Permanently Going Back and Forth between the ``Quadratic World' and the ``Convexity World' in Optimization

The objective of this work is twofold. Firstly, we propose a review of different results concerning convexity of images by quadratic mappings, putting them in a chronological perspective. Next, we enlighten these results from a geometrical point of view in order to provide new and comprehensive proofs and to immerse them in a more general and abstract context. \keywords{Quadratic functions, Range convexity, Joint positive definiteness.} \amsclass{90C20, 52A20, 15A60.} Accepted 27 July 2001. Online publication 7 January 2002.     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

   Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

Remarks on a Mean Field Equation on $\mathbb{S}^2$

In this note, we study symmetry of solutions of the elliptic equation\begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2},\end{equation*} that arises in the consideration of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.     

New Results for the BBM Equation

The BBM equation posed on $\mathbb{R}$ and $\mathbb{R}^+$ is revisited. Improving on earlier results, global well-posedness and bounds for the growth in time of relevant norms of solutions corresponding to very general auxiliary data are derived.     

Optimal control of a one-dimensional storage process

We consider the discounted and ergodic optimal control problems related to a one-dimensional storage process. The existence and uniqueness of the corresponding Bellman equation and the regularity of the optimal value is established. Using the Bellman equation an optimal feedback control is constructed. Finally we show that under this optimal control the origin is reachable.This work was supported by the National Science Foundation under Grant No. MCS 8121940.     

Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.     

带非局部阻尼项的粘弹性方程解的衰减估计

In this paper,we consider the following viscoelastic equation u tt- △u +∫t 0 g(t-s)△u(s)ds + a(x)u t + u |u|r = 0 with initial condition and Dirichlet boundary condition.The decay property of the energy function closely depends on the properties of the relaxation function g(t) at infinity.In the previous works of [3,7,11],it was required that the relaxation function g(t) decay exponentially or polynomially as t → +∞.In the recent work of Messaoudi [12,13],it was shown that the energy decays at a similar rate of decay of the relaxation function,which is not necessarily dacaying in a polynomial or exponential fashion.Motivated by [12,13],under some assumptions on g(x),a(x) and r,and by introducing a new perturbed energy,we also prove the similar results for the above equation.     

Landau-Type Theorems for Solutions of a Quasilinear Differential Equation

In this paper, we study solutions of the quasilinear differential equation $\bar{z}\partial_{\bar{z}}f(z)+z\partial_{z}f(z)+(1-|z|^2)\partial_{z}\partial_{\bar{z}}f(z)=f(z)$. We utilize harmonic mappings to obtain an explicit representation of solutions of this equation. By this result, we give two versions of Landau-type theorem under proper normalization conditions.     

拟线性退化抛物方程Cauchy问题的解

对来自金融数学领域的方程_xxu+u_yu-_tu=c（x,y,t,u）,（x,y,t）∈Q_T=R²×[0,T)的Cauchy问题,给出了一种新的熵解的定义,得到了其适定性结果.可以证明所得到的解还是强解,即方程中所出现的各阶偏导数几乎处处连续.最后讨论了解的爆破性质以及与解的间断点相关的几何性质.     

Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

We investigate the initial boundary value problem of the pseudo-parabolic equation $u_{t} - \triangle u_{t} - \triangle u = \phi_{u}u + |u|^{p - 1}u,$ where $\phi_{u}$ is the Newtonian potential, which was studied by Zhu et al. (Appl. Math. Comput., 329 (2018) 38-51), and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels. We in this note determine the upper and lower bounds for the blow-up time. While estimating the upper bound of blow-up time, we also find a sufficient condition of the solution blowing-up in finite time at arbitrary initial energy level. Moreover, we also refine the upper bounds for the blow-up time under the negative initial energy.     

Convergence Properties of Projection and Contraction Methods for Variational Inequality Problems

In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method), where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system of the inequality constrained VI problem. \keywords{Variational inequality, Projection and contraction method, Predictor-corrector stepsize, Convergence property.} \amsclass{90C30, 90C33, 65K05.} Accepted 5 September 2000. Online publication 16 January 2001.     

Gradient Estimates for a Nonlinear Heat Equation Under Finsler-geometric Flow

This paper considers a compact Finsler manifold $(M^n, F(t), m)$ evolving under a Finsler-geometric flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation $$\partial_{t}u(x,t)=\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\in M\times[0,T],$$where $\Delta_{m}$ is the Finsler-Laplacian. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Our results generalize and correct the work of S. Lakzian, who established similar results for the Finsler-Ricci flow. Our results are also natural extension of similar results on Riemannian-geometric flow, previously studied by J. Sun. Finally, we give an application to the Finsler-Yamabe flow.     

A Simple Proof of the Theorem Concerning Optimality in a One-Dimensional Ergodic Control Problem

We give a simple proof of the theorem concerning optimality in a one-dimensional ergodic control problem. We characterize the optimal control in the class of all Markov controls. Our proof is probabilistic and does not need to solve the corresponding Bellman equation. This simplifies the proof. Accepted 24 March 1998     

Some Results on Risk-Sensitive Control with Full Observation

The Bellman equation of the risk-sensitive control problem with full observation is considered. It appears as an example of a quasi-linear parabolic equation in the whole space, and fairly general growth assumptions with respect to the space variable x are permitted. The stochastic control problem is then solved, making use of the analytic results. The case of large deviation with small noises is then treated, and the limit corresponds to a differential game. Accepted 25 March 1996     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Viscosity Solutions of an Infinite-Dimensional Black—Scholes—Barenblatt Equation

We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman equations.     

Long Time Asymptotic Behavior of Solution of Difference Scheme for a Semilinear Parabolic Equation

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as $t\rightarrow\infty$. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.     

Long Time Well-Posedness of the MHD Boundary Layer Equation in Sobolev Space

In this paper, we study the long time well-posedness of 2-D MHD boundary layer equation. It was proved that if the initial data satisfies■,then the life span of the solution is at least of order ε~(2-η) for η 0.