DNA damage by lipid peroxidation products: implications in cancer, inflammation and autoimmunity

Fabrizio Gentile; Alessia Arcaro; Stefania Pizzimenti; Martina Daga; Giovanni Paolo Cetrangolo; Chiara Dianzani; Alessio Lepore; Maria Graf; Paul R. J. Ames; Giuseppina Barrera; Fabrizio Gentile; Alessia Arcaro; Stefania Pizzimenti; Martina Daga; Giovanni Paolo Cetrangolo; Chiara Dianzani; Alessio Lepore; Maria Graf; Paul R. J. Ames; Giuseppina Barrera

doi:10.3934/genet.2017.2.103

AIMS Genetics

2017, Volume 4, Issue 2: 103-137. doi: 10.3934/genet.2017.2.103

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Review

DNA damage by lipid peroxidation products: implications in cancer, inflammation and autoimmunity

1.
Department of Medicine and Health Sciences “V. Tiberio”, University of Molise, Campobasso, Italy
2.
Department of Clinical and Biological Sciences, University of Torino, Torino, Italy
3.
Department of Drug Science and Technology, University of Torino, Torino, Italy
4.
Department of Molecular Medicine and Medical Biotechnologies, University of Naples Federico II, Naples, Italy
5.
CEDOC, NOVA Medical School, Universidade NOVA de Lisboa, Lisboa, Portugal, and Department of Haematology, Dumfries Royal Infirmary, Dumfries, Scotland, UK

Received: 14 December 2016 Accepted: 12 April 2017 Published: 18 April 2017

Oxidative stress and lipid peroxidation (LPO) induced by inflammation, excess metal storage and excess caloric intake cause generalized DNA damage, producing genotoxic and mutagenic effects. The consequent deregulation of cell homeostasis is implicated in the pathogenesis of a number of malignancies and degenerative diseases. Reactive aldehydes produced by LPO, such as malondialdehyde, acrolein, crotonaldehyde and 4-hydroxy-2-nonenal, react with DNA bases, generating promutagenic exocyclic DNA adducts, which likely contribute to the mutagenic and carcinogenic effects associated with oxidative stress-induced LPO. However, reactive aldehydes, when added to tumor cells, can exert an anticancerous effect. They act, analogously to other chemotherapeutic drugs, by forming DNA adducts and, in this way, they drive the tumor cells toward apoptosis. The aldehyde-DNA adducts, which can be observed during inflammation, play an important role by inducing epigenetic changes which, in turn, can modulate the inflammatory process.
The pathogenic role of the adducts formed by the products of LPO with biological macromolecules in the breaking of immunological tolerance to self antigens and in the development of autoimmunity has been supported by a wealth of evidence. The instrumental role of the adducts of reactive LPO products with self protein antigens in the sensitization of autoreactive cells to the respective unmodified proteins and in the intermolecular spreading of the autoimmune responses to aldehyde-modified and native DNA is well documented. In contrast, further investigation is required in order to establish whether the formation of adducts of LPO products with DNA might incite substantial immune responsivity and might be instrumental for the spreading of the immunological responses from aldehyde-modified DNA to native DNA and similarly modified, unmodified and/or structurally analogous self protein antigens, thus leading to autoimmunity.

Keywords:

Citation: Fabrizio Gentile, Alessia Arcaro, Stefania Pizzimenti, Martina Daga, Giovanni Paolo Cetrangolo, Chiara Dianzani, Alessio Lepore, Maria Graf, Paul R. J. Ames, Giuseppina Barrera. DNA damage by lipid peroxidation products: implications in cancer, inflammation and autoimmunity[J]. AIMS Genetics, 2017, 4(2): 103-137. doi: 10.3934/genet.2017.2.103

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Abstract

1. Introduction

Let $\Omega$ be a bounded $C^3$ domain in $\mathbb{R}^n$ and $u\in C^{3}(\overline{\Omega})$ . In this paper, we will establish a priori gradient estimates for solutions of the prescribed $k$ -curvature equation with the prescribed contact angle boundary value

$\begin{equation} \left\{ \begin{aligned} \sigma_{k}(\kappa)& = f(x,u),\quad \text{in}\ \ \Omega,\\ \frac{\partial u}{\partial \nu}& = \phi(x)\sqrt{1+|Du|^2},\quad \text{on} \ \partial \Omega. \end{aligned} \right. \end{equation}$

(1.1)

where $\kappa = (\kappa_1, \cdots, \kappa_n)$ are the principal curvatures of the graph $\mathcal{M} = \{(x, u(x))\in\mathbb{R}^{ n+1}| x\in\Omega\}$ , $n \geq2$ , $f$ is a smooth, positive function in $\Omega$ and $\phi$ is a smooth function on $\overline{\Omega}$ such that $-1 < \phi < 1$ . And for any $k = 1, 2, \cdots, n$ ,

$\begin{align} \sigma_{k}(\lambda) = \sum\limits_{1\leq i_{1} < i_{2} < \cdots < i_{k}\leq n}\lambda_{i_{1}}\lambda_{i_{2}}\cdots\lambda_{i_{k}}, \end{align}$

(1.2)

the $k$ -order fundamental symmetric function of $\lambda\in \mathbb{R}^n$ . For $k = 1$ , $n$ , the (1.1) is the mean curvature and Gaussian curvature equation respectively.

The gradient estimate for the prescribed mean curvature equation has been extensively studied. The interior gradient estimate, for the minimal surface equation, was obtained in the case of two variables by Finn ^[2]. Bombieri-De Giorgi-Miranda ^[1] obtained the estimate for high dimensional cases. For the general mean curvature equation, such an estimate had also been obtained by Ladyzhenskaya and Ural'tseva ^[10], Trudinger ^[17] and Simon ^[13]. All their methods were used by test function argument and a resulting Sobolev inequality. A more detailed history could be found in Gilbarg and Trudinger ^[3]. In 1983, Korevaar ^[5] introduced the normal variation technique and got the maximum principle proof for the interior gradient estimate on the minimal surface equation, then in 1987 Korevaar ^[6] got the interior gradient estimates for the higher order curvature equations. Trudinger ^[18] also studied the curvature equations and got the interior gradient estimates for a class curvature equation. In 1998, Wang ^[19] gave new proof for the interior gradient estimates on the general $k$ -curvature equation via the standard Bernstein technique. In 2012, Sheng-Trudinger-Wang ^[16] also gave a new proof for the general Weingarten curvatures equation by the moving frame on the hypersurface.

For the mean curvature equation with the Neumann boundary value problem, Ma and Xu ^[11] used the technique developed by Spruck ^[14], Lieberman ^[7], Wang ^[19] and Jin-Li-Li ^[4] to get the global gradient estimates. As a consequence, they obtained an existence theorem for a class of mean curvature equations with the Neumann boundary value. For a fully nonlinear elliptic equation with Neumann boundary value or oblique derivative problem, we recommend Lieberman ^[8] to readers. Recently, Ma and Wang ^[12] used the technique developed by Sheng-Trudinger-Wang ^[16] to give a simpler new proof of the gradient estimates for the mean curvature equation with Neumann boundary value and prescribed contact angle boundary value. In this paper, we use the same technique to get the global gradient estimates for the $k$ -curvature equation with the prescribed contact angle boundary value. Precisely, we have the following theorem.

Theorem 1.1. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with a $C^3$ boundary, $n\geq2$ , $\nu$ is the unit inner normal to $\partial\Omega$ . Suppose $f\in C^1(\overline{\Omega}\times[-M_0, M_0])$ satisfies that $f_z\geq0$ with $\inf_{\Omega}f\geq f_0 > 0$ , and $\phi\in C^3(\overline{\Omega})$ and $-1 < \phi < 1$ . If $u\in C^2(\overline{\Omega})\cap C^3(\Omega)$ is a bounded $k$ -admissible solution of the $k$ -curvature equation (1.1), then we have

$\begin{align} \sup\limits_{\overline{\Omega}}|Du|\leq C, \end{align}$

(1.3)

where $C$ is a positive constant depending on $n$ , $k, f_0$ , $\Omega$ , $|f|_{C^1(\overline{\Omega}\times[-M_0, M_0])}$ , $|\phi|_{C^3(\overline{\Omega})}$ , and $M_0 = |u|_{C^0(\overline{\Omega})}$ .

Remark 1.2. We define the Garding's cone as $\Gamma_k = \{\lambda\in\mathbb{R}^n|\sigma_i(\lambda) > 0, \ 1\leq i\leq k\}$ . Then we say a function $u$ is $k$ -admissible if $\lambda(D^2u)\in\Gamma_k$ , where $\lambda(D^2u) = (\lambda_1, \cdots, \lambda_n)$ are eigenvalues of the Hessian matrix $D^2u$ .

Remark 1.3. In order to prove the existence theorem for the $k$ -curvature equations with the prescribed contact angle boundary value problem, we still need global estimates for second-order derivatives. In another paper, we had gotten the global gradient estimates for the $k$ -curvature equation with the Neumann boundary value problem.

The rest of the paper is organized as follows. In Section 2, we first give the definitions and some notations. We also give some basic properties of the fundamental symmetric functions. In Section 3, we prove the main Theorem 1.1 by the moving frame on the hypersurface.

2. Preliminary

$A$ , $B$ , $\cdots$ will be from $1$ to $n + 1$ and $i$ , $j$ , $\alpha$ , $\cdots$ from $1$ to $n$ , the repeated indices denote summation over the indices.

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and $u\in C^{\infty}(\overline{\Omega})$ . Then the graph of $u$ is a hypersurface in $\mathbb{R}^{n+1}$ , denoted by $\mathcal{M}$ , given by the smooth embedding $X : \Omega \rightarrow \mathbb{R}^{ n+1}$ ,

$\begin{align} X(x_1 ,\cdots,x_n ) = (x_1 ,\cdots,x_n ,u(x_1 ,\cdots,x_n )). \end{align}$

(2.1)

Denote $u_i = u_{x_i}$ , $u_{ij} = u_{x_ix_j}$ , and $Du = (u_1, \cdots, u_n)$ . Then the downward unit normal of $\mathcal{M}$ is

$\begin{align} N = \frac{(Du,-1)}{\sqrt{1+|Du|^2}}. \end{align}$

(2.2)

Let $\{\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_{n+1}\}$ be the standard orthonormal basis in $\mathbb{R}^{n+1}$ . We choose an orthonormal frame in $\mathbb R^{n+1}$ such that $\{e_1, e_2, ..., e_n\}$ are tangent to $\mathcal{M}$ and $e_{n+1} = N$ is the downward unit normal. Let the corresponding coframe be denoted by $\{\omega_A\}$ and the connection forms by $\{\omega_{A, B}\}$ . The pullback of them through the embedding are still denoted by $\{\omega_A \}$ , $\{\omega_{A, B}\}$ in the abuse of notation. Therefore on $\mathcal{M}$

$\begin{eqnarray*} \omega_{n+1} = 0. \end{eqnarray*}$

The second fundamental form is defined by the symmetry matrix $\{h_{ij}\}$ with

$\begin{align} \omega_{i,n+1} = h_{ij}\omega_{j}. \end{align}$

(2.3)

The principal curvatures $\kappa = (\kappa_1, \kappa_2, \cdots, \kappa_n)$ are the eigenvalues of the second fundamental form $(h_{ij})$ .

The first and second-order covariant derivatives will be denoted by $\nabla_i$ , $\nabla_i\nabla_j$ respectively. We recall the following fundamental formulas of a hypersurface in ${\mathbb R}^{n+1}$ .

$\begin{align} \nabla_j\nabla_iX = \nabla_{j}e_i = & -h_{ij} e_{n+1}, \quad \text {(Gauss formula)} \end{align}$

(2.4)

$\begin{align} \nabla_ie_{n+1} = &\ h_{ij}e_{j}. \quad \text {(Weingarten equation)} \end{align}$

(2.5)

We denote

$\begin{align*} d(x) = dist(x,\partial\Omega),\nonumber\\ \Omega_{\mu} = \{x\in\Omega|\ d(x) < \mu\}. \end{align*}$

It is well known that there exists a small positive universal constant $\mu_0$ such that $d(x)\in C^3(\overline{\Omega_{\mu}}), \ \forall0 < \mu\leq\mu_0$ , provided $\partial\Omega\in C^3$ . As in Simon-Spruck ^[15] or Lieberman ^[7] (on page 331), we can extend $\nu$ by $\nu = D d$ in $\Omega_{\mu}$ and note that $\nu$ is a $C^3(\overline{\Omega_{\mu}})$ vector field. As mentioned in the book ^[7], we also have the following formulas

$\begin{align} \begin{split} |D\nu|+|D^2\nu|\leq C(n,\Omega),\quad\text{in}\ \Omega_{\mu}, \\ \sum\limits_{i = 1}^n\nu^iD_j\nu^i = \sum\limits_{i = 1}^n\nu^iD_i\nu^j = \sum\limits_{i = 1}^nd_id_{ij} = 0,\ |\nu| = |Dd| = 1,\quad\text{in}\ \Omega_{\mu}. \end{split} \end{align}$

(2.6)

Lemma 2.1. Denote $v = \sqrt{1+|Du|^2}$ and $e_A^B = \langle e_A, \varepsilon_B\rangle$ for $A, B = 1, \cdots, n+1$ . We have

$\begin{align} \nabla_iv = &\ v^2h_{ir}\nabla_ru, \end{align}$

(2.7)

$\begin{align} \nabla_j\nabla_iv = &\ 2v^3h_{jr}\nabla_ruh_{is}\nabla_su+v^2\nabla_rh_{ij}\nabla_ru+vh_{ir}h_{jr}, \end{align}$

(2.8)

$\begin{align} \nabla_iu_r = &\ vh_{is}(u_r\nabla_su+e_s^r), \end{align}$

(2.9)

$\begin{align} \nabla_j\nabla_iu_r = &\ 2v^2h_{jp}\nabla_puh_{iq}(u_r\nabla_qu+e_q^r)+v\nabla_ph_{ij}(u_r\nabla_pu+e_p^r). \end{align}$

(2.10)

Proof. Note that $u = \langle X, \varepsilon_{n+1}\rangle$ . Using the Gauss formula and Weingarten equation above, we obtain

$\begin{align} \begin{split}\nonumber \nabla_{i}v = &\ \nabla_{i}(-\frac{1}{\langle e_{n+1},\varepsilon_{n+1}\rangle }) \\ = &\ \frac{1}{\langle e_{n+1},\varepsilon_{n+1}\rangle ^{2}}\nabla_{i}\langle e_{n+1},\varepsilon_{n+1}\rangle \\ = &\ v^{2}h_{il}\langle e_{l},\varepsilon_{n+1}\rangle \\ = &\ v^{2}h_{il}\nabla_{l}u. \end{split} \end{align}$

Similarly, we have

$\begin{align} \begin{split}\nonumber \nabla_j\nabla_iv = &\ \nabla_j\left(v^{2}h_{il}\langle e_{l},\varepsilon_{n+1}\rangle\right) \\ = &\ 2v^3h_{jr}\nabla_ruh_{is}\nabla_su+v^2\nabla_rh_{ij}\nabla_ru+vh_{ir}h_{jr} \end{split} \end{align}$

It follows that, recall $u_l = v\langle e_{n+1}, \varepsilon_l\rangle = ve_{n+1}^l$ ,

$\begin{align} \begin{split}\nonumber \nabla_{i}u_{l} = &\ \nabla_i(v\langle e_{n+1},\varepsilon_l\rangle) = v^{2}h_{ir}\nabla_{r}u e_{n+1}^{l}+vh_{ir}e_{r}^{l} \\ = &\ vh_{ir}(u_{l}\nabla_{r}u+e_{r}^{l}). \end{split} \end{align}$

Furthermore, we have

$\begin{align*} \nabla_{j}\nabla_{i}u_{l} = &\ \nabla_{j}\big(vh_{ir}(u_{l}\nabla_{r}u+e_{r}^{l})\big) \\ = &\ v^{2}h_{js}\nabla_{s}uh_{ir}(u_{l}\nabla_{r}u+e_{r}^{l}) +v\nabla_{j}h_{ir}(u_{l}\nabla_{r}u+e_{r}^{l}) \\ &\ +vh_{ir}\big(u_{l}\nabla_{j}\nabla_{r}u+v\nabla_{r}uh_{js} (u_{l}\nabla_{s}u+e_{s}^{l})+\nabla_{j}e_{r}^{l}\big), \end{align*}$

noting that

$\begin{align} \nabla_{j}e_{r}^{l} = -h_{jr}e_{n+1}^{l} = -u_l\langle\nabla_j\nabla_rX,\varepsilon_{n+1}\rangle = -u_{l}\nabla_{j}\nabla_{r}u, \end{align}$

then, since $\nabla_jh_{ir} = \nabla_rh_{ij}$ (Codazzi equation),

$\begin{align} \nabla_{j}\nabla_{i}u_{l} = 2v^{2}h_{js}\nabla_{s}uh_{ir}(u_{l}\nabla_{r}u+e_{r}^{l}) + v\nabla_{r}h_{ij}(u_{l}\nabla_{r}u+e_{r}^{l}). \end{align}$

□

Now we give some basic properties of elementary symmetric functions, which could be found in ^[9].

First, we denote by $\sigma_{k}(\lambda|i)$ the symmetric function with $\lambda_{i} = 0$ and $\sigma_{k}(\lambda|ij)$ the symmetric function with $\lambda_{i} = \lambda_{j} = 0$ .

Proposition 2.2. Let $\lambda = (\lambda_{1}, \cdots, \lambda_{n})\in \mathbb{R}^{n}$ and $k = 1, \cdots, n$ , then

$\begin{align} &\ \sigma_{k}(\lambda) = \sigma_{k}(\lambda|i)+\lambda_{i}\sigma_{k-1}(\lambda|i),\quad \forall 1\leq i\leq n,\\ &\ \sum\limits_{i = 1}^{n}\lambda_{i}\sigma_{k-1}(\lambda|i) = k\sigma_{k}(\lambda),\\ &\ \sum\limits_{i = 1}^{n}\sigma_{k}(\lambda|i) = (n-k)\sigma_{k}(\lambda). \end{align}$

Recall that Garding's cone is defined as

$\begin{eqnarray*} \Gamma_{k} = \{\lambda\in\mathbb{R}^{n} : \sigma_{i} > 0, \forall \ 1\leq i\leq k\}. \end{eqnarray*}$

Proposition 2.3. Let $\lambda \in\Gamma_{k}$ and $k\in\{1, 2, \cdots, n\}$ . Suppose that

$\begin{eqnarray*} \lambda_{1}\geq\cdots\geq\lambda_{k}\geq\cdots\geq\lambda_{n}, \end{eqnarray*}$

then we have

$\begin{align} \sigma_{k-1}(\lambda|n)\geq \cdots \geq \sigma_{k-1}(\lambda|k) \geq \cdots \geq \sigma_{k-1}(\lambda |1) > 0. \end{align}$

(2.11)

Then the $k$ -curvature equation (1.1) is elliptic if the principal curvatures $\kappa\in\Gamma_k$ .

3. Gradient estimate for prescribed contact angle

We consider the following $k$ -curvature equation with the prescribed angle condition and obtain a gradient estimate of $k$ -admissible solution. We state it again in the following theorem.

Theorem 3.1. Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with $C^{3}$ boundary. $f\in C^1(\Omega\times[-M_0, M_0])$ satisfies that $f_z\geq0$ . Assume $u$ is a $k$ -admissible solution of the equation

$\begin{equation} \left\{ \begin{aligned} \sigma_{k}(h_{ij})& = f(x,u), &\quad\mathit{{in}}\ \Omega& \\ \frac{\partial u}{\partial\nu}& = \phi(x)\sqrt{1+|Du|^{2}}, &\quad\mathit{{on}}\ \partial\Omega,& \end{aligned} \right. \end{equation}$

(3.1)

where $\nu$ be the unit inner normal vector on $\partial\Omega$ and $\phi\in C^3\left(\bar \Omega, (-1, 1)\right)$ . We have

$\begin{align} \sup\limits_{\overline{\Omega}}|Du|\leq C. \end{align}$

(3.2)

Proof. Denote $v = \sqrt{1+|Du|^{2}}$ and $M_{0} = \sup_{\overline{\Omega}}|u|$ . Let

$\begin{align} w: = &\ v-u_{\nu}\phi, \end{align}$

(3.3)

$\begin{align} \psi(u): = &\ \alpha_{1}(1+M_{0}+u). \end{align}$

(3.4)

The constant $\alpha_{1}$ will be determined later. Fix a small $0 < \mu\leq\mu_0$ and consider the auxiliary function

$\begin{align} G(x): = \log\log w+\psi(u)+ d,\quad x\in\overline{\Omega}_{\mu}. \end{align}$

(3.5)

There are three cases to be considered.

Case 1. $G(x)$ attains maximum at $x_{0}\in\partial\Omega_{\mu}\cap\Omega$ .

By the interior gradient estimates of Korevaar ^[6] and Wang ^[19], we have

$\begin{align} \sup\limits_{\overline{\Omega}}|Du|\leq C. \end{align}$

(3.6)

Case 2. $G(x)$ attains maximum at $x_{0}\in\partial \Omega$ .

Assume $U\subset\mathbb{R}^{n}$ be a neighborhood of $x_{0}$ . We choose a geodesic coordinate $\{x_{i}\}_{i}^{n-1}$ on $U\cap\partial\Omega$ centered at $x_0$ . We let $\partial_{x_n} = \nu$ at $x_0$ . In the following, we take all calculations at $x_0$ .

Denote $(b_{ij})$ the second fundamental form of $\partial\Omega$ with respect to $\nu$ . We have

$\begin{align} G_{n} = \frac{w_{n}}{w\log w}+\alpha_{1}u_{n}+ 1\leq0, \end{align}$

(3.7)

and

$\begin{align} G_{j} = \frac{w_{j}}{w\log w}+\alpha_{1}u_{j} = 0, \quad j = 1, 2, \cdots, n-1. \end{align}$

(3.8)

Denote $a = w\log w$ for simplicity. Note that

$\begin{align} u_{n} = &\ \phi v,\ \end{align}$

(3.9)

$\begin{align} w_{l} = &\ v_{l}-u_{nl}\phi-u_{n}\phi_{l}, \quad l = 1, 2, \cdots, n, \end{align}$

(3.10)

$\begin{align} v_{n} = &\ \frac{1}{v}{\sum\limits_{i = 1}^{n-1}u_{i}u_{in}}+\frac{u_{n}u_{nn}}{v}\\ = &\ \frac{1}{v}{\sum\limits_{i = 1}^{n-1}u_{i}u_{in}}+\phi u_{nn}. \end{align}$

(3.11)

Choose $l = n$ in (3.10), then plug into (3.11) to get

$\begin{align} \begin{split} w_{n} = &\ \frac{1}{v}\sum\limits_{i = 1}^{n-1}u_{i}u_{in}-u_{n}\phi_{n} \\ = &\ \frac{1}{v}\sum\limits_{i = 1}^{n-1}u_{i}u_{ni}-\frac{1}{v}\sum\limits_{i = 1}^{n-1}u_{i}b_{ij}u_{j}-v\phi\phi_{n} \\ \geq&\ \frac{1}{v}\sum\limits_{i = 1}^{n-1}u_{i}u_{ni}-Cv. \end{split} \end{align}$

(3.12)

By (3.8) and (3.10),

$\begin{align} v_{i} = u_{ni}\phi+u_{n}\phi_{i}-\alpha_{1}au_{i}. \end{align}$

(3.13)

From the boundary data $u_n = \phi v$ and (3.13), we have

$\begin{align} \begin{split} u_{ni} = (\phi v)_{i} = &\ \phi_{i}v+v_{i}\phi\\ = &\ \phi_{i}v-\alpha_{1}au_{i}\phi+u_{ni}\phi^{2}+u_{n}\phi_{i}\phi. \end{split} \end{align}$

(3.14)

It follows that

$\begin{align} \begin{split} u_{ni} = &\ \frac{1}{1-\phi^{2}}(\phi_{i}v-\alpha_{1}au_{i}\phi+v\phi_{i}\phi^{2}) \\ \geq&\ -\frac{\alpha_{1}a\phi u_{i}}{1-\phi^{2}}-Cv. \end{split} \end{align}$

(3.15)

Plugging (3.15) into (3.12), we get

$\begin{align} \begin{split} w_{n}\geq&\ -\frac{\alpha_{1}a\phi}{(1-\phi^{2})v}\sum\limits_{i = 1}^{n-1}u_{i}^{2}-Cv \\ = &\ \frac{\alpha_{1}a\phi}{(1-\phi^{2})v}-\alpha_{1}a\phi v-Cv. \end{split} \end{align}$

(3.16)

Here we use the fact

$\begin{align} v^{2}-1 = \sum\limits_{i = 1}^{n-1}u_{i}^{2}+u_{n}^{2} = \sum\limits_{i = 1}^{n-1}u_{i}^{2}+v^{2}\phi^{2}\quad\text{at}\ x_0\in\partial\Omega. \end{align}$

(3.17)

Putting (3.17) into (3.7), we have

$\begin{align} 0\geq&\ -\alpha_{1}\phi v+\frac{\alpha_{1}\phi}{(1-\phi^{2})v}-\frac{Cv}{w\log w}+\alpha_{1}u_{n}+ 1\\ = &\ \frac{\alpha_{1}\phi}{(1-\phi^{2})v}-\frac{Cv}{w\log w}+ 1. \end{align}$

(3.18)

Thus we have $v\leq C$ .

Case 3. $G(x)$ attains its maximum at $x_0\in\Omega_{\mu}$ .

Direct computation shows that

$\begin{align} \nabla_i G = \frac{\nabla_i w}{w\log w}+\alpha_1\nabla_i u+\nabla_id, \end{align}$

(3.19)

and

$\begin{align} \nabla_j\nabla_i G = &\ \frac{\nabla_j\nabla_i w}{w\log w}-\frac{\nabla_iw\nabla_j w}{(w\log w)^2}(1+\log w)+\alpha_1\nabla_j\nabla_i u+\nabla_j\nabla_id. \end{align}$

(3.20)

From (2.7)–(2.10) and (3.3), we have

$\begin{align} \nabla_iw = &\ \nabla_i(v-u_rd_r\phi)\\ = &\ v^2h_{ir}\nabla_ru-vh_{is}(u_r\nabla_su+e_s^r)d_r\phi-u_r\nabla_i(d_r\phi)\\ = &\ v(v-u_rd_r\phi)h_{ir}\nabla_ru-vh_{ir}\nabla_rd\phi-u_r\nabla_i(d_r\phi)\\ = &\ vwh_{ir}\nabla_ru-vh_{ir}\nabla_rd\phi-u_r\nabla_i(d_r\phi), \end{align}$

(3.21)

and

$\begin{align} \nabla_j\nabla_iw = &\ \nabla_j\nabla_iv-\nabla_j\nabla_iu_rd_r\phi-\nabla_iu_r\nabla_j(d_r\phi)-\nabla_ju_r\nabla_i(d_r\phi)\\ &\ -u_r\nabla_j\nabla_i(d_r\phi)\\ = &\ 2v^3h_{jr}\nabla_ruh_{is}\nabla_su+v^2\nabla_rh_{ij}\nabla_ru+vh_{ir}h_{jr}\\ &\ -2v^2h_{jp}\nabla_puh_{iq}(u_r\nabla_qu+e_q^r)d_r\phi-v\nabla_ph_{ij}(u_r\nabla_pu+e_p^r)d_r\phi\\ &\ -vh_{is}(u_r\nabla_su+e_s^r)\nabla_j(d_r\phi)-vh_{js}(u_r\nabla_su+e_s^r)\nabla_i(d_r\phi)-u_r\nabla_j\nabla_i(d_r\phi)\\ = &\ 2v^2wh_{jr}\nabla_ruh_{is}\nabla_su+vw\nabla_rh_{ij}\nabla_ru+vh_{ir}h_{jr}\\ &\ -2v^2h_{jp}\nabla_puh_{iq}\nabla_qd\phi-v\nabla_ph_{ij}\nabla_pd\phi\\ &\ -vh_{is}(u_r\nabla_su+e_s^r)\nabla_j(d_r\phi)-vh_{js}(u_r\nabla_su+e_s^r)\nabla_i(d_r\phi)-u_r\nabla_j\nabla_i(d_r\phi). \end{align}$

(3.22)

By selecting a suitable moving frame, we assume $(h_{ij})$ is diagonal at $x_0$ . At the maximum point $x_0\in\Omega_{\mu}$ , from (3.19) and $\nabla G = 0$ , we see that

$\begin{align} -\frac{\nabla_i w}{w\log w} = \alpha_1\nabla_i u+\nabla_id. \end{align}$

(3.23)

Together with (3.21), we also have

$\begin{align} -w\log w(\alpha_1\nabla_i u+\nabla_id) = vwh_{ii}\nabla_iu-vh_{ii}\nabla_id\phi-u_r\nabla_i(d_r\phi). \end{align}$

(3.24)

We divide the indexes $i\in I = \{1, 2, \cdots, n\}$ into two subsets as follows.

$\begin{align} &\ J = \{i\in I:|\alpha_1\nabla_iu|\leq 9\}, \end{align}$

(3.25)

$\begin{align} &\ K = I\backslash J = \{i\in I:|\alpha_1\nabla_iu| > 9\}. \end{align}$

(3.26)

For $i\in J$ , recall $|\nabla d|\leq 1$ ,

$\begin{align} |\alpha_1\nabla_iu+\nabla_id|^2\leq 100. \end{align}$

(3.27)

For $i\in K$ , we have

$\begin{align} \alpha_1|\nabla_iu| > \frac{4}{5}|\alpha_1\nabla_iu+\nabla_i d| > 6, \end{align}$

(3.28)

Now we assume $v$ , $w$ large enough, i.e., $v\geq\max\{\frac{\alpha_1}{\epsilon}, \exp{\frac{2}{\epsilon}}\}$ , then

$\begin{align} |vh_{ii}\nabla_id\phi||\leq\epsilon |vwh_{ii}\nabla_iu|, \end{align}$

(3.29)

and

$\begin{align} |u_r\nabla_i(d_r\phi)|\leq\epsilon w\log w|\alpha_1\nabla_iu+\nabla_id|, \end{align}$

(3.30)

for a small positive number $\epsilon$ . Then, using (3.24), we get

$\begin{align} h_{ii}\leq 0,\quad \text{for}\ i\in K, \end{align}$

(3.31)

and

$\begin{align} \frac{1+\epsilon}{\log w}v|h_{ii}\nabla_iu|\geq(1-\epsilon)|\alpha_1\nabla_iu+\nabla_id|. \end{align}$

(3.32)

Thus, plugging (3.28) into (3.32), we have

$\begin{align} -\alpha_1vh_{ii}|\nabla_iu|^2\geq\frac{3\log w}{4}|\alpha_1\nabla_iu+\nabla_id|^2, \quad\text{for}\ i\in K, \end{align}$

(3.33)

provided we set $\epsilon = \frac{1}{100}$ .

By the maximum principle, by (3.20), we have

$\begin{align} 0\geq&\ F^{ij}\nabla_j\nabla_iG\\ = &\ \frac{1}{w\log w}F^{ii}\nabla_i\nabla_iw-\frac{1+\log w}{(w\log w)^2}F^{ii}|\nabla_iw|^2+\alpha_1F^{ii}\nabla_i\nabla_iu+F^{ii}\nabla_i\nabla_id\\ \geq&\ \frac{1}{w\log w}F^{ii}\nabla_i\nabla_iw-\frac{9\log w}{8}F^{ii}(\alpha_1\nabla_iu+\nabla_id)^2+\frac{\alpha_1f}{v}-C\mathcal{F}. \end{align}$

(3.34)

We use the fact $\nabla_i\nabla_ju = \frac{h_{ij}}{v}$ in the last inequality and assume $v$ large.

From (3.22) we obtain

$\begin{align} F^{ii}\nabla_i\nabla_iw = &\ 2v^2wF^{ii}h_{ii}^2|\nabla_iu|^2+vwF^{ii}\nabla_rh_{ii}\nabla_ru+vF^{ii}h_{ii}^2-2v^2F^{ii}h^2_{ii}\nabla_iu\nabla_id\phi\\ &\ -vF^{ii}\nabla_rh_{ij}\nabla_rd\phi-2vF^{ii}h_{ii}(\nabla_iuu_r+e_i^r)\nabla_i(d_r\phi)-u_rF^{ii}\nabla_j\nabla_i(d_r\phi)\\ = &\ 2vF^{ii}h_{ii}\nabla_iu(vwh_{ii}\nabla_ru-vh_{ii}\nabla_id\phi-u_r\nabla_i(d_r\phi))+vF^{ii}h_{ii}^2\\ &\ +vw\langle\nabla f,\nabla u\rangle +vwf_z|\nabla u|^2-v\phi\langle\nabla f,\nabla d\rangle \\ &\ -2vF^{ii}h_{ii}e_i^r\nabla_i(d_r\phi)-u_rF^{ii}\nabla_j\nabla_i(d_r\phi). \end{align}$

(3.35)

By (3.24), $f_z\geq0$ , $|\langle\nabla f, \nabla u\rangle |\leq \frac{|Df|}{v}$ , and the Cauchy-Schwartz inequality, we have

$\begin{align} F^{ii}\nabla_i\nabla_iw = &\ -2vw\log wF^{ii}h_{ii}\nabla_iu(\alpha_1\nabla_iu+\nabla_id)+vF^{ii}h_{ii}^2\\ &\ +vw\langle\nabla f,\nabla u > -v\phi\langle\nabla f,\nabla d > -2vF^{ii}h_{ii}e_i^r\nabla_i(d_r\phi)-u_rF^{ii}\nabla_j\nabla_i(d_r\phi)\\ \geq&\ -2\alpha_1 vw\log wF^{ii}h_{ii}|\nabla_iu|^2-2vw\log wF^{ii}h_{ii}\nabla_iu\nabla_id+vF^{ii}h_{ii}^2\\ &\ -2vF^{ii}h_{ii}e_i^r\nabla_i(d_r\phi)-Cv(1+\mathcal{F})\\ \geq&\ -2\alpha_1 vw\log wF^{ii}h_{ii}|\nabla_iu|^2-2vw\log wF^{ii}h_{ii}\nabla_iu\nabla_id+vF^{ii}h_{ii}^2\\ &\ -\epsilon_1vF^{ii}h_{ii}^2-\frac{Cv}{\epsilon_1}\mathcal{F}-Cv(1+\mathcal{F}). \end{align}$

(3.36)

Multiplying $\nabla_id$ with both sides of (3.24), we have

$\begin{align} -w\log w(\alpha_1\nabla_i u+\nabla_id)\nabla_id = vwh_{ii}\nabla_iu\nabla_id-vh_{ii}|\nabla_id|^2\phi-u_r\nabla_i(d_r\phi)\nabla_id. \end{align}$

It follows that, by the Cauchy-Schwartz inequality,

$\begin{align} 2vwF^{ii}h_{ii}\nabla_iu\nabla_id = &\ 2\phi vF^{ii}h_{ii}|\nabla_id|^2+2F^{ii}u_r\nabla_i(d_r\phi)\nabla_id\\ &\ -2w\log wF^{ii}(\alpha_1\nabla_i u+\nabla_id)\nabla_id\\ \geq&\ -\epsilon_1\frac{ v}{\log w}F^{ii}h_{ii}^2–\frac{\phi^2}{\epsilon_1}\log w\mathcal{F}-Cv\mathcal{F}\\ &\ -\epsilon_2w\log w F^{ii}(\alpha_1\nabla_i u+\nabla_id)^2-\frac{w\log w}{\epsilon_2}\mathcal{F}. \end{align}$

(3.37)

Similarly, multiplying $\nabla_iu$ with both sides of (3.24), we have

$\begin{align} -w\log w(\alpha_1\nabla_i u+\nabla_id)\nabla_iu = vwh_{ii}|\nabla_iu|^2-vh_{ii}\nabla_id\nabla_iu\phi-u_r\nabla_i(d_r\phi)\nabla_iu. \end{align}$

By the Cauchy-Schwartz inequality, recall $|\alpha_1\nabla_iu|\leq9, \ \text{for}\ i\in J$ ,

$\begin{align} -\sum\limits_{i\in J}2\alpha_1vwF^{ii}h_{ii}|\nabla_iu|^2 = &\ -\sum\limits_{i\in J}2\alpha_1\phi vF^{ii}h_{ii}\nabla_id\nabla_iu-\sum\limits_{i\in J}2\alpha_1F^{ii}u_r\nabla_i(d_r\phi)\nabla_iu\\ &\ +\sum\limits_{i\in J}2\alpha_1w\log wF^{ii}(\alpha_1\nabla_i u+\nabla_id)\nabla_iu\\ \geq&\ -\epsilon_1\frac{ v}{\log w}F^{ii}h_{ii}^2–\frac{81\phi^2}{\epsilon_1}\log w\mathcal{F}-Cv\mathcal{F}\\ &\ -\epsilon_2w\log w F^{ii}(\alpha_1\nabla_i u+\nabla_id)^2-\frac{81w\log w}{\epsilon_2}\mathcal{F}. \end{align}$

(3.38)

Here we point out that $C$ is independent of $\alpha_1$ .

Plugging (3.37) and (3.38) into (3.36), we have

$\begin{align} F^{ii}\nabla_i\nabla_iw \geq&\ -\sum\limits_{i\in K}2\alpha_1 vw\log wF^{ii}h_{ii}|\nabla_iu|^2+v(1-3\epsilon_1)F^{ii}h_{ii}^2-\frac{82\phi^2}{\epsilon_1}|\log w|^2\mathcal{F}\\ &\ -2\epsilon_2 w|\log w|^2F^{ii}(\alpha_1\nabla_iu+\nabla_id)^2-\frac{82w|\log w|^2}{\epsilon_2}\mathcal{F}-Cv\log w\mathcal{F}\\ &\ \frac{Cv}{\epsilon_1}\mathcal{F}-Cv(1+\mathcal{F}). \end{align}$

(3.39)

Set $\epsilon_1 = \frac{1}{3}$ , $\epsilon_2 = \frac{1}{16}$ , and assume $v$ is sufficiently large, then

$\begin{align} F^{ii}\nabla_i\nabla_iw\geq&\ -2\alpha_1 vw\log wF^{ii}h_{ii}|\nabla_iu|^2-\frac{1}{8} w|\log w|^2F^{ii}(\alpha_1\nabla_iu+\nabla_id)^2\\ &\ -Cw|\log w|^2\mathcal{F}, \end{align}$

(3.40)

Putting (3.40) into (3.34), we get

$\begin{align} 0\geq&\ F^{ij}\nabla_j\nabla_iG\\ \geq&\ -\sum\limits_{i\in K}2\alpha_1 vF^{ii}h_{ii}|\nabla_iu|^2-\frac{5\log w}{4}F^{ii}(\alpha_1\nabla_iu+\nabla_id)^2\\ &\ -C\log w\mathcal{F}-C\mathcal{F}. \end{align}$

(3.41)

In view of (3.28), we see that

$\begin{align} -\sum\limits_{i\in I}\frac{5\log w}{4}F^{ii}(\alpha_1\nabla_iu+\nabla_id)^2\geq&\ -\sum\limits_{i\in K}\frac{5\log w}{4}F^{ii}(\alpha_1\nabla_iu+\nabla_id)^2\\ &\ -C\log w\mathcal{F}. \end{align}$

(3.42)

On the other hand, by (3.31) and (3.33),

$\begin{align} -\sum\limits_{i\in K}2\alpha_1 vF^{ii}h_{ii}|\nabla_iu|^2 \geq&\ \sum\limits_{i\in K}\frac{6\log w}{4}F^{ii}(\alpha_1\nabla_iu+\nabla_id)^2. \end{align}$

(3.43)

Particularly, there is $i_0\in K$ , say $i_0 = 1$ , such that $|\nabla_1u|\geq\frac{1}{2\sqrt{n}}$ and $F^{11}\geq\frac{1}{n}\mathcal{F}$ .

Plugging (3.42) and (3.43) into (3.41) and choosing $\alpha_1 = \max\{4\sqrt{n}, 16nC\}$ , we have

$\begin{align} 0\geq&\ \frac{\log w}{4}F^{11}(\alpha_1\nabla_1u+\nabla_1d)^2-C\log w\mathcal{F}-C\mathcal{F}\\ \geq&\ \big(\frac{\alpha_1^2\log w}{128n^2}-C\big)\mathcal{F}. \end{align}$

(3.44)

Thus we have $(1-|\phi|)v\leq w\leq\exp{\frac{128n^2C}{\alpha_1^2}}$ and finish the proof.

□

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The research of the first author was supported by China Postdoctoral Science Foundation 2022M722474. The research of the second author was supported by NSFC 12141105, NSFC 11871255 and National Key Research and Development Project SQ2020YFA070080.

Conflict of interest

The authors declare no conflict of interest.

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