Regularity of locally convex surfaces

Abstract

Interior estimates are derived for the C2, µ-Hölder norm of the radius vector X ∈ C1, 1 (Ω) of a locally convex surface Σ in terms of the first fundamental form IΣ, the Gauss curvature K and the integral ∫ |H| dσ. Here H is the mean curvature of Σ. The coefficients gij of IΣ are assumed to belong to the Hölder class C2, µ (Ω) for some μ, 0 < μ < 1. A boundary condition is discussed which ensures an estimate for ∫ | H | dσ.