# The Stochastic Logarithmic Norm for Stability Analysis of Stochastic
Differential Equations

Sk. Safique Ahmad<EMAIL_ADDRESS>Nagalinga Rajan
<EMAIL_ADDRESS>Soumyendu Raha<EMAIL_ADDRESS>
###### Abstract

To analyze the stability of Itô stochastic differential equations with
multiplicative noise, we introduce the stochastic logarithmic norm. The
logarithmic norm was originally introduced by G. Dahlquist in 1958 as a tool
to study the growth of solutions to ordinary differential equations and for
estimating the error growth in discretization methods for their approximate
solutions. We extend the concept to the stability analysis of Itô stochastic
differential equations with multiplicative noise. Stability estimates for
linear Itô SDEs using the one, two and $\infty$-norms in the $l$-th mean,
where $1\leq l<\infty$, are derived and the application of the stochastic
logarithmic norm is illustrated with examples.

###### keywords:

Logarithmic norms, Stochastic differential equations.

and and ††thanks: Scientific Computation Laboratory, Supercomputer Education
and Research Centre, Indian Institute of Science, Bangalore 560012, India.
The last author is the corresponding author. ††thanks: This work was partially
supported by the Naval Research Board, DRDO, Government of India with grant
number DNRD/05/4003/NRB/88.

##  1 Introduction

For $A\in{\mathbb{C}}^{n\times n}$ and $X(t)\in{\mathbb{C}}^{n},$ we consider
the ordinary differential equation (ODE) $dX_{t}=AX_{t}dt,X(0)=x_{0}.$ Then we
have $\|X(t)\|\leq\|x_{0}\|e^{\mu(A)t}$ where $\mu(A)$ is the logarithmic norm
of the matrix $A$ [4, 19, 20]. If $\mu(A)<0$, then the ODE is asymptotically
stable. Also, $\mu(A)$ using the matrix $2$-norm gives an estimate [13] for
the pseudospectrum [15] of $A$:
$\max\Re\lambda_{\epsilon}(A)-\epsilon\leq\mu(A)$ where $1\gg\epsilon>0$.
Since the pseudospectrum captures the stability of the numerical solution of
the ODE over a finite number of time steps under the effect of local stiffness
and nonnormality of $A$ (we shall refer to this as the numerical stability)
[6, 7], having $\mu(A)<0$ implies numerical stability in addition to the
asymptotic stability of the ODE. As is already shown in [6], transient
numerical stability affects the computation and choice of methods for
numerical integration of the ODE.

In this paper we extend the classical logarithmic norm to the stability
analysis of Itô stochastic differential equations (SDE) and introduce the
stochastic logarithmic norm for estimating the numerical stability of an SDE
in order to facilitate the selection of stiff and balanced stochastic
numerical integration schemes. Letting $A,B\in{\mathbb{C}}^{n\times
n},X(t)\in{\mathbb{C}}^{n},~{}B\in{\mathbb{C}}^{n\times n}$, the Itô SDE with
a single channel of multiplicative noise is considered in the form of
$dX_{t}=AX_{t}dt+BX_{t}dW_{t}$ given the initial condition that $X(0)=x_{0}$
with probability (w.p.) $1$, where $W$ is the one dimensional Wiener process
such that $\int_{0}^{s}dW_{t}\sim N(0,s)$, i.e., is standard Gaussian
distributed with mean $0$ and variance $s$. In our definition of the
stochastic logarithmic norm we shall use the the matrix $p$-norm induced by
the vector $p$-norm and the expectation of the $l$-th raw moment, i.e., the
$l$-th mean of the solution to the linear multiplicative SDE. The stochastic
logarithmic norm is computed over the sample paths as the expected logarithmic
norm of the system in the sense of the existence of a generalized derivative
of the Wiener process which itself is not obligatory differentiable with
respect to time.

Throughout the paper the following standard assumptions are made as in [9].
Let there be a common probability space $(\Omega,{\mathcal{A}},P)$ with index
$t\in{\mathcal{T}}\subset{\mathbb{R}}$ on which the stochastic process $X(t)$
is a collection of random variables. The Wiener process $W=\\{W_{t},t\geq
t_{0}\\}$ is associated with an increasing family of $\sigma$-algebras
$\\{{\mathcal{A}}_{t},t\geq t_{0}\\}$. For the general case of multi-
dimensional noise, each component of $\\{W_{t}^{(i)}\\}$ is
${\mathcal{A}}_{t}$-measurable with ${{\bf E}}(W(t_{0}))=0$ w.p. $1$,$~{}{{\bf
E}}(W(t)|{\mathcal{A}}_{t_{0}})=0$, $~{}~{}{{\bf
E}}((W^{(i)}_{t}-W^{(i)}_{s})(W^{(j)}_{t}-W^{(j)}_{s})|{\mathcal{A}}_{s})=\delta_{i,j}(t-s)$
for $t_{0}\leq s\leq t$ and $\Delta W=W(t_{n+1})-W(t_{n})$, the component wise
increments of the multi-dimensional Wiener process, are independent of each
other at all points in the partition of the time interval ${\mathcal{T}}$:
$t_{0}\leq t_{1}\leq t_{2}\ldots\leq t_{r}\leq t_{r+1}\leq\ldots\leq
t_{N}=t_{f}$. The initial value $X_{0}$ is assumed to be
$\mathcal{A}_{t_{0}}$-measurable with $\left\|X_{0}\right\|_{p}<\infty$ w.p.
$1$. All expectations on a function $\phi(X_{t})$ are evaluated as ${\bf
E}(\phi(X_{t})|{\mathcal{A}}_{t})$ unless otherwise stated. Inequalities and
equalities involving random variables hold almost surely where applicable.

The stability analysis in [17] uses test equations with scalar and $2$-by-$2$
matrix coefficients having multiplicative noise of dimension one [16]. Some of
the analysis uses the classical logarithmic norm to establish the stability of
the moment equations (derived from the SDE) which are deterministic. The
present approach with stochastic logarithmic norm generalizes the stability
analysis of SDEs as found in [16, 17].

###  1.1 Classical Logarithmic Norm

For $1\leq p\leq\infty,$ the $p$-norm on ${\mathbb{C}}^{n}$ is given by

$\displaystyle\|x\|_{p}:=\left\\{\begin{array}[]{ll}\left(\sum^{n}_{j=1}|x_{j}|^{p}\right)^{1/p},&\mbox{
for }1\leq p<\infty,\\\ \max_{1\leq j\leq n}|x_{j}|,&\mbox{ for
}p=\infty.\end{array}\right.$

Obviously, $\|x\|_{2}:=(x^{H}x)^{1/2}$ is the $2$-norm on ${\mathbb{C}}^{n}$.
For $A\in{\mathbb{C}}^{n\times n},$ the spectrum $\Lambda(A)$ of $A$ is given
by $\Lambda(A):=\\{\lambda\in{\mathbb{C}}:{\bf rank}(A-\lambda I)<n\\}.$ We
denote a matrix $p$-norm on ${\mathbb{C}}^{n\times n}$ induced by the vector
$p$-norm as $\|\cdot\|_{p}$ for $p=1,2,\infty$ and define these norms as
$\|A\|_{2}:=\max_{j}\\{\sqrt{\lambda_{j}}:\lambda_{j}\ \in\Lambda(AA^{H})\\},$
$\;\|A\|_{1}:=\max_{j=1,\dots,n}\left(\sum_{i=1}^{n}|a_{ij}|\right)\,\mbox{and}\,\|A\|_{\infty}:=\max_{i=1,\dots,n}\left(\sum_{j=1}^{n}|a_{ij}|\right)$.
Then the logarithmic norm for a single matrix is defined as

$\mu_{p}(A):=\lim_{h\rightarrow 0^{+}}\frac{\|I+hA\|_{p}-1}{h}.$

For the $1,2$ and $\infty$-norms, the classical logarithmic norms,
respectively, are computed ([5],Vol 1) as
$\mu_{1}(A):=\max_{j}\left(\Re(a_{jj})+\sum_{j\not=i}^{n}|a_{ij}|\right),\,\mu_{2}(A):={\displaystyle{\frac{\lambda_{\max}(A+A^{H})}{2}}}\,$
and as
$\mu_{\infty}(A):=\max_{i}\left(\Re(a_{ii})+\sum_{i\not=j}^{m}|a_{ij}|\right)$.

###  1.2 Stability of the SDE

The stability of the vector SDE with single channel multiplicative noise is
defined as follows.

###### Definition 1

[2, 3] The equilibrium solution $X_{t}\equiv 0$ to
$dX_{t}=AX_{t}dt+BX_{t}dW_{t}$ is stochastically stable in the $l$-th mean
($l$ is a finite integer $\geq$ 1) using a vector $p$-norm if
$\forall\epsilon>0$, $\exists\delta>0$ such that

${\bf E}(\|X(t)\|_{p}^{l})<\epsilon\quad\forall\,t\geq t_{0}\quad
and\quad\|X(t_{0})\|_{p}<\delta~{}w.p.~{}1$ (2)

and is asymptotically stable in $l$-th mean if in addition,
$\exists\delta_{0}>0$ such that

$\lim_{t\to\infty}{\bf
E}(\|X_{t}\|_{p}^{l})=0\quad\forall\,\,\|X(t_{0})\|_{p}<\delta_{0}~{}w.p.~{}1.$
(3)

##  2 Background

In the following items we review the existing stability analysis of linear
stochastic differential equations with multiplicative noise.

  * $(a)$

In [16] scalar stochastic differential equations of the form $dX_{t}=\lambda
X_{t}dt+\beta X_{t}dW_{t}$ with $X_{0}=1$ w.p. $1$, where $\lambda$ and
$\beta$ are constants have been considered and it is shown that the above
stochastic differential equation is mean square stable using the $2$-norm if
$2\Re(\lambda)+|\beta|^{2}\leq 0$ when $\lambda,\beta$ are complex scalars.

  * $(b)$

In [17] vector stochastic differential equations, with single channel
multiplicative noise, of the form $dX_{t}=DX_{t}dt+BX_{t}dW_{t},$ where
$D=\left(\matrix{\lambda_{1}&0\cr 0&\lambda_{2}}\right)$ and
$B=\left(\matrix{\alpha_{1}&\beta_{1}\cr\beta_{2}&\alpha_{2}}\right)$ have
been analyzed for mean square stability. It is shown that the above stochastic
differential equation is mean square stable in the $\infty$ norm if
$\max\\{2\lambda_{1}+(|\alpha_{1}|+|\beta_{1}|)^{2},2\lambda_{2}+(|\alpha_{2}|+|\beta_{2}|)^{2}\\}<0.$

  * $(b_{1})$

In [16] for the same stochastic differential equation as in $(a)$ the
Mil’stein scheme [11] $X_{n+1}=X_{n}+\lambda X_{n}h+\mu X_{n}\Delta
W+\frac{(\Delta W)^{2}-h}{2}\mu^{2}X_{n}$ with $O(h^{1.5})$ root mean square
error is analyzed for stability in the mean square sense. The mean square
stability function $R(h)=|1+h\lambda|^{2}+|h\mu^{2}|+\frac{1}{2}|h^{2}\
\mu^{4}|$ is derived and it is shown that the SDE is stochastically
asymptotically stable in the mean square when $R(h)<1$.

  * $(b_{2})$

The reference [16] also considered the SDE as given in $(b)$ and applied the
Euler-Maruyama numerical integration scheme as
$X_{n+1}=X_{n}+hDX_{n}+BX_{n}\Delta W,$ where $h$ and $\Delta W$ stand for
step-size and the increment of the Wiener process, respectively. Then the
discretized SDE is shown to be stochastically asymptotically stable if

$\max\\{(1+\lambda_{1}h)^{2}+(|\alpha_{1}|+|\beta_{1}|)^{2},(1+\lambda_{2}h)^{2}+(|\alpha_{2}|+|\beta_{2}|)^{2}\\}<1.$

Next we briefly review a few essential aspects of the logarithmic norm that
are used in the stability analysis of ODEs.

  * $(c)$

In [4, 10] the logarithmic norm was introduced in order to derive error bounds
for the solution of initial value ODE problems using differential inequalities
that distinguish between forward and reverse time integration. This led to
requirements for the stability of initial value and boundary value ODE
problems. The classical analysis, using the vector norm $\|\cdot\|{}$ on
${\mathbb{C}}^{n}$ and the sub-ordinate matrix norm on ${\mathbb{C}}^{n\times
n}$, defined the logarithmic norm of a matrix $A$ as

$\mu_{p}(A):=\lim_{h\rightarrow 0^{+}}\frac{\|I+hA\|_{p}-1}{h}.$

  * $(d)$

More recently Söderlind [20] considered $f:D\subset X\rightarrow X$ and
defined least upper bounds (lub) and greatest lower bounds (glb) Lipschitz
constants by

$L[f]=\sup_{u\not=v}\frac{f(u)-f(v)}{|u-v|},~{}~{}l[f]=\inf_{u\not=v}\frac{f(u)-f(v)}{|u-v|},$

for $u$ and $v\in D,$ where the domain is path connected and

$l[f]|u-v|\leq|f(u)-f(v)|\leq L[f]|u-v|.$

If $l[f]>0,$ then $f(u)\rightarrow f(v)$ implies that $u\rightarrow v.$ Then
$f$ is an injection, with an inverse on $f(D),$ the same as a matrix
$A\in{\mathbb{C}}^{n\times n}$ being invertible if its glb is strictly
positive. Also then, $L[f^{-1}]=l[f^{-1}],$ where $L[f^{-1}]$ is defined over
$f(D).$ If $f=A$ is a linear map then $L[A]=\|A\|.$ Hence $L[.]$ is left and
right $G$-differentiable for the class of Lipschitz maps. This allows one to
define the lub and glb logarithmic Lipschitz constant, by

$M[f]=\lim_{h\rightarrow 0^{+}}\frac{L[I+hf]-1}{h},m[f]=\lim_{h\rightarrow
0^{-}}\frac{L[I+hf]-1}{h}.$

The lub logarithmic Lipschitz generalizes the classical logarithmic norm for
every matrix $A,$ so that, $M[A]=\mu(A).$

In the following section we develop the stochastic logarithmic norm as an
upper bound estimate of the rate of growth of the solution of a multiplicative
noise linear SDE. The rate of growth is analyzed as a Dini derivative of the
$l$-th mean in vector $p$-norm of the solution. This approach may be seen as a
special case of the modern definition of logarithmic norm in [20] and as an
extension of the classical definition as in [4, 10, 19]. Other works [8] deal
with logarithmic norm of matrix pencils with one invertible matrix. However in
our current treatment we do not use matrix pencils for defining the stochastic
logarithmic norm.

##  3 Definition of the Stochastic Logarithmic Norm

A linear Itô SDE with a single channel of multiplicative noise is written as

$dX_{t}=AX_{t}dt+BX_{t}dW_{t},$ (4)

where $A,B\in{\mathbb{C}}^{n\times n},$ are constant matrices. For the non-
linear Itô SDE with a single channel multiplicative noise given by

$dX_{t}=f(X,t)dt+g(X,t)dW_{t},$ (5)

the strong order $1.0$ Itô-Taylor expansion (Mil’stein scheme when used in
numerical integration) of $X_{t}$ at $t=t_{n+1}$ is given as

$X_{n+1}=X_{n}+f(X_{n},t_{n})h+g(X_{n},t_{n})\Delta
W+\left(g\frac{\partial{g}}{\partial{x}}\right)_{(X_{n},t_{n})}\frac{((\Delta
W)^{2}-h)}{2}+R,$ (6)

where $h=t_{n+1}-t_{n}$ is the step-size, $W(t_{n+1})-W(t_{n})=:\Delta W\sim
N(0,h)$ and $R$ are the $O(h^{1.5})$ remainder terms in the root mean square
sense. When linearized at $t=t_{n}$, $A:=\left(\frac{\partial f}{\partial
x}\right)_{n}$ and $B:=\left(\frac{\partial g}{\partial x}\right)_{n}$. We
shall define the stochastic logarithmic norm by studying the growth of the
$l$-th mean of the solution $X$ of the linear SDE (4) (linearized SDE in case
of SDE (5)) w. r. t. time. We seek an upper bound for the growth rate of ${\bf
E}(\|X\|_{p}^{l})$ using the upper-right Dini derivative which for any
function $\Xi(t)$ w.r.t $t$ is defined as

$D_{+}\left(\Xi(t)\right)=\lim_{h\to 0+}\frac{\Xi(t+h)-\Xi(t)}{h}.$ (7)

The strong order $1.0$ Itô-Taylor expansion (linearized at $t$ in case of the
non-linear SDE (5)) applied to $X_{t}$ over the time interval $[t,t+h]$ gives

$X(t+h)=X(t)+hAX(t)+\Delta WBX(t)+\frac{(\Delta W)^{2}-h}{2}B^{2}X(t)+R,$ (8)

with the remainder terms $R$ being $O(h^{1.5})$ in the root mean square sense.
In the above expansion (8) $\Delta W:=\left\\{\Delta W(t),t\geq 0\right\\}$
are the independent increments of a Wiener process over the interval
$[t,t+h]$. Taking the $p$-norms and raising both sides of (8) to the power of
$l$, we can write the following inequality:

$\|X(t+h)\|_{p}^{l}\leq\|I+hA+\Delta WB+\frac{(\Delta
W)^{2}-h}{2}B^{2}+R_{x}\|_{p}^{l}\|X(t)\|_{p}^{l},$ (9)

where $R_{x}$ are root mean square $O(h^{1.5})$ remainder terms. For the
$l$-th mean using a $p$-norm, we apply expectation to both sides of the above
inequality and get, almost surely (a.s.),

${\bf E}\|X(t+h)\|_{p}^{l}\leq{\bf E}\left(\|I+hA+\Delta WB+\frac{(\Delta
W)^{2}-h}{2}B^{2}+R_{x}\|_{p}^{l}\right){\bf E}\|X(t)\|_{p}^{l}.$ (10)

observing that $X(t)$ is independent of the Wiener increment $\Delta W$ since
the Wiener process is a non-anticipative process. Then, we estimate the
expected rate of growth as

${\bf E}\left(D_{+}\|X(t)\|_{p}^{l}\right)\leq\lim_{h\to 0^{+}}\frac{{\bf
E}(\|I+hA+\Delta WB+\frac{(\Delta W)^{2}-h}{2}B^{2}+R_{x}\|_{p}^{l})-1}{h}{\bf
E}(\|X(t)\|_{p}^{l}).$

Based on the limit term on the right hand side above we introduce the
stochastic logarithmic norm.

###### Definition 2

The stochastic logarithmic norm of a square matrix pair of same dimensions,
$(A,B)$, in the $l$-th mean using a matrix $p$-norm is defined as

$\nu_{p}^{l}\left(A,B\right)=\lim_{h\to 0^{+}}\frac{{\bf E}\left(\|I+hA+\Delta
WB+\frac{(\Delta W)^{2}-h}{2}B^{2}\|_{p}^{l}\right)-1}{h},$ (11)

where the limit is taken in the sense of the existence of the generalized
derivative of the Wiener process and $\|A\|_{p},\|B\|_{p}$ are assumed to be
finite.

From the above definition the expected rate of growth of the solution can be
estimated as ${\bf
E}\left(D_{+}\|X(t)\|_{p}^{l}\right)\leq\nu_{p}^{l}\left(A,B\right){\bf
E}(\|X(t)\|_{p}^{l})$ since $R_{x}$ are of root mean square $O(h^{1.5})$.
Obviously the weaker estimate

$D_{+}{\bf E}\|X(t)\|_{p}^{l}\leq\nu_{p}^{l}\left(A,B\right){\bf
E}(\|X(t)\|_{p}^{l})$

also holds so that ${\bf E}\|X(t)\|_{p}^{l}\leq
e^{\nu_{p}^{l}\left(A,B\right)t}\|X(t_{0})\|_{p}^{l}$ for the linear SDE (4).
The linear SDE (4) is stochastically stable in the $l$-th mean using a
$p$-norm when $\nu_{p}^{l}\left(A,B\right)\leq 0$ and is asymptotically stable
in the $l$-th mean using a $p$-norm when $\nu_{p}^{l}\left(A,B\right)<0$. The
linearized SDE (5) (and hence the linear SDE (4)) is numerically
stochastically stable in the $l$-th mean using the $p$-norm if
$\nu_{p}^{l}\leq 0$ since over $k~{}(k\ll\infty)$ time steps each of size
sufficiently small $h_{i}>0$ almost surely we have ${\bf
E}\|X_{t+kh}\|_{p}^{l}\leq e^{\sum_{i=1}^{k}\nu_{p}^{l}(A_{i},B_{i})h}{\bf
E}\|X_{t}\|_{p}^{l}$ when each $\nu_{p}^{l}(A_{i},B_{i})\leq 0$. If ${\bf
E}\|X_{t}\|_{p}^{l}$ is finitely bounded w.p. $1$, then ${\bf
E}\|X_{t+kh}\|_{p}^{l}$ is almost surely finitely bounded. Later in the paper
we relate the stochastic logarithmic norm to the pseudospectrum of the Ito
stability matrix $A-\frac{1}{2}B^{2}$ for the linear SDEs with multiplicative
noise.

It may be remarked that for an SDE with additive noise and for an ODE, the
stochastic logarithmic norm is given as $\nu_{p}^{1}(A,0)=\mu_{p}(A)$.

##  4 Some Estimates of the Stochastic Logarithmic Norm

In this section we derive some estimates of the stochastic logarithmic norm
using the matrix $p$-norm, where $p=1,2,\infty.$ The estimates show the
incremental behavior of the stochastic logarithmic norm under perturbations
and also the effect of noise on the deterministic ODE.

Let $\lambda_{\max}(A)$ be the largest eigenvalue of any square matrix $A$. We
state the following Lemma from pp.62, [1] which we shall use in estimating the
stochastic logarithmic norm while using the matrix $2$-norm.

###### Lemma 1

Let $A,B\in{\mathbb{C}}^{n\times n}$ be Hermitian matrices. Then

$\lambda_{\max}{(A)}+\lambda_{\min}{(B)}\leq\lambda_{\max}(A+B)\leq\lambda_{\max}(A)+\lambda_{\max}(B).$

The equality holds when $B=kI,$ where $k$ is a scalar constant and $I$ is an
identity matrix.

For the matrix $2$-norm and in the $l^{th}$ mean we can estimate the following
from (11).

###### Theorem 5

The stochastic logarithmic norm in the $l^{th}$ mean, where $1\leq l<\infty$
using the matrix $2$-norm satisfies the following bounds in the sense of
existence of a generalized derivative for the Wiener process as $\zeta
dt=dW_{t},~{}\zeta\sim N(0,1)$.

$\displaystyle\nu_{2}^{l}(A,B)$ $\displaystyle\leq$
$\displaystyle\frac{l}{2}\lambda_{\max}(A+A^{H})+\frac{l}{4}\left(\lambda_{\max}(B+B^{H})+\lambda_{\max}(-B-B^{H})\right)$
(12)
$\displaystyle+\frac{l}{2}\lambda_{\max}(B^{H}B)+\frac{l(l-2)}{8}\lambda_{\max}^{2}(B+B^{H}),~{}~{}l>2,$
$\displaystyle\nu_{2}^{l}(A,B)$ $\displaystyle\leq$
$\displaystyle\frac{l}{2}\lambda_{\max}(A+A^{H})+\frac{l}{4}\left(\lambda_{\max}(B+B^{H})+\lambda_{\max}(-B-B^{H})\right)$
(13) $\displaystyle+\frac{l}{2}\lambda_{\max}(B^{H}B),~{}l\leq 2,$
$\displaystyle\nu_{2}^{l}(A,I)$ $\displaystyle=$
$\displaystyle\frac{l}{2}\lambda_{\max}(A+A^{H})+\frac{l}{2}+\frac{l(l-2)}{2},$
(14)

where $A,B\in{\mathbb{R}}^{n\times n}$ are the (linearized) drift and
diffusion coefficient matrices in a vector SDE with a single channel of Wiener
process noise.

Proof: For $1\leq l<\infty$ we have from (11)

$\nu_{2}^{l}\left(A,B\right)=\lim_{h\to 0^{+}}\frac{{\bf E}\left(\|I+hA+\Delta
WB+\frac{(\Delta W)^{2}-h}{2}B^{2}\|_{2}^{l}\right)-1}{h}=$ $\lim_{h\to
0^{+}}\frac{{\bf E}\left(\sqrt{\lambda_{\max}\left((I+hA+\Delta
WB+\frac{(\Delta W)^{2}-h}{2}B^{2})^{H}(I+hA+\Delta WB+\frac{(\Delta
W)^{2}-h}{2}B^{2})\right)}\right)^{l}}{h}.$

Then we may write

$\displaystyle(I+hA+\Delta WB+\frac{(\Delta W)^{2}-h}{2}B^{2})^{H}(I+hA+\Delta
WB+\frac{(\Delta W)^{2}-h}{2}B^{2})$ $\displaystyle=(I+hA^{H}+\Delta
WB^{H}+\frac{(\Delta W)^{2}-h}{2}(B^{H})^{2})(I+hA+\Delta WB+\frac{(\Delta
W)^{2}-h}{2}B^{2})=$ $\displaystyle I+hA+\Delta WB+\frac{(\Delta
W)^{2}-h}{2}B^{2}+hA^{H}+\Delta WB^{H}+(\Delta W)^{2}B^{H}B$
$\displaystyle+\frac{((\Delta
W)^{2}-h)(B^{H})^{2}}{2}+\cdots=I+h(A+A^{H})+\Delta W(B+B^{H})+(\Delta
W)^{2}B^{H}B+$ $\displaystyle\frac{(\Delta
W)^{2}-h}{2}(B^{2}+(B^{H})^{2})+\cdots.$

It is possible to write $C=h(A+A^{H})+\Delta W(B+B^{H})+(\Delta
W)^{2}B^{H}B+\frac{(\Delta W)^{2}-h}{2}(B^{2}+(B^{H})^{2})$ in a series in the
normalized time step size $h\ll 1$. We use the identities
$\lambda_{\max}(cA)=|c|\lambda_{\max}(\mathrm{sign}(c)A)$ (where $c$ is a
constant, $\mathrm{sign}(c)={\displaystyle{\frac{z}{|z|}}},z\not=0$) and
$\lambda_{\max}^{2}(A)=\lambda_{\max}(A^{2})$ for estimating the terms in the
series. Using the triangle inequality and the identity
$[\lambda_{\max}(I+C)]^{\frac{l}{2}}=(1+\lambda_{\max}(C))^{\frac{l}{2}}$, the
following can be written:

$\displaystyle(1+\lambda_{\max}(C))^{\frac{l}{2}}=1+\frac{l}{2}\lambda_{\max}(C)+\frac{l(l-2)}{8}\lambda_{\max}(C)^{2}+\dots=1+$
$\displaystyle\frac{l}{2}\lambda_{\max}\left(h(A+A^{H})+\Delta
W(B+B^{H})+(\Delta W)^{2}B^{H}B+\frac{(\Delta
W)^{2}-h}{2}(B^{2}+(B^{H})^{2})\right)+$
$\frac{l(l-2)}{8}\lambda_{\max}\left(h(A+A^{H})+\Delta W(B+B^{H})+(\Delta
W)^{2}B^{H}B+\frac{(\Delta W)^{2}-h}{2}(B^{2}+(B^{H})^{2})\right)^{2}$
$\displaystyle+\cdots\leq
1+\frac{l}{2}h\lambda_{\max}(A+A^{H})+\frac{l}{2}\lambda_{\max}(\Delta
W(B+B^{H}))+\frac{l}{2}(\Delta W)^{2}\lambda_{\max}(B^{H}B)$
$\displaystyle+((\Delta
W)^{2}-h)\frac{l}{4}\lambda_{\max}(B^{2}+(B^{H})^{2})+(\Delta
W)^{2}\frac{l(l-2)}{8}\lambda_{\max}^{2}(B+B^{H})+\cdots.$ (15)

The equality holds when $B=I$ and follows from Lemma 1. Taking expectation on
both sides and in the sense of generalized derivative of the Wiener process
which is the Gaussian white noise $\zeta$ we have ${\bf
E}(\lambda_{\max}(I+C))^{\frac{l}{2}}={\bf
E}(1+\lambda_{\max}(C))^{\frac{l}{2}}\leq
1+\frac{l}{2}h\lambda_{\max}(A+A^{H})+{\bf
E}(|\zeta|)h\frac{l}{4}(\lambda_{\max}(B+B^{H})+\lambda_{\max}(-B-B^{H}))+{\bf
E}((\Delta W)^{2})\frac{l}{2}\lambda_{\max}(B^{H}B)+({\bf E}((\Delta
W)^{2})-h)\frac{l}{4}\lambda_{\max}(B^{2}+(B^{H})^{2})+{\bf E}((\Delta
W)^{2})\frac{l(l-2)}{8}\lambda_{\max}^{2}(B+B^{H})+O(h^{1.5})=1+h\frac{l}{2}\lambda_{\max}(A+A^{H})+h\frac{l}{4}(\lambda_{\max}(B+B^{H})+\lambda_{\max}(-B-B^{H}))+\frac{l}{2}h\lambda_{\max}(B^{H}B)-h\frac{l(l-2)}{8}\lambda_{\max}^{2}(B+B^{H})+O(h^{1.5}).$
For $h\to 0^{+}$ we obtain

$\lim_{h\rightarrow 0^{+}}\frac{{\bf
E}(\lambda_{\max}(1+C))^{\frac{l}{2}}-1}{h}\leq\frac{l}{2}\lambda_{\max}(A+A^{H})+\frac{l}{4}(\lambda_{\max}(B+B^{H})+\lambda_{\max}(-B-B^{H}))$
$+\frac{l}{2}\lambda_{\max}(B^{H}B)+\frac{l(l-2)}{8}\lambda_{\max}^{2}(B+B^{H}).$

$(b)$ When $B=I,$ we have

$\displaystyle\lim_{h\rightarrow 0^{+}}\frac{{\bf
E}(\lambda_{\max}(I+C))^{\frac{l}{2}}-1}{h}$ $\displaystyle=$
$\displaystyle\frac{l}{2}\lambda_{\max}(A+A^{H})+\frac{l}{2}\lambda_{\max}(B^{H}B)+$
$\displaystyle\frac{l(l-2)}{8}\lambda_{\max}^{2}(B+B^{H})$
$\displaystyle\Rightarrow\nu_{2}^{l}(A,B)$ $\displaystyle=$
$\displaystyle\frac{l}{2}\lambda_{\max}(A+A^{H})+\frac{l}{2}+\frac{l(l-2)}{2}.\blacksquare$

###### Corollary 1

For a positive integer $l>2$,

$\displaystyle\nu_{2}^{l}(A,B)$ $\displaystyle\leq$ $\displaystyle
l\left(\mu_{2}(A)+\frac{1}{2}\|B\|_{2}^{2}+\frac{1}{2}(\mu_{2}(B)+\mu_{2}(-B))+\frac{l-2}{2}(\mu_{2}(B))^{2}\right)$
(16)

For $l\leq 2$,

$\displaystyle\nu_{2}^{l}(A,B)$ $\displaystyle\leq$ $\displaystyle
l\left(\mu_{2}(A)+\frac{1}{2}(\mu_{2}(B)+\mu_{2}(-B))+\frac{1}{2}\|B\|_{2}^{2}\right)$
(17)

Proof: The results (16) and (17) follow directly from (15) in Theorem 5 by
Lemma 1 and the Itô isometry for the expectation of the Itô
integrals.$\blacksquare$
The inequality (16) has been used for mean square stability estimates in [17].

###### Corollary 2

If $l=1,2$ and $p=2$, that is, in the mean and in the mean square and using
the matrix two norm, the following results hold when $B$ is the identity
matrix $I$.

$\displaystyle\nu_{2}^{1}(A,I)=\lambda_{\max}\left(\frac{A+A^{H}}{2}\right)+\frac{1}{2}-\frac{4}{8}=\lambda_{\max}\left(\frac{A+A^{H}}{2}\right)=\mu_{2}(A),$
(18)
$\displaystyle\nu_{2}^{2}(A,I)=\lambda_{\max}\left(A+A^{H}\right)+1=2\mu_{2}(A)+1,$
(19)

Proof: The results follow from Theorem 5, by substituting $p=2,l=1$ and
$p=2,l=2.\blacksquare$

Some properties of the stochastic logarithmic norm under perturbation is
obtained in the following result. In the generalized sense of the Wiener
process derivative a couple of useful inequalities showing how the Wiener
process perturbs the deterministic ODE is also given.

###### Theorem 6

For any square matrices of same dimensions, $B,A,\Delta A,\Delta B$ and a real
number $\alpha>0$ the stochastic logarithmic norm has the following
properties.

$\displaystyle\nu_{p}^{l}(\alpha A,\sqrt{\alpha}B)$ $\displaystyle=$
$\displaystyle\alpha\nu_{p}^{l}(A,B)$ (20) $\displaystyle\nu_{p}^{1}(A+\Delta
A,B+\Delta B)$ $\displaystyle\leq$
$\displaystyle\nu_{p}^{1}(A,\sqrt{2}B)+\nu_{p}^{1}(\Delta A,\sqrt{2}\Delta
B)+\frac{1}{\sqrt{2}}\|(B-\Delta B)^{2}\|_{p}$ (21)
$\displaystyle\nu_{p}^{l}(A+\Delta A,B+\Delta B)$ $\displaystyle\leq$
$\displaystyle\nu_{p}^{l}(A,\frac{B+\Delta B}{\sqrt{2}})+\nu_{p}^{l}(\Delta
A,\frac{B+\Delta B}{\sqrt{2}})$ (22) $\displaystyle\nu_{p}^{l}(A,0)$
$\displaystyle=$ $\displaystyle l\mu_{p}(A)$ (23)

Further, in the sense of the existence of a generalized derivative of the
Wiener process which is a Gaussian white noise, the following estimate holds
for a positive integer $l$ and for any of the matrix $p$-norms.

$\displaystyle\nu^{l}_{p}(A,B)$ $\displaystyle\leq$ $\displaystyle
l\mu_{p}(A)+\frac{l}{2}\mu_{p}(-B^{2})+\frac{l(l+1)}{4}\|B\|^{2}_{p}+l\|B\|_{p}$
(24) $\displaystyle\leq$ $\displaystyle
l\mu_{p}(A)+l\|B\|_{p}\left(1+\frac{l+3}{4}\|B\|_{p}\right)\leq
l\mu_{p}(A)+l\left(1+\frac{l+3}{4}\|B\|_{p}\right)^{2}$ (25)

Proof: We can scale the Wiener process as $\sqrt{\alpha}$ and write

$\displaystyle\nu_{p}^{l}(\alpha A,\sqrt{\alpha}B)$ $\displaystyle=$
$\displaystyle\lim_{h\to 0^{+}}\alpha\frac{{\bf E}\left(\|I+(\alpha
h)A+\Delta(\sqrt{\alpha}W)B+\frac{(\Delta\sqrt{\alpha}W)^{2}-\alpha
h}{2}B^{2}\|_{p}^{l}\right)-1}{\alpha h}$ $\displaystyle=$
$\displaystyle\alpha\nu_{p}^{l}(A,B)$

for any positive integer $l$ and any of the matrix $p$-norms.

From the definition of the stochastic logarithmic norm and scaling the Wiener
process, we can write

$\nu_{p}^{1}(A+\Delta A,B+\Delta B)\leq\lim_{h\to 0^{+}}\bigg{(}\frac{{\bf
E}(\|I+(2h)A+(\sqrt{2}\Delta
W)(\sqrt{2}B)+\frac{(\Delta\sqrt{2}W)^{2}-2h}{2}(\sqrt{2}B)^{2}\|_{p})-1}{2h}$
$+\frac{{\bf E}(\|I+(2h)\Delta A+(\sqrt{2}\Delta W)(\sqrt{2}\Delta
B)+\frac{(\Delta\sqrt{2}W)^{2}-2h}{2}(\sqrt{2}\Delta B)^{2}\|_{p})-1}{2h}+$
$\frac{{\bf E}|\int_{0}^{h}W_{u}dW_{u}|}{h}\|B\Delta B+\Delta BB-B^{2}-(\Delta
B)^{2}\|_{p}\bigg{)}\\\ \leq\nu_{p}^{1}(A,\sqrt{2}B)+\nu_{p}^{1}(\Delta
A,\sqrt{2}\Delta B)+\frac{1}{\sqrt{2}}\|(B-\Delta B)^{2}\|_{p}$

since $\lim_{h\to 0^{+}}\frac{{\bf
E}|\int_{0}^{h}W_{u}dW_{u}|}{h}=h/(\sqrt{2}h)=\frac{1}{\sqrt{2}}$. For any
positive integer $l$, the above can be re-written as

$\displaystyle\nu_{p}^{l}(A+\Delta A,B+\Delta B)$ $\displaystyle\leq$
$\displaystyle\lim_{h\to 0^{+}}\bigg{(}\bigg{(}2^{l-1}\frac{1}{2^{l-1}}{\bf
E}\|I+(2h)A+(\sqrt{2}\Delta W)(\sqrt{2}\frac{B+\Delta B}{2})$
$\displaystyle+\frac{(\Delta\sqrt{2}W)^{2}-2h}{2}(\sqrt{2}\frac{B+\Delta
B}{2})^{2}\|_{p}^{l}-1\bigg{)}/(2h)$
$\displaystyle+\bigg{(}2^{l-1}\frac{1}{2^{l-1}}{\bf E}\|I+(2h)\Delta
A+(\sqrt{2}\Delta W)(\sqrt{2}\frac{B+\Delta B}{2})$
$\displaystyle+\frac{(\Delta\sqrt{2}W)^{2}-2h}{2}(\sqrt{2}\frac{B+\Delta
B}{2})^{2}\|_{p}^{l}-1\bigg{)}/(2h)\bigg{)}$
$\displaystyle\leq\nu_{p}^{l}(A,\frac{B+\Delta
B}{\sqrt{2}})+\nu_{p}^{l}(\Delta A,\frac{B+\Delta B}{\sqrt{2}}).$

Applying (21) recursively to $\nu_{p}^{l}(A+0,B/2+B/2)$ and using (20), we
obtain
$\nu_{p}^{l}(A,B)\leq\lim_{n\to\infty}(\nu_{p}^{l}(A,\frac{1}{(\sqrt{2})^{n}}B)+(2^{n}-1)\nu_{p}^{l}(0,\frac{1}{(\sqrt{2})^{n}}B))=\nu_{p}^{l}(A,0)+\lim_{n\to\infty}(2^{n}/2^{n})\nu^{l}_{p}(0,B)+0=\nu_{p}^{l}(A,0)+\nu_{p}^{l}(0,B)$.
For $l=1$, the inequality reduces to
$\nu_{p}^{1}(A,B)\leq\mu_{p}(A)+\nu_{p}^{1}(0,B)$.

For the deterministic case with no noise, we have

$\displaystyle\nu_{p}^{l}(A,0)$ $\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}\frac{\|I+hA\|^{l}_{p}-1}{h}$ $\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}l\|I+hA\|^{l-1}_{p}D_{+,h}\|I+hA\|_{p}$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\to 0^{+}}l\frac{\|I+\epsilon
A\|_{p}-1}{\epsilon}$ $\displaystyle=$ $\displaystyle l\mu_{p}(A).$

For the stochastic logarithmic norm of the diffusion coefficient $B$ it is
possible to write

$\displaystyle\nu_{p}^{l}(0,B)$ $\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}\frac{{\bf E}\|I+\Delta WB+\frac{(\Delta
W)^{2}-h}{2}B^{2}\|_{p}^{l}-1}{h}$ $\displaystyle\leq$
$\displaystyle\lim_{h\to
0^{+}}\left(\frac{\|I-(2h)\frac{1}{2}B^{2}\|_{p}^{l}-1}{(2h)}+\frac{{\bf
E}\|I+2B\Delta W+B^{2}(\Delta W)^{2}\|^{l}_{p}-1}{(2h)}\right)$
$\displaystyle\leq$
$\displaystyle\nu_{p}^{l}(-\frac{1}{2}B^{2},0)+\frac{l}{2}\|B\|^{2}_{p}+l\|B\|_{p}+\frac{l(l-1)}{4}\|B\|^{2}_{p}$

in the sense of the existence of a generalized derivative of the Wiener
process so that $\int_{0}^{h}\xi dt=\int_{0}^{h}dW_{s}$ where $\xi\sim N(0,1)$
is a Gaussian white noise. Further we have
$\nu_{p}^{l}(-\frac{1}{2}B^{2},0)\\\ =\frac{l}{2}\mu_{p}(-B^{2})$ and
$\frac{l}{2}|\mu_{p}(-B^{2})|\leq\frac{l}{2}\|B\|_{p}^{2}$ from the properties
of the logarithmic norm [5]. Hence the estimates (24) and (25) .
$\blacksquare$

We remark that the stochastic logarithmic norm does not satisfy the triangle
inequality property nor the that of the multiplication by a scalar in the same
way as the (deterministic) logarithmic norm. However, it is consistent with
Itô calculus in its property (20) of multiplication with a scalar. Again, Itô
calculus makes the noise “redistribute“ for any additive perturbation to the
drift and diffusion coefficients as found in (22).

The logarithmic norm coincides with the stochastic logarithmic norm in special
cases. One such case is obtained from Theorem 5 by setting $l=1$ and $B=I$
(and also by setting $B=0$ in the trivial case) for $p=2$.

###### Theorem 7

$\max_{\lambda}\Re(\lambda(A))\leq\frac{1}{2}\nu_{2}^{2}(A,I)-\frac{1}{2}$
where $I$ is the identity matrix.

Proof: From (19) we have $\nu_{2}^{2}(A,I)=2\mu_{2}(A)+1$. The logarithmic
norm has the lower bound property [5]:
$\max_{\lambda}\Re(\lambda(A))\leq\mu_{2}(A)$. Hence the inequality.
$\blacksquare$

##  8 Conditions for Mean and Mean Square Stability

We recall that the linear SDE (4) is stochastically stable in the $l$-th mean
using a $p$-norm if $\nu_{p}^{l}(A,B)\leq 0.$ To estimate of the stability of
a linear SDE one of the upper bounds derived in the last section may be used,
especially, when it is needed to compute incrementally the effect of adding
noise to an ODE or an SDE with known stability estimates. In practice, when
using the upper bounds for estimating the stability of an SDE incrementally, a
small positive number (determined by the stochastic stability region of the
numerical integrator) is used as cut-off rather than a very small tolerance or
zero so that the effect of non-normality and stiffness [6] in the SDE’s (both
linear and locally linearized) transient numerical behavior in the stochastic
logarithmic norm is taken into account.

The following results are a couple of special cases of the stochastic
logarithmic norm approach to the stability of SDEs with multiplicative noise.

  * •

We consider the scalar case $A=\alpha\in{\mathbb{C}}$ and
$B=\beta\in{\mathbb{C}}$ in (4). Then we apply Theorem 5$(a)$ and consider
(16) for $l=1,2$ the condition $\nu_{p}^{l}(A,B)<0,$ so that for $p=2$, we get
the condition for stochastic stability in the mean as

$\displaystyle\Re(\alpha)+\frac{1}{2}|\beta|^{2}\leq 0$ (26)

and in the mean square as

$\displaystyle 2\Re(\alpha)+|\beta|^{2}\leq 0.$ (27)

The above conditions are essentially the same but in practice the $0$ on the
right hand side is replaced by $TOL$ which is a small positive real number
[13] and thus $l$ becomes significant for estimates in higher moments. These
results are the same as in review $(a)$ of Section 2.

  * •

For the matrices $A=\left(\matrix{\lambda_{1}&0\cr 0&\lambda_{2}}\right)$ and
$B=\left(\matrix{\alpha_{1}&\beta_{1}\cr\beta_{2}&\alpha_{2}}\right),$ the SDE
(4) is mean square stable in the $\infty$-norm if

$\displaystyle
2\max\\{\lambda_{1},\lambda_{2}\\}+2\left(\frac{5}{4}\max\\{|\alpha_{1}|+|\beta_{1}|,|\alpha_{2}|+|\beta_{2}|\\}+1\right)^{2}\leq
0.$ (28)

The above condition is obtained using (25) in which

$\displaystyle\nu^{2}_{\infty}(A,0)$ $\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}\frac{\left(\max\\{|1+h\lambda_{1}|,|1+h\lambda_{2}|\\}\right)^{2}-1}{h}$
$\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}2\max\\{|1+h\lambda_{1}|,|1+h\lambda_{2}|\\}\max\\{\lambda_{1},\lambda_{2}\\}$
$\displaystyle=$ $\displaystyle 2\max_{i=1,2}{\lambda_{i}}$

and
$\|B\|_{\infty}=\max\\{|\alpha_{1}|+|\beta_{1}|,|\alpha_{2}|+|\beta_{2}|\\}.$

##  9 Direct Computation of the Stochastic Logarithmic Norm and Sharper Bounds

###### Theorem 10

In the sense of the existence of a generalized derivative for the Wiener
process, i.e., $\zeta dt=dW_{t},~{}\zeta\sim N(0,1)$, we can compute the
stochastic logarithmic norm directly as follows.

$\nu_{p}^{l}(A,B)=l{\bf
E}\left(\mu_{p}\left(A-\frac{1}{2}B^{2}+B\zeta\right)\right)$

Proof:

$\displaystyle\nu_{p}^{l}(A,B)$ $\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}\frac{{\bf E}\|I+hA-\frac{1}{2}B^{2}h+B\Delta W+\frac{1}{2}B^{2}(\Delta
W)^{2}\|_{p}^{l}-1}{h}$ (29) $\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}{\bf E}\bigg{(}l\|I+hA-\frac{1}{2}B^{2}h+B\Delta
W+\frac{1}{2}B^{2}(\Delta W)^{2}\|_{p}^{l-1}$ $\displaystyle\times
D_{+,h}\|I+hA-\frac{1}{2}B^{2}h+B\Delta W+\frac{1}{2}B^{2}(\Delta
W)^{2}\|_{p}\bigg{)}$ $\displaystyle=$ $\displaystyle l\lim_{\epsilon\to
0^{+}}\frac{{\bf E}\|I+(A-\frac{1}{2}B^{2})\epsilon+B\int_{0}^{\epsilon}\zeta
ds+\frac{1}{2}B^{2}\left(\int_{0}^{\epsilon}\zeta
ds\right)^{2}\|-1}{\epsilon}$ $\displaystyle=$ $\displaystyle l{\bf
E}\left(\mu_{p}\left(A-\frac{1}{2}B^{2}+B\zeta\right)\right),$

where the last equality follows from considering the Itô formula along with
the generalized derivative of the Wiener process. $\blacksquare$

The above result shows that the stochastic logarithmic norm is the expected
logarithmic norm behavior of the SDE. Consequently we can derive the following
inequalities.

###### Corollary 3

$\displaystyle\nu_{p}^{l}(A,B)$ $\displaystyle\leq$ $\displaystyle
l\mu_{p}(A)+\frac{l}{2}\left(\mu_{p}(-B^{2})+\mu_{p}(B)+\mu_{p}(-B)\right)$
(30) $\displaystyle\nu_{p}^{l}(A,B)$ $\displaystyle\geq$ $\displaystyle
l\mu_{p}(A)-\frac{l}{2}\left(\mu_{p}(B^{2})+\mu_{p}(B)+\mu_{p}(-B)\right)$
(31)

Proof: From (29) we have $\nu_{p}^{l}(A,B)\leq
l\mu_{p}(A)+\frac{l}{2}\left(\mu_{p}(-B^{2})+{\bf E}(\mu_{p}(B\zeta))\right)$
in view of the triangular inequality of the logarithmic norm. Since $\zeta\sim
N(0,1)$, we have ${\bf E}(\mu_{p}(B\zeta))={\bf
E}|\zeta|(\mu_{p}(B)+\mu_{p}(-B))/2=\left(\mu_{p}(B)+\mu_{p}(-B)\right)/2$.
Again,
$\mu_{p}(A)=\mu_{p}(A-\frac{1}{2}B^{2}+B\zeta+\frac{1}{2}B^{2}-B\zeta)\leq\mu_{p}(A-\frac{1}{2}B^{2}+B\zeta)+\mu_{p}(\frac{1}{2}B^{2})+\mu_{p}(-B\zeta)$
so that $l\mu_{p}(A)-l\mu_{p}(\frac{1}{2}B^{2})-l{\bf
E}\mu_{p}(-B\zeta)\leq\nu_{p}^{l}(A,B)$. Hence the inequalities.
$\blacksquare$

###### Corollary 4

The stochastic logarithmic norm can be bounded as follows.

$\left|\nu_{p}^{l}(A,B)\right|\leq
l\left|\mu_{p}(A-\frac{1}{2}B^{2})\right|+l{\bf
E}\left|\mu_{p}(B\zeta)\right|\leq
l\left\|A-\frac{1}{2}B^{2}\right\|_{p}+l\left\|B\right\|_{p}$ (32)

Proof: By Theorem 10 we have $\left|\nu_{p}^{l}(A,B)\right|=l\left|{\bf
E}\left(\mu_{p}\left(A-\frac{1}{2}B^{2}+B\zeta\right)\right)\right|$. Then, by
Jensen inequality one obtains, a.s.,

$l\left|{\bf
E}\left(\mu_{p}\left(A-\frac{1}{2}B^{2}+B\zeta\right)\right)\right|\leq l{\bf
E}\left|\left(\mu_{p}\left(A-\frac{1}{2}B^{2}+B\zeta\right)\right)\right|.$

From the triangular inequality we have

$\left|\mu_{p}\left(A-\frac{1}{2}B^{2}+B\zeta\right)\right|\leq\left|\mu_{p}\left(A-\frac{1}{2}B^{2}\right)\right|+\left|\mu_{p}(B\zeta)\right|$

and the bound property of the logarithmic norm leads to

$\left|\nu_{p}^{l}(A,B)\right|\leq
l\left\|A-\frac{1}{2}B^{2}\right\|_{p}+l{\bf E}\left\|B\zeta\right\|_{p}\leq
l\left\|A-\frac{1}{2}B^{2}\right\|_{p}+l\left\|B\right\|_{p}.\blacksquare$

##  11 Examples

###  11.1 Stabilization of an inverted pendulum

It is well known [18, 14] that a vertical inverted pendulum can be stabilized
around a mean vertical position by application of a suitable highly
oscillatory excitation in the form of an appropriate noise. We compute the
stochastic logarithmic norm of such a system to show how an appropriate noise
stabilizes the system. The equation of the inverted pendulum may be written as

$\displaystyle d\theta$ $\displaystyle=$ $\displaystyle v~{}dt+\epsilon
v~{}dW_{t},~{}1\gg\epsilon>0$ (33) $\displaystyle dv$ $\displaystyle=$
$\displaystyle\frac{g}{l}\theta~{}dt+b\theta~{}dW_{t},$ (34)

where $g$ is acceleration due to gravity, $l>0$ is the effective length of the
pendulum and $\theta$ is the small angular displacement from the mean
verticalposition, i.e, $\theta=0$, so that $A=\pmatrix{0&1\cr\frac{g}{l}&0}$
and $B=\pmatrix{0&\epsilon\cr b&0}$. It may be noted that
$\mu_{2}(A)=\frac{1}{2}+\frac{g}{2l}$ and
$\Re\lambda_{\max}(A)=\frac{g}{l}>0$. Thus the system without the Wiener
process excitation is unstable. The stochastic logarithmic norm of the Wiener
process excited system may be computed as

$\nu_{2}^{2}(A,B)={\bf
E}\left(\max\left\\{1+\frac{g}{l}+(b+\epsilon)\zeta-b\epsilon,-\left(1+\frac{g}{l}+(b+\epsilon)\zeta+b\epsilon\right)\right\\}\right),$

where $\zeta\sim N(0,1)$. For stabilizing the pendulum around the mean
position $\theta=0$ we require $\nu_{2}^{2}(A,B)\leq 0$ and get the condition
that

$b\geq\frac{1}{\epsilon}\left(1+\frac{g}{l}\right)$ (35)

in which $\epsilon/(2(1+g/l))$ can be interpreted physically as the amplitude
of a very wide band vertical excitation (as an approximation to Wiener
process) at the base of the pendulum. Since a Karhunen-Loeve expansion of the
Wiener process [9] contains the high frequency terms, this also shows how the
above result is consistent with the result (in [14]) that a small amplitude
highly oscillatory wide band vertical excitation stabilizes a vertical
inverted pendulum.

###  11.2 Nonnormality

From [7] we take this linear SDE in the form of (4) in which
$A:=\pmatrix{-1&b\cr
0&-1},B:=\pmatrix{0&\sigma\cr-\sigma&0},b\in{\mathbb{R}}$. Obviously the
system without any Wiener process excitation, i.e, the deterministic ODE is
asymptotically stable but
$\mu_{2}(A)=\max\\{\frac{b}{2}-1,-\left(\frac{b}{2}+1\right)\\}$ due to non-
normality of $A$ and the ODE system tends to be numerically unstable in its
transient behavior when $|b|>2$. Formulating the system as a multiplicative
noise SDE we compute the stochastic logarithmic norm directly from (29) in the
mean square and using the $2$-norm as in [7]:

$\nu_{2}^{2}=\max\\{\sigma^{2}-2\pm b\\}$

whence it is required that $\sigma^{2}\leq\min\\{2\mp b\\}$ so that
$\nu_{2}^{2}\leq 0$ for the mean square stability of the SDE. If
$\sigma\in{\mathbb{R}}$ and $|b|>2$, then it is not possible to numerically
stabilize the SDE in the mean square by choosing an appropriate $\sigma$. For
$|b|>2$ and $\sigma\in{\mathbb{C}}$, $\sigma^{2}\leq 2-|b|$ would be
sufficient for exploiting the noise towards numerically stabilizing the SDE
system in the mean square. If $\sigma,b\in{\mathbb{R}}$ and
$\sigma:=b^{-\frac{1}{4}}$, then the SDE system is stochastically stable when
$1\geq b\geq\frac{3-\sqrt{5}}{2}$.

###  11.3 Numerical examples

###### Example 1

We consider

  * $(a)$

$A=\left(\matrix{-100&0\cr 0&-200}\right),B=\left(\matrix{5&0\cr 0&6}\right).$

  * $(b)$

$A=\left(\matrix{-100&0\cr 200&-200}\right),B=\left(\matrix{5&2\cr
0&6}\right).$

  * $(c)$

$A=\left(\matrix{-100&20\cr 0&-200}\right),B=\left(\matrix{5&2\cr
0&6}\right).$

  * $(d)$

$A=\left(\matrix{-100+20i&0\cr 2&-200+i}\right),B=\left(\matrix{5+i&0\cr
2i&-6-10i}\right).$

  * $(e)$

$A=\left(\matrix{-100&20\cr 7&-200}\right),B=\left(\matrix{5&2\cr
4&6}\right).$

  * $(f)$

$A=-100,B=10$ ([16]).

  * $(g)$

$A:=\left(\matrix{A_{1}&A_{12}\cr
0&A_{2}}\right);B:=\left(\matrix{B_{1}&B_{12}\cr 0&B_{2}}\right)$ where
$A_{1}:=\left(\matrix{0.1&4&20\cr 0&0.1&5\cr 0&0&0.1}\right),\\\
A_{2}:=\left(\matrix{-0.2&3&100\cr 0&-0.2&50\cr
0&0&-0.2}\right),B_{1}:=\left(\matrix{2&30&10\cr 0&2&50\cr
0&0&2}\right),B_{2}:=\left(\matrix{4&6&20\cr 0&4&40\cr 0&0&4}\right),$
$A_{12}=\left(\matrix{2.2857\times 10^{-2}&-2.3547\times 10^{-2}&-6.8279\times
10^{-2}\cr 9.3914\times 10^{-2}&-9.6719\times 10^{-2}&-2.8049\times 10^{-1}\cr
2.8585\times 10^{-1}&-2.9443\times 10^{-1}&-8.5382\times 10^{-1}}\right),$
and $B_{12}=\left(\matrix{1.2606\times 10^{-1}&-4.6007\times
10^{-1}&7.0963\times 10^{-3}\cr 1.8156\times 10^{-1}&-6.6259\times
10^{-1}&1.0235\times 10^{-2}\cr 1.4481\times 10^{-1}&-5.2845\times
10^{-1}&8.1625\times 10^{-3}}\right).$

  * $(h)$

$A=100\times C_{100\times 100},B=100\times D_{100\times
100};~{}C_{ij},D_{ij}\sim U(0,1)$.

  * $(i)$

$\left(\matrix{-100&0\cr 0&-1}\right).$ $B=\left(\matrix{0&2\cr 2&0}\right)$
[17]

Example | Lbound | $\nu_{2}^{2}(A,B)$ | Ubound
---|---|---|---
$(a)$ | $-1.1239\times 10^{2}$ | $-1.0470\times 10^{2}$ | $-4.0393\times 10^{1}$
$(b)$ | $-1.1919\times 10^{2}$ | $-1.1468\times 10^{2}$ | $-3.1393\times 10^{1}$
$(c)$ | $-2.4082\times 10^{2}$ | $-2.2415\times 10^{2}$ | $-1.5302\times 10^{2}$
$(d)$ | $-2.2490\times 10^{2}$ | $-2.2354\times 10^{2}$ | $-5.9075\times 10^{1}$
$(e)$ | $-2.6837\times 10^{2}$ | $-2.3232\times 10^{2}$ | $-1.21915\times 10^{2}$
$(f)$ | $-3.0000\times 10^{2}$ | $-3.0026\times 10^{2}$ | $-1.0000\times 10^{2}$
$(g)$ | $-9.1852\times 10^{2}$ | $9.2453\times 10^{2}$ | $4.8398\times 10^{3}$
$(h)$ | $-2.5191\times 10^{7}$ | $1.2369\times 10^{5}$ | $2.5330\times 10^{7}$
$(i)$ | $-6.0000$ | $-5.91409$ | $-2.0000$
Table 1:  The table gives values $\nu_{2}^{2}(A,B)$ and compares them with the
estimates $Ubound$ and $Lbound$.

In Table 1 Ubound is computed using the right hand side in (30) with
$p=2,~{}l=2$ and Lbound is computed using the right hand side in (31). The
stochastic logarithmic norm is computed using (29).

##  12 Extension to Multiple Noise Channels

The definition of the stochastic logarithmic norm can be extended to the
vector SDE with multiple channels of multiplicative noise (i.e., with multi-
dimensional Wiener process). Considering the Itô-Taylor strong order 1.0
(Mil’stein) scheme for the SDE

$\displaystyle dX_{t}=AX_{t}dt+\sum_{j=1}^{m}B^{(j)}X_{t}dW_{t}^{(j)},$ (36)

where each $W^{(j)}$ is an independent component Wiener process, we can define
the following.

###### Definition 3

The stochastic logarithmic norm for a tuple of $m+1$ square matrices of same
dimensions, $(A,B^{(1)},B^{(2)},\cdots,B^{(m)})$, i.e.,
$\left(A,B^{(1:m)}\right)$, which are the (linearized) drift and diffusion
coefficients of a (non-linear) vector SDE with $m$ channels of multiplicative
noise, is defined as

$\nu_{p}^{l}\left(A,B^{(1:m)}\right)=\lim_{h\to 0^{+}}\frac{{\bf
E}\|I+hA+{\displaystyle{\sum_{j=1}^{m}B^{(j)}\Delta
W^{(j)}+\sum_{i=1}^{m}\sum_{j=1}^{m}B^{(i)}B^{(j)}}}\int_{0}^{h}\int_{0}^{s}dW_{u}^{(i)}dW_{s}^{(j)}\|^{l}_{p}-1}{h},$

where the limit is taken in the sense of the existence of the generalized
derivative of the Wiener process and $\|A\|_{p}$ and each $\|B^{(i)}\|_{p}$
are assumed to be finite.

For $p=2,l\geq 2$ it is easy to obtain the following estimate after proceeding
as in the estimate in (16).

$\displaystyle\nu_{2}^{l}(A,B^{(1:m)})$ $\displaystyle\leq$ $\displaystyle
l\mu_{2}(A)+\frac{l}{2}\sum_{j=1}^{m}\|B^{(j)}\|_{2}^{2}$ (37)
$\displaystyle+\frac{l}{2}\sum_{j=1}^{m}(\mu_{2}(B^{(j)})+\mu_{2}(-B^{(j)}))+\frac{l(l-2)}{2}\sum_{i=1}^{m}(\mu_{2}(B^{(i)}))^{2}$

In general, for any $p$, it is possible to estimate $\nu_{p}^{l}(0,B^{(1:m)})$
in the sense of the existence of a generalized derivative of the Wiener
process such that $dW^{(i)}=\zeta^{(i)}dt,~{}\zeta^{(i)}\sim N(0,1),~{}{\bf
E}(\zeta^{(i)}\zeta^{(j)})=0$ for $i\neq j$:

$\nu_{p}^{l}(0,B^{(1:m)})=\lim_{h\to 0^{+}}\frac{{\bf
E}\|I+\sum_{j=1}^{m}B^{(j)}\Delta
W^{(j)}+{\displaystyle{\sum_{i=1}^{m}\sum_{k=1}^{m}}}B^{(i)}B^{(k)}\int_{0}^{h}\int_{0}^{s}dW^{(i)}_{u}dW^{(k)}_{s}\|_{p}^{l}-1}{h}$
$\leq\lim_{h\to
0^{+}}\bigg{(}\frac{\|I-(2h){\displaystyle{\sum_{i=1}^{m}}}\frac{1}{2}{B^{(i)}}^{2}\|_{p}^{l}-1}{(2h)}+$
$\frac{{\bf E}\|I+2\Delta W{\displaystyle{\sum_{i=1}^{m}B^{(i)}+(\Delta
W)^{2}\sum_{i=1}^{m}{B^{(i)}}^{2}+2\sum_{i=1}^{m}\sum_{j=1,~{}i\neq
j}^{m}}}B^{(i)}B^{(j)}\int_{0}^{h}\int_{0}^{s}dW_{u}^{(i)}dW_{s}^{(j)}\|^{l}_{p}-1}{(2h)}\bigg{)}$
$\leq\frac{l}{2}\mu_{p}(-\sum_{i=1}^{m}B^{(i)})+l\sum_{i=1}^{m}\|B^{(i)}\|_{p}+\frac{l}{2}\sum_{i=1}^{m}\|B^{(i)}\|^{2}_{p}+\frac{l}{\sqrt{2}}\sum_{i=1}^{m}\sum_{j=1,~{}i\neq
j}^{m}\|B^{(i)}B^{(j)}\|_{p}.$

Then we can upper bound the stochastic logarithmic norm as

$\displaystyle\nu_{p}^{l}(A,B^{(1:m)})$ $\displaystyle\leq$ $\displaystyle
l\mu_{p}(A)-\frac{l}{2}\mu_{p}(\sum_{i=1}^{m}B^{(i)})+l\sum_{i=1}^{m}\|B^{(i)}\|_{p}+\frac{l}{2}\sum_{i=1}^{m}\|B^{(i)}\|^{2}_{p}+$
(38) $\displaystyle\frac{l}{\sqrt{2}}\sum_{i=1}^{m}\sum_{j=1,~{}i\neq
j}^{m}\|B^{(i)}B^{(j)}\|_{p}.$

Similar to (29) we can compute the stochastic logarithmic norm for the multi-
channel case with

$\displaystyle\nu_{p}^{l}(A,B^{(1:m)})=l{\bf
E}\left(\mu_{p}\left(A-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}+\sum_{i=1}^{m}B^{(i)}\zeta^{(i)}\right)\right),$
(39)

and bound it as in (32) as

$\displaystyle\left|\nu_{p}^{l}(A,B^{(1:m)}\right|\leq
l\left\|A-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}\right\|_{p}+l\sum_{i=1}^{m}\left\|B^{(i)}\right\|_{p}.$
(40)

From (39) and as in Corollary 3, the stochastic logarithmic norm for the
multiplicative multiple channel noise can be bounded as

$\displaystyle
l\mu_{p}(A)-\frac{l}{2}\sum_{i=1}^{m}\left(\mu_{p}({B^{(i)}}^{2})+\mu_{p}(B^{(i)})+\mu_{p}(-B^{(i)})\right)\leq\nu_{p}^{l}(A,B^{(1:m)})$
$\displaystyle\leq
l\mu_{p}(A)+\frac{l}{2}\sum_{i=1}^{m}\left(\mu_{p}(-{B^{(i)}}^{2})+\mu_{p}(B^{(i)})+\mu_{p}(-B^{(i)})\right).$
(41)

##  13 Relationship with Pseudospectrum

We have mentioned in the introduction that the logarithmic norm as a bound on
the pseudospectrum of the stability matrix provides an estimate of the finite
time interval numerical stability of an ODE. The finite time interval
numerical stability differs from the asymptotic stability in capturing the
effect of nonnormality of the stability matrices and local stiffness that
affect the computation of the numerical integration. In SDEs with
multiplicative noise the diffusion coefficients may significantly affect this
transient numerical stability of an SDE. It may be recalled that balanced
methods have been designed [12] to overcome difficulties arising from
stiffness in both drift and diffusion. The stochastic logarithmic norm relates
to the pseudospectrum of the drift coefficients by way of diffusion
coefficients acting as perturbations and thus captures the expected transient
stability of the SDE. The estimate of stability based on the stochastic
logarithmic norm can then be used for selecting an appropriate stiff
stochastic integrator.

Let
$\|-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}+\sum_{i=1}^{m}B^{(i)}\zeta^{(i)}\|_{2}=\beta$.
From the definition of pseudospectrum [15], we may write:

$\displaystyle{\bf E}\max_{\lambda}\Re\lambda_{\beta}(A)$ $\displaystyle=$
$\displaystyle{\bf
E}\max_{\lambda}\Re{\lambda\left(A-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}+\sum_{i=1}^{m}B^{(i)}\zeta^{(i)}\right)}$
(42) $\displaystyle\leq{\bf
E}\left(\mu_{2}\left(A-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}+\sum_{i=1}^{m}B^{(i)}\zeta^{(i)}\right)\right)$
$\displaystyle~{}~{}=\frac{1}{2}\nu_{2}^{2}(A,B^{(1:m)})$

using (39). For small noise with $1\gg\beta>0$, obviously, the stochastic
logarithmic norm gives an upper bound estimate of the mean transient numerical
stability behavior of the deterministic ODE $dX_{t}=AX_{t}dt$. Denoting
$\gamma=\|\sum_{i=1}^{m}B^{(i)}\zeta^{(i)}\|_{2}$, and proceeding in a similar
fashion as in (42) one obtains

$\displaystyle{\bf
E}\max_{\lambda}\Re\lambda_{\gamma}\left(A-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}\right)$
$\displaystyle\leq$ $\displaystyle{\bf
E}\left(\mu_{2}\left(A-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}+\sum_{i=1}^{m}B^{(i)}\zeta^{(i)}\right)\right)$
(43) $\displaystyle=\frac{1}{2}\nu_{2}^{2}(A,B^{(1:m)}).$

In the above inequalities the stochastic logarithmic norm appears as an upper
bound on the mean maximum real part of the perturbed spectrum of
$A-\frac{1}{2}\sum_{i=1}^{m}{B^{(i)}}^{2}$ which is significant for the
stochastic asymptotic stability of the SDE (36).

##  14 Conclusion

This paper has extended the classical logarithmic norm to define the
stochastic logarithmic norm for the numerical stability analysis of vector Itô
stochastic differential equations with multi-dimensional multiplicative noise.
Incremental estimates of the stochastic logarithmic norm due to perturbations
and bounds with respect to logarithmic norm of the drift and diffusion
coefficient matrices have been studied. Further investigation relating the
stochastic logarithmic norm to the pseudo-spectrum and stiffness, both in
drift and diffusion, is needed since the stochastic logarithmic norm gives an
upper bound on the mean maximum real part of the pseudospectrum of a matrix
perturbed by the noise. This last property may be used in choosing stiff and
balanced numerical integrators and a detailed study for various class of
integrators in this respect remains to be done. Detailed study of application
of the stochastic logarithmic norm to the stability analysis of non-linear SDE
is necessary too.

## References

  * [1] R. Bhatia, Matrix Analysis, Springer-Verlag, 1997.
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  * [15] L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, 2005.
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# Saddle solutions for Allen-Cahn type equations involving the prescribed mean
curvature operator

Renan J. S. Isneri
Unidade Acadêmica de Matemática
Universidade Federal de Campina Grande,
58429-970, Campina Grande - PB - Brazil<EMAIL_ADDRESS>
###### Abstract.

The goal of this paper is to investigate the existence of saddle solutions for
some classes of elliptic partial differential equations of the Allen-Cahn
type, formulated as follows:

$-div\left(\frac{\nabla u}{\sqrt{1+|\nabla
u|^{2}}}\right)+A(x,y)V^{\prime}(u)=0\leavevmode\nobreak\ \leavevmode\nobreak\
\text{ in }\leavevmode\nobreak\ \leavevmode\nobreak\ \mathbb{R}^{2}.$

Here, the function $A:\mathbb{R}^{2}\to\mathbb{R}$ exhibits periodicity in all
its arguments, while $V:\mathbb{R}\to\mathbb{R}$ characterizes a double-well
symmetric potential with minima at $t=\pm\alpha$.

###### Key words and phrases:

Saddle solutions, Quasilinear elliptic equations, Mean curvature operator,
Allen-Cahn equations

###### 2020 Mathematics Subject Classification:

Primary: 34C37, 35J62, 35J93

##  1\. Introduction

Over the past 50 years, in view of several concrete real-world applications,
considerable attention has been devoted to addressing the existence and
classification of bounded solutions of the elliptic Allen-Cahn equations,
whose prototype is given by

(1.1) $-\Delta u+V^{\prime}(u)=0\quad\text{in}\quad\mathbb{R}^{N},$

where the standard model of $V:\mathbb{R}\to\mathbb{R}$ is the classical
double well Ginzburg-Landau potential

(1.2) $V(u)=\frac{1}{4}(u^{2}-1)^{2}.$

The concept of “heteroclinic solutions” has been extensively investigated in
equations of the (1.1) type, leading to the emergence of numerous
constructions documented in the literature. This type of solutions was the
starting point to provide a rich amount of differently shaped families of
solutions such as saddle, stationary layered and multibump solutions. Readers
seeking a more comprehensive understanding of these solution types are
encouraged to explore further in references such as [1, 3, 2, 4, 18, 19].

Equation (1.1) represents a reaction-diffusion type equation originating from
mathematical physics. The initial model, encompassing (1.1)-(1.2), was first
proposed in 1979 by S. Allen and J. Cahn [5] to delineate the phenomenon of
phase separation within multi-component alloy systems. In this framework, the
solution $u$ serves as a phase parameter, characterizing the material’s state
at each point, with the trivial solutions $u=-1$ and $u=+1$ corresponding to
the pure phases of the system. Since then much progress has been made on
(1.1)-(1.2), and moreover, its variations have been broadly treated in the PDE
literature in the last years. These variations are mostly of one of the two
types:

  * (1)

The Laplacian operator is replaced by more general operators;

  * (2)

(1.1) is changed to a class of more general semilinear elliptic equations
whenever (1.2) is replaced by a class of potentials with different or even
similar geometry.

Typically, both cases can occur simultaneously. We would like to cite some
works involving generalizations and variations concerning problem (1.1)-(1.2).
In [6] and [14], discussions revolve around results concerning the fractional
Laplacian operator. On the other hand, [10] and [11] delve into outcomes
associated with the $\Phi$-Laplacian operator, encompassing the classical
$p$-Laplacian. Other interesting works involving the prescribed mean curvature
operator are found in [7, 13]. Lastly, for a recent comprehensive overview of
a fourth-order equation, we refer the reader to [12].

In this work, we will show that the following class of prescribed mean
curvature equations

(1.3) $-div\left(\frac{\nabla u}{\sqrt{1+|\nabla
u|^{2}}}\right)+A(x,y)V^{\prime}(u)=0\leavevmode\nobreak\ \leavevmode\nobreak\
\text{ in }\leavevmode\nobreak\ \leavevmode\nobreak\ \mathbb{R}^{2},$

where $A(x,y)$ is $1$-periodic in all its variables, has a saddle solution
whenever the distance between the roots of the double well symmetric potential
$V$ is small. Moreover, in the particular case, when $A(x,y)$ is a positive
constant, we will show that there is an infinite number of geometrically
distinct saddle-type solutions for (1.3). Throughout this paper, the
oscillatory factor $A(x,y)$ satisfies precisely the following assumptions:

  * ($A_{1}$)

$A$ is continuous and $A(x,y)>0$ for each $(x,y)\in\mathbb{R}^{2}$;

  * ($A_{2}$)

$A(x,y)=A(-x,y)=A(x,-y)$ for all $(x,y)\in\mathbb{R}^{2}$;

  * ($A_{3}$)

$A(x,y)=A(x+1,y)=A(x,y+1)$ for any $(x,y)\in\mathbb{R}^{2}$;

  * ($A_{4}$)

$A(x,y)=A(y,x)$ for all $(x,y)\in\mathbb{R}^{2}$.

An interesting model for $A$ is described by the expression:

$A(x,y)=\cos(2\pi x)\cos(2\pi y)+c,$

where $c$ is a constant with $c>1$. From now on, let’s consider
$\mathcal{V}=\\{V_{\alpha}\\}_{\alpha>0}$ as a class of symmetric potentials
$V_{\alpha}:\mathbb{R}\to\mathbb{R}$ that satisfy the following conditions:

  * ($V_{1}$)

$V_{\alpha}(t)\geq 0$ for all $t\in\mathbb{R}$ and
$V_{\alpha}(t)=0\Leftrightarrow t=-\alpha,\alpha$ for $\alpha>0$.

  * ($V_{2}$)

$V_{\alpha}(-t)=V_{\alpha}(t)$ for any $t\in\mathbb{R}$ and
$V^{{}^{\prime\prime}}_{\alpha}(\pm\alpha)>0$.

With respect to class $\mathcal{V}$, we will also assume the following local
uniform estimate involving the root $\alpha$ of $V_{\alpha}$:

  * ($V_{3}$)

Given $\lambda>0$, there exists $C=C(\lambda)>0$ independent of
$\alpha\in(0,\lambda)$ such that

$\displaystyle\max_{|t|\in[0,\alpha]}|V^{\prime}_{\alpha}(t)|\leq
C,\quad\forall
V_{\alpha}\in\mathcal{V}\quad\text{with}\quad\alpha\in(0,\lambda).$

An important example of a family $\mathcal{V}$ of potentials $V_{\alpha}$, for
which conditions $(V_{1})$-$(V_{3})$ are satisfied is given by:

$V_{\alpha}(t)=(t^{2}-\alpha^{2})^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha>0.$

Recently, Kurganov and Rosenau in [15] reported an interesting physical
motivation for investigating transition-type solutions to Allen-Cahn type
equations involving the prescribed mean curvature, as exemplified by equation
(1.3). Motivated by [15], Bonheure, Obersnel and Omari in [13] were led to
minimization problems defined in certain function classes in
$BV_{\text{loc}}(\mathbb{R})$ to investigate the existence of a basic
heteroclinic solution, which naturally connect the stationary points $\pm 1$,
of the one-dimensional equation

(1.4)
$-\left(\frac{q^{\prime}}{\sqrt{1+(q^{\prime})^{2}}}\right)^{\prime}+a(t)V^{\prime}(q)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R},$

where $V$ a double-well potential with minima at $t=\pm 1$ and $a$ is
asymptotic at infinity to a positive periodic function with
$0<\displaystyle\operatorname{ess}\inf_{t\in\mathbb{R}}a(t)$. After, Alves and
Isneri in [7] also explored the existence of basic heteroclinic solution for
(1.4) in the case where $a\in L^{\infty}(\mathbb{R})$ is an even non-negative
function with $0<\displaystyle\inf_{t\geq M}a(t)$ for some $M>0$. In that
paper, inspired to the one introduced by Obersnel and Omari in [17], the
authors used cutting techniques on the mean curvature operator to build up a
variational framework on the usual Orlicz-Sobolev space
$W^{1,\Phi}_{\text{loc}}(\mathbb{R})$ to show that (1.4) has a solution $q$
such that

$\lim_{t\to\pm\infty}q(t)=\pm\alpha$

whenever the distance between the roots $\pm\alpha$ of the symmetric potential
$V$ is small. When $a(t)=1$, it was shown in the recent work by Alves, Isneri
and Montecchiari [9] that under certain conditions in potential $V$, there
exists $\alpha_{0}>0$ such that for each $\alpha\in(0,\alpha_{0})$ the
equation (1.4) has, up to translation, a unique twice differentiable
heteroclinic solution $q$ in $C^{1,\gamma}_{\text{loc}}(\mathbb{R})$ for some
$\gamma\in(0,1)$ satisfying the following exponential decay estimates

$0<\alpha-q(t)\leq\theta_{1}e^{-\theta_{2}t}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
0<q^{\prime}(t)\leq\beta_{1}e^{-\beta_{2}t}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }t\geq 0$

and

$0<\alpha+q(t)\leq\theta_{3}e^{\theta_{4}t}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
0<q^{\prime}(t)\leq\beta_{3}e^{\beta_{4}t}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }t\leq 0,$

for some real numbers $\theta_{i},\beta_{i}>0$. When
$V(t)=\left(t^{2}-\alpha^{2}\right)^{2}$ the exponential estimates above are
refined to

$\alpha\tanh\left(\alpha\sqrt{2}t\right)\leq
q(t)\leq\alpha\tanh\left(\frac{\alpha\sqrt{2}t}{\kappa}\right)\leavevmode\nobreak\
\text{ for }\leavevmode\nobreak\ t\geq 0,$

where $\kappa$ is a positive constant that depends on
$\|q^{\prime}\|_{L^{\infty}(\mathbb{R})}$. Heteroclinic-type solutions are
also observed in planar scenarios. For instance, in [8], Alves and Isneri show
the existence of such solutions for (1.3). These solutions, denoted by
$u(x,y)$, exhibit periodicity in $y$ with a particular asymptotic behavior:

$u(x,y)\rightarrow\alpha\text{ as }x\rightarrow-\infty\text{ and
}u(x,y)\rightarrow\beta\text{ as }x\rightarrow+\infty,\text{ uniformly in
}y\in\mathbb{R},$

where $\alpha$ and $\beta$ represent the minima of a nonsymmetric double-well
potential $V$. This phenomenon occurs under the condition that $A(x,y)$
displays oscillatory behavior with respect to the variable $y$ and satisfies
some geometric constraints on $x$, such as periodicity, asymptotic periodicity
at infinity, and asymptotically away from zero at infinity.

In this study, we show that equation (1.3) exhibits additional transition-type
solutions, including the saddle solution, which are more intricate in nature.
The key theorems of this paper are outlined below.

######  Theorem 1.1.

Assume ($A_{1}$)-($A_{4}$). Given $L>0$, there exists $\delta>0$ such that for
each $V_{\alpha}\in\mathcal{V}\cap C^{2}(\mathbb{R},\mathbb{R})$ with
$\alpha\in(0,\delta)$, the prescribed mean curvature equation

$-div\left(\frac{\nabla u}{\sqrt{1+|\nabla
u|^{2}}}\right)+A(x,y)V_{\alpha}^{\prime}(u)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ in }\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{2},$

possesses a weak solution $v_{\alpha,L}$ in
$C^{1,\gamma}_{\text{loc}}(\mathbb{R}^{2})$, for some $\gamma\in(0,1)$,
satisfying the following properties:

  * (a)

$0<v_{\alpha,L}(x,y)<\alpha$ on the fist quadrant in $\mathbb{R}^{2}$,

  * (b)

$v_{\alpha,L}(x,y)=-v_{\alpha,L}(-x,y)=-v_{\alpha,L}(x,-y)$ for all
$(x,y)\in\mathbb{R}^{2}$,

  * (c)

$v_{\alpha,L}(x,y)=v_{\alpha,L}(y,x)$ for any $(x,y)\in\mathbb{R}^{2}$,

  * (d)

$v_{\alpha,L}(x,y)\rightarrow\alpha$ as $x\rightarrow\pm\infty$ and
$y\rightarrow\pm\infty$,

  * (e)

$v_{\alpha,L}(x,y)\rightarrow-\alpha$ as $x\rightarrow\mp\infty$ and
$y\rightarrow\pm\infty$,

  * (f)

$\|\nabla v_{\alpha,L}\|_{L^{\infty}(\mathbb{R}^{2})}\leq\sqrt{L}$.

The items $(d)$ and $(e)$ of the theorem above tells us that along directions
parallel to the axes, the saddle solution $v_{\alpha,L}$ is uniformly
asymptotic to stationary solutions $\pm\alpha$. Moreover, when $A(x,y)$ is a
positive constant, we obtain an infinite number of geometrically distinct
saddle-type solutions to the equation (1.3). This observation is detailed in
the ensuing result.

######  Theorem 1.2.

Assume that $A(x,y)$ is a positive constant $b$. Then, given $L>0$, there
exists $\delta>0$ such that for each $V_{\alpha}\in\mathcal{V}\cap
C^{2}(\mathbb{R},\mathbb{R})$ with $\alpha\in(0,\delta)$ and for each $j\geq
2$, the prescribed mean curvature equation

$-div\left(\frac{\nabla u}{\sqrt{1+|\nabla
u|^{2}}}\right)+bV_{\alpha}^{\prime}(u)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ in }\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{2},$

possesses a weak solution $v_{\alpha,L,j}$ in
$C^{1,\gamma}_{\text{loc}}(\mathbb{R}^{2})$, for some $\gamma\in(0,1)$, such
that if
$\tilde{v}_{\alpha,L,j}(\rho,\theta)=v_{\alpha,L,j}(\rho\cos(\theta),\rho\sin(\theta))$,
then

  * (a)

$0<\tilde{v}_{\alpha,L,j}(\rho,\theta)<\alpha$ for any
$\theta\in[\frac{\pi}{2}-\frac{\pi}{2j},\frac{\pi}{2})$ and $\rho>0$,

  * (b)

$\tilde{v}_{\alpha,L,j}(\rho,\frac{\pi}{2}+\theta)=-\tilde{v}_{\alpha,L,j}(\rho,\frac{\pi}{2}-\theta)$
for all $(\rho,\theta)\in[0,+\infty)\times\mathbb{R}$,

  * (c)

$\tilde{v}_{\alpha,L,j}(\rho,\theta+\frac{\pi}{j})=-\tilde{v}_{\alpha,L,j}(\rho,\theta)$
for all $(\rho,\theta)\in[0,+\infty)\times\mathbb{R}$,

  * (d)

$\tilde{v}_{\alpha,L,j}(\rho,\theta)\rightarrow(-\alpha)^{k+1}$ as
$\rho\rightarrow+\infty$ whenever
$\theta\in\left(\frac{\pi}{2}+k\frac{\pi}{j},\frac{\pi}{2}+(k+1)\frac{\pi}{j}\right)$
for $k=0,\ldots,2j-1$,

  * (e)

$\|\nabla v_{\alpha,L,j}\|_{L^{\infty}(\mathbb{R}^{2})}\leq\sqrt{L}$.

In other words, the saddle-type solution $\tilde{v}_{\alpha,L,j}$ is
antisymmetric with respect to the half-line $\theta=\frac{\pi}{2}$,
$\frac{2\pi}{j}$-periodic in the angle variable and has $L^{\infty}$-norm less
than or equal to $\alpha$ with the asymptotic behavior at infinity described
in item $(d)$. Moreover, the solutions $v_{\alpha,L,j}$ described in the
theorem above are characterized by the fact that, along different directions
parallel to the end lines, they are uniformly asymptotic to $\pm\alpha$ and
such solutions may appropriately be termed “pizza solutions”.

In order to establish Theorems 1.1 and 1.2, it was necessary improve the
findings presented in the works of Alves and Isneri [10, 11], particularly
concerning saddle-type solutions, for a larger class of $N$-functions. For a
comprehensive understanding of these advancements, readers are encouraged to
refer directly to Section 2 of this work. Moreover, as far as we know,
Theorems 1.1 and 1.2 are the first results in the literature on saddle-type
solutions for some stationary Allen-Cahn-type equations involving the
prescribed mean curvature operator in the whole plane. Transition-type
solutions to equations involving the prescribed mean curvature operator is an
extremely fascinating field of mathematics and there are still many open
questions one can work on. For example, a possible extension to Theorems 1.1
and 1.2 would be to study the existence of a saddle solution for (1.3) without
requiring that the distance between the absolute minima $\pm\alpha$ of $V$ be
small. We believe that a natural approach to solve such a problem would be to
look for minima of an action functional on a convex subset of the space of
functions of bounded variation $BV_{\text{loc}}(\mathbb{R})$.

The structure of this manuscript unfolds as follows: In Section 2, we delve
into the analysis of the existence of saddle-type solutions for a specific
class of quasilinear elliptic equations that complements and extends the
investigations presented in [10, 11]. Taking into account the study in Section
2, in Section 3 we will present the proof of our main results. Lastly, in
Section 4, we will make some final comments about our study.

##  2\. Existence of saddle solutions for quasilinear equations

We will show in this section that the main results about saddle-type solutions
in works [10, 11] are extended to a larger class of $N$-functions than the
class that was presented there. We will start with a brief review of what has
been done in [11] for saddle solutions. To recapitulate, in that paper, the
authors studied the existence of transition-type solutions for the following
class of quasilinear elliptic equations given by

(2.1) $-\Delta_{\Phi}u+A(x,y)V^{\prime}(u)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ in }\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{2},$

where the conditions $(A_{1})$-$(A_{4})$ were assumed on $A$ and on $V$,
$(V_{1})$-$(V_{2})$ and

  * ($V_{4}$)

There are $\delta_{\alpha}\in(0,\alpha)$ and $w_{1},w_{2}>0$ such that

$w_{1}\Phi(|t-\alpha|)\leq V(t)\leq w_{2}\Phi(|t-\alpha|)\leavevmode\nobreak\
\leavevmode\nobreak\ \forall
t\in(\alpha-\delta_{\alpha},\alpha+\delta_{\alpha}).$

  * ($V_{5}$)

There are $\omega_{1},\omega_{2},\omega_{3},\omega_{4},\tau>0$ such that

$-\omega_{3}\phi(\omega_{4}|\alpha-t|)(\alpha-t)t\leq
V^{\prime}(t)\leq-\omega_{1}\phi(\omega_{2}|\alpha-t|)(\alpha-t)t\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t\in[0,\alpha+\tau].$

With respect to $\Phi:\mathbb{R}\to[0,+\infty)$, $\Phi$ is designated as an
$N$-function, that is, it satisfies precisely the following properties

  * (a)

$\Phi$ is continuous, convex and even,

  * (b)

$\Phi(t)=0$ if and only if $t=0$,

  * (c)

$\displaystyle\lim_{t\rightarrow 0}\dfrac{\Phi(t)}{t}=0$ and
$\displaystyle\lim_{t\rightarrow+\infty}\dfrac{\Phi(t)}{t}=+\infty$.

Specifically, $\Phi$ has the form

$\Phi(t)=\int_{0}^{|t|}\phi(s)sds,$

with $\phi\in C^{1}([0,+\infty),[0,+\infty))$ satisfying

  * ($\phi_{1}$)

$\phi(t)>0$ and $(\phi(t)t)^{\prime}>0$ for any $t>0$.

  * ($\phi_{2}$)

There are $l,m\in\mathbb{R}$ with $1<l\leq m$ such that

$l-1\leq\dfrac{(\phi(t)t)^{\prime}}{\phi(t)}\leq m-1\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
t>0.$

  * ($\phi_{3}$)

There exist constants $c_{1},c_{2},\eta>0$ and $s>1$ satisfying

$c_{1}t^{s-1}\leq\phi(t)t\leq c_{2}t^{s-1}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
t\in[0,\eta].$

  * ($\phi_{4}$)

$\phi$ is non-decreasing on $(0,+\infty)$.

Finally, $\Delta_{\Phi}$ emerges as a quasilinear operator in divergence form,
succinctly expressed as:

$\Delta_{\Phi}u=\text{div}(\phi(|\nabla u|)\nabla u).$

The conditions $(\phi_{1})$-$(\phi_{3})$ establish the groundwork for ensuring
the existence of a weak solution $u_{0}$ to (2.1) with specific properties.
Firstly, $u_{0}$ is $1$-periodic in the variable $y$, odd symmetry with
respect to $x$, and furthermore, it must satisfy

$0<u_{0}(x,y)<\alpha\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ x>0.$

Moreover, $u_{0}$ is a heteroclinic type solution, that is,

$u_{0}(x,y)\rightarrow-\alpha\text{ as }x\rightarrow-\infty\text{ and
}u_{0}(x,y)\rightarrow\alpha\text{ as }x\rightarrow+\infty\text{ uniformly in
}y\in\mathbb{R}.$

The introduction of condition $(\phi_{4})$ serves a crucial purpose, allowing
for the utilization of exponential decay estimates specifically for
$u_{0}\pm\alpha$. This condition facilitates the derivation of an inequality
crucial for further analysis, presented as

(2.2)
$\phi(|\zeta^{\prime}(x)|)\leq\phi(\omega_{2}\zeta(x))\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
x\in\mathbb{R},$

which involves the real function $\zeta:\mathbb{R}\to\mathbb{R}$ given by

(2.3)
$\zeta(x)=\delta_{\alpha}\frac{\cosh\left(a\left(x-\frac{j-\frac{1}{4}+L}{2}\right)\right)}{\cosh\left(a\frac{j-\frac{1}{4}-L}{2}\right)},$

where $\omega_{2}$, $L$ and $j$ are chosen properly and the constant $a$ is
small enough. The role of inequality (2.2) is pivotal as it permits direct
calculations leading to the desired exponential-type estimates. For a
comprehensive understanding of the intricate details and rigorous
verification, readers are directed to Lemmas 4.4, 4.5, and 4.6 in the
referenced work [11]. This study of the asymptotic behavior at infinity of
heteroclinic solutions plays a fundamental role in the search of saddle-type
solutions to (2.1), as outlined in the methodology employed in [10, 11].

Our goal now is to replace $(\phi_{4})$ with another condition on $\phi$ that
includes a larger number of examples for $\phi$, including the monotonically
non-decreasing functions. Specifically, we introduce the following assumption

  * $(\tilde{\phi}_{4})$

There exist positive constants $\kappa_{1}$ and $\kappa_{2}$ such that the
inequality

$\phi(|\zeta^{\prime}(t)|)\leq\kappa_{1}\phi(\kappa_{2}\zeta(t))\quad\text{for
all}\quad t\in\mathbb{R}$

holds whenever $a>0$ is sufficiently small in (2.3).

The reader is invited to conduct a meticulous verification of the exponential
decay estimates relative to the heteroclinic solutions as delineated in
references [10, 11], specifically observing their occurrence when substituting
$(\phi_{4})$ by $(\tilde{\phi}_{4})$. It should be noted that functions $\phi$
satisfying $(\phi_{4})$ also satisfy $(\tilde{\phi}_{4})$. However, it’s
important to observe that the converse is not true, as certain functions,
compliant with $(\tilde{\phi}_{4})$, may not necessarily satisfy $(\phi_{4})$,
thereby establishing $(\tilde{\phi}_{4})$ as a broader framework compared to
$(\phi_{4})$. In the following section, we provide an explicit example
elucidating this distinction.

Under our current assumptions on $\phi$, we provide below a result that
generalizes Theorem 1.2 of [11].

######  Theorem 2.1.

Assume ($\phi_{1}$)-($\phi_{3}$), ($\tilde{\phi}_{4}$), $V\in
C^{1}(\mathbb{R},\mathbb{R})$, ($V_{1}$)-($V_{2}$), ($V_{4}$)-($V_{5}$) and
($A_{1}$)-($A_{4}$). Then, there is $v\in
C^{1,\gamma}_{\text{loc}}(\mathbb{R}^{2})$ for some $\gamma\in(0,1)$ such that
$v$ is a weak solution of (2.1) that verifies the following:

  * (a)

$0<v(x,y)<\alpha$ on the fist quadrant in $\mathbb{R}^{2}$,

  * (b)

$v(x,y)=-v(-x,y)=-v(x,-y)$ for all $(x,y)\in\mathbb{R}^{2}$,

  * (c)

$v(x,y)=v(y,x)$ for any $(x,y)\in\mathbb{R}^{2}$,

  * (d)

$v(x,y)\rightarrow\alpha$ as $x\rightarrow\pm\infty$ and
$y\rightarrow\pm\infty$,

  * (e)

$v(x,y)\rightarrow-\alpha$ as $x\rightarrow\mp\infty$ and
$y\rightarrow\pm\infty$.

When $A(x,y)$ is a positive constant, additional conditions regarding the
geometric properties of the potential $V$ were deemed necessary within the
framework presented in [10] to establish the existence of infinitely many
geometrically distinct saddle-type solutions of equation (2.1). Specifically,
the following conditions on $V$ were explored ($V_{1}$)-($V_{2}$),
($V_{4}$)-($V_{5}$) and

  * $(V_{6})$

There exists $\delta_{0}>0$ such that $V^{\prime}$ is increasing on
$(\alpha-\delta_{0},\alpha)$.

  * $(V_{7})$

There exist $\gamma,\epsilon>0$ such that

$\tilde{\Phi}(V^{\prime}(t))\leq\gamma\Phi(|\alpha-t|),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t\in(\alpha-\epsilon,\alpha).$

Therefore, motivated by Theorem 2.1, we can follow the steps of work [10] to
derive the subsequent result.

######  Theorem 2.2.

Assume ($\phi_{1}$)-($\phi_{3}$), ($\tilde{\phi}_{4}$), ($V_{1}$)-($V_{2}$),
($V_{4}$)-($V_{7}$) and that $A(x,y)$ is a positive constant. Then, For each
$j\geq 2$, there exists $v_{j}\in C^{1,\gamma}_{\text{loc}}(\mathbb{R}^{2})$
for some $\gamma\in(0,1)$ such that $v_{j}$ is a weak solution of (2.1).
Moreover, by setting
$\tilde{v}_{j}(\rho,\theta)=v_{j}(\rho\cos(\theta),\rho\sin(\theta))$, we have
that

  * (a)

$0<\tilde{v}_{j}(\rho,\theta)<\alpha$ for any
$\theta\in[\frac{\pi}{2}-\frac{\pi}{2j},\frac{\pi}{2})$ and $\rho>0$,

  * (b)

$\tilde{v}_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde{v}_{j}(\rho,\frac{\pi}{2}-\theta)$
for all $(\rho,\theta)\in[0,+\infty)\times\mathbb{R}$,

  * (c)

$\tilde{v}_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde{v}_{j}(\rho,\theta)$ for all
$(\rho,\theta)\in[0,+\infty)\times\mathbb{R}$,

  * (d)

$\tilde{v}_{j}(\rho,\theta)\rightarrow(-\alpha)^{k+1}$ as
$\rho\rightarrow+\infty$ whenever
$\theta\in\left(\frac{\pi}{2}+k\frac{\pi}{j},\frac{\pi}{2}+(k+1)\frac{\pi}{j}\right)$
for $k=0,\ldots,2j-1$.

In fact, the conditions on $V$ in the aforementioned theorem can be refined;
namely, conditions $(V_{6})$ and $(V_{7})$ can be omitted. To illustrate this,
one can combine several arguments from the paper [11] and implement the
concept of partitioning $\mathbb{R}^{2}$ into $2j$ disjoint triangular sets,
where $j\geq 2$, as described in the construction of saddle solutions
presented in [10, Section 6]. As a result, we can present the following
theorem, with the details of its verification left to the reader.

######  Theorem 2.3.

Assume ($\phi_{1}$)-($\phi_{3}$), ($\tilde{\phi}_{4}$), ($V_{1}$)-($V_{2}$),
($V_{4}$)-($V_{5}$) and that $A(x,y)$ is a positive constant. Then, For each
$j\geq 2$, there exists $v_{j}\in C^{1,\gamma}_{\text{loc}}(\mathbb{R}^{2})$
for some $\gamma\in(0,1)$ such that $v_{j}$ is a weak solution of (2.1).
Moreover, by setting
$\tilde{v}_{j}(\rho,\theta)=v_{j}(\rho\cos(\theta),\rho\sin(\theta))$, we have
that

  * (a)

$0<\tilde{v}_{j}(\rho,\theta)<\alpha$ for any
$\theta\in[\frac{\pi}{2}-\frac{\pi}{2j},\frac{\pi}{2})$ and $\rho>0$,

  * (b)

$\tilde{v}_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde{v}_{j}(\rho,\frac{\pi}{2}-\theta)$
for all $(\rho,\theta)\in[0,+\infty)\times\mathbb{R}$,

  * (c)

$\tilde{v}_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde{v}_{j}(\rho,\theta)$ for all
$(\rho,\theta)\in[0,+\infty)\times\mathbb{R}$,

  * (d)

$\tilde{v}_{j}(\rho,\theta)\rightarrow(-\alpha)^{k+1}$ as
$\rho\rightarrow+\infty$ whenever
$\theta\in\left(\frac{\pi}{2}+k\frac{\pi}{j},\frac{\pi}{2}+(k+1)\frac{\pi}{j}\right)$
for $k=0,\ldots,2j-1$.

##  3\. Saddle solution of the prescribed mean curvature equation

Our primary goal in this section is to establish the existence of a saddle
solution to the prescribed mean curvature equation (1.3) under the condition
that the global minima of the potential $V$ are sufficiently close. For this
reason, we will index the number $\alpha$ in $V$, denoting it as $V_{\alpha}$,
and throughout this section, we will assume that $V_{\alpha}\in\mathcal{V}\cap
C^{2}(\mathbb{R},\mathbb{R})$. To achieve this, we first study an auxiliary
problem of the form (2.1), proving the existence of a saddle solution within
this context. The idea here is similar to those presented in work [7], so the
exposition will be brief. For our purposes, we employ a truncation scheme for
the prescribed mean curvature operator, defined for each $L>0$ as follows:

$\varphi_{L}(t)=\left\\{\begin{array}[]{lll}\dfrac{1}{\sqrt{1+t}},&\mbox{if}\quad
t\in[0,L],\\\ x_{L}(t-L-1)^{2}+y_{L},&\mbox{if}\quad t\in[L,L+1],\\\
y_{L},&\mbox{if}\quad t\in[L+1,+\infty),\end{array}\right.$

where the numbers $x_{L}$ and $y_{L}$ are expressed by

$x_{L}=\frac{\sqrt{1+L}}{4(1+L)^{2}}\leavevmode\nobreak\ \text{ and
}\leavevmode\nobreak\ y_{L}=(4L+3)x_{L}.$

As a consequence, for each fixed $L>0$, we get the following quasilinear
equation

(3.1) $-\Delta_{\Phi_{L}}u+A(x,y)V^{\prime}_{\alpha}(u)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{2},$

where $\Phi_{L}:\mathbb{R}\rightarrow[0,+\infty)$ is an $N$-function of the
form

$\Phi_{L}(t)=\int_{0}^{|t|}\phi_{L}(s)sds\leavevmode\nobreak\
\leavevmode\nobreak\ \text{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\phi_{L}(t)=\varphi_{L}(t^{2}).$

The following two lemmas will be frequently referenced throughout this work
and they can be found in [7, 9].

######  Lemma 3.1.

For each $L>0$, the functions $\phi_{L}$ and $\Phi_{L}$ satisfy the following
properties:

  * (a)

$\phi_{L}$ is $C^{1}$.

  * (b)

$y_{L}\leq\phi_{L}(t)\leq 1$ for each $t\geq 0$.

  * (c)

$\dfrac{y_{L}t^{2}}{2}\leq\Phi_{L}(t)\leq\dfrac{t^{2}}{2}$ for any $t\geq 0$.

  * (d)

$\Phi_{L}$ is a convex function.

  * (e)

$\left(\phi_{L}\left(t\right)t\right)^{\prime}>0$ for all $t>0$.

######  Lemma 3.2.

For each $L>0$, it turns out that $\phi_{L}$ satisfies
$(\phi_{1})$-$(\phi_{3})$.

A direct check shows that $\phi_{L}$ exhibits a monotonically non-increasing
behavior over the interval $(0,+\infty)$. Consequently, $\phi_{L}$ does not
satisfy $(\phi_{4})$. However, each $\phi_{L}$ satisfies $(\tilde{\phi}_{4})$,
as the following lemma says.

######  Lemma 3.3.

For each $L>0$, the function $\phi_{L}$ satisfies $(\tilde{\phi}_{4})$.

###### Proof.

Indeed, according to Lemma 3.1-$(b)$ we have that

$y_{L}\leq\phi_{L}(t)\leq 1\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ t\geq 0.$

In particular,

$y_{L}\leq\phi_{L}(|\zeta^{\prime}(t)|),\phi_{L}(\zeta(t))\leq
1\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\
\leavevmode\nobreak\ t\in\mathbb{R},$

and therefore,

$\phi_{L}(|\zeta^{\prime}(t)|)\leq\frac{1}{y_{L}}\phi_{L}(\zeta(t))\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
t\in\mathbb{R}.$

Finally, since there is no restriction for the constant $a>0$ in (2.3), the
lemma follows. ∎

The evidence provided in Lemma 3.3 leads to the following observation.

######  Remark 1.

By the argument of the previous lemma, we can conclude that any function
$\phi:[0,+\infty)\to[0,+\infty)$ satisfying

$c_{1}\leq\phi(t)\leq c_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ t\geq 0$

for some constants $c_{1},c_{2}>0$ satisfies condition $(\tilde{\phi}_{4})$.

To find saddle solutions for the equation (1.3), our approach involves
examining the existence of these solutions within the framework of the
auxiliary problem (3.1). To begin with, note that the condition
$V_{\alpha}\in\mathcal{V}\cap C^{2}(\mathbb{R},\mathbb{R})$ implies that
$V_{\alpha}$ satisfies the conditions $(V_{4})$ with respect to $\Phi_{L}$ and
$(V_{5})$ concerning $\phi_{L}$. Indeed, by $(V_{2})$ it is possible to find
numbers $\rho_{1},\rho_{2},d_{1},d_{2},d_{3}>0$ such that

$|V_{\alpha}^{\prime}(t)|\leq d_{1}|t-\alpha|\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
t\in[\alpha-\rho_{1},\alpha+\rho_{1}]$

and

$d_{2}|t-\alpha|^{2}\leq V_{\alpha}(t)\leq
d_{3}|t-\alpha|^{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\
t\in(\alpha-\rho_{2},\alpha+\rho_{2}),$

from which it follows by Lemma 3.1 that

$|V_{\alpha}^{\prime}(t)|\leq\frac{d_{1}}{y_{L}}\phi_{L}(|t-\alpha|)|t-\alpha|\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
t\in[\alpha-\rho_{1},\alpha+\rho_{1}]$

and

$2d_{2}\Phi_{L}(|t-\alpha|)\leq
V_{\alpha}(t)\leq\frac{2d_{3}}{y_{L}}\Phi_{L}(|t-\alpha|)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
t\in(\alpha-\rho_{2},\alpha+\rho_{2}).$

Hence, we may use Theorem 2.1 to obtain, for each $L>0$, a saddle solution
denoted as $v_{\alpha,L}:\mathbb{R}^{2}\to\mathbb{R}$ to equation (3.1). This
solution, $v_{\alpha,L}$, is a weak solution to (3.1) belonging to
$C^{1,\gamma}_{\text{loc}}(\mathbb{R}^{2})$, for some $\gamma\in(0,1)$,
satisfying

$|v_{\alpha,L}(x,y)|\leq\alpha\leavevmode\nobreak\ \leavevmode\nobreak\ \text{
for all }\leavevmode\nobreak\ \leavevmode\nobreak\ (x,y)\in\mathbb{R}^{2}.$

Moreover, it preserves the same sign pattern as $xy$, odd in both the
variables $x$ and $y$, symmetric with respect to the diagonals $y=\pm x$ and
presents the asymptotic behavior described in items $(d)$ and $(e)$ of Theorem
2.1. We are now going to use this information together with condition
$(V_{3})$ to establish an estimate involving the functions $v_{\alpha,L}$.

######  Lemma 3.4.

There exists $\delta>0$ such that for each $V_{\alpha}\in\mathcal{V}$ with
$\alpha\in(0,\delta)$, the function $v_{\alpha,L}$ satisfies following the
estimate below

(3.2) $\|v_{\alpha,L}\|_{C^{1}(B_{1}(z))}<\sqrt{L},$

where $B_{1}(z)$ denotes any open ball in $\mathbb{R}^{2}$ with radius 1.

###### Proof.

To establish this lemma, we employ a proof by contradiction. Let us assume the
contrary. Then for each $n\in\mathbb{N}$ there exists a potential
$V_{\alpha_{n},\beta_{n}}\in\mathcal{V}\cap C^{2}(\mathbb{R},\mathbb{R})$ with
$\alpha\in\left(0,\frac{1}{n}\right)$ and
$z_{n}=(z_{n,1},z_{n,2})\in\mathbb{R}^{2}$ such that the function
$v_{\alpha_{n},L}$ satisfies

(3.3)
$\|v_{\alpha_{n},L}\|_{C^{1}(B_{1}(z_{n}))}\geq\sqrt{L}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
n\in\mathbb{N}.$

For the purpose of our analysis, we will study the regularity of specific
solutions to the following elliptic equation:

(3.4) $-\Delta_{\Phi_{L}}u+B_{n}(x,y)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ in }\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{2},$

where the scalar measurable function $B_{n}:\mathbb{R}^{2}\to\mathbb{R}$ is
defined as:

$B_{n}(x,y)=A(x+z_{n,1},y+z_{n,2})V_{\alpha_{n}}^{\prime}(v_{\alpha_{n},L}(x,y)).$

Under condition $(V_{3})$, $B_{n}$ satisfies the subsequent estimate:

(3.5) $|B_{n}(x,y)|\leq
C\|A\|_{L^{\infty}(\mathbb{R}^{2})}\leavevmode\nobreak\ \leavevmode\nobreak\
\forall(x,y)\in\mathbb{R}^{2}\text{ and }\forall n\geq n_{0},$

for some $n_{0}\in\mathbb{N}$ and $C>0$, where $C$ is independent of $n$.
Indeed, it’s worth noting that the condition $\alpha_{n}\to 0$ as
$n\to+\infty$ implies the existence of $n_{0}\in\mathbb{N}$ such that
$\alpha_{n}\in(0,1)$ for all $n\geq n_{0}$. Consequently, by $(V_{3})$, it
follows that there exists a constant $C=C(1)>0$ independent of
$\alpha\in(0,1)$ satisfying

$\displaystyle\max_{|t|\in[0,\alpha]}|V^{\prime}_{\alpha}(t)|\leq
C,\quad\forall V_{\alpha}\in\mathcal{V}\quad\text{with}\quad\alpha\in(0,1).$

Now, since $\alpha_{n}\in(0,1)$ for all $n\geq n_{0}$ and

$|v_{\alpha_{n},L}(x,y)|\leq\alpha_{n}\quad\forall(x,y)\in\mathbb{R}^{2}\quad\forall
n\in\mathbb{N},$

we obtain

$|V^{\prime}_{\alpha_{n}}(v_{\alpha_{n},L}(x,y))|\leq C,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall(x,y)\in\mathbb{R}^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\forall n\geq n_{0},$

which is sufficient to ensure inequality (3.5). A particularly relevant weak
solution to (3.4) is given by:

$u_{n}(x,y)=v_{\alpha_{n},L}(x+z_{n,1},y+z_{n,2})\leavevmode\nobreak\ \text{
for }\leavevmode\nobreak\ (x,y)\in\mathbb{R}^{2}.$

Now, utilizing inequality (3.5), we can employ elliptic regularity estimates
on $u_{n}$, as developed by Lieberman in [16, Theorem 1.7]. This yields
$u_{n}\in C^{1,\gamma_{0}}_{\text{loc}}(\mathbb{R}^{2})$ for some
$\gamma_{0}\in(0,1)$, and

$\|u_{n}\|_{C^{1,\gamma_{0}}_{\text{loc}}(\mathbb{R}^{2})}\leq
R\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\
\leavevmode\nobreak\ n\geq n_{0},$

where $R$ is a positive constant independent of $n$. Consequently, from the
above estimate, employing Arzelà-Ascoli’s theorem, we deduce the existence of
$u\in C^{1}(B_{1}(0))$ and a subsequence of $(u_{n})$, still denoted by
$(u_{n})$, such that

(3.6) $u_{n}\to u\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ in
}\leavevmode\nobreak\ \leavevmode\nobreak\ C^{1}(B_{1}(0)).$

Since $\|v_{\alpha_{n},L}\|_{L^{\infty}(\mathbb{R}^{2})}$ goes to $0$ as
$\alpha_{n}\to 0$,

$\|u_{n}\|_{L^{\infty}(\mathbb{R}^{2})}\to 0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\ \leavevmode\nobreak\
n\to+\infty,$

thus from the convergence (3.6), we naturally infer that $u=0$ on $B_{1}(0)$.
Therefore, there exists $\tilde{n}\in\mathbb{N}$ such that:

$\|u_{n}\|_{C^{1}\left(B_{1}(0)\right)}<\sqrt{L}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
n\geq\tilde{n},$

which, arising from the definition of $u_{n}$, implies:

$\|v_{\alpha_{n},L}\|_{C^{1}(B_{1}(z_{n}))}<\sqrt{L}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
n\geq\tilde{n},$

which contradicts (3.3). Hence, the proof of the lemma is complete. ∎

The estimate (3.2) will play a key role in the theorem below, facilitating the
investigation of the existence of saddle solutions for the prescribed mean
curvature equation (1.3) whenever the roots $\pm\alpha$ of $V$ are
sufficiently close.

The proof of Theorem 1.1.

We claim that for each fixed $L>0$ there exists $\delta>0$ such that for each
$V_{\alpha}\in\mathcal{V}\cap C^{2}(\mathbb{R},\mathbb{R})$ with
$\alpha\in(0,\delta)$, the function $v_{\alpha,L}$ satisfies the estimate
below

(3.7) $\|\nabla v_{\alpha,L}\|_{L^{\infty}(\mathbb{R}^{2})}\leq\sqrt{L}.$

In fact, given any $(x,y)\in\mathbb{R}^{2}$ we can choose a point
$z\in\mathbb{R}^{2}$ such that $(x,y)\in B_{1}(z)$. By virtue of Lemma 3.4,
there exists $\delta>0$ such that if $V_{\alpha}\in\mathcal{V}\cap
C^{2}(\mathbb{R},\mathbb{R})$ and $\alpha\in(0,\delta)$ $\alpha\in(0,\delta)$
one gets

$\|v_{\alpha,L}\|_{C^{1}(B_{1}(z))}<\sqrt{L}.$

In particular,

$\|\nabla v_{\alpha,L}\|_{L^{\infty}(B_{1}(z))}\leq\sqrt{L}.$

Therefore, the claim (3.7) is valid from the arbitrariness of
$(x,y)\in\mathbb{R}^{2}$, and thus, the estimate (3.7) guarantees that
$v_{\alpha,L}$ is a saddle solution to the equation (1.3) satisfying items
$(a)$ to $(f)$, thanks to the study developed in the auxiliary problem (3.1).
This completes the proof. $\square$

In the case where $A(x,y)$ is a positive constant, the existence of infinitely
many saddle-type solutions for the prescribed mean curvature equation (1.3)
can be established. These solutions, bearing a geometry reminiscent of pizzas,
may be named as ”pizza solutions”. Employing the strategy outlined in proving
Theorem 1.1, we can prove the Theorem 1.2, which is a multiplicity result
concerning saddle-type solutions to (1.3).

##  4\. Final remarks

In this last section, we aim to highlight that although we have made
advancements over the results presented in works [10, 11], particularly
concerning saddle solutions to quasilinear elliptic equations represented by

(4.1) $-\Delta_{\Phi}u+A(x,y)V^{\prime}(u)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ in }\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{2}$

by imposing condition $(\tilde{\phi}_{4})$ on $\phi$ instead of $(\phi_{4})$,
a significant challenge remains unresolved. Specifically, the case
$\phi(t)=t^{p-2}$ with $p\in(1,2)$, associated with the $N$-function

$\Phi(t)=\frac{|t|^{p}}{p}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\ p\in(1,2),$

presents an open problem because, in this case, $\phi$ does not satisfy
$(\tilde{\phi}_{4})$. This fact arises from the observation that through a
direct calculation, we have

$\frac{\phi(|\zeta^{\prime}(t)|)}{\phi(\kappa_{2}\zeta(t))}\to+\infty\leavevmode\nobreak\
\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\ \leavevmode\nobreak\
|t|\to\frac{j-\frac{1}{4}+L}{2},$

for any positive constant $\kappa_{2}$. This divergence is evident from the
behavior of

$\left|\frac{\sinh(t)}{\cosh(t)}\right|=\left|\frac{e^{2t}-1}{e^{2t}+1}\right|,$

which tends to zero as $|t|\to 0$, resulting in

$\left|\frac{\sinh(t)}{\cosh(t)}\right|^{p-2}\to+\infty\leavevmode\nobreak\
\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\ \leavevmode\nobreak\ |t|\to
0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\
\leavevmode\nobreak\ p\in(1,2).$

Finally, it’s worth noting that exploring exponential decay type estimates to
determine saddle solutions in cases where (4.1) involves the $p$-Laplacian
operator with $1<p<2$ is potentially difficult and interesting.

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