Xu-Jia Wang

Neumann problems of semilinear elliptic equations involving critical Sobolev exponents

Abstract In this paper we study the existence of positive solutions to the equation Δu + up + f(x, u) = 0 under the Neumann boundary condition D gg u + α(x)u = 0, where p = (n + 2) (n − 2) , f(x, u) is a lower order perturbation of up at infinity. When α(x) = 0, we prove the existence of a positive solution provided lim u → 0 f(x, u) u = a(x) ⩽ 0, a(x) − 0, and f(x, u) ⩾ −Au − Bu q for some constants A, B ⩾ 0, q ϵ (1, n (n − 2) ) . For general α(x), we prove the existence under an additional assumption on the boundary ∂Ω .

Hessian measures II

In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain Q in Euclidean n-space, k = 1,... , n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider k-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of k-convex functions.

ON THE WEAK CONTINUITY OF ELLIPTIC OPERATORS AND APPLICATIONS TO POTENTIAL THEORY

In this paper, we establish weak continuity results for quasilinear elliptic and subelliptic operators of divergence form, acting on corresponding classes of subharmonic functions. These results are analogous to our earlier results for fully nonlinear k-Hessian operators. From the weak continuity, we derive various potential theoretic results including capacity estimates, potential estimates and the Wiener criterion for regular boundary points. Our methods make substantial use of Harnack inequalities for solutions.

On the second boundary value problem for Monge-Ampère type equations and optimal transportation

This paper is concerned with the existence of globally smooth so- lutions for the second boundary value problem for certain Monge-Amp` ere type equations and the application to regularity of potentials in optimal transportation. In particular we address the fundamental issue of determining conditions on costs and domains to ensure that optimal mappings are smooth diffeomorphisms. The cost functions satisfy a weak form of the condition (A3), which was introduced in a recent paper with Xi-nan Ma, in conjunction with interior regularity. Our condition is optimal and includes the quadratic cost function case of Caffarelli and Urbas as well as the various examples in our previous work. The approach is through the derivation of global estimates for second derivatives of solutions.

On the design of a reflector antenna

The reflector antenna design problem requires to solve a second boundary value problem for a complicated Monge-Ampere equation, for which the traditional discretization methods fail. In this paper we reduce the problem to that of finding a minimizer or a maximizer of a linear functional subject to a linear constraint. Therefore it becomes an linear optimization problem and algorithms in linear programming apply.

On the Monge mass transfer problem

Abstract. The Monge mass transfer problem, as proposed by Monge in 1781, is to move points from one mass distribution to another so that a cost functional is minimized among all measure preserving maps. The existence of an optimal mapping was proved by Sudakov in 1979, using probability theory. A proof based on partial differential equations was recently found by Evans and Gangbo. In this paper we give a more elementary and shorter proof by constructing an optimal mapping directly from the potential functions of Monge and Kantorovich.

Hessian measures I

Alternatively we may write (1.3) Fk[u] = [Du]k, where [A]k denotes the sum of the k× k principal minors of an n× n matrix A. Our purpose in this paper is to extend the definition of the Fk to corresponding classes of continuous functions so that Fk[u] is a Borel measure and to consider the Dirichlet problem in this setting. A function u ∈ C(Ω) is called k-convex (uniformly k-convex) in Ω if Fj [u] ≥ 0 (> 0) for j = 1, . . . , k. The operator Fk

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

The k-hessian equation

Abstract The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a gradient flow method we prove a Sobolev type inequality for k-admissible functions vanishing on the boundary, and study the corresponding variational problems. We also extend the definition of k-admissibility to non-smooth functions and prove a weak continuity of the k-Hessian operator. The weak continuity enables us to deduce a Wolff potential estimate. As an application we prove the Holder continuity of weak solutions to the k-Hessian equation. These results are mainly from the papers [CNS2, W2, CW1, TW2, Ld] in the references of the paper.

The Dirichlet Problem for degenerate Hessian Equations

Abstract In this paper, we study the Dirichlet problem for a class of fully nonlinear degenerate elliptic equations which depend only on the eigenvalues of the Hessian matrix. We provide a new and simpler approach to the estimation of second derivatives, through differentiation with respect to vector fields with skew-symmetric Jacobians.