Feida Jiang

On the Neumann Problem for Monge-Ampère Type Equations

In this paper, we study the global regularity for regular Monge-Amp\`ere type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of the Neumann boundary value problem is proved under natural conditions. The techniques build upon the delicate and intricate treatment of the standard Monge-Amp\`ere case by Lions, Trudinger and Urbas in 1986 and the recent barrier constructions and second derivative bounds by Jiang, Trudinger and Yang for the Dirichlet problem. We also consider more general oblique boundary value problems in the strictly regular case.

Oblique boundary value problems for augmented Hessian equations I

In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge–Ampère type operators in optimal transportation and geometric optics, the general theory here embraces Neumann problems arising from prescribed mean curvature problems in conformal geometry as well as general oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations.

Gradient estimates for Neumann boundary value problem of Monge–Ampère type equations

This paper concerns the gradient estimates for Neumann problem of a certain Monge–Ampere type equation with a lower order symmetric matrix function in the determinant. Under a one-sided quadratic structure condition on the matrix function, we present two alternative full discussions of the global gradient bound for the elliptic solutions.

WEAK SOLUTIONS OF MONGE-AMPÈRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION

Abstract This paper concerns the weak solutions of some Monge-Ampere type equations in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Ampere type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.

On the Second Boundary Value Problem for Monge–Ampère Type Equations and Geometric Optics

In this paper, we prove the existence of classical solutions to second boundary value problems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics. Our results depend in particular on a priori second derivative estimates recently established by the authors under weak co-dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables, (A3w). We also avoid domain deformations by using the convexity theory of generating functions to construct unique initial solutions for our homotopy family, thereby enabling application of the degree theory for nonlinear oblique boundary value problems.

Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

Abstract In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war u t - Δ ∞ β ⁢ u = f ⁢ ( x , t ) , {u_{t}}-\Delta_{\infty}^{\beta}u=f(x,t), where β is a fixed constant and Δ ∞ β {\Delta_{\infty}^{\beta}} is the β-biased infinity Laplacian operator Δ ∞ β ⁢ u = Δ ∞ N ⁢ u + β ⁢ | D ⁢ u | \Delta_{\infty}^{\beta}u=\Delta_{\infty}^{N}u+\beta\lvert Du\rvert related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when f = 0 {f=0} , we show some explicit solutions.

On the semilinear reaction diffusion system arising from nuclear reactors

In this work, the initial-boundary value problem for a class of semilinear reaction-diffusion systems is considered. By an abstract fixed point theorem on positive cone together with an accurate a priori estimate, we establish the coexistence of the positive stationary solutions and the uniqueness of ordered positive stationary solutions. Next, we study the global existence and blowup of positive solutions and obtain a threshold result. Finally, we give the blowup rate estimate of positive blowup solutions.