# Variational problems in $L^{\infty}$ involving semilinear second order
differential operators

Nikos Katzourakis
Department of Mathematics and Statistics University of Reading  Whiteknights
Pepper Lane  Reading  RG6 6AX  UK. E-mail<EMAIL_ADDRESS>Roger
Moser
Department of Mathematical Sciences University of Bath  Bath BA2 7AY  UK.
E-mail<EMAIL_ADDRESS>
###### Abstract

For an elliptic, semilinear differential operator of the form
$S(u)=A:D^{2}u+b(x,u,Du)$, consider the functional
$E_{\infty}(u)=\operatorname*{ess\,sup}_{\Omega}|S(u)|$. We study minimisers
of $E_{\infty}$ for prescribed boundary data. Because the functional is not
differentiable, this problem does not give rise to a conventional Euler-
Lagrange equation. Under certain conditions, we can nevertheless give a system
of partial differential equations that all minimisers must satisfy. Moreover,
the condition is equivalent to a weaker version of the variational problem.

##  1 Introduction

Variational problems involving an $L^{\infty}$-norm tend to present challenges
not shared by more conventional variational problems. Indeed, the underlying
functionals are typically not differentiable, not even in the Gateaux sense,
and therefore, the usual derivation of an Euler-Lagrange equation does not
work. Sometimes it is possible to derive an associated partial differential
equation nevertheless (the Aronsson equation [2] is an example) but such an
equation is typically only degenerate elliptic and may have discontinuous
coefficients [17]. Indeed, for higher order problems (as studied in this
paper), the equations may be fully nonlinear and not elliptic at all [23].
Moreover, while the interesting functionals in the calculus of variations in
$L^{\infty}$ typically enjoy a certain degree of convexity, they are not
strictly convex. Therefore, minimisers are not usually expected to be unique.

All these difficulties notwithstanding, there are some problems in the theory
that are understood very well. This applies in particular to a class of
problems involving first order derivatives of scalar functions. More
precisely, let $\Omega\subseteq\mathbb{R}^{n}$ be a bounded Lipschitz domain
and consider functions $u\colon\Omega\to\mathbb{R}$ with fixed boundary data.
For a function
$F\colon\Omega\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$, consider the
problem of minimising $\operatorname*{ess\,sup}_{x\in\Omega}F(x,u(x),Du(x))$.
Under certain conditions on $F$, there is a good theory giving, for example,
existence of solutions with good properties [1, 4, 3], uniqueness [15, 16],
and (for $F(x,y,z)=|z|^{2}$) regularity [9, 10, 11].

Recently, the authors have studied a different, second order variational
problem in $L^{\infty}$ and established good properties of its solutions [21].
Suppose now that we wish to minimise a quantity such as
$\operatorname*{ess\,sup}_{x\in\Omega}F(x,\Delta u(x))$. Assuming that
$F\colon\Omega\times\mathbb{R}\to\mathbb{R}$ satisfies some growth and
convexity conditions, and that $\Omega$, $F$, and the boundary data are
sufficiently regular, it turns out that there exists a unique minimiser, which
satisfies a certain system of partial differential equations. Conversely, any
solution of that system corresponds to a minimiser. Some of the underlying
ideas go back to earlier work [25, 26], and similar tools have in the meantime
also been used for other problems [22, 24, 12].

In the current paper, we study extensions of these results. This is one of a
pair of works that examine two different types of generalisations. Here we
replace $\Delta u$ by more general, semilinear differential operators, while
restricting our attention to $F(x,\xi)=|\xi|$ (but $x$-dependence is still
included implicitly, because the coefficients of the differential operator
need not be constant any more). In a companion paper [20], we study a quantity
of the form $\operatorname*{ess\,sup}_{x\in\Omega}F(x,u(x),\Delta u(x))$ for a
fairly general class of functions
$F\colon\Omega\times\mathbb{R}^{2}\to\mathbb{R}$. There is of course some
overlap between the two settings, and it would be interesting to have a common
framework, but this appears to be difficult for technical reasons.

We consider the following situation. Let $\Omega\subseteq\mathbb{R}^{n}$, as
above, be a bounded Lipschitz domain. (For some of our results we will need to
impose additional regularity assumptions on $\Omega$.) Let $A\in
C^{2}(\overline{\Omega};\mathbb{R}^{n\times n})$ and $b\in
C^{2}(\overline{\Omega}\times\mathbb{R}\times\mathbb{R}^{n})$. We assume that
there exists $\lambda>0$ such that for every $x\in\Omega$, the matrix $A(x)$
is symmetric and satisfies $A(x):\zeta\otimes\zeta\geq\lambda|\zeta|^{2}$ for
all $\zeta\in\mathbb{R}^{n}$, where the colon denotes the Frobenius inner
product. Define the semilinear differential operator

$S(u)=A:D^{2}u+b(x,u,Du)$ (1)

for $u\colon\Omega\to\mathbb{R}$, where $Du$ is the gradient and $D^{2}u$ is
the Hessian of $u$. We are interested in the functional

$E_{\infty}(u)=\operatorname*{ess\,sup}_{\Omega}|S(u)|.$

So far we have not mentioned the space on which this functional is defined. In
order to obtain a good theory, we need to work in a Sobolev space that may
appear unconventional, but is quite natural for the problem. We define

$\mathcal{W}^{2,\infty}(\Omega)=\bigcap_{1<q<\infty}\left\\{u\in
W^{2,q}(\Omega)\mid A:D^{2}u\in L^{\infty}(\Omega)\right\\}.$

Furthermore,

$\mathcal{W}_{0}^{2,\infty}(\Omega)=\mathcal{W}^{2,\infty}(\Omega)\cap
W_{0}^{2,2}(\Omega).$

Given $u_{0}\in\mathcal{W}^{2,\infty}(\Omega)$, we wish to study minimisers of
$E_{\infty}$ in $u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$. For other
variational problems, the critical points would also be of interest, but in
this case, the concept is not useful as $E_{\infty}$ is not differentiable. We
therefore work with the following idea instead.

######  Definition 1 (Almost-minimiser).

A function $u\in\mathcal{W}^{2,\infty}(\Omega)$ is called an _almost-
minimiser_ of $E_{\infty}$ if there exists $M\in\mathbb{R}$ such that

$E_{\infty}(u)\leq E_{\infty}(u+\phi)+M\|\phi\|_{W^{1,\infty}(\Omega)}^{2}$

for every $\phi\in\mathcal{W}_{0}^{2,\infty}(\Omega)$.

Intuitively, the definition provides a substitute for the idea that the Taylor
expansion of $E_{\infty}$ has a vanishing first order term at $u$. Since
$E_{\infty}$ is not differentiable, it does of course not have a Taylor
expansion. Instead, we use a quadratic form
$Q\colon\mathcal{W}^{2,\infty}(\Omega)\to\mathbb{R}$ such that the graph of
$Q$ touches the graph of $E_{\infty}({\mkern 2.0mu\cdot\mkern
2.0mu}+u)-E_{\infty}(u)$ from below. This is related to the notion of second
order subjets or subdifferentials that appears in the theory of viscosity
solutions of partial differential equations (discussed, e.g., in a survey
article by Crandall, Ishii, and Lions [6] or the introductory text by the
first author [18]). In the case of Definition 1, the condition is formulated
in terms of the norm of $W^{1,\infty}(\Omega)$, so it may be best to think of
$E_{\infty}$ as a functional defined on $W^{1,\infty}(\Omega)$ in this
context, with $E_{\infty}(v)=\infty$ when
$v\not\in\mathcal{W}^{2,\infty}(\Omega)$. There is, however, some flexibility
here. Our main results remain true if $\|{\mkern 2.0mu\cdot\mkern
2.0mu}\|_{W^{1,\infty}(\Omega)}$ is replaced, e.g., by $\|{\mkern
2.0mu\cdot\mkern 2.0mu}\|_{W^{2,q}(\Omega)}$ for any $q<\infty$.

We show below that almost-minimisers can be characterised in terms of a system
of partial differential equations. In order to formulate this, we use the
formal linearisation of the operator $S$ at a point
$u\in\mathcal{W}^{2,\infty}(\Omega)$, denoted $S_{u}^{\prime}$, and its formal
$L^{2}$-adjoint, denoted $S_{u}^{*}$. We use the notation $(x,y,z)$ for the
variables in $\Omega\times\mathbb{R}\times\mathbb{R}^{n}$. We further write
$b_{y}$ for the partial derivative of $b$ with respect to $y$, and $b_{z}$ for
the vector comprising the partial derivatives with respect to
$z_{1},\dotsc,z_{n}$. Then

$S_{u}^{\prime}\phi=A:D^{2}\phi+b_{z}(x,u,Du)\cdot D\phi+b_{y}(x,u,Du)\phi$

and

$S_{u}^{*}f=\operatorname{div}\operatorname{div}(fA)-\operatorname{div}\bigl{(}fb_{z}(x,u,Du)\bigr{)}+fb_{y}(x,u,Du).$

In the div-div term, the divergence is applied once column-wise and once row-
wise.

We are interested in the equation

$S_{u}^{*}f=0.$ (2)

We can make sense of this for $f\in L^{1}(\Omega)$: if

$\int_{\Omega}fS_{u}^{\prime}\phi\,dx=0$ (3)

holds for all $\phi\in C_{0}^{\infty}(\Omega)$, then we say that $f$ is a
_weak solution_ of the equation.

Our first main result is as follows.

######  Theorem 2.

Let $u_{\infty}\in\mathcal{W}^{2,\infty}(\Omega)$ be such that there exist
$e_{\infty}\geq 0$ and $f_{\infty}\in L^{1}(\Omega)\setminus\\{0\\}$
satisfying

$|f_{\infty}|S(u_{\infty})=e_{\infty}f_{\infty}$ (4)

almost everywhere in $\Omega$ and

$S_{u_{\infty}}^{*}f_{\infty}=0$ (5)

weakly. Then $u_{\infty}$ is an almost-minimiser of $E_{\infty}$.

The converse is also true, provided that we impose some additional regularity
on $\partial\Omega$ and on the boundary data, and provided that we restrict
our attention to differential operators that permit certain $L^{p}$-estimates.

######  Definition 3.

For a differential operator $S$ as in (1), we say that $S$ is _admissible_ if
there exists $p_{0}>1$ such that for any $p\geq p_{0}$, the following holds
true. Suppose that $u_{0}\in\mathcal{W}^{2,\infty}(\Omega)$ and $\Lambda>0$.
Then there exists $C>0$ such that for any $u\in
u_{0}+\mathcal{W}_{0}^{2,p}(\Omega)$, if

$\|S(u)\|_{L^{p}(\Omega)}\leq\Lambda,$

then

$\|u\|_{W^{2,p}(\Omega)}\leq C.$

Now we can formulate our second main result.

######  Theorem 4.

Suppose that $\partial\Omega$ is of class $C^{3}$ and $u_{0}\in
C^{2}(\overline{\Omega})$. Further suppose that $S$ is admissible. If
$u_{\infty}\in u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$ is an almost-
minimiser of $E_{\infty}$, then there exist $f_{\infty}\in
L^{1}(\Omega)\setminus\\{0\\}$ and $e_{\infty}>0$ such that (4) holds almost
everywhere in $\Omega$ and (5) holds weakly.

For admissible operators, we can furthermore be certain that minimisers of
$E_{\infty}$ exist for prescribed boundary data. A minimiser is in particular
an almost-minimiser, and thus we are guaranteed that the system (4), (5) has a
non-trivial solution.

######  Proposition 5.

If $S$ is admissible, then $E_{\infty}$ attains its minimum in
$u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$ for any
$u_{0}\in\mathcal{W}^{2,\infty}(\Omega)$.

The proof relies on standard arguments and in particular on the direct method.
For completeness, we outline these arguments anyway.

###### Proof.

Let $(u_{k})_{k\in\mathbb{N}}$ be a minimising sequence. Then
$\|S(u_{k})\|_{L^{\infty}(\Omega)}$ is obviously bounded. Hence if $p_{0}$ is
the number from Definition 3, then it follows that $(u_{k})_{k\in\mathbb{N}}$
is bounded in $W^{2,p}(\Omega)$ for any $p\in[p_{0},\infty)$. Therefore, we
may assume (extracting a subsequence if necessary) that we have the
convergence $u_{k}\rightharpoonup u_{\infty}$ weakly in $W^{2,p}(\Omega)$ for
every $p<\infty$. Moreover, the limit belongs to $u_{0}+W_{0}^{2,p}(\Omega)$.

Using the Sobolev embedding theorem, we further conclude that
$(u_{k})_{k\in\mathbb{N}}$ is bounded in $C^{1,\alpha}(\overline{\Omega})$ as
well for every $\alpha\in(0,1)$. The Arzelà-Ascoli theorem implies that
$u_{k}\to u_{\infty}$ in $C^{1}(\overline{\Omega})$. Hence
$b(x,u_{k},Du_{k})\to b(x,u_{\infty},Du_{\infty})$ uniformly.

Since $(S(u_{k}))_{k\in\mathbb{N}}$ is bounded in $L^{\infty}(\Omega)$, we may
assume that $S(u_{k})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\sigma$
weakly* in $L^{\infty}(\Omega)$ for some $\sigma\in L^{\infty}(\Omega)$. Using
the above convergence, we conclude that $\sigma=S(u_{\infty})$. Now it follows
that

$E_{\infty}(u_{\infty})\leq\liminf_{k\to\infty}E_{\infty}(u_{k}),$

and thus $u_{\infty}$ is a minimiser. ∎

Summarising Theorem 2 and Theorem 4, we can say that the system comprising
equations (4) and (5) is _equivalent_ to the almost-minimising condition under
certain assumptions. If we accept that the latter is a reasonable substitute
for critical points, then we may think of (4) and (5) as a substitute for an
Euler-Lagrange equation.

For a variety of other variational problems in $L^{\infty}$, a corresponding
differential equation has been identified by quite different methods. In the
case of the optimal Lipschitz extension problem, the result is the Aronsson
equation [2]. Formally, Aronsson’s calculations can be carried out for the
above problem as well. They give rise to the equation $S(u)D(S(u))=0$. The
connections between this equation and the variational problem are not explored
in this work, but we observe that the former is satisfied by solutions of (4).

The ‘Aronsson equation’ $S(u)D(S(u))=0$ is of third order, quasilinear, and in
non-divergence form. It is not elliptic in any reasonable sense. It allows
neither weak nor viscosity solutions, but there is another approach that does
apply and has produced some results on equations such as this (see, e.g., the
papers of the first author and coauthors [19, 7, 23]). In this paper, however,
we follow the alternative approach outlined above and consider solutions to
the system (4), (5) instead. This can be seen as a divergence-form (or div-div
form) alternative to the Aronsson equation.

The comparison with other variational problems in $L^{\infty}$ makes another
aspect of the above results remarkable. We note that the system (4), (5) is
local in the sense that if it holds in $\Omega$, then it is automatically
satisfied in any open subdomain $\Omega^{\prime}\subseteq\Omega$. Under the
conditions of Theorem 4, it follows that the almost-minimising condition is
also local in a similar sense. Indeed, if $\partial\Omega$ is of class $C^{3}$
and $u_{0}\in C^{2}(\overline{\Omega})$, then for any almost-minimiser
$u_{\infty}\in u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$, Theorem 4 applies.
Given a Lipschitz subdomain $\Omega^{\prime}\subseteq\Omega$, we can then use
Theorem 2 in $\Omega^{\prime}$, concluding that $u_{\infty}$ is an almost-
minimiser in $\Omega^{\prime}$ as well. This is in stark contrast to the
optimal Lipschitz extension problem and similar variational problems, where
locality must be imposed in order to obtain solutions with good properties.
These solutions are then known as absolute minimisers.

For the special case $S(u)=\Delta u$ (and for certain other problems), a
previous paper [21] gives stronger results. It is shown that solutions of (4),
(5) are unique under the boundary condition $u_{\infty}\in
u_{0}+\mathcal{W}^{2,\infty}(\Omega)$ and correspond to unique minimisers of
$E_{\infty}$. For similar problems involving nonlinear operators, however, we
do not have uniqueness in general [24]. For semilinear operators as in this
paper, the question is open.

In the next section, we discuss second order elliptic, linear equations in
div-div form, of which (5) is an example. We consider solutions in
$L^{1}(\Omega)$ and derive some interior regularity that we will need for the
proofs of our main results. We will also show that weak solutions can be
tested with functions in $\mathcal{W}_{0}^{2,\infty}(\Omega)$. We then prove
Theorem 2 and Theorem 4 in the following two sections. In the final section,
we discuss some specific differential operators of the form $S(u)=\Delta
u+g(u)$ for a function $g\colon\mathbb{R}\to\mathbb{R}$. In particular, we
give some conditions that imply that the operator is admissible in the sense
of Definition 3. The purpose of this section is not just to show that Theorem
4 is not vacuous, but also to give an idea of the nonlinearities allowed.

##  2 Elliptic equations in div-div form

In this section, we prove some properties of weak solutions of an equation of
the form

$\operatorname{div}\operatorname{div}(fA)+\operatorname{div}(fB)+fc=\operatorname{div}G+g,$
(6)

where $f$ is assumed to be in $L^{1}(\Omega)$ or even a Radon measure on
$\Omega$. The matrix $A$ will be the same as in the introduction and will be
fixed throughout. The properties of the coefficients $B$ and $c$ are described
below. We eventually apply these results to equations such as
$S_{u}^{*}f=\operatorname{div}G+g$ for some $u\in W^{2,p}(\Omega)$, or even to
$S_{u_{\infty}}^{*}f=0$ for a function
$u_{\infty}\in\mathcal{W}^{2,\infty}(\Omega)$, but we formulate them more
generally here.

First we prove some interior regularity for weak solutions of the equation. We
duplicate some results from a more general theory here (see, e.g., the survey
article of Bogachev, Krylov, and Röckner [5]). In order to make the paper
self-contained, we include a proof nevertheless.

######  Lemma 6.

For any $\Omega^{\prime}\Subset\Omega$ and any $p\in(n,\infty)$, there exists
a constant $C>0$ such that the following holds true. Let
$p^{\prime}=\frac{p}{p-1}$ be the exponent conjugate to $p$. Suppose that
$B\in L^{\infty}(\Omega;\mathbb{R}^{n})$, $c\in L^{\infty}(\Omega)$, $g\in
L^{1}(\Omega)$, and $G\in L^{p^{\prime}}(\Omega;\mathbb{R}^{n})$. Set

$\Gamma=\|B\|_{L^{\infty}(\Omega)}+\|c\|_{L^{\infty}(\Omega)}+1.$

Suppose that $\mu\in(C_{0}^{0}(\Omega))^{*}$ is a distributional solution of

$\operatorname{div}\operatorname{div}(A\mu)+\operatorname{div}(B\mu)+c\mu=\operatorname{div}G+g,$

meaning that

$\int_{\Omega}(A:D^{2}\phi-B\cdot
D\phi+c\phi)\,d\mu=\int_{\Omega}(g\phi-G\cdot D\phi)\,dx$ (7)

for every $\phi\in C_{0}^{\infty}(\Omega)$. Then $\mu$ is absolutely
continuous with respect to the Lebesgue measure. Its Radon-Nikodym derivative
$f$ belongs to $W_{\mathrm{loc}}^{1,p^{\prime}}(\Omega)$ and satisfies

$\|f\|_{W^{1,p^{\prime}}(\Omega^{\prime})}\leq
C\Gamma^{2}\left(|\mu|(\Omega)+\|G\|_{L^{p^{\prime}}(\Omega)}+\|g\|_{L^{1}(\Omega)}\right).$

###### Proof.

Consider a function $\chi\in C_{0}^{\infty}(\Omega)$. Define a functional
$\alpha\in(C_{0}^{1}(\Omega))^{*}$ by

$\alpha(\psi)=\int_{\Omega}\chi\psi\,d\mu,\quad\psi\in C_{0}^{1}(\Omega).$

(This means that $\alpha$ corresponds to the measure $\chi\mu$, but we regard
it as a functional on $C_{0}^{1}(\Omega)$ at first.) Choose an open,
precompact set $\Omega^{\prime\prime}\Subset\Omega$ with smooth boundary and
with $\operatorname{supp}\chi\subseteq\Omega^{\prime\prime}$. Given $\psi\in
C_{0}^{1}(\Omega)$, we can solve the equation

$A:D^{2}\phi=\psi$

in $W^{2,p}(\Omega^{\prime\prime})\cap W_{0}^{1,p}(\Omega^{\prime\prime})$ by
[13, Theorem 9.15]. Moreover, [13, Theorem 9.19] implies that $\phi\in
C^{2}(\Omega^{\prime\prime})$. By approximation, we then see that (7) is
satisfied for the test function $\chi\phi$. Thus we obtain

$\begin{split}\alpha(\psi)&=\int_{\Omega}\chi A:D^{2}\phi\,d\mu\\\
&=\int_{\Omega}\bigl{(}\chi B\cdot D\phi+\phi B\cdot
D\chi-c\chi\phi-A:(2D\chi\otimes D\phi+\phi D^{2}\chi)\bigr{)}\,d\mu\\\
&\quad+\int_{\Omega}(\chi g\phi-\chi G\cdot D\phi-\phi G\cdot
D\chi)\,dx.\end{split}$

Note that [13, Lemma 9.17] implies that

$\|\phi\|_{W^{2,p}(\Omega^{\prime\prime})}\leq C_{1}\|\psi\|_{L^{p}(\Omega)}$

for a constant $C_{1}=C_{1}(n,\Omega^{\prime\prime},p,A)$. Hence

$\|\phi\|_{C^{1}(\overline{\Omega^{\prime\prime}})}\leq
C_{2}\|\psi\|_{L^{p}(\Omega)}$

for a constant $C_{2}$ with the same dependence. Therefore, we find a constant
$C_{3}=C_{3}(n,\Omega^{\prime\prime},p,A,\chi)$ such that

$|\alpha(\psi)|\leq
C_{3}\left(\Gamma|\mu|(\Omega)+\|g\|_{L^{1}(\Omega)}+\|G\|_{L^{p^{\prime}}(\Omega)}\right)\|\psi\|_{L^{p}(\Omega)}.$

In particular, the functional $\alpha$ has a continuous linear extension to
$L^{p}(\Omega)$. It follows that there exists $\tilde{f}\in
L^{p^{\prime}}(\Omega)$ such that

$\int_{\Omega}\chi\psi\,d\mu=\int_{\Omega}\tilde{f}\psi\,dx$

for all $\psi\in C_{0}^{1}(\Omega)$. Since $C_{0}^{1}(\Omega)$ is dense in
$C_{0}^{0}(\Omega)$, this means that $\chi\mu$ is absolutely continuous with
respect to the Lebesgue measure and $\tilde{f}$ is the Radon-Nikodym
derivative. Since these arguments work for any $\chi\in
C_{0}^{\infty}(\Omega)$, the measure $\mu$ is absolutely continuous as well
and has a Radon-Nikodym derivative $f\in
L_{\mathrm{loc}}^{p^{\prime}}(\Omega)$. Choosing $\chi$ such that $\chi\equiv
1$ in $\Omega^{\prime}$, we also obtain the inequality

$\|f\|_{L^{p^{\prime}}(\Omega^{\prime})}\leq
C_{3}\left(\Gamma|\mu|(\Omega)+\|G\|_{L^{p^{\prime}}(\Omega)}+\|g\|_{L^{1}(\Omega)}\right).$
(8)

We now conclude that (7) holds true for every $\phi\in W^{2,p}(\Omega)$ with
compact support in $\Omega$.

For $i\in\\{1,\dotsc,n\\}$, we next consider the functional
$\beta_{i}\in(C_{0}^{1}(\Omega))^{*}$ with

$\beta_{i}(\psi)=\int_{\Omega}f\chi\psi_{x_{i}}\,dx,\quad\psi\in
C_{0}^{1}(\Omega)$

(corresponding to a distributional derivative of $\chi f$). Given a fixed
$\psi\in C_{0}^{1}(\Omega)$, solve

$A:D^{2}\phi+(\operatorname{div}A)\cdot D\phi=\psi_{x_{i}}$

in $W^{2,p}(\Omega^{\prime\prime})\cap W_{0}^{1,p}(\Omega^{\prime\prime})$. If
we write $e_{i}$ for the $i$-th standard unit vector in $\mathbb{R}^{n}$, then
the equation can alternatively be represented in the form

$\operatorname{div}(AD\phi)=\operatorname{div}(\psi e_{i}).$

Standard $L^{p}$-estimates for weak solutions thus give the estimate

$\|\phi\|_{W^{1,p}(\Omega^{\prime\prime})}\leq C_{4}\|\psi\|_{L^{p}(\Omega)}$

for a constant $C_{4}=C_{4}(n,\Omega^{\prime\prime},p,A)$. The function
$\chi\phi$ is again a suitable test function for (7). We obtain

$\begin{split}\beta_{i}(\psi)&=\int_{\Omega}f\bigl{(}\chi(\operatorname{div}A)\cdot
D\phi-A:(2D\chi\otimes D\phi+\phi D^{2}\chi)\bigr{)}\,dx\\\
&\quad+\int_{\Omega}f\bigl{(}B\cdot(\chi D\phi+\phi
D\chi)-c\chi\phi\bigr{)}\,dx\\\ &\quad+\int_{\Omega}(\chi g\phi-\chi G\cdot
D\phi-\phi G\cdot D\chi)\,dx.\end{split}$

Hence there exists a constant $C_{5}=C_{5}(n,\Omega^{\prime\prime},p,A,\chi)$
such that

$|\beta_{i}(\psi)|\leq
C_{5}\left(\Gamma\|f\|_{L^{p^{\prime}}(\Omega^{\prime\prime})}+\|G\|_{L^{p^{\prime}}(\Omega)}+\|g\|_{L^{1}(\Omega)}\right)\|\psi\|_{L^{p}(\Omega)}.$

Therefore, there exists $h_{i}\in L^{p^{\prime}}(\Omega)$ such that

$\int_{\Omega}\chi f\psi_{x_{i}}\,dx=\int_{\Omega}h_{i}\psi\,dx$

for all $\psi\in C_{0}^{1}(\Omega)$. This is true for $i=1,\dots,n$, so the
function $\chi f$ has weak derivatives in $L^{p^{\prime}}(\Omega)$, which
satisfy

$\left\|(\chi f)_{x_{i}}\right\|_{L^{p^{\prime}}(\Omega)}\leq
C_{5}\left(\Gamma\|f\|_{L^{p^{\prime}}(\Omega^{\prime\prime})}+\|G\|_{L^{p^{\prime}}(\Omega)}+\|g\|_{L^{1}(\Omega)}\right).$
(9)

Since $\chi\in C_{0}^{\infty}(\Omega)$ can be chosen arbitrarily, it follows
that $f\in W_{\mathrm{loc}}^{1,p^{\prime}}(\Omega)$. Moreover, we obtain the
desired inequality if we combine (8) with (9). ∎

We will also require the following statement, which says that a weak solution
of an equation of the form

$\operatorname{div}\operatorname{div}(fA)+\operatorname{div}(fB)+cf=0$ (10)

can be tested with functions from the space
$\mathcal{W}_{0}^{2,\infty}(\Omega)$.

######  Lemma 7.

Suppose that $f\in L^{1}(\Omega)$ is a weak solution of equation (10). Then

$\int_{\Omega}f(A:D^{2}\phi-B\cdot D\phi+c\phi)\,dx=0$

for all $\phi\in\mathcal{W}_{0}^{2,\infty}(\Omega)$.

###### Proof.

Given $\phi\in\mathcal{W}_{0}^{2,\infty}(\Omega)$, we first construct a family
of approximations $(\phi_{\epsilon})_{\epsilon\in(0,\epsilon_{0}]}$ in
$C_{0}^{\infty}(\Omega)$ such that $\phi_{\epsilon}\to\phi$ in
$W^{2,q}(\Omega)$ for any $q<\infty$ and, at the same time, such that
$A:D^{2}\phi_{\epsilon}$ remains bounded in $L^{\infty}(\Omega)$.

For this purpose, we extend $\phi$ by $0$ outside of $\Omega$. Choose a finite
open cover $\\{G_{1},\dotsc,G_{L}\\}$ of $\overline{\Omega}$ with the property
that there exist $R>0$ and there exist open cones
$C_{1},\dotsc,C_{L}\subseteq\mathbb{R}^{n}$ such that for any
$x\in\partial\Omega\cap G_{\ell}$,

$C_{\ell}\cap B_{R}(0)\cap(x-\overline{\Omega})=\emptyset.$

(Hence $(x-C_{\ell})\cap B_{R}(x)$ is an exterior cone to
$\overline{\Omega}$.) This is possible, because $\Omega$ is a bounded
Lipschitz domain.

For every $\ell=1,\dotsc L$, choose $\eta_{\ell}\in
C_{0}^{\infty}(C_{\ell}\cap B_{1}(0))$ with $\eta_{\ell}\geq 0$ and

$\int_{B_{1}(0)}\eta_{\ell}(x)\,dx=1.$

For $\epsilon\in(0,R]$, set

$\eta_{\ell\epsilon}(x)=\frac{1}{\epsilon^{n}}\eta_{\ell}\left(\frac{x}{\epsilon}\right)$

and

$\phi_{\ell\epsilon}=\phi*\eta_{\ell\epsilon}.$

Then for $x\in\partial\Omega\cap G_{\ell}$,

$\phi_{\ell\epsilon}(x)=\int_{C_{\ell}\cap
B_{\epsilon}(0)\cap(x-\Omega)}\eta_{\ell\epsilon}(y)\phi(x-y)\,dy=0.$

Moreover, the function $\phi_{\ell\epsilon}$ vanishes in a neighbourhood of
any such point $x$.

Now choose a partition of unity $\chi_{1},\dotsc,\chi_{L}$ in $\Omega$ with
$\chi_{\ell}\in C_{0}^{\infty}(G_{\ell})$ for $\ell=1,\dotsc,L$. Set

$\phi_{\epsilon}=\sum_{\ell=1}^{L}\chi_{\ell}\phi_{\ell\epsilon}.$

Then it is clear that $\phi_{\epsilon}\in C_{0}^{\infty}(\Omega)$ and that
$\phi_{\epsilon}\to\phi$ in $W^{2,q}(\Omega)$, for any $q<\infty$, as
$\epsilon\searrow 0$.

For $x\in\Omega$, we compute

$\begin{split}A(x):D^{2}\phi_{\ell\epsilon}(x)&=\int_{B_{\epsilon}(0)}\eta_{\ell\epsilon}(y)A(x):D^{2}\phi(x-y)\,dy\\\
&=\int_{B_{\epsilon}(0)}\eta_{\ell\epsilon}(y)(A(x)-A(x-y)):D^{2}\phi(x-y)\,dy\\\
&\quad+\int_{B_{\epsilon}(0)}\eta_{\ell\epsilon}(y)A(x-y):D^{2}\phi(x-y)\,dy.\end{split}$

Using an integration by parts, we find that

$\int_{B_{\epsilon}(0)}\eta_{\ell\epsilon}(y)(A(x)-A(x-y)):D^{2}\phi(x-y)\,dy\\\
\begin{aligned}
&=\int_{B_{\epsilon}(0)}\eta_{\ell\epsilon}(y)\operatorname{div}A(x-y)\cdot
D\phi(x-y)\,dy\\\
&\quad+\int_{B_{\epsilon}(0)}(A(x)-A(x-y)):D\eta_{\ell\epsilon}(y)\otimes
D\phi(x-y)\,dy.\end{aligned}$

For $y\in B_{\epsilon}(0)$, we have the inequality

$|A(x)-A(x-y)|\leq\epsilon\|A\|_{C^{1}(\overline{\Omega})}.$

Hence there exists a universal constant $C_{1}$ such that

$\left|\int_{B_{\epsilon}(0)}\eta_{\ell\epsilon}(y)(A(x)-A(x-y)):D^{2}\phi(x-y)\,dy\right|\\\
\leq
C_{1}\|A\|_{C^{1}(\overline{\Omega})}\bigl{(}1+\|D\eta_{\ell}\|_{L^{1}(B_{1}(0))}\bigr{)}\|D\phi\|_{L^{\infty}(\Omega)}.$

It is clear that

$\left|\int_{B_{\epsilon}(0)}\eta_{\ell\epsilon}(y)A(x-y):D^{2}\phi(x-y)\,dy\right|\leq\|A:D^{2}\phi\|_{L^{\infty}(\Omega)}.$

Thus we obtain a uniform estimate for
$\|A:D^{2}\phi_{\ell\epsilon}\|_{L^{\infty}(\Omega)}$. A similar estimate for
$\phi_{\epsilon}$ is then easy to prove.

It follows that there exists a sequence $\epsilon_{k}\searrow 0$ such that
$A:D^{2}\phi_{\epsilon_{k}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}g$,
weakly* in $L^{\infty}(\Omega)$, for some $g\in L^{\infty}(\Omega)$. For any
$\psi\in C_{0}^{\infty}(\Omega)$, we then compute

$\begin{split}\int_{\Omega}\psi g\,dx&=\lim_{k\to\infty}\int_{\Omega}\psi
A:D^{2}\phi_{\epsilon_{k}}\,dx\\\
&=\lim_{k\to\infty}\int_{\Omega}\operatorname{div}\operatorname{div}(\psi
A)\phi_{\epsilon_{k}}\,dx\\\
&=\int_{\Omega}\operatorname{div}\operatorname{div}(\psi A)\phi\,dx\\\
&=\int_{\Omega}\psi A:D^{2}\phi\,dx.\end{split}$

It follows that $g=A:D^{2}\phi$. It then also follows that
$A:D^{2}\phi_{\epsilon}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}A:D^{2}\phi$,
i.e, it is not necessary to take a subsequence.

Now the claim of the lemma is proved by approximation with $\phi_{\epsilon}$
and with standard arguments. ∎

##  3 Sufficiency of the equations

We now show that a non-trivial solution of the system (4), (5) gives rise to
an almost-minimiser of $E_{\infty}$.

###### Proof of Theorem 2.

We consider $u_{\infty}\in\mathcal{W}^{2,\infty}(\Omega)$ and assume that
there exist $e_{\infty}\geq 0$ and $f_{\infty}\in L^{1}\setminus\\{0\\}$ such
that (4) is satisfied almost everywhere and (5) weakly in $\Omega$. We are
required to show that $u_{\infty}$ is an almost-minimiser of $E_{\infty}$. It
suffices, however, to prove the inequality in Definition 1 under the
assumption that $\|\phi\|_{W^{1,\infty}(\Omega)}\leq 1$, because otherwise,

$E_{\infty}(u_{\infty})\leq
E_{\infty}(u_{\infty}+\phi)+M\|\phi\|_{W^{1,\infty}(\Omega)}^{2}$

for the number $M=E_{\infty}(u_{\infty})$.

We first note that $f_{\infty}\in W_{\mathrm{loc}}^{1,q}(\Omega)$ for some
$q>1$ by Lemma 6. Hence the equation $S_{u_{\infty}}^{*}f_{\infty}=0$ can be
written in the form

$\operatorname{div}\bigl{(}ADf_{\infty}+f_{\infty}\operatorname{div}A-f_{\infty}b_{z}(x,u_{\infty},Du_{\infty})\bigr{)}+f_{\infty}b_{y}(x,u_{\infty},Du_{\infty})=0.$

With standard regularity theory for elliptic equations, we then obtain higher
regularity, in particular $f_{\infty}\in W_{\mathrm{loc}}^{2,p}(\Omega)$ for
every $p<\infty$, and the results of Hardt and Simon [14] on the structure of
the nodal set apply. It follows that $f_{\infty}\not=0$ almost everywhere.

If $e_{\infty}=0$, then (4) implies that $E_{\infty}(u_{\infty})=0$, and
$u_{\infty}$ is in fact a global minimiser. Thus it suffices to consider
$e_{\infty}>0$.

Fix $\phi\in\mathcal{W}_{0}^{2,\infty}(\Omega)$ with
$\|\phi\|_{W^{1,\infty}(\Omega)}\leq 1$. For $t\in\mathbb{R}$, note that

$\frac{\partial}{\partial
t}S(u_{\infty}+t\phi)=S_{u_{\infty}+t\phi}^{\prime}\phi$

and

$\begin{split}\frac{\partial^{2}}{\partial
t^{2}}S(u_{\infty}+t\phi)&=\phi^{2}b_{yy}(x,u_{\infty}+t\phi,Du_{\infty}+tD\phi)\\\
&\quad+2\phi D\phi\cdot b_{yz}(x,u_{\infty}+t\phi,Du_{\infty}+tD\phi)\\\
&\quad+(D\phi\otimes
D\phi):b_{zz}(x,u_{\infty}+t\phi,Du_{\infty}+tD\phi).\end{split}$

Thus Taylor’s theorem, applied to the function $t\mapsto
S(u_{\infty}+t\phi)(x)$ for each $x\in\Omega$, implies there exists a function
$\tau\colon\Omega\to[0,1]$ such that

$S(u_{\infty}+\phi)=S(u_{\infty})+S_{u_{\infty}}^{\prime}\phi+B_{\tau}$ (11)

almost everywhere in $\Omega$, where

$B_{\tau}=\frac{1}{2}\phi^{2}b_{yy}(x,u_{\infty}+\tau\phi,Du_{\infty}+\tau
D\phi)+\phi D\phi\cdot b_{yz}(x,u_{\infty}+\tau\phi,Du_{\infty}+\tau D\phi)\\\
+\frac{1}{2}(D\phi\otimes D\phi):b_{zz}(x,u_{\infty}+\tau\phi,Du_{\infty}+\tau
D\phi).$

Therefore,

$(S(u_{\infty}+\phi))^{2}=(S(u_{\infty}))^{2}+2\bigl{(}S(u_{\infty})+B_{\tau}\bigr{)}S_{u_{\infty}}^{\prime}\phi+(S_{u_{\infty}}^{\prime}\phi)^{2}+2S(u_{\infty})B_{\tau}+B_{\tau}^{2}.$
(12)

Formula (11) implies that $B_{\tau}$ is measurable. Since
$\|\phi\|_{W^{1,\infty}(\Omega)}\leq 1$, we have the estimate

$\|u_{\infty}+\tau\phi\|_{L^{\infty}(\Omega)}+\|Du_{\infty}+\tau
D\phi\|_{L^{\infty}(\Omega)}\leq C_{1}$

for a constant $C_{1}$ that is independent of $\phi$. Hence there exists a
constant $C_{2}$, also independent of $\phi$, such that

$\|B_{\tau}\|_{L^{\infty}(\Omega)}\leq
C_{2}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}.$ (13)

We now claim that

$E_{\infty}(u_{\infty}+\phi)\geq
E_{\infty}(u_{\infty})-2C_{2}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}.$ (14)

Once this inequality is established, the proof is complete.

If $2C_{2}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}>E_{\infty}(u_{\infty})$, then
(14) is obvious. Thus we assume that
$2C_{2}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}\leq E_{\infty}(u_{\infty})$.

If $S_{u_{\infty}}^{\prime}\phi=0$ almost everywhere, then (12) and (13) imply
that

$(S(u_{\infty}+\phi))^{2}\geq(S(u_{\infty}))^{2}+2S(u_{\infty})B_{\tau}\geq(S(u_{\infty}))^{2}-2C_{2}E_{\infty}(u_{\infty})\|\phi\|_{W^{1,\infty}(\Omega)}^{2}$

almost everywhere. Therefore,

$(E_{\infty}(u_{\infty}+t\phi))^{2}\geq(E_{\infty}(u_{\infty}))^{2}-2C_{2}E_{\infty}(u_{\infty})\|\phi\|_{W^{1,\infty}(\Omega)}^{2}.$
(15)

If $S_{u_{\infty}}^{\prime}\phi\neq 0$ in a set of positive measure, then we
test equation (5) with $\phi$. (This is possible in view of Lemma 7.) We
obtain

$\int_{\Omega}f_{\infty}\,S_{u_{\infty}}^{\prime}\phi\,dx=0.$

Recall that $f_{\infty}\neq 0$ almost everywhere. Therefore, there exists a
set $\Omega_{+}\subseteq\Omega$ of positive measure such that
$f_{\infty}\,S_{u_{\infty}}^{\prime}\phi>0$ in $\Omega_{+}$. As $f_{\infty}$
has the same sign as $S(u_{\infty})$ almost everywhere by (4), this means that
$S(u_{\infty})\,S_{u_{\infty}}^{\prime}\phi>0$ almost everywhere in
$\Omega_{+}$. Equation (4) also implies that $|S(u_{\infty})|=e_{\infty}$
almost everywhere. As
$\|\phi\|_{W^{1,\infty}(\Omega)}\leq\sqrt{e_{\infty}/(2C_{2})}$ by the above
assumption, inequality (13) implies that $S(u_{\infty})+B_{\tau}$ has the same
sign as $S(u_{\infty})$ almost everywhere. Hence

$\bigl{(}S(u_{\infty})+B_{\tau}\bigr{)}S_{u_{\infty}}^{\prime}\phi>0$

almost everywhere in $\Omega_{+}$. With the help of (12) and (13), we conclude
that

$(S(u_{\infty}+\phi))^{2}\geq
e_{\infty}^{2}-2C_{2}e_{\infty}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}$

in $\Omega_{+}$. Hence we obtain (15) in this case as well.

Finally, from (15) we now obtain the estimate

$\begin{split}E_{\infty}(u_{\infty}+\phi)&\geq
E_{\infty}(u_{\infty})\sqrt{1-\frac{2C_{2}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}}{E_{\infty}(u_{\infty})}}\\\
&\geq
E_{\infty}(u_{\infty})\left(1-\frac{2C_{2}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}}{E_{\infty}(u_{\infty})}\right)\\\
&=E_{\infty}(u_{\infty})-2C_{2}\|\phi\|_{W^{1,\infty}(\Omega)}^{2}.\end{split}$

This proves (14) and completes the proof. ∎

##  4 Necessity of the equations

In this section, we prove Theorem 4. For this purpose, we require the
following lemma, which is an extension of a result proved by the authors in a
previous paper [21, Lemma 8]. This is where the extra regularity assumptions
in Theorem 4 are used. For $r>0$, we use the notation
$\Omega_{r}=\left\\{x\in\Omega\mid\operatorname{dist}(x,\partial\Omega)<r\right\\}$
here.

######  Lemma 8.

Suppose that $\partial\Omega$ is of class $C^{3}$ and $u_{0}\in
C^{2}(\overline{\Omega})$. Let $g\in C^{0}(\overline{\Omega})$ and $u\in
C^{1}(\overline{\Omega})$. Given $\epsilon>0$, there exist $r>0$ and $v\in
C^{2}(\overline{\Omega})$, with $v=u_{0}$ and $Dv=Du_{0}$ on $\partial\Omega$,
such that $\|S_{u}^{\prime}v-g\|_{L^{\infty}(\Omega_{r})}\leq\epsilon$.

###### Proof.

Let $\delta>0$. Choose $\tilde{u}_{0}\in C^{4}(\overline{\Omega})$ such that

$\|u_{0}-\tilde{u}_{0}\|_{C^{2}(\overline{\Omega})}\leq\delta$

and choose $\tilde{g}\in C^{2}(\overline{\Omega})$ such that

$\bigl{\|}g-\tilde{g}-u_{0}b_{y}(x,u,Du)-Du_{0}\cdot
b_{z}(x,u,Du)\bigr{\|}_{C^{0}(\overline{\Omega})}\leq\delta.$

Consider a number $r_{0}>0$ such that the function
$x\mapsto\operatorname{dist}(x,\partial\Omega)$ is of class $C^{3}$ in
$\overline{\Omega}_{2r_{0}}$. We can construct a function $\rho\in
C^{3}(\overline{\Omega})$ such that
$\rho(x)=\operatorname{dist}(x,\partial\Omega)$ for $x\in\Omega_{r_{0}}$. Note
that there exists $c>0$ such that $A:D\rho\otimes D\rho\geq c$ in
$\Omega_{r_{0}}$ by the fact that $A$ is uniformly positive definite and
$|D\rho|=1$. Define $\lambda\in C^{2}(\overline{\Omega})$ such that
$\lambda=2A:D\rho\otimes D\rho$ in $\Omega_{r_{0}}$ and $\lambda>0$
everywhere. Now define

$h=\tilde{g}-A:D^{2}\tilde{u}_{0}$

and

$v=u_{0}+\frac{\rho^{2}h}{\lambda}.$

Then

$Dv=Du_{0}+\frac{2\rho
h}{\lambda}D\rho+\rho^{2}D\left(\frac{h}{\lambda}\right)$

and

$D^{2}v=D^{2}u_{0}+\frac{2h}{\lambda}D\rho\otimes D\rho\\\ +\frac{2\rho
h}{\lambda}D^{2}\rho+2\rho D\rho\otimes D\left(\frac{h}{\lambda}\right)+2\rho
D\left(\frac{h}{\lambda}\right)\otimes
D\rho+\rho^{2}D^{2}\left(\frac{h}{\lambda}\right).$

Thus there exist $\Phi\in C^{0}(\overline{\Omega};\mathbb{R}^{n})$ and
$\Psi\in C^{0}(\overline{\Omega};\mathbb{R}^{n\times n})$ such that

$Dv=Du_{0}+\rho\Phi$

and

$D^{2}v=D^{2}u_{0}+\frac{2h}{\lambda}D\rho\otimes D\rho+\rho\Psi.$

In $\Omega_{r_{0}}$, it follows that

$\begin{split}S_{u}^{\prime}v&=A:D^{2}u_{0}+\frac{2h}{\lambda}A:D\rho\otimes
D\rho+\rho A:\Psi\\\
&\quad+\left(u_{0}+\frac{\rho^{2}h}{\lambda}\right)b_{y}(x,u,Du)+\left(Du_{0}+\rho\Phi\right)\cdot
b_{z}(x,u,Du)\\\
&=A:D^{2}u_{0}+\tilde{g}-A:D^{2}\tilde{u}_{0}+u_{0}b_{y}(x,u,Du)+Du_{0}\cdot
b_{z}(x,u,Du)+\rho X,\end{split}$

where

$X=A:\Psi+\frac{\rho h}{\lambda}b_{y}(x,u,Du)+\Phi\cdot b_{z}(x,u,Du).$

By the choice of $\tilde{u}_{0}$ and $\tilde{g}$, we conclude that

$\|S_{u}^{\prime}v-g\|_{L^{\infty}(\Omega_{r})}\leq(\|A\|_{L^{\infty}(\Omega)}+1)\delta+r\|X\|_{L^{\infty}(\Omega)}.$

Choosing $\delta$ and $r$ sufficiently small, we obtain the desired
inequality.

The boundary conditions are readily checked as well. ∎

###### Proof of Theorem 4.

We assume that $\partial\Omega$ is of class $C^{3}$ and $u_{0}\in
C^{2}(\overline{\Omega})$. We further assume that $S$ is admissible. Suppose
that $u_{\infty}\in u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$ is an almost-
minimiser of $E_{\infty}$. We wish to show that there exist $e_{\infty}\geq 0$
and $f_{\infty}\in L^{1}(\Omega)\setminus\\{0\\}$ such that (4) is satisfied
almost everywhere and (5) weakly.

If $E_{\infty}(u_{\infty})=0$, then we choose $e_{\infty}=0$. By a consequence
of the Fredholm alternative [8, Theorem 6.2.4], we can either find a
nontrivial solution of $S_{u_{\infty}}^{*}f_{\infty}=0$ in
$W_{0}^{1,2}(\Omega)$, or we can solve the boundary value problem

$\displaystyle S_{u_{\infty}}^{*}f_{\infty}$ $\displaystyle=0$
$\displaystyle\text{in $\Omega$},$ $\displaystyle f_{\infty}$
$\displaystyle=1$ $\displaystyle\text{on $\partial\Omega$},$

in $W^{1,2}(\Omega)$. In either case, we conclude that $f_{\infty}\in
L^{1}(\Omega)$ and does not vanish identically. Hence the required conditions
are satisfied.

We now assume that $E_{\infty}(u_{\infty})>0$. Because $u_{\infty}$ is an
almost-minimiser, there exists $M\in\mathbb{R}$ such that

$E_{\infty}(u_{\infty})\leq
E_{\infty}(v)+M\|u_{\infty}-v\|_{W^{1,\infty}(\Omega)}^{2}$

for all $v\in u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$. Choose $p_{0}>n$ such
that the statement from Definition 3 applies. Note that by [13, Lemma 9.17],
there exists a constant $C$ such that for all $\phi\in
W_{0}^{2,p_{0}}(\Omega)$, the inequality

$\|\phi\|_{W^{2,p_{0}}(\Omega)}\leq C\|A:D^{2}\phi\|_{L^{p_{0}}(\Omega)}$

holds true. In conjunction with the Sobolev embedding theorem, this implies
that there exists $\mu>0$ such that

$E_{\infty}(u_{\infty})\leq
E_{\infty}(v)+\mu\|A:D^{2}(u_{\infty}-v)\|_{L^{p_{0}}(\Omega)}^{2}$

under the above assumptions.

For $p<\infty$, we consider the functionals

$E_{p}(u)=\left(\fint_{\Omega}|S(u)|^{p}\,dx\right)^{\frac{1}{p}}.$

Furthermore, we fix $\sigma>0$ and define

$E_{p}^{\sigma}(u)=E_{p}(u)+\sigma\|A:D^{2}(u_{\infty}-u)\|_{L^{p_{0}}(\Omega)}^{2}.$

For every $p\geq p_{0}$, choose a minimiser $u_{p}\in
u_{0}+W_{0}^{2,p}(\Omega)$ of $E_{p}^{\sigma}$. (This can be found with the
direct method. Coercivity of the functional is a consequence of the assumption
that $S$ is admissible, similarly to the proof of Proposition 5. Lower
semicontinuity of $E_{p}$ with respect to weak convergence in
$W^{2,p}(\Omega)$ is also proved analogously to Proposition 5, and the lower
semicontinuity of the additional term follows from its convexity.) For
$p_{0}\leq p\leq q$, the minimality of $E_{p}(u_{p})$ and Hölder’s inequality
imply that

$E_{p}^{\sigma}(u_{p})\leq E_{p}^{\sigma}(u_{q})\leq E_{q}^{\sigma}(u_{q})\leq
E_{q}^{\sigma}(u_{\infty})=E_{q}(u_{\infty})\leq E_{\infty}(u_{\infty}).$ (16)

Because $S$ is admissible, we infer that

$\limsup_{p\to\infty}\|u_{p}\|_{W^{2,q}(\Omega)}<\infty$ (17)

for any $q<\infty$. We may therefore choose a sequence $p_{k}\to\infty$ and
find $w_{\infty}\in u_{0}+\bigcap_{q<\infty}W_{0}^{2,q}(\Omega)$ such that
$u_{p_{k}}\rightharpoonup w_{\infty}$ weakly in $W^{2,q}(\Omega)$ for every
$q<\infty$. Then we also have the strong convergence $u_{p_{k}}\to w_{\infty}$
in $W^{1,\infty}(\Omega)$. It further follows that
$S(u_{p_{k}})\rightharpoonup S(w_{\infty})$ weakly in $L^{q}(\Omega)$ for
every $q<\infty$.

By the lower semicontinuity of the $L^{q}$-norm with respect to weak
convergence and by (16), we have the inequalities

$\begin{split}E_{\infty}^{\sigma}(w_{\infty})&=\lim_{q\to\infty}E_{q}^{\sigma}(w_{\infty})\\\
&\leq\limsup_{q\to\infty}\liminf_{k\to\infty}E_{q}^{\sigma}(u_{p_{k}})\\\
&\leq\liminf_{k\to\infty}E_{p_{k}}^{\sigma}(u_{p_{k}})\\\ &\leq
E_{\infty}(u_{\infty}).\end{split}$ (18)

Hence $w_{0}\in u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$. As $u_{\infty}$ is
an almost-minimiser of $E_{\infty}$, we also know that

$E_{\infty}(u_{\infty})\leq E_{\infty}^{\mu}(w_{\infty}).$

Hence

$(\sigma-\mu)\|A:D^{2}(w_{\infty}-u_{\infty})\|_{L^{p_{0}}(\Omega)}\leq 0,$

which implies that $w_{\infty}=u_{\infty}$ if $\sigma>\mu$. As this fixes the
limit, we can in fact conclude that $u_{p}\rightharpoonup u_{\infty}$, as
$p\to\infty$, weakly in $W^{2,q}(\Omega)$ for every $q<\infty$. It also
follows from (18) and (16) that $E_{p}^{\sigma}(u_{p})\to
E_{\infty}(u_{\infty})$ as $p\to\infty$.

We further observe that for any sequence $p_{k}\to\infty$,

$E_{\infty}(u_{\infty})+\sigma\liminf_{k\to\infty}\|A:D^{2}(u_{p_{k}}-u_{\infty})\|_{L^{p_{0}}(\Omega)}^{2}\\\
\begin{aligned}
&=\lim_{q\to\infty}E_{q}(u_{\infty})+\sigma\liminf_{k\to\infty}\|A:D^{2}(u_{p_{k}}-u_{\infty})\|_{L^{p_{0}}(\Omega)}^{2}\\\
&\leq\limsup_{q\to\infty}\liminf_{k\to\infty}E_{q}(u_{p_{k}})+\sigma\liminf_{k\to\infty}\|A:D^{2}(u_{p_{k}}-u_{\infty})\|_{L^{p_{0}}(\Omega)}^{2}\\\
&\leq\limsup_{q\to\infty}\liminf_{k\to\infty}E_{q}^{\sigma}(u_{p_{k}}).\end{aligned}$

For every fixed $q<\infty$, Hölder’s inequality gives
$E_{q}^{\sigma}(u_{p_{k}})\leq E_{p_{k}}^{\sigma}(u_{p_{k}})$ whenever $k$ is
sufficiently large. Hence

$\liminf_{k\to\infty}E_{q}^{\sigma}(u_{p_{k}})\leq\liminf_{k\to\infty}E_{p_{k}}^{\sigma}(u_{p_{k}}),$

and we conclude that

$E_{\infty}(u_{\infty})+\sigma\liminf_{k\to\infty}\|A:D^{2}(u_{p_{k}}-u_{\infty})\|_{L^{p_{0}}(\Omega)}^{2}\leq\liminf_{k\to\infty}E_{p_{k}}^{\sigma}(u_{p_{k}})=E_{\infty}(u_{\infty}).$

It follows that

$\lim_{p\to\infty}\|A:D^{2}(u_{p}-u_{\infty})\|_{L^{p_{0}}(\Omega)}=0.$

Set $e_{\infty}=E_{\infty}(u_{\infty})$ and $e_{p}=E_{p}(u_{p})$. Then it
follows that

$e_{\infty}=\lim_{p\to\infty}e_{p}.$

Set furthermore

$a_{p}=\|A:D^{2}(u_{p}-u_{\infty})\|_{L^{p_{0}}(\Omega)}.$

Then the Euler-Lagrange equation for $u_{p}$ is

$e_{p}^{1-p}S_{u_{p}}^{*}\bigl{(}|S(u_{p})|^{p-2}S(u_{p})\bigr{)}\\\
+2\sigma|\Omega|a_{p}^{2-p_{0}}\operatorname{div}\operatorname{div}\bigl{(}|A:D^{2}(u_{p}-u_{\infty})|^{p_{0}-2}(A:D^{2}(u_{p}-u_{\infty}))A\bigr{)}=0.$

Set

$f_{p}=e_{p}^{1-p}|S(u_{p})|^{p-2}S(u_{p})$

and

$\phi_{p}=a_{p}^{2-p_{0}}|A:D^{2}(u_{p}-u_{\infty})|^{p_{0}-2}A:D^{2}(u_{p}-u_{\infty}).$

Then we have the system

$\displaystyle|f_{p}|^{\frac{p-2}{p-1}}S(u_{p})$ $\displaystyle=e_{p}f_{p},$
(19) $\displaystyle
S_{u_{p}}^{*}f_{p}+2\sigma|\Omega|\operatorname{div}\operatorname{div}(\phi_{p}A)$
$\displaystyle=0.$ (20)

We compute

$\fint_{\Omega}|f_{p}|^{p/(p-1)}\,dx=e_{p}^{-p}\fint_{\Omega}|S(u_{p})|^{p}\,dx=1.$
(21)

Hence we can find a sequence $p_{k}\to\infty$ such that $f_{p_{k}}$ converges,
in the weak* sense in $(C^{0}(\overline{\Omega}))^{*}$, to a Radon measure
$F_{\infty}$ on $\overline{\Omega}$. Testing equation (20) with $\eta\in
C_{0}^{\infty}(\Omega)$, we see that

$\begin{split}\int_{\Omega}S_{u_{\infty}}^{\prime}\eta\,dF_{\infty}&=\lim_{k\to\infty}\int_{\Omega}f_{p_{k}}S_{u_{\infty}}^{\prime}\eta\,dx\\\
&=\lim_{k\to\infty}\int_{\Omega}f_{p_{k}}S_{u_{p_{k}}}^{\prime}\eta\,dx\\\
&=-2\sigma|\Omega|\lim_{k\to\infty}\int_{\Omega}\phi_{p_{k}}A:D^{2}\eta\,dx.\end{split}$

(In the second step, we have used the fact that $b_{y}(x,u_{p},Du_{p})\to
b_{y}(x,u_{\infty},Du_{\infty})$ and $b_{z}(x,u_{p},Du_{p})\to
b_{z}(x,u_{\infty},Du_{\infty})$ uniformly as $p\to\infty$.) If
$p_{0}^{\prime}$ is the exponent conjugate to $p_{0}$, then

$\|\phi_{p}\|_{L^{p_{0}^{\prime}}(\Omega)}=a_{p}^{2-p_{0}}\left(\int_{\Omega}|A:D^{2}(u_{p}-u_{\infty})|^{p_{0}}\,dx\right)^{\frac{p_{0}-1}{p_{0}}}=a_{p}\to
0$ (22)

as $p\to\infty$. Hence $F_{\infty}$ is a distributional solution of
$S_{u_{\infty}}^{*}F_{\infty}=0$. According to Lemma 6, its restriction to
$\Omega$ is absolutely continuous with respect to the Lebesgue measure, and
the Radon-Nikodym derivative $f_{\infty}\in L^{1}(\Omega)$ belongs to
$W_{\mathrm{loc}}^{1,q}(\Omega)$ for all $q\in(1,\frac{n}{n-1})$ (but
$F_{\infty}$ may have a part supported on $\partial\Omega$ as well).
Obviously, we now have equation (5) in the weak sense.

Set $h_{p}=f_{p}+2\sigma|\Omega|\phi_{p}$. Then (20) can be written in the
form

$S_{u_{p}}^{*}h_{p}=2\sigma|\Omega|\left(\phi_{p}b_{y}(x,u_{p},Du_{p})-\operatorname{div}\bigl{(}\phi_{p}b_{z}(x,u_{p},Du_{p})\bigr{)}\right).$

We already know that the functions $b_{z}(x,u_{p},Du_{p})$ and
$b_{y}(x,u_{p},Du_{p})$ are uniformly bounded in $L^{\infty}(\Omega)$. Because
of (22), we have uniform bounds for $\phi_{p}b_{y}(x,u_{p},Du_{p})$ and for
$\phi_{p}b_{z}(x,u_{p},Du_{p})$ in $L^{p_{0}^{\prime}}(\Omega)$. The
coefficients of $S_{u_{p}}^{*}$ are also bounded in $L^{\infty}(\Omega)$.
According to Lemma 6, this means that we have a uniform bound for
$\|h_{p}\|_{W^{1,p_{0}^{\prime}}(\Omega^{\prime})}$ for any precompact open
set $\Omega^{\prime}\Subset\Omega$. In particular, we can choose the above
sequence $p_{k}\to\infty$ such that $(h_{p_{k}})_{k\in\mathbb{N}}$ converges
in $L^{p_{0}^{\prime}}(\Omega^{\prime})$. Since we have the convergence
$\phi_{p}\to 0$ in the same space by (22), we must have
$L^{p_{0}^{\prime}}$-convergence for $(f_{p_{k}})_{k\in\mathbb{N}}$. The limit
is of course $f_{\infty}$.

We may assume that $f_{p_{k}}\to f_{\infty}$ almost everywhere in
$\Omega^{\prime}$ and, at the same time, that
$\|f_{p_{k}}-f_{\infty}\|_{L^{p_{0}^{\prime}}(\Omega^{\prime})}\leq 2^{-k}$
for every $k\in\mathbb{N}$ (otherwise we choose a further subsequence with
this property). If we set

$\theta=|f_{\infty}|+\sum_{k=1}^{\infty}|f_{p_{k}}-f_{\infty}|,$

then this guarantees that $\theta\in L^{p_{0}^{\prime}}(\Omega^{\prime})$.
Moreover, we see that $|f_{p_{k}}|\leq\theta$ for every $k\in\mathbb{N}$. We
therefore obtain the pointwise inequality

$\left||f_{p_{k}}|^{\frac{p_{k}-2}{p_{k}-1}}-|f_{\infty}|\right|^{p_{0}^{\prime}}\leq\left(1+2\theta\right)^{p_{0}^{\prime}}$

(provided that $p_{k}>2$), and the dominated convergence theorem implies that

$|f_{p_{k}}|^{\frac{p_{k}-2}{p_{k}-1}}\to|f_{\infty}|$

in $L^{p_{0}^{\prime}}(\Omega^{\prime})$. Since we have the weak convergence
of $S(u_{p})$ to $S(u_{\infty})$ in $L^{q}(\Omega)$ for any $q<\infty$, we
conclude that

$|f_{p_{k}}|^{\frac{p_{k}-2}{p_{k}-1}}S(u_{p_{k}})\rightharpoonup|f_{\infty}|S(u_{\infty})$

weakly in $L_{\mathrm{loc}}^{1}(\Omega)$. Recall that $e_{p}\to e_{\infty}$ as
$p\to\infty$. Thus passing to the limit in (19), we obtain (4).

It remains to show that $f_{\infty}\not\equiv 0$.

For any $p\geq p_{0}$, we compute

$|\Omega|e_{p}=e_{p}^{1-p}\int_{\Omega}|S(u_{p})|^{p}\,dx=\int_{\Omega}f_{p}S(u_{p})\,dx=\int_{\Omega}f_{p}\left(S_{u_{p}}^{\prime}u_{p}-g_{p}\right)\,dx,$

where

$g_{p}=-b(x,u_{p},Du_{p})+u_{p}b_{y}(x,u_{p},Du_{p})+Du_{p}\cdot
b_{z}(x,u_{p},Du_{p}).$

We define $g_{\infty}$ by the analogous formula as well. Given $\epsilon>0$,
we choose $r>0$ and $v\in u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$ with

$\|S_{u_{\infty}}^{\prime}v-g_{\infty}\|_{L^{\infty}(\Omega_{r})}\leq\epsilon$

with the help of Lemma 8. Then, by (20),

$\begin{split}|\Omega|e_{p}&=\int_{\Omega}f_{p}\left(S_{u_{p}}^{\prime}u_{p}-g_{p}\right)\,dx\\\
&=\int_{\Omega}f_{p}\left(S_{u_{p}}^{\prime}v-g_{p}\right)\,dx-2\sigma|\Omega|\int_{\Omega}\phi_{p}A:D^{2}(u_{p}-v)\,dx\\\
&=\int_{\Omega_{r}}f_{p}\left(S_{u_{\infty}}^{\prime}v-g_{\infty}\right)\,dx+\int_{\Omega_{r}}f_{p}\left(S_{u_{p}}^{\prime}v-S_{u_{\infty}}^{\prime}v-g_{p}+g_{\infty}\right)\,dx\\\
&\quad+\int_{\Omega\setminus\Omega_{r}}f_{p}\left(S_{u_{p}}^{\prime}v-g_{p}\right)\,dx-2\sigma|\Omega|\int_{\Omega}\phi_{p}A:D^{2}(u_{p}-v)\,dx\\\
&\leq\epsilon|\Omega|+\int_{\Omega_{r}}f_{p}\left(S_{u_{p}}^{\prime}v-S_{u_{\infty}}^{\prime}v-g_{p}+g_{\infty}\right)\,dx\\\
&\quad+\int_{\Omega\setminus\Omega_{r}}f_{p}\left(S_{u_{p}}^{\prime}v-g_{p}\right)\,dx+2\sigma|\Omega|a_{p}\,\|A:D^{2}(u_{p}-v)\|_{L^{p_{0}}(\Omega)}.\end{split}$

Letting $p\to\infty$, we note that $S_{u_{p}}^{\prime}v\to
S_{u_{\infty}}^{\prime}v$ and $g_{p}\to g_{\infty}$ uniformly in $\Omega$,
while

$\limsup_{p\to\infty}\|f_{p}\|_{L^{1}(\Omega)}<\infty$

by (21). Furthermore, we know that $a_{p}\to 0$, while $A:D^{2}u_{p}$ is
uniformly bounded in $L^{p_{0}}(\Omega)$. We further know that $f_{p_{k}}\to
f_{\infty}$ in $L^{1}(\Omega\setminus\Omega_{r})$. Hence

$|\Omega|(e_{\infty}-\epsilon)\leq\int_{\Omega\setminus\Omega_{r}}f_{\infty}(S_{u_{\infty}}^{\prime}v-g_{\infty})\,dx.$

Choosing $\epsilon<e_{\infty}$, we conclude that the integral on the right-
hand side does not vanish. Hence $f_{\infty}\not\equiv 0$, and this concludes
the proof. ∎

##  5 Examples

The condition for admissible operators in Definition 3 often follows from
standard estimates for linear operators. For semilinear ones, some additional
arguments are sometimes required. In this section, we consider two examples.
We restrict our attention to operators of the form

$S(u)=\Delta u+g(u)$

for some function $g\in C^{2}(\mathbb{R})$ here. We show that $S$ is
admissible if either $g$ has the correct sign or satisfies a suitable growth
condition.

######  Proposition 9.

If $yg(y)\leq 0$ for all $y\in\mathbb{R}$, then $S$ is admissible.

###### Proof.

Choose any $p_{0}>n/2$. Given $u_{0}\in\mathcal{W}^{2,\infty}(\Omega)$, let

$\alpha=\sup_{x\in\partial\Omega}|u_{0}(x)|.$

Suppose that $u\in u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$ satisfies
$\|S(u)\|_{L^{p}(\Omega)}\leq\Lambda$ for some $p\geq p_{0}$. Then
$\|S(u)\|_{L^{p_{0}}(\Omega)}\leq|\Omega|^{1/p_{0}-1/p}\Lambda$ by Hölder’s
inequality. We now solve the boundary value problem

$\displaystyle\Delta w$ $\displaystyle=-|S(u)|$ $\displaystyle\text{in
$\Omega$},$ $\displaystyle w$ $\displaystyle=\alpha$ $\displaystyle\text{on
$\partial\Omega$}.$

Then $w\geq\alpha$ by the maximum principle. Furthermore, using
$L^{p}$-estimates for the Laplacian and the Sobolev embedding theorem, we see
that $w$ is bounded by a constant that depends only on $n$, $\Omega$, $p_{0}$,
$u_{0}$, and $\Lambda$.

In the set $\left\\{x\in\Omega\mid u(x)>0\right\\}$, we have the inequality

$\Delta u=S(u)-g(u)\geq S(u)\geq\Delta w.$

Hence the comparison principle implies that $u\leq w$ in $\Omega$.

Similarly, we show that $u$ is bounded from below by a constant depending only
on $n$, $\Omega$, $p_{0}$, $u_{0}$, and $\Lambda$. The condition from
Definition 3 then follows with standard elliptic estimates. ∎

For our second example, we assume that

$\lim_{y\to\pm\infty}\frac{yg(y)}{\int_{0}^{y}g(t)\,dt}=\alpha.$ (23)

That is, the function $g$ has asymptotic growth at $\pm\infty$ like $y\mapsto
cy^{\alpha-1}$ for some $c\in\mathbb{R}$.

######  Proposition 10.

Let $n\geq 3$. Suppose that $g$ satisfies (23) for some
$\alpha\in[2,\frac{2n}{n-2})$. Then $S$ is admissible.

###### Proof.

Define

$G(y)=\int_{0}^{y}g(t)\,dt.$

Given $\beta>\alpha$, inequality (23) implies that $yG^{\prime}(y)<\beta G(y)$
when $|y|$ is sufficiently large. The Grönwall inequality implies that there
exists $C_{1}>0$ with

$G(y)\leq C_{1}(|y|^{\beta}+1)$

for all $y\in\mathbb{R}$. Using (23) again, we find another constant $C_{2}$
such that

$|g(y)|\leq C_{2}(|y|^{\beta-1}+1)$ (24)

for all $y\in\mathbb{R}$.

Now suppose that $u_{0}\in\mathcal{W}^{2,\infty}(\Omega)$ and consider $u\in
u_{0}+\mathcal{W}_{0}^{2,\infty}(\Omega)$. We write $\nu$ for the outer normal
vector on $\partial\Omega$ and $\sigma$ for the surface measure on
$\partial\Omega$. Then an integration by parts yields the identity

$\int_{\Omega}(|Du|^{2}-ug(u))\,dx=\int_{\partial\Omega}u_{0}\nu\cdot
Du_{0}\,d\sigma-\int_{\Omega}uS(u)\,dx.$ (25)

We furthermore compute

$\operatorname{div}\left((x\cdot
Du)Du-\left(\frac{1}{2}|Du|^{2}-G(u)\right)x\right)=(x\cdot
Du)S(u)-\frac{n-2}{2}|Du|^{2}+nG(u).$

Hence

$\int_{\Omega}\left(\frac{n-2}{2}|Du|^{2}-nG(u)\right)\,dx=\int_{\Omega}(x\cdot
Du)S(u)\,dx\\\ -\int_{\partial\Omega}\left((x\cdot Du_{0})(\nu\cdot
Du_{0})-\left(\frac{1}{2}|Du_{0}|^{2}-G(u_{0})\right)x\cdot\nu\right)\,d\sigma.$
(26)

Fix $\beta\in(\alpha,\frac{2n}{n-2})$. Then the combination of (25) and (26)
implies that

$\left(\frac{1}{\beta}-\frac{n-2}{2n}\right)\int_{\Omega}|Du|^{2}\,dx=\int_{\Omega}\left(\frac{ug(u)}{\beta}-G(u)\right)\,dx\\\
-\int_{\Omega}\left(\frac{1}{\beta}uS(u)+\frac{1}{n}(x\cdot
Du)S(u)\right)\,dx+\frac{1}{\beta}\int_{\partial\Omega}u_{0}\nu\cdot
Du_{0}\,d\sigma\\\ +\frac{1}{n}\int_{\partial\Omega}\left((x\cdot
Du_{0})(\nu\cdot
Du_{0})-\left(\frac{1}{2}|Du_{0}|^{2}-G(u_{0})\right)x\cdot\nu\right)\,d\sigma.$
(27)

Under the assumptions of the proposition, we know that $yg(y)\leq\beta G(y)$
whenever $|y|$ is sufficiently large. Hence there exists some constant
$C_{3}$, depending only on $g$, such that

$\int_{\Omega}\left(\frac{ug(u)}{\beta}-G(u)\right)\,dx\leq C_{3}|\Omega|.$

The boundary integrals in (27) depend only on $n$, $\Omega$, $u_{0}$, $g$, and
$\beta$. Hence there exists $C_{4}=C_{4}(n,\Omega,u_{0},g,\alpha)$ such that

$\int_{\Omega}|Du|^{2}\,dx\leq
C_{4}\left(\|S(u)\|_{L^{2}(\Omega)}\|u\|_{W^{1,2}(\Omega)}+1\right).$

From this and the Poincaré inequality, we derive the estimate

$\|u\|_{W^{1,2}(\Omega)}\leq C_{5}\left(\|S(u)\|_{L^{2}(\Omega)}+1\right)$

for a constant $C_{5}$ with the same dependence.

Fix $p_{0}>n$ and consider $p\geq p_{0}$. If we assume that

$\|S(u)\|_{L^{p}(\Omega)}\leq\Lambda,$

then we can now use standard bootstrapping arguments to derive higher
estimates. Since the growth of $g$ described in (24) is subcritical for the
purpose of such estimates, we will eventually obtain a bound for
$\|u\|_{W^{2,p}(\Omega)}$ that depends only on $n$, $\Omega$, $u_{0}$, $g$,
$p$, and $\Lambda$. Hence $S$ is admissible. ∎

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11institutetext: Worcester Polytechnic Institute, Worcester MA 01609, USA
22institutetext: University of Luxembourg, Luxembourg

# CRNNet: Copy Recurrent Neural Network Structure Network

Xiaofan Zhou 11 Xunzhu Tang 22

###### Abstract

The target of Electronic Health Record (EHR) coding is to find the diagnostic
codes according to the EHRs. In previous research, researchers have preferred
to do multi-classification on the EHR coding task; most of them encode the EHR
first and then process it to get the probability of each code based on the EHR
representation. However, the question of complicating diseases is neglected
among all these methods. In this paper, we propose a novel EHR coding
framework, which is the first attempt at detecting complicating diseases,
called Copy Recurrent Neural Network Structure Network (CRNNet). This method
refers to the idea of adversarial learning; a Path Generator and a Path
Discriminator are designed to more efficiently finish the task of EHR coding.
We propose a copy module to detect complicating diseases; by the proposed copy
module and the adversarial learning strategy, we identify complicating
diseases efficiently. Extensive experiments show that our method achieves a
57.30% ratio of complicating diseases in predictions, demonstrating the
effectiveness of our proposed model. According to the ablation study, the
proposed copy mechanism plays a crucial role in detecting complicating
diseases.

###### Keywords:

EHR codingPath generationAdversarial learningRecurrent Neural Networks.

##  1 Introduction

#####

In the contemporary era of healthcare, the abundance of biomedical data has
increasingly become crucial, making coding for Electronic Health Records (EHR)
a significant task [19]. EHR coding is a classification task that generates
International Classification of Diseases (ICD) codes for EHRs, which plays a
crucial role in various areas, e.g., search, data mining, billing,
epidemiology, and disease detection for healthcare manufacturers [1, 6]. Deep
learning methods for EHR coding are popular due to their higher accuracy and
ability to handle large volumes of data efficiently within a short time frame.

#####

EHR coding remains a challenging task today, and multi-label classification
has been the main trend in recent years’ research on the EHR coding task [13,
22]. Among these methods, deep learning techniques, e.g., multilayer
perceptron (MLP) [23], and sequential models like long short-term memory and
recurrent neural networks [18, 10], generate probabilities for each code.
Codes with probabilities higher than a threshold are then used as labels for
the EHRs. However, these methods face several shortcomings. Firstly, codes
exist in a very high-dimensional space; in ICD-9-CM, there are more than
18,000 codes, and these codes have a severely unbalanced distribution.
Secondly, existing methods often neglect the fact that some codes may form a
group of complications; their only target is to classify the codes directly.
For example, people with diabetes mellitus may also have diabetic retinopathy
and cataracts caused by diabetes. Moreover, determining the appropriate
threshold in these methods is challenging. A second family of EHR coding
methods generates a path where the code in this path is treated as the
classification result for the EHR. This approach, called RPGNet, reduces the
candidate space of ICD codes and uses each node as a prediction result for
EHRs [26]. However, this method still cannot effectively address the problem
of identifying complicating diseases.

#####

In this paper, we concentrate on EHR coding methods aimed at identifying
groups of complicating diseases. To detect these complicating diseases, our
new method generates ICD codes using Recurrent Neural Networks (RNNs). We
employ the odds ratio to evaluate whether two nodes might be complications.
Subsequently, we use existing ICD codes to generate new ICD codes based on
their complication information. This formulation allows for the discovery of
ICD codes that are based on complication information. Furthermore, each output
in the RNN can be considered as a prediction result, enabling the RNN method
to be used for multi-label prediction without the need to determine a
threshold for labels.

##  2 RELATED WORK

###  2.1 Automatic EHRs coding

Deep learning has been proven efficient in various fields. The field of
automatic EHR coding has recently seen a surge in publications, encompassing a
range of approaches and techniques. Several recent studies address this
question using sequential models. Li and Yu leverage a residual structure and
a series of convolutional layers to extract more efficient representations for
EHRs [16]. [6] and [3] used recurrent neural networks (RNNs) to extract the
representation of EHRs.

Other works introduce medical knowledge into EHR coding. To utilize the
information from the ICD code definitions, they encode the descriptions
provided by medical workers for the ICD codes for more efficient
representation through CNNs and RNNs [22, 20]. Additionally, [27] proposed a
tree-of-sequences LSTM model that can capture both the semantics of each code
and the hierarchical relationships of codes. [24] introduced various methods
to leverage the hierarchical knowledge of ICD codes by adding refined loss
functions.

Recently, the RAC model [14] proposed a Read, Attend, and Code (RAC) framework
that can predict ICD codes more efficiently, greatly improving the task. In
JointLAAT [25], a developed label-aware attention mechanism is leveraged in a
bidirectional LSTM. Hypercore [4] considers the hierarchical structure as a
hyperbolic space and leverages a graph-based neural network to capture co-
occurrences of codes. ISD [31] proposes a self-distillation learning mechanism
to alleviate the long-tail code distribution and uses shared representation
extraction for codes of different frequencies. EffectiveCAN [17] developed a
network with a squeeze-and-excitation (SE) mechanism connected by residual
connections and extracted representations from all encoder layers for label
attention. They also introduced focal loss to address data imbalance issues.
However, these recent models aim to find the relationship between EHR
representations and code representations. In contrast, our work emphasizes the
relationship between the codes, applying a copy module to identify
complicating diseases.

###  2.2 Adversarial learning in healthcare

Generative Adversarial Networks (GANs) have demonstrated impressive results in
generative tasks, especially in computer vision. A GAN comprises a
discriminator and a generator, where the generator creates outputs to deceive
the discriminator, and the discriminator is designed to classify which outputs
are from the generator and which are from the ground truth. In the healthcare
domain, various GAN models have excelled in Electronic Health Record (EHR)
tasks. RGAN and RCGAN [8] generate realistic time-series data, while medGAN
[7] and mtGAN [11] produce convincing EHRs. Furthermore, [29] deals with rare
disease detection using semi-supervised GANs.

Distinct from previous research, our approach employs seqGAN concepts [30],
integrating GAN with LSTM to generate multi-label outputs and calculate
rewards for every generated path. The discriminator evaluates the legitimacy
of the path, using the outcomes as rewards for each generated path, thereby
guiding our model’s learning process.

##  3 METHOD

The aim of this study is to formulate a series of diagnostic codes, denoted as
$Y=[c^{1},c^{2},\ldots,c^{j},\ldots]$, which correspond to Electronic Health
Records (EHRs). Each $c^{i}$ in this series signifies an International
Classification of Diseases (ICD) code and where i means the position of code
in the path, and $c^{i}\in C$, where $C$ represents code dictionary. For the
sake of simplification, we treat an EHR, denoted as $x$, as unstructured text.
In mathematical terms, $x=[w_{1},w_{2},\ldots,w_{i},\ldots]$, where $w_{i}\in
W$, W is the words token dictionary. The methods we propose to achieve this
objective will be elaborated in the subsequent sections of this paper.

Figure 1: Overview of CRNNet

###  3.1 Overview

In CRNNet, there are two cruicial parts, the first part is a path
generator$G_{\theta}$, the path generator is used to generate a list of ICD
paths, the ICD paths will be seen as the classification result for the given
EHR $x$, in the path generator$G_{\theta}$, firstly, a EHR encoder generate a
feature representation using convolutional neraul network and a token
embedding matrix $I^{n*d}$ where $n$ represents the count of the token while
$d$ represents embedding matrix’s dimension. Then, we propose a EHR-ICD fusion
module, to combine the information of previous generated code, EHR embedding.
To consider the long-short term relation, we leverage LSTM module to generate
the probability of every code at each timestamp and choose the code with
largest probability as the code at each timestamp, finally, a copy module is
used to help with finding the complication disease based on the preceding code
$c_{t-1}$. We use the probability given by LSTM and copy module as the
classification result. The length of LSTM is a preset hyper-parameter.

To classify whether the code path generated by the path generator $G_{\theta}$
is a feasible code path, we apply the idea of GAN to design a path
discriminator $D_{\phi}$, which output a probability score $r_{t}$, $r_{t}$
means the probability of truth of the generated path. The input of $D_{\phi}$
is the embedding of EHR $x$, and the ICD path is generated by path
generator$G_{\theta}$. By the way, we split the ICD path generated by path
generator$G_{\theta}$ into $t$ parts, where $t$ is the valid codes number in
the path, so that we can enlarge the sample number fed into the $D_{\phi}$.
This path discriminator helps to better train the path generator, and updated
in a certain round.

###  3.2 Long Short Term Memory Networks (LSTM)

#####

Recurrent Neural Network (RNN) architecture the main part of the path
generator, specialized for processing sequential data. Unlike traditional
feedforward neural networks, RNNs are designed with interconnected hidden
states that allow them to capture dynamic temporal patterns within sequences.
Conceptually, they can be thought of as an advancement over hidden Markov
models, but with the incorporation of nonlinear transition functions. This
structure makes RNNs particularly adept at identifying long-range temporal
dependencies. Compared to other RNNs, Long Short-Term Memory (LSTM) networks
is special. Each neuron in LSTMs contains three gates: input$i$,forget$f$, and
output $o$. These gating mechanisms bestow LSTMs with three distinct
advantages: firstly, they excel in learning long-term sequence dependencies;
secondly, they facilitate easier optimization, as the gates effectively route
the input signal through recurrent hidden states $r(t)$ without causing
undesired alterations in the output; and thirdly, they mitigate the well-known
exploding and vanishinggradient encountered during the training of
conventional RNNs.

The mathematical representation of the LSTM is given by:

$\displaystyle f_{t}$ $\displaystyle=\sigma(W_{f}\cdot[h_{t-1},c_{t}]+b_{f})$
(1) $\displaystyle i_{t}$
$\displaystyle=\sigma(W_{i}\cdot[h_{t-1},c_{t}]+b_{i})$
$\displaystyle\tilde{C}_{t}$
$\displaystyle=relu\left(W_{C}\cdot[h_{t-1},c_{t}]+b_{C}\right)$
$\displaystyle C_{t}$ $\displaystyle=f_{t}\times
C_{t-1}+i_{t}\times\tilde{C}_{t}$ $\displaystyle o_{t}$
$\displaystyle=\sigma(W_{o}\cdot[h_{t-1},c_{t}]+b_{o})$ $\displaystyle h_{t}$
$\displaystyle=o_{t}\times\tanh(C_{t})$

$\sigma$ denotes sigmoid function, $\cdot$ represents element-wise product,
and the $W_{*}$ matrix is weight matrix, $b_{*}$ is bias vector, $h_{t-1}$
represents hidden representation before. In this paper, rectified linear units
(ReLU) us used for activation function.

###  3.3 Path generator $G_{\theta}$

The path generator is contituted by four parts: we first get the
representation $x$ by multichannel CNN encoder from raw text, then we use PMP
to consider the connection between the codes in the path and get $v$ as
representation, after put the $v$ into lstm, then we use the former code in
the path $c$ and $o$ into the copy module to get the final codes which can
help with finding the complications. We will introduce this progress step by
step.

####  3.3.1 EHR-ICD fusion module

We use the EHR-ICD fusion module to process the hidden state $h_{t}$ obtained
from LSTM by considering the relationship between EHR $x$ and the generated
Codes path, the relationship between the current code and all other codes.
Next, we will talk about how EHR-ICD fusion module is used to address the
question.

##### relationship between EHR and the generated Codes path

In this step, we fuse the representation of EHR $x$ and the representation of
generated code path, and we represent the result in this step as $o_{t}$,
which is shown below:

$\displaystyle o_{t}=tanh(W_{o}(u_{t}))$ (2)

where $W_{o}$ is weight matrix and $u_{t}$ is calculated by a series of
operation on the representation of $x$ and $o_{t}$, such operation is shown
below:

$\displaystyle u_{t}=x\oplus c_{t}\oplus(x\odot
c_{t})\oplus(x+c_{t})\oplus(x-c_{t})\oplus(c_{t}-x)$ (3)

The $\oplus$ represents concatenation operation, $\odot$ represents product.

##### Former-to-later code relationship

This part aims to capture the relationship of previous ICD codes $c_{t-1}$ and
subsequent ICD codes $c_{t}$. To find the how generated code and all codes in
relation, we use the $u_{t}$ generated by considering the relationship between
an EHR and the generated Codes path as input, we use the representation of all
codes to do the task before:

$\displaystyle v_{t}=o_{t}\cdot c_{t}^{c}$ (4)

$c_{t}^{c}$ is the representation of each code, $\cdot$ is the element-wise
product operation.

####  3.3.2 Copy module for complication detection

A code vocabulary $\mathcal{V_{c}}=\\{v_{1},...,v_{N}\\}$ is designed for code
$c$ as its potential complication disease and use UNK for any code have low
probability to be complication. As $\mathcal{C}$ is the whole code dictionary,
considering code of $\mathcal{C}$ let this module consider non-comlication
diseases. Then Given the decoder lstm state $h_{t-1}$ at time $t-1$, we
generate the ’mixture’ probability of code $c_{t}$ as follows:

$\displaystyle p\left(c_{t}\mid\mathbf{h}_{t},c_{t-1}\right)$ (5)
$\displaystyle=p\left(c_{t},c\mid\mathbf{h}_{t},c_{t-1},\mathcal{V_{c_{t-1}}}\right)+p\left(c_{t},g\mid\mathbf{h}_{t},c_{t-1},\mathcal{C}\right)$

where $g$ represents the generate-mode, and $c$ denotes the copy mode. The
probabilities for these two modes are given respectively by:

$\begin{gathered}p\left(c_{t},c\mid\cdot\right)=\left\\{\begin{array}[]{cc}\frac{1}{Z}e^{\psi_{c}\left(h_{t}\right)},&y_{t}\in\mathcal{V}\\\
0,&y_{t}\in\mathcal{X}\cap\bar{V}\\\ \end{array}\right.\end{gathered}$ (6)

$\begin{gathered}p\left(c_{t},g\mid\cdot\right)=\frac{1}{Z}\sum_{j:c_{j}=y_{t}}e^{\psi_{g}\left(h_{j}\right)}\\\
\end{gathered}$ (7)

$\psi_{g}\left(.\right)$ and $\psi_{c}\left(.\right)$ are rating methods, and
we treat $Z$ as the normalization term
$Z=\sum_{v\in\mathcal{V}\cup{\mathrm{UNK}}}e^{\psi_{g}(v)}+\sum_{x\in
X}e^{\psi_{c}(x)}$. Caused by this normalization term, above two modes
essentially compete through a softmax function. The score of each mode is
calculated:

##### Generate-Mode:

The rating function used is same as in the generic RNN encoder-decoder [2],

$\psi_{g}\left(y_{t}=v_{i}\right)=\mathbf{v}_{i}^{\top}\mathbf{W}_{o}\mathbf{h}_{t},\quad
c_{i}\in\mathcal{C}$ (8)

where $\mathbf{W}_{o}\in\mathbb{R}^{(N+1)\times d_{s}}$ is a weight matrix and
$v_{i}$ is one-hot representation of $c_{i}$.

##### Copy-Mode:

The rating for copy module is calculated as

$\begin{gathered}\psi_{c}\left(c_{t}=x_{j}\right)=\sigma\left(\mathbf{h}_{j}^{\top}\mathbf{W}_{c}\right)\mathbf{h}_{t},\quad
x_{j}\in\mathcal{V}\end{gathered}$ (9)

In this equation, $\mathbf{W}{c}\in\mathbb{R}^{d{h}\times d_{s}}$ is the
weight matrix, and $\sigma$ is activation function. Empirically, we discovered
that non-linearity activation tanh performs better than other activation
function, and we utilize the one-hot complicating disease vector for $h_{t}$.

###  3.4 EHR encoder

In this section, we extract the representation $x$ of each EHR for the sake of
later processing by multi-channel CNN[5]. The first step in this encoder is
embedding the tokens in the EHR into a embedding matrix $X\in I^{n\times d}$
$n$ is the total number of tokens among all EHR and $d$ is the dimension of
the embedding matrix. In the convolutional layers, we apply $c_{i}$ different
filters to each kernal size $k_{i},1\leq i\leq m$. Each filter $f_{ij}$,
$1\leq j\leq c_{i}$, is a matrix $\in\mathbb{R}^{k_{i}\times d}$, and it is
used for $k_{i}$ words in text to produce a feature map
$\mathbf{F}_{ij}\in\mathbb{R}^{(n-k_{i}+1)\times 1}$:

$\begin{array}[]{ll}\mathbf{F}_{ij}[p]=\text{ReLU}(\mathbf{f}_{ij}\cdot\mathbf{X}_{p:p+k_{i}-1}+b_{ij}),\end{array}$
(10)

where $\cdot$ denotes the dot product, $\mathbf{X}_{p:p+k_{i}-1}$ is the
$p$-th to $(p+k_{i}-1)$-th row of $\mathbf{X}$, $b_{ij}$ is a bias term, and
ReLU is nonlinear activation function.

Next, we utilize max-pooling to each feature map $\mathbf{F}_{ij}$ to obtain a
scalar $s_{ij}$:

$\begin{array}[]{ll}s_{ij}=\max_{p}\mathbf{F}_{ij}[p].\end{array}$ (11)

Finally, we concatenate all the $s_{ij}$ as the final representation of the
text:

$\begin{array}[]{ll}\mathbf{x}=[s_{11},s_{12},\ldots,s_{mc_{m}}].\end{array}$
(12)

####  3.4.1 Orderless Recurrent Models

As code for each EHR is orderless, we need to find a way to compare our
generated code which is in order, we adopt the concept from [28] to align them
with the network’s predictions before calculating the loss. We use the
predicted label alignment (PLA) to achieve this goal.

The core idea behind PLA is when predicted label is true, we prefer to keep
it. Which leads to the following optimization problem:

$\begin{array}[]{ll}\mathcal{L}=\qquad\min\limits_{T}&\operatorname{tr}(T\log(P))\\\
\text{ subject to }&T_{tj}\in\\{0,1\\},\sum_{j}T_{tj}=1\\\ &T_{tj}=1\text{ if
}\hat{l}_{t}\in L\text{ and }j=\hat{c}_{t}\\\ &\sum_{t}T_{tj}=1\ \forall j\in
L\\\ &\sum_{t}T_{tj}=0\ \forall j\notin L\end{array}$ (13)

$\hat{c}_{t}$ represents the model’s predicted label at time step $t$.
Initially, we secure the elements in matrix $T$ corresponding to predictions
that are verified by the ground truth set $L$. Subsequently, we employ the
Hungarian algorithm to allocate remaining labels. This approach offers greater
alignment with the labels that the LSTM model actually predicts.

###  3.5 Path Discriminator $D_{\phi}$

For the sake of training the path generator in a more effiecient way, we refer
from the idea of GAN to design a path discriminator module $D_{\phi}$,
$D_{\phi}$ is designed to generate a rewart $r_{t}$ for every pairs of EHR
representation $x$ and path representation generated by a path encoder, ICD
code path $y_{t}=(c_{1},c_{2},...,c_{k},...,c_{t})$ by path generator we
generate the path representation by lstm encoder:

$\begin{array}[]{ll}h_{k}&=LSTM(h_{k-1},c_{k}),\end{array}$ (14)

$h_{k-1}$ is the hidden state at timestamp $k-1$; $c_{k}$ is the $k$-th ICD
code representation. Then the calculation of $r_{t}$ is shown below:

$\begin{array}[]{ll}r_{t}&=R_{\left(s_{t},x\right),a_{t}}\\\
&=p_{D}\left(y_{t},\boldsymbol{x}\right)\\\
&=\sigma\left(\boldsymbol{W}_{r}\left(\boldsymbol{h}_{t}\oplus\boldsymbol{x}\right)\right),\end{array}$
(15)

$\oplus$ represents concatenation, $\boldsymbol{W}_{r}$ represents weight of a
linear layer. The path $y_{t}$ is obtained by recurrently applying an LSTM to
the ICD code path $y_{t}=(c_{1},c_{2},...,c_{k},...,c_{t})$, and $\sigma$
represents the sigmoid function.

We train the path discriminator using adversarial training strategy. Here, the
paths generated by the path generator are treated as negative samples, the
ground truth paths are considered as positive samples. We optimize a cross-
entropy function to improve the effectiveness of $D_{\phi}$:

$\begin{array}[]{ll}L_{D}&=-\sum_{\left(y_{t},x\right)\in Q^{+}}\log
p_{D}\left(y_{t},x\right)\\\ &-\sum_{\left(y_{t},x\right)\in
Q^{-}}\log\left(1-p_{D}\left(y_{t},x\right)\right),\end{array}$ (16)

$Q^{+}$ and $Q^{-}$ represent a pair of positive and negative samples,
respectively; $p_{D}\left(y_{t},x\right)$ is the probability of sample
($y_{t},x$) belonging to a positive sample.

##  4 Experimental Design

###  4.1 Research Questions

We plan to solve the following research problems:(RQ1)How well CRNNet preform
on find the complicating disease. (RQ2) What are the effects of different
components? Where do the improvements of CRNNet come from?

###  4.2 Dataset

We use MIMIC-III as the dataset for comparison, which is a large EHR dataset.
To further elaborate, we preprocess the EHRs by first removing any personally
identifiable information, such as patient and hospital names, as well as other
irrelevant information like medications and procedures not related to the
patient’s diagnosis. We also tokenize the remaining text and use stopword
removal and stemming to preprocess the EHR. To simplify the problem, we use a
top-50 label setting to preprocess the dataset.

For the top-50 label setting, we only use the most frequent 50 codes from the
dataset and filter out instances that do not contain at least one of these
codes. This results in a subset of the dataset that is more focused and
relevant to our task.

The split ratio of the train, test, and validation datasets from the resulting
dataset is 4:1:1, with each set containing approximately 80%, 10%, and 10% of
the instances, respectively. This allows us to evaluate the performance of our
models on unseen data while ensuring the training set is large enough for
effective learning.

###  4.3 Baselines

To demonstrate the performance and functionality of CRNNet, several methods
are re-implemented for comparison. Specifically, we compare CRNNet with
following methods:

##### CAML[20]

CAML leverages Text-CNN[5] to generate representations for each Electronic
Health Record (EHR). It then employs attention module to learn salient
features for codeing. This tailored representation forms the basis for binary
classification tasks.

##### RPGNet[7]

RPGNet approaches EHR coding by employing a dual structure consisting of a
Path Generator (PG) and a Path Discriminator (PD). They propose PMP module to
encode the path. In addition, the PD component calculates a reward value for
every generated path. The network is trained using adverserial learning
strategy.

##### PLM-ICD[12]

PLM-ICD utilizes Pretrained Language Models to obtain context-rich
representations. It use transformer methods the encode the EHR content. It
currently stands as the state-of-the-art method for ICD classification.

###  4.4 Metric

In our study, we utilize a comprehensive set of evaluation metrics, frequently
employed in related work [20]. Our evaluation framework incorporates both
micro- and macro-averaged metrics. The micro-averaged approach treats each
instance as a separate prediction during computation, whereas the macro-
averaged method averages metrics for each individual label, making it
especially relevant for assessing performance on rare labels. More
specifically, we examine four primary metrics: Precision, Recall, F1 Score,
and the Area Under the Curve (AUC) [9]. The F1 Score is the harmonic mean of
Precision and Recall, and the AUC provides a summary of the model’s
performance across various thresholds. Additionally, we employ the Jaccard
Similarity Coefficient as an overlap measure between the sets of predicted and
actual ICD codes [21]. This coefficient is defined as

$Jaccard=\frac{1}{m}\sum_{i=1}^{m}\frac{|Y_{i}\cap\hat{Y}_{i}|}{|Y_{i}\cup\hat{Y}_{i}|},$

$m$ denotes the number of code in the dataset, while $Y_{i}$ is the set of
predicted codes, and $\hat{Y}_{i}$ represents the set of actual codes.
Furthermore, we gauge the models’ effectiveness in identifying complications
by evaluating the proportion of complications in each disease combination
present in the predictions. The formula for computing the complication ratio
is as follows:

$\begin{array}[]{ll}complication=\frac{\sum_{1<=i<j<=n}I(c_{i},c_{j})}{\binom{2}{n}}\end{array}$
(17)

where $I(c_{i},c_{j})$ represent whether code $i$ and code $j$ is a pair of
complication, $n$ represent the number of codes in the prediction.

###  4.5 Implementation details

We use 300 as the dimension for the representation of EHR and path, and the
dimension of the token and code embedding is set as 100(Eq. 3 and Eq. 4). For
the EHR encoder, the filter size of convolutional layers is 100, while the
kernal size of these layers are 3, 4, 5, respectively. while the the drop
ratio of the dropout is 0.5 (Eq. 11). For EHR-ICD fusion module, we set the
hidden size of LSTM gate as 300. In training stage, we use 200 as the max
train iteration, and the batch size is set as 32. We train path generator and
path discriminator in every batch alternately. All model parameters are
randomly initialized. Adam optimizer[15] is employed to train the objective
function, learning rate chosen as 0.0001 and default momentum parameters of
$\beta_{1}$ = 0.9 and $\beta_{2}$ = 0.999. The code is implemented in PyTorch
and we train it on an NVIDIA T4 Tensor Core GPU.

Table 1: Results(%) on dataset, bold face indicates the best result.

|  |  | Precision | Recall | F1 | AUC
---|---|---|---|---|---|---
Method | Jaccard | Complication | Micro | Macro | Micro | Macro | Micro | Macro | Micro | Macro
CAML(CNN) | 27.68 | 24.57 | 58.59 | 38.35 | 40.25 | 33.50 | 47.63 | 33.59 | 68.31 | 66.35
RPGNet(CNN) | 33.34 | 36.31 | 41.20 | 33.59 | 56.81 | 44.51 | 48.35 | 35.40 | 75.14 | 70.50
CRNNet(CNN) | 33.79 | 57.30 | 54.67 | 34.74 | 43.59 | 32.01 | 48.45 | 31.31 | 69.58 | 65.36
PLM-ICD(Bert) | 54.87 | 42.29 | 68.26 | 64.24 | 71.25 | 66.74 | 69.72 | 65.46 | 92.64 | 90.14

Table 2: Ablation study

|  |  | Precision | Recall | F1 | AUC
---|---|---|---|---|---|---
Method | Jaccard | Complication | Micro | Macro | Micro | Macro | Micro | Macro | Micro | Macro
CRNNet | 33.79 | 57.30 | 54.67 | 34.74 | 43.59 | 32.01 | 48.45 | 31.31 | 69.58 | 65.36
CRNNet(no-copy) | 35.05 | 51.86 | 54.64 | 35.03 | 45.39 | 31.93 | 49.52 | 31.28 | 70.39 | 65.15
CRNNet(no-ARL) | 34.09 | 55.80 | 54.57 | 34.10 | 44.73 | 31.79 | 49.08 | 30.94 | 70.08 | 65.10

##  5 Experimental Result

###  5.1 RQ1.1

We assessed the performance of various models, including Caml, RPGnet, CRNNet,
and PLM-ICD. We evaluated these models using the top-50 label setting and
trained the first three models with the same preprocessing method.
Additionally, we incorporated a pretrained PLM-ICD (BERT) model in our
analysis. The results presented in Table 1 lead to the following observations:

Among the compared models, CRNNet achieved the highest complication ratio.
Notably, CRNNet’s complication ratio was 15.01% higher than PLM-ICD, the
state-of-the-art method for EHR coding. This indicates that CRNNet effectively
identifies complications associated with other codes. This may be attributed
to CRNNet’s exploration of ICDs through a sequence of paths, as well as its
utilization of information from previous codes to generate new ones.

###  5.2 Ablation study(RQ1.2)

We conducted an ablation study to evaluate the effects of modules in CRNNet,
shown in Table 2. Followings are ablation of CRNNet that we considered: (1) No
copy represents CRNNet removed the copy module. We removed the copy module and
used only the RNN structure model to generate codes. (2) No ARL represents
CRNNet removed adversarial learning strategy. We removed the adversarial
learning process to train our model. From the results, we obtained the
following finding:

First, the ability of CRNNet to identify complication diseases decreased
dramatically after removing the copy module. Specifically, the complication
ratio in the result of CRNNet(no-copy) dropped by 5.44% respectively. The
result shows that copy module plays a significant role in our method to
identify complicating diseases. The copy module extracts information related
to complicating diseases based on the former ICD in the path, which helps to
generate complicating diseases.

Second, the ability of CRNNet to identify complicating diseases is still
working better than PLM-ICD when copy module is not used. The complication
ratio of PLM-ICD is 42.29% and that of CRNNet(no-copy) is 51.86%, which
indicates that the RNN structure is also helpful in identifying complicating
diseases, as the ICD on timestamp $t+1$ is generated based on the ICD on
timestamp $t$.

Finally, the adversarial reinforcement learning (ARL) does not significantly
improve the performance of CRNNet, but it slightly improves its ability to
identify complicating diseases. This is due to adversarial reinforcement
learning helps CRNNet generate codes paths similar to the ground truth paths.

##  6 CONCLUSION AND FUTURE WORK

In this paper, we propose an innovative approach for Electronic Health Record
(EHR) coding, specifically targeting the identification of complicating
diseases. We frame this as a path generation task and introduce the Copy
Recurrent Neural Network Structure Network (CRNNet). CRNNet consists of a EHR-
ICD fusion module designed to fuse the representation of EHRs and code path.
An adversarial learning framework is used for training stage. Our experimental
results on the MIMIC-III dataset reveal that CRNNet surpasses existing methods
in detecting complicating diseases, and its performance excels when using the
CNN encoder method. The performance improvement is attributed to the
implementation of the copy module and the adversarial reinforcement learning
mechanisms.

Nonetheless, CRNNet’s limitations lie in its improvement of recall and AUC.
While it performs well in detecting complications, it does not focus on
enhancing basic performance metrics. Additionally, although it is based on a
CNN encoder, transformer methods are now considered a more efficient way to
encode EHRs. In future research, we will focus on integrating transformer
methods and improving performance with the aid of complication detection,
which is a crucial task in the EHR coding area.

##  7 DATA AVAILABILITY

We make our code and dataset publicly available at:

https://anonymous.4open.science/r/CRNNNet-D780

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