Fully Nonlinear Elliptic Equations and Applications to Geometry

Joel Spruck
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Joel Spruck: Johns Hopkins University

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 1145-1152 from Springer

Abstract: Abstract In this paper we will describe some recent advances in the theory of fully nonlinear elliptic equations that are motivated by some basic geometric problems. For example, one can ask, when does a smooth Jordan curve inR3 bound a surface of positive constant Gauss curvature? The theme of this talk is roughly that such geometric problems often suggest the proper formulation of purely analytic partial differential equation (PDE) results. As an example, in 1984 [2] it was shown that the classical Monge-Ampère boundary value problem * $$ \left\{ {_{{\text{ }}u{\text{ = }}\phi {\text{ on }}\partial \Omega }^{{\text{det }}{u_{ij{\text{ = }}\psi {\text{(}}x){\text{ in }}\Omega }}}} \right.$$ , where φ, ψ, Ω smooth, ψ0 = infΩ ψ > 0, and Ωstrictly convex, always has a (unique) strictly convex solution $$ u \in {C^\infty }(\Omega )$$ .

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_107

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