图像修复中BSCB模型的粘性分析

考虑图像修复中BSCB方程和变形的BSCB方程组的粘性问题.运用半群理论,得到粘性BSCB方程光滑解的存在唯一性.此外,利用粘性消失方法还得到:当粘性系数ν→0时,粘性变形的BSCB方程组的解在经典意义下收敛到变形的BSCB方程组的解.     

图像修复中BSCB模型的粘性分析

考虑图像修复中BSCB方程和变形的BSCB方程组的粘性问题.运用半群理论,得到粘性BSCB方程光滑解的存在唯一性.此外,利用粘性消失方法还得到:当粘性系数v→0时,粘性变形的BSCB方程组的解在经典意义下收敛到变形的BSCB方程组的解.     

一类非齐次弹性力学方程组整体熵解的存在性

考虑非线性、非齐次弹性力学方程组在一定条件下整体熵解的存在性问题.由不变区域理论导出粘性解的L~∞模的先验估计,利用粘性消失法结合补偿列紧理论给出粘性解的收敛性,即广义解的存在性.     

奇异Hamilton-Jacobi-Bellman方程粘性解的存在唯一性

研究系数在边界点有奇性的一类Hamilt on- Jacobi- Bellman (HJB)方程的粘性解的存在唯一性问题及解的渐近估计,这类问题包括波动系数振荡或爆破的情况.奇异HJB方程在随机最优控制和金融数学等许多领域都有重要的应用,包括金融数学中的随机利率模型.应用粘性上下解理论建立了一类奇异HJB方程的比较原理,给出了粘性解存在唯一性的条件.     

平面平行管道中的MHD方程组的边界层

该文研究平面平行管道中不可压缩MHD方程组的边界层问题.利用多尺度分析和精细的能量方法,证明了当粘性系数与磁耗散系数趋近于0时,粘性与磁耗散MHD方程组的解收敛到理想MHD方程组的解.     

非线性守恒律高阶谱粘性法的收敛性

讨论守恒型方程周期边界问题的高阶谱粘性方法逼近解的收敛性.在逼近解一致有界的假设下,通过建立其高阶导数的上界估计,证明了高阶谱粘性方法逼近解具有同二阶谱粘性方法逼近解相类似的高频衰减性质.以此为基础,用补偿列紧法证明了高阶谱粘性方法逼近解收敛于守恒型方程的物理解.     

偶数维空间耗散波动方程解的衰减估计

研究偶数维空间带粘性的波动方程柯西问题解的逐点估计.通过对格林函数的精细分析,得到解的大时间状态.解呈现出惠更斯现象.     

偶数维空间耗散波动方程解的衰减估计

研究偶数维空间带粘性的波动方程柯西问题解的逐点估计.通过对格林函数的精细分析,得到解的大时间状态.解呈现出惠更斯现象.     

二阶非线性椭圆偏微分方程Dirichlet问题的粘性解

研究了一类二阶非线性椭圆偏微分方程Dirichlet问题粘性解的存在性与唯一性。首先建立粘性解的比较定理，确保了解的唯一性，然后运用Perron方法构造出解。从而解决了这类问题的粘性解的存在性与唯一性。     

非紧空间上折现Hamilton-Jacobi方程粘性解的存在性讨论*

当底空间紧时, 初始函数为连续函数的Lax-Oleinik型粘性解是局部半凹的,所以是相应的Hamilton-Jacobi\ (以下简称为H-J) 演化方程(简称为接触H-J方程)的粘性解.当底空间非紧时, 对于H-J方程和接触H-J方程, 其Lax-Oleinik型解的下确界未必能取到.文章将探讨在非紧空间上, 折现H-J方程粘性解有限性的条件, 并给出了在此假设下粘性解的表达式.     

Viscosity solutions for elliptic-parabolic problems

We study an elliptic-parabolic problem appearing in the theory of partially saturated flows in the framework of viscosity solutions. This is part of current investigation to understand the theory of viscosity solutions for PDE problems involving free boundaries. We prove that the problem is well posed in the viscosity setting and compare the results with the weak theory. Dirichlet or Neumann boundary conditions are considered.     

On viscosity solutions of fully nonlinear equations with measurable ingredients

We study fully nonlinear, uniformly elliptic equations with measurable ingredients. Caffarelli's recent work on W^2,p estimates for viscosity solutions has led to significant progress in this area. Here we present a unified treatment of this theory based on an appropriate notion of viscosity solution. For instance, it is shown that strong solutions are viscosity solutions, that viscosity solutions are twice differentiable a.e., and that the pointwise derivatives satisfy the equation a.e. An important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations. This extends results of this type due to Evans and Krylov. Our work is to some extent expository, the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients. © 1996 John Wiley & Sons, Inc.     

On Aronsson Equation and Deterministic Optimal Control

When Hamiltonians are nonsmooth, we define viscosity solutions of the Aronsson equation and prove that value functions of the corresponding deterministic optimal control problems are solutions if they are bilateral viscosity solutions of the Hamilton-Jacobi-Bellman equation. We characterize such a property in several ways, in particular it follows that a value function which is an absolute minimizer is a bilateral viscosity solution of the HJB equation and these two properties are often equivalent. We also determine that bilateral solutions of HJB equations are unique among absolute minimizers with prescribed boundary conditions. This research was partially supported by MIUR-Prin project “Metodi di viscosità, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari”.     

Semi-concave singularities and the Hamilton-Jacobi equation

We study the Cauchy problem for the Hamilton-Jacobi equation with a semiconcave initial condition. We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one.     

Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.     

Forward dynamic utility functions: A new model and new results

A major obstacle in the existing models of forward dynamic utilities and investment performance evaluation is to establish the existence and uniqueness of the optimal solutions. Consequently, we present a new model of forward dynamic utilities. In doing so, we establish the existence and uniqueness of the solutions for a general (smooth) utility function, and we show that the assumptions needed for such solutions are similar to those under the backward formulation. Moreover, we provide unique viscosity solutions. We also provide discontinuous viscosity solutions. In addition, we introduce Hausdorff-continuous viscosity solutions to the portfolio model.     

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions. When ...     

On travelling-wave solutions to systems of conservation laws with singular viscosity

Abstract. The method of vanishing artificial viscosity is used to obtain smooth, largedata travelling-wave solutions to a class of conservation laws with semidefinite viscosity. The one-dimensional Navier-Stokes equations serve as an illustrating example.     

Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems

Pessimistic bilevel optimization problems are not guaranteed to have a solution even when restricted classes of data are involved. Thus, we propose a concept of viscosity solution, which satisfactorily obviates the lack of optimal solutions since it allows to achieve in appropriate conditions the security value. Differently from the viscosity solution concept for optimization problems, introduced by Attouch (SIAM J Optim 6:769–806, 1996) and defined through a viscosity function that aims at regularizing the objective function, viscosity solutions for pessimistic bilevel optimization problems are defined through regularization families of the solutions map to the lower-level optimization. These families are termed “inner regularizations” since they approach the optimal solutions map from the inside. First, we investigate, in Banach spaces, several classical regularizations of parametric constrained minimum problems giving sufficient conditions for getting inner regularizations; then, we establish existence results for the corresponding viscosity solutions under possibly discontinuous data.