Neil S. Trudinger

On the Dirichlet problem for Hessian equations

in domains f~ in Euclidean n-space, R n, where f is a given symmetric function on R n, A denotes the eigenvalues A1, ..., An of the Hessian matrix of second derivatives D2u and r is a given function in f t • n. Equations of this type were treated by Calfarelli, Nirenberg and Spruek [2], for the case r 1 6 2 who demonstrated the existence of classical solutions for the Dirichiet problem, under various hypotheses on the function f and the domain ft. Their results extended their previous work [1], and that of Krylov [13], Ivochkina [8] and others, on equations of Monge-Amp~re type,

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 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

Harnack inequalities for quasi-minima of variational integrals

Abstract In his fundamental work on linear elliptic equations, De Giorgi established local bounds and Holder estimates for functions satisfying certain integral inequalities. The main result of this paper is that the Harnack inequality can be proved directly for functions in the De Giorgi classes. This implies that every non-negative Q-minimum (in the terminology of Giaquinta and Giusti) satisfies a Harnack inequality.

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.

Isoperimetric inequalities for quermassintegrals

Abstract The isoperimetric inequalities of Aleksandrov and Fenchel for quermassintegrals (cross sectional measures) of convex domains in Euclidean space are established for non-convex domains, subject to natural curvature conditions. The techniques are new and draw upon the theory of Monge-Ampere type equations related to previous work of the author on equations of prescribed curvature.

On some inequalities for elementary symmetric functions

In this note, we prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems.