As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo cite{Guo} proved that $n$-di
mensional compact self-shrinkers in $mathbb{R}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $mathbb{S}^n(sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in $mathbb{R}^{n+1}$ with constant scalar curvature are isometric to the round sphere $mathbb{S}^n(sqrt{n})$(see also cite{Hui1}). Complete classifications of $n$-dimensional translating solitons in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature were given by Mart{i}n, Savas-Halilaj and Smoczykcite{MSS} and Ancari and Chengcite{AC}, respectively. In this paper we give complete classifications of $n$-dimensional complete self-shrinkers in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Mart{i}n, Savas-Halilaj and Smoczyk cite{MSS} and Ancari and Chengcite{AC}.
We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer degree is well
defined. Secondly, we compute the Brouwer degree case by case. As consequences, we give new proofs of some known existence results for the Kazdan-Warner equation on a connected finite graph.
Let $Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $Sigma$, $rho$ and $alpha$ real numbers. In this paper, we study a generalized mean field equation begin{align*} -Delta u=rholeft(dfrac{he^u}{int_Sigma he^u}-dfrac{1}{mathrm
{Area}left(Sigmaright)}right)+alphaleft(u-fint_{Sigma}uright), end{align*} where $Delta$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $rhoin (8kpi, 8(k+1)pi)$ for some non-negative integer number $kin mathbb{N}$ and $alpha otinmathrm{Spec}left(-Deltaright)setminusset{0}$. Then we obtain existence results for $alpha<lambda_1left(Sigmaright)$ by using the Leray-Schauder degree theory and the minimax method, where $lambda_1left(Sigmaright)$ is the first positive eigenvalue for $-Delta$.
In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ begin{align*} -Delta u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}}-1right) end{align*} on a closed Riemann surface whose area is equal to one. The s
olutions are the critical points of the functional $J_{8pi}$ which is defined by begin{align*} J_{8pi}(u)=frac{1}{16pi}int_{Sigma}| abla u|^2+int_{Sigma}u-lnleft|int_{Sigma}he^{u}right|,quad uin H^1left(Sigmaright). end{align*} We prove the existence of minimizer of $J_{8pi}$ by assuming begin{equation*} Delta ln h^++8pi-2K>0 end{equation*}at each maximum point of $2ln h^++A$, where $K$ is the Gaussian curvature, $h^+$ is the positive part of $h$ and $A$ is the regular part of the Green function. This generalizes the existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), 230-248] to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence $u_{varepsilon}$ of critical points of $J_{8pi-varepsilon}$ with $int_{Sigma}he^{u_{varepsilon}}=1, limlimits_{varepsilonsearrow 0}J_{8pi-varepsilon}left(u_{varepsilon}right)<infty$ and obtain the following identity during the blow-up process begin{equation*} -varepsilon=frac{16pi}{(8pi-varepsilon)h(p_varepsilon)}left[Delta ln h(p_varepsilon)+8pi-2K(p_varepsilon)right]lambda_{varepsilon}e^{-lambda_{varepsilon}}+Oleft(e^{-lambda_{varepsilon}}right), end{equation*}where $p_varepsilon$ and $lambda_varepsilon$ are the maximum point and maximum value of $u_varepsilon$, respectively. Moreover, $p_{varepsilon}$ converges to the blow-up point which is a critical point of the function $2ln h^{+}+A$.
In this note, we study minimal Lagrangian surfaces in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian free boundary on $mathbb{S}^3$ mus
t be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$ must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).
We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(Sigma,g)$ begin{align*} -Delta_{g}u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}{rm d}mu_{g}}-frac{1}{int_{Sigma}{rm d}mu_{g}}right) end{align*} w
here the prescribed function $hgeq0$ and $max_{Sigma}h>0$. We prove the global existence and convergence under additional assumptions such as begin{align*} Delta_{g}ln h(p_0)+8pi-2K(p_0)>0 end{align*} for any maximum point $p_0$ of the sum of $2ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $Sigma$. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145 (2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1 (1997), no. 2, 230-248] to the non-negative prescribed function case.
In a directional modulation (DM) network, the issues of security and privacy have taken on an increasingly important role. Since the power allocation of confidential message and artificial noise will make a constructive effect on the system performan
ce, it is important to jointly consider the relationship between the beamforming vectors and the power allocation (PA) factors. To maximize the secrecy rate (SR), an alternating iterative structure (AIS) between the beamforming and PA is proposed. With only two or three iterations, it can rapidly converge to its rate ceil. Simulation results indicate that the SR performance of proposed AIS is much better than the null-space projection (NSP) based PA strategy in the medium and large signal-to-noise ratio (SNR) regions, especially when the number of antennas at the DM transmitter is small.
Medium-scale or large-scale receive antenna array with digital beamforming can be employed at receiver to make a significant interference reduction, but leads to expensive cost and high complexity of the RF-chain circuit. To deal with this issue, a c
lassic analog-and-digital beamforming (ADB) structure was proposed in the literature for greatly reducing the number of RF-chains. Based on the ADB structure, we in this paper propose a robust hybrid ADB scheme to resist directions of arrival (DOAs) estimation errors. The key idea of our scheme is to employ null space projection (NSP) in analog beamforming domain and diagonal loading (DL) method in digital beamforming domain. Simulation results show that the proposed scheme performs more robustly, and moreover, has a significant improvement on the receive signal to interference plus noise ratio (SINR) compared to NSP ADB scheme and DL method.
