-\varphi_{\infty}|-\pi,$ (41)

whete $r_{0}$ is the distance of closest approach and

$\varphi(r_{0})=\varphi_{\infty}+\int_{r_{0}}^{\infty}\left[\frac{A(r)}{C(r)}\right]^{1/2}\left[\frac{C(r)B(r_{0})}{C(r_{0})B(r)}-1\right]^{-1/2}dr.$
(42)

Substituting Eq. (37) into Eq. (42), we have

$\varphi(r_{0})-\varphi_{\infty}=\int_{0}^{1}\frac{dx}{\sqrt{1-x^{2}}}\left[1+\frac{GM}{r_{0}}\frac{1}{1+x}-\frac{G\mu^{2}}{4r_{0}^{4}}x^{4}+\frac{\mu^{2}}{2\beta^{2}r_{0}^{6}}(1+x^{2}+x^{4})+\cdots\right],$
(43)

where $x=r_{0}/r$. The integral yields

$\varphi(r_{0})-\varphi_{\infty}=\frac{\pi}{2}+\frac{2GM}{r_{0}}-\frac{3\pi}{64}\frac{G\mu^{2}}{r_{0}^{4}}+\frac{15\pi}{32}\frac{\mu^{2}}{\beta^{2}r_{0}^{6}}+\cdots.$
(44)

Inserting Eq. (44) in Eq. (41), we have

$\Delta\varphi=\frac{4GM}{r_{0}}-\frac{3\pi}{32}\frac{G\mu^{2}}{r_{0}^{4}}+\frac{15\pi}{16}\frac{\mu^{2}}{\beta^{2}r_{0}^{6}}.$
(45)

The first two terms having the gravitational constant come from the general
relativistic effects while the third term reflects the nonlinear
electrodynamic effects. The relative sign of the second term is different from
the first and the third terms. This means that the contribution of the
magnetic dipole to the total bending angle by nontrivial geodesic is repulsive
while the contribution by nonlinear electrodynamic effect is attractive. For
the mode polarized parallel to the magnetic field ($z$-mode), $\beta$ is
replaced by $\gamma$ in Eq. (45). In the weak deflection limit where
$r_{0}\simeq b$, the third term in Eq. (45) exactly matches the leading order
computed from geometric optics Eq. (22).

##  IV Order-of-magnitude estimation for magnetars

The strongest magnetic field in the present observed universe is observed at
highly magnetized neutron stars known as magnetars. The magnetic field on the
surface of magnetars is estimated to reach up to the order $10^{11}\rm T$ [26,
27]. Let us apply the results to magnetars for the order-of-magnitude
estimation. For this purpose we decompose Eq. (45) according to the order of
the impact parameter $b$ as

$\Delta\varphi=\Delta\varphi_{1}-\Delta\varphi_{2}+\Delta\varphi_{3},$ (46)

where

$\Delta\varphi_{1}=\frac{4GM}{b},~{}~{}\Delta\varphi_{2}=\frac{3\pi}{32}\frac{G\mu^{2}}{b^{4}},~{}~{}\Delta\varphi_{3}=\frac{15\pi}{16}\frac{\mu^{2}}{\beta^{2}b^{6}}.$
(47)

From Eq. (5), in the limit $\beta_{0}\to\infty$, $\Delta\varphi_{3}$ can be
written as

$\Delta\varphi_{3}=\frac{\pi}{3}\frac{\alpha^{2}}{m_{e}^{4}}\frac{\mu^{2}}{b^{6}}.$
(48)

Restoring all normalized units, $\Delta\varphi_{1},\Delta\varphi_{2}$ and
$\Delta\varphi_{3}$ can be written as

$\Delta\varphi_{1}=\frac{4GM}{c^{2}b},$ (49)
$\Delta\varphi_{2}=\frac{3\pi}{32}\frac{G}{c^{4}}\frac{\mu_{0}}{4\pi}\frac{\mu^{2}}{b^{4}},$
(50)
$\Delta\varphi_{3}=\frac{\pi}{3}\frac{\alpha^{2}}{m_{e}^{4}}\left(\frac{\hbar^{3}\epsilon_{0}}{c^{3}}\right)\left(\frac{\mu_{0}}{4\pi}\right)^{2}\frac{\mu^{2}}{b^{6}}.$
(51)

It is more useful to express $\Delta\varphi_{2}$ and $\Delta\varphi_{3}$ in
terms of the magnetic induction on the surface of the magnetar and QED
critical field strength as

$\Delta\varphi_{2}=\frac{3\pi}{32}\frac{G}{c^{4}}\frac{4\pi}{\mu_{0}}B_{s}^{2}\frac{r_{s}^{6}}{b^{4}},$
(52)
$\Delta\varphi_{3}=\frac{\alpha}{12}\frac{B_{s}^{2}}{B_{c}^{2}}\frac{r_{s}^{6}}{b^{6}},$
(53)

where $r_{s}$ is the radius of the magnetar,
$B_{s}=(\mu_{0}/4\pi)(\mu/r_{s}^{3})$, $B_{c}=m_{e}^{2}c^{2}/e\hbar=4.4\times
10^{9}{\rm T}$, and $\alpha=e^{2}/4\pi\epsilon_{0}\hbar c$.

For the order-of-magnitude estimation for magnetar, we consider the mass and
radius of the typical neutron star, $M=1.4M_{\rm sun}=2.8\times 10^{30}{\rm
kg}$, $r_{s}=10{\rm km}$, and the maximal surface magnetic field
$B_{s}=10^{11}\rm T$. For these values the Schwarzschild radius is about
$4.2{\rm km}$. The possible maximal bending angle is obtained from a light ray
glancing ($b\simeq r_{s}$) the equator of the magnetar as

$\Delta\varphi_{1}=8.30\times 10^{-1}{\rm
rad},~{}~{}\Delta\varphi_{2}=2.42\times 10^{-8}{\rm
rad},~{}~{}\Delta\varphi_{3}=3.14\times 10^{-1}{\rm rad}.$ (54)

In table 1, we list the numerical estimations of $\Delta\varphi_{1}$,
$\Delta\varphi_{2}$ and $\Delta\varphi_{3}$ for certain values of impact
parameter. Near the equator of the magnetar, the bending caused by nonlinear
electrodynamic effect $\Delta\varphi_{3}$ is comparable to gravitational
lensing by mass $\Delta\varphi_{1}$. Because $\Delta\varphi_{1}\propto b^{-1}$
while $\Delta\varphi_{3}\propto b^{-6}$, we can neglect the lensing by
nonlinear electrobynamic effect away from the magnetar. For example, at
$b=5r_{s}$, we have

$\Delta\varphi_{1}=1.67\times 10^{-1}{\rm
rad},~{}~{}\Delta\varphi_{2}=3.87\times 10^{-11}{\rm
rad},~{}~{}\Delta\varphi_{3}=2.01\times 10^{-5}{\rm rad}.$ (55)

To see the effect of the finiteness of the classical Born-Infeld parameter, we
plot the bending angle $|\Delta\varphi_{\bot}|$ given by Eq. (20) and the
bending angle $\Delta\varphi_{3}$ given by Eq. (53), corresponding to Euler-
Heisenberg limit, compared with the gravitational bending angle
$\Delta\varphi_{1}$ in Fig. 3. We also plot $|\Delta\varphi_{\bot}|$ and the
total bending angle $\Delta\varphi(\simeq\Delta\varphi_{1}+\Delta\varphi_{3})$
in Eq. (45) to see the dependence on mass in Fig. 4.

Table 1: Numerical estimation of each bending angles in Eq. (47) for certain
values of $r_{s}\leq b\leq 10r_{s}$ with $M=1.4M_{\rm sun}$, $r_{s}=10{\rm
km}$ and $B_{s}=10^{11}\rm T$.  Figure 3: The plots of $\Delta\varphi_{1}$
(solid), $|\Delta\varphi_{\bot}|$ (dashed), and $\Delta\varphi_{3}$ (dotted)
for $\beta_{0}=5\times 10^{11}\rm T$ with the same $M$, $r_{s}$ and $B_{s}$ as
in Table 1. As $\beta_{0}$ becomes larger the dashed curve comes closer to the
dotted curve.  Figure 4: The plots of $|\Delta\varphi_{\bot}|$ (dashed) and
$\Delta\varphi$ (solid) for varying mass. Here $M=2M_{\rm sun},1.4M_{\rm
sun},M_{\rm sun}$ (top to bottom) and $\beta_{0}=5\times 10^{11}\rm T$ with
the same $r_{s}$ and $B_{s}$ as in Table 1. As $M\to 0$, the solid curve
approaches to the dotted curve of Fig. 3, corresponding to the bending by
Euler-Heisenberg Lagrangian.

Here we consider the bending of light by magnetic dipole for the simple case
when the light ray is passing on the equator. For a magnetic dipole located at
the origin with arbitrary orientation, the bending can occur in both the
horizontal and the vertical directions. Taking the direction of the incoming
light as $x$-axis, the bending angle of each direction in the Euler-Heisenberg
electrodynamics can be written in terms of the directional cosines
${\hat{\alpha}}={\hat{\mu}}\cdot{\hat{x}},~{}{\hat{\beta}}={\hat{\mu}}\cdot{\hat{y}},~{}{\hat{\gamma}}={\hat{\mu}}\cdot{\hat{z}}$
as [16]

$\Delta\varphi_{3,y}=\frac{\pi}{3\cdot
2^{7}}a\frac{\alpha^{2}}{m_{e}^{4}}\left(\frac{\hbar^{3}\epsilon_{0}}{c^{3}}\right)\left(\frac{\mu_{0}}{4\pi}\right)^{2}\frac{\mu^{2}}{b^{6}}(15{\hat{\alpha}}^{2}+41{\hat{\beta}}^{2}+16{\hat{\gamma}}^{2}),$
(56) $\Delta\varphi_{3,z}=\frac{5\pi}{3\cdot
2^{6}}a\frac{\alpha^{2}}{m_{e}^{4}}\left(\frac{\hbar^{3}\epsilon_{0}}{c^{3}}\right)\left(\frac{\mu_{0}}{4\pi}\right)^{2}\frac{\mu^{2}}{b^{6}}{\hat{\beta}}{\hat{\gamma}},$
(57)

where $a=8$ ($14$) for the perpendicular (parallel) polazation.

##  V Conclusion

We consider the bending of light when it passes in the vicinity of a compact
astrophysical object with strong magnetic field in the generalized Born-Infeld
electrodynamics. To be specific, we compute the weak bending angle when the
light is passing in the equatorial plane of the magnetic dipole. First we
compute the bending angle using the trajectory equation based on geometric
optics and show that it corresponds with the result from the Euler-Heisenberg
Lagrangian in the appropriate limit.

Then we consider the light bending by a massive object possessing magnetic
dipole moment for the general relativistic computation. We compute the bending
angle using the geodesic equation of the static axially symmetric solution for
the metric and the four-potential. We obtain the bending angle as a function
of impact parameter. In the massless limit, we confirm that the result agree
with the bending angle obtained from the trajectory equation. As an
application to astrophysics, we estimate the order of magnitude of the
possible maximum bending angle for magnetars. When the impact parameter is
large, the bending by mass term dominates. However, for magnetars with surface
magnetic field of the order $10^{11}\rm T$, the bending by nonlinear
electromagnetic effects can be comparable to the bending by mass term when the
ray is passing close to the magnetar. There are other nonlinear electrodynamic
consequences by such extremely intense field that can be comparable to those
by mass. For example, it has been shown that the nonlinear electrodynamic
shift from Euler-Heisenberg dipole for strong enough background field can
reach values that are of the order of the gravitational redshift [28].

As mentioned before the classical Born-Infeld parameter $\beta$ is estimated
much larger than the critical field of QED $B_{c}$. Then the nonlinear
electrodynamic bending of light is essentially the same as the bending by
Euler-Heisenberg Lagrangian. Because the bending angle obtained from the
Euler-Heisenberg Lagrangian is polarization-dependent while the bending angle
from general relativistic effect $\Delta\varphi_{1}$ is isotropic,
birefringence of $\Delta\varphi_{3}$ may play an important role for
observation.

Recently astronomers, using the Event Horizon Telescope, have observed the
signature of magnetic field at black holes [29]. If the magnetic field of
black hole is intrinsic and strong enough, although this violates the no-hair
theorem, one might consider the light bending by nonlinear electrodynamic
effects. Close to the event horizon where the bending is strong one should
compute the trajectory numerically from the geodesic equation. Not close to
the event horizon where the bending is weak the same formalism used in this
paper can be applied to compute the bending angle. We leave the extensions of
our results to future work.

## Acknowledgements

This work was supported by Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education,
Science and Technology (NRF-2019R1F1A1060409).

## References

  * [1] M. Born, Proc. R. Soc. London A 143, 410 (1934).
  * [2] M. Born and M. Infeld, Proc. R. Soc. London A 144, 425 (1934).
  * [3] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
  * [4] J. Schwinger, Phys. Rev. 82, 664 (1951).
  * [5] J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, 1998).
  * [6] R. Cameron et al., Phys. Rev. D 47, 3707 (1993).
  * [7] F. Della Valle et al., Phys. Rev. D 90, 092003 (2014).
  * [8] A. Cadene, P. Berceau, M. Fouche, R. Battesti, and C. Rizzo, Eur. Phys. J. D 68, 16 (2014).
  * [9] D. A. Rasheed, arXiv:hep-th/9702087.
  * [10] R. Garcia-Salcedo and N. Breton, Int. J. Mod. Phys. A 15, 4341 (2000).
  * [11] N. Breton, Phys. Rev. D 67,124004 (2003).
  * [12] C. S. Camara, M. R. de Garcia Maia, J. C. Carvalho, and J. A. S. Lima, Phys. Rev. D 69,123504 (2004).
  * [13] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Oxford University, New York, 1983).
  * [14] V. I. Denisov, I. P. Denisova, and S. I. Svertilov, Dokl. Akad. Nauk. Ser. Fiz. 380, 435 (2001).
  * [15] J. Y. Kim and T. Lee, Mod. Phys. Lett. A 26, 1481 (2011).
  * [16] J. Y. Kim and T. Lee, J. Cosmol. Astropart. Phys. 11, 017 (2011) .
  * [17] J. Y. Kim, Eur. Phys. J. C 81, 508 (2021).
  * [18] J. F. Plebanski, Lectures on Nonlinear Electrodynamics (Nordita, Copenhagen, 1970).
  * [19] M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, Phys. Rev. D 61, 045001 (2000).
  * [20] N. Breton, Class. Quantum Grav. 19, 601 (2002).
  * [21] S. I. Kruglov, J. Phys. A 43, 375402 (2010).
  * [22] A. W. Martin and P. L. Pritchett, J. Math. Phys. 9, 593 (1968) .
  * [23] J. S. Heyl and L. Hernquist, J. Phys. A 30, 6475 (1997).
  * [24] V. A. De Lorenci, N. Figueiredo, H. H. Fliche, and M. Novello, Astron. Astrophys. 369, 690 (2001).
  * [25] E. F. Eiroa, Phys. Rev. D 73, 043002 (2006).
  * [26] R. C. Duncan and C. Thompson, Astrophys. J. 408, 194 (1993).
  * [27] C. Thompson and R. C. Duncan, Mon. Not. R. Astron. Soc. 275, 255 (1995).
  * [28] L. Bonetti, S. E. P. Bergliaffa, and A. D. A. M. Spallicci, The Fourteenth Marcel Grossmann Meeting, pp. 3531-3536, (World Scientific, 2017).
  * [29] K. Akiyama et al., Astrophys. J. 910, L13 (2021).