Neumann-Type Boundary Conditions for Hamilton-Jacobi Equations in Smooth Domains

Neumann or oblique derivative boundary conditions for viscosity solutions of Hamilton-Jacobi equations are considered. As developed by P.L. Lions, such boundary conditions are naturally associated with optimal control problems for which the state equations employ "Skorokhod" or reflection dynamics to ensure that the state remains in a prescribed set, assumed here to have a smooth boundary. We develop connections between the standard formulation of viscosity boundary conditions and an alternative formulation using a naturally occurring discontinuous Hamiltonian which incorporates the reflection dynamics directly. (This avoids the dependence of such equivalence on existence and uniqueness results, which may not be available in some applications.) At points of differentiability, equivalent conditions for the boundary conditions are given in terms of the Hamiltonian and the geometry of the state trajectories using optimal controls.     

自然增长条件下非线性椭圆方程的全局BMO估计

本文研究自然增长条件下一类具有H?lder连续系数的椭圆方程弱解梯度的全局BMO估计.在系数矩阵A为H?lder连续并满足一致椭圆条件下,利用极大函数方法,获得非线性Calderón-Zygmund型全局BMO估计.     

Existence Results for Superlinear Elliptic Equations with Nonlinear Boundary Value Conditions

In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + u = {{\left| u \right|}^{r - 2}}u}&{in\;\Omega ,\;\;} \\ {\frac{{\partial u}}{{\partial v}} = {{\left| u \right|}^{q - 2}}u}&{on\;\partial \Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN, N ≥ 3 is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both q and r are subcritical; (ii) r is critical and q is subcritical; (iii) r is subcritical and q is critical; (iv) Both q and r are critical.     

Hamilton-Jacobi Equations on Graph and Applications

This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand’s transport inequality.     

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.     

解Hamilton-Jacobi方程的MUSCL格式

本文利用单调数值通量和分片线性重构导数的方法构造了一种求HJ方程数值解的有限差分格式:MUSCL格式,并证明该格式具有TVB稳定性.数值实验表明该格式具有二阶精度,能避免产生伪振荡,尤其在类似"角点"的间断处有较好的分辩率.     

Periodic Solutions of Superlinear Convex Autonomous Hamiltonian Systems

Using the Szulkin's variant of Mountain Pass Theorem, we prove the existence of nontrivial orbits with prescribed period for autonomous Hamiltonian systems in infinite dimen-sional Hilbert spaces.     

Hypercontractivity of Solutions to Hamilton-Jacobi Equations

We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.     

Hamilton-Jacobi Equations with Discontinuous Source Terms

We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinuous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function.     

一类具有次二次增长条件和L^1资料的拟线性椭圆方程

该文研究了关于梯度具有次二次增长条件,右端项在L^1空间的一类拟线性椭圆型方程熵解的存在性．     

Quantitative Stochastic Homogenization of Viscous Hamilton-Jacobi Equations

We prove explicit estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi equations, assuming the coefficients satisfy a finite range of dependence. In particular, we obtain an algebraic rate of convergence with overwhelming probability under certain structural conditions on the Hamiltonian.     

Nonlinear Nonhomogeneous Dirichlet Equations Involving a Superlinear Nonlinearity

We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carathéodory function which is (p?1)-superlinear but does not satisfy the Ambrosetti–Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang (Ann Inst H Poincaré Anal Non Linéaire 8(1):43–57, 1991). Subsequently, by imposing additional conditions on the nonlinearity \({f(x,\cdot)}\), we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special case of (p, 2)-equations we prove the existence of a second nodal solution for a total of six nontrivial solutions given with complete sign information. Finally, we study a nonlinear eigenvalue problem and we show that the problem has at least two nontrivial positive solutions for all parameters \({\lambda > 0}\) sufficiently small where one solution vanishes in the Sobolev norm as \({\lambda \to 0^+}\) and the other one blows up (again in the Sobolev norm) as \({\lambda \to 0^+}\).     

Oscillation for First Order Superlinear Delay Differential Equations

Some almost sharp sufficient conditions of oscillation and nonoscillationare obtained for the superlinear delay differential equation x'(t)+p(t)[x((t))]=0, tt₀     

Homogenization of Monotone Systems of Non-Coercive Hamilton-Jacobi Equations

In this article, we study homogenization for a class of monotone systems of first-order timedependent Hamilton-Jacobi equations in the case of non-coercive Hamiltonians. And we prove the uniform convergence of the solution of oscillating systems to the solution of the homogenized systems.     

Conditions for Superlinear Convergence in l1 and l{infty} Solutions of Overdetermined Non-linear Equations

Let n be the number of unknowns in an overdetermined systemof non-linear equations. If fewer than n equations are satisfiedin an l₁ solution or if fewer than (n+1) maximum residuals occurin an lsolution, then iterative methods of calculation convergesuperlinearly only if some second derivative information isused. This paper establishes some conditions on second derivativeestimates that are necessary and sufficient for superlinearconvergence.     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints

We study Hamilton-Jacobi equations in a bounded domain with the state constraint boundary condition. We establish a general convergence result for viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with the state constraint boundary condition to asymptotic solutions as time goes to infinity.