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,

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On the Dirichlet problem for Hessian equations

We derive a priori second-order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under structure conditions which are close to optimal. We treat both equations on closed manifolds and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary. These estimates yield regularity and existence results, some of which are new even for equations in Euclidean space.

Second-order estimates and regularity for fully nonlinear elliptic equations on riemannian manifolds

The general notion of fully nonlinear second-order elliptic equation is given. Its relation to so-called Bellman equations is investigated. A general existence theorem for the equations like Pm(uxixj) = JJTM^1 ck(x)Pk(uxixj) *s obtained as an example of an application of the general notion of fully nonlinear elliptic equations. We will be dealing with the following question: Given an equation F(uxixi(x), uxi(x), u(x), x) = 0 in a domain D cW1, under what conditions and in what class of twice continuously differentiable functions u do we call it an elliptic equation? It may look strange that we address such a question. Indeed, there are even books [9], [17] and many articles about the general theory of fully nonlinear elliptic equations. Therefore, very general results are available in the theory. However, on the other hand, it turns out that if an unexperienced reader meets a fully nonlinear second-order partial differential equation in his investigations and tries to get any information about its solvability from the literature, then almost certainly he fails to find what he needs, unless he considers an equation that is exactly one which had already been treated. The point is that in the general theory we consider nonlinear equations only of a special type, say, such that for any x e D, £ e Rd, uij, w,, u e R S\\{\\2 < F(uij+^,z)-F(uij, z) < S~x\\c]\\2, where z = (w,, u, x) and S is a positive constant. Obviously, even for the simplest Monge-Ampere equation, when F = det(w,;) f(x), conditions of this type are not satisfied (for instance, for d > 2 the function det(M,;) grows much faster than linearly). Nevertheless, the general theory applies to this and many other special equations, and the reason for this is that in an appropriate class of functions they can be reduced to those considered in the general theory. We apply some techniques to include into our theory concrete equations such as the Monge-Ampere equation (real or complex) or more general Weingarten equations. These techniques are different in different cases, and by answering our main question here we want to show, in particular, what should be done for Received by the editors October 1, 1993. 1991 Mathematics Subject Classification. Primary 35J60. The work was partially supported by NSF Grant DMS-9302516. © 1995 American Mathematical Society 857 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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On the general notion of fully nonlinear second-order elliptic equations

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.

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On some inequalities for elementary symmetric functions

on a Riemannian manifold ( M n, g), wheref is a symmetric function of λ ∈ R, κ is a constant, ∇2u denotes the Hessian of a function u on M and, for a (0,2) tensorh on M , λ(h) = (λ1, · · · , λn) denotes the eigenvalues of h with respect to the metricg. The Dirichlet problem for equations of type (1.1) in R, with κ = 0, under various hypothesis, is studied by Caffarelli, Nirenberg and Spruck [3], Krylov [15], Trudinger [21] as well as in [16], [22], [24], [4] and [7] etc. Other boundary value problems have also been considered. In [19], Trudinger treated the Dirichlet and Neumann problems in balls for the degenerate case, while Urbas [23] studied nonlinear oblique boundary value problems in two dimensions. Our motivation to study such equations on Riemannian manifolds comes from their close connection with problems in differential geometry, such as the Minkowski-Christoffel problems (see, for example, [5] and [18] for details). In [16], Y. Y. Li treated similar equations on compact manifolds (without boundary) of nonnegative sectional curvature. The Dirichlet problems for Monge-Amp ère equations are treated by Guan and Spruck [10] on S and by Atallah and Zuily [1] and Guan and Li [9] (see also [8]) on general Riemannian manifolds. As in [3], we assumef is a smooth function defined in an open, convex, symmetric coneΓ ⊂ R with vertex at the origin,Γ + ⊆ Γ / = R where Γ + ≡ {λ ∈ R : each component λi > 0},

The Dirichlet problem for Hessian equations on Riemannian manifolds

Solvability conditions for curvature equations of order which are sufficient, and almost necessary, are obtained, and theorems concerning the existence of solutions in , , , are proved. The first-order curvature equation coincides with the curvature equation of order , and the curvature equation of order with the Monge-Ampere equation.Bibliography: 18 titles.

http://iopscience.iop.org/article/10.1070/SM1990v067n02ABEH002089/pdf

N M Ivochkina

Papers

Solution of the Dirichlet Problem for Curvature Equations of Order m