# Curvature estimates for $p$-convex hypersurfaces of prescribed curvature

Weisong Dong  School of Mathematics, Tianjin University, 135 Yaguan Road,
Jinnan, Tianjin, China, 300354<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish curvature estimates for $p$-convex hypersurfaces
in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq\frac{n}{2}$. The
existence of a star-shaped hypersurface of prescribed curvature is obtained.
We also prove a type of interior $C^{2}$ estimates for solutions to the
Dirichlet problem of the corresponding equation.

_Mathematical Subject Classification (2010):_ 53C42; 53C21; 35J60.

_Keywords:_ curvature estimates; $p$-convex hypersurfaces; interior estimates.

##  1\. Introduction

Let $M\subset\mathbb{R}^{n+1}$ be a closed hypersurface and let
$\kappa(X)=(\kappa_{1},\cdots,\kappa_{n})$ be the principal curvatures of $M$
at $X$. Given $1\leq p\leq n$, a $C^{2}$ regular hypersurface $M$ is called
_$p$ -convex_ if, at each $X\in M$, $\kappa(X)$ satisfies

$\kappa_{i_{1}}+\cdots+\kappa_{i_{p}}\geq 0,\>\forall\;1\leq
i_{1}<\cdots<i_{p}\leq n.$

In other words, the sum of the $p$ smallest principal curvatures is
nonnegative at each point of $M$. The notion of $p$-convexity goes back to Wu
[36] and has been studied extensively by Wu [36], Sha [29, 30] and Harvey-
Lawson [19, 20].

In this paper, we are interested in finding a $p$-convex hypersurface
$M\subset\mathbb{R}^{n+1}$ of prescribed curvature as below

(1.1) $\Pi_{1\leq i_{1}<\cdots<i_{p}\leq
n}(\kappa_{i_{1}}+\cdots+\kappa_{i_{p}})=f(X,\nu(X)),\;\forall\;X\in M,$

where $\nu(X)$ is the unit outer normal of $M$ at $X$, the function
$f(X,\nu)\in C^{2}(\Gamma)$ is positive and $\Gamma$ is an open neighborhood
of unit normal bundle of $M$ in $\mathbb{R}^{n+1}\times\mathbb{S}^{n}$. The
Gaussian curvature equation, that corresponds to $p=1$ in (1.1), was studied
by Oliker [24]. The mean curvature equation corresponding to $p=n$ in (1.1)
was studied by Bakelman-Kantor [1] and Treibergs-Wei [34]. For general
curvature equations, see Caffarelli-Nirenberg-Spruck [4] and Gerhardt [11].
When $p=n-1$, the equation was studied by Chu-Jiao [7] and, in complex
settings, it is related to the Gauduchon conjecture which was solved by
Székelyhidi-Tosatti-Weinkove [31]. For some previous work on this topic, see
Tosatti-Weinkove [32, 33] and Fu-Wang-Wu [9, 10].

It is of great interest in geometry and PDEs to derive a $C^{2}$ estimate for
equation (1.1) for general $f(X,\nu(X))$. We have the following main result.

######  Theorem 1.1.

Suppose $M\subset\mathbb{R}^{n+1}$ is a closed star-shaped $p$-convex
hypersurface with $p\geq\frac{n}{2}$ satisfying the curvature equation (1.1).
Then, there is a positive constant $C$ such that

(1.2) $\sup_{X\in M,i=1,\cdots,n}|\kappa_{i}(X)|\leq C,$

where $C$ depends on $n$, $p$, $|M|_{C^{1}}$, $\inf f$ and $|f|_{C^{2}}$.

We remark that in the theorem only a few conditions are assumed on $f$.
Usually, to derive $C^{2}$ estimate for elliptic equations which are not
strictly elliptic there should be some extra assumptions on $f$ due to the
dependency on $\nu(X)$. We refer the reader to Ivochkina [21, 22], Guan-Guan
[13], Guan-Lin-Ma [16], Guan-Li-Li [15] and Guan-Jiao [14] for more details.
Moreover, Guan-Ren-Wang in [17] showed that for the following curvature
equation

$\frac{\sigma_{k}}{\sigma_{l}}(\kappa(X))=f(X,\nu(X)),\;\forall X\in M,$

where $0<l<k\leq n$ and $\sigma_{k}$ is the $k$-th elementary symmetric
function, estimate (1.2) fails generally, though it may hold for special $f$
as in Guan-Guan [13]. When $l=0$, some results are known for general $f$. For
instance, estimate (1.2) was proved for $k=n$ by Caffarelli-Nirenberg-Spruck
[3] and, for $2\leq k\leq n$, Guan-Ren-Wang [17] obtained the estimate for
convex solutions. For $k=n-1$ and $n-2$, estimate (1.2) was established by
Ren-Wang [25, 26]. They also conjectured that the estimate still holds for
$k>\frac{n}{2}$ in [26, 27]. When $f$ is independent of $\nu$, Caffarelli-
Nirenberg-Spruck [4] proved the $C^{2}$ estimate for a general class of fully
nonlinear curvature equations. For hypersurfaces of prescribed curvature in
Riemannian manifolds and Minkowski space, see [5] and [35, 28]. We also refer
the reader to Guan-Zhang [18] and references therein for a class of curvature
equations arising from convex geometry.

To obtain the existence of a $p$-convex hypersurface satisfying the prescribed
curvature equation (1.1), we assume the following two conditions on $f$. The
first one is that there exists two positive constants $r_{1}<1<r_{2}$ such
that

(1.3) $\displaystyle f\Big{(}X,\frac{X}{|X|}\Big{)}\geq$
$\displaystyle\;\frac{p^{C_{n}^{p}}}{r_{1}^{C_{n}^{p}}},\;\mbox{for}\;|X|=r_{1};$
$\displaystyle f\Big{(}X,\frac{X}{|X|}\Big{)}\leq$
$\displaystyle\;\frac{p^{C_{n}^{p}}}{r_{2}^{C_{n}^{p}}},\;\mbox{for}\;|X|=r_{2}.$

This condition is used to derive $C^{0}$ estimates. The second one is that for
any fixed unit vector $\nu$,

(1.4) $\frac{\partial}{\partial\rho}\Big{(}\rho^{C_{n}^{p}}f(X,\nu)\Big{)}\leq
0,\;\mbox{where}\;\rho=|X|,$

and will be used to derive $C^{1}$ estimates. Actually, with suitable
assumptions of $f$, Li [23] proved that the interior gradient estimate holds.

By the continuity method argument as in [4], we can obtain the following
result.

######  Theorem 1.2.

Let $f\in C^{2}((\overline{B_{r_{2}}}\backslash
B_{r_{1}})\times\mathbb{S}^{n})$ be a positive function satisfying (1.3) and
(1.4). Then equation (1.1) has a unique $C^{3,\alpha}$ star-shaped $p$-convex
solution $M$ in $\\{X\in\mathbb{R}^{n+1}:r_{1}\leq|X|\leq r_{2}\\}$ for any
$\alpha\in(0,1)$ as long as $p\geq\frac{n}{2}$.

The method of proving Theorem 1.1 can be applied to obtain an interior $C^{2}$
estimate for the Dirichlet problem of the corresponding equation in the
Euclidean space. Suppose that $\Omega$ is a bounded domain in
$\mathbb{R}^{n}$. For a function $u\in C^{2}(\Omega)$, denote by
$\lambda(D^{2}u)=(\lambda_{1},\cdots,\lambda_{n})$ the eigenvalues of the
Hessian $D^{2}u$. We say that $u\in C^{2}(\Omega)$ is _$p$ -plurisubharmonic_
if the eigenvalues of $D^{2}u$ satisfy
$\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}\geq 0$, for all $1\leq
i_{1}<\cdots<i_{p}\leq n$ (see [20]). Given a $C^{2}$ $p$-plurisubharmonic
function $v$ on $\bar{\Omega}$, consider the following Dirichlet problem,

(1.5) $\Pi_{1\leq i_{1}<\cdots<i_{p}\leq
n}(\lambda_{i_{1}}+\cdots+\lambda_{i_{p}})=f(x,u,Du),\;\mbox{in}\;\Omega$

with boundary data

$u=v,\;\mbox{on}\;\partial\Omega,$

where $f\in C^{2}(\bar{\Omega}\times\mathbb{R}\times\mathbb{R}^{n})$ is a
positive function. By the same argument as that of Theorem 1.1, we can prove
the following interior estimate.

######  Theorem 1.3.

Suppose that a $p$-plurisubharmonic function $u\in C^{4}(\Omega)\cap
C^{1,1}(\bar{\Omega})$ is a solution to the Dirichlet problem of equation
(1.5) and satisfies $u<v$ in $\Omega$. Then, there exist constants $C$ and
$\beta$ depending only on $n,p$, $|u|_{C^{1}}$, $|v|_{C^{1}}$, $\inf f$,
$|f|_{C^{2}}$ and $\Omega$ such that

(1.6) $\sup_{\Omega}(v-u)^{\beta}\Delta u\leq C$

as long as $p\geq\frac{n}{2}$.

######  Remark 1.4.

As a byproduct of the proof of the above theorem, one can conclude the
following global $C^{2}$ estimate for equation (1.5)

$\sup_{\Omega}|D^{2}u|\leq C(1+\sup_{\partial\Omega}|D^{2}u|),$

where $C$ is a constant as in Theorem 1.3.

We shall only give an outline for the proof of Theorem 1.3, as it is almost
the same as that of Theorem 1.1. The estimate (1.6) can also be seen in some
sense as a generalization of Theorem 0.4 in [8], since there the right hand
side function $f$ does not depend on $Du$. Such an estimate for the
$k$-Hessian equation $\sigma_{k}(\lambda)=f(x,u)$ has been proved by Chou-Wang
[6]. The function $f$ depending on $\nu$ or $Du$ creates substantial
difficulties to derive a $C^{2}$ estimate, as the bad term $-Ch_{11}$ or
$-Cu_{11}$ appears when one applies the maximum principle to the test
function. One way to overcome this is like in [17, 25, 26] to control the bad
third order terms, which is very hard even for the $k$-Hessian equation.
Another way is as in [7] to control $-Ch_{11}$ firstly by good terms (see
Lemma 3.2), and then the bad third order terms can be eliminated easily by
Lemma 3.4. Thanks to Dinew [8], where many properties of the operator have
been proved, we can follow the argument in [7] to prove the estimate.

The paper is organized as follows. In Section 2, we recall some properties of
the operator from [8]. In Section 3, we prove the curvature estimate. In
Section 4 we derive the gradient estimate and in Section 5 we apply the
continuity method to prove Theorem 1.2. Finally, we give an outline of the
proof of Theorem 1.3 in Section 6.

Acknowledgements: We would like to thank the anonymous referees for helpful
comments and especially for pointing out a mistake. This work was partially
supported by the National Natural Science Foundation of China, No. 11801405\.

##  2\. Preliminaries

Let $\mathbb{S}^{n}$ be the unit sphere in $\mathbb{R}^{n+1}$ and let $\nabla$
be the connection on it. Assume that $M$ is star-shaped with respect to the
origin, i.e. the position vector $X$ of $M$ can be written as $X(x)=\rho(x)x$,
where $x\in\mathbb{S}^{n}$. Then the unit outer normal of $M$ is given by

$\nu=\frac{\rho x-\nabla\rho}{\sqrt{\rho^{2}+|\nabla\rho|^{2}}}.$

Let $\\{e_{1},\cdots,e_{n}\\}$ be a smooth local orthonormal frame on
$\mathbb{S}^{n}$. Then the metric of $M$ is given by
$g_{ij}=\rho^{2}\delta_{ij}+\rho_{i}\rho_{j}$, and the second fundamental form
of $M$ is

(2.1)
$h_{ij}=\frac{\rho^{2}\delta_{ij}+2\rho_{i}\rho_{j}-\rho\rho_{ij}}{\sqrt{\rho^{2}+|\nabla\rho|^{2}}}.$

The principal curvatures $\kappa=(\kappa_{1},\cdots,\kappa_{n})$ are the
eigenvalues of $h_{ij}$ with respect to $g_{ij}$.

At a point $X_{0}$ in $M$, choose a local orthonormal frame
$\\{e_{1},e_{2}\cdots,e_{n}\\}$. The following geometric formulas are well
known:

(2.2) $\displaystyle X_{ij}=$ $\displaystyle\;-h_{ij}\nu\;\;(\mbox{Gauss
formula}),$ $\displaystyle(\nu)_{i}=$
$\displaystyle\;h_{ij}e_{j}\;\;(\mbox{Weingarten equation}),$ $\displaystyle
h_{ijk}=$ $\displaystyle\;h_{ikj}\;\;(\mbox{Codazzi formula}),$ $\displaystyle
R_{ijkl}=$ $\displaystyle\;h_{ik}h_{jl}-h_{il}h_{jk}\;\;(\mbox{Gauss
equation}),$

where $R_{ijkl}$ is the $(4,0)$-Riemannian curvature tensor and the formula

(2.3)
$h_{ijkl}=h_{klij}+h_{mk}(h_{mj}h_{il}-h_{ml}h_{ij})+h_{mi}(h_{mj}h_{kl}-h_{ml}h_{kj}).$

We recall the $p$-convex cones introduced by Harvey and Lawson [20].

######  Definition 2.1.

Let $p\in\\{1,\cdots,n\\}$. The cone $\mathcal{P}_{p}$ is defined by

$\mathcal{P}_{p}=\\{(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{R}^{n}|\;\forall\;1\leq
i_{1}<i_{2}<\cdots<i_{p}\leq n,\;\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}>0\\}.$

Associated to $\mathcal{P}_{p}$ is the cone of symmetric $n\times n$ matrices
defined by

$P_{p}=\\{A|\;\forall\;1\leq i_{1}<i_{2}<\cdots<i_{p}\leq
n,\;\lambda_{i_{1}}(A)+\cdots+\lambda_{i_{p}}(A)>0\\}.$

We call $A$ is $p$-positive if $A\in P_{p}$.

For convenience, we introduce the following notations

$F(h_{ij}):=F(\kappa)=\Pi_{1\leq i_{1}<\cdots<i_{p}\leq
n}(\kappa_{i_{1}}+\cdots+\kappa_{i_{p}})\;\mbox{and}\;\tilde{F}=F^{\frac{1}{C_{n}^{p}}},$

where $C_{n}^{p}=\frac{n!}{p!(n-p)!}$. Equation (1.1) then can be written as

(2.4) $\tilde{F}(h_{ij}):=\tilde{F}(\kappa)=\tilde{f}(X,\nu(X)),$

where $\kappa=(\kappa_{1},\cdots,\kappa_{n})$ and
$\tilde{f}=f^{\frac{1}{C_{n}^{p}}}$. Denote

$F^{ij}=\frac{\partial F}{\partial
h_{ij}},\;F^{ij,kl}=\frac{\partial^{2}F}{\partial h_{ij}\partial
h_{kl}},\;\mbox{and}\;\mathcal{F}=\sum F^{ii}.$

Direct calculations show that

$\tilde{F}^{ij}=\frac{1}{C_{n}^{p}}F^{\frac{1}{C_{n}^{p}}-1}F^{ij}$

and

$\tilde{F}^{ij,kl}=\frac{1}{C_{n}^{p}}F^{\frac{1}{C_{n}^{p}}-1}F^{ij,kl}+\frac{1}{C_{n}^{p}}\Big{(}\frac{1}{C_{n}^{p}}-1\Big{)}F^{\frac{1}{C_{n}^{p}}-2}F^{ij}F^{kl}.$

We remark that $\tilde{F}$ is concave with respect to $h_{ij}$ by Lemma 1.13
and Corollary 1.14 in [8]. And the equation is elliptic as the
matrix$\\{\frac{\partial\tilde{F}}{\partial h_{ij}}\\}$ is positive definite
for $\\{h_{ij}\\}\in P_{p}$.

Now we do some basic calculations which will be used in the next section. Our
calculations are carried out at a point $X_{0}$ on the hypersurface $M$, and
we use coordinates such that at this point $\\{h_{ij}\\}$ is diagonal and its
eigenvalues with respect to $g_{ij}$ are ordered as
$\kappa_{1}\geq\kappa_{2}\geq\cdots\geq\kappa_{n}$. Note that $F^{ij}$ is also
diagonal at $X_{0}$ and we have the following formulas

$F^{kk}=\frac{\partial
F}{\partial\kappa_{k}}=\sum_{k\in\\{i_{1},\cdots,i_{p}\\}}\frac{F(\kappa)}{\kappa_{i_{1}}+\cdots+\kappa_{i_{p}}},$

for which we refer to Lemma 1.10 in [8]. We also have formulas for the second
order derivatives of $F$ at $X_{0}$ as below

$F^{kk,ll}=\frac{\partial^{2}F}{\partial\kappa_{k}\partial\kappa_{l}}=\sum_{\begin{subarray}{c}k\in\\{i_{1},\cdots,i_{p}\\}\\\
l\in\\{j_{1},\cdots,j_{p}\\}\\\
\\{i_{1},\cdots,i_{p}\\}\neq\\{j_{1},\cdots,j_{p}\\}\end{subarray}}\frac{F(\kappa)}{(\kappa_{i_{1}}+\cdots+\kappa_{i_{p}})(\kappa_{j_{1}}+\cdots+\kappa_{j_{p}})},$

and, for $k\neq r$,

$F^{kr,rk}=\frac{F^{kk}-F^{rr}}{\kappa_{k}-\kappa_{r}}=-\sum_{\begin{subarray}{c}k\notin\\{i_{1},\cdots,i_{p}\\}\ni
r\\\ r\notin\\{j_{1},\cdots,j_{p}\\}\ni k\\\
\\{i_{1},\cdots,i_{p}\\}\backslash\\{r\\}=\\{j_{1},\cdots,j_{p}\\}\backslash\\{k\\}\end{subarray}}\frac{F(\kappa)}{(\kappa_{i_{1}}+\cdots+\kappa_{i_{p}})(\kappa_{j_{1}}+\cdots+\kappa_{j_{p}})}.$

Otherwise, we have $F^{ij,kl}=0$. See Lemma 1.12 in [8] for the above
formulas. These formulas can also be easily obtained from Theorem 5.5 in [2].
The following properties of the function $F$ which are very similar to the
properties of $\sigma_{k}$ were proved by Dinew [8].

######  Lemma 2.2 ([8]).

Suppose that the diagonal matrix $A=diag(\lambda_{1},\cdots,\lambda_{n})$
belongs to $P_{p}$ and that $\lambda_{1}\geq\cdots\geq\lambda_{n}$. Then,

  * (1)

$\tilde{F}^{11}(A)\lambda_{1}\geq\frac{1}{n}\tilde{F}(A);$

  * (2)

$\sum_{k=1}^{n}\tilde{F}^{kk}(A)\geq p;$

  * (3)

$\sum_{k=1}^{n}F^{kk}(A)\lambda_{k}=C_{n}^{p}F(A);$

  * (4)

there is a constant $\theta=\theta(n,p)$ such that $F^{jj}(A)\geq\theta\sum
F^{ii}$ for all $j\geq n-p+1.$

For the reader’s convenience, we provide a short proof of the above lemma in
the appendix.

##  3\. Curvature Estimates

Set $u=\langle X,\nu\rangle$, which is the support function of the
hypersurface $M$. Clearly, we have

$u=\frac{\rho^{2}}{\sqrt{\rho^{2}+|\nabla\rho|^{2}}}.$

There exists a positive constant $C$ depending on $\inf_{M}\rho$ and
$|\rho|_{C^{1}}$ such that

$\frac{1}{C}\leq\inf_{M}u\leq u\leq\sup_{M}u\leq C.$

In order to prove Theorem 1.1, we consider the following auxiliary function

$G=\log\kappa_{max}-\log(u-a)+\frac{A}{2}|X|^{2}$

where $\kappa_{max}$ is the largest principal curvature,
$a=\frac{1}{2}\inf_{M}u>0$ and $A\geq 1$ is a large constant to be determined.
Suppose the maximum of $G$ is achieved at a point $X_{0}\in M$. Choose a local
orthonormal frame $\\{e_{1},\cdots,e_{n}\\}$ around $X_{0}$ such that

$h_{ij}=\delta_{ij}h_{ii}\;\mbox{and}\;h_{11}\geq h_{22}\geq\cdots\geq
h_{nn}\;\mbox{at}\;X_{0}.$

Since $\kappa_{max}$ may not be differentiable, we define a new function
$\hat{G}$ near $X_{0}$ by

$\hat{G}=\log h_{11}-\log(u-a)+\frac{A}{2}|X|^{2}.$

It is easy to see $\hat{G}$ achieves a maximum at $X_{0}$. Now,
differentiating $\hat{G}$ at $X_{0}$ twice yields that

(3.1) $0=\frac{h_{11i}}{h_{11}}-\frac{u_{i}}{u-a}+A\langle X,e_{i}\rangle$

and

(3.2)
$0\geq\frac{h_{11ii}}{h_{11}}-\Big{(}\frac{h_{11i}}{h_{11}}\Big{)}^{2}-\frac{u_{ii}}{u-a}+\Big{(}\frac{u_{i}}{u-a}\Big{)}^{2}+A(1+\langle
X,X_{ii}\rangle).$

Contracting (3.2) with $\tilde{F}^{ii}$, we get

(3.3)
$0\geq\frac{\tilde{F}^{ii}h_{11ii}}{h_{11}}-\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}-\frac{\tilde{F}^{ii}u_{ii}}{u-a}+\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}+A\tilde{F}^{ii}(1+\langle
X,X_{ii}\rangle).$

######  Lemma 3.1.

We have

(3.4) $\displaystyle 0\geq$ $\displaystyle\;-\frac{2}{h_{11}}\sum_{i\geq
2}\tilde{F}^{1i,i1}h_{11i}^{2}-\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}-Ch_{11}$
$\displaystyle\;+\frac{a\tilde{F}^{ii}h_{ii}^{2}}{u-a}+\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}+A\sum\tilde{F}^{ii}-CA.$

###### Proof.

From the formula (2.3), we have

(3.5)
$\displaystyle\tilde{F}^{ii}h_{11ii}=\tilde{F}^{ii}h_{ii}h_{11}^{2}-\tilde{F}^{ii}h_{ii}^{2}h_{11}+\tilde{F}^{ii}h_{ii11}.$

Differentiating equation (1.1) twice at $X_{0}$, we obtain

(3.6)
$\tilde{F}^{ii}h_{iik}=(d_{X}\tilde{f})(e_{k})+h_{kk}(d_{\nu}\tilde{f})(e_{k})$

and

(3.7)
$\displaystyle\tilde{F}^{ii}h_{iikk}+\tilde{F}^{pq,rs}h_{pqk}h_{rsk}\geq-C-
Ch_{11}^{2}+\sum_{l}h_{lkk}(d_{\nu}\tilde{f})(e_{l}).$

By the concavity of $\tilde{F}$ and the Codazzi formula, we have

(3.8) $-\tilde{F}^{pq,rs}h_{pq1}h_{rs1}\geq-2\sum_{i\geq
2}\tilde{F}^{1i,i1}h_{11i}^{2}.$

Note that by Lemma 2.2 (3) we have $\tilde{F}^{ii}h_{ii}=\tilde{f}$. Hence, we
see that

(3.9) $\tilde{F}^{ii}h_{11ii}\geq-2\sum_{i\geq
2}\tilde{F}^{1i,i1}h_{11i}^{2}-\tilde{F}^{ii}h_{ii}^{2}h_{11}+\sum_{l}h_{l11}(d_{\nu}\tilde{f})(e_{l})-C-Ch_{11}^{2}.$

We now compute the term $\tilde{F}^{ii}u_{ii}$. By (2.2), we have

$u_{i}=h_{ii}\langle X,e_{i}\rangle\;\mbox{and}\;u_{ii}=\sum_{k}h_{iik}\langle
X,e_{k}\rangle-uh_{ii}^{2}+h_{ii}.$

Hence, we obtain

(3.10) $\displaystyle\tilde{F}^{ii}u_{ii}=$
$\displaystyle\;\sum_{k}\tilde{F}^{ii}h_{iik}\langle
X,e_{k}\rangle-u\tilde{F}^{ii}h_{ii}^{2}+\tilde{f}$ $\displaystyle\leq$
$\displaystyle\;\sum_{k}h_{kk}(d_{\nu}\tilde{f})(e_{k})\langle
X,e_{k}\rangle-u\tilde{F}^{ii}h_{ii}^{2}+C.$

By the Gauss formula, we have

(3.11) $\langle X,X_{ii}\rangle=-h_{ii}\langle X,\nu\rangle=-h_{ii}u.$

Substituting (3.9), (3.10) and (3.11) in (3.3), we obtain that

(3.12) $\displaystyle 0\geq$ $\displaystyle\,-\frac{2}{h_{11}}\sum_{i\geq
2}\tilde{F}^{1i,i1}h_{11i}^{2}-\tilde{F}^{ii}h_{ii}^{2}+\frac{1}{h_{11}}\sum_{l}h_{l11}(d_{\nu}\tilde{f})(e_{l})-Ch_{11}$
$\displaystyle\;-\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}-\frac{\sum_{k}h_{kk}(d_{\nu}\tilde{f})(e_{k})\langle
X,e_{k}\rangle}{u-a}+\frac{u\tilde{F}^{ii}h_{ii}^{2}}{u-a}-\frac{C}{u-a}$
$\displaystyle\;+\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}+A\sum\tilde{F}^{ii}-CA.$

By the Codazzi formula, $u_{k}=h_{kk}\langle X,e_{k}\rangle$ and (3.1), we
have

$\frac{1}{h_{11}}\sum_{k}h_{k11}(d_{\nu}\tilde{f})(e_{k})-\frac{h_{kk}(d_{\nu}\tilde{f})(e_{k})\langle
X,e_{k}\rangle}{u-a}\geq-CA.$

Therefore, we arrive at

(3.13) $\displaystyle 0\geq$ $\displaystyle\,-\frac{2}{h_{11}}\sum_{i\geq
2}\tilde{F}^{1i,i1}h_{11i}^{2}-\tilde{F}^{ii}h_{ii}^{2}-Ch_{11}-\frac{C}{u-a}$
$\displaystyle\;-\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}+\frac{u\tilde{F}^{ii}h_{ii}^{2}}{u-a}+\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}+A\sum\tilde{F}^{ii}-CA,$

which is just the inequality (3.4).

∎

Next, we deal with the bad term $-Ch_{11}$.

######  Lemma 3.2.

Suppose $p\geq\frac{n}{2}$. If $h_{11}$ is large enough, we have

(3.14)
$Ch_{11}\leq\frac{a\tilde{F}^{ii}h_{ii}^{2}}{2(u-a)}+\frac{A}{2}\sum\tilde{F}^{ii}$

for sufficiently large $A$.

###### Proof.

Note that

$\kappa_{n-p+1}+\kappa_{n-p+2}+\cdots+\kappa_{n}>0.$

We divide the proof into two cases.

Case 1. Suppose $\kappa_{n}\leq-\delta\kappa_{1}$, where $\delta>0$ is a small
constant to be determined later. By Lemma 2.2 we see
$\tilde{F}^{nn}\geq\theta\sum\tilde{F}^{ii}\geq\theta p$. We then obtain that

$\tilde{F}^{nn}h_{nn}^{2}\geq\delta^{2}\kappa_{1}^{2}\tilde{F}^{nn}\geq\theta
p\delta^{2}\kappa_{1}^{2}.$

Therefore, for sufficiently large $\kappa_{1}$, we have

$Ch_{11}\leq\frac{a\tilde{F}^{nn}h_{nn}^{2}}{2(u-a)}.$

Case 2. Now $\kappa_{n}\geq-\delta\kappa_{1}$. We further divide this case
into two cases.

Subcase 2.1. Suppose
$\kappa_{n-p+1}+\kappa_{n-p+2}+\cdots+\kappa_{n}<\frac{\delta}{\kappa_{1}}$.
Since

$F^{nn}\geq\frac{F(\kappa)}{\kappa_{n-p+1}+\kappa_{n-p+2}+\cdots+\kappa_{n}},$

we see that

$\tilde{F}^{nn}=\frac{1}{C_{n}^{p}}F^{\frac{1}{C_{n}^{p}}-1}F^{nn}\geq\frac{F^{\frac{1}{C_{n}^{p}}}}{C_{n}^{p}}\frac{\kappa_{1}}{\delta}=\frac{\tilde{f}}{C_{n}^{p}}\frac{\kappa_{1}}{\delta}.$

Choosing $\delta$ sufficiently small, we obtain that

$C\kappa_{1}\leq\tilde{F}^{nn}.$

Subcase 2.2. Suppose
$\kappa_{n-p+1}+\kappa_{n-p+2}+\cdots+\kappa_{n}\geq\frac{\delta}{\kappa_{1}}$.
For a fixed $(p-1)$-tuple $2\leq i_{1}<\cdots<i_{p-1}\leq n$, we have

$\kappa_{1}+\kappa_{i_{1}}+\cdots+\kappa_{i_{p-1}}\geq(1-(p-1)\delta)\kappa_{1}.$

Hence, we have

$\displaystyle F^{nn}\geq$ $\displaystyle\;\mathop{\Pi}\limits_{2\leq
i_{1}<\cdots<i_{p-1}\leq
n}(\kappa_{1}+\kappa_{i_{1}}+\cdots+\kappa_{i_{p-1}})$
$\displaystyle\times\mathop{\Pi}\limits_{\begin{subarray}{c}2\leq
i_{1}<\cdots<i_{p}\leq n\\\
(i_{1},\cdots,i_{p})\neq(n-p+1,\cdots,n)\end{subarray}}(\kappa_{i_{1}}+\kappa_{i_{2}}+\cdots+\kappa_{i_{p}})$
$\displaystyle\geq$
$\displaystyle\;[(1-(p-1)\delta)\kappa_{1}]^{C_{n-1}^{p-1}}[\frac{\delta}{\kappa_{1}}]^{C_{n-1}^{p}-1}.$

For $p\geq\frac{n}{2}$, a direct calculation shows that

$C_{n-1}^{p-1}-C_{n-1}^{p}=\frac{(n-1)\cdots(n-p+1)}{(p-1)!}\Big{(}1-\frac{n-p}{p}\Big{)}\geq
0.$

Therefore, we obtain

$F^{nn}\geq c_{\delta}\kappa_{1},$

where $c_{\delta}=[(1-(p-1)\delta)]^{C_{n-1}^{p-1}}\delta^{C_{n-1}^{p}-1}$. It
then follows that, for sufficiently large $A$,

$C\kappa_{1}\leq\frac{A}{2}\tilde{F}^{nn}.$

∎

By the above Lemma, (3.4) becomes

(3.15) $\displaystyle 0\geq$ $\displaystyle\,-\frac{2}{h_{11}}\sum_{i\geq
2}\tilde{F}^{1i,i1}h_{11i}^{2}-\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}+\frac{a\tilde{F}^{ii}h_{ii}^{2}}{2(u-a)}$
$\displaystyle\;+\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}+\frac{A}{2}\sum\tilde{F}^{ii}-CA.$

######  Lemma 3.3.

For $\kappa_{1}$ sufficiently large, we have

$|\kappa_{n-p+1}|,\cdots,|\kappa_{n}|\leq CA.$

###### Proof.

By the critical equation (3.1) and the Cauchy-Schwarz inequality, we have

(3.16)
$-\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}\geq-(1+\varepsilon)\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}-\Big{(}1+\frac{1}{\varepsilon}\Big{)}A^{2}\tilde{F}^{ii}\langle
X,e_{i}\rangle^{2}.$

From (3.15) and $-\tilde{F}^{1i,i1}\geq 0$, we see that

(3.17) $\displaystyle 0\geq$
$\displaystyle\;\frac{a\tilde{F}^{ii}h_{ii}^{2}}{2(u-a)}-\frac{\varepsilon\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}-\frac{CA^{2}}{\varepsilon}\sum\tilde{F}^{ii}-CA.$

Using $u_{i}=h_{ii}\langle X,e_{i}\rangle$ and choosing $\varepsilon$
sufficiently small, we obtain from (3.17) that

(3.18) $\displaystyle 0\geq$
$\displaystyle\;\frac{a\tilde{F}^{ii}h_{ii}^{2}}{4(u-a)}-\frac{CA^{2}}{\varepsilon}\sum\tilde{F}^{ii},$

where we also used $\sum\tilde{F}^{ii}\geq p$. By Lemma 2.2, we now arrive at

(3.19) $\displaystyle 0\geq$
$\displaystyle\;\frac{a\theta}{4(u-a)}\Big{(}\sum\tilde{F}^{ii}\Big{)}\Big{(}\sum_{i\geq
n-p+1}h_{ii}^{2}\Big{)}-\frac{CA^{2}}{\varepsilon}\sum\tilde{F}^{ii},$

which implies that

$\sum_{i\geq n-p+1}h_{ii}^{2}\leq CA^{2}.$

∎

######  Lemma 3.4.

Given $1>\delta>0$, there is an $\epsilon=\epsilon(p,\delta)>0$ such that,

$-2F^{1i,i1}+2\frac{F^{11}}{\kappa_{1}}\geq(1+\delta)\frac{F^{ii}}{\kappa_{1}},\;i=2,3,\cdots
n,$

for $\kappa_{1}$ sufficiently large.

###### Proof.

Recall that $\kappa_{1}\geq\kappa_{2}\geq\cdots\geq\kappa_{n}$. By the formula
$F^{1i,i1}=\frac{F^{11}-F^{ii}}{\kappa_{1}-\kappa_{i}}$, we see that

$\frac{F^{ii}}{\kappa_{1}}=\frac{\kappa_{1}-\kappa_{i}}{\kappa_{1}}(-F^{1i,i1})+\frac{F^{11}}{\kappa_{1}}.$

Since $\kappa_{i}>0$ for $i\leq n-p+1$, we obtain

$\frac{F^{ii}}{\kappa_{1}}\leq-F^{1i,i1}+\frac{F^{11}}{\kappa_{1}},\;\mbox{for}\;i=2,\cdots,n-p+1.$

By Lemma 3.3, we can assume that $|\kappa_{i}|\leq\epsilon\kappa_{1}$ for
$i\geq n-p+2$ for sufficiently small $\epsilon$ and large $\kappa_{1}$. Hence,
we have

$\frac{F^{ii}}{\kappa_{1}}\leq-(1+\epsilon)F^{1i,i1}+\frac{F^{11}}{\kappa_{1}},\;\mbox{for}\;i=n-p+2,\cdots,n.$

By the above two inequalities, we get the desired inequality. ∎

By Lemma 3.4, (3.15) becomes

(3.20) $\displaystyle 0\geq$ $\displaystyle\,\sum_{i\geq
2}\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}-\frac{\tilde{F}^{ii}h_{11i}^{2}}{h_{11}^{2}}-2\sum_{i\geq
2}\frac{\tilde{F}^{11}h_{11i}^{2}}{h_{11}^{2}}+\frac{a\tilde{F}^{ii}h_{ii}^{2}}{2(u-a)}$
$\displaystyle\;+\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}+\frac{A}{2}\sum\tilde{F}^{ii}-CA.$

By the critical equation (3.1), the Cauchy-Schwarz inequality and
$u_{i}=h_{ii}\langle X,e_{i}\rangle$, we see that

(3.21) $\displaystyle-\frac{\tilde{F}^{11}h_{111}^{2}}{h_{11}^{2}}\geq$
$\displaystyle\;-(1+\varepsilon)\frac{\tilde{F}^{11}u_{1}^{2}}{(u-a)^{2}}-\Big{(}1+\frac{1}{\varepsilon}\Big{)}A^{2}\tilde{F}^{11}\langle
X,e_{1}\rangle^{2}$ $\displaystyle\geq$
$\displaystyle\;-\frac{\tilde{F}^{11}u_{1}^{2}}{(u-a)^{2}}-C\varepsilon\frac{\tilde{F}^{11}h_{11}^{2}}{(u-a)^{2}}-\frac{CA^{2}}{\varepsilon}\tilde{F}^{11}$

and

$2\sum_{i\geq 2}\frac{\tilde{F}^{11}h_{11i}^{2}}{h_{11}^{2}}\leq C\sum_{i\geq
2}\tilde{F}^{11}h_{ii}^{2}+CA^{2}F^{11}.$

Note that

$F^{11}=\sum_{1\notin\\{i_{1},\cdots,i_{p-1}\\}}\frac{F(\kappa)}{\kappa_{1}+\kappa_{i_{1}}+\cdots+\kappa_{i_{p-1}}}\leq\frac{C}{\kappa_{1}}.$

Hence, by Lemma 3.2, we obtain

(3.22) $2\sum_{i\geq
2}\frac{\tilde{F}^{11}h_{11i}^{2}}{h_{11}^{2}}\leq\frac{a\tilde{F}^{ii}h_{ii}^{2}}{4(u-a)}+\frac{A}{4}\sum\tilde{F}^{ii}+CA^{2}\tilde{F}^{11}.$

Substituting (3.21) and (3.22) into (3.20), we have

(3.23) $\displaystyle 0\geq$
$\displaystyle\,\frac{a\tilde{F}^{ii}h_{ii}^{2}}{4(u-a)}-\frac{\tilde{F}^{11}u_{1}^{2}}{(u-a)^{2}}-C\varepsilon\frac{\tilde{F}^{11}h_{11}^{2}}{(u-a)^{2}}-\frac{CA^{2}}{\varepsilon}\tilde{F}^{11}$
$\displaystyle+$
$\displaystyle\;\frac{\tilde{F}^{ii}u_{i}^{2}}{(u-a)^{2}}+\frac{A}{4}\sum\tilde{F}^{ii}-CA^{2}\tilde{F}^{11}-CA.$

Choosing $\varepsilon$ sufficiently small and assuming $h_{11}$ sufficiently
large, we derive that

(3.24) $\displaystyle 0\geq$
$\displaystyle\,\frac{a\tilde{F}^{ii}h_{ii}^{2}}{8(u-a)}+\frac{A}{4}\sum\tilde{F}^{ii}-CA.$

It then follows that

$\sum\tilde{F}^{ii}\leq C.$

Next we prove that under this condition, one have

$\tilde{F}^{11}\geq\frac{1}{C}.$

Since $\sum F^{ii}\leq C$, in particular we have

$\frac{F(\kappa)}{\kappa_{n-p+1}+\cdots+\kappa_{n}}\leq C.$

This implies that

$\kappa_{n-p+1}+\cdots+\kappa_{n}\geq\frac{1}{C},$

where $C$ also depends on $\inf f$. This yields

$\tilde{F}^{ii}\geq\frac{1}{C},\;\forall\;1\leq i\leq n.$

Substituting the above inequality into (3.24), we obtain

$0\geq\frac{h_{11}^{2}}{C}-CA,$

from which we can derive an upper bound for $h_{11}$. Theorem 1.1 is proved.

##  4\. Gradient Estimates

Before we apply the continuity method to obtain a solution to equation (1.1),
we need to derive a $C^{1}$ estimate for the equation. We show that there
exists a positive constant $C$ depending on $n$, $p$, $\inf\rho$, $\sup\rho$,
$\inf f$ and $|f|_{C^{1}}$ such that

$|\nabla\rho|\leq C,$

where $\nabla$ denotes the connection on $\mathbb{S}^{n}$. Note that

$u=\frac{\rho^{2}}{\sqrt{\rho^{2}+|\nabla\rho|^{2}}}.$

We only need to derive a positive lower bound of $u$. As in [16] we consider
the following quantity

$w=-\log u+\gamma(|X|^{2}),$

where the function $\gamma(\cdot)$ will be determined later. Suppose the
maximum of $w$ is achieved at $X_{0}\in M$. If at $X_{0}$, $X$ is parallel to
$\nu$, we have

$u=\langle X,\nu\rangle=\rho\geq\inf_{M}\rho,$

which gives a lower bound since $\rho$ is assumed to have a positive lower
bound. If at $X_{0}$ , $X$ is not parallel to $\nu$, we can choose a local
orthonormal frame $\\{e_{1},e_{2},\cdots,e_{n}\\}$ such that

$\langle X,e_{1}\rangle\neq 0\;\mbox{and}\;\langle
X,e_{i}\rangle=0\;\mbox{for}\;i\geq 2.$

Differentiate $w$ at $X_{0}$ to obtain that

(4.1) $0=w_{i}=-\frac{u_{i}}{u}+2\gamma^{\prime}\langle
X,e_{i}\rangle=-\frac{h_{i1}\langle X,e_{1}\rangle}{u}+2\gamma^{\prime}\langle
X,e_{i}\rangle,$

where in the last equality we used the Weingarten equation. Hence, we have

$h_{11}=2\gamma^{\prime}u\;\mbox{and}\;h_{1i}=0\;\mbox{for}\;i\geq 2.$

Without loss of generality, we can assume $\\{h_{ij}\\}$ is diagonal at
$X_{0}$. Differentiating $w$ at $X_{0}$ a second time and contracting with
$\\{\tilde{F}^{ij}\\}$, we obtain that

(4.2)
$0\geq\tilde{F}^{ii}\Big{(}-\frac{u_{ii}}{u}+\frac{u_{i}^{2}}{u^{2}}+\gamma^{\prime\prime}(|X|^{2})_{i}^{2}+\gamma^{\prime}(|X|^{2})_{ii}\Big{)}.$

Combining (4.1) with the above inequality, we arrive at

(4.3) $0\geq-\frac{\tilde{F}^{ii}u_{ii}}{u}+4(\gamma^{\prime
2}+\gamma^{\prime\prime})F^{11}\langle
X,e_{1}\rangle^{2}+\gamma^{\prime}\tilde{F}^{ii}(|X|^{2})_{ii}.$

By (3.10), we have

(4.4) $\displaystyle\tilde{F}^{ii}u_{ii}=\langle
X,e_{1}\rangle((d_{X}\tilde{f})(e_{1})+h_{11}(d_{\nu}\tilde{f})(e_{1}))-u\tilde{F}^{ii}h_{ii}^{2}+\tilde{f}.$

Also, we have

$\tilde{F}^{ii}(|X|^{2})_{ii}=2\sum\tilde{F}^{ii}-2u\tilde{f},$

where we used $\tilde{F}^{ij}h_{ij}=\tilde{f}$. Recall that
$h_{11}=2\gamma^{\prime}u$. Substituting the above two equalities into (4.3)
we get

(4.5) $\displaystyle 0\geq$ $\displaystyle\;-\frac{1}{u}\Big{(}\langle
X,e_{1}\rangle(d_{X}\tilde{f})(e_{1})+\tilde{f}\Big{)}-2\langle
X,e_{1}\rangle\gamma^{\prime}(d_{\nu}\tilde{f})(e_{1})$
$\displaystyle\;+\tilde{F}^{ii}h_{ii}^{2}+4(\gamma^{\prime
2}+\gamma^{\prime\prime})F^{11}\langle
X,e_{1}\rangle^{2}+2\gamma^{\prime}(\sum\tilde{F}^{ii}-u\tilde{f}).$

At $X_{0}$, we see that $X=\langle X,e_{1}\rangle e_{1}+\langle
X,\nu\rangle\nu$. It then follows that

$(d_{X}\tilde{f})(X)=\langle X,e_{1}\rangle(d_{X}\tilde{f})(e_{1})+\langle
X,\nu\rangle(d_{X}\tilde{f})(\nu).$

From (1.4), we see that

$\displaystyle 0\geq$
$\displaystyle\;\frac{\partial}{\partial\rho}\Big{(}\rho^{C_{n}^{p}}f(X,\nu)\Big{)}=\frac{\partial}{\partial\rho}\Big{(}\rho^{C_{n}^{p}}\tilde{f}^{C_{n}^{p}}(X,\nu)\Big{)}$
$\displaystyle=$
$\displaystyle\;{C_{n}^{p}}(\rho\tilde{f})^{{C_{n}^{p}}-1}\Big{(}\tilde{f}+(d_{X}\tilde{f})(X)\Big{)}$
$\displaystyle=$
$\displaystyle\;{C_{n}^{p}}(\rho\tilde{f})^{{C_{n}^{p}}-1}\Big{(}\tilde{f}+\langle
X,e_{1}\rangle(d_{X}\tilde{f})(e_{1})+\langle
X,\nu\rangle(d_{X}\tilde{f})(\nu)\Big{)}.$

We therefore obtain

$-(\tilde{f}+\langle X,e_{1}\rangle(d_{X}\tilde{f})(e_{1}))\geq\langle
X,\nu\rangle(d_{X}\tilde{f})(\nu)=u(d_{X}\tilde{f})(\nu).$

Substituting this into (4.5) we obtain

(4.6) $\displaystyle 0\geq$ $\displaystyle\;(d_{X}\tilde{f})(\nu)-2\langle
X,e_{1}\rangle\gamma^{\prime}(d_{\nu}\tilde{f})(e_{1})+\tilde{F}^{ii}h_{ii}^{2}$
$\displaystyle\;+4(\gamma^{\prime 2}+\gamma^{\prime\prime})F^{11}\langle
X,e_{1}\rangle^{2}+2\gamma^{\prime}(\sum\tilde{F}^{ii}-u\tilde{f}).$

Now we choose $\gamma(t)=\frac{\alpha}{t}$, where $\alpha$ is a large constant
to be determined later. Recall that $h_{11}=2\gamma^{\prime}u$ at $X_{0}$,
which implies that $h_{11}(X_{0})<0$. This means that
$h_{11}\in\\{\kappa_{n-p+2},\kappa_{n-p+3},\cdots,\kappa_{n}\\}$ and therefore
by Lemma 2.2

$F^{11}\geq\theta\sum F^{ii}.$

Similar to [7], we can assume $\langle
X,e_{1}\rangle^{2}\geq\frac{1}{2}\inf_{M}\rho^{2}$. Now we arrive at

(4.7) $\displaystyle 0\geq$
$\displaystyle\;\Big{(}\frac{\alpha^{2}}{C}-C\alpha\Big{)}\sum\tilde{F}^{ii}-C\alpha.$

Choosing $\alpha$ sufficiently large, we obtain a contradiction. Therefore,
$X_{0}$ is parallel to $\nu$, and $u$ has a positive lower bound.

##  5\. Existence of a solution

We use the continuity method as in [4] to prove Theorem 1.2. Consider the
following family of functions

$f^{t}(X,\nu)=tf(X,\nu)+(1-t)p^{C_{n}^{p}}[\frac{1}{|X|^{C_{n}^{p}}}+\varepsilon(\frac{1}{|X|^{C_{n}^{p}}}-1)],$

where $\varepsilon$ is a small positive constant such that

$\min_{r_{1}\leq\rho\leq
r_{2}}[\frac{1}{\rho^{C_{n}^{p}}}+\varepsilon(\frac{1}{\rho^{C_{n}^{p}}}-1)]\geq
c_{0}>0,$

for some positive constant $c_{0}$. It is easy to see that$f^{t}(X,\nu)$
satisfies (1.3) and (1.4) with strict inequalities for $0\leq t<1$.

Let $M_{t}$ be the solution of the equation

$F(\kappa)=f^{t}(X_{t},\nu_{t}),$

where $X_{t}$ and $\nu_{t}$ are position vector and unit outer normal of
$M_{t}$ respectively. Clearly, when $t=0$, we have $M_{0}=\mathbb{S}^{n}$ and
$X_{0}=x$. For $t\in(0,1)$, suppose $\rho_{t}=|X_{t}|$ attains its maximum at
the point $x_{0}$. At this point, by (2.1), we have

$g_{ij}=\rho_{t}^{2}\delta_{ij}\;\mbox{and}\;h_{ij}=-(\rho_{t})_{ij}+\rho_{t}\delta_{ij}\geq\rho_{t}\delta_{ij}$

under a smooth local orthonormal frame on $\mathbb{S}^{n}$. Then, we have

$F(\kappa)\geq
F(\frac{1}{\rho_{t}}(1,\cdots,1))=\frac{p^{C_{n}^{p}}}{\rho_{t}^{C_{n}^{p}}}.$

On the other hand, at $x_{0}$, the unit outer normal $\nu_{t}$ is parallel to
$X_{t}$. If $\rho_{t}(x_{0})=r_{2}$, we obtain

$\frac{p^{C_{n}^{p}}}{{r_{2}}^{C_{n}^{p}}}\leq
F(\kappa)=f^{t}(X_{t},\nu_{t})<\frac{p^{C_{n}^{p}}}{{r_{2}}^{C_{n}^{p}}},$

which is a contradiction. So we have $\sup_{M_{t}}\rho_{t}\leq r_{2}$.
Similarly argument at the minimum point of $\rho_{t}$ gives that
$\inf_{M_{t}}\rho_{t}\geq r_{1}$ on $M_{t}$. Hence, $C^{0}$ estimate follows.
Combining our $C^{1}$ estimate, our $C^{2}$ estimate, the Evans-Krylov theorem
with the argument in [4], we get the existence and uniqueness of solution to
equation (1.1). Theorem 1.2 is proved.

##  6\. Proof of Theorem 1.3

By Lemmas 3.1, 3.2, 3.3 and 3.4 one can prove Theorem 1.3. For completeness,
we include an outline here.

###### Proof.

We consider the following function

$G(x,\xi)=\log u_{\xi\xi}+\frac{a}{2}|\nabla
u|^{2}+\frac{A}{2}|x|^{2}+\beta\log(v-u),$

where $a,A$, and $\beta$ are constants to be determined later. Suppose that
$G$ achieves its maximum at $(x_{0},\xi_{0})$. Around $x_{0}$, we a choose
coordinate system such that $\xi_{0}=e_{1}$ and $u_{ij}(x_{0})$ is diagonal
such that

$u_{11}\geq u_{22}\geq\cdots\geq u_{nn}\;\mbox{at}\;x_{0}.$

This can be done as in [12]. Thus, the new function defined by

$\hat{G}(x)=\log u_{11}+\frac{a}{2}|\nabla
u|^{2}+\frac{A}{2}|x|^{2}+\beta\log(v-u)$

also attains its maximum at $x_{0}$. Differentiate it once to obtain

(6.1)
$0=\frac{u_{11i}}{u_{11}}+au_{i}u_{ii}+Ax_{i}+\frac{\beta(v-u)_{i}}{v-u}.$

Differentiating it twice and by similar computations as Lemma 3.1 and Lemma
3.2 we can arrive at

(6.2) $\displaystyle 0\geq$ $\displaystyle\,-\frac{2}{u_{11}}\sum_{i\geq
2}\tilde{F}^{1i,i1}u_{11i}^{2}-\frac{\tilde{F}^{ii}u_{11i}^{2}}{u_{11}^{2}}+\frac{a\tilde{F}^{ii}u_{ii}^{2}}{2}$
$\displaystyle\;+\frac{A}{2}\sum\tilde{F}^{ii}-CA-\frac{\beta\tilde{F}^{ii}(v-u)_{i}^{2}}{(v-u)^{2}}-\frac{C\beta}{v-u}.$

We remark that in the above inequality we used

$\displaystyle\sum_{k}F^{kk}v_{kk}=$
$\displaystyle\;\sum_{k}\sum_{k\in\\{i_{1},\cdots,i_{p}\\}}\frac{F(\lambda)}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}v_{kk}$
$\displaystyle=$ $\displaystyle\;\sum_{1\leq i_{1}<i_{2}<\cdots<i_{p}\leq
n}\frac{F(\lambda)}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}(v_{i_{1}i_{1}}+\cdots+v_{i_{p}i_{p}})$

which is nonnegative since $v$ is $p$-plurisubharmonic. Using the same
argument as in Lemma 3.3, we obtain

$|u_{ii}|\leq\frac{C}{v-u}\;\mbox{for}\;i\geq n-p+1,$

where $C$ depends on $a,A$ and $\beta$.

By (6.1) and the Cauchy-Schwarz inequality, we have

$-\frac{\tilde{F}^{11}u_{111}^{2}}{u_{11}^{2}}\geq-
Ca^{2}\tilde{F}^{11}u_{11}^{2}-CA^{2}\tilde{F}^{11}-\frac{C\beta^{2}\tilde{F}^{11}}{(v-u)^{2}}$

and

$-\sum_{i\geq
2}\frac{\beta\tilde{F}^{ii}(v-u)_{i}^{2}}{(v-u)^{2}}\geq-\frac{3}{\beta}\sum_{i\geq
2}\frac{\tilde{F}^{ii}u_{11i}^{2}}{u_{11}^{2}}-\frac{Ca^{2}}{\beta}\sum_{i\geq
2}\tilde{F}^{ii}u_{ii}^{2}-\frac{CA^{2}}{\beta}\sum_{i\geq 2}\tilde{F}^{ii}.$

Substituting the above two inequalities into (6.2) we have

(6.3) $\displaystyle 0\geq$ $\displaystyle\,-\frac{2}{u_{11}}\sum_{i\geq
2}\tilde{F}^{1i,i1}u_{11i}^{2}-\Big{(}1+\frac{3}{\beta}\Big{)}\sum_{i\geq
2}\frac{\tilde{F}^{ii}u_{11i}^{2}}{u_{11}^{2}}+\Big{(}\frac{a}{2}-\frac{Ca^{2}}{\beta}\Big{)}\tilde{F}^{ii}u_{ii}^{2}$
$\displaystyle\;+\Big{(}\frac{A}{2}-\frac{CA^{2}}{\beta}\Big{)}\sum\tilde{F}^{ii}-Ca^{2}\tilde{F}^{11}u_{11}^{2}-CA^{2}\tilde{F}^{11}-\frac{C\beta^{2}\tilde{F}^{11}}{(v-u)^{2}}$
$\displaystyle\;-CA-\frac{2\beta\tilde{F}^{11}u_{1}^{2}}{(v-u)^{2}}-\frac{C\beta}{v-u}.$

By Lemma 3.2, similar to (3.22), we can get

(6.4) $2\sum_{i\geq
2}\tilde{F}^{11}\frac{u_{11i}^{2}}{u_{11}^{2}}\leq\frac{a}{8}\tilde{F}^{ii}u_{ii}^{2}+\frac{A}{8}\sum\tilde{F}^{ii}+CA^{2}\tilde{F}^{11}+\frac{C\beta^{2}}{(v-u)^{2}}\tilde{F}^{11}$

for sufficiently large $(v-u)u_{11}$ and $A$. Combining Lemma 3.4 with (6.4)
and choosing $a$ sufficiently small and $\beta$ sufficiently large such that
$\delta\geq\frac{3}{\beta}$, we get from (6.3) that

(6.5) $\displaystyle 0\geq$
$\displaystyle\,\frac{a}{4}\tilde{F}^{11}u_{11}^{2}-\frac{C}{(v-u)^{2}}\tilde{F}^{11}-CA-\frac{C}{v-u}$
$\displaystyle\geq$
$\displaystyle\,\frac{a}{8}\tilde{F}^{11}u_{11}^{2}-\frac{C}{v-u},$

where in the second inequality we assumed $(v-u)u_{11}$ is large enough.

By Lemma 2.2 (1) we have that $\tilde{F}^{11}u_{11}\geq c_{0}$, where
$c_{0}>0$ depends on $\inf\tilde{f}$. From (6.5), we then obtain

$(v-u)u_{11}\leq C$

which implies the estimate (1.6).

∎

##  7\. Appendix

In this appendix, we include a proof of Lemma 2.2. For $A\in P_{p}$, recall
the notations

$F(A):=F(\lambda(A))=\Pi_{1\leq i_{1}<\cdots<i_{p}\leq
n}(\lambda_{i_{1}}+\cdots+\lambda_{i_{p}})\;\mbox{and}\;\tilde{F}=F^{\frac{1}{C_{n}^{p}}},$

where $\lambda(A)=(\lambda_{1},\cdots,\lambda_{n})$ are the eigenvalues of $A$
and $P_{p}$ is defined in Definition 2.1. Suppose that the diagonal matrix
$A=diag(\lambda_{1},\cdots,\lambda_{n})$ belongs to $P_{p}$ and
$\lambda_{1}\geq\cdots\geq\lambda_{n}$.

######  Lemma .1.

$\tilde{F}^{11}(A)\lambda_{1}\geq\frac{1}{n}\tilde{F}(A)$

###### Proof.

We have that

$\displaystyle\tilde{F}^{11}(A)=$
$\displaystyle\;\frac{1}{C_{n}^{p}}[F(A)]^{\frac{1}{C_{n}^{p}}-1}\sum_{1\in\\{i_{1},\cdots,i_{p}\\}}\frac{F(A)}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}$
$\displaystyle\geq$
$\displaystyle\;\frac{1}{C_{n}^{p}}[F(A)]^{\frac{1}{C_{n}^{p}}-1}C_{n-1}^{p-1}\frac{F(A)}{p\lambda_{1}}$
$\displaystyle=$
$\displaystyle\;\frac{1}{n\lambda_{1}}[F(A)]^{\frac{1}{C_{n}^{p}}},$

where in the inequality we used $\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}\leq
p\lambda_{1}$. ∎

######  Lemma .2.

$\sum_{k=1}^{n}\tilde{F}^{kk}(A)\geq p$.

###### Proof.

We have that

$\displaystyle\sum_{k=1}^{n}\tilde{F}^{kk}(A)=$
$\displaystyle\;\frac{1}{C_{n}^{p}}[F(A)]^{\frac{1}{C_{n}^{p}}-1}\sum_{k=1}^{n}\sum_{k\in\\{i_{1},\cdots,i_{p}\\}}\frac{F(A)}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}$
$\displaystyle=$
$\displaystyle\;\frac{nC_{n-1}^{p-1}}{[C_{n}^{p}]^{2}}[F(A)]^{\frac{1}{C_{n}^{p}}}\sum_{1\leq
i_{1}<\cdots<i_{p}\leq n}\frac{1}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}$
$\displaystyle\geq$
$\displaystyle\;\frac{nC_{n-1}^{p-1}}{[C_{n}^{p}]^{2}}[F(A)]^{\frac{1}{C_{n}^{p}}}\frac{C_{n}^{p}}{[F(A)]^{\frac{1}{C_{n}^{p}}}}=p,$

where the inequality of arithmetic and geometric means was used in the
inequality. ∎

######  Lemma .3.

$\sum_{k=1}^{n}F^{kk}(A)\lambda_{k}=C_{n}^{p}F(A)$.

###### Proof.

Observe that

$\displaystyle\sum_{k=1}^{n}F^{kk}(A)\lambda_{k}=$
$\displaystyle\;F(A)\sum_{k=1}^{n}\sum_{k\in\\{i_{1},\cdots,i_{p}\\}}\frac{\lambda_{k}}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}$
$\displaystyle=$ $\displaystyle\;F(A)\frac{nC_{n-1}^{p-1}}{p}=C_{n}^{p}F(A).$

∎

######  Lemma .4.

$\sum_{k=1}^{n}\tilde{F}^{kk}(A)\lambda_{k}=\tilde{F}(A).$

###### Proof.

Observe that

$\sum_{k=1}^{n}\tilde{F}^{kk}(A)\lambda_{k}=\frac{1}{C_{n}^{p}}[F(A)]^{\frac{1}{C_{n}^{p}}-1}\sum_{k=1}^{n}F^{kk}(A)\lambda_{k}=\tilde{F}(A).$

∎

######  Lemma .5.

There is a constant $\theta=\theta(n,p)$ such that, for all $j\geq n-p+1$,

$F^{jj}(A)\geq\theta\sum_{i=1}^{n}F^{ii}(A).$

###### Proof.

Note that, for $j\geq n-p+1$,

$F^{jj}(A)\geq\frac{F(A)}{\lambda_{n-p+1}+\cdots+\lambda_{n}}\geq\frac{1}{C_{n}^{p}}\sum_{1\leq
i_{1}<\cdots<i_{p}\leq n}\frac{F(A)}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}},$

and

$\displaystyle\sum_{k=1}^{n}F^{kk}(A)=$
$\displaystyle\;\sum_{k=1}^{n}\sum_{k\in\\{i_{1},\cdots,i_{p}\\}}\frac{F(A)}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}$
$\displaystyle=$ $\displaystyle\;\frac{nC_{n-1}^{p-1}}{C_{n}^{p}}\sum_{1\leq
i_{1}<\cdots<i_{p}\leq n}\frac{F(A)}{\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}}.$

Thus the desired inequality is proved. ∎

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# Accelerated Proximal Iterative re-Weighted $\ell_{1}$ Alternating
Minimization for Image Deblurring

Tarmizi Adam  Faculty of Computing
Universiti Teknologi Malaysia
Jalan Iman, 81310, Skudai, Johor, Malaysia
<EMAIL_ADDRESS>Alexander Malyshev  Department of Mathematics,
University of Bergen,
Postbox 7803, 5020 Bergen, Norway
<EMAIL_ADDRESS>Mohd Fikree Hassan  School of Information
Technology,
Monash University Malaysia,
Jalan Lagoon Selatan, Bandar Sunway,
47500 Subang Jaya, Selangor, Malaysia
<EMAIL_ADDRESS>Nur Syarafina Mohamed  Department of
Mathematical Sciences, Faculty of Sciences,
Universiti Teknologi Malaysia
81310, Skudai, Johor, Malaysia
<EMAIL_ADDRESS>Md Sah Hj Salam  Faculty of Computing
Universiti Teknologi Malaysia
Jalan Iman, 81310, Skudai, Johor, Malaysia
<EMAIL_ADDRESS>
###### Abstract

The quadratic penalty alternating minimization (AM) method is widely used for
solving the convex $\ell_{1}$ total variation (TV) image deblurring problem.
However, quadratic penalty AM for solving the nonconvex nonsmooth $\ell_{p}$,
$0<p<1$ TV image deblurring problems is less studied. In this paper, we
propose two algorithms, namely proximal iterative re-weighted $\ell_{1}$ AM
(PIRL1-AM) and its accelerated version, accelerated proximal iterative re-
weighted $\ell_{1}$ AM (APIRL1-AM) for solving the nonconvex nonsmooth
$\ell_{p}$ TV image deblurring problem. The proposed algorithms are derived
from the proximal iterative re-weighted $\ell_{1}$ (IRL1) algorithm and the
proximal gradient algorithm. Numerical results show that PIRL1-AM is effective
in retaining sharp edges in image deblurring while APIRL1-AM can further
provide convergence speed up in terms of the number of algorithm iterations
and computational time.

###### Index Terms:

total variation, convex optimization, deblurring, nonconvex optimization,
alternating minimization

##  I Introduction

In this paper, we are interested in the image deblurring problem obtained from
the following image degradation model

$\mathbf{f}=\mathbf{Ku}+\mathbf{n},$ (1)

where $\mathbf{f}\in\mathbb{R}^{n}$ is the observed noisy and blurred image,
$\mathbf{K}\in\mathbb{R}^{n\times n}$ is a blur kernel,
$\mathbf{u}\in\mathbb{R}^{n}$ is the uncorrupted image to be estimated, and
$\mathbf{n}\in\mathbb{R}^{n}$ is additive Gaussian noise.

One way to solve (1) for the image $\mathbf{u}$ is by minimizing the nonconvex
nonsmooth composite optimization problem

$\underset{\mathbf{u}\in\mathbb{R}^{n}}{\text{min}}\,\frac{1}{2}\|\mathbf{Ku}-\mathbf{f}\|_{2}^{2}+\mu\|\nabla\mathbf{u}\|_{p}^{p},$
(2)

where $0<p<1$ and $\nabla\in\mathbb{R}^{n\times n}$ are the discrete
difference operator [1]. If $p=1$, problem (2) results in the convex
$\ell_{1}$-norm total variation (TV) image restoration [2].

Operator splitting methods such as the alternating direction method of
multipliers (ADMM) and the alternating minimization (AM) [3, 4] for solving
the convex $\ell_{1}$-norm TV problem produces the sublinear convergence rate
of $\mathcal{O}\left(\frac{1}{k}\right)$ which is quite slow in practice[5,
6]. Therefore, this has motivated researchers to accelerate these methods to
improved the convergence rate to $\mathcal{O}\left(\frac{1}{k^{2}}\right)$.
However, the majority of the work in this direction is focused on convex
optimization.

This paper, focuses on the model (2) when $0<p<1$ i.e., $\ell_{p}$ quasi-norm
hence, the nonconvex nonsmooth $\ell_{p}$-norm TV. Our motivation mainly stems
from the advantage of nonconvex nonsmooth penalties in restoring even sharper
image quality compared to the $\ell_{1}$-norm TV [7]. Furthermore, this paper
is further motivated by applying the quadratic penalty AM which is an operator
splitting method for solving nonconvex nonsmooth image deblurring problems
(2).

By this, we propose a proximal iterative re-weighted $\ell_{1}$ alternating
minimization (AM) algorithm along with its accelerated version. The proposed
algorithm uses the ideas of proximal operators and the iterative re-weighted
$\ell_{1}$ (IRL1) method [8] in combination with the alternating minimization
method [3]. To accelerate the convergence of the proposed method, Nesterov
acceleration is also used [9, 10].

##  II Related Works

The alternating minimization (AM) method for solving the $\ell_{1}$ TV image
deblurring problem was initially proposed in [3]. An accelerated version
employing acceleration techniques from [10] was further investigated in [11].

Specifically, it was shown that the $\ell_{1}$-norm sub minimization problem
of the AM can be seen as a proximal gradient step. Hence, amenable to
acceleration via the proximal gradient method [10]. However, the accelerated
AM assumes the minimization problem to be convex.

The interesting link between AM and the accelerated proximal gradient method
shown in [11] suggests that AM may have links with nonconvex proximal gradient
type methods. For example, the re-weighted $\ell_{1}$ method that was
originally proposed in [8] for nonconvex $\ell_{p}$ sparse recovery problems
along with its proximal version [12, 13].

The proximal re-weighted $\ell_{1}$ algorithm with acceleration is relatively
new and studied in [14, 15, 16] and has shown promising results in minimizing
problems of the form (2). However, their applications have been mainly
restricted to sparse and low-rank matrix recovery problems. Furthermore, their
connections with operator splitting methods such as AM for image deblurring
have not been to our knowledge explored.

##  III Problem Formulation and Algorithm

In this paper, we focus on image deblurring by minimizing the nonconvex
nonsmooth optimization problem (2). However, we restrict our results and
discussion to a value of $p=0.1$. By adding a quadratic penalty term to (2),
we have

$\underset{\mathbf{u}}{\text{min}}\,\frac{1}{2}\|\mathbf{Ku}-\mathbf{f}\|_{2}^{2}+\mu\|\mathbf{z}\|^{p}_{p}+\frac{\beta}{2}\|\mathbf{z}-\nabla\mathbf{u}\|_{2}^{2}.$
(3)

Indeed, problem (3) is equivalent to problem (2) with constraint
$\mathbf{z}=\nabla\mathbf{u}$ [17]. To minimize (3), we can fix $\mathbf{u}$
with the current value and minimize with respect to $\mathbf{z}$ and vice-
versa

$\begin{cases}\mathbf{z}_{k+1}=\underset{\mathbf{z}}{\operatorname{arg\,min}}\,\frac{\beta}{2}\|\mathbf{z}-\nabla\mathbf{u}\|_{2}^{2}+\mu\|\mathbf{z}\|^{p}_{p},\\\
\mathbf{u}_{k+1}=\underset{\mathbf{u}}{\operatorname{arg\,min}}\,\frac{1}{2}\|\mathbf{Ku}-\mathbf{f}\|_{2}^{2}+\frac{\beta}{2}\|\mathbf{z}-\nabla\mathbf{u}\|_{2}^{2}.\\\
\end{cases}$ (4)

Quadratic penalty AM scheme (4) is a classic method and commonly used in the
image and signal processing literature [17, 3].

Note that the minimization problem concerning $\mathbf{z}$ in (4) is the
nonconvex nonsmooth $\ell_{p}$ minimization problem hence, amenable to the
iterative re-weighted $\ell_{1}$ minimization algorithm [8]. If $p=1$ in (4),
this minimization problem is convex and solvable using soft thresholding [18].
Consequently, it was shown that it is equivalent to the proximal gradient
update [11]

$\displaystyle\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\text{argmin}}\,f\left(\mathbf{y}_{k}\right)+$
$\displaystyle\langle\nabla
f\left(\mathbf{y}_{k}\right),\,\mathbf{z}-\mathbf{y}_{k}\rangle$ (5)
$\displaystyle+\frac{L}{2}\|\mathbf{z}-\mathbf{y}_{k}\|_{2}^{2}+\mu\|\mathbf{z}\|_{1},$

where $\mathbf{y}_{k}=\nabla\mathbf{u}_{k}$, $L$ is the Lipschitz constant of
$f\left(\mathbf{y}_{k}\right)=\frac{\beta}{2}\|\mathbf{z}-\mathbf{y}_{k}\|_{2}^{2}$,
and $\nabla
f\left(\mathbf{y}_{k}\right)=\beta\left(\mathbf{z}-\nabla\mathbf{u}\right)$.
Due to this equivalence, the $\mathbf{z}$ sub problem can be accelerated by
the fast iterative shrinkage and thresholding algorithm (FISTA) [10].

###  III-A Proximal Iterative re-Weighted $\ell_{1}$ Alternating Minimization

With previous foundations in place, coming back to the nonconvex nonsmooth
$\ell_{p}$ minimization subproblem $\mathbf{z}$ in (4), we have

$\mathbf{z}_{k+1}=\underset{\mathbf{z}}{\operatorname{arg\,min}}\,\frac{1}{2}\|\mathbf{z}-\nabla\mathbf{u}_{k}\|_{2}^{2}+\frac{\mu}{\beta}\|\mathbf{z}\|^{p}_{p},$
(6)

after a simple re-arrangement of $\beta$. The IRL1 approximately solves (6) by
[8, 19]

$\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\operatorname{arg\,min}}\,\frac{1}{2}\|\mathbf{z}-\nabla\mathbf{u}_{k}\|_{2}^{2}+\frac{\mu}{\beta}\sum_{i}w^{i}|z^{i}|,$
(7)

where $w^{i}=p\left(|z^{i}_{k}|+\epsilon\right)^{p-1}$, and $z^{i}$ are the
weights and entries of vector $\mathbf{z}$ respectively. Note that in (7) the
nonconvex nonsmooth $\ell_{p}$ minimization problem is approximated into a
convex weighted $\ell_{1}$-norm minimization hence, IRL1 is a convex
relaxation method for nonconvex optimization problems.

By introducing a diagonal weight matrix
$\mathbf{W}_{k}=\text{diag}\left(w^{i}_{k},\cdots w_{k}^{n}\right)$, IRL1
problem (7) can be written as

$\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\operatorname{arg\,min}}\,\frac{1}{2}\|\mathbf{z}-\nabla\mathbf{u}_{k}\|_{2}^{2}+\frac{\mu}{\beta}\|\mathbf{W}_{k}\mathbf{z}\|_{1}.$
(8)

Problem (8) can be interpreted as solving a proximal linearization of the term
$f\left(\mathbf{z}\right)=\frac{1}{2}\|\mathbf{z}-\bar{\mathbf{z}_{k}}\|_{2}^{2}$
at $\bar{\mathbf{z}_{k}}=\nabla\mathbf{u}_{k}$ i.e.,

$\displaystyle\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\operatorname{arg\,min}}\,f\left(\bar{\mathbf{z}_{k}}\right)+$
$\displaystyle\langle\nabla
f\left(\bar{\mathbf{z}_{k}}\right),\,\mathbf{z}-\bar{\mathbf{z}_{k}}\rangle$
(9)
$\displaystyle+\frac{L}{2}\|\mathbf{z}-\bar{\mathbf{z}_{k}}\|_{2}^{2}+\frac{\mu}{\beta}\|\mathbf{W}_{k}\mathbf{z}\|_{1},$

hence, by ignoring constant terms can be written as [20, 10]

$\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\operatorname{arg\,min}}\,\frac{1}{2}\|\mathbf{z}-\mathbf{v}_{k}\|_{2}^{2}+\lambda\|\mathbf{W}_{k}\mathbf{z}\|_{1},$
(10)

with $\mathbf{v}_{k}=\bar{\mathbf{z}_{k}}-\frac{1}{L}\nabla
f\left(\bar{\mathbf{z}_{k}}\right)$ and $\lambda=\frac{\mu}{L\beta}$.
Furthermore, by combining the weight entries with $\lambda$ in the weight
matrix $\mathbf{W}_{k}$ we finally arrive at

$\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\operatorname{arg\,min}}\,\frac{1}{2}\|\mathbf{z}-\mathbf{v}_{k}\|_{2}^{2}+\frac{p\lambda}{\left(|\bar{z}^{i}_{k}|+\epsilon\right)^{1-p}}\|\mathbf{z}\|_{1},$
(11)

where $\bar{z}^{i}_{k}$ is the $i^{\text{th}}$ entry of $\bar{\mathbf{z}}_{k}$
at the $k^{\text{th}}$ iteration and $\epsilon$ a very small number to avoid
division by zero. Equation (11) can be solved in a closed form via the soft-
thresholding operation as

$\mathbf{z}_{k+1}=\text{sgn}\left(\bar{z}_{k}^{i}\right)\text{max}\left(0,\,|\bar{z}_{k}^{i}|-\frac{p\lambda}{\left(|z^{i}_{k}|+\epsilon\right)^{1-p}}\right).$
(12)

From (12), it can be seen that solving the original nonconvex subproblem (6)
boils down to a series of proximal weighted $\ell_{1}$ minimization problem.

The idea of proximal re-weighted $\ell_{1}$ minimization has been proposed in
[12] and its convergence behavior analyzed in [13]. However, the applications
are mainly confined to sparse signal recovery and its use as a sub-
minimization problem in the AM scheme (4) for image deblurring to our
knowledge has not been studied.

Next, the sub-minimization problem for $\mathbf{u}$

$\mathbf{u}_{k+1}=\underset{\mathbf{u}}{\operatorname{arg\,min}}\,\frac{1}{2}\|\mathbf{Ku}-\mathbf{f}\|_{2}^{2}+\frac{\beta}{2}\|\mathbf{z}_{k+1}-\nabla\mathbf{u}\|_{2}^{2},$
(13)

is a convex quadratic problem that can be solved by solving the following
linear system

$\mathbf{u}_{k+1}=\left(\mathbf{K}^{\top}\mathbf{K}+\beta\nabla^{\top}\nabla\right)^{-1}\left(\mathbf{K}^{\top}\mathbf{f}+\beta\nabla^{\top}\mathbf{z}_{k+1}\right).$
(14)

Taking into account equations (12) and (14), the complete listing for the
proximal iterative re-weighted $\ell_{1}$ alternating minimization (PIRL1-AM)
is listed as Algorithm 1.

1

Initialize: $\mathbf{u}_{0}$, $\mathbf{z}_{0}$, $\mu>0$, $\beta>0$, $p=0.1$,
$L=1$, and $k=0$

2

3while  _not converged_ do

4

5
$\mathbf{z}_{k+1}=\text{sgn}\left(\bar{z}_{k}^{i}\right)\text{max}\left(0,\,|\bar{z}_{k}^{i}|-\frac{p\lambda}{\left(|z^{i}_{k}|+\epsilon\right)^{1-p}}\right)$,

6

7
$\mathbf{u}_{k+1}=\left(\mathbf{K}^{\top}\mathbf{K}+\beta\nabla^{\top}\nabla\right)^{-1}\left(\mathbf{K}^{\top}\mathbf{f}+\beta\nabla^{\top}\mathbf{z}_{k+1}\right)$,

8 $k=k+1$

Algorithm 1 Proximal iterative re-weighted $\ell_{1}$ alternating minimization
(PIRL1-AM)

###  III-B Accelerated Proximal Iterative re-Weighted $\ell_{1}$ Alternating
Minimization

To accelerate the proximal iterative re-weighted $\ell_{1}$ AM for the
nonconvex nonsmooth $\ell_{p}$ TV image deblurring problem, accelerated
techniques from [9, 10] can be used. Consider the sub-minimization problem
(11), due to the convexity of (11) and its equivalence to minimizing a
proximal linearization of $\frac{1}{2}\|\mathbf{z}-\mathbf{v}_{k}\|_{2}^{2}$
as discussed earlier, we have

$\displaystyle\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\text{argmin}}\,f\left(\mathbf{v}_{k}\right)+$
$\displaystyle\langle\nabla
f\left(\mathbf{v}_{k}\right),\,\mathbf{z}-\mathbf{v}_{k}\rangle$ (15)
$\displaystyle+\frac{L}{2}\|\mathbf{z}-\mathbf{v}_{k}\|_{2}^{2}+\tau\|\mathbf{z}\|_{1},$

where $\tau=\frac{p\lambda}{\left(|\bar{z}^{i}_{k}|+\epsilon\right)^{1-p}}$.
Then, the acceleration strategies along the lines of [10, 9] can be employed,
which involves the following iterative scheme

$\displaystyle\begin{cases}\mathbf{z}_{k+1}=\underset{\mathbf{z}\in\mathbb{R}^{n}}{\text{argmin}}\,f\left(\mathbf{y}_{k}\right)+\langle\nabla
f\left(\mathbf{y}_{k}\right),\,\mathbf{z}-\mathbf{y}_{k}\rangle\\\
\qquad\qquad\qquad\qquad\qquad+\frac{L}{2}\|\mathbf{z}-\mathbf{y}_{k}\|_{2}^{2}+\tau\|\mathbf{z}\|_{1},\\\
t_{k}=\frac{k-1}{k+2},\\\
\mathbf{y}_{k+1}=\mathbf{z}_{k+1}+t_{k}\left(\mathbf{z}_{k+1}-\mathbf{z}_{k}\right).\\\
\end{cases}$ (16)

The $\mathbf{z}$ minimization in (16) as disccused previously has a closed
form solution of

$\mathbf{z}_{k+1}=\text{sgn}\left(\bar{z}_{k}^{i}\right)\text{max}\left(0,\,|\bar{z}_{k}^{i}|-\tau\right).$
(17)

In the iterative scheme (16), the scalar $t_{k}$ is known as the Nesterov
momentum coefficient and changes in each iteration $k$. The step
$\mathbf{y}_{k}$ is called the extrapolation step. Also, recall that
$\mathbf{z}_{k}=\nabla\mathbf{u}_{k}$. Acceleration techniques of this form
are shown to match the theoretical lower bound of
$\mathcal{O}\left(\frac{1}{k^{2}}\right)$ for smooth first-order convex
optimization.

Having shown the equivalence between (11) and the proximal linearization step
(15) along with its acceleration, the iterative scheme of the accelerated
proximal iterative re-weighted $\ell_{1}$ AM (APIRL1-AM) taking account the
sub minimization problem for $\mathbf{u}$ is as follows

$\begin{cases}\mathbf{z}_{k+1}=\text{sgn}\left(\bar{z}_{k}^{i}\right)\text{max}\left(0,\,|\bar{z}_{k}^{i}|-\tau\right),\\\
t_{k}=\frac{k-1}{k+2},\\\
\mathbf{y}_{k+1}=\mathbf{z}_{k+1}+t_{k}\left(\mathbf{z}_{k+1}-\mathbf{z}_{k}\right),\\\
\mathbf{u}_{k+1}=\left(\mathbf{K}^{\top}\mathbf{K}+\beta\nabla^{\top}\nabla\right)^{-1}\left(\mathbf{K}^{\top}\mathbf{f}+\beta\nabla^{\top}\mathbf{z}_{k+1}\right).\\\
\end{cases}$ (18)

The complete algorithm is listed as Algorithm 2.

1

Initialize: $\mathbf{u}_{0}$, $\mathbf{z}_{0}$, $\mu>0$, $\beta>0$, $p=0.1$,
$L=1$, and $k=0$

2

3while  _not converged_ do

4

5
$\mathbf{z}_{k+1}=\text{sgn}\left(\bar{z}_{k}^{i}\right)\text{max}\left(0,\,|\bar{z}_{k}^{i}|-\tau\right)$

6 $t_{k}=\frac{k-1}{k+2}$

7
$\mathbf{y}_{k+1}=\mathbf{z}_{k+1}+t_{k}\left(\mathbf{z}_{k+1}-\mathbf{z}_{k}\right)$

8
$\mathbf{u}_{k+1}=\left(\mathbf{K}^{\top}\mathbf{K}+\beta\nabla^{\top}\nabla\right)^{-1}\left(\mathbf{K}^{\top}\mathbf{f}+\beta\nabla^{\top}\mathbf{z}_{k+1}\right)$.

9 $k=k+1$

Algorithm 2 Accelerated proximal iterative re-weighted $\ell_{1}$ alternating
minimization (APIRL1-AM)

In Algorithms 1 and 2, the Lipschitz constant $L$ is fixed to 1. In real
applications, the value of $L$ is usually unknown. The values of $\mu$ and
$\beta$ are used for computing $\lambda=\frac{\mu}{L\beta}$. Additionally, the
linear system for solving $\mathbf{u}$ can be solved very fast using the two-
dimensional fast Fourier transforms (FFT) with complexity
$\mathcal{O}\left(n^{n}\log n\right)$ [3] while the $\mathbf{z}$ problem is
only of linear complexity $\mathcal{O}\left(n\right)$.

##  IV Results

In this section, we apply the two proposed algorithms on the nonconvex
nonsmooth image deblurring problem (2) and discuss the results of the proposed
algorithms without and with acceleration.

###  IV-A Experimental Setup

For the experiments, the blurred and noisy images are obtained by the model
(1). The blur kernel used is a Gaussian blur of size $17\times 17$ pixel with
$\sigma=7$. Two blur levels were used namely, blur signal-to-noise ratio
(BSNR) 30 and 20 [21]. The values for $\beta$ are $\beta=0.01$ and
$\beta=0.009$ for BSNR 20 and 30 respectively. The deblurring on each image
was done 10 times and the average was taken.

All images are of size $512\times 512$ pixels. Experiments were conducted
using MATLAB111Codes at https://github.com/tarmiziAdam2005/PIRL1-AM on an
Intel Core i3-10105 CPU operating at 3.70 GHz with a memory of 4 Gb.

TABLE I: Image deblurring Results. Value of $\mu=30,\,60$ for Peppers and Cameraman respectively for BSNR 30. For BSNR 20, Value of $\mu=100$ for both Peppers and Cameraman |  |  | APIRL1-AM |  |  |  |  | PIRL1-AM |  |
---|---|---|---|---|---|---|---|---|---|---
BSNR lvl |  | PSNR | SSIM | Itr | T |  | PSNR | SSIM | Itr | T
| Peppers | 27.39 | 0.759 | 269 | 11.20 |  | 27.39 | 0.759 | 364 | 14.71
30 | Cameraman | 26.03 | 0.763 | 228 | 9.50 |  | 26.04 | 0.762 | 324 | 13.12
| Peppers | 25.52 | 0.625 | 141 | 6.03 |  | 25.52 | 0.625 | 186 | 7.63
20 | Cameraman | 24.49 | 0.570 | 225 | 9.64 |  | 24.49 | 0.570 | 244 | 10.02

(a)

(b)

Figure 1: Convergence of the relative errors
$\frac{\|\mathbf{u}_{k}-\mathbf{u}_{k+1}\|_{2}}{\|\mathbf{u}_{k}\|_{2}}\leq
1\times 10^{-8}$, produced by the iterates of APIRL1-AM and PIRL1-AM for
imagem (a) Peppers (BSNR $=30$ ) and (b) Cameraman (BSNR $=20$).

(a)

(b)

(c)

(d)

Figure 2: Deblurred image Peppers BSNR$=30$. (a) Original. (b) Blurred (PSNR =
22.97, SSIM = 0.659). (c) Zoomed. (d) APIRL1-AM deblurred.

(a)

(b)

(c)

(d)

Figure 3: Deblurred image Cameraman BSNR$=20$. (a) Original. (b) Blurred (PSNR
= 21.65, SSIM = 0.499). (c) Zoomed. (d) APIRL1-AM deblurred.

###  IV-B Results and discussion

Table I shows the results of the two proposed algorithms. In terms of image
quality of PSNR and SSIM [22], the algorithms gave almost similar results.
Some222Only deblurred images of APIRL1-AM are shown due to both algorithms
giving very similar PSNR and SSIM values. of the deblurred images are shown in
Figures 2 and 3. For BSNR 30 (less noise corruption), Figure 2 shows the
ability of the methods in preserving sharp images.

However, there is a difference in terms of the number of iterations to
converge to the required relative error value. For both BSNR levels tested,
the acceleration technique used in APIRL1-AM manages to decrease the number of
iterations to converge. Additionally, the time to converge also improves for
APIRL1-AM compared to PIRL1-AM.

Figure 1 compares the convergence between APIRL1-AM and PIRL1-AM. It can be
seen that APIRL1-AM converges faster than PIRL1-AM and exhibits acceleration
ripples akin to first-order accelerated techniques [23]. Taking into account
similar results in terms of PSNR and SSIM of the two algorithms, acceleration
gives an additional advantage for arriving at similar deblurring quality at a
lower number of iterations and CPU time.

##  V Conclusion

In this paper, two algorithms PIRL1-AM and APIRL1-AM were proposed for
nonconvex nonsmooth $\ell_{p}$ TV image deblurring. The algorithms were
derived by showing the links between the proximal gradient method and the
proximal iterative re-weighted $\ell_{1}$. Both algorithms were able to retain
sharp images for image deblurring. For algorithm convergence, APIRL1-AM
exhibits the optimal $\mathcal{O}\left(\frac{1}{k^{2}}\right)$ rate of
convergence and improves the CPU time and the number of iterations to
converge. Future works include establishing the convergence rate of both
algorithms and applying them to different problems.

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