# A change of variable for Dahlberg-Kenig-Pipher operators.

Joseph Feneuil  Joseph Feneuil. Université Paris-Saclay, Laboratoire de
mathématiques d’Orsay, 91405, Orsay, France  joseph.feneuil@universite-paris-
saclay.fr

###### Abstract.

In the present article, we give a method to deal with Dahlberg-Kenig-Pipher
(DPK) operators in boundary value problems on the upper half plane.

We give a nice subclass of the weak DKP operators that generates the full
class of weak DKP operators under the action of bi-Lipschitz changes of
variable on $\mathbb{R}^{n}_{+}$ that fix the boundary $\mathbb{R}^{n-1}$.
Therefore, if one wants to prove a property on DKP operators which is stable
by bi-Lipschitz transformations, one can directly assume that the operator
belongs to the subclass. Our method gives an alternative proof to some past
results and self-improves others beyond the existing literature.

This article was written during the author’s stay at the Université Paris-
Saclay in France, where he was supported by the Simons Foundation grant
601941, GD

Key words: Boundary value problems, Elliptic operators with rough
coefficients, Dahlberg-Kenig-Pipher operators, Carleson perturbations.

AMS classification: 35J25.

###### Contents

  1. 1 Introduction
  2. 2 Proof of Proposition 1.7
  3. 3 Applications
    1. 3.1 Regularity problem
    2. 3.2 Dirichlet problem in higher codimension.

##  1\. Introduction

In the Euclidean setting, boundary value problems for elliptic operators have
been studied for ages by many researchers, who worked to find the relationship
between the smoothness of the boundary data, the geometry of the boundary, and
the smoothness of the coefficients of the elliptic operator under
consideration.

Let us skip the early literature on the cases where the boundary datum is
continuous or where the ambient space is the complex plane. In the higher
dimensional Euclidean setting, the Dirichlet value problem in $L^{p}$ was
studied for the Laplacian and rough boundaries in for instance [Da], [BJ],
[DJ], [Ken], [HM], [HMU], [AHM3TV], [AHMNT], [AHMMT], [Azz]. When the operator
is not the Laplacian, negative results on the solvability of the Dirichlet
problem in $L^{p}$ are in [CFK], [MM], and positive results are for either
$t$-independent operators on
$\mathbb{R}^{n}_{+}:=\\{(x,t)\in\mathbb{R}^{n-1}\times(0,\infty)\\}$ (see for
instance [JK], [KKPT], [HKMP]), or for operators that verify some conditions
described in terms of Carleson measures (see [FKP], [KP1], [KP3], [DPP],
[DP]). Of course, it would be impossible to mention all the articles that
could be linked to the topic, in particular knowing that the present article
is meant to be brief, hence the aforementioned papers are only a small sample
of the considerable work in the research area. We refer to the introduction of
[DFM3] and [FP] for a better description of the current state of the art.

The operators that fall into the scope of the present article are in the
second form, i.e. related to Carleson measures. The Dahlberg-Kenig-Pipher
operators - or DKP operators for short - are roughly the uniformly elliptic
operators in the form $L=-\operatorname{div}\mathcal{B}\nabla$ on a domain
$\Omega$ for which

(1.1) $\operatorname{dist}(X,\partial\Omega)|\nabla\mathcal{B}|^{2}dX\text{ is
a Carleson measure on $\Omega$.}$

More precise definitions will be given later. For the DKP operators (and
sufficiently smooth boundaries), the Dirichlet problem is solvable in $L^{p}$
for sufficiently large $p\in(1,\infty)$. This assertion was claimed by
Dahlberg and proved by Kenig and Pipher in [KP3], hence the name ‘DKP’.
Kenig’s and Pipher’s proof is based on the following simple yet powerful
observation. Consider a matrix $\mathcal{B}$ such that $\nabla\mathcal{B}$ is
at least in $L^{1}_{loc}$. We write $\mathcal{B}$ as a matrix by blocs

(1.2) $\mathcal{B}:=\begin{pmatrix}\mathcal{B}_{1}&\mathcal{B}_{2}\\\
\mathcal{B}_{3}&b\end{pmatrix},$

where $b$ is a scalar function on $\Omega$, and so $\mathcal{B}_{1}$ is a
matrix function of order $n-1$, and $\mathcal{B}_{2}$ and $\mathcal{B}_{3}$
are respectively vertical and horizontal vector functions of length $n-1$. We
construct $\widetilde{\mathcal{B}}$ and $\mathcal{T}$ as

$\widetilde{\mathcal{B}}:=\begin{pmatrix}b^{-1}\mathcal{B}_{1}&b^{-1}(\mathcal{B}_{2}+\mathcal{B}_{3}^{T})\\\
0&1\end{pmatrix}\quad\text{ and
}\quad\mathcal{T}:=\begin{pmatrix}0&\mathcal{B}_{3}^{T}\\\
-\mathcal{B}_{3}&0\end{pmatrix},$

and we notice that we have the formal computation

(1.3)
$\begin{split}-\mathop{\operatorname{div}}\mathcal{B}\nabla&=-b\mathop{\operatorname{div}}\Big{(}\frac{\mathcal{B}}{b}\nabla\Big{)}-\frac{\nabla
b}{b}\cdot\mathcal{B}\nabla\\\
&=-b\mathop{\operatorname{div}}\Big{(}\frac{\mathcal{B}+\mathcal{T}}{b}\nabla\Big{)}+b\mathop{\operatorname{div}}\Big{(}\frac{\mathcal{T}}{b}\nabla\Big{)}-\frac{\nabla
b}{b}\cdot\mathcal{B}\nabla\\\
&=-b\mathop{\operatorname{div}}\Big{(}\widetilde{\mathcal{B}}\,\nabla\Big{)}+b\underbrace{\left[\mathop{\operatorname{div}}\Big{(}\frac{\mathcal{T}}{b}\Big{)}-\mathcal{B}^{T}\frac{\nabla
b}{b^{2}}\right]}_{\mathcal{D}}\cdot\nabla,\end{split}$

where the divergence of a matrix $\mathcal{A}$ is a vector containing the
divergence of the columns of $\mathcal{A}$. If $\mathcal{B}$ is uniformly
elliptic - see (1.6) below - and if $u$ is a weak solution to
$-\mathop{\operatorname{div}}\mathcal{B}\nabla u=0$, then $u$ is also a weak
solution to the elliptic operator with drift
$(-\mathop{\operatorname{div}}\widetilde{\mathcal{B}}\nabla+\mathcal{D}\cdot\nabla)u=0$.
Following [KP3], this transformation has been used by numerous authors to turn
an elliptic operator into another elliptic operator with (for instance) a nice
last line, at the price of adding a drift (see for instance [DP], [DPP],
[DPR], [FMZ]). The same trick can be used to get 0 in the top right corner of
$\mathcal{B}$ (instead of the bottom right corner) or to turn $\mathcal{B}$
into a symmetric matrix, always at the price of adding a drift. The identity
(1.3) is so easy that it may not even be discussed by the authors that used
it, and may often be only a couple of lines in the deepest part of the proofs.
But we did not know any way to avoid it, that is, before the present article,
we were not aware of any method that allows people to avoid drifts when
dealing with DKP operators. The apparently innocent observation (1.3), which
allows us to trivialize one line (or column) of the matrix $\mathcal{B}$, has
been crucial in many proofs. However, it is obviously limited to cases where
we can handle drifts.

In this paper, we purpose an alternative method to the transformation (1.3),
which similarly allows us to assume that one line of $\mathcal{B}$ is simple,
and the price to pay this time is a “Carleson perturbation”. Carleson
perturbations are a class of perturbations on the matrix $\mathcal{A}$ of a
uniformly elliptic operator $L=\mathop{\operatorname{div}}\mathcal{A}\nabla$
which will preserve the solvability of the Dirichlet problem in $L^{p}$ for
large $p$ ([FKP]), and they are even optimal in some sense (see [FKP] again).
Morally speaking, Carleson perturbations are objects that fit very well in the
theory of boundary value problems with data in $L^{p}$, and many results in
the recent literature include them. So we are saying that we purpose here an
alternative method that trivializes a line of $\mathcal{B}$ by adding a term
which is often free (because already included in the theory, because natural
in the theory). Of course, we do not say that the argument presented below is
strictly better than using (1.3). Both have pros and cons that we shall
discuss later. Our aim is really to introduce the reader to a substitute
method, and to show that this other method can be used to self improve some
existing results in situations where using (1.3) does not work.

We shall present our method in a fairly simple case. Yet, it is meant to be
adapted to more complicated scenarios, some of them will be discussed, but of
course we cannot anticipate all of them. Our main choice towards
simplification is that our domain will always be the upper half plane
$\mathbb{R}^{n}_{+}$. In this context, we define the Carleson measures and the
Carleson measure condition as follows.

######  Definition 1.4.

We say that a quantity $f$ (a scalar, vector, or matrix function) defined on
$\mathbb{R}^{n}_{+}$ satisfies the Carleson measure condition if
$|f|^{2}\frac{dt}{t}dx$ is a Carleson measure, that is if for any
$x\in\mathbb{R}^{d}$ and any $r>0$,

(1.5) $\int_{B(x,r)}\int_{0}^{r}|f(y,t)|^{2}\,\frac{dt}{t}\,dy\leq Mr^{d}.$

We write $f\in CM$ \- or $f\in CM(M)$ when we want to refer to the constant in
(1.5) - when $f$ satisfies the Carleson measure condition.

In addition, we write that $f\in CM_{sup}$ if $f_{sup}\in CM$, where $f_{sup}$
is the function on $\mathbb{R}^{d+1}_{+}$ defined as

$f_{sup}(y,t):=\sup\\{|f(z,s)|,\,s\in(t/2,2t),\,z\in B(y,t)\\}.$

Note that $f\in CM_{sup}$ is equivalent to $f_{sup}\in CM_{sup}$, and implies
that $f\in CM$ and $f\in L^{\infty}(\mathbb{R}^{n}_{+})$.

Let us turn to the operators under consideration. An operator
$L:=-\operatorname{div}\mathcal{A}\nabla$ is said to be uniformly elliptic if
there exists $C_{A}$ such that

(1.6) $\mathcal{A}(x,t)\xi\cdot\xi\geq(C_{A})^{-1}|\xi|^{2}\text{ and
}|\mathcal{A}(x,t)\xi\cdot\zeta|\leq C_{A}|\xi||\zeta|\qquad\text{ for
}(x,t)\in R^{n}_{+},\,\xi,\zeta\in\mathbb{R}^{n}.$

The Dahlberg-Kenig-Pipher operators are the uniformly elliptic operators on
$\mathbb{R}^{n}_{+}$ for which $|t\nabla\mathcal{A}|\in CM_{sup}$ or
$|t\nabla\mathcal{A}|\in CM\cap L^{\infty}$, depending on the context. Recall
that the solvability of the Dirichlet problem in $L^{p}$ is stable under
Carleson perturbation on the coefficients ([FKP]), therefore it means that a
larger class of DKP-type operators on $\mathbb{R}^{n}_{+}$ can be given by
assuming that the matrix $\mathcal{A}$ can be decomposed as
$\mathcal{A}=\mathcal{B}+\mathcal{C}$ with
$|t\nabla\mathcal{B}|+|\mathcal{C}|\in CM_{sup}$.

Our main result is given as follows.

######  Proposition 1.7.

Let $L:=-\mathop{\operatorname{div}}\mathcal{A}\nabla$ be a uniformly elliptic
operator on $\mathbb{R}^{n}_{+}$ such that $\mathcal{A}$ can be decomposed as
$\mathcal{A}=\mathcal{B}+\mathcal{C}$ where
$|t\nabla\mathcal{B}|+|\mathcal{C}|\in CM_{sup}(M)$. Then there exists a bi-
Lipschitz change of variable $\rho$ that maps $\mathbb{R}^{n}_{+}$ to itself
and fixes the boundary $\mathbb{R}^{n-1}$ such that the conjugate operator
$L_{\rho}:=-\mathop{\operatorname{div}}\mathcal{A}_{\rho}\nabla$ verifies that
$\mathcal{A}_{\rho}$ can be decomposed as
$\mathcal{A}_{\rho}=\mathcal{B}_{\rho}+\mathcal{C}_{\rho}$ where
$\mathcal{B}_{\rho}$ has the form

(1.8)
$\mathcal{B}_{\rho}=\begin{pmatrix}\mathcal{B}_{1,\rho}&\mathcal{B}_{2,\rho}\\\
0&1\end{pmatrix}$

and $t|\nabla\mathcal{B}_{\rho}|+|\mathcal{C}_{\rho}|\in
CM_{sup}(M^{\prime})$. We can give a bound on $M^{\prime}/M$ and on the
Lipschitz constants in $\rho$ and that depends only on $n$ and the ellipticity
constant $C_{A}$.

In the proposition, by conjugate operator, we mean the operator $L_{\rho}$ for
which $v:=u\circ\rho$ is a weak solution to $L_{\rho}v=0$ whenever $u$ is a
weak solution to $Lu=0$.

Remarks:

  1. (i)

We insist on the fact that the aim of our paper is to present the method, and
not to give the most general result possible. In future articles, we intend to
adapt the argument in other contexts by explaining the differences with the
proof given here (instead of repeating all the details).

  2. (ii)

The precise type of conditions on $\mathcal{B}$ and $\mathcal{C}$ do not
really matter, as long as they are stable by bi-Lipschitz transformation. For
instance, assuming $|t\nabla\mathcal{B}|+|\mathcal{C}|\in CM$ (instead of
$CM_{sup}$) would give the analogous conclusion where
$t|\nabla\mathcal{B}_{\rho}|+|\mathcal{C}_{\rho}|\in CM$ (instead of
$CM_{sup}$).

  3. (iii)

We even only need a Carleson condition on the last line of $\mathcal{B}$ to
get the simplification of the last line of $\mathcal{B}_{\rho}$ (but if we
only assume the Carleson condition on the last line of $\mathcal{B}$, we
cannot conclude anymore that $|t||\nabla\mathcal{B}_{\rho}|\in CM_{sup}$).

  4. (iv)

With the same technique, we could find $\rho$ so that $\mathcal{B}_{\rho}$ is
zero in the top right corner (instead of the bottom left one). And if
$\mathcal{B}$ is symmetric, then $\mathcal{B}_{\rho}$ would be diagonal by
blocs.

  5. (v)

The proof can immediately be adapted to operators on
$\mathbb{R}^{n}\setminus\mathbb{R}^{d}:=\\{(x,t)\in\mathbb{R}^{d}\times\mathbb{R}^{n-d},\,t\neq
0\\}$ ($d<n-1$) in the form
$L=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}\nabla]$ with $\mathcal{A}$ as in
(1.6). Here $\mathbb{R}^{n}\setminus\mathbb{R}^{d}$ is the analogue of
$\mathbb{R}^{n}_{+}$ in domains with higher codimensional boundaries. The
elliptic theory associated to such operators was studied in [DFM1] (see also
[FKS]) and the solvability of the Dirichlet problem was studied under various
situations in for instance [DFM2], [FMZ], [Fen1], [DM], [MP], [Fen2]. We
really hesitated to include this case in the proposition, but we decided that
it was not worth complicating the argument further. A quick discussion about
the differences is given in the proof of Theorem 3.16 below.

  6. (vi)

For elliptic operators with complex coefficients (see [DP], [FMZ]), our method
allows us to remove the real part of the last line of $\mathcal{B}$, but not
the imaginary part. Yet using (1.3) yields the same reduction and the same
limitation.

  7. (vii)

Unfortunately, contrary to the transformation (1.3), our method does not seem
easily adaptable to systems of elliptic equations.

  8. (viii)

On the other hand, Proposition 1.7 is well adapted to the regularity problem
and the Neumann problem, where a theory on Carleson perturbations exists (see
[KP2]). We give an application of Proposition 1.7 in Corollary 3.11 below.

  9. (ix)

The method is dependent on the invariance of the domain by tangential
translations and by dilatations. However, there are many ways to reduce the
domain to $\mathbb{R}^{n}_{+}$. For instance, in a paper under preparation
[FLM], we will use a variant of the method to prove a result on uniformly
rectifiable domains.

  10. (x)

To the best of the author’s knowledge, the only time where a change of
variable has been used to simplify the coefficients of an elliptic operator is
in [HKMP]. But the context of [HKMP] ($t$-independent operators) is different
from our context.

In the rest of the article, we prove Proposition 1.7, and we give two examples
where the method extends the results from the existing literature (Corollary
3.11 and Theorem 3.16).

The notation $A\lesssim B$ is employed when $A\leq CB$ for a constant that
depends only on parameters that will be either recalled or obvious from
context. We shall also use $A\approx B$ when $A\lesssim B$ and $A\gtrsim B$.

##  2\. Proof of Proposition 1.7

Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be like in the assumptions
of the proposition, that is $\mathcal{A}$ is uniformly elliptic with a
constant $C_{A}$, $\mathcal{A}:=\mathcal{B}+\mathcal{C}$, and
$|t\nabla\mathcal{B}|+|\mathcal{C}|\in CM_{sup}(M)$.

Step 1: Construction of $\rho_{v,h}$. For a positive scalar function $h$ and a
vector function $v$, we define

(2.1) $\rho(y,t)=\rho_{v,h}(y,t):=(y+tv(y,t),h(y,t)t),$

which fixes the boundary $\mathbb{R}^{n-1}$ and which is a Lipschitz map
whenever $v$, $h$, $|t\nabla v|$, and $|t\nabla h|$ are uniformly bounded. Its
Jacobian is

(2.2) $Jac_{\rho}=\begin{pmatrix}I+t\nabla_{x}v&t\nabla_{x}h\\\
v+t\partial_{t}v&h+t\partial_{t}h\end{pmatrix}.$

Let $J_{\rho}$ be the matrix function

(2.3) $J_{\rho}:=\begin{pmatrix}I&0\\\ v&h\end{pmatrix}.$

Since $|Jac_{\rho}-J_{\rho}|\lesssim|t\nabla h|+|t\nabla v|$, we can find
$\epsilon_{0}>0$ small enough and $C_{0}$ large enough (both depending on $d$,
$n$ and $\|h^{-1}\|_{\infty}$) so that, if

(2.4) $\|t\nabla h\|_{\infty}+\|t\nabla v\|_{\infty}<\epsilon_{0},$

then for any $(y,t)\in\Omega_{0}$, $Jac_{\rho}(y,t)$ is invertible,

(2.5) $|\det(Jac_{\rho}(y,t))-h(y,t)|\leq(2\|h^{-1}\|_{\infty})^{-1},$

and

(2.6) $|(Jac_{\rho})^{-1}(y,t)-(J_{\rho})^{-1}(y,t)|\leq C_{0}(|t\nabla
h(y,t)|+|t\nabla v(y,t)|).$

If $h$ and $v$ satisfy (2.4), which we now assume, then $\rho$ is a bi-
Lipschitz change of variable. So we can look for the conjugate of the operator
$L=-\operatorname{div}\mathcal{A}\nabla$ by $\rho$. We let the reader check
that if $u$ is a weak solution to $L=-\operatorname{div}\mathcal{A}\nabla$,
then $u\circ\rho$ is solution to
$L_{\rho}=-\operatorname{div}\mathcal{A}_{\rho}\nabla$ where

(2.7)
$\mathcal{A}_{\rho}=\det(Jac_{\rho})(Jac_{\rho})^{-T}(\mathcal{A}\circ\rho)\,(Jac_{\rho})^{-1}.$

To lighten the notation, we use $\mathcal{O}(f)$ to denote any
scalar/vector/matrix function on $\Omega$ which can be bounded by $Cf$ (where
$C$ depends only on $d$, $n$, $\|h^{-1}\|_{\infty}$ and later $C_{A}$). In
particular, the quantity $\mathcal{O}(f)$ can change from one line to another.
We use this notation to rewrite (2.6) as

$(Jac_{\rho})^{-1}=(J_{\rho})^{-1}+\mathcal{O}(|t\nabla h|+|t\nabla
v|)\quad\text{ and
}\quad(Jac_{\rho})^{-T}=(J_{\rho})^{-T}+\mathcal{O}(|t\nabla h|+|t\nabla v|).$

Of course, we also have

$\det(Jac_{\rho})=h+\mathcal{O}(|t\nabla h|+|t\nabla v|).$

Remember that the matrix $\mathcal{A}$ equals
$\mathcal{A}=\mathcal{B}+\mathcal{C}=\mathcal{B}+\mathcal{O}(|\mathcal{C}|)$,
so if we write $\mathcal{B}$ as the matrix by blocs

(2.8) $\mathcal{B}=\begin{pmatrix}B_{1}&B_{2}\\\ B_{3}&b\end{pmatrix},$

then

(2.9) $\begin{split}\mathcal{A}_{\rho}&=h\begin{pmatrix}I&-h^{-1}v^{T}\\\
0&h^{-1}\end{pmatrix}\begin{pmatrix}B_{1}\circ\rho&B_{2}\circ\rho\\\
B_{3}\circ\rho&b\circ\rho\end{pmatrix}\begin{pmatrix}I&0\\\
-h^{-1}v&h^{-1}\end{pmatrix}+\mathcal{O}(|\mathcal{C}\circ\rho|+|t\nabla
h|+|t\nabla v|)\\\
&=\begin{pmatrix}*&B_{2}\circ\rho-(b\circ\rho)h^{-1}v^{T}\\\
B_{3}\circ\rho-(b\circ\rho)h^{-1}v&(b\circ\rho)h^{-1}\end{pmatrix}+\mathcal{O}(|\mathcal{C}\circ\rho|+|t\nabla
h|+|t\nabla v|)\\\ &=\underbrace{\begin{pmatrix}*&B_{2}-bh^{-1}v^{T}\\\
B_{3}-bh^{-1}v&bh^{-1}\end{pmatrix}}_{\mathcal{B}_{\rho}}+\mathcal{O}(|\mathcal{B}\circ\rho-\mathcal{B}|+|\mathcal{C}\circ\rho|+|t\nabla
h|+|t\nabla v|),\end{split}$

where the top left corner of $\mathcal{B}$ is not given to lighten the
computations, but depends only on products, quotients, differences, and sums
of coefficients of $\mathcal{B}$, $h$, and $v$. We write $\mathcal{C}_{\rho}$
for $\mathcal{A}_{\rho}-\mathcal{B}_{\rho}$, and we have that

(2.10)
$|\mathcal{C}_{\rho}|\lesssim|\mathcal{B}\circ\rho-\mathcal{B}|+|\mathcal{C}\circ\rho|+|t\nabla
h|+|t\nabla v|.$

Step 2: In the change of variable $\rho_{v,h}$ constructed in Step 1, we
really want to take $h:=b$ and $v:=B_{3}$. However, with those choices,
nothing guarantees that (2.4) is satisfied … and we may not have a change of
variable. Still, we can see that it works when $t\nabla b$ and $t\nabla B_{3}$
are small enough. So the strategy will be to do multiple changes of variables,
each of them canceling only a small portion of $b$ and $B_{3}$, but altogether
reducing $b$ to 1 and $B_{3}$ to 0.

For this, we want to do an induction on the size of $t\nabla b$ and $t\nabla
B_{3}$. The issue is that we do not have a control on those terms, hence no
control on the number of change of variable that we need (so ultimately no
control on the Carleson constant $M^{\prime}$ in the conclusion of the
Proposition). That is a bit annoying, so we need the following observation:
without loss of generality, we can assume that

(2.11) $\mathcal{B}$ satisfies (1.6) with the constant $C_{A}$

and

(2.12) $\|t\nabla\mathcal{B}\|_{\infty}\leq CC_{A},$

where $C>0$ depends only on the dimension (via our choice for $\theta$ below).
To prove this, we shall construct $\widetilde{\mathcal{B}}$ which satisfies
(2.11), (2.12), and $|t\nabla\widetilde{\mathcal{B}}|\in CM_{sup}(CM)$ such
that $\widetilde{\mathcal{C}}:=\mathcal{A}-\widetilde{\mathcal{B}}\in
CM_{sup}(CM)$.

We choose once for all the proof a bump function $\theta\in
C^{\infty}_{0}(\mathbb{R}^{n})$ supported in $B(0,1/2)$, that is
$0\leq\theta\leq 1$ and $\iint_{\mathbb{R}^{n}}\theta\,dX=1$. We construct
$\theta_{y,t}(z,s)=t^{-n}\theta\big{(}\frac{z-y}{t},\frac{s-t}{t}\big{)}$,
which satisfies $\iint_{\mathbb{R}^{n}}\theta_{y,t}=1$. Note, if
$\Theta(X):=X\theta(X)$, that

(2.13)
$t\nabla_{y,t}\theta_{y,t}(z,s)=-t^{-n}\nabla\theta\Big{(}\frac{z-y}{t},\frac{s-t}{t}\Big{)}-t^{-n}\mathop{\operatorname{div}}(\Theta)\Big{(}\frac{z-y}{t},\frac{s-t}{t}\Big{)},$

in particular $\|t\nabla_{y,t}\theta_{y,t}\|_{L^{1}}\leq C_{\theta}$. We
define

(2.14)
$\widetilde{\mathcal{B}}(y,t):=\iint_{\mathbb{R}^{n}}\mathcal{A}(z,s)\,\theta_{y,t}(z,s)\,dz\,ds.$

Since $\widetilde{\mathcal{B}}$ is an average of $\mathcal{A}$, then having
(2.11) for $\widetilde{\mathcal{B}}$ is immediate. Observe that

(2.15)
$|t\nabla\widetilde{\mathcal{B}}(y,t)|\leq\||\mathcal{A}|\|_{\infty}\iint_{\mathbb{R}^{n}}|t\nabla_{y,t}\theta_{y,t}(z,s)|\,dz\,ds\lesssim
C_{A}.$

Moreover, using that $\mathcal{A}=\mathcal{B}+\mathcal{C}$ in (2.14), we have
by (2.13) that

$\begin{split}|t\nabla\widetilde{\mathcal{B}}(y,t)|&\leq
t^{-n}\left|\iint_{\mathbb{R}^{n}}[\mathcal{B}(z,s)+\mathcal{C}(z,s)]\left[\nabla\theta\Big{(}\frac{z-y}{t},\frac{s-t}{t}\Big{)}+\mathop{\operatorname{div}}(\Theta)\Big{(}\frac{z-y}{t},\frac{s-t}{t}\Big{)}\right]\,dz\,ds\right|\\\
&\leq
t^{-n}\iint_{\mathbb{R}^{n}}|t\nabla\mathcal{B}(z,s)|\left|\theta\Big{(}\frac{z-y}{t},\frac{s-t}{t}\Big{)}\right|\,dz\,ds\\\
&\qquad+t^{-n}\iint_{\mathbb{R}^{n}}|\mathcal{C}(z,s)|\left|\nabla\theta\Big{(}\frac{z-y}{t},\frac{s-t}{t}\Big{)}+\mathop{\operatorname{div}}(\Theta)\Big{(}\frac{z-y}{t},\frac{s-t}{t}\Big{)}\right|\,dz\,ds\\\
&\lesssim\sup_{\begin{subarray}{c}z\in B(y,t)\\\
t/2<s<2t\end{subarray}}|t\nabla\mathcal{B}(z,s)|+|\mathcal{C}(z,s)|\end{split}$

where, for the second inequality, we used an integration by part to move the
divergence and the gradient from $\theta$ and $\Theta$ to $\mathcal{B}$. We
conclude that $t\nabla\widetilde{\mathcal{B}}\in CM_{sup}$ as desired. It
remains to verify that $\widetilde{\mathcal{C}}\in CM_{sup}$. Indeed,

$\begin{split}|\widetilde{\mathcal{C}}(y,s)|&=|\mathcal{A}(y,t)-\widetilde{\mathcal{B}}(y,t)|=\iint_{\mathbb{R}^{n}}\Big{[}|\mathcal{B}(y,t)-\mathcal{B}(z,s)|+|\mathcal{C}(y,t)|+|\mathcal{C}(z,t)|\Big{]}\,\theta_{y,t}(z,s)\,dz\,ds\\\
&\lesssim\sup_{\begin{subarray}{c}z\in B(y,t)\\\
t/2<s<2t\end{subarray}}|\mathcal{B}(y,t)-\mathcal{B}(z,s)|+|\mathcal{C}(z,s)|\lesssim\sup_{\begin{subarray}{c}z\in
B(y,t)\\\
t/2<s<2t\end{subarray}}|t\nabla\mathcal{B}(z,s)|+|\mathcal{C}(z,s)|\in
CM_{sup}.\end{split}$

Step 2 follows.

Step 3: We write $\mathcal{B}$ as the matrix by blocs

(2.16) $\mathcal{B}=\begin{pmatrix}B_{1}&B_{2}\\\ B_{3}&b\end{pmatrix},$

where $b$ is a scalar function, so $B_{1}$ is a matrix of order $n-1$, $B_{2}$
and $B_{3}$ are respectively a vertical and a horizontal vector of length
$n-1$. According to Step 2, we can assume that $\mathcal{B}$ satisfies (2.11)
and (2.12), so

(2.17)
$\|b\|_{\infty}+\|b^{-1}\|_{\infty}+\|\mathcal{B}_{3}\|_{\infty}+\|t\nabla
b\|_{\infty}+\|t\nabla B_{3}\|_{\infty}\lesssim 1,$

where the constants depends only on $n$ and $C_{A}$. We want to use the change
of variable (2.1) with $h=b^{1/N}$ and $v_{k}=\frac{1}{N}b^{-k/N}B_{3}$
($0\leq k\leq N$) for a large enough $N$. So we need to verify (2.4). We have

$\begin{split}\|t\nabla
h\|_{\infty}\lesssim\frac{1}{N}\|b^{-1}\|_{\infty}\|t\nabla
b\|_{\infty}\lesssim\frac{1}{N}\end{split}$

by (2.17). Moreover, similar computations yield, for $0\leq k\leq N$,

$\begin{split}\|t\nabla
v_{k}\|_{\infty}&\lesssim\frac{1}{N}(\|b^{-2}\|_{\infty}\|B_{3}\|_{\infty}\|t\nabla
b\|_{\infty}+\|b^{-1}\|_{\infty}\|t\nabla
B_{3}\|_{\infty})\lesssim\frac{1}{N}.\end{split}$

Altogether, we have for any $0\leq k\leq N$

(2.18) $\|t\nabla h\|_{\infty}+\|t\nabla
v_{k}\|_{\infty}\leq\frac{C(C_{A},n)}{N}<\epsilon_{0}$

for some large enough $N$ that depends only on $C_{A}$ and $n$, that we fix
for the rest of the proof. By possibly taking $N$ slightly bigger (still
depending only on $C_{A}$ and $n$), we can even assume that $N$ is such that

(2.19) $\frac{1}{2}<h<2\quad\text{ and }\quad\sup_{0\leq k\leq N}|v_{k}|\leq
1$

to simplify the incoming computations.

By (2.18), the map $\rho_{N-1}$ constructed in (2.1) with of $h$ and $v_{N-1}$
is a bi-Lipschitz change of variable. Therefore, thanks to (2.9) and (2.10),
the conjugate operator $L_{N-1}=-\operatorname{div}\mathcal{A}_{N-1}\nabla$ of
$L$ by $\rho_{N-1}$ is such that $\mathcal{A}_{N-1}$ can be decomposed as

$\mathcal{A}_{N-1}:=\underbrace{\begin{pmatrix}*&*\\\
\frac{N-1}{N}B_{3}&b^{(N-1)/N}\end{pmatrix}}_{\mathcal{B}_{N-1}}+\
\mathcal{C}_{N-1}$

where $|t\nabla\mathcal{B}_{N-1}|\in CM_{sup}$ (because $\mathcal{B}_{N-1}$ is
the product, quotient, difference or sum of coefficients of $\mathcal{B}$) and
$\mathcal{C}_{N-1}\in CM_{sup}$ as well because

$\begin{split}|\mathcal{C}_{N-1}|\lesssim|\mathcal{B}\circ\rho-\mathcal{B}|+|\mathcal{C}\circ\rho|+|t\nabla
h|+|t\nabla v|\end{split}$

by (2.10),

$|t\nabla h|+|t\nabla v|\lesssim|t\nabla\mathcal{B}|\in CM_{sup},$

and since $\rho(y,t)\in B(y,t)\times(t/2,2t)$ by (2.19),

$|\mathcal{C}\circ\rho(y,t)|+|\mathcal{B}\circ\rho(y,t)-\mathcal{B}(y,t)|\lesssim\sup_{\begin{subarray}{c}|z-y|<t\\\
t/2<s<2t\end{subarray}}\Big{[}|\mathcal{C}(z,s)|+|t\nabla\mathcal{B}(z,s)|\Big{]}\in
CM_{sup}.$

We iterate the process, and so the conjugate operator $L_{k-1}$ of $L_{k}$ by
$\rho_{k-1}$ (constructed with $h$ and $v_{k-1}$) is
$-\operatorname{div}\mathcal{A}_{k-1}\nabla$ where $\mathcal{A}_{k-1}$ can be
decomposed as $\mathcal{A}_{k-1}=\mathcal{B}_{k-1}+\mathcal{C}_{k-1}$ where

$\mathcal{B}_{k-1}=\begin{pmatrix}*&*\\\
\frac{k-1}{N}B_{3}&b^{(k-1)/N}\end{pmatrix},$

and $|\mathcal{C}_{k-1}|+|t\nabla\mathcal{B}_{k-1}|\in CM_{sup}$. The
proposition follows when we reach $k=0$.

##  3\. Applications

###  3.1. Regularity problem

The first applications that we can think of are related to the Dirichlet and
regularity problems, that we shall now introduce properly.

Given a uniformly elliptic operator $L$ on $\mathbb{R}^{n}_{+}$, we can
construct an associated elliptic measure
$\\{\omega^{X}_{L}\\}_{X\in\mathbb{R}^{n}_{+}}$ on the boundary
$\mathbb{R}^{n-1}$. That is, for any $g\in C_{c}(\mathbb{R}^{n-1})$ \- the set
of continuous and compactly supported functions on $\mathbb{R}^{n-1}$ \- the
function $u_{g}$ defined for all $X\in\mathbb{R}^{n}_{+}$ as

(3.1) $u_{g}(X):=\int_{\mathbb{R}^{n-1}}g(y)\,d\omega^{X}_{L}(y)\quad\text{
for }X\in\mathbb{R}^{n}_{+}$

is the (unique) solution to $Lu=0$ which is bounded, continuous up to the
boundary, and satisfies $u=g$ on the boundary $\mathbb{R}^{n-1}$. We can use
the construction (3.1) when $g\in L^{\infty}(\mathbb{R}^{n-1})$ or even
$BMO(\mathbb{R}^{n-1})$. In those cases $u_{g}$ will still be a solution to
$Lu=0$ but the fact that $u=g$ on $\mathbb{R}^{n-1}$ has to be taken in a
weaker sense (for instance non-tangential limit almost everywhere). But we
cannot guarantee that (3.1) makes sense if $g$ lies only in $L^{p}$ for
$p<\infty$. This issue is called the Dirichlet problem.

######  Definition 3.2 (Dirichlet problem).

We say that the Dirichlet problem for $L$ is solvable in $L^{p}$ if there
exists a constant $C>0$ such that for any $g\in C_{c}(\mathbb{R}^{n-1})$, the
solution $u_{g}$ constructed in (3.1) satisfies

(3.3) $\|N(u_{g})\|_{p}\leq C\|g\|_{p},$

where $N$ is the non-tangentially maximal function defined as

(3.4) $N(u)(x):=\sup_{(y,t)\in\Gamma(x)}u$

and $\Gamma(x):=\\{(y,t)\in\mathbb{R}^{n}_{+},\,|x-y|<t\\}$ is the vertical
cone with vertex in $x$ and aperture $1$.

When the Dirichlet problem is solvable in the sense given in the above
definition, then we can use the density of $C_{c}(\mathbb{R}^{n-1})$ in
$L^{p}$ and (3.3) to say that a solution can be constructed using (3.1) for
any $g\in L^{p}(\mathbb{R}^{n-1})$, hence proving the existence of solutions
for any data in $L^{p}$.

The regularity problem is similar to the Dirichlet problem, but on the
gradients.

######  Definition 3.5 (Regularity problem).

We say that the regularity problem for $L$ is solvable in $L^{q}$ if there
exists a constant $C>0$ such that for any $g\in C_{c}(\mathbb{R}^{n-1})$, the
solution $u_{g}$ constructed in (3.1) satisfies

(3.6) $\|\widetilde{N}(\nabla u_{g})\|_{q}\leq C\|\nabla g\|_{q},$

where $\widetilde{N}$ is the averaged non-tangentially maximal function
defined as

(3.7) $\widetilde{N}(\nabla u)(x):=\sup_{(y,t)\in\Gamma(x)}\left(\fint_{z\in
B(y,t)}\fint_{t/2<s<2t}|\nabla u|^{2}\,ds\,dz\right)^{\frac{1}{2}}.$

Such averaged version of $N$ is needed here because, contrary to $u_{g}$, the
quantity $\nabla u_{g}$ is not necessarily locally bounded.

A first result on the regularity problem, which is key to our argument, is:

######  Proposition 3.8.

The regularity problem in $L^{q}$ is stable under a bi-Lipschitz change of
variable $\rho:\,\mathbb{R}^{n}_{+}\to\mathbb{R}^{n}_{+}$ that fixes the
boundary $\partial\mathbb{R}^{n}_{+}=\mathbb{R}^{n-1}$.

In addition, the ratio between the constants in (3.6) before and after the
change of variable depends only on the dimension $n$ and the bi-Lipschitz
constants of $\rho$.

###### Proof.

Let $L$ be a uniformly elliptic operator and $L_{\rho}$ be its conjugate by
$\rho$. By uniqueness of the harmonic measure, we have
$\omega_{L_{\rho}}^{X}=\omega_{L}^{\rho(X)}$, thus the solutions $u_{g}$ and
$v_{g}$ to respectively $Lu=0$ and $L_{\rho}v=0$ constructed by (3.1) verify
$v_{g}=u_{g}\circ\rho$.

From there, we only need to show that $\|\widetilde{N}(\nabla
u)\|_{q}\approx\|\widetilde{N}(\nabla[u\circ\rho])\|_{q}$, and even

(3.9) $\|\widetilde{N}(\nabla[u\circ\rho])\|_{q}\lesssim\|\widetilde{N}(\nabla
u)\|_{q}.$

because the proof of other bound is identical by using $\rho^{-1}$ instead of
$\rho$.

Let $x\in\mathbb{R}^{n-1}=\partial\mathbb{R}^{n}_{+}$, and take
$(y,t)\in\Gamma(x)$. The map $\rho$ is bi-Lipschitz, so the Jacobian matrix
$\operatorname{Jac}_{\rho}$ and $1/\det(\operatorname{Jac}_{\rho})$ are
uniformly bounded. Therefore,

$\fint_{z\in
B(y,t)}\fint_{t/2<s<2t}|\nabla[u\circ\rho]|^{2}\,ds\,dz=\fint_{z\in
B(y,t)}\fint_{t/2<s<2t}|\operatorname{Jac}_{\rho}(\nabla
u)\circ\rho|^{2}\,ds\,dz\\\ \lesssim\fint_{z\in
B(y,t)}\fint_{t/2<s<2t}|(\nabla u)\circ\rho|^{2}\,ds\,dz\lesssim
t^{-n}\iint_{\rho(B(y,t)\times(t/2,2t))}|\nabla
u|^{2}\,ds^{\prime}\,dz^{\prime}\\\
\lesssim\sup_{(y^{\prime},t^{\prime})\in\rho(B(y,t)\times(t/2,2t))}\fint_{z\in
B(y^{\prime},t^{\prime})}\fint_{t^{\prime}/2<s<2t^{\prime}}|\nabla
u|^{2}\,ds\,dz,$

where the last line holds because we can cover the image
$\rho(B(y,t)\times(t/2,2t))$ by at most $N$ sets of the form
$B(y^{\prime},t^{\prime})\times(t^{\prime}/2,2t^{\prime})$, with $N$ depending
only on the bi-Lipschitz constants of $\rho$. Since $\rho(x)=x$ (and again
$\rho$ is bi-Lipschitz), we also know that the image
$\rho(B(y,t)\times(t/2,2t))$ is included in the cone
$\Gamma_{*}(x):=\\{(y^{\prime},t^{\prime})\in\mathbb{R}^{n}_{+},\,|y^{\prime}-x|<K_{*}t^{\prime}\\}$
with a large enough aperture $K_{*}$ that depends only on the bi-Lipschitz
constants of $\rho$. We deduce that

$\widetilde{N}(\nabla[u\circ\rho])(x)\lesssim\widetilde{N}_{*}(\nabla u)(x),$

where $\widetilde{N}_{*}(\nabla u)(x)$ is defined with the cone
$\Gamma_{*}(x)$ instead of $\Gamma(x)$, and hence

$\|\widetilde{N}(\nabla[u\circ\rho])\|_{q}\lesssim\|\widetilde{N}_{*}(\nabla
u)\|_{q}.$

The claim (3.9) is now a consequence of $\|\widetilde{N}_{*}(\nabla
u)\|_{q}\approx\|\widetilde{N}(\nabla u)\|_{q}$, which is a classical result
in real analysis, see for instance Chapter II, equation (25) of [Ste]. ∎

The regularity problem is solvable for the Laplacian on $\mathbb{R}^{n}_{+}$
and Carleson perturbations preserve the solvability of the regularity problem
in $L^{q}$ for small $q>1$, as shown in [KP2, Theorem 2.1], so we have

######  Theorem 3.10.

If $\mathcal{C}$ is a matrix function on $\mathbb{R}^{n}_{+}$ such that
$\mathcal{C}\in CM_{sup}$ and the operator
$L:=-\operatorname{div}(I+\mathcal{C})\nabla$ is uniformly elliptic - see
(1.6) - then there exists a $q>1$ such that the regularity problem for $L$ is
solvable in $L^{q}$.

We shall prove the following corollary.

######  Corollary 3.11.

Let $b$ be a scalar function on $\mathbb{R}^{n}_{+}$ which satisfies
$C_{b}\leq b\leq C_{b}$ and $|t\nabla b|\in CM_{sup}(M)$ for some positive
constants $C_{b}$ and $M$. We construct the operator
$L:=-\mathop{\operatorname{div}}\mathcal{B}\nabla$, where

$\mathcal{B}:=\begin{pmatrix}b^{-1}I&0\\\ 0&b\end{pmatrix}.$

Then there exists $q>1$ that depends only on $C_{b}$ and $M$ such that the
regularity problem for $L$ is solvable in $L^{q}$.

The corollary is not optimized at all, but the result is still new when
$|t\nabla b|\in CM_{sup}(M)$ for a large Carleson constant $M$. So our method
enlarge the class of operators for which the solvability of the regularity
problem in $L^{q}$ is known.

Proof of the corollary:  We cannot use Proposition 1.7 directly here, but we
can adapt the proof to fit our situation. Let us sketch it.

We shall use (2.1) with $v=0$, so we have

(3.12) $\rho(y,t):=(y,h(y,t)t),\quad
Jac_{\rho}=\begin{pmatrix}I&t\nabla_{x}h\\\ 0&h+t\partial_{t}h\end{pmatrix}.$

If $2|t\nabla h|\leq h$, the map $\rho$ is a change of variable. Moreover, if
$\mathcal{A}$ has the form

$\mathcal{A}:=\begin{pmatrix}a^{-1}I&0\\\ 0&a\end{pmatrix}$

then
$\mathcal{A}_{\rho}:=\det(Jac_{\rho})(Jac_{\rho})^{-T}(\mathcal{A}\circ\rho)\,(Jac_{\rho})^{-1}$
can be decomposed as

(3.13) $\mathcal{A}_{\rho}=\underbrace{\begin{pmatrix}ha^{-1}&0\\\
0&h^{-1}a\end{pmatrix}}_{\mathcal{B}_{\rho}}+\ \mathcal{O}(|t\nabla
h|+|a\circ\rho-a|).$

We write $\mathcal{C}_{\rho}$ for $\mathcal{A}_{\rho}-\mathcal{B}_{\rho}$.

We know that $|t\nabla b|\in CM_{sup}(M)$. Therefore $\|t\nabla
b\|_{\infty}\lesssim M$ and there exists $N$ (depending only on $M$ and
$C_{b}$) such that $\|t\nabla(b^{1/N})\|_{\infty}\leq C_{b}/2$. We use $\rho$
with $h=b^{1/N}$, which is a change of variable, so the conjugate of the
operator $L$ by $\rho$ is the operator
$L_{1}=-\operatorname{div}\mathcal{A}_{1}\nabla$ where (3.13),

$\mathcal{A}_{1}=\underbrace{\begin{pmatrix}b^{-(N-1)/N}&0\\\
0&b^{(N-1)/N}\end{pmatrix}}_{\mathcal{B}_{1}}+\ \mathcal{C}_{1},$

with

$|\mathcal{C}_{1}|\lesssim|t\nabla b|+|b\circ\rho-b|\lesssim|t\nabla b|\in
CM_{sup}.$

By iteration, we find that the conjugate of $L$ by $\rho^{k}$ is
$L_{k}=-\operatorname{div}\mathcal{A}_{k}\nabla$ where $\mathcal{A}_{k}$ can
be decomposed as

$\mathcal{A}_{k}=\underbrace{\begin{pmatrix}b^{-(N-k)/N}&0\\\
0&b^{(N-k)/N}\end{pmatrix}}_{\mathcal{B}_{k}}+\ \mathcal{C}_{k},$

where $|\mathcal{C}_{k}|\in CM_{sup}$. Taking $k=N$ yields that
$|\mathcal{A}_{N}-I|\in CM_{sup}$ so Theorem 3.10 entails that there exists
$q>1$ such that the regularity problem for $L_{N}$ is solvable in $L^{q}$.
Proposition 3.8 concludes the proof. $\square$

###  3.2. Dirichlet problem in higher codimension.

As we said in remark after Proposition 1.7, our method can be adapted to
degenerate elliptic operators on
$\mathbb{R}^{n}\setminus\mathbb{R}^{d}:=\\{(x,t)\in\mathbb{R}^{d}\times\mathbb{R}^{n-d},\,t\neq
0\\}$, $d<n-1$. In this domain, the ‘degenerate’ uniformly elliptic operators
are the ones that can be written as
$L=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}\nabla]$, where $\mathcal{A}$
satisfies the classical elliptic condition (1.6). A rich elliptic theory was
developed in [DFM1] by Guy David, Svitlana Mayboroda, and the author, and
provides for the aforementioned operators a notion of elliptic measure similar
to (3.1), and then a notion of Dirichlet and regularity problem. The Carleson
measure condition (1.5) has also to be adapted, but the adaptation is fairly
straightforward.

Without entering into much details, we know from [FMZ] and [Fen2] that the
Dirichlet problem is solvable in $L^{p}$ for sufficiently large $p$ when
$\mathcal{A}$ can be written as $\mathcal{A}=\mathcal{B}+\mathcal{C}$ where
$\mathcal{B}$ can be written as a bloc matrix in the form

(3.14) $\mathcal{B}:=\begin{pmatrix}B_{1}&B_{2}\\\ B_{3}&bI\end{pmatrix}$

with $B_{1}$ being a $d\times d$ matrix, $(B_{2})^{T}$ and $B_{3}$ being
$(n-d)\times d$ matrices, $b$ being a scalar function, $I$ being the identity
matrix on $\mathbb{R}^{n-d}$, and

(3.15) $|\mathcal{C}|+|t||\nabla
b|+|t\nabla_{x}B_{3}|+|t|^{n-d}|\operatorname{div}_{t}[|t|^{d+1-n}B_{3}]|\in
CM_{sup}.$

We recall that $\operatorname{div}_{t}A$ is a vector containing the divergence
of each column of $A$. One can think of operators in the form
$L=-\operatorname{div}[|t|^{d+1-n}\mathcal{B}\nabla]$ with $\mathcal{B}$ as
above to be the higher codimension analogue of the Dahlberg-Kenig-Pipher
operators. Indeed, the proof of the solvability of the Dirichlet problem for
those operators rely on methods similar to the ones used by Kenig-Pipher, in
particular the use of a transformation similar to (1.3).

By using our method based on a change of variable, the condition
$|t|^{n-d}|\operatorname{div}_{t}[|t|^{d+1-n}B_{3}]|$ would be replaced by the
much nicer $|t||\nabla_{t}B_{3}|$, leading to the following alternative:

######  Theorem 3.16.

Let $L=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}\nabla]$ be an operator for
which $\mathcal{A}$ satisfies (1.6). Assume that $\mathcal{A}$ can be
decomposed as $\mathcal{A}=\mathcal{B}+\mathcal{C}$ where

$\mathcal{B}=\begin{pmatrix}B_{1}&B_{2}\\\ B_{3}&bI\end{pmatrix}$

and

(3.17) $|\mathcal{C}|+|t||\nabla b|+|t\nabla B_{3}|\in CM_{sup}.$

Then there exists a large $p<+\infty$ such that the Dirichlet problem for $L$
is solvable in $L^{p}$.

######  Remark 3.18.

Both (3.15) and (3.17) are conditions that can appear naturally. The condition
(3.15) is the one that we would get by operators constructed from DKP
operators as in [DFM3, (4.6)], while the condition (3.17) can appear when we
flatten the graph of Lipschitz function in higher codimension (see Remark 1.25
in [Fen2]).

Proof: The proof of the above result is not much different from the proof of
Proposition 1.7, but let us highlight the small changes. We use the maps
$\rho$ defined as

$\rho(y,t):=(y+tv(y,t),h(y,t)t)$

where $v(y,t)$ is a $(n-d)\times d$ matrix and $h$ is a scalar function (we
see values in $\mathbb{R}^{d}$ and $\mathbb{R}^{n-d}$ as horizontal vectors).
As before, the map $\rho$ is a change of variable if $|t||\nabla v|+|t||\nabla
h|$ is small enough. The conjugate operator $L_{\rho}$ is such that
$L_{\rho}:=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}_{\rho}\nabla]$ where

(3.19)
$\mathcal{A}_{\rho}=h^{d+1-n}\det(Jac_{\rho})(Jac_{\rho})^{-T}(\mathcal{A}\circ\rho)\,(Jac_{\rho})^{-1},$

the extra term $h$ being due to the weight $|t|^{d+1-n}$. But it is fine
because it will cancel out with $\det(Jac_{\rho})\approx h^{n-d}$ to give,
similarly to the codimension 1 case, that

(3.20) $\begin{split}\mathcal{A}_{\rho}=\underbrace{\begin{pmatrix}*&*\\\
B_{3}-bh^{-1}v&bh^{-1}I\end{pmatrix}}_{\mathcal{B}_{\rho}}+\
\mathcal{O}(|\mathcal{B}\circ\rho-\mathcal{B}|+|\mathcal{C}\circ\rho|+|t||\nabla
h|+|t||\nabla v|).\end{split}$

The rest of the proof does not change from Proposition 1.7, which gives the
existence of a bi-Lipschitz change of variable $\rho^{\prime}$ that fixes the
boundary $\mathbb{R}^{d}$ and for which the matrix
$\mathcal{B}_{\rho^{\prime}}$ after the change of variable is in the form

$\mathcal{B}_{\rho^{\prime}}=\begin{pmatrix}*&*\\\ 0&I\end{pmatrix},$

and thus can be treated by the previous literature. Going back from the
solvability of the Dirichlet problem for $L_{\rho^{\prime}}$ to the one of $L$
is then similar to Proposition 3.8. $\square$

######  Remark 3.21.

One may think that we can use a matrix $H$ of order $n-1$ \- instead of the
scalar $h$ \- in the change of variable $\rho$ to try to reduce the bottom
right corner of $\mathcal{B}$ to identity. But doing so would change the
expression (3.19) to

$\mathcal{A}_{\rho}=\left(\frac{|Ht|}{|t|}\right)^{d+1-n}\det(Jac_{\rho})(Jac_{\rho})^{-T}(\mathcal{A}\circ\rho)\,(Jac_{\rho})^{-1}$

and $\left(\frac{|Ht|}{|t|}\right)^{d+1-n}$ is killed by $\det(Jac_{\rho})$
only when $H$ has the form $H=hI$ \- i.e. $H$ is the identity matrix times a
scalar function111and in few other cases not worth mentioning because they are
way more complicated.. Therefore, the bottom right corner of the Jacobian of
the change of variable $\rho$ will always be close - up to Carleson measures -
to a scalar function times identity, and can only cancel out terms in the same
form. That is why, in Theorem 3.16, we only consider a bottom right corner of
$\mathcal{B}$ which is a scalar function times identity.

## References

  * [Azz] J. Azzam. Semi-uniform domains and the A∞ property for harmonic measure. Int. Math. Res. Not. 2021 (2021), no. 9, 6717–6771.
  * [AHM3TV] J. Azzam, S. Hoffman, M. Mourgoglou, J. M. Martell, S. Mayboroda, X. Tolsa, A. Volberg. Rectifiability of harmonic measure, Geom. Funct. Anal. 26 (2016), no. 3, 703–728.
  * [AHMMT] J. Azzam, S. Hofmann, J. M. Martell, M. Mourgoglou, X. Tolsa. Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$-solvability of the Dirichlet problem. Invent. Math. 222 (2020), no. 3, 881–993.
  * [AHMNT] J. Azzam, S. Hofmann, J.M. Martell, K. Nyström, T. Toro. A new characterization of chord-arc domains, J. Eur. Math. Soc. 19 (2017), no. 4, 967–981.
  * [BJ] C. Bishop, P. Jones. Harmonic measure and arclength, Ann. of Math. (2), 132 (1990), 511–547.
  * [CFK] L. Caffarelli, E. Fabes, C. Kenig. Completely singular elliptic-harmonic measures. Indiana Univ. Math. J., 30 (1981), no. 6, 917–924.
  * [Da] B. E. J. Dahlberg, _Estimates of harmonic measure_. Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288.
  * [DJ] G. David, D. Jerison. Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals. Indiana Univ. Math. J., 39 (1990), no. 3, 831–845.
  * [DM] G. David, S. Mayboroda. Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets. Preprint, arXiv:2006.14661.
  * [DFM1] G. David, J. Feneuil, S. Mayboroda. Elliptic theory for sets with higher co-dimensional boundaries. Mem. Am. Math. Soc., accepted (arXiv:1702.05503).
  * [DFM2] G. David, J. Feneuil, S. Mayboroda. Dahlberg’s theorem in higher co-dimension. J. Funct. Anal. 276 (2019), no. 9 2731–2820.
  * [DFM3] G. David, J. Feneuil, S. Mayboroda. A new elliptic measure in lower dimensional sets. Acta Math.Sinica 35 (2019), no 6, 876–902.
  * [DPP] M. Dindoš, S. Petermichl, J. Pipher. The $L^{p}$ Dirichlet problem for second order elliptic operators and a $p$-adapted square function. J. Funct. Anal. 249 (2007), no. 2, 372–392.
  * [DPR] M. Dindoš, J. Pipher, D. Rule. Boundary Value Problems for Second-Order Elliptic Operators Satisfying a Carleson Condition. Comm. Pure Appl. Math. 70 (2017), no. 7, 1316-1365.
  * [DP] M. Dindoš, J. Pipher. Regularity theory for solutions to second order elliptic operators with complex coefficients and the Lp Dirichlet problem. Adv. Math. 341 (2019), 255–298.
  * [FKP] R. Fefferman, C. Kenig, J. Pipher. The theory of weights and the Dirichlet problem for elliptic equations. Ann. of Math. (2) 134 (1991), no. 1, 65–124.
  * [FKS] E. Fabes, C. Kenig, R. Serapioni. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7 (1982), no. 1, 77–116.
  * [Fen1] J. Feneuil. Absolute continuity of the harmonic measure on low dimensional rectifiable sets. Preprint, arXiv:2006.03118.
  * [Fen2] J. Feneuil. The Green function with pole at infinity applied to the study of the elliptic measure. Anal. PDE., accepted (arXiv:2010.04034).
  * [FMZ] J. Feneuil, S. Mayboroda, Z. Zihui. The Dirichlet problem in domains with lower dimensional boundaries. Rev. Mat. Iberoam. 37 (2021), no. 3, 821–910.
  * [FLM] J. Feneuil, L. Li, S. Mayboroda. Comparison between Green functions and smooth distances. Under preparation.
  * [FP] J. Feneuil, B. Poggi. Generalized Carleson perturbations of elliptic operators and applications. Preprint, arXiv:2011.06574.
  * [HKMP] S. Hofmann, C. Kenig, S. Mayboroda, J. Pipher. _Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators_ , J. Amer. Math. Soc. 28 (2015), no. 2, 483–529.
  * [HM] S. Hofmann, J.M. Martell. Uniform rectifiability and harmonic measure I: uniform rectifiability implies Poisson kernels in $L^{p}$. Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), no. 3, 577–654.
  * [HMU] S. Hofmann, J.M. Martell, I. Uriarte-Tuero. Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^{p}$ imply uniform rectifiability. Duke Math. J., 163 (2014), no. 8, 1601–1654.
  * [JK] D. Jerison and C. Kenig. The Dirichlet problem in nonsmooth domains. Ann. of Math. (2) 113 (1981), no. 2, 367–382.
  * [Ken] C. E. Kenig. Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics, 83. Amer. Math. Soc., Providence, RI, 1994.
  * [KKPT] C. Kenig, H. Koch, J. Pipher, T. Toro. A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math., 153 (2000), no. 2, 231–298.
  * [KP1] C. Kenig, J. Pipher. The Neumann problem for elliptic equations with non-smooth coefficients. Invent. Math. 113 (1993), 447–509.
  * [KP2] C. Kenig, J. Pipher. The Neumann problem for elliptic equations with non-smooth coefficients: Part II. Duke Math. J. 81 (1995), no. 1, 227–250.
  * [KP3] C. Kenig, J. Pipher. The Dirichlet problem for elliptic equations with drift terms. Publ. Mat., 45 (2001), no. 1, 199–217.
  * [MP] S. Mayboroda, B. Poggi. Carleson perturbations of elliptic operators on domains with low dimensional boundaries. J. Funct. Anal. 280 (2021), no 8, 108930.
  * [MM] L. Modica, S. Mortola. Construction of a singular elliptic-harmonic measure. Manuscripta Math. 33 (1980/81), no 1, 81–98.
  * [Ste] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43. Princeton University Press, Princeton, N.J., 1993.


# Enhanced photogalvanic effect in the two-dimensional MgCl2/ZnBr2 vertical
heterojunction by inhomogenous tensile stress

Liyu Qian  Department of Physics, Shanghai Normal University, 100 Guilin Road,
Shanghai 200234, P. R. China  Juan Zhao  Department of Physics, Shanghai
Normal University, 100 Guilin Road, Shanghai 200234, P. R. China  Yiqun Xie
<EMAIL_ADDRESS>Department of Physics, Shanghai Normal University, 100
Guilin Road, Shanghai 200234, P. R. China

###### Abstract

The photogalvanic effect (PGE) occurring in noncentrosymmetric materials
enables the generation of a dc photocurrent at zero bias with a high
polarization sensitivity, which makes it very attractive in photodetection.
However, the magnitude of the PGE photocurrent is usually small, leading to a
low photoresponsivity, and therefore hampers its practical application in
photodetection. Here, we propose an approach to largely enhancing the PGE
photocurrent by applying an inhomogenous mechanical stretch, based on quantum
transport simulations. We model a two-dimensional photodetector consisting of
the wide-bandgap MgCl2/ZnBr2 vertical van der Waals heterojunction with the
noncentrosymmetric $C_{3v}$ symmetry. Polarization-sensitive PGE photocurrent
is generated under the vertical illumination of linearly polarized light. By
applying inhomogenous mechanical stretch on the lattice, the photocurrent can
be largely increased by up to 3 orders of magnitude due to the significantly
increased device asymmetry. Our results propose an effective way to enhance
the PGE by inhomogenous mechanical strain, showing the potential of the
MgCl2/ZnBr2 vertical heterojunction in the low-power UV photodetection.

##  I INTRODUCTION

Ultraviolet (UV) photodetectors (PDs) have been widely utilized in military
and civilian fields, such as missile detection, environmental monitoring, and
space optical communications AM2016-H.Chen ; AP2019-Q.Zhang ;
ACSNano2018-D.Guo . Robust photoresponsivity, high polarization sensitivity,
low-energy power consumption, ultra-small size and good flexibility are
important figure of merits for the UV photodetections. In recent years, UV-PDs
based on two-dimensional (2D) semiconductors have shown great potential to
fulfill these demands. There has been considerable research on UV-PDs based on
the 2D wide bandgap (2D-WBG) semiconductors, such as MgO with an energy gap
(Eg) of 7.3 eV ACSAMI2018-W.Zhang-MgO  few-layered h-BN ACSAMI2018-W.Zhang-BN
and FePS3 Nanotechnology2018-Y.Guo . Outstanding performances have been
achieved via such 2D-WBG UV-PDs. For example, the graphene/ZnS/4H-SiC
photodetector has demonstrated an ultrafast photoresponse to 250 nm light with
a 28 ms rise time ACSAMI2019-H.Kan .

Various mechanisms are responsible for the generation of the photocurrent in
these photodetectors, which include the photovoltaic effect, photogating,
photoconducting, as well as photothermal effects. Of particular importance is
the photogalvanic effect (PGE) that occurs in materials without the space
inversion symmetry under the illumination of polarized light Xiao2020 ;
PRB1981-R.von ; PRA1978-V.I.Belinicher ; JPCM2003-S.D.Ganichev ; PRB2020-X.Tao
; Nanoscale2020-M.M.R.Moayed ; AcsNano2020-M.Chen ; SciAdv2020-C.Guo . The PGE
is able to generate a dc photocurrent that is highly polarization sensitive,
without the need for any external electric fields or a pn-junction. Therefore,
the PGE is a perfect mechanism for the self-powered and polarization sensitive
UV photodetection with a low dark current. In our previous work, we have
proposed the UV photodetection based on the PGE by using the photodetector
consisting of the wide-bandgap 2D materials and their heterostructures
JMCA2019-Y.Luo . A recent experiment has also demonstrated that the PGE is an
effective mechanism for the UV photodetection with a high polarization
sensitivity and low dark current ACIE2020-Y.Peng .

However, the magnitude of the PGE photocurrent is usually small, which leads
to a relatively low photoresponsivity typically of mA/W. Therefore, it is of
practical importance to find a mechanism to enhance the magnitude of the PGE
photocurrent. Several recent experiments have suggested that the PGE
photocurrent can be largely enhanced under mechanical strains
NATURE2019-Y.J.Zhang ; NatPhoto2016-J.E.Spanier ; APL2016-J.Li . It has been
shown that, by applying an appropriate mechanical stress gradient on the
SrTiO3 single crystal, the PGE photocurrent can be enhanced by orders of
magnitude Science2018-M.-M.Yang , which is known as the flexophotovoltaic
efect, though the controversy was raised regarding on the underlying mechanism
ACSnano2019-H.Zou . Direct experimental evidence for the efective enhancement
of the PGE by mechanical strain has recently been achieved in the Fe-doped
LiNbO3 single crystal, in which the photocurrent was increased by 75$\%$ even
under a tiny uniaxial compressive stress (0.005$\%$) on the lattice
Sciadv2010-S.Nadupalli . Motivated by these experimental findings, we have
investigated the influence of uniform mechanical tensile stress on the PGE for
a phosphonene photodetector by using quantum transport simulations, and
proposed that the large enhancement of the PGE photocurrent is mainly due to
the strain-increased device asymmetry ZHAOJUAN-PRA .

In this work, using quantum transport simulations, we present a theoretical
study on the PGE in the photodetector based on the 2D MgCl2/ZnBr2 vertical van
der Waals heterojunction (vdWH). By applying an inhomogenous mechanical
tensile stress on the photodetector, the PGE photocurrent can be largely
enhanced under the illumination of linearly polarized light. Our results
propose an effective approach to enhancing the PGE, suggesting its promising
application in the low-power UV photodetection.

##  II Model and Methods

The 2D monolayer MgCl2 and ZnBr2 have recently been predicted with a stable
configuration NatNanotechnol2018-N.Mounet . Their primitive cells contain two
haloid atoms (Cl, Br) and a metal atom (Mg, Zn), as shown in Figs.1(a) and
1(b). The lattice constant is 3.645 Å and 3.761 Å for MgCl2 and ZnBr2, with
the bandgap of 6.0 eV and 3.41 eV, respectively, according to our previous
first-principle calculations JMCA2019-Y.Luo . Since the monolayer MgCl2 and
ZnBr2 have a centrosymmetric $D_{3d}$ symmetry, the PGE cannot be generated.
Therefore, to obtain the PGE, we construct a vdWH using the monolayer MgCl2
and ZnBr2, as shown in Fig.1. This vdWH has the noncentrosymmetric $C_{3v}$
symmetry, and therefore the PGE can be generated under the vertical
illumination of linearly polarized light.

The electronic bandstructure of the MgCl2/ZnBr2 vdWH are calculated using the
Vienna Ab-initio Simulation Package (VASP) PRB1996-G.Kresse ; PRB1999-G.Kresse
. The plane wave basis set with the projector augmented wave (PAW)
pseudopotential is adopted. The exchange correlation potentials are
approximated using the generalized gradient approximation (GGA) functional as
parameterized by Perdew, Burke and Ernzerhof (PBE) functional
PRL1996-J.P.Perdew . Brillouin zone integration is performed with a
15$\times$1$\times$15 k-point mesh for geometry optimization and self-
consistent electronic structure calculations. The kinetic energy cutoff for
the plane wave basis is 500 eV. All the atoms in the unit cell are fully
relaxed until the force on each atom is less than 0.001 eV/Å and the
convergence criteria for energy in the self-consistent field cycle was
$10^{-6}$ eV. The supercell is periodic in the $xz$-plane and is separated by
a 20 Å vacuum layer in the $z$-direction to avoid the interaction between the
imaginary cells.

The photocurrent is calculated using the first-principles quantum transport
package NanoDcal NANODCAL , where density functional theory is used combined
with nonequilibrium Green’s formalism (NEGF-DFT). The photocurrent is
calculated based on linear response theory JAP2002-L.E.Henrickson .
Specifically, for linearly polarized light, the photocurrent injecting into
the left lead (electrode) can be written as PRA2018-Y.Xie ;
Nanotechnology2015-Y.Xie ; JMCA2019-Y.Luo ,

$\displaystyle J_{L}^{(ph)}$ $\displaystyle=$ $\displaystyle{ie\over
h}\int\\{\cos^{2}\theta\textrm{Tr}\\{\Gamma_{L}[G^{<(ph)}_{1}+f_{L}(G^{>(ph)}_{1}-G^{<(ph)}_{1})]\\}$
$\displaystyle+$
$\displaystyle\sin^{2}\theta\textrm{Tr}\\{\Gamma_{L}[G^{<(ph)}_{2}+f_{L}(G^{>(ph)}_{2}-G^{<(ph)}_{2})]\\}$
$\displaystyle+$
$\displaystyle{2\sin(2\theta)}\textrm{Tr}\\{\Gamma_{L}[G^{<(ph)}_{3}$
$\displaystyle+$ $\displaystyle
f_{L}(G^{>(ph)}_{3}-G^{<(ph)}_{3})]\\}\\}\textrm{d}E,$

where $G^{>/<(ph)}_{1,2,3}$ denotes the greater/lesser Green’s functions with
electron-photon interaction, which are determined by the symmetry, photon
frequence and polarization vector e. For linearly polarized light, the
polarization vector e=$\cos\theta$e1+$\sin\theta$e2, where $\theta$ is the
angle formed by the polarization direction with respect to the vector
$\textbf{e}_{1}$. The vectors $\textbf{e}_{1}$ and $\textbf{e}_{2}$ are set
along the zigzag and armchair directions, respectively. The photocurrent can
be normalized as, $I_{ph}=J_{L}^{(ph)}/eI_{\omega}$ which still has dimensions
of area of $a_{0}^{2}$/photon, where $a_{0}$ is the Bohr radius. In the
numerical calculations, the DZP atomic orbital basis is used to expand all the
physical quantities; the exchange and correlation were treated at the level of
the GGA functional as parameterized by the PBE approximation; atomic cores are
determined by the standard norm conserving nonlocal pseudopotentials, and
16$\times$1$\times$1 k-points are used. These calculation details are verified
to obtain converged results.

According to our previous work ZHAOJUAN-PRA , the magnitude of the PGE
photocurrent is determined by the $|J_{L}^{(ph)}|=\Delta J=|J_{+}-|J_{-}||$,
where $J_{-}={ie\over
h}\int^{0}_{-\infty}\textrm{Tr}[\Gamma_{L}G^{>(ph)}]\textrm{d}E<0$ denotes the
current flowing out the center region to the left lead, while $J_{+}={ie\over
h}\int_{0}^{+\infty}\textrm{Tr}[\Gamma_{L}G^{<(ph)}]$ $\textrm{d}E$ $>0$
denotes the current flowing into the center region from the lead. Therefore we
can define a factor $A$ to describe the device asymmetry as

$\textit{A}=\left|\frac{J_{+}-|J_{-}|}{J_{+}}\right|.$ (1)

For the device with the space inversion symmetry the current flowing into the
center region equals that out of the center region, that is, there is no net
current flowing through the device. In this case, $J_{+}=|J_{-}|$, so that
$J_{L}^{(ph)}=0$, and $A$=0. For the device without the inversion symmetry,
there is a net photocurrent and $A$ is nonzero. The magnitude of the
photocurrent can then be rewritten as,

$|J_{L}^{(ph)}|=AJ_{+}.$ (2)

In the below, we will show that the device asymmetry plays a dominant role in
enhancing the photocurrent under inhomogenous mechanical stress.

##  III Results and discussion

We consider three configurations of the 2D MgCl2/ZnBr2 vertical vdWHs, namely,
AA, AB and AC, as shown in Fig.1. The lattice constant of the ZnBr2 (3.761 Å)
is used for the heterostructure, which means a 3.18$\%$ stretch in the lattice
of the MgCl2. For the AA structure, the Zn atoms in the top layer of ZnBr2 lie
directly above the Mg atoms in the bottom layer of MgCl2, as shown in
Fig.1(a). For the AB structure, the Zn atoms of the top ZnBr2 layer lie
directly above the Br atoms in the top-sublayer of the MgCl2 (Fig.1(b)). For
the AC structure, Zn atoms of the top ZnBr2 layer are directly above the Br
atoms of the bottom MgCl2 sublayer (Fig.1(c)). We calculate the binding energy
of the three heterojunctions, which is defined as
$E_{B}$=$E_{1}+E_{2}$-$E_{vdWH}$. Here, $E_{vdWH}$, $E_{1}$ and $E_{2}$
represent the total energy of the heterojunction, monoalyer MgCl2 and ZnBr2,
respectively. Considering the negative signs of the energies, a higher (more
positive) binding energy means a more strongly bounded and stable system. The
AA structure is the most stable one among the three vdWHs, as indicated in
Tab.1. Therefore, in the following we mainly investigate electronic properties
and the PGE of the AA MgCl2/ZnBr2 vdWH.

Figure 1: Three kinds of stacking patterns of the 2D MgCl2/ZnBr2 vertical vdWH. (a), (b) and (c) are top views for AA, AB and AC configurations, respectively, and (d), (e) and (f) are corresponding side views. The red arrows in (a) denote the lattice constants of the primitive cell.  Table 1: Total energy ($E_{vdWH}$), binding energy ($E_{B}$), interlayer spacing $d$ and lattice constant $a$ for three kinds of 2D MgCl2/ZnBr2 vertical vdWHs.  Stacking |  $E_{vdWH}$(eV) |  $E_{B}$(eV) |  $d$(Å) |  $a$(Å)
---|---|---|---|---
AA | -5.976 | 0.222 | 3.221 | 3.761
AB | -5.969 | 0.214 | 3.410 | 3.761
AC | -5.920 | 0.166 | 3.666 | 3.761

The electronic bandstructure of the AA MgCl2/ZnBr2 vertical vdWH is shown in
Fig.2(a). It has an indirect bandgap of 3.47 eV, with the valence band maximum
(VBM) located at Gamma point, while the conduction band minimum (CBM) is
located at M point. Figure 2(b) shows the absorption coefficient
$\alpha(\omega)$ of the AA MgCl2/ZnBr2 vdWH. There are two prominent
absorption peaks located in the ultraviolet range, and the first peak appears
at around 5.1 eV. We will focus on the PGE for the photon energy lower than
6.0 eV, which covers the first absorption peak.

Figure 2: (a) The electronic bandstructure and (b) optical absorption
coefficient of the AA MgCl2/ZnBr2 vdWH.

We model an UV-PD by using the AA MgCl2/ZnBr2 vdWH, which contains: left and
right electrodes (leads), and the central region, as indicated in Fig.3(a).
The whole device is periodical in the $x$-direction, and two electrodes are
extended into $z=\pm\infty$, respectively. When the center region is
illuminated by linearly polarized light, the PGE is generated due to the
$C_{3v}$ symmetry, giving rise to a robust photocurrent that flows along the
$z$ direction. The photon energy considered is from 3.5 to 6.0 eV, which is
greater than the bandgap of the vdWH and covers the first optical absorption
peak.

Figure 3: Atomic model for the photodetector composed of the AA MgCl2/ZnBr2
vertical vdWH. (a) and (b) are top and side views of the photodetector,
respectively. The blue curved arrows in (b) represent the vertically incident
linearly polarized light on the center region.

Our calculations show that the PGE photocurrent has a form $\beta*\cos
2\theta+I_{0}$, which is determined by the $C_{3v}$ symmetry, in agreement
with the phenomenological theory of the PGE SovPhysUsp1980-V.I.Belinicher ;
APL2000-S.D.Ganichev . Here, $\beta$ and $I_{0}$ are determined by the
symmetry and photon energy. Fig. 4(a) gives the photocurrent for photon
energies of 4.7 eV, 4.9 and 5.9 eV as examples, which shows an evident cosine
dependence on the polarization angle $\theta$. We can then obtain the maximum
photocurrent ($I_{max}$) at either $\theta=0^{\circ}$ or $\theta=90^{\circ}$
for each photon energy. The $I_{max}$ for different photon energies is shown
in Fig. 4(b). For photon energies of 3.5, 3.6, and 3.7 eV, the photocurrent is
very small with a magnitude of $10^{-4}$ (not shown), and from 3.8 eV the
photocurrent begins to increase. The photocurrent reaches the maximum (1.06)
at 5.4 eV. Overall, the photocurrent in the higher photon energy range ($>$
4.5 eV) is larger than that in the low photon energy range (3.5-4.5 eV), which
should be partly attributed to a stronger optical absorption for high photon
energies, as shown in Fig.2(b).

Figure 4: (a) Variation of the photocurrent with the polarization angle
$\theta$ for photon energies of 4.7, 4.9 and 5.9 eV. (b) The maximum
photocurrent for different photon energies.

Next, we investigate the influence of mechanical tension stress on the PGE
photocurrent. Three types of tensile stress are exerted on the lattice. The
first one is a uniform 5$\%$ stretch on the entire device along the
z-direction. The second one is a partial stretch, in which only the right part
of the device is stretched by 5$\%$ along the z-direction. The third is a
gradient stretch: The left electrode and the left 2/7 of the center region are
invariant; the following 3/7 of the center region is stretched by 3$\%$ along
the z-direction; the rest 2/7 of the center region and the right electrode are
stretched by 5$\%$. For the uniform stretch, the $C_{3v}$ symmetry of the
photodetector is not changed, so that the photocurrent remains the cosine
dependence on the 2$\theta$, as shown in Fig.5(a). For the partial and
gradient stretch, the symmetry of the photodetector is reduced from the
$C_{3v}$ to $C_{s}$, while the photocurrent still holds the cosine form under
the $C_{s}$ symmetry, as shown in Fig.5(b), which is in a good agreement with
the phenomenological theory of the PGE. The maximum photocurrent for different
photon energies is shown in Figs.5(c). It can be seen that the photocurrent
under partial and gradient stretches (the blue and red spheres) is much larger
than that of the uniform stretch. Here, we calculate the increase ratio
$R_{I}$ of the photocurrent, which is the ratio of the photocurrent under
mechanical strain with respect to that without any strain, as shown in
Figs.5(d) The maximum $R_{I}$ is 1372 at 3.8 eV for the gradient stretch, and
is 849 at 3.8 eV for the partial stretch. For a number of photon energies,
e.g., 4.1 to 4.4 eV, $R_{I}$ is greater than 100, and moreover, for all photon
energies, $R_{I}$ is greater than 1 for both the partial and gradient
stretches. In contrast, for uniform stretch the photocurrent only increases
for a few of low photon energies, and is even decreased for some high
energies. These results show that the magnitude of the photocurrent can be
largely enhanced by partial and gradient stretches. The reason is that the
device is reduced from the $C_{3v}$ symmetry to the $C_{s}$ symmetry under
partial and gradient stretches, which means that the device becomes more
asymmetric, in other words, the device asymmetry is increased. The magnitude
of the photocurrent has a linear dependence on the device asymmetry, as
proposed in our previous work ZHAOJUAN-PRA , and therefore, the photocurrent
is largely enhanced. In contrast, for the uniform stretch the symmetry of the
photodetector is not changed, so that the magnitude of the photocurrent does
not change considerably.

Figure 5: Variation of the photocurrent with the polarization angle $\theta$
for (a) the uniform stretch and (b) gradient stretch. (c) The maximum
photocurrent at different photon energies for three types of stretches. (d)
The increase ratio $R_{I}$ of the photocurrent under stretch with respect to
that without the stretch.

Then, we calculate the device asymmetry $A$, and also the increase ratio
$R_{A}$ of the device asymmetry with respect to that without any strain. Fig.
6(a) plots the device symmetry $A$ at different photon energies for different
mechanical strain, which shows that the device asymmetry is considerably
increased, as compared to that without any strain (the pink squares).
Furthermore, we compare $R_{I}$ with $R_{A}$ in Figs. 6(b) and 6(c) at
different photon energies for different mechanical strain. It can be seen that
the overall trend of the $R_{I}$ agrees very well with that of the $R_{A}$ for
almost all of the photon energies. We also examined the photocurrent under
gradient compression for several photon energies, and found that the
photocurrent is also significantly enhanced, with a largest $R_{I}$ of 128 at
5.0 eV, as shown in Figs. 5(c) and 5(d)(see the green circles). This is also
attributed to the largely increased device asymmetry (see Figs. 6(a) and
6(c)). These results strengthen that the inhomogenous strain including partial
stretch, gradient stretch and gradient compression can effectively enhance the
PGE photocurrent in the photodetector because of the reduced device symmetry.

Figure 6: (a) The device symmetry $A$ at different photon energies for
different mechanical strain. (b) Comparison of the $R_{I}$ with the increase
ratio of the device asymmetry ($R_{A}$) for partial stretch, and (c) for
gradient stretch and compression.

##  IV CONCLUSIONS

In conclusion, we have studied the PGE in the photodetector composed of the 2D
MgCl2/ZnBr2 vertical vdWH by using quantum transport simulations. PGE
photocurrent is generated at zero bias under the illumination of linearly
polarized light in the UV range for photon energies from 3.8 eV to 6.0 eV,
owing to the noncentrosymmetric $C_{3v}$ symmetry. By applying partial stretch
(5$\%$) and gradient stretch along the transport direction on the device, the
PGE photocurrent is largely enhanced by up to 3 orders of magnitude for all of
the photon energies. This is because the partial and gradient stretches reduce
the device symmetry from the $C_{3v}$ to the $C_{s}$, which increases the
device asymmetry, and therefore the photocurrent. Our results propose an
effective way to largely enhance the PGE by inhomogenous tensile stress, and
suggest a potential application of the 2D MgCl2/ZnBr2 vertical vdWH in the
low-power UV photodetection.

## ACKNOWLEDGMENT

This work is supported by the National Natural Science Foundation of China
under Grant No.51871156.

## References

  * (1) H. Chen, H. Liu, Z. Zhang, K. Hu and X. Fang, Adv. Mater, 28, 403-433 (2016).
  * (2) Q. Zhang, X. Li, Z. He, M. Xu, C. Jin and X. Zhou, J. Phys. D, 52, 303002 (2019).
  * (3) D. Guo, Y. Su, H. Shi, P. Li, N. Zhao, J. Ye, S. Wang, A. Liu, Z. Chen, C. Li and W. Tang, ACS Nano, 12, 12827-12835 (2018).
  * (4) W. Zheng, R. Lin, Y. Zhu, Z. Zhang, X. Ji and F. Huang, ACS Appl. Mater. Interfaces, 10, 20696-20702 (2018).
  * (5) W. Zheng, R. Lin, Z. Zhang and F. Huang, ACS Appl. Mater. Interfaces, 10, 27116-27123 (2018).
  * (6) Y. Gao, S. Lei, T. Kang, L. Fei, C. L. Mak, J. Yuan, M. Zhang, L. Shaojuan, Q. Bao and Z. Zeng, Nanotechnology, 29, 244001 (2018).
  * (7) H. Kan, W. Zheng, R. C. Lin, M. Li, C. Fu, H. B. Sun, M. Dong, C. H. Xu, J. T. Luo, Y. Q. Fu and F. Huang, ACS Appl. Mater. Interfaces, 11, 8412-8418 (2019).
  * (8) Y. Gao, Y. Zhang, and D. Xiao, Phys. Rev. Lett., 124, 077401 (2020).
  * (9) R. von Baltz and W. Kraut, Phys. Rev. B, 23, 5590-5596 (1981).
  * (10) V. I. Belinicher, Phys. Lett. A, 66, 213-214 (1978).
  * (11) S. D. Ganichev and W. Prettl, J. Phys.: Condens. Matter, 15, R935 (2003).
  * (12) X. Tao, P. Jiang, H. Hao, X. Zheng, L. Zhang and Z. Zeng, Phys. Rev. B, 102, 081402 (2020).
  * (13) M. M. R. Moayed, F. Li, P. Beck, J.-C. Schober and C. Klinke, Nanoscale, 12 6256-6252 (2020).
  * (14) M. Chen, K. Lee, J. Li, L. Cheng, Q. Wang, K. Cai, E. E. M. Chia, H. Cheng and H. Yang, Acs Nano, 14 3539-3545 (2020).
  * (15) C. Guo, Y. Hu, G. Chen, D. Wei, L. Zhang, Z. Chen, W. Guo, H. Xu, C.-N. Kuo, C. S. Lue, X. Bo, X. Wan, L. Wang, A. Politano, X. Chen and W. Lu, Sci. Adv., 6 eabb6500 (2020).
  * (16) Y. Luo, Y. Hu and Y. Xie, J. Mater. Chem. A, 7, 27503-27513 (2019).
  * (17) Y. Peng, X. T. Liu, Z. H. Sun, C. M. Ji, L. N. Li, Z. Y. Wu, S. S. Wang, Y. P. Yao, M. C. Hong and J. H. Luo, Angew. Chem. Int. Ed, 59, 3933-3937 (2020).
  * (18) Y. J. Zhang, T. Ideue, M. Onga, F. Qin, R. Suzuki, A. Zak, R. Tenne, J. H. Smet and Y. Iwasa, Nature, 570, 349-353 (2019).
  * (19) J. E. Spanier, V. M. Fridkin, A. M. Rappe, A. R. Akbashev, A. Polemi, Y. Qi, Z. Gu, S. M. Young, C. J. Hawley, D. Imbrenda, G. Xiao, A. L.Bennett-Jackson and C. L. Johnson, Nat. Photonics, 10, 611-616 (2016).
  * (20) J. Li and P. M. Haney, Appl. Phys. Lett., 109, 193903 (2016).
  * (21) M.-M. Yang, D. J. Kim and M. Alexe, Science, 360, 904-907 (2018).
  * (22) H. Zou, C. Zhang, H. Xue, Z. Wu and Z. L. Wang, ACS nano, 13, 12259-12267 (2019).
  * (23) S. Nadupalli, J. Kreisel and T. Granzow, Sci. adv., 5, eaau9199 (2019).
  * (24) J Zhao, Y Hu, Y Xie, L Zhang and Y Wang, Phys. Rev. Appl., 14, 064003 (2020).
  * (25) N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Cepellotti and G. Pizzi, Nat. Nanotechnol., 13, 246-252 (2018).
  * (26) G. Kresse and J. Furthm$\ddot{u}$ller, Phys. Rev. B, 54, 11169-11186 (1996).
  * (27) G. Kresse and D. Joubert, Phys. Rev. B, 59, 1758-1775 (1999).
  * (28) J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 77, 3865-3868 (1996).
  * (29) J. Taylor, H. Guo and J. Wang, Phys. Rev. B, 63, 245407 (2001).
  * (30) L. E. Henrickson, J. Appl. Phys., 91, 6273-6281 (2002).
  * (31) Y. Xie, M. Chen, Z. Wu, Y. Hu, Y. Wang, J. Wang and H.Guo, Phys. Rev. Appl., 10, 034005 (2018).
  * (32) Y. Xie, L. Zhang, Y. Zhu, L. Liu and H. Guo, Nanotechnology, 26, 455202 (2015).
  * (33) V. I. Belinicher and B. I. Sturman, Sov. Phys. Usp., 23, 199-223 (1980).
  * (34) S. D. Ganichev, H. Ketterl, W. Prettl, E. L. Ivchenko and L. E. Vorobjev, Appl. Phys. Lett., 77, 3146-3148 (2000).


YufeiWang MengyueWu

# Evaluation of data inconsistency for multi-modal sentiment analysis

###### Abstract

Emotion semantic inconsistency is an ubiquitous challenge in multi-modal
sentiment analysis (MSA). MSA involves analyzing sentiment expressed across
various modalities like text, audio, and videos. Each modality may convey
distinct aspects of sentiment, due to subtle and nuanced expression of human
beings, leading to inconsistency, which may hinder the prediction of
artificial agents. In this work, we introduce a modality conflicting test set
and assess the performance of both traditional multi-modal sentiment analysis
models and multi-modal large language models (MLLMs). Our findings reveal
significant performance degradation across traditional models when confronted
with semantically conflicting data and point out the drawbacks of MLLMs when
handling multi-modal emotion analysis. Our research presents a new challenge
and offer valuable insights for the future development of sentiment analysis
systems.

###### keywords:

multi-modal sentiment analysis, multi-modal large language model, data
inconsistency

##  1 Introduction

Multi-modal sentiment analysis has become a research focus in the field of
artificial intelligence over the last few years in human-machine interaction
[1, 2]. Its primary aim is to decipher the sentiment expressed by individuals
across a spectrum of modalities, including language, facial expressions,
speech, posture and sometimes physiological signals [3]. However, despite the
rapid advancements in multimodal learning, the task of detecting human
emotions remains challenging, given the intricate nature of emotion expression
[4], that emotion expression is always subtle and nuanced. This subtlety,
which is complicated yet very common, comes from the intricate interplay of
cultural, societal, and personal factors that shape how individuals perceive
and convey their feelings. Thus, human emotions expressed from different
modalities tends to be inconsistent. An illustration of conflicting emotion
sample is shown in Figure 1.

In multi-modal sentiment analysis, the three most commonly utilized modalities
are text, audio and visual [5]. Textual data plays a pivotal role, as it
carries significant semantic context embedded in language [6]. The work [7]
combines BERT [8] and CNN and acquires remarkable performance on movie review
emotion classification dataset. However, depending solely on textual data may
not suffice, as individuals also rely on non-verbal expressions to convey
emotions, like acoustic and visual. By incorporating audio and visual cues,
models can better capture the interplay between these modalities and have a
broader recognition of the context. In Geometric Multimodal Contrastive (GMC)
learning [9], they compare the model performance using only uni-modal modality
with multiple modalities and prove that multiple modalities can significantly
improve the accuracy. The works [10, 11] introduce a message hub to exchange
information with each modality to merge information from uni-modals and result
a better performance compared with uni-modal classification. The recent
benchmark [12] compares the uni-modal and multimodal emotion classification
results for multiple datasets using different feature extractors.

Figure 1: An example of multi-modal data conflicting samples. "M", "V", "A",
"T" represent multimodal, visual, acoustic, textual label respectively.

With the rise of large language models (LLMs), embedding other modalities into
LLMs becoming popular. Multimodal Large Language Models (MLLMs) accommodate
the audio-video-text multi-modal data together with LLMs [13]. Video-LLaMA
[14] enables video-based comprehension by integrating audio-visual signals to
LLaMA [15] and shows the ability to perceive and comprehend video content and
generate meaningful responses. AffectGPT [13] pre-trains audio, video and text
modalities together and handles the explainable multimodal emotion reasoning.

The evolution of traditional multimodal sentiment analysis systems can be
categorized by their approach to modality fusion. The deep neural networks
usually necessitate intricate fusion mechanisms to integrate complementary but
also (possibly) redundant information across different modalities. Based on
the summation of the work [16], we introduce different fusion mechanisms.
Early fusion, also named as feature-level fusion, extracts the various
features from multiple modalities and fuses them at the input level. Late
fusion, on the contrary, first conducts sentiment analysis based on each
modality and then incorporates uni-modal sentiment decisions into the final
prediction. Hybrid fusion combines early fusion and late fusion together.
Tensor-based fusion obtains the multimodal sentence representation by tensor
product of uni-modal representations. Translation-based fusion converts one
modality into another to capture meaningful semantics across modalities.
Memory-based fusion highlights the sequential interactions of each modality
over time.

Despite significant advances in multi-modal models, multi-modal emotion
analysis encounters contradictions from single-modality emotion prediction.
This obstacle stems from the existence of potentially semantically conflicting
information within each modality. Human emotions are inherently expressed
diversely across different modalities. Occasionally, these modalities may
convey incongruent meanings. This phenomena complicates the integration and
interaction of different modalities, inevitably posing difficulty for models.
Consequently, the integration of these diverse modalities becomes a challenge
and combating such inconsistency needs higher-level of multimodal reasoning.
However, currently there lacks such a data-inconsistent benchmark dataset and
no proper investigation into multi-modal emotion recognition models on
handling inconsistent situations. Our work fills the gap and contributes
primarily on two key aspects:

  * •

We propose a clearly defined setting termed ``multi-modal conflicting data
sentiment analysis" and introduce a standardized benchmark test set named
DiffEmo for evaluation purposes with properly defined conflicted data. This
initiative aims to establish a structured framework for researchers to assess
and compare the performance of different models in handling modal conflicts
effectively.

  * •

We conduct a comprehensive evaluation of multiple models, including multi-
modal large language models. Both deep neural networks with different modality
fusion strategies and MLLMs fail to output reliable emotion label when facing
inconsistent sentiment data. This extensive evaluation enables researchers to
gain deeper insights into the strengths and limitations of various models when
confronted with modal conflicts. Our results show that more complex reasoning
ability is needed in multimodal sentiment analysis.

##  2 DiffEmo Dataset Construction

In multi-modal sentiment analysis, data conflict refers to the scenario in
which a single data sample displays discordant emotional expressions across
its three modalities: text, audio, and video. Various multi-modal sentiment
datasets have been constructed to meet the growing demand for human sentiment
perception [17, 18, 19, 20, 21].Among them, CH-SIMS v2.0 [20] is a Chinese
multi-modal sentiment analysis dataset. It contains a total of 4,402 video
clips with both uni-modal and multi-modal annotations. Each instance in the
dataset is annotated with sentiment scores ranging from _Strong Negative
(-1,-0.8)_ , _Weak Negative (-0.6,-0.4,-0.2)_ , _Neutral (0)_ , _Weak Positive
(0.2, 0.4, 0.6)_ , to _Strong Positive (0.8, 1.0)_ for uni-modal and multi-
modal labels. From this dataset, we extract conflicting data samples based on
their uni-modal annotations. A conflicting sample is defined as:

$\displaystyle\text{Conflicting sample}=\left\\{\begin{array}[]{ll}1,&\text{if
}|m_{1}-m_{2}|>1\\\ 0,&\text{otherwise}\end{array}\right.$ (3)

where $m_{1}$ and $m_{2}$ represent the annotations from two different
modalities. The absolute difference $|m_{1}-m_{2}|$ measures the inconsistency
between these uni-modals. If the absolute difference exceeds 1, indicating
strong inconsistency, the sample is labeled as conflicting (1); otherwise, it
is labeled as consistent (0).

Through this process, we collected a total of 661 conflicting data samples. We
present the label distribution of these samples in Figure 2. We find that text
contains most neutral labels while video has higher distribution of positive
sentiment. To further investigate the conflicting label distribution, we list
the portion of each modality that is inconsistent with multi-modal labels when
multi-modal label is classified as positive, neutral and negative in Table 1.
We observe that textual modality is highly probably tends to be inconsistent
with multi-modal label compared with other two modalities. Following manual
scrutiny of the conflicting data, we identify and select a test set of 173
samples which encapsulates more severe semantic conflicts.

Figure 2: Distribution of uni-modal and multi-modal labels for conflicting data. Table 1:  Illustration of modality inconsistency proportion, when a multi-modal label is categorized as positive, neutral, or negative, the portion of uni-modal label that contradicts the multi-modal label.  Modality | Multi-modal Label
---|---
Positive | Neutral | Negative
T | 25.87% | 8.78% | 27.23%
A | 14.52% | 7.56% | 13.91%
V | 16.79% | 8.01% | 15.27%

In order to compare the performance of various models on conflicting data, we
resample CH-SIMS v2.0 dataset and make a DiffEmo Dataset, consisting of three
different test settings, namely Mixed Set, Conflicting Set and Aligned Set,
aiming to validate that dealing with conflicting data indeed presents a more
challenging setting. An illustration of data split for DiffEmo Dataset is
shown in Table 2.

  1. 1.

Mixed Set, contains 2722 train samples, including 488 conflicting samples; 346
test samples, including 173 conflicting samples.

  2. 2.

Conflicting Set, contains the same train set as Mixed Set; 173 conflicting
samples for test set.

  3. 3.

Aligned Set, contains the same train set as Mixed Set; 173 coherent data
samples for test set.

Table 2: Illustration of data split in DiffEmo Dataset.  Set Name | Train | Test
---|---|---
| Align | Conflict | Align | Conflict
Mixed | 2234 | 488 | 173 | 173
Conflicting | 2234 | 488 | 0 | 173
Aligned | 2234 | 488 | 173 | 0

##  3 Evaluation Protocol

In this section, we elaborate on the evaluation methodology utilized to
address several inquiries concerning the performance of models on conflicting
data. Initially, we validate that dealing with conflicting data indeed
presents a more challenging setting compared to standard multi-modal emotion
recognition. Subsequently, we conduct an analysis of various model
performances and compare their fusion methods. Lastly, we assess the
effectiveness of multi-modal large language models. Additionally, we perform
an ablation study aimed at distinguishing between consistent data samples and
conflicting data samples.

Existence of conflicting data challenge By comparing the model performance
between the Mixed Set, Conflicting Set and Aligned Set , we can demonstrate
the existence of the conflicting data problem.

Different fusion method comparison We conduct evaluations on multiple models
to scrutinize the effectiveness of different fusion methods. A illustration is
shown in Table 3. We list the fusion method, whether it is multi-task, and
performance of different models on original CH-SIMS v2.0 with evaluation
metric of binary classification accuracy (Acc). They are mainstream multimodal
emotion analysis models and exhibit good performance on the original dataset.

Table 3: An overview of the models for evaluation. Model | Details
---|---
| Fusion | Multi-task | Acc
EF-LSTM [22] | early | ✗ | 78.94
SELF-MM [23] | early | ✓ | 79.69
LF-DNN [24] | late | ✗ | 79.36
MLF_DNN [25] | late | ✓ | 76.59
MMIM [26] | hybrid | ✗ | 76.6
TETFN [27] | hybrid | ✓ | 79.73
TFN [28] | tensor | ✗ | 80.14
MTFN [29] | tensor | ✓ | 80.43
MLMF [30] | tensor | ✓ | 77.23
MulT [31] | translation | ✗ | 80.68
MFN [32] | memory | ✗ | 81.14
Video-LLaMA [14] | ✗ | ✓ | ✗

Performance of multi-modal large language model We conduct an evaluation of
the performance of multi-modal large language models on multi-modal
conflicting data samples. This evaluation assesses the model's effectiveness
in integrating and synthesizing information from various modalities within a
specific context. Our objective is to test the model's capability to manage
modal conflicts and generate coherent responses despite encountering
conflicting semantics across different modalities.

Ablation study: Distinguish conflicting data To explore the distinguish-
ability between modal-conflicting data and modal-consistent data, we conduct
an ablation study employing the Video-LLaMA for classification. This aims to
ascertain whether instances of modal conflict exhibit discernible
characteristics that set them apart from modal-consistent data.

##  4 Experiments and Results

In this part, we entail our experimental results for investigating traditional
multimodal emotion models and LLMs under different data settings.

###  4.1 Experimental details

For traditional neural networks, we adopt consistent parameter settings with
the approach outlined in prior work [33] and evaluate them on Mixed Set,
Conflicting Set and Aligned Set. All experiments are conducted on one NVIDIA
A10 GPU.

Regarding Video-LLaMA, to leverage the knowledge of MLLM, we implement prompt
and in-context learning strategies. Due to GPU memory limitations and the
imperative need to incorporate acoustic, visual and textual modalities, we
choose Video-LLaMA-2-7B to test. To evaluate its emotion recognition ability,
we employ prompt and in-context learning to unveil the latent knowledge
embedded within the model. An illustration of the method is as shown in Figure
3. To facilitate comprehension of LLaMA, we use a translation tool [34] to
translate Chinese text to English, as the translation process in AffectGPT.
Subsequently, we provide commands to prompt MLLM to follow the examples
provided for in-context learning. These examples are randomly sampled from the
dataset. We give it four examples for each test sample. Finally, we evaluate
the model's ability on 200 randomly sampled test samples.

Figure 3: Prompt and in-context learning for Video-LLaMA.

###  4.2 Evaluation metrics

We adopt the commonly-used metrics to evaluate the models. Binary
classification accuracy (Acc) reflects the correctness of basic sentiment
polarity prediction. Mean Absolute Error (MAE) is recorded for fine grained
prediction evaluation.

###  4.3 Results

Table 4: Performance comparison between different dataset settings. The numbers are shown as percentages. Model Name | Mixed Set | Conflicting Set | Aligned Set
---|---|---|---
Acc | MAE | Acc | MAE | Acc | MAE
EF-LSTM | $77.48\pm 1.43$ | $37.97\pm 0.7$ | $67.62\pm 2.34$ | $47.53\pm 2.05$ | $85.9\pm 1.99$ | $31.07\pm 7.6$
SELF-MM | $77.36\pm 1.07$ | $37.84\pm 1.01$ | $65.48\pm 2.16$ | $49.36\pm 2.08$ | $86.82\pm 1.06$ | $26.93\pm 0.57$
LF-DNN | $75.07\pm 3.27$ | $37.67\pm 2.93$ | $62.98\pm 4.94$ | $50.39\pm 3.46$ | $83.24\pm 8.99$ | $29.62\pm 7.24$
MLF_DNN | $75.78\pm 3.68$ | $37.75\pm 3.99$ | $67.38\pm 2.3$ | $45.6\pm 1.62$ | $84.63\pm 8.24$ | $28.78\pm 7.83$
MMIM | $72.2\pm 1.29$ | $42.06\pm 0.52$ | $61.79\pm 1.97$ | $49.99\pm 1.72$ | $83.59\pm 2.89$ | $32.0\pm 1.01$
TETFN | $76.42\pm 0.8$ | $37.39\pm 0.51$ | $66.07\pm 1.36$ | $48.8\pm 1.74$ | $89.37\pm 0.28$ | $23.83\pm 0.66$
TFN | $67.98\pm 7.47$ | $43.49\pm 6.5$ | $64.52\pm 3.72$ | $47.94\pm 3.58$ | $75.15\pm 8.72$ | $35.77\pm 8.7$
MTFN | $76.95\pm 0.71$ | $35.84\pm 0.95$ | $62.86\pm 3.36$ | $49.32\pm 2.11$ | $84.51\pm 8.62$ | $28.37\pm 7.86$
MLMF | $69.91\pm 5.05$ | $43.21\pm 4.51$ | $62.98\pm 2.18$ | $49.52\pm 1.81$ | $83.7\pm 8.06$ | $30.39\pm 7.38$
MulT | $77.19\pm 1.19$ | $35.95\pm 0.97$ | $65.6\pm 1.48$ | $48.17\pm 1.74$ | $89.13\pm 1.18$ | $24.45\pm 0.74$
MFN | $73.55\pm 5.23$ | $38.15\pm 4.55$ | $66.07\pm 1.06$ | $46.96\pm 1.66$ | $81.39\pm 8.08$ | $30.67\pm 6.61$

#### Performance comparison under different settings

As shown in Table 4, all models have a severe degradation compared between
Conflicting Set and Aligned Set. We observe that for early and hybrid fusion,
multi-task models always perform better compared with single-task ones, while
for tensor-based and late fusion, single-task models achieve the best score,
better than multi-task ones. Testing on Conflicting Set, EF-LSTM and MLF_DNN
are best models, indicating that simple fusion networks sometimes may be
effective for semantically non-aligned samples.

As for the relatively degradation between three datasets, MulT performs
relatively steady and achieves competitively performance on all three
constructed dataset, which indicating the effectiveness of transformer
architecture for capturing the intricate semantics for long-range series.

#### Multimodal LLM on DiffEmo

Due to the nature of LLM, they may occasionally produce irrelevant outputs,
indicating a potential lack of contextual understanding. We categorize the
output of Video-LLaMA into three categories: ``True" if the output aligns with
emotion label, ``False" if it deviates from the label, and ``None" if it is
uncertain and gives irrelevant information. We compare under different
combination examples for in-context learning. The results are shown in Table
5.

Table 5: Comparison between different in-context learning settings. The numbers are shown as percentages. In-context | Test | Ratio
---|---|---
True | False | None
Conflicting | Conflicting | 43.4 | 39.8 | 16.6
Conflicting | Aligned | 43.3 | 37.5 | 19.2
Aligned | Aligned | 52.1 | 26.7 | 21.2
Mixed | Mixed | 43.8 | 37.9 | 18.3

We observe that MLLM is struggling in emotion reasoning and fails to give the
proper answer. When aligned data is given as the prompt, it also shows better
results. Due to complex human emotion expression, multimodal conflicting
sentiment analysis is a harder setting for MLLM. Compared with deep neural
networks, Video-LLaMA gets worse results. This may come from the insufficiency
in video-based emotion entailment pre-training for MLLMs.

###  4.4 Ablation Study

In order to distinguish the conflicting data with aligned ones, we test MLLM's
ability to distinguish the inconsistency between various modalities. Following
previous settings, we change the prompt with ``Follow examples, decide whether
the sentiment of the man in visual, acoustic and textual modalities shows
inconsistent. Omit explanations". For in-context learning, we update the video
and input that ``In the video, the speaker says: <utterance>. Based on the
video, audio and speech of the speaker, the sentiment in visual, acoustic and
textual modalities is <inconsistent> or <coherent>." The results is shown in
Table 6. Video-LLaMA also encounters a great difficulty in distinguishing the
conflicting data samples with those coherent ones. This indicates that
conflicting data naturally shared a deceptive appearance with aligned ones.
MLLM, interestingly, tends to recognize all samples as conflicting ones. They
lack a basic coherent recognition of the sentiment. Further training is needed
on this aspect to enhance MLLMs' ability in emotion reasoning under conflicted
conditions.

Table 6: Results for distinguishing conflicting data with consistent data. The numbers are shown as percentages. In-context | Test | Ratio
---|---|---
True | False | None
mixed | mixed | 53.68 | 42.10 | 3.16

##  5 Conclusion

In this paper, driven by the ambiguity and subtlety in human emotions, we
introduce a new setting called ``multimodal sentiment analysis with
conflicting data'' and construct a well-split test set to explore the
performance of traditional multimodal feature fusion based models and
multimodal large language model. Compared with the performance of these models
on normal setting, we observe a great degradation, which illustrates the
difficulty of this challenge. We observe that simple fusion methods may be
effective for conflicting data. Adding multi-setting learning to models may
impact the performance depending on the fusion methods. Transformer is an
effective method to capture semantics for conflicting data. For MLLMs, due to
the complicated nature of human emotion reasoning, they suffer great
degradation in detailing the emotion conveyed in videos. Distinguishing the
semantically conflicting data samples with aligned ones may also be a
challenge for MLLMs. In the future, video emotion entailment datasets can be
constructed to enhance the emotion recognition ability for MLLMs.

Limitations Due to the GPU memory limitations and the necessity of using
textual, acoustic and visual modalities, we only select Video-LLaMA to test
for MLLMs, which may be insufficient.

## References

  * [1] R. Kaur and S. Kautish, ``Multimodal sentiment analysis: A survey and comparison,'' _Research Anthology on Implementing Sentiment Analysis Across Multiple Disciplines_ , pp. 1846–1870, 2022.
  * [2] K. Yang, H. Xu, and K. Gao, ``Cm-bert: Cross-modal bert for text-audio sentiment analysis,'' in _Proceedings of the 28th ACM international conference on multimedia_ , 2020, pp. 521–528.
  * [3] J. Li _et al._ , ``Hybrid multimodal feature extraction, mining and fusion for sentiment analysis,'' in _Proceedings of the 3rd International on Multimodal Sentiment Analysis Workshop and Challenge_ , 2022, pp. 81–88.
  * [4] R. Franceschini _et al._ , ``Multimodal emotion recognition with modality-pairwise unsupervised contrastive loss,'' in _2022 26th International Conference on Pattern Recognition (ICPR)_. IEEE, 2022, pp. 2589–2596.
  * [5] D. Ghosal _et al._ , ``Contextual inter-modal attention for multi-modal sentiment analysis,'' in _proceedings of the 2018 conference on empirical methods in natural language processing_ , 2018, pp. 3454–3466.
  * [6] Y. Wu _et al._ , ``A text-centered shared-private framework via cross-modal prediction for multimodal sentiment analysis,'' in _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , 2021, pp. 4730–4738.
  * [7] P. Huang _et al._ , ``Text sentiment analysis based on bert and convolutional neural networks,'' in _Proceedings of the 2021 5th International Conference on Natural Language Processing and Information Retrieval_ , 2021, pp. 1–7.
  * [8] J. D. M.-W. C. Kenton and L. K. Toutanova, ``Bert: Pre-training of deep bidirectional transformers for language understanding,'' in _Proceedings of naacL-HLT_ , vol. 1, 2019, p. 2.
  * [9] P. Poklukar _et al._ , ``Geometric multimodal contrastive representation learning,'' in _International Conference on Machine Learning_. PMLR, 2022, pp. 17 782–17 800.
  * [10] F. Lv _et al._ , ``Progressive modality reinforcement for human multimodal emotion recognition from unaligned multimodal sequences,'' in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 2554–2562.
  * [11] J. Sun _et al._ , ``Layer-wise fusion with modality independence modeling for multi-modal emotion recognition,'' in _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , 2023, pp. 658–670.
  * [12] Z. Lian _et al._ , ``Merbench: A unified evaluation benchmark for multimodal emotion recognition,'' _arXiv preprint arXiv:2401.03429_ , 2024.
  * [13] Z. Lian, L. Sun, M. Xu, H. Sun, K. Xu, Z. Wen, S. Chen, B. Liu, and J. Tao, ``Explainable multimodal emotion reasoning,'' _arXiv preprint arXiv:2306.15401_ , 2023.
  * [14] H. Zhang, X. Li, and L. Bing, ``Video-llama: An instruction-tuned audio-visual language model for video understanding,'' _arXiv preprint arXiv:2306.02858_ , 2023.
  * [15] H. Touvron _et al._ , ``Llama: Open and efficient foundation language models,'' _arXiv preprint arXiv:2302.13971_ , 2023.
  * [16] L. Zhu _et al._ , ``Multimodal sentiment analysis based on fusion methods: A survey,'' _Information Fusion_ , vol. 95, pp. 306–325, 2023.
  * [17] S. Poria _et al._ , ``Meld: A multimodal multi-party dataset for emotion recognition in conversations,'' _arXiv preprint arXiv:1810.02508_ , 2018.
  * [18] A. Zadeh _et al._ , ``Multimodal sentiment intensity analysis in videos: Facial gestures and verbal messages,'' _IEEE Intelligent Systems_ , vol. 31, no. 6, pp. 82–88, 2016.
  * [19] A. B. Zadeh _et al._ , ``Multimodal language analysis in the wild: Cmu-mosei dataset and interpretable dynamic fusion graph,'' in _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , 2018, pp. 2236–2246.
  * [20] Y. Liu _et al._ , ``Make acoustic and visual cues matter: Ch-sims v2. 0 dataset and av-mixup consistent module,'' in _Proceedings of the 2022 International Conference on Multimodal Interaction_ , 2022, pp. 247–258.
  * [21] Z. Lian _et al._ , ``Mer 2023: Multi-label learning, modality robustness, and semi-supervised learning,'' in _Proceedings of the 31st ACM International Conference on Multimedia_ , 2023, pp. 9610–9614.
  * [22] J. Williams _et al._ , ``Recognizing emotions in video using multimodal dnn feature fusion,'' in _Proceedings of Grand Challenge and Workshop on Human Multimodal Language (Challenge-HML)_. Association for Computational Linguistics, 2018, pp. 11–19.
  * [23] W. Yu, H. Xu, Z. Yuan, and J. Wu, ``Learning modality-specific representations with self-supervised multi-task learning for multimodal sentiment analysis,'' in _Proceedings of the AAAI conference on artificial intelligence_ , vol. 35, no. 12, 2021, pp. 10 790–10 797.
  * [24] E. Cambria _et al._ , ``Benchmarking multimodal sentiment analysis,'' in _Computational Linguistics and Intelligent Text Processing: 18th International Conference, CICLing 2017, Budapest, Hungary, April 17–23, 2017, Revised Selected Papers, Part II 18_. Springer, 2018, pp. 166–179.
  * [25] W. Yu, H. Xu, F. Meng, Y. Zhu, Y. Ma, J. Wu, J. Zou, and K. Yang, ``CH-SIMS: A Chinese multimodal sentiment analysis dataset with fine-grained annotation of modality,'' in _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , D. Jurafsky, J. Chai, N. Schluter, and J. Tetreault, Eds. Online: Association for Computational Linguistics, Jul. 2020, pp. 3718–3727. [Online]. Available: https://aclanthology.org/2020.acl-main.343
  * [26] W. Han, H. Chen, and S. Poria, ``Improving multimodal fusion with hierarchical mutual information maximization for multimodal sentiment analysis,'' in _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , 2021, pp. 9180–9192.
  * [27] D. Wang _et al._ , ``Tetfn: A text enhanced transformer fusion network for multimodal sentiment analysis,'' _Pattern Recognition_ , vol. 136, p. 109259, 2023.
  * [28] A. Zadeh _et al._ , ``Tensor fusion network for multimodal sentiment analysis,'' _arXiv preprint arXiv:1707.07250_ , 2017.
  * [29] W. Yu _et al._ , ``Ch-sims: A chinese multimodal sentiment analysis dataset with fine-grained annotation of modality,'' in _Proceedings of the 58th annual meeting of the association for computational linguistics_ , 2020, pp. 3718–3727.
  * [30] Z. Liu _et al._ , ``Efficient low-rank multimodal fusion with modality-specific factors,'' _arXiv preprint arXiv:1806.00064_ , 2018.
  * [31] Y.-H. H. Tsai _et al._ , ``Multimodal transformer for unaligned multimodal language sequences,'' in _Proceedings of the conference. Association for Computational Linguistics. Meeting_ , vol. 2019. NIH Public Access, 2019, p. 6558.
  * [32] A. Zadeh _et al._ , ``Memory fusion network for multi-view sequential learning,'' in _Proceedings of the AAAI conference on artificial intelligence_ , vol. 32, no. 1, 2018.
  * [33] H. Mao _et al._ , ``M-sena: An integrated platform for multimodal sentiment analysis,'' _arXiv preprint arXiv:2203.12441_ , 2022.
  * [34] Y. Tang _et al._ , ``Multilingual translation with extensible multilingual pretraining and finetuning,'' _arXiv preprint arXiv:2008.00401_ , 2020.