# Traveling vortex pairs for 2D Boussinesq equations

Daomin Cao, Shanfa Lai, Guolin qin  Institute of Applied Mathematics, Chinese
Academy of Sciences, Beijing 100190, and University of Chinese Academy of
Sciences, Beijing 100049, P.R. China<EMAIL_ADDRESS>Institute of Applied
Mathematics, Chinese Academy of Sciences, Beijing 100190, and University of
Chinese Academy of Sciences, Beijing 100049, P.R. China<EMAIL_ADDRESS>Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190,
and University of Chinese Academy of Sciences, Beijing 100049, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the existence and asymptotic properties of the
traveling vortex pairs for the two-dimensional inviscid incompressible
Boussinesq equations. We construct a family of traveling vorticity pairs,
which constitutes the de-singularization of a pair of point vortices with
equal intensity but opposite sign. Using the improved vorticity method, we
also give limiting position of the supports of vorticities.

_Keywords_. Boussinesq equations; Traveling vortex pair solutions; Variational
method; Asymptotic behavior

##  1\. Introduction and main results

In this paper we are concerned with the two-dimensional Boussinesq equations

$\displaystyle\partial_{t}\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}+\nabla
P=\nu\Delta\mathbf{v}+\rho\mathbf{e}_{2},$ (1.1)
$\displaystyle\partial_{t}\rho+(\mathbf{v}\cdot\nabla)\rho=\vartheta\Delta\rho,$
(1.2) $\displaystyle\nabla\cdot\mathbf{v}=0.$ (1.3)

where $\mathbf{v}=(v_{1},v_{2})$ is the velocity field, $P$ is the scalar
pressure, and $\rho$ is the density of the fluid. In the above equation,
$\nu\geq 0$ is the viscosity coefficient, $\vartheta\geq 0$ is the thermal
conductivity coefficient, and $\rho\mathbf{e}_{2}$ is the buoyancy force.

The Boussinesq equations has very important mathematical implications,
especially in two dimensions. In fact, the two-dimensional Boussinesq system
retains some key properties of the three-dimensional Euler equations and the
three-dimensional Navier-Stokes equations, such as the vortex stretching
mechanism. There are many studies of the global/local well-posedness for this
system, e.g., [4, 12, 13, 21, 31, 37] and the references therein. The
regularity and finite time blow up of solutions are well-known open problems
for two-dimensional inviscid Boussinesq system $(\nu\equiv\vartheta\equiv 0)$.
Recently, T. Hmidi and K. Choi et al. have done some work on such issues for
reference [14, 25, 26, 27, 24].

In this paper, we consider hydrostatic equilibrium of the 2D inviscid
Boussinesq system $(\nu\equiv\vartheta\equiv 0)$ in the whole plane. Although
there are many results for the above unsteady cases, there seems to be little
work for the relative equilibrium cases. Our main aim is to construct a family
of traveling vortex pairs for two-dimensional Boussinesq equations with
general vorticity and density functions, which constitute the
desingularization of a pair of point vortices with equal strength but opposite
sign.

Now we deduce the vorticity-density formulation of the system
$(\nu\equiv\vartheta\equiv 0)$. Let
$\omega=\nabla\times\mathbf{v}=\partial_{1}v_{2}-\partial_{2}v_{1}$ be the
corresponding vorticity of the flow. By virtue of Eq. (1.3), there exists a
Stokes stream function $\Psi(x,t)$ such that

$\mathbf{v}=\nabla^{\perp}\Psi:=\left(\partial_{x_{2}}\Psi,-\partial_{x_{1}}\Psi\right).$

Note that by the definitions of $\Psi$ and $\omega$ we have
$-\Delta\Psi=\omega$. Thus the velocity field $\mathbf{v}$ can be recovered
from the vorticity function $\omega$ through the Biot-Savart law

$\mathbf{v}=\nabla^{\perp}(-\Delta)^{-1}\omega=-\frac{1}{2\pi}\frac{x^{\perp}}{|x|^{2}}*\omega.$

Taking the curl of (1.1), we get the following vorticity-density formulation:

$\displaystyle\partial_{t}{\omega}+(\mathbf{v}\cdot\nabla)\omega=\partial_{{1}}\rho,$
(1.4) $\displaystyle\partial_{t}\rho+(\mathbf{v}\cdot\nabla)\rho=0,$ (1.5)
$\displaystyle\mathbf{v}=\nabla^{\perp}\Psi,\Psi=(-\Delta)^{-1}\omega.$ (1.6)

Vorticity localization in nature consists of a finite number of small regions,
outside which vorticity either is zero or rapidly decreases to zero.
Mathematically we can consider the singular solution of the Boussinesq system

$\omega=\sum_{i=1}^{N}\kappa_{i}\delta\left(x-\Im_{i}(t)\right).$ (1.7)

The corresponding velocity can be described as

$\mathbf{v}(x)=-\sum_{i=1}^{N}\frac{\kappa_{i}}{2\pi}\frac{\left(x-\Im_{i}(t)\right)^{\perp}}{\left|x-\Im_{i}(t)\right|^{2}},$

and the position of the vortices $\Im_{i}:\mathbb{R}\rightarrow\mathbb{R}^{2}$
satisfies

$\frac{d}{dt}\Im_{i}(t)=-\sum_{j=1,j\neq
i}^{N}\frac{\kappa_{j}}{2\pi}\frac{\left(\Im_{i}(t)-\Im_{j}(t)\right)^{\perp}}{\left|\Im_{i}(t)-\Im_{j}(t)\right|^{2}}.$
(1.8)

Here $\delta(x)$ denotes the standard Dirac mass at the origin, and the
constants $\kappa_{i}$ are called the intensities of the vortices
$\Im_{i}(t)$. The Hamiltonian system (1.8) is now generally referred to as the
point vortex model or the Kirchhoff-Routh model (see [32]).

In this paper, we study solutions with sharply concentrated vorticity around
the points $\Im_{1}(t),\ldots,\Im_{N}(t)$ which are idealized as regular
solutions that approximate the point vortex model. Finding these regular
solutions is a classical vortex desingularization problem, see [15, 7, 8, 9]
and the references therein. Under certain appropriate conditions, the system
(1.8) allows a simple traveling solution of the form

$\Im_{i}(t)=b_{i}+Wt\mathbf{e}_{1},\quad i=1,\ldots,N,$ (1.9)

where $b_{1},\ldots,b_{N}$ are points in $\mathbb{R}^{2}$, the constant $W>0$
is the speed, and $\mathbf{e}_{1}=(1,0)$. In this paper, we focus on traveling
solutions of the form (1.9). We consider the traveling vortex pair, namely the
solution with $N=2$, and

$b_{1}=-b_{2}=r^{*}\mathbf{e}_{2},\quad\mathbf{e}_{2}=(0,1),\quad\kappa_{1}=-\kappa_{2}=\kappa,\quad
W>0$ (1.10)

where $W,\kappa>0$ are two given constants and $r^{*}>0$ will be suitably
chosen later. It is easy to check that such a vortex pair is indeed a solution
of system (1.8).

To study vortex desingularization problem of the traveling vortex pair, we can
look for traveling solutions to (1.4)-(1.6) by requiring that

$\omega(x,t)=\zeta(x_{1}-Wt,x_{2}),\ \rho(x,t)=\eta(x_{1}-Wt,x_{2}),$ (1.11)

for some profile functions $(\zeta,\eta)$ defined on $\mathbb{R}^{2}$. Let
$\Psi\left(x_{1},x_{2},t\right)=\phi\left(x_{1}-Wt,x_{2}\right)$ with
$\phi=(-\Delta)^{-1}\zeta$. Thus the system (1.4)-(1.6) is reduced to a
stationary problem

$\displaystyle\nabla^{\perp}(\phi-Wx_{2})\cdot\nabla\zeta=\partial_{1}\eta,$
(1.12) $\displaystyle\nabla^{\perp}(\phi-Wx_{2})\cdot\nabla\eta=0,$ (1.13)
$\displaystyle\phi=(-\Delta)^{-1}\zeta.$ (1.14)

We consider axisymmetric flows about $x_{1}$, and limit our attention
henceforth to the half-plane $\Pi=\left\\{x\in\mathbb{R}^{2}\mid
x_{2}>0\right\\}$. Specifically, we may assume

$\zeta\left(x_{1},x_{2}\right)=-\zeta\left(x_{1},-x_{2}\right),$

so that $\phi\left(x_{1},x_{2}\right)=-\phi\left(x_{1},-x_{2}\right)$. Suppose
that there exists some function $b(s)$ satisfying

$\eta=b(\phi-Wx_{2}).$ (1.15)

Taking (1.15) into (1.12), we get

$\nabla^{\perp}(\phi-Wx_{2})\cdot\nabla(\zeta+x_{2}b^{\prime}(\phi-
Wx_{2}))=0.$

Thus, if there exists some function $a(s)$ such that

$\zeta=a(\phi-Wx_{2})-x_{2}b^{\prime}(\phi-Wx_{2})$

then (1.12) is satisfied automatically. Thus, we are led to the following
nonlinear elliptic equation:

$-\Delta\phi=\lambda(f(\phi-Wx_{2}-\mu)+x_{2}g(\phi-Wx_{2}-\mu))\ \text{in}\
\Pi,$ (1.16)

For technical reasons, we make the following assumptions on $f$ and $g$.

  1. (H1)

$f\in C(\mathbb{R}\setminus\\{0\\})$ and $g\in C(\mathbb{R})$ are both
nonnegative and nondecreasing.

Set

$i(x_{2},t)=f(t)+x_{2}g(t)$

and $I(x_{2},t)=\int_{0}^{t}i(x_{2},s)ds$. Let $J(x_{2},\cdot)$, the modified
conjugate function to $I(x_{2},\cdot)$, be defined by

$J(x_{2},s)=\sup_{t}[st-I(x_{2},t)]\ \text{ if }s\geq 0;\ J(x_{2},s)=0\text{
if }s<0.$ (1.17)

We further require the following:

  1. (H2)

For each $x_{2}>0,i(x_{2},t)=0$ if $t\leq 0$, and $i(x_{2},t)$ is strictly
increasing (w.r.t. t) in $[0,+\infty)$.

  2. (H3)

For each $d>0$, there exist $\delta_{0}\in(0,1)$ and $\delta_{1}>0$ such that

$I(x_{2},t)\leq\delta_{0}i(x_{2},t)t+\delta_{1}i(x_{2},t),\quad\forall\
t>0,\forall\ 0<x_{2}\leq d$

  3. (H4)

For all $\tau>0$, there holds

$\lim_{t\rightarrow+\infty}i(x_{2},t)e^{-\tau t}=0,\quad\forall\ x_{2}>0.$

######  Remarks 1.1.

We give some remarks on the above assumptions.

  1. (1)

Assumption (H2) excludes the cases where $g\equiv 0$ and $f$ is a step
function. This case has been studied in [7, 10, 22].

  2. (2)

Assumption (H3) is of Ambrosetti-Rabinowitz type, cf. condition $(p5)$ in [1].
This assumption implies that $i(r,t)\rightarrow+\infty\text{ as
}t\rightarrow+\infty$ (see[34]). Note that $i(x_{2},\cdot)$ and
$\partial_{s}J(x_{2},\cdot)$ are inverse graphs (see [36]), one can check that
(H3) is actually equivalent to

  3. $(H3)^{\prime}$

:

$J(x_{2},s)\geq(1-\delta_{0})\partial_{s}J(x_{2},s)s-\delta_{1}s,\forall\
t>0,\forall\ 0<x_{2}\leq d.$

  4. (3)

The assumption (H4) relaxes the condition (1.6) in [3]. In such problems,
proper assumptions on the growth at infinite are usually necessary.

It is not difficult to find some $(f,g)$ that satisfies hypothesis (H1)-(H4).
For example, Lamb found an explicit solution of (1.16) in [29], considering
the case $f(s)=s_{+}^{p}$, $g(s)\equiv 0$ and $p=1$. The explicit solution
obtained in [29] is often called the Lamb dipole or Chaplygin-Lamb dipole, see
[33]. For the type above, de Valeriola and Van Schaftingen in [16] investigate
the case where $p>1$. Turkington also gives a desingularization solution in
[39] for $f(s)=\alpha\chi_{\\{s>0\\}},g(s)=s_{+}$, which can be taken as
$p=0$.

Exact solutions of (1.16) are known only in special cases. Besides those exact
solutions, the existence of steady vortex pairs has been rigorously proved.
Turkington studied in [38] when $f$ is the Heaviside function and constructed
a family of desingularization solutions for (1.16). Burton [5] and Badiani [2]
studied the existence of steady vortex pairs with a prescribed distribution of
the vorticity. Inspired by Burton’s work, Elcrat and Miller [19, 20] extended
Turkington’s result to the context of rearrangements of vorticity and to more
general exterior plane domains. Recently, Hmidi and Mateu [23] established the
existence of corotating and counter-rotating vortex pairs of simply connected
patches for the active scalar equations by using the contour dynamics
equations. For numerical studies, see, e.g., [17, 28].

As mentioned above, for some special nonlinearities $f$, there are already
some desingularization results. To the best of our knowledge, no related paper
has established the solution of the two-dimensional Boussinesq equations to
the desingularization of traveling vortex pairs of the general vorticity and
density functions. Our goal is to generalize these special cases to more
general nonlinear terms and thus enrich the known solutions to pairs of steady
vortices.

Now, we turn to state the main results. To do this, let’s introduce some known
facts. Let

$J_{G}(s)=\sup_{t}\left[st-G(t)\right]\ \text{ if }s\geq 0;\ J_{G}(s)=0\text{
if }s<0,$

where $G(t)=\int_{0}^{t}g(s)ds$. We introduce the limiting profile
$V^{\kappa}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ defined as the unique
radially symmetric solution of the problem

$\left\\{\begin{array}[]{l}-\Delta V^{\kappa}=g(V^{\kappa}),\quad
x\in\mathbb{R}^{2},\\\
\int_{\mathbb{R}^{2}}g(V^{\kappa})dx=\kappa.\end{array}\right.$ (1.18)

According to the Pohozaev identity, $V^{\kappa}$ has the following properties

$\int_{B_{r}(0)}G(V^{\kappa}(z))dz=\frac{\pi}{2}((V^{\kappa})^{\prime}|_{\partial
B_{r}(0)})^{2}r^{2},\ \ \
\int_{B_{r}(0)}g(V^{\kappa})dz=2\pi(-(V^{\kappa})^{\prime}|_{\partial
B_{r}(0)})r,$ (1.19)

where $supp(V_{+}^{\kappa})=B_{r}(0)$. By the above formula, we can calculate
directly

$\begin{split}\mathcal{C}_{g,\kappa}:&=\int_{\mathbb{R}^{2}}\Big{(}V^{\kappa}(z)g(V^{\kappa}(z))-J_{G}\Big{(}g(V^{\kappa}(z))\Big{)}\Big{)}dz\\\
&=\int_{\mathbb{R}^{2}}G(V^{\kappa}(z))dz=\frac{\kappa^{2}}{8\pi}.\end{split}$
(1.20)

And from this we find an interesting fact that $\mathcal{C}_{g,\kappa}$ does
not change with the choice of $g$.

We similarly define $J_{F}$ as a convex conjugate of $F$, where
$F(t)=\int_{0}^{t}f(s)ds$, and introduce the function
$U^{\kappa}:\mathbb{R}^{2}\to\mathbb{R}$ defined as the unique radially
symmetric solution of the problem

$\left\\{\begin{array}[]{l}-\Delta U^{\kappa}=f(U^{\kappa}),\quad
x\in\mathbb{R}^{2}\\\
\int_{\mathbb{R}^{2}}f(U^{\kappa})dx=\kappa.\end{array}\right.$ (1.21)

Let

$\mathcal{W}(t)=\frac{\kappa^{2}}{4\pi}\log\frac{1}{2t}+{\kappa
Wt}+\int_{\mathbb{R}^{2}}\frac{1+\alpha r^{*}}{1+\alpha
t}J_{F}\left(\frac{1+\alpha t}{1+\alpha r^{*}}f(U^{\kappa}(z))\right)dz,\ \
t>0.$ (1.22)

where $r^{*}$ is the unique solution of (2.2) on $(0,\infty)$.

Our first main result concerns the special case where $g=\alpha f$ for some
$\alpha\geq 0$ and read as follows.

######  Theorem 1.2.

Let $\alpha>0$ be given, $W$ and $\kappa>0$ satisfy
$W\geq\frac{(2-\alpha)_{+}^{2}\kappa}{32\pi}$. Consider the traveling vortex
pair given by (1.10). Suppose that $(f,\alpha f)$ satisfies (H1)-(H4). Then
for $\lambda>0$ large there is a traveling solution
$\left(\omega^{\lambda},\rho^{\lambda},\Psi^{\lambda}\right)$ to the
Boussinesq system (1.4)-(1.6) given by (1.11) and

$\begin{split}\omega^{\lambda}(x,t)=\zeta^{\lambda}\left(x_{1}-Wt,x_{2}\right)&=\lambda(1+\alpha
x_{2})f\left(\Psi^{\lambda}\left(x_{1}-Wt,x_{2}\right)-Wx_{2}-\mu^{\lambda}\right),\\\
\rho^{\lambda}(x,t)=\eta^{\lambda}(x_{1}-Wt,x_{2})&=-\lambda\alpha
F(\Psi^{\lambda}(x_{1}-Wt,x_{2})-Wx_{2}-\mu^{\lambda}),\end{split}$

for all $\left(x_{1},x_{2}\right)\in\Pi$, which satisfies

$\begin{split}&supp(\zeta^{\lambda})\cap\Pi=supp(\eta^{\lambda})\cap\Pi\subset
B_{L\lambda^{-\frac{1}{2}}}(x^{\lambda}),\ \text{for some}\
x^{\lambda}=(0,x_{2}^{\lambda})\in\Pi,\\\
&\zeta^{\lambda}(x)\rightharpoonup\kappa\delta\left(x-b_{1}\right)-\kappa\delta\left(x-b_{2}\right),\
\ b_{1}=-b_{2}=r^{*}\mathbf{e}_{2},\ \ \text{ as
}\lambda\rightarrow+\infty,\end{split}$ (1.23)

where the convergence is in the sense of measures and $L>0$ is a constant
independent of $\lambda$. Moreover, one has

$\zeta^{\lambda}\left(x_{1},x_{2}\right)=-\zeta^{\lambda}\left(x_{1},-x_{2}\right)=\zeta^{\lambda}\left(-x_{1},x_{2}\right),\
\ \int_{\Pi}\zeta^{\lambda}dx=\kappa,$ (1.24)

and

$\frac{d\mathcal{W}(x_{2}^{\lambda})}{dt}\to 0,\ \ x_{2}^{\lambda}\to r^{*},\
\ \text{as}\ \lambda\to\infty.$

######  Remarks 1.3.

For $\alpha=0$, the above theorem is reduced to a desingularization result for
the Euler equation. One can see from the definition of $r^{*}$ that the
limiting position is dependent not only on $W$ and $\kappa$ but also on the
vorticity-density function $(f,g)$. This is a special phenomenon of the
Boussinesq system which does not occur for the Euler equation.

Recently, D. Cao, J. Wan and W. Zhan [11] has obtained a result of
desingularization for the case of $f(s)=g(s)=s_{+}^{p}$ in a bounded domain.
Using the above theorem, we can verify that such special $(f,g)$ satisfies all
assumptions (H1)-(H4) and hence as a corollary, we can easily get the
existence of traveling vortex pairs with the same density function $(f,g)$ as
[11].

Our next two results are concerned with the case where $f$ and $g$ are both
general functions but one of them is vanishing as $\lambda\to+\infty$.

######  Theorem 1.4.

Suppose that $f(0^{+})=0$, $f(s)>0$ for $s>0$ and $(f,\delta_{\lambda}g)$
satisfies (H1)-(H4) with the constants $\delta_{0}$ and $\delta_{1}$
independent of $\lambda$ in (H3), where $\delta_{\lambda}>0\ \text{and}\
\delta_{\lambda}\to 0\ \text{as}\ \lambda\to\infty$. Then for $\lambda>0$
large there is a traveling solution
$\left(\omega^{\lambda},\rho^{\lambda},\Psi^{\lambda}\right)$ to the
Boussinesq system (1.4)-(1.6) given by (1.11) and

$\begin{split}\zeta^{\lambda}\left(x_{1}-Wt,x_{2}\right)&=\lambda\Big{(}f\left(\Psi^{\lambda}\left(x_{1}-Wt,x_{2}\right)-Wx_{2}-\mu^{\lambda}\right)\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \
+x_{2}\delta_{\lambda}g\left(\Psi^{\lambda}\left(x_{1}-Wt,x_{2}\right)-Wx_{2}-\mu^{\lambda}\right)\Big{)},\\\
\eta^{\lambda}(x_{1}-Wt,x_{2})&=-\lambda\delta_{\lambda}G(\Psi^{\lambda}(x_{1}-Wt,x_{2})-Wx_{2}-\mu^{\lambda}),\end{split}$

for all $\left(x_{1},x_{2}\right)\in\Pi$, which satisfies (1.24) and

$\begin{split}&supp(\zeta^{\lambda})\cap\Pi=supp(\eta^{\lambda})\cap\Pi\subset
B_{L\lambda^{-\frac{1}{2}}}(x^{\lambda}),\ \text{for some}\
x^{\lambda}=(0,x_{2}^{\lambda})\in\Pi,\\\
&\zeta^{\lambda}(x)\rightharpoonup\kappa\delta\left(x-b_{1}\right)-\kappa\delta\left(x-b_{2}\right),\
\ b_{1}=-b_{2}=r_{1}^{*}\mathbf{e}_{2},\ \ \text{ as
}\lambda\rightarrow+\infty.\end{split}$ (1.25)

where the convergence is in the sense of measures and $L>0$ is a constant
independent of $\lambda$. Moreover, one has

$\frac{d\mathcal{W}_{1}(x_{2}^{\lambda})}{dt}\to 0,\ x_{2}^{\lambda}\to
r_{1}^{*},\ \ \text{as}\ \lambda\to\infty.$

where $r_{1}^{*}=\frac{\kappa}{4\pi W}$ and $\mathcal{W}_{1}$ be defined as
follows

$\mathcal{W}_{1}(t):=\frac{\kappa^{2}}{4\pi}\log\frac{1}{2t}+{\kappa Wt}.$

######  Theorem 1.5.

Suppose that $W>0$ and $\kappa>0$. Let $g\in C^{1}(\mathbb{R}^{2})$,
$g^{\prime}(s)>0$ for $s>0$ and $(\delta_{\lambda}f,g)$ satisfies (H1)-(H4)
with the constants $\delta_{0}$ and $\delta_{1}$ independent of $\lambda$ in
(H3), where $\delta_{\lambda}>0\ \text{and}\ \delta_{\lambda}\to 0\ \text{as}\
\lambda\to\infty$. Then for $\lambda>0$ large there is a traveling solution
$\left(\omega^{\lambda},\rho^{\lambda},\Psi^{\lambda}\right)$ to the
Boussinesq system (1.4)-(1.6) given by (1.11) and

$\begin{split}\zeta^{\lambda}\left(x_{1}-Wt,x_{2}\right)&=\lambda\Big{(}\delta_{\lambda}f\left(\Psi^{\lambda}\left(x_{1}-Wt,x_{2}\right)-Wx_{2}-\mu^{\lambda}\right)\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \
+x_{2}g\left(\Psi^{\lambda}\left(x_{1}-Wt,x_{2}\right)-Wx_{2}-\mu^{\lambda}\right)\Big{)},\\\
\eta^{\lambda}(x_{1}-Wt,x_{2})&=-\lambda
G(\Psi^{\lambda}(x_{1}-Wt,x_{2})-Wx_{2}-\mu^{\lambda}),\end{split}$

for all $\left(x_{1},x_{2}\right)\in\Pi$, which satisfies (1.24) and

$\begin{split}&supp(\zeta^{\lambda})\cap\Pi=supp(\eta^{\lambda})\cap\Pi\subset
B_{L\lambda^{-\frac{1}{2}}}(x^{\lambda}),\ \text{for some}\
x^{\lambda}=(0,x_{2}^{\lambda})\in\Pi,\\\
&\zeta^{\lambda}(x)\rightharpoonup\kappa\delta\left(x-b_{1}\right)-\kappa\delta\left(x-b_{2}\right),\
\ b_{1}=-b_{2}=r_{2}^{*}\mathbf{e}_{2},\ \ \text{ as
}\lambda\rightarrow+\infty.\end{split}$ (1.26)

where the convergence is in the sense of measures and $L>0$ is a constant
independent of $\lambda$. Moreover, one has

$\frac{d\mathcal{W}_{2}(x_{2}^{\lambda})}{dt}\to 0,\ x_{2}^{\lambda}\to
r_{2}^{*},\ \ \text{as}\ \lambda\to\infty.$

where $r_{2}^{*}=\frac{\kappa}{8\pi W}$ and $\mathcal{W}_{2}$ is defined as

$\mathcal{W}_{2}(t)=\frac{\kappa^{2}}{4\pi}\log\frac{1}{2t}+{\kappa
Wt}+\frac{r_{2}^{*}}{t}\int_{\mathbb{R}^{2}}J_{G}\Big{(}\frac{t}{r_{2}^{*}}g(V^{\kappa}(z))\Big{)}dz.$
(1.27)

######  Remarks 1.6.

What we want to emphasize is that it’s not a necessary condition for $g\in
C^{1}(\mathbb{R}^{2})$. For example, for some special functions
$g(t)=1_{\\{t>0\\}}$, $\kappa=\pi$, we can calculate
$\int_{\mathbb{R}^{2}}G(V^{\kappa}(z))dz=\frac{\kappa^{2}}{8\pi}$, but for $s$
greater than $1$, we have $J_{G}(s)=+\infty$, so the integral in
$C_{g,\kappa}$ may diverge. However, we also believe that this type of
characteristic function can be obtained by the method with some modifications
in this paper.

##  2\. Proof of Theorem 1.2

In this section, we will give the proof for Theorem 1.2. Let $G(\cdot,\cdot)$
be the Green function for $-\Delta$ in $\Pi$ with zero Dirichlet data, namely,

$G(x,y)=\frac{1}{2\pi}\log\frac{|\bar{x}-y|}{|x-y|},$

where $\bar{x}=(x_{1},-x_{2})$ denotes reflection of $x=(x_{1},x_{2})$ in the
$x_{1}$-axis. Define the Green’s operator $\mathcal{G}$ as follows

$\mathcal{G}\zeta(x)=\int_{\Pi}G(x,y)\zeta(y)dy.$

Let $J$ be the conjugate function to $I$ defined by (1.17), and $J_{F}$ be
defined as

$J_{F}(s)=\sup_{t\in\mathbb{R}}\left[st-F(t)\right]\ \text{ if }s\geq 0;\
J_{F}(s)=0\text{ if }s<0,$

where $F(t)=\int_{0}^{t}f(s)ds$. Then $J$ has the well-known properties

$J(x_{2},s)=i(x_{2},t)t-I(x_{2},t),\ \
\partial_{ss}J(x_{2},s)=[\partial_{t}i(x_{2},t)]^{-1},$ (2.1)

with $s=i(x_{2},t)$ or, equivalently, $t=\partial_{s}J(x_{2},s)$. Similarly,
$J_{F}$ has the same property. Moreover, when $g=\alpha f$, it holds

$\displaystyle J(x_{2},s)$
$\displaystyle=\sup_{t\in\mathbb{R}}(st-I(x_{2},t))=(1+\alpha
x_{2})\sup_{t\in\mathbb{R}}\Big{(}\frac{s}{1+\alpha x_{2}}t-F(t)\Big{)}$
$\displaystyle=(1+\alpha x_{2})J_{F}\Big{(}\frac{s}{1+\alpha x_{2}}\Big{)}.$

###  2.1. Variational problem

In this paper, we consider this problem by using an improved vorticity method.

For fixed $W>0$ and $\kappa>0$, we consider the energy functional given by

$\mathcal{E}_{\varepsilon}(\zeta)=\frac{1}{2}\int_{\Pi}{\zeta\mathcal{G}\zeta}dx-{W}\int_{\Pi}x_{2}\zeta
dx-\frac{1}{\varepsilon^{2}}\int_{\Pi}J(x_{2},\varepsilon^{2}\zeta)dx.$

Let

$D=\\{x\in\Pi~{}|~{}-1<x_{1}<1,\ r^{*}/2<x_{2}<2r^{*}\\},$

where $r^{*}>0$ is uniquely determined by the following

$-\frac{\kappa^{2}}{4\pi r^{*}}+\kappa W+\frac{\alpha}{1+\alpha
r^{*}}\frac{\kappa^{2}}{8\pi}=0,$ (2.2)

and

$\mathcal{A}_{\varepsilon,\Lambda}=\\{\zeta\in
L^{\infty}(\Pi)~{}|~{}0\leq\zeta\leq\frac{\Lambda}{\varepsilon^{2}}~{}\text{a.e.},\int_{\Pi}\zeta
dx\leq\kappa,~{}supp(\zeta)\subseteq D\\},$

where $\varepsilon>0$ is a small parameter, $f(0^{+})=\lim_{t\to 0^{+}}f(t)$
and $\Lambda>\max\\{1,f(0^{+})\\}$ is an appropriate constant (whose value
will be determined later, see Lemma 2.8 below). We will seek maximizers of
$\mathcal{E}_{\varepsilon}$ relative to $\mathcal{A}_{\varepsilon,\Lambda}$.

Let ${\zeta}^{*}$ be the Steiner symmetrization of $\zeta$ with respect to the
line $x_{1}=0$ in $\Pi$ (see Appendix I of [5, 35]).

For the existence of maximizers for $\mathcal{E}_{\varepsilon}$ over
$\mathcal{A}_{\varepsilon,\Lambda}$ we have

######  Lemma 2.1.

There exists
$\zeta^{\varepsilon,\Lambda}=(\zeta^{\varepsilon,\Lambda})^{*}\in\mathcal{A}_{\varepsilon,\Lambda}$
such that

$\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon,\Lambda})=\max_{\tilde{\zeta}\in\mathcal{A}_{\varepsilon,\Lambda}}\mathcal{E}_{\varepsilon}(\tilde{\zeta})<+\infty.$

###### Proof.

It is easy to verify that $\mathcal{E}_{\epsilon}$ is bounded from above over
$\mathcal{A}_{\varepsilon,\Lambda}$. We may take a sequence
$\\{\zeta_{k}\\}\subset\mathcal{A}_{\varepsilon,\Lambda}$ such that as
$k\to+\infty$

$\begin{split}\mathcal{E}_{\varepsilon}(\zeta_{k})&\to\sup\\{\mathcal{E}_{\varepsilon}(\tilde{\zeta})~{}|~{}\tilde{\zeta}\in\mathcal{A}_{\varepsilon,\Lambda}\\},\\\
\zeta_{k}&\to\zeta\in L^{\infty}({D})~{}~{}\text{weakly-star}.\end{split}$

Clearly $\zeta\in\mathcal{A}_{\varepsilon,\Lambda}$. Using the standard
arguments (see [5]), we may assume that $\zeta_{k}=(\zeta_{k})^{*}$, and hence
$\zeta=\zeta^{*}$. Since $G(\cdot,\cdot)\in L^{1}(D\times D)$, we have

$\lim_{k\to+\infty}\int_{D}{\zeta_{k}\mathcal{G}\zeta_{k}}dx=\int_{D}{\zeta\mathcal{G}\zeta}dx.$

On the other hand, we have the lower semicontinuity for each of the rest terms
in $\mathcal{E}_{\varepsilon}$, namely,

$\begin{split}\liminf_{k\to+\infty}\int_{D}x_{2}\zeta_{k}dx&\geq\int_{D}x_{2}\zeta
dx,\\\
\liminf_{k\to+\infty}\int_{D}J(x_{2},\varepsilon^{2}\zeta_{k})dx&\geq\int_{D}J(x_{2},\varepsilon^{2}\zeta)dx.\end{split}$

Consequently, we conclude that
$\mathcal{E}_{\varepsilon}(\zeta)=\lim_{k\to+\infty}\mathcal{E}_{\varepsilon}(\zeta_{k})=\sup_{\mathcal{A}_{\varepsilon,\Lambda}}\mathcal{E}_{\varepsilon}$,
with $\zeta\in\mathcal{A}_{\varepsilon,\Lambda}$, which completes the proof. ∎

Since $\zeta^{\varepsilon,\Lambda}\in L^{\infty}(\Pi)$, it follows that
$\mathcal{G}\zeta^{\varepsilon,\Lambda}\in C^{1}_{\text{loc}}(\Pi)$. Moreover,
by $\zeta^{{\varepsilon,\Lambda}}=(\zeta^{\varepsilon,\Lambda})^{*}$ we can
conclude that $\mathcal{G}\zeta^{\varepsilon,\Lambda}$ is symmetric decreasing
in $x_{1}$ and $\partial_{x_{1}}\mathcal{G}\zeta^{\varepsilon,\Lambda}(x)<0$
if $\zeta^{\varepsilon,\Lambda}\not\equiv 0$ and $x_{1}>0$. Therefore, every
level set of $\mathcal{G}\zeta^{{\varepsilon,\Lambda}}-{Wx_{2}}$ has measure
zero by the implicit function theorem.

######  Lemma 2.2.

Let $\zeta^{{\varepsilon,\Lambda}}$ be a maximizer as in Lemma 2.1, then there
exists a Lagrange multiplier $\mu^{{\varepsilon,\Lambda}}\geq 0$ such that

$\zeta^{{\varepsilon,\Lambda}}=\frac{1}{\varepsilon^{2}}i\left(x_{2},\psi^{{\varepsilon,\Lambda}}\right)\chi_{A_{\varepsilon,\varLambda}}+\frac{\varLambda}{\varepsilon^{2}}\chi_{B_{\varepsilon,\varLambda}}\
\ a.e.\ \text{in}\ D,$ (2.3)

where

$\psi^{{\varepsilon,\Lambda}}=\mathcal{G}\zeta^{{\varepsilon,\Lambda}}-{Wx_{2}}-\mu^{{\varepsilon,\Lambda}},$
(2.4)

and

$A_{\varepsilon,\varLambda}:=\\{(x_{1},x_{2})\in\Pi~{}|~{}0<\psi^{{\varepsilon,\Lambda}}<\partial_{s}J(x_{2},\varLambda)\\},$
$B_{\varepsilon,\varLambda}:=\\{(x_{1},x_{2})\in\Pi~{}|~{}\psi^{{\varepsilon,\Lambda}}\geq\partial_{s}J(x_{2},\varLambda)\\}.$

Moreover, it holds $\int_{D}\zeta^{\varepsilon,\Lambda}dx=\kappa$ provided
$\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon,\Lambda})>0$ and every
$\mu^{\varepsilon,\Lambda}$ satisfying (2.3) is positive.

###### Proof.

Note that we may assume $\zeta^{\varepsilon,\Lambda}\not\equiv 0$, otherwise
the conclusion is obtained by letting $\mu^{\varepsilon,\Lambda}=0$. We
consider a family of variations of $\zeta^{\varepsilon,\Lambda}$

$\zeta_{(t)}=\zeta^{\varepsilon,\Lambda}+t(\tilde{\zeta}-\zeta^{\varepsilon,\Lambda}),\
\ \ t\in[0,1],$

defined for arbitrary $\tilde{\zeta}\in\mathcal{A}_{\varepsilon,\Lambda}$.
Since $\zeta^{\varepsilon,\Lambda}$ is a maximizer, we have

$\begin{split}0&\geq\frac{d}{dt}\mathcal{E}_{\varepsilon}(\zeta_{(t)})|_{t=0^{+}}\\\
&=\int_{D}(\tilde{\zeta}-\zeta^{\varepsilon,\Lambda})\left[\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}}-\partial_{s}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon,\Lambda})\right]dx.\end{split}$

This implies that for any $\tilde{\zeta}\in\mathcal{A}_{\varepsilon,\Lambda}$,
there holds

$\begin{split}\int_{D}\zeta^{\varepsilon,\Lambda}&\left[\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}}-\partial_{s}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon,\Lambda})\right]dx\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \
\geq\int_{D}\tilde{\zeta}\left[\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}}-\partial_{s}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon,\Lambda})\right]dx.\end{split}$

By an adapted bathtub principle (see Lieb-Loss [30], §1.14), we obtain that
for any point in $D$, it holds

$\displaystyle\begin{cases}~{}\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}}-\mu^{\varepsilon,\Lambda}\geq&\partial_{s}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon,\Lambda})\
\ \ \ \ \mbox{whenever}\
\zeta^{\varepsilon,\Lambda}=\frac{\Lambda}{\varepsilon^{2}},\\\
~{}\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}}-\mu^{\varepsilon,\Lambda}=&\partial_{s}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon,\Lambda})\
\ \ \ \ \mbox{whenever}\
0<\zeta^{\varepsilon,\Lambda}<\frac{\Lambda}{\varepsilon^{2}},\\\
~{}\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}}-\mu^{\varepsilon,\Lambda}\leq&\partial_{s}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon,\Lambda})\
\ \ \ \ \mbox{whenever}\ \zeta^{\varepsilon,\Lambda}=0,\end{cases}$ (2.5)

for some $\mu^{\varepsilon,\Lambda}\geq 0$. It follows that
$\\{0<\zeta^{\varepsilon,\Lambda}\leq
f(0^{+})\varepsilon^{-2}\\}\subseteq\\{\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}}=\mu^{\varepsilon,\Lambda}\\}$.
Since
$-\Delta(\mathcal{G}\zeta^{\varepsilon,\Lambda}-{Wx_{2}})=\zeta^{\varepsilon,\Lambda}$
almost everywhere in $D$, we conclude that
$\left|\\{0<\zeta^{\varepsilon,\Lambda}\leq
f(0^{+})\varepsilon^{-2}\\}\right|=0$. Recall that for $t>0$ and $s>f(0^{+})$,
there holds

$t=\partial_{s}J(x_{2},s)\ \ \ \text{if and only if}\ \ \ s=i(x_{2},t).$

Now the stated form $\eqref{2-1}$ follows from (2.5) immediately. Moreover,
when $\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon,\Lambda})>0$ and
$\mu^{\varepsilon,\Lambda}>0$, by the bathtub principle [30], we have
$\int_{D}\zeta^{\varepsilon,\Lambda}d\nu=\kappa$. The proof is thus complete.
∎

###  2.2. Asymptotic behavior of the maximizers

In the following, we study the asymptotic behavior of
$\zeta^{\varepsilon,\Lambda}$ when $\varepsilon\to 0^{+}$. In the sequel, we
will use the same $C$ to denote positive constants independent of
$\varepsilon$.

To begin with, we give a lower bound of the energy.

######  Lemma 2.3.

We have

$\mathcal{E}_{\varepsilon}(\zeta^{{\varepsilon,\Lambda}})\geq\frac{\kappa^{2}}{4\pi}\log\frac{1}{\varepsilon}-C.$

###### Proof.

The key idea is to select a suitable test function. Let $x_{0}\in D$ be a
fixed point and

$\tilde{\zeta}_{1}^{\varepsilon}:=\frac{1}{\varepsilon^{2}}\chi_{{}_{B_{\varepsilon\sqrt{{\kappa}/{\pi}}}(x_{0})}},$

where $\chi_{{}_{A}}$ denotes the characteristic function of a set $A$. It is
clear that
$\tilde{\zeta}_{1}^{\varepsilon}\in\mathcal{A}_{\varepsilon,\Lambda}$ if
$\varepsilon$ is sufficiently small. By a simple calculation, we get

$\begin{split}\mathcal{E}(\tilde{\zeta}_{1}^{\varepsilon})\geq\frac{\kappa^{2}}{4\pi}\log\frac{1}{\varepsilon}-C.\end{split}$

Since $\zeta^{\varepsilon,\Lambda}$ is a maximizer, we have
$\mathcal{E}(\zeta^{\varepsilon,\Lambda})\geq\mathcal{E}(\tilde{\zeta}_{1}^{\varepsilon})$
and the proof is thus complete. ∎

We now turn to estimate the Lagrange multiplier $\mu^{\varepsilon,\Lambda}$.

######  Lemma 2.4.

There holds

$\displaystyle\mu^{\varepsilon,\Lambda}$
$\displaystyle\geq\frac{2\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon,\Lambda})}{\kappa}+\frac{W}{\kappa}\int_{D}x_{2}\zeta^{\varepsilon,\Lambda}dx-|2\delta_{0}-1|\kappa^{-1}\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx$
$\displaystyle\ \ \ \ \
-(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\right)^{\frac{1}{2}}\right).$

###### Proof.

For convenience, let us abbreviate
$(\zeta^{\varepsilon,\Lambda},\psi^{\varepsilon,\Lambda},\mu^{\varepsilon,\Lambda})$
to $(\zeta,\psi,\mu)$ here. Recall that from $(H3)^{\prime}$

$J(x_{2},s)\geq(1-\delta_{0})\partial_{s}J(x_{2},s)s-\delta_{1}s,\ \ \
\forall~{}t>0,\ \forall~{}0<x_{2}\leq 2r^{*}.$

By (2.4) and $(H3)^{\prime}$, we then have

$\displaystyle 2\mathcal{E}_{\varepsilon}(\zeta)$
$\displaystyle=\int_{D}{\zeta\mathcal{G}\zeta}dx-{{2W}}\int_{D}x_{2}\zeta
dx-\frac{2}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle=\int_{D}{\zeta\left(\mathcal{G}\zeta-{W}x_{2}-\mu\right)}dx-\frac{2}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ -{W}\int_{D}x_{2}\zeta
dx+\mu\int_{D}\zeta dx$ $\displaystyle\leq\int_{D}\zeta\psi
dx-2(1-\delta_{0})\int_{D}\partial_{s}J(x_{2},\varepsilon^{2}\zeta)\zeta
dx+2\delta_{1}\int_{D}\zeta dx$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
-{{W}}\int_{D}x_{2}\zeta dx+\mu\int_{D}\zeta dx.$

Thus we get

$2\mathcal{E}_{\varepsilon}(\zeta)\leq\int_{D}\zeta\psi
dx-2(1-\delta_{0})\int_{D}\partial_{s}J(x_{2},\varepsilon^{2}\zeta)\zeta
d\nu-{{W}}\int_{D}x_{2}\zeta d\nu+\mu\kappa+2\delta_{1}\kappa.$ (2.6)

If we take $\psi_{+}\in H_{0}^{1}(\Pi)$ as a test function, we then obtain

$\int_{\Pi}{|\nabla\psi_{+}|^{2}}dx=\int_{D}\zeta\psi dx.$ (2.7)

Let $U:=\left(\psi-\partial_{s}J(x_{2},\Lambda)\right)_{+}$. Recalling (2.5),
by Hölder’s inequality and Sobolev’s inequality, we have

$\displaystyle\int_{D}\zeta\psi dx$
$\displaystyle=\int_{D}\zeta\left(\psi-\partial_{s}J(x_{2},\Lambda)\right)_{+}dx+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle\leq\frac{\Lambda}{\varepsilon^{2}}|\\{\zeta=\Lambda\varepsilon^{-2}\\}|^{\frac{1}{2}}\left(\int_{D}U^{2}dx\right)^{\frac{1}{2}}+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle\leq\frac{C\Lambda}{\varepsilon^{2}}|\\{\zeta=\Lambda\varepsilon^{-2}\\}|^{\frac{1}{2}}\left(\int_{D}U^{2}dx\right)^{\frac{1}{2}}+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle\leq\frac{C\Lambda}{\varepsilon^{2}}|\\{\zeta=\Lambda\varepsilon^{-2}\\}|^{\frac{1}{2}}\left(\int_{D}|\nabla
U|dx+\int_{D}|U|dx\right)+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle\leq\frac{C\Lambda}{\varepsilon^{2}}|\\{\zeta=\Lambda\varepsilon^{-2}\\}|^{\frac{1}{2}}\left(\int_{\\{\zeta=\Lambda\varepsilon^{-2}\\}}|\nabla\psi|dx+\int_{\\{\zeta=\Lambda\varepsilon^{-2}\\}}|\nabla\partial_{s}J(x_{2},\Lambda)|dx\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{C\Lambda}{\varepsilon^{2}}|\\{\zeta=\Lambda\varepsilon^{-2}\\}|^{\frac{1}{2}}\int_{\\{\zeta=\Lambda\varepsilon^{-2}\\}}\psi
dx+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle\leq\frac{C\Lambda}{\varepsilon^{2}}|\\{\zeta=\Lambda\varepsilon^{-2}\\}|\left(\int_{D}{|\nabla\psi|^{2}}dx\right)^{\frac{1}{2}}+o_{\Lambda\varepsilon^{-2}}(1)\int_{D}\zeta\psi
dx+o_{\Lambda\varepsilon^{-2}}(1)+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx$
$\displaystyle\leq
C\left(\int_{D}{|\nabla\psi_{+}|^{2}}dx\right)^{\frac{1}{2}}+o_{\Lambda\varepsilon^{-2}}(1)\int_{D}\zeta\psi
dx+o_{\Lambda\varepsilon^{-2}}(1)+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx.$

Combining (2.7) and above estimate, we get

$\int_{D}\zeta\psi dx\leq C\left(\int_{D}\zeta\psi
dx\right)^{\frac{1}{2}}+o_{\Lambda\varepsilon^{-2}}(1)\int_{D}\zeta\psi
dx+o_{\Lambda\varepsilon^{-2}}(1)+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx,$

which implies

$\int_{D}\zeta\psi
dx\leq(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx\right)^{\frac{1}{2}}\right)+\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx.$
(2.8)

Combining (2.6) and (2.8), we obtain

$\begin{split}2\mathcal{E}_{\varepsilon}(\zeta)\leq&(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\zeta\partial_{s}J(x_{2},\varepsilon^{2}\zeta)dx\right)^{\frac{1}{2}}\right)+|2\delta_{0}-1|\int_{D}\partial_{s}J(x_{2},\varepsilon^{2}\zeta)\zeta
dx\\\ &-{{W}}\int_{D}x_{2}\zeta dx+\mu\kappa+2\delta_{1}\kappa,\end{split}$

which clearly implies the desired result. The proof is thus complete. ∎

Combining Lemmas 2.3 and 2.4, we immediately get the following estimate.

######  Lemma 2.5.

$\begin{split}\mu^{\varepsilon,\Lambda}&\geq\frac{\kappa}{2\pi}\log{\frac{1}{\varepsilon}}+\frac{W}{\kappa}\int_{D}x_{2}\zeta^{\varepsilon,\Lambda}dx-|2\delta_{0}-1|\kappa^{-1}\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\\\
&\ \ \ \ \
-(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\right)^{\frac{1}{2}}\right).\end{split}$

It is easy to see that
$\partial_{s}J(x_{2},\Lambda)\leq\partial_{s}J(2r^{*},\Lambda)$ for $x_{2}\leq
2r^{*}$. Hence
$\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\leq C^{*}$.
As a consequence of Lemmas 2.2, 2.3 and 2.5, we have

######  Corollary 2.6.

For each fixed $\Lambda$, there holds
$\int_{D}\zeta^{\varepsilon,\Lambda}dx=\kappa$ when $\varepsilon$ is
sufficiently small.

The following lemma shows that $\psi^{\varepsilon,\Lambda}$ has a priori upper
bound with respect to $\Lambda$.

######  Lemma 2.7.

Let $\Lambda$ be fixed. Then for all sufficiently small $\varepsilon$, we have

$\displaystyle\psi^{\varepsilon,\Lambda}(x)\leq$ $\displaystyle\
|2\delta_{0}-1|\kappa^{-1}\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}d\nu$
$\displaystyle+(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}d\nu\right)^{\frac{1}{2}}\right)+C\log\Lambda,\
\ \ \ a.e.\ \text{in}\ D.$

###### Proof.

For any $x=(x_{1},x_{2})\in supp\,(\zeta^{\varepsilon,\Lambda})$, by Lemma 2.2
, we have

$\begin{split}\psi^{\varepsilon,\Lambda}(x)&=\mathcal{G}\zeta^{\varepsilon,\Lambda}(x)-{Wx_{2}}-\mu^{\varepsilon,\Lambda}\\\
&=\int_{D}G(x,y)\zeta^{\varepsilon,\Lambda}(y)dy-
Wx_{2}-\mu^{\varepsilon,\Lambda}\\\
&\leq\frac{1}{2\pi}\int_{D}\log{\frac{1}{|x-y|}}\zeta^{\varepsilon,\Lambda}(y)dy-
Wx_{2}-\mu^{\varepsilon,\Lambda}+C\\\
&\leq\frac{1}{2\pi}\frac{\varLambda}{\varepsilon^{2}}\int_{B_{\varepsilon\sqrt{\kappa/\varLambda\pi}}(x)}\log\frac{1}{|x-y|}\zeta^{\varepsilon,\varLambda}(y)dy-
Wx_{2}-\mu^{\varepsilon,\Lambda}+C\\\
&\leq\frac{\kappa}{2\pi}\log\frac{1}{\varepsilon}+\frac{\kappa}{4\pi}\log{\varLambda}-Wx_{2}-\mu^{\varepsilon,\Lambda}+C\\\
&\leq\frac{\kappa}{2\pi}\log\frac{1}{\varepsilon}-\mu^{\varepsilon,\Lambda}+C(1+\log{\varLambda}),\\\
\end{split}$

where the positive number $C$ does not depend on $\varepsilon$ and $\Lambda$.

On the other hand, by Lemma 2.5 we have

$\begin{split}\mu^{\varepsilon,\Lambda}&\geq\frac{\kappa}{2\pi}\log{\frac{1}{\varepsilon}}+\frac{W}{\kappa}\int_{D}x_{2}\zeta^{\varepsilon,\Lambda}dx-|2\delta_{0}-1|\kappa^{-1}\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\\\
&\ \ \ \ \
-(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\right)^{\frac{1}{2}}\right).\end{split}$

Thus we have

$\begin{split}\psi^{\varepsilon,\Lambda}(x)\leq&\
|2\delta_{0}-1|\kappa^{-1}\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\\\
&+(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\right)^{\frac{1}{2}}\right)+C\log\Lambda,\end{split}$

and the proof is thus complete. ∎

Now we can choose a suitable $\Lambda$ such that the patch part in (2.3) will
vanish when $\varepsilon$ is suffciently small.

######  Lemma 2.8.

There exists $\Lambda_{0}>\max\\{1,f(0^{+})\\}$ such that

$\zeta^{\varepsilon,\Lambda_{0}}=\frac{1}{\varepsilon^{2}}i(x_{2},\psi^{\varepsilon,\Lambda_{0}}),\
\ \ \ \ a.e.\ \text{in}\ D,$

provided $\varepsilon>0$ is sufficiently small.

###### Proof.

Notice that on the patch part
$\\{\zeta^{\varepsilon,\Lambda}={\Lambda\varepsilon^{-2}}\\}$, we have

$\psi^{\varepsilon,\Lambda}(x_{1},x_{2})\geq\partial_{s}J(x_{2},\Lambda)=\partial_{s}J(r^{*},\Lambda)+o_{\Lambda\varepsilon^{-2}}(1).$
(2.9)

On the other hand, by Lemma 2.7, we have

$\begin{split}\psi^{\varepsilon,\Lambda}&\leq|2\delta_{0}-1|\kappa^{-1}\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\\\
&\ \ \ \ \ \
+(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\int_{D}\partial_{s}J(x_{2},\Lambda)\zeta^{\varepsilon,\Lambda}dx\right)^{\frac{1}{2}}\right)+C\log\Lambda\\\
&\leq|2\delta_{0}-1|\partial_{s}J(r^{*},\Lambda)+(C+o_{\Lambda\varepsilon^{-2}}(1))\left(1+\left(\partial_{s}J(r^{*},\Lambda)\right)^{\frac{1}{2}}\right)+C\log\Lambda,\
\ a.e.\ \text{in}\ D.\end{split}$ (2.10)

By assumption $(H4)$, we see that for all $\tau>0$

$\lim_{s\to+\infty}\left(\partial_{s}J(r^{*},s)-\tau\log s\right)=+\infty.$

Hence we can choose $\Lambda_{0}>\max\\{1,f(0^{+})\\}$ such that

$(1-|2\delta_{0}-1|)\partial_{s}J(r^{*},\Lambda_{0})-(C+1)\left(1+\left(\partial_{s}J(r^{*},\Lambda_{0})\right)^{\frac{1}{2}}\right)-C\log\Lambda_{0}>1.$
(2.11)

From (2.9),(2.10) and (2.11), we conclude that
$|\\{\zeta^{\varepsilon,\Lambda_{0}}={\Lambda_{0}\varepsilon^{-2}}\\}|=0$ when
$\varepsilon$ is sufficiently small, which completes the proof. ∎

For simplicity, we will abbreviate
$(\zeta^{\varepsilon,\Lambda_{0}},\psi^{\varepsilon,\Lambda_{0}},\mu^{\varepsilon,\Lambda_{0}})$
to $(\zeta^{\varepsilon},\psi^{\varepsilon},\mu^{\varepsilon})$ in the sequel.

######  Lemma 2.9.

The following asymptotic expansions hold as $\varepsilon\to 0^{+}$:

$\displaystyle\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon})$
$\displaystyle=\frac{\kappa^{2}}{4\pi}\ln\frac{1}{\varepsilon}+O(1),$ (2.12)
$\displaystyle\mu^{\varepsilon}$
$\displaystyle=\frac{\kappa}{2\pi}\ln\frac{1}{\varepsilon}+O(1).$ (2.13)

###### Proof.

Using Riesz’s rearrangement inequality, we obtain

$\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon})\leq\frac{\kappa^{2}}{4\pi}\ln\frac{1}{\varepsilon}+C.$

Combining this and Lemma 2.3, we get (2.12). Using the same argument as in the
proof of Lemma 2.5, we can easily get (2.13). The proof is complete. ∎

Now we turn to estimate the size of the supports of $\zeta^{\varepsilon}$ for
$\varepsilon$ small. To this end, we first recall an auxiliary lemma.

######  Lemma 2.10 ([8], Lemma 2.8).

Let $\Omega\subset D$, $0<\varepsilon<1$, $\eta\geq 0$, and let $\xi$ be a
non-negative function satisfying $\xi\in L^{1}(D)$, $\int_{D}\xi(x)dx=1$ and
$||\xi||_{L^{p}(D)}\leq C_{1}\varepsilon^{-2(1-{1}/{p})}$ for some
$1<p\leq+\infty$ and $C_{1}>0$. Suppose for any $x\in\Omega$, there holds

$(1-\eta)\log\frac{1}{\varepsilon}\leq\int_{D}\log\frac{1}{|x-y|}\xi(y)dy+C_{2},$
(2.14)

where $C_{2}$ is a positive constant. Then there exists some constant $R>1$
such that

$diam(\Omega)\leq R\varepsilon^{1-2\eta}.$

The constant $R$ may depend on $C_{1}$, $C_{2}$, but not on $\eta$,
$\varepsilon$.

Note that

$G(x,y)\leq\frac{1}{2\pi}\log\frac{1}{|x-y|}+C\ \ \ \text{in}\ D\times D,$

and

$\text{supp}(\zeta^{\varepsilon})\subseteq\\{x\in
D\mid\mathcal{G}\zeta^{\varepsilon}(x)-Wx_{2}\geq\mu^{\varepsilon}\\}.$

Using Lemmas 2.5 and 2.10, we immediately get the following estimate.

######  Lemma 2.11.

Let $\zeta^{\varepsilon}$ be a maximizer as in Lemma 2.1, then for
$\varepsilon$ small it holds $\text{supp}(\zeta^{\varepsilon})\leq
C\varepsilon$.

We proceed to determine the limiting location and the asymptotic shape of
$\operatorname{supp}\left(\zeta^{\varepsilon}\right)$. Define the center of
$\zeta^{\varepsilon}$ by

$x^{\varepsilon}:=\frac{1}{\kappa}\int_{\Pi}x\zeta^{\varepsilon}(x)dx.$

From now on for the remainder of the discussion we fix a sequence
$\varepsilon=\varepsilon_{j}\rightarrow 0^{+}$such that

$x^{\varepsilon}\rightarrow(0,\bar{r}),\text{ as
}\varepsilon=\varepsilon_{j}\rightarrow 0^{+},\text{where}\
\bar{r}\in[r^{*}/2,2r^{*}].$ ($\ast$)

Next we study the limiting behavior of the corresponding stream functions
$\psi^{\varepsilon}$. Define the scaled versions of $\psi^{\varepsilon}$ as
follows

$\Psi^{\varepsilon}(y):=\psi^{\varepsilon}(x^{\varepsilon}+\varepsilon y),\ \
\ \ y\in\Pi^{\varepsilon}:=\left\\{y\in\mathbb{R}^{2}\mid
x^{\varepsilon}+\varepsilon y\in\Pi\right\\}.$

Thus, we have

$-\Delta\Psi^{\varepsilon}=(1+\alpha(x_{2}^{\varepsilon}+\varepsilon
y_{2}))f(\Psi^{\varepsilon})\ \ \text{ in }\Pi^{\varepsilon},$ (2.15)
$\int_{\Pi^{\varepsilon}}(1+\alpha(x_{2}^{\varepsilon}+\varepsilon
y_{2}))f(\Psi^{\varepsilon})dy=\kappa.$ (2.16)

Note that $\left\\{x\in D\mid\Psi^{\varepsilon}(x)>0\right\\}\subset
B_{R_{0}}(0)$. As mentioned in section 1, we introduce the limiting profile
$U^{\kappa}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ defined as the unique
radially symmetric solution of the problem

$\left\\{\begin{array}[]{l}-\Delta U^{\kappa}=f(U^{\kappa}),\quad
x\in\mathbb{R}^{2}\\\
\int_{\mathbb{R}^{2}}f(U^{\kappa})dx=\kappa.\end{array}\right.$ (2.17)

######  Lemma 2.12.

As $\varepsilon\rightarrow 0^{+}$, we have $\Psi^{\varepsilon}(x)\rightarrow
U^{\kappa}\left(\left(1+\alpha\bar{r}\right)^{\frac{1}{2}}x\right)$ in
$C_{\text{loc }}^{1,\gamma}\left(\mathbb{R}^{2}\right)$, where $\bar{r}$ is
the same as in ($\ast$ ‣ 2.2)

###### Proof.

Note that
$\left\\{\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon
y)\right\\}$ is bounded in $L^{\infty}\left(\Pi^{\varepsilon}\right)$. Thus,
by classical elliptic estimates, the sequence
$\left\\{\Psi^{\varepsilon}\right\\}$ is bounded in $W_{\text{loc
}}^{2,p}\left(\Pi^{\varepsilon}\right)$ for every $1\leq p<+\infty$. By the
Sobolev embedding theorem, we may conclude
$\left\\{\Psi^{\varepsilon}\right\\}$ is compact in $C_{\text{loc
}}^{1,\gamma}\left(\Pi^{\varepsilon}\right)$ for every $0<\gamma<1$. Up to a
subsequence we may assume
$\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon
y)\rightarrow\zeta(y)$ weakly-star in $L^{\infty}\left(\mathbb{R}^{2}\right)$
and $\Psi^{\varepsilon}\rightarrow\Psi$ in $C_{\text{loc
}}^{1,\gamma}\left(\mathbb{R}^{2}\right)$. Let
$\tilde{\Psi}(x):=\Psi\left(x/\left(1+\alpha\bar{r}\right)^{\frac{1}{2}}\right)$.
By virtue of (2.15),(2.16) and $f$ is non-negative, we get

$-\Delta\tilde{\Psi}=f(\tilde{\Psi})\text{ in
}\mathbb{R}^{2},\quad\int_{\mathbb{R}^{2}}f(\tilde{\Psi})dx=\kappa.$

In view of Lemma 2.9, we know that $\zeta$ is a radially non-increasing
function. Using this fact, (2.13) and Green’s representation, it is not hard
to find that $\Psi$ and $\tilde{\Psi}$ are also radial. Therefore, we have
$\tilde{\Psi}\equiv U^{\kappa}$. Thus the proof is complete. ∎

######  Remark 2.13.

With the above results in hand, one can further study the shape of the free
boundary $\\{x\in D\mid\psi^{\varepsilon}(x)=0\\}$ by the standard scaling
techniques (see [8, 38]). The vortex core will be approximately a disk.

As a consequence of Lemma 2.12, we can obtain as follows.

######  Corollary 2.14.

As $\varepsilon\rightarrow 0^{+}$, one has
$\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon
x)\rightarrow\left(1+\alpha\bar{r}\right)f\Big{(}U^{\kappa}\big{(}\left(1+\alpha\bar{r}\right)^{\frac{1}{2}}x\big{)}\Big{)}$
in $L^{\infty}\left(\mathbb{R}^{2}\right)$.

We have shown that the vorticies $\\{\zeta^{\varepsilon}\\}$ would shrink to
some point in $\bar{D}$ when $\varepsilon\to 0$. Now we investigate this
limiting location. Let

$\mathcal{W}(t)=\frac{\kappa^{2}}{4\pi}\log\frac{1}{2t}+{\kappa
Wt}+\int_{\mathbb{R}^{2}}\frac{1+\alpha r^{*}}{1+\alpha
t}J_{F}\left(\frac{1+\alpha t}{1+\alpha r^{*}}f(U^{\kappa}(z))\right)dz,\ \
t>0.$ (2.18)

We want to show that $r^{*}$ is the unique minimum point
$\mathcal{W}(r^{*})=\min_{t>0}\mathcal{W}(t)$. By calculating

$\begin{split}\frac{d\mathcal{W}(t)}{dt}&=-\frac{\kappa^{2}}{4\pi t}+\kappa
W-\int_{\mathbb{R}^{2}}{\alpha}\frac{1+\alpha r^{*}}{(1+\alpha
t)^{2}}J_{F}\Big{(}\frac{1+\alpha t}{1+\alpha
r^{*}}f\big{(}U^{\kappa}(z)\big{)}\Big{)}dz\\\ &\ \ \ \ \ \ \
+\frac{\alpha}{1+\alpha
t}\int_{\mathbb{R}^{2}}\partial_{s}J_{F}\Big{(}\frac{1+\alpha t}{1+\alpha
r^{*}}f\big{(}U^{\kappa}(z)\big{)}\Big{)}f\big{(}U^{\kappa}(z)\big{)}dz.\\\
\end{split}$

Let

$K(s):=\partial_{s}J_{F}(s)s-J_{F}(s).$

By (2.1), we note that

$K(s)=J_{F}^{*}(\sigma)=F(\sigma),\ \text{for}\ s>0,$

where $J_{F}^{*}$ be the conjugate function to $J_{F}$ and
$\sigma=\partial_{s}J_{F}(s)$ is non-decreasing with respect to $s$, then we
get $K$ is non-decreasing. Recall that $r^{*}$ defined by (2.2). Thus we have

$\displaystyle\frac{d\mathcal{W}(t)}{dt}=\frac{1}{{(1+\alpha t)^{2}}}\Big{(}$
$\displaystyle M(t)-M(r^{*})+{\alpha}{(1+\alpha r^{*})}\times$
$\displaystyle\int_{\mathbb{R}^{2}}\Big{(}K\Big{(}\frac{1+\alpha t}{1+\alpha
r^{*}}f\big{(}U^{\kappa}(z)\big{)}\Big{)}-K(f\big{(}U^{\kappa}(z)\big{)})\Big{)}dz\Big{)},$

where

$M(t)=(1+\alpha t)^{2}(-\frac{\kappa^{2}}{4\pi t}+\kappa W).$

We notice that $W\geq\frac{(2-\alpha)_{+}^{2}\kappa}{32\pi}$, so

$M^{\prime}(t)=\kappa\frac{(1+\alpha t)}{(4\pi t^{2})}(8\pi
Wt^{2}-(2-\alpha)\kappa t+\kappa)\geq 0.\ $

$M^{\prime}(t)$ is zero at most finitely many points, so $r^{*}$ is the only
minimum point. Hence we obtain $\mathcal{W}(r^{*})=\min_{t>0}\mathcal{W}(t)$.

The following lemma show that the limiting location is $b_{1}$.

######  Lemma 2.15.

One has

$dist\left(supp(\zeta^{\varepsilon}),b_{1}\right)\to 0,\ \ \text{as}\
\varepsilon\to 0$

###### Proof.

Take $(0,x^{\varepsilon}_{2})\in\text{supp}(\zeta^{\varepsilon})$. Suppose (up
to a subsequence) $x_{2}^{\varepsilon}\to\bar{r}>0$, as $\varepsilon\to 0$. We
now show that $\bar{r}=r^{*}$. Set
$\tilde{\zeta}^{\varepsilon}=\zeta^{\varepsilon}(\cdot-r^{*}\mathbf{e}_{2}+\bar{r}\mathbf{e}_{2})\in\mathcal{A}_{\varepsilon,\Lambda_{0}}$.
Then

$\int_{D}\int_{D}{\tilde{\zeta}^{\varepsilon}(x)\log\frac{1}{|x-x^{\prime}|}\tilde{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}=\int_{D}\int_{D}{{\zeta}^{\varepsilon}(x)\log\frac{1}{|x-x^{\prime}|}{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}.$

By Corollary 2.14 , we have

$\displaystyle\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon})dx$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{D}(1+\alpha
x_{2})J_{F}\Big{(}\frac{\varepsilon^{2}\zeta^{\varepsilon}}{1+\alpha
x_{2}}\Big{)}dx$ $\displaystyle=\int_{\Pi^{\varepsilon}}(1+\alpha
x_{2}^{\varepsilon}+\varepsilon\alpha
y_{2})J_{F}\Big{(}\frac{\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon
y)}{1+\alpha x_{2}^{\varepsilon}+\varepsilon\alpha y_{2}}\Big{)}dy$
$\displaystyle=\int_{\mathbb{R}^{2}}J_{F}(f(U^{\kappa}(z)))dz+o(1).$

Similarly, we can obtain

$\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\tilde{\zeta}^{\varepsilon})dx=\int_{\mathbb{R}^{2}}\frac{1+\alpha
r^{*}}{1+\alpha\bar{r}}J_{F}\left(\frac{1+\alpha\bar{r}}{1+\alpha
r^{*}}f(U^{\kappa}(z))\right)dz+o(1).\\\ $

Since
$\mathcal{E}_{\varepsilon}(\tilde{\zeta}^{\varepsilon})\leq\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon})$,
it follows that

$\begin{split}&\frac{1}{4\pi}\int_{D}\int_{D}{{\zeta}^{\varepsilon}(x)\log\frac{1}{|\bar{x}-x^{\prime}|}{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}+W\int_{D}x_{2}{\zeta}^{\varepsilon}dx+\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon})dx\\\
&\ \ \ \
\leq\frac{1}{4\pi}\int_{D}\int_{D}{\tilde{\zeta}^{\varepsilon}(x)\log\frac{1}{|\bar{x}-x^{\prime}|}\tilde{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}+W\int_{D}x_{2}\tilde{\zeta}^{\varepsilon}dx+\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\tilde{\zeta^{\varepsilon}})dx.\end{split}$

Letting $\varepsilon\to 0$, we get

$\mathcal{W}(\bar{r})\leq\mathcal{W}(r^{*}).$

This implies $\bar{r}=r^{*}$ and the proof is thus complete. ∎

Combining Lemma 2.11 and Lemma 2.15, we get

######  Lemma 2.16.

For all sufficiently small $\varepsilon$, it holds
$dist\left(supp(\zeta^{\varepsilon}),\partial D\right)>0$.

######  Lemma 2.17.

For all sufficiently small $\varepsilon$, one has

$\zeta^{\varepsilon,\Lambda_{0}}=\frac{1}{\varepsilon^{2}}i(x_{2},\psi^{\varepsilon,\Lambda_{0}}),\
\ \ \ \ a.e.\ \text{in}\ \Pi.$

###### Proof.

By Lemma 2.8, it suffices to show that $\psi^{\varepsilon}\leq 0$ a.e. on
$\Pi\backslash D$. By Lemma 2.16, we have

$\mathcal{G}\zeta^{\varepsilon}(x)\leq C,\quad\forall x\in\Pi\backslash D$

when $\varepsilon$ is small enough. In view of Lemma 2.5, it follows that for
$\varepsilon$ small, we have $\psi^{\varepsilon}\leq 0$ a.e. on $\Pi\backslash
D$ and the proof is thus complete. ∎

We now turn to prove Theorem 1.2.

###### Proof of Theorem 1.2.

It follows from the above lemmas by letting $\lambda=1/\varepsilon^{2}$. ∎

##  3\. Proofs of Theorems 1.4 and 1.5

In order to complete the proof of Theorems 1.4 and 1.5, we only need to re-
determine the limit position where the vortex core will converge. Before
proving Theorem 1.4, we need some preliminary lemmas.

We first study the limiting behavior of the corresponding stream functions
$\psi^{\varepsilon}$. Define the scaling form of the stream function as
follows

$\Psi_{1}^{\varepsilon}(y):=\psi^{\varepsilon}(x^{\varepsilon}+\varepsilon
y),\ \ \ \ y\in\Pi^{\varepsilon}:=\left\\{y\in\mathbb{R}^{2}\mid
x^{\varepsilon}+\varepsilon y\in\Pi\right\\}.$

By the properties of the stream function, we have

$-\Delta\Psi_{1}^{\varepsilon}=f(\Psi_{1}^{\varepsilon})+(x_{2}+\varepsilon
y_{2})\delta_{\varepsilon}g(\Psi_{1}^{\varepsilon})\ \ \text{ in
}\Pi^{\varepsilon},$ (3.1)
$\int_{\Pi^{\varepsilon}}f(\Psi_{1}^{\varepsilon})+(x_{2}+\varepsilon
y_{2})\delta_{\varepsilon}g(\Psi_{1}^{\varepsilon})dy=\kappa.$ (3.2)

Similar to Lemma 2.12 for the limiting behavior of $\Psi_{1}^{\varepsilon}$,
we have

######  Lemma 3.1.

As $\varepsilon\rightarrow 0^{+}$, $\Psi_{1}^{\varepsilon}(x)\rightarrow
U^{\kappa}\left(x\right)$ in $C_{\text{loc
}}^{1,\gamma}\left(\mathbb{R}^{2}\right).$

Since the proof process is similar, we omit it here.

From the above lemma, we have the following

######  Corollary 3.2.

As $\varepsilon\rightarrow 0^{+}$,
$\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon x)\rightarrow
f(U^{\kappa}(x))$ in $L^{\infty}\left(\mathbb{R}^{2}\right)$.

By Lemma 2.11, we obtain that the vortex will converge to some point
$(0,\bar{r})$ in $\bar{D}$. Now let’s figure out where the limit point should
be. Let

$\mathcal{W}_{1}(t)=\frac{\kappa^{2}}{4\pi}\log\frac{1}{2t}+{\kappa Wt}.$
(3.3)

To prove Theorem 1.4, instead of using (2.2) as the definition of $r^{*}$, we
choose $r_{1}^{*}=\frac{\kappa}{4\pi W}$. It’s easy to see that
$\mathcal{W}_{1}(r_{1}^{*})=\min_{t>0}\mathcal{W}_{1}(t)$.

The following lemma shows that the limiting position is indeed
$b_{1}=r_{1}^{*}\mathbf{e}_{2}$.

######  Lemma 3.3.

One has

$dist\left(supp(\zeta^{\varepsilon}),b_{1}\right)\to 0,\ \ \text{as}\
\varepsilon\to 0$

###### Proof.

Take $(0,x^{\varepsilon}_{2})\in\text{supp}(\zeta^{\varepsilon})$. Suppose (up
to a subsequence) that $x_{2}^{\varepsilon}\to\bar{r}>0$, as $\varepsilon\to
0$. We now show that $\bar{r}=r_{1}^{*}$. Set
$\tilde{\zeta}^{\varepsilon}=\zeta^{\varepsilon}(\cdot-
r_{1}^{*}\mathbf{e}_{2}+\bar{r}\mathbf{e}_{2})\in\mathcal{A}_{\varepsilon,\Lambda_{0}}$.
Then

$\int_{D}\int_{D}{\tilde{\zeta}^{\varepsilon}(x)\log\frac{1}{|x-x^{\prime}|}\tilde{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}=\int_{D}\int_{D}{{\zeta}^{\varepsilon}(x)\log\frac{1}{|x-x^{\prime}|}{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}.$

By Corollary 3.2 , we have

$\begin{split}\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon})dx&=\int_{\Pi^{\varepsilon}}J(x_{2}^{\varepsilon}+\varepsilon
y_{2},\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon y))dy\\\
&=\int_{\Pi^{\varepsilon}}\sup_{t}\Big{(}\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon
y)t-\Big{[}\int_{0}^{t}f(s)+(x_{2}^{\varepsilon}+\varepsilon
y_{2})\delta_{\varepsilon}g(s)ds\Big{]}\Big{)}dy\\\
&=\int_{\mathbb{R}^{2}}\sup_{t}(f(U^{\kappa})t-F(t))dy+o(1)\\\
&=\int_{\mathbb{R}^{2}}J_{F}(f(U^{\kappa}))dy+o(1).\\\ \end{split}$

Similarly, we can obtain

$\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\tilde{\zeta}^{\varepsilon}(x))dx=\int_{\mathbb{R}^{2}}J_{F}(f(U^{\kappa}))dy+o(1).$

Since
$\mathcal{E}_{\varepsilon}(\tilde{\zeta}^{\varepsilon})\leq\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon})$,
it follows that

$\begin{split}&\frac{1}{4\pi}\int_{D}\int_{D}{{\zeta}^{\varepsilon}(x)\log\frac{1}{|\bar{x}-x^{\prime}|}{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}+W\int_{D}x_{2}{\zeta}^{\varepsilon}dx+\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon})dx\\\
&\ \ \ \
\leq\frac{1}{4\pi}\int_{D}\int_{D}{\tilde{\zeta}^{\varepsilon}(x)\log\frac{1}{|\bar{x}-x^{\prime}|}\tilde{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}+W\int_{D}x_{2}\tilde{\zeta}^{\varepsilon}dx+\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\tilde{\zeta^{\varepsilon}})dx.\end{split}$

Letting $\varepsilon\to 0$, we get

$\mathcal{W}_{1}(\bar{r})\leq\mathcal{W}_{1}(r_{1}^{*}).$

This implies $\bar{r}=r_{1}^{*}$ and the proof is thus complete. ∎

###### Proof of Theorem 1.4.

Combining Lemma 2.1 to Lemma 2.11 in section 2 and Lemma 3.3 in section 3, we
prove that Theorem 1.4 is true by taking $\lambda=\frac{1}{\varepsilon^{2}}$.
∎

Finally we turn to proving Theorem 1.5. We modify the area slightly as follows

$D=\\{x\in\Pi~{}|~{}-1<x_{1}<1,\ c_{1}r_{2}^{*}<x_{2}<r_{2}^{*}/c_{1}\\},$

where $c_{1}\in(0,1)$ will be given later by (3.10), and $r_{2}^{*}$ will be
given by (3.8).

Similar to the previous process, we first study the limiting behavior of the
corresponding stream function $\psi^{\varepsilon}$. We define the scaling form
of the stream function below

$\Psi_{2}^{\varepsilon}(y):=\psi^{\varepsilon}(x^{\varepsilon}+\varepsilon
y),\ \ \ \ y\in\Pi^{\varepsilon}:=\left\\{y\in\mathbb{R}^{2}\mid
x^{\varepsilon}+\varepsilon y\in\Pi\right\\},$

where $\Psi_{2}$ satisfies

$-\Delta\Psi_{2}^{\varepsilon}=\delta_{\varepsilon}f(\Psi_{2}^{\varepsilon})+(x_{2}^{\varepsilon}+\varepsilon
y_{2})g(\Psi_{2}^{\varepsilon})\ \ \text{ in }\Pi^{\varepsilon},$ (3.4)
$\int_{\Pi^{\varepsilon}}\delta_{\varepsilon}f(\Psi_{2}^{\varepsilon})+(x_{2}^{\varepsilon}+\varepsilon
y_{2})g(\Psi_{2}^{\varepsilon})dy=\kappa.$ (3.5)

Let’s introduce the limiting profile
$V^{\kappa}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ defined as the unique
radially symmetric solution of the problem

$\left\\{\begin{array}[]{l}-\Delta V^{\kappa}=g(V^{\kappa}),\quad
x\in\mathbb{R}^{2}\\\
\int_{\mathbb{R}^{2}}g(V^{\kappa})dx=\kappa.\end{array}\right.$ (3.6)

Thus, the asymptotic behavior of $\Psi_{2}^{\varepsilon}$ is described as
follows

######  Lemma 3.4.

As $\varepsilon\rightarrow 0^{+}$, we have
$\Psi_{2}^{\varepsilon}(x)\rightarrow V^{\kappa}\left(\sqrt{\bar{r}}x\right)$
in $C_{\text{loc }}^{1,\gamma}\left(\mathbb{R}^{2}\right).$

As a corollary to the above theorem, we have the following result

######  Corollary 3.5.

As $\varepsilon\rightarrow 0^{+}$, one has
$\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon
x)\rightarrow\bar{r}g(V^{\kappa}(\sqrt{\bar{r}}x))$ in
$L^{\infty}\left(\mathbb{R}^{2}\right)$.

According to Lemma 2.11, we know that the vortex will converge to some point
$(0,\bar{r})$ in $\bar{D}$, where
$\bar{r}\in[c_{1}r_{2}^{*},r_{2}^{*}/c_{1}]$. We want to find the position of
the limit of vortex convergence, so we define the following function

$\mathcal{W}_{2}(t)=\frac{\kappa^{2}}{4\pi}\log\frac{1}{2t}+{\kappa
Wt}+\frac{r_{2}^{*}}{t}\int_{\mathbb{R}^{2}}J_{G}\Big{(}\frac{t}{r_{2}^{*}}g(V^{\kappa}(z))\Big{)}dz.$
(3.7)

Here we define $r_{2}^{*}>0$ as follows

$r_{2}^{*}=\frac{1}{\kappa
W}\Big{(}\frac{\kappa^{2}}{4\pi}-\mathcal{C}_{g,\kappa}\Big{)}=\frac{\kappa}{8\pi
W},$ (3.8)

with

$\begin{split}\mathcal{C}_{g,\kappa}&=\int_{\mathbb{R}^{2}}\Big{(}V^{\kappa}(z)g(V^{\kappa}(z))-J_{G}\Big{(}g(V^{\kappa}(z))\Big{)}\Big{)}dz\\\
&=\int_{\mathbb{R}^{2}}G(V^{\kappa}(z))dz=\frac{\kappa^{2}}{8\pi}.\end{split}$
(3.9)

We want to get
$\mathcal{W}_{2}(r_{2}^{*})=\min_{t\in[c_{1}r_{2}^{*},r_{2}^{*}/c_{1}]}\mathcal{W}_{2}(t)$.
So by taking the derivative of $\mathcal{W}_{2}$, we have

$\begin{split}\frac{d\mathcal{W}_{2}(t)}{dt}&=-\frac{\kappa^{2}}{4\pi
t}+\kappa
W-\frac{r_{2}^{*}}{t^{2}}\int_{\mathbb{R}^{2}}J_{G}\Big{(}\frac{t}{r_{2}^{*}}g(V^{\kappa}(z))\Big{)}dz\\\
&\ \ \ \ \
+\frac{1}{t}\int_{\mathbb{R}^{2}}\partial_{s}J_{G}\Big{(}\frac{t}{r_{2}^{*}}g(V^{\kappa}(z))\Big{)}g(V^{\kappa}(z))dz.\end{split}$

For brevity, let’s define $L(s):=\partial_{s}J_{G}(s)s-J_{G}(s)$, and it’s
also a non-decreasing function. By definition (3.8) of $r_{2}^{*}$, we can
abbreviate it as follows

$\begin{split}\frac{d\mathcal{W}_{2}(t)}{dt}&=-\frac{\kappa^{2}}{4\pi
t}+\kappa
W+\frac{r_{2}^{*}}{t^{2}}\int_{\mathbb{R}^{2}}L\Big{(}\frac{t}{r_{2}^{*}}g(V^{\kappa}(z))\Big{)}dz\\\
&=-\frac{\kappa^{2}}{4\pi t}+\kappa
W+\frac{r_{2}^{*}}{t^{2}}\int_{\mathbb{R}^{2}}G(\partial_{s}J_{G}(\frac{t}{r_{2}^{*}}g(V^{\kappa}(z))))dz\\\
&=\frac{1}{t^{2}}\Big{(}N(t)-N(r_{2}^{*})\Big{)},\end{split}$

where

$N(t)=-\frac{\kappa^{2}}{4\pi}t+\kappa
Wt^{2}+r_{2}^{*}\int_{\mathbb{R}^{2}}G(\partial_{s}J_{G}(\frac{t}{r_{2}^{*}}g(V^{\kappa}(z))))dz.$

We need to determine whether $N$ is monotonic at $r_{2}^{*}$, and we can
compute the derivative of $N$ as follows

$\begin{split}N^{\prime}(t)=-\frac{\kappa^{2}}{4\pi}+2\kappa
Wt+\int_{\mathbb{R}^{2}}g(\partial_{s}J_{G}(\frac{t}{r_{2}^{*}}g(V^{\kappa})))\partial_{ss}J_{G}(\frac{t}{r_{2}^{*}}g(V^{\kappa}))g(V^{\kappa})dz.\\\
\end{split}$

Thus we have

$N^{\prime}(r_{2}^{*})=\int_{\mathbb{R}^{2}}g(V^{\kappa})\partial_{ss}J_{G}(g(V^{\kappa}))g(V^{\kappa})dz>0.$
(3.10)

According to the above formula, we can find a constant $0<c_{1}<1$ , such that
$N^{\prime}(t)>0$ on the set $[c_{1}r_{2}^{*},r_{2}^{*}/c_{1}]$. Using the
sign of the derivative of $\mathcal{W}_{2}$, we obtain
$\mathcal{W}_{2}(r_{2}^{*})=\min_{t\in[c_{1}r_{2}^{*},r_{2}^{*}/c_{1}]}\mathcal{W}_{2}(t)$
and $r_{2}^{*}$ is the only minimum point in this range.

Next, we want to obtain the specific position of vortex convergence.

######  Lemma 3.6.

One has

$dist\left(supp(\zeta^{\varepsilon}),b_{1}\right)\to 0,\ \ \text{as}\
\varepsilon\to 0,$

where $b_{1}=r_{2}^{*}\mathbf{e}_{2}$.

###### Proof.

Take $(0,x^{\varepsilon}_{2})\in\text{supp}(\zeta^{\varepsilon})$. Suppose (up
to a subsequence)
$x_{2}^{\varepsilon}\to\bar{r}\in[c_{1}r_{2}^{*},r_{2}^{*}/c_{1}]$, as
$\varepsilon\to 0$. We now show that $\bar{r}=r_{2}^{*}$. Set
$\tilde{\zeta}^{\varepsilon}=\zeta^{\varepsilon}(\cdot-
r_{2}^{*}\mathbf{e}_{2}+\bar{r}\mathbf{e}_{2})\in\mathcal{A}_{\varepsilon,\Lambda_{0}}$.
Then

$\int_{D}\int_{D}{\tilde{\zeta}^{\varepsilon}(x)\log\frac{1}{|x-x^{\prime}|}\tilde{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}=\int_{D}\int_{D}{{\zeta}^{\varepsilon}(x)\log\frac{1}{|x-x^{\prime}|}{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}.$

By Corollary 3.5 , we have

$\begin{split}\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon})dx&=\int_{\Pi^{\varepsilon}}J(x_{2}^{\varepsilon}+\varepsilon
y_{2},\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon y))dy\\\
&=\int_{\Pi^{\varepsilon}}\sup_{t}(\varepsilon^{2}\zeta^{\varepsilon}(x^{\varepsilon}+\varepsilon
y)t-\Big{[}\int_{0}^{t}\delta_{\varepsilon}f(s)+(x_{2}^{\varepsilon}+\varepsilon
y_{2})g(s)ds\Big{]})dy\\\
&=\int_{\mathbb{R}^{2}}\sup_{t}(\bar{r}g(V^{\kappa}(\sqrt{\bar{r}}y))t-\bar{r}G(t))dy+o(1)\\\
&=\int_{\mathbb{R}^{2}}J_{G}(g(V^{\kappa}(z)))dz+o(1).\end{split}$

Similarly, we can obtain

$\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\tilde{\zeta}^{\varepsilon}(x))dx=\frac{r_{2}^{*}}{\bar{r}}\int_{\mathbb{R}^{2}}J_{G}\Big{(}\frac{\bar{r}}{r_{2}^{*}}g(V^{\kappa}(z))\Big{)}dz+o(1).$

Since
$\mathcal{E}_{\varepsilon}(\tilde{\zeta}^{\varepsilon})\leq\mathcal{E}_{\varepsilon}(\zeta^{\varepsilon})$,
it follows that

$\begin{split}&\frac{1}{4\pi}\int_{D}\int_{D}{{\zeta}^{\varepsilon}(x)\log\frac{1}{|\bar{x}-x^{\prime}|}{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}+W\int_{D}x_{2}{\zeta}^{\varepsilon}dx+\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\zeta^{\varepsilon})dx\\\
&\ \ \ \
\leq\frac{1}{4\pi}\int_{D}\int_{D}{\tilde{\zeta}^{\varepsilon}(x)\log\frac{1}{|\bar{x}-x^{\prime}|}\tilde{\zeta}^{\varepsilon}}(x^{\prime})dxdx^{\prime}+W\int_{D}x_{2}\tilde{\zeta}^{\varepsilon}dx+\frac{1}{\varepsilon^{2}}\int_{D}J(x_{2},\varepsilon^{2}\tilde{\zeta^{\varepsilon}})dx.\end{split}$

Letting $\varepsilon\to 0$, we get

$\mathcal{W}_{2}(\bar{r})\leq\mathcal{W}_{2}(r_{2}^{*}).$

Because $\bar{r}\in[c_{1}r_{2}^{*},r_{2}^{*}/c_{1}]$, and
$\mathcal{W}_{2}(r_{2}^{*})=\min_{t\in[c_{1}r_{2}^{*},r_{2}^{*}/c_{1}]}\mathcal{W}_{2}(t)$,
which implies $\bar{r}=r_{2}^{*}$ and the proof is thus complete. ∎

###### Proof of Theorem 1.5.

Combining Lemma 2.1 to Lemma 2.11 in section 2 and Lemma 3.6 in section 3, we
can obtain Theorem 1.5 by taking $\lambda=\frac{1}{\varepsilon^{2}}$. ∎

Acknowledgments. This work was supported by NNSF of China Grant 11831009.

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$\displaystyle\geq\hat{\mathcal{P}}({\mathcal{S}}^{\prime})=p^{*}\geq\hat{\mathcal{P}}({\mathcal{S}}^{*})\geq\mathcal{P}({\mathcal{S}}^{*})-2n\kappa=\mathcal{P}({\mathcal{S}}^{*})-\epsilon
P_{0}\geq(1-\epsilon)\mathcal{P}({\mathcal{S}}^{*}),$

where first inequality follows because the rewards are rounded down and the
penalties are rounded up in calculation of $\hat{\mathcal{P}}$, second
inequality follows because ${\mathcal{S}}^{\prime}$ is the optimal set for
objective $\hat{\mathcal{P}}$, the third inequality follows because
$|{\mathcal{S}}^{*}|\leq n$ and $T\leq n$, and the last inequality follows
from (10) that $\mathcal{P}({\mathcal{S}}^{*})\geq P_{0}$. ∎

It remains to find $P_{0}$ which satisfies (10). Since
$\mathcal{P}({\mathcal{S}}^{*})\leq\bar{P}$, we can enumerate $P_{0}$ from
$\bar{P}/2,\bar{P}/4,\bar{P}/8,\ldots$, and one of them must satisfy (10). The
FPTAS is presented as Algorithm 10.

Algorithm 10 FPTAS for MPBKP-S in $\mathcal{O}(n^{2}\log n/\epsilon)$

1:$P_{0}\leftarrow{\bar{P}}$

2:$p^{*}\leftarrow 0$

3:while $p^{*}<(1-\epsilon)P_{0}$ do

4: $P_{0}\leftarrow\frac{P_{0}}{2}$

5: Run Algorithm 9 with the current $P_{0}$.

6:
$p^{*}\leftarrow\max_{\left\\{\begin{subarray}{c}p\in\left\\{0,\ldots,\left\lceil\frac{4n}{\epsilon}\right\rceil\right\\}\cdot\kappa\\\
\hat{A}(n,p)>-\infty\end{subarray}\right\\}}p$

7:end while

######  Theorem 6.

Algorithm 10 is a fully polynomial approximation scheme for the MPBKP-S, which
achieves $(1-\epsilon)$ factor of optimal with running time
$\mathcal{O}\left(\frac{n^{2}\log n}{\epsilon}\right)$.

###### Proof of Theorem 6.

Time complexity:  When $P_{0}$ satisfies (10), by Lemma 15 we have that

$p^{*}\geq(1-\epsilon)\mathcal{P}({\mathcal{S}}^{*})\geq(1-\epsilon)P_{0}.$

Thus, the “while” loop terminates when $P_{0}$ satisfies (10), if not before
$P_{0}$ satisfies (10). When $P_{0}$ satisfies (10), we would also have
$\mathcal{P}({\mathcal{S}}^{*})/2\leq
P_{0}\leq\mathcal{P}({\mathcal{S}}^{*})$. Therefore, the number of iterations
is upper bounded by

$\text{number of
iterations}\leq\log\frac{\bar{P}/2}{\mathcal{P}({\mathcal{S}}^{*})/2}\leq\log
n,$

where we have used the fact that $\bar{P}\leq nP\leq
n\mathcal{P}({\mathcal{S}}^{*})$. Since each iteration takes time
$\mathcal{O}\left(n\cdot\left\lceil\frac{4n}{\epsilon}\right\rceil\right)$ we
get a total time complexity of $\mathcal{O}\left(\frac{n^{2}\log
n}{\epsilon}\right)$.

Approximation ratio: When Algorithm 10 terminates, it returns the last $p^{*}$
and the solution set $\mathcal{S}^{\prime}$ corresponding to
$\hat{A}(n,p^{*})$. If the “while” loop terminates when
$P_{0}>\mathcal{P}({\mathcal{S}}^{*})$, i.e., it stops before $P_{0}$ falls
below $\mathcal{P}({\mathcal{S}}^{*})$, then we have that

$\mathcal{P}({\mathcal{S}}^{\prime})\geq
p^{*}\geq(1-\epsilon)P_{0}>(1-\epsilon)\mathcal{P}({\mathcal{S}}^{*}).$

Otherwise, from the time complexity analysis, we know that the “while” loop
must terminate when $P_{0}$ first falls below
$\mathcal{P}({\mathcal{S}}^{*})$, which implies that the last $P_{0}$
satisfies (10). Then by Lemma 15 we again have that

$\mathcal{P}({\mathcal{S}}^{\prime})\geq(1-\epsilon)\mathcal{P}({\mathcal{S}}^{*}).$

In either case, the solution we obtained from Algorithm 10 achieves
$(1-\epsilon)$ optimal. $(1-\epsilon)$ factor of
$\mathcal{P}({\mathcal{S}}^{*})$. ∎