The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian. (English) Zbl 0654.35031

This paper is a continuation of parts I and II [Commun. Pure Appl. Math. 37, 369-402 (1984; Zbl 0598.35047) and 38, 209-252 (1985; Zbl 0598.35048)]. Here is studied the solvability of Dirichlet’s problem in a bounded domain \(\Omega \subset R^ n\) with smooth boundary \(\partial \Omega:\) \[ F(D^ 2u)=\psi \quad in\quad \Omega;\quad u=\phi \quad on\quad \partial \Omega, \] where the function F is defined by a smooth symmetric function \(f(\lambda_ 1,...,\lambda_ n)\) of the eigenvalues \(\lambda =(\lambda_ 1,...,\lambda_ n)\) of the Hessian matrix \(D^ 2u=\{u_{ij}\}\). It is assumed that the equation is elliptic, i.e. \(\partial f/\partial x_ i>0\), for all i, and that f is a concave function.
Reviewer: P.Drábek


35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI


[1] Caffarelli, L., Nirenberg, L. &Spruck, J., The Dirichlet problem for nonlinear second order elliptic equations, I: Monge-Ampère equations.Comm. Pure Appl. Math. 37 (1984), 369–402. · Zbl 0598.35047
[2] Caffarelli, L., Kohn, J. J., Nirenberg, L. &Spruck, J.. The Dirichlet problem for nonlinear second order elliptic equations, II: Complex Monge-Ampère, and uniformly elliptic equations.Comm. Pure Appl. Math., 38 (1985), 209–252. · Zbl 0598.35048
[3] Gårding, L., An inequality for hyperbolic polynomials.J. Math. Mech., 8 (1959), 957–965. · Zbl 0090.01603
[4] Harvey, R. &Lawson jr, H. B., Calibrated geometries.Acta Math., 148 (1982), 47–157. · Zbl 0584.53021
[5] Ivočkina, N. M., The integral method of barrier functions and the Dirichlet problem for equations with operators of Monge-Ampère type.Mat. Sb. (N.S.), 112 (1980), 193–206 (Russian);Math. USSR-Sb., 40 (1981), 179–192 (English).
[6] Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations in a domain.Izv. Akad. Nauk SSSR, 47 (1983), 75–108.
[7] Marcus, M., An eigenvalue inequality for the product of normal matrices.Amer. Math. Monthly, 63 (1956), 173–174. · Zbl 0070.01304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.