<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%~MoulinÃ© par MaN_auto v.0.33.5 2025-03-26 09:41:18
\documentclass[CRMATH,Unicode,XML,biblatex]{cedram}

\TopicFR{\'Equations aux dÃ©rivÃ©es partielles}
\TopicEN{Partial differential equations}

\addbibresource{CRMATH_Matias_20250007.bib}
% https://studenttheses.uu.nl/bitstream/handle/20.500.12932/41293/MasterThesisFinal.pdf

\usepackage{mathtools}

\DeclareMathOperator*{\supess}{ess\;sup}
\DeclareMathOperator*{\BV}{\mathit{BV}}
\DeclareMathOperator*{\SBV}{\mathit{SBV}}
\DeclareMathOperator*{\SBVzero}{\mathit{SBV_{\mathrm{0}}}}
\DeclareMathOperator*{\PC}{\mathit{PC}}

\newcommand{\deriv}{\mathrm{D}}

\newcommand{\R}[1]{\mathbb{R}^{#1}}
\newcommand{\cI}{\mathcal I}
\newcommand{\loc}{\mathrm{loc}}

\renewcommand{\leq}{\leqslant}

\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}

%%%---------------------------------------------------------------------------

%%% Changer la largeur des adresses
% Jouer avec \hsize :
\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
%  \hspace*{0pt}%
\begingroup\hsize=10.75cm\advance\hsize\abstractmarginL\raggedright
\noindent
%\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm}
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother
%~ La premiÃ¨re page avait trop d'espaces blanc avec les adresses larges.

%%%---------------------------------------------------------------------------

%%% Better widebar macro
\makeatletter
\let\save@mathaccent\mathaccent
\newcommand*\if@single[3]{%
  \setbox0\hbox{${\mathaccent"0362{#1}}^H$}%
  \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}%
  \ifdim\ht0=\ht2 #3\else #2\fi
  }
%The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath:
\newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
%If there's a superscript following the bar, then no negative kern may follow the bar;
%an additional {} makes sure that the superscript is high enough in this case:
\newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}}
%Use a separate algorithm for single symbols:
\newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}}
\newcommand*\wide@bar@[3]{%
  \begingroup
  \def\mathaccent##1##2{%
%Enable nesting of accents:
    \let\mathaccent\save@mathaccent
%If there's more than a single symbol, use the first character instead (see below):
    \if#32 \let\macc@nucleus\first@char \fi
%Determine the italic correction:
    \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}%
    \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}%
    \dimen@\wd\tw@
    \advance\dimen@-\wd\z@
%Now \dimen@ is the italic correction of the symbol.
    \divide\dimen@ 3
    \@tempdima\wd\tw@
    \advance\@tempdima-\scriptspace
%Now \@tempdima is the width of the symbol.
    \divide\@tempdima 10
    \advance\dimen@-\@tempdima
%Now \dimen@ = (italic correction / 3) - (Breite / 10)
    \ifdim\dimen@&gt;\z@ \dimen@0pt\fi
%The bar will be shortened in the case \dimen@&lt;0 !
    \rel@kern{0.6}\kern-\dimen@
    \if#31
      \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}%
      \advance\dimen@0.4\dimexpr\macc@kerna
%Place the combined final kern (-\dimen@) if it is &gt;0 or if a superscript follows:
      \let\final@kern#2%
      \ifdim\dimen@&lt;\z@ \let\final@kern1\fi
      \if\final@kern1 \kern-\dimen@\fi
    \else
      \overline{\rel@kern{-0.6}\kern\dimen@#1}%
    \fi
  }%
  \macc@depth\@ne
  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
  \macc@set@skewchar\relax
  \let\mathaccentV\macc@nested@a
%The following initialises \macc@kerna and calls \mathaccent:
  \if#31
    \macc@nested@a\relax111{#1}%
  \else
%If the argument consists of more than one symbol, and if the first token is
%a letter, use that letter for the computations:
    \def\gobble@till@marker##1\endmarker{}%
    \futurelet\first@char\gobble@till@marker#1\endmarker
    \ifcat\noexpand\first@char A\else
      \def\first@char{}%
    \fi
    \macc@nested@a\relax111{\first@char}%
  \fi
  \endgroup
}
\makeatother

\let\oldbar\bar
\renewcommand*{\bar}[1]{{\mathchoice{\widebar{#1}}{\widebar{#1}}{\widebar{#1}}{\oldbar{#1}}}}

% /!\ Il y a parfois des soucis avec cette macro lorsqu'il y a des caractÃ¨res en minuscule en subscript (et dans les captions). Dans ce cas, faire avec \overline et des espaces manuels.

%%% Double bar
% \overline{\overline{\!a}}

%%%---------------------------------------------------------------------------


\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\let\oldtilde\tilde
\renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}}
\let\oldhat\hat
\renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}}

\title{Lower semicontinuity of nonlocal $L^\infty$ energies on $SBV_0(I)$}
\alttitle{Semi-continuitÃ© infÃ©rieure des Ã©nergies $L^\infty$ non locales sur $SBV_0(I)$}

\author{\firstname{Jos\'e} \lastname{Matias}}
\address{Departamento de Matem\'atica, Instituto Superior T\'ecnico, University of Lisbon, Portugal}
\email{jose.c.matias@tecnico.ulisboa.pt}

\author{\firstname{Pedro Miguel} \lastname{Santos}}
\address[1]{Departamento de Matem\'atica, Instituto Superior T\'ecnico, University of Lisbon, Portugal}
\email{pedro.m.santos@tecnico.ulisboa.pt}

\author{\firstname{Elvira} \lastname{Zappale}}
\address{Department of Basic and Applied Sciences for Engineering, Sapienza-University of Rome, via A.\ Scarpa 16, 00161 Roma, Italy}
\address{CIMA, Universidade de \'Evora, Portugal}
\email{elvira.zappale1@uniroma1.it}

\subjclass{49J45, 26B25, 26A45, 26A51}

\thanks{The research of J.~M.\ was funded by FCT/Portugal through project UIDB/04459/2020 with DOI identifier 10-54499/UIDP/04459/2020. E.~Z.\ acknowledges the support of Piano Nazionale di Ripresa e Resilienza (PNRR) -- Missione 4 ``Istruzione e Ricerca'' -- Componente C2 Investimento 1.1, ``Fondo per il Programma Nazionale di Ricerca e Progetti di Rilevante Interesse Nazionale (PRIN)'' -- CUP 853D23009360006. She is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica ``F.~Severi'' (INdAM). She also acknowledges the partial support of the INdAM--GNAMPA Project ``Metodi variazionali per problemi dipendenti da operatori frazionari isotropi e anisotropi'' coordinated by Alessandro Carbotti, CUP ES324001950001}
\CDRGrant[FCT/Portugal]{UIDB/04459/2020}
\CDRGrant[Piano Nazionale di Ripresa e Resilienza (PNRR)]{853D23009360006}
\CDRGrant[INdAM--GNAMPA]{ES324001950001}

\keywords{\kwd{Cartesian submaximality} \kwd{nonlocal $L^\infty$ energies} \kwd{interfacial energy} \kwd{lower semicontinuity} \kwd{nonlocal gradients} \kwd{supremal functionals}}

\altkeywords{\kwd{Sous-maximalitÃ© cartÃ©sienne} \kwd{Ã©nergies $L^\infty$ non locales} \kwd{Ã©nergie interfaciale} \kwd{semi-continuitÃ© infÃ©rieure} \kwd{gradients non locaux} \kwd{fonctionnelles suprÃ©males}}

\begin{DefTralics}
\def\coloneqq{:=}
\DeclareMathOperator{\supess}{ess\;sup}
\newcommand{\supesss}[1]{\supess_{#1}}
\end{DefTralics}

\newcommand{\supesss}[1]{\supess_{#1}}

\begin{abstract}
We characterize the lower-semicontinuity of nonlocal one-dimensional energies of the type
\[
\supesss{(s,t) \in I\times I} h\bigl([u](s), [u](t)\bigr),
\]
where $I$ is an open and bounded interval in the real line, $u \in \mathit{SBV_{\mathrm{0}}}(I)$ and $[u](r)\coloneqq u(r^+)- u(r^-)$, with $r\in I$.
\end{abstract}

\begin{altabstract}
Nous caractÃ©risons la semi-continuitÃ© infÃ©rieure des Ã©nergies non locales unidimensionnelles
\[
\supesss{(s,t) \in I\times I} h\bigl([u](s), [u](t)\bigr),
\]
oÃ¹ I est un intervalle ouvert et bornÃ© dans l'axe rÃ©el, $u \in \mathit{SBV_{\mathrm{0}}}(I)$ et $[u](r)\coloneqq u(r^+)- u(r^-)$ avec $r\in I$.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

In recent years a great attention has been devoted to nonlocal functionals in the form of double integrals because of their many different applications; see, for instance~\cite{BMCP, BP, BDM, BN}, and the bibliography contained therein, among a wide literature. New mathematical topics, inspired by machine learning issues, presented models involving supremal (or $L^\infty$-)functionals in the nonlocal setting, see~\cite{GZ, KS, KRZ, KZ, RB, S} where energies are either of the form
\[
\supess_{(x,y) \in \Omega\times \Omega} f\bigl(u(x), u(y)\bigr),
\]
with $\Omega \subset \mathbb R^N $, $f\colon\mathbb R^m \times \mathbb R^m \to \mathbb R$ a suitable lower semicontinuous and coercive function, and $u \in L^\infty(\Omega;\mathbb R^m)$, possibly satisfying a PDE's constraint, or
\[
\supess_{x \in \Omega} f\bigl(x, u(x), \nabla^\alpha u(x)\bigr),
\]
with $f\colon\Omega \times \mathbb R^m\times \mathbb R^{N\times m}\to \mathbb R $ a suitable integrand and $u \in \mathcal{S}^{\alpha,\infty}_{u_0}(\Omega;\mathbb R^m)$, with $\alpha \in (0,1)$, $\nabla^\alpha u$ being the Riesz weak fractional gradient and the space $\mathcal{S}^{\alpha,\infty}_{u_0}(\Omega;\mathbb R^m) = \bigl\{u \in L^\infty (\mathbb R^m), \nabla^\alpha u \in L^\infty(\mathbb R^{m\times N}), u= u_0 \in \mathbb R^N\setminus\Omega\bigr\}$.

The interest towards supremal functionals started with the pioneering works by Aronsson, see~\cite{A1,A2,A3,A4}, and has been later developed by many authors, see~\cite{ABP, BJW}, both in connection with the so-called power-law approximation (see, for instance, \cite{CDP,BEZ} for developments in wider contexts than classical Sobolev spaces) and in relation with the $\infty$-laplacian type equations. We refer to~\cite{AK, KM1, KM2} for recent advances on this topic.

$L^\infty$ energies defined at jump points of piecewise continuous functions on the real line were first analyzed in~\cite{ABC}. Indeed, the $L^1$-lower semicontinuity of the functional
\begin{equation}\label{Gdef}
\supess_{t \in I} g\bigl([u](t)\bigr),
\end{equation}
where $I$ is a bounded open interval of the real line, $u$ is a piecewise constant function in the sense of~\cite{AFP} and
\begin{equation}\label{applim}
[u](r)\coloneqq \lim_{\varrho \to r^+} u(\varrho)- \lim_{\varrho \to r^-}u(\varrho),
\end{equation}
was characterized in terms of the lower semicontinuity and the \emph{submaximality} of the density $g$, which can be seen as the supremal counterpart of $\BV$-ellipticity introduced by~\cite{AB} to characterize the lower semicontinuity of elliptic integrands defined on piecewise constant functions.

More specifically, we recall that a function $g \colon \mathbb R\setminus \{0\} \to \mathbb R$ is said to be submaximal if the following inequality holds true for all $x_1, x_2 \in \mathbb R \setminus \{0\}$ such that $x_1 \not= -x_2$:
\begin{equation*}
g(x_1 + x_2)\leq \max\bigl\{g(x_1), g(x_2)\bigr\}.
\end{equation*}
It is also possible to consider $g$ as defined in whole $\mathbb R$, even though its value in $0$ is never taken into account, by setting $g(0) = \inf g$. Note that, in~\cite{ABC}, $g$ is submaximal and lower semicontinuous in $\mathbb R \setminus \{0\}$ if and only if its extension to $0$ is submaximal and lower semicontinuous.

Taking into account the models described before, we consider here the following nonlocal energy
\[
H(u)\coloneqq\supess_{(s,t) \in S(u)\times S(u)} h\bigl([u](s), [u](t)\bigr),
\]
where, as in~\eqref{Gdef}, $u$ is a piecewise constant function on $I$ and $S(u)$ denotes the set of discontinuity points of $u$.

In fact, under natural assumptions on the supremand $h\colon(\mathbb R \setminus\{0\})\times (\mathbb R\setminus \{0\})\to \mathbb R$, which we refer to Definition~\ref{defsymdiag} for, we detect necessary and sufficient conditions on $h$ for the $L^1$-lower semicontinuity of $H$, namely, we prove the following result:

\begin{theo}\label{asprop3.1ABC}
Let $h\colon(\mathbb R\setminus \{0\}) \times (\mathbb R\setminus \{0\}) \to \mathbb R$ be diagonal and symmetric and let $H$ be the functional defined by~\eqref{Hdef}.
$H$ is lower semicontinuous with respect to $L^1(I)$ convergence if and only if $h$ is lower semicontinuous and Cartesian submaximal.
\end{theo}

\section{Notation and preliminaries}

For the reader's convenience, we mainly adopt the notation in~\cite{ABC}. Let $I \subset \R{}$ be open and bounded. We recall that a function $u \in L^1(I)$ is a function of bounded variation if its distributional derivative $\deriv u$ is a measure in $I$. In this case $u$ is approximately differentiable almost everywhere and its a.e.\ derivative is denoted by $u'$; moreover the set of approximate discontinuity points (or jump set) of $u$, denoted by $S(u)$, is at most countable and at each point $t \in S(u)$ there exist the right-hand and left-hand approximate limits, denoted by $u(t^\pm)$. We denote by $[u](t)$ the difference between the two sides approximate limits, i.e.\ $[u](t)\coloneqq u(t^+)- u(t^-)$, the so-called jump of $u$ at $t,$ as in~\eqref{applim}. With this notation, the distributional derivative $\deriv u$ admits the decomposition $\deriv u = u'\lfloor {\mathcal L}^1 + \deriv^{\mathrm s}u = u'\lfloor {\mathcal L}^1 +
\sum_{t \in S(u)} \bigl(u(t^+) - u(t^-)\bigr)\delta_t + \deriv^{\mathrm c}u$, where $\mathcal L^1$ stands for the one-dimensional Lebesgue measure, $\delta_t$ is the Dirac delta at $t$, and $\deriv^{\mathrm c}u$, the Cantor part of $\deriv u$, is a non-atomic measure which is orthogonal to the Lebesgue measure. The notation $\deriv^{\mathrm s}u$ stands for the singular part of the measure $\deriv u$ (with respect to the Lebesgue measure). The space of functions of bounded variation in $I$ is denoted by $\BV (I).$ Its subset, constituted by functions whose $\deriv^{\mathrm c}u$ is null, is denoted by $\SBV(I)$ and the ones where $u'$ is also null is constituted by functions which take countably many constant values, i.e.\ such that $\abs{\deriv^j u}(I) &lt; +\infty$. Furthermore, when the set where $u$ takes constant values has finite perimeter, we say, as in~\cite{BC}, that $u \in \SBVzero(I)$, namely the singular set of $u$ is a Caccioppoli's partition, equivalently the set $S(u)$ has finite perimeter, i.e.\ $\mathcal H^0\bigl(S(u)\bigr)&lt;+\infty$. In the subsequent analysis, we will focus on this class of functions, which, as proven in~\cite{AFP} can be identified with the set of piecewise constant functions, also denoted by $\PC(I)$. In the following, as in~\cite{ABC}, we will also adopt this notation.

For the sake of exposition, $I\coloneqq(a,b)$, $a,b \in \mathbb R$, $a&lt;b$.

\begin{defi}\label{PCfunct_def}
A function $u \colon (a, b) \to \mathbb R$ is said to be piecewise constant on $(a, b)$ if there exist points $a = t_0 &lt; t_1 &lt; \dots &lt; t_N &lt; t_{N+1} = b$ such that
\begin{equation}\label{3.11}
u(t) \;\text{ is constant a.e.\ on }\;(t_{i-1}, t_i) \;\text{ for all }\;i = 1, 2, \dots,N + 1.
\end{equation}
\end{defi}

Then $S(u)$ can be seen as the minimal set $\{t_1, t_2, \dots, t_N\}\subset (a, b)$ such that~\eqref{3.11} holds. The subspace of $L^1(a, b)$ of all such $u$ is denoted by $\PC(a, b)$.

We aim at studying the lower semicontinuity with respect to the $L^1$ topology of the following nonlocal functional $H(u) \colon \PC(I) \to \mathbb R$ of the form
\begin{equation}\label{Hdef}
H(u) = \supess_{(t,s) \in S(u)\times S(u)}h\bigl([u](t), [u](s)\bigr),
\end{equation}
where $h \colon (\mathbb R \setminus\{0\})\times (\mathbb R\setminus \{0\}) \to \bar{\mathbb R}.$

We observe that there is no loss of generality in assuming $h$ bounded, since, as emphasized in~\cite{ABP}, the analysis regarding the behaviour of the functional $H$ is not affected by replacing $h$ by the composition $f \circ h$, with $f$ strictly increasing, such as $f(t)= \arctan t$.

As mentioned earlier, the goal is to characterize the lower semicontinuity of the functional $H$.

It is also possible to define $h$ at points of the type $(\zeta,0)$ or $(0,\theta)$, with an abuse of notation, imposing that $h(\zeta,0)=h(0,\theta)\coloneqq\inf_{(\xi, \eta) \in (\mathbb R\setminus\{0\}) \times (\mathbb R \setminus \{0\})}h(\xi,\eta).$ Clearly, it is possible to consider $h$ defined in all $\mathbb R \times \mathbb R$, tacitly assuming the above values in $(\zeta, 0)$ and $(0,\theta)$, $\zeta, \theta \in \mathbb R$. With this choice, the functional $H$ in~\eqref{Hdef} can be replaced by
\begin{equation*}
H(u) = \supess_{(t,s) \in I\times I}h\bigl([u](t), [u](s)\bigr),
\end{equation*}
without affecting the analysis that we will present below.

\section{Lower semicontinuity}

We start by showing some intermediate results that are important for the proof of Theorem~\ref{asprop3.1ABC} below.

As first observed in the integral setting in~\cite{P1,P2} when dealing with nonlocal functionals, and later expanded in~\cite{KZ, KRZ} in the supremal framework, some symmetry of the densities involved in the nonlocal models appear naturally.

For every $m \in \mathbb N$, given a set $E \subset \mathbb R^m \times \mathbb R^m$, following~\cite[(1.6)]{KZ}, we introduce
\begin{align*}\label{hat E}
\hat E\coloneqq\bigl\{(\xi, \eta) \in E: (\xi,\xi), (\eta, \eta), (\eta,\xi) \in E\bigr\}.
\end{align*}

Let $f\colon\mathbb R \times \mathbb R\to \bar{\mathbb R}$, and $c \in \mathbb R$. The $c$-(sub)level set of $f$ is defined as
\begin{equation*}
L_c(f)\coloneqq\bigl\{(\xi,\eta)\in \mathbb R\times \mathbb R: f(\xi,\eta)\leq c\bigr\}.
\end{equation*}

Given $h\colon(\mathbb R\setminus \{0\}) \times (\mathbb R\setminus \{0\}) \to \mathbb R$, define
\begin{equation}\label{hath}
\hat{h}(\zeta, \eta) \coloneqq \inf \bigl\{ c \in \mathbb{R}: (\zeta, \eta) \in \widehat{L_c(h)},\; (\zeta, \eta) \in \mathbb{R} \times \mathbb{R} \bigr\}.
\end{equation}

\begin{defi}\label{defsymdiag}
A function $h\colon(\mathbb R\setminus \{0\}) \times (\mathbb R\setminus \{0\}) \to \bar{\mathbb R}$ is said to be \emph{symmetric} if $h(\xi,\eta)= h(\eta,\xi)$ for every $\xi, \eta \in \mathbb R\setminus \{0\}$.

A symmetric function $h\colon(\mathbb R\setminus \{0\}) \times (\mathbb R\setminus \{0\}) \to \bar{\mathbb R}$ is said to be \emph{diagonal} if $h(\xi,\eta)= \hat{h}(\xi,\eta)$ for every $\xi, \eta \in \mathbb R\setminus \{0\}$.
\end{defi}
\color{black} Observe that, as in~\cite{ABC}, the functional $H$ in~\eqref{Hdef} coincides with
\begin{equation*}
\mathcal H^0-\supess_{(s,t) \in S(u)\times S(u)} h\bigl([u](s), [u](t)\bigr) \in \bar{\mathbb R}.
\end{equation*}

We stress that there is no loss of generality in assuming $h$ diagonal and symmetric when evaluating $H$ in~\eqref{Hdef}, and, as observed before, that $h$ is finite-valued.

First, as proven in~\cite[equation~(2.8)]{KRZ}, $\hat h$ in~\eqref{hath} admits the following equivalent representation:
\begin{equation}\label{hhat}
\hat h(\xi, \eta) = \max\bigl\{ h(\xi,\xi), h(\eta,\eta), h(\eta,\xi), h(\xi,\eta)\bigr\}.
\end{equation}

\begin{prop}\label{propdiagsym}
Let $h\colon(\mathbb R\setminus \{0\}) \times (\mathbb R\setminus \{0\}) \to \mathbb R$ and let $\hat h$ be as in~\eqref{hath}. Then
\begin{equation*}
\hat H(u)\coloneqq \mathcal H^0-\supess_{(t,s) \in S(u)\times S(u)} \hat h\bigl([u](s), [u](t)\bigr)=H(u).
\end{equation*}
\end{prop}
\begin{proof}
It can be easily verified that the symmetry assumption is not restrictive. It remains to check the diagonality. To this end, it immediately results that $h(\xi, \eta)\leq \hat h(\xi, \eta)$, hence $H \leq \hat H$. To prove the opposite inequality, we can observe that ${\mathcal H}^0- \supess$ in~\eqref{Hdef} is indeed a maximum and is evaluated in a finite set of values, thus it will be a value of the type $h\bigl([u](\bar{s}), [u](\bar t)\bigr)$ with $\bar s, \bar t \in S(u)$, but, clearly by the very definition of~\eqref{Hdef}, it is greater than $h\bigl([u](\bar t), [u](\bar s)\bigr)$, $h\bigl([u](\bar t), [u](\bar t)\bigr)$, $h\bigl([u](\bar s), [u](\bar s)\bigr)$, hence $h\bigl([u](\bar s), [u](\bar t)\bigr) = \hat h\bigl([u](\bar s), [u](\bar t)\bigr)$, hence $\hat H \leq H$.
\end{proof}

In view of the above result, in the sequel, when not explicitly mentioned, we will assume that $h$ is diagonal and symmetric.

\begin{defi}\label{Csubmaxdef}
Let $h\colon(\mathbb R\setminus \{0\}) \times (\mathbb R\setminus \{0\}) \to \mathbb R $ be symmetric and diagonal. We say that $h$ is\/ \emph{Cartesian submaximal} if
\begin{equation}\label{new condition}
h(w_1 + w_2, y) \leq \max \bigl\{ h(w_1, y), h(w_2, y), h(w_1, w_2)\bigr\},
\end{equation}
for every $w_1, w_2, y \in \mathbb R$.
\end{defi}

\begin{rema}\label{new rem}
It is also possible to consider Cartesian submaximal functions $h \colon\mathbb R^+\times \mathbb R^+ \to \mathbb R$. Then a Cartesian submaximal extension $\tilde h$ of such functions to $(\mathbb R\setminus\{0\})\times (\mathbb R \setminus \{0\})$ could be the following
\begin{equation*}
\tilde{h}(\xi,\eta)\coloneqq
\begin{cases}
h(\xi,\eta) &amp; \text{ if }\; (\xi,\eta) \in \mathbb R^+ \times \mathbb R^+,\\
\supess_{(\xi,\eta) \in \mathbb R^+ \times \mathbb R^+} h(\xi,\eta) &amp;\text{ otherwise},
\end{cases}
\end{equation*}
\end{rema}

Condition~\eqref{new condition} characterizes the $L^1(I)$-lower semicontinuity of $H$ in~\eqref{Hdef}. In fact, the following result holds.

\begin{proof}[Proof of Theorem~\ref{asprop3.1ABC}]

Let $H$ be $L^1(I)$-lower semicontinuous, then, taking two sequences $w_n \not = v_n \in \mathbb R$, converging to $w \not =v \in \mathbb R$ respectively, and defining for every $t \in I$ and $n \in \mathbb N\cup\{+\infty\}$,
\begin{equation*}
u_n(t)\coloneqq
\begin{cases}
z &amp;\text{ if }\; t \leq t_0,\\
z + w_n &amp; \text{ if }\; t_0 &lt; t \leq t_1,\\
z +w_n + v_n &amp; \text{ if }\; t &gt; t_1,
\end{cases}
\qquad\text{ and }\qquad
u_\infty(t)\coloneqq
\begin{cases}
z &amp;\text{ if }\; t \leq t_0,\\
z + w &amp; \text{ if }\; t_0 &lt; t \leq t_1,\\
z +w + v &amp; \text{ if }\; t &gt; t_1,
\end{cases}
\end{equation*}
with $t_0 &lt; t_1 \in I$, and taking into account that $u_n \to u_\infty$ in $L^1(I)$, $H(u_n)= h(w_n, v_n)$, and $H(u_\infty)= h(w,v)$, it results that $h(w,v)\leq \liminf_n h(w_n, u_n)$, hence $h$ is lower semicontinuous.

Moreover $h$ turns out to be Cartesian submaximal, indeed, consider $y, w_1,w_2 \in \mathbb R$ such that we can consider the sequence
\[
u_n(t)\coloneqq
\begin{cases}
z &amp;\text{ if }\; t \leq t_0,\\
z + y &amp; \text{ if }\; t_0 &lt; t \leq t_1,\\
z + y+ w_1&amp; \text{ if }\; t_1 \leq t \leq t_1+ \frac{1}{n},\\
z+ y+ w_1+w_2 &amp;\text{ if }\; t &gt; t_1 + \frac{1}{n}.
\end{cases}
\]
Clearly $u_n \to u_\infty$ in $L^1(I)$, where
\[
u_\infty(t)=
\begin{cases}
z &amp;\text{ if }\; t \leq t_0,\\
z + y &amp; \text{ if }\; t_0 &lt; t \leq t_1,\\
z + y +w_1+w_2 &amp; \text{ if }\; t &gt; t_1.
\end{cases}
\]
Then, by the $L^1(I)$-lower semicontinuity of $H$,
%\begin{align*}
%\supess_{s, t \in S(u)}h\bigl([u_\infty](s), [u_\infty](t)\bigr)= &amp; h(y, w_1+w_2)\leq \\
%\liminf_{n \to +\infty}\supess_{s, t \in S(u)}h\bigl([u_n](s), [u_n](t)\bigr)= &amp;\max\bigl\{h(y, w_1), h(y, w_2), h(w_1,w_2)\bigr\},
%\end{align*}
\begin{multline*}
\supess_{s, t \in S(u)}h\bigl([u_\infty](s), [u_\infty](t)\bigr)= h(y, w_1+w_2)\\
\leq\liminf_{n \to +\infty}\supess_{s, t \in S(u)}h\bigl([u_n](s), [u_n](t)\bigr)= \max\bigl\{h(y, w_1), h(y, w_2), h(w_1,w_2)\bigr\},
\end{multline*}
where in the above equalities we have exploited the symmetry and diagonality of $h$.

In order to prove the opposite implication, assume that $h$ is Cartesian submaximal and lower semicontinuous.

With the same construction as in~\cite[Proof of Theorem~3.1]{ABC}, take $u_n$ converging to $u_\infty$ in $L^1_{\loc}(I)$, hence, up to a not relabelled subsequence, also converging a.e.\ in $I$. Since $u_\infty \in \PC(I)$, there exists $\varepsilon &gt; 0$ such that $\varepsilon &lt; \inf\bigl\{\abs{t-s} : t, s \in S(u_\infty), t \not= s\bigr\}$. Fix $t, t' \in S(u_\infty)$, we can suppose that $u_n(t \pm \varepsilon) \to u_\infty(t \pm \varepsilon) = u_\infty(t\pm)$, $u_n(t' \pm \varepsilon) \to u_\infty(t' \pm \varepsilon) = u_\infty(t'\pm)$ and that, for all $n$, $(t \pm \varepsilon), (t' \pm \varepsilon) \not \in S(u_n).$

By the lower semicontinuity of $h$ it results
\[
h\bigl(\mk[u_\infty](t), [u_\infty](t')\mk\bigr) \!=\! h\bigl(\mk[u_\infty](t \pm \varepsilon), [u_\infty](t'\pm \varepsilon)\mk\bigr) \leq \liminf_{n\to +\infty} h\bigl(\mk u_n(t + \varepsilon)- u_n(t-\varepsilon), u_n(t'+\varepsilon)- u_n(t'-\varepsilon)\mk\bigr).
\]
By the Cartesian submaximality, the right-hand side can be estimated as
%\begin{align*}
%h\bigl(u_n(t + \varepsilon)- u_n(t-\varepsilon), u_n(t'+\varepsilon)- u_n(t'-\varepsilon)\bigr)\leq \\
%\sup_{s, s'\in S_{u_n,\varepsilon, t, t'}} h\bigl([u_n](s), [u_n](s')\bigr) \leq
%\sup_{s,s' \in S(u_n)} h\bigl([u_n](s), [u_n](s')\bigr),
%\end{align*}
\begin{equation*}
\begin{split}
h\bigl(u_n(t + \varepsilon)- u_n(t-\varepsilon), u_n(t'+\varepsilon)- u_n(t'-\varepsilon)\bigr)&amp;\leq
\sup_{s, s'\in S_{u_n,\varepsilon, t, t'}} h\bigl([u_n](s), [u_n](s')\bigr)\\
&amp;\leq \sup_{s,s' \in S(u_n)} h\bigl([u_n](s), [u_n](s')\bigr),
\end{split}
\end{equation*}
where the set $S_{u_n,\varepsilon, t, t'}\coloneqq\bigl(S(u_n)\cap (t-\varepsilon, t+\varepsilon)\bigr)\cup \bigl(S(u_n)\cap (t'-\varepsilon, t' +\varepsilon)\bigr)$. Consequently, taking the supremum as $t, t' \in S(u)$ on the left-hand side, we have the desired inequality:
\[
\sup_{t, t'\in S(u)}h\bigl([u](t), [u](t')\bigr)\leq \liminf_{n\to +\infty} \sup_{t, t' \in S(u_n)} h\bigl([u_n](t), [u_n](t')\bigr).
\]
This concludes the proof.
\end{proof}

We conclude by making some comments and presenting some examples devoted to a better understanding of the notion of Cartesian submaximality.

\begin{rema}
\mbox{}
\begin{itemize}
\item The composition of a submaximal function with a subadditve one is not Cartesian submaximal as the following example shows $h(x,y)\coloneqq\frac{\abs{\sin (x+y)}}{x+y}$: indeed $h= g \circ +$, with $g\colon\mathbb R\setminus \{0\} \to \mathbb R$ defined as \smash{$g(x)\coloneqq\frac{\abs{\sin x}}{x}$} and $+\colon\mathbb R\times \mathbb R\to \mathbb R$; it suffices to consider $x,y,w=\pi/2$ to verify $1=h(x+y,w)&gt; \max\bigl\{h(x,y), h(y,w), h(x,w)\bigr\}= 0$.
\item Each function $h \colon \mathbb R\times \mathbb R \to (0,+\infty)$ which is separately submaximal, i.e.\ such that $h(y, w_1+w_2)\leq \max \bigl\{h(y, w_1), h(y,w_2)\bigr\}$ is also Cartesian submaximal. In particular, if a symmetric function $h\colon \mathbb R^+ \times \mathbb R^+ \to (0,+\infty)$ is such that $h(\,\cdot\,, y)$ is decreasing for a.e.\ $y$ and we extend it~as
\begin{equation}\label{hstar}
h^\star(w, y) =
\begin{cases}
h(w,y), &amp;\text{ if }\; (w, y)\in \mathbb R^+\times\mathbb R^+,\\
0 &amp;\text{ if }\; w =0 \;\text{ or }\; y =0,\\%\times(\mathbb R \setminus \{0\}),\\
+\infty &amp;\text{ otherwise},
\end{cases}
\end{equation}
then $h^\star$ is separately submaximal. As an example one can consider $h(w,y)=\frac{1}{w+y}$ for $w,y \in \mathbb R^+$ and $h^\star$ as in~\eqref{hstar}. The same holds if $h$ is the ratio of a subadditive and superadditive function in each variable.
\item Examples of separately submaximal functions defined in $\mathbb R \times \mathbb R$ with values in $\bar{\mathbb R}$ (and hence Cartesian submaximal) resulting immediately from~\cite[Example~3.3]{ABC}:
\[
h(w, y) =
\begin{cases}
\max\biggl\{ \frac{\abs{\sin w}}{w}, \frac{\abs{\sin y}}{y} \biggr\}, &amp;\text{ if }\; (w,y)\in \mathbb R^+\times\mathbb R^+,\\
0 &amp;\text{ if }\; w=0 \;\text{ or }\; y=0, \\
+\infty &amp;\text{ otherwise},
\end{cases}
\]
and
\[
h_\alpha(w,y) =
\begin{cases}
\alpha\frac{\abs{\sin w}}{w} + (1 - \alpha) \frac{\abs{\sin y}}{y} &amp;\text{ if }\; (w,y)\in \mathbb R^+\times\mathbb R^+, \\
0 &amp;\text{ if }\; w=0 \;\text{ or }\; y=0,\\
+\infty &amp;\text{ otherwise},
\end{cases}
\]
$\alpha \in (0,1)$.
\end{itemize}
\end{rema}

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\printbibliography
\end{document}</pre></body></html>