Neumann and second boundary value problems for Hessian and Gauß curvature flows

Abstract

We consider the flow of a strictly convex hypersurface driven by the Gauß curvature. For the Neumann boundary value problem and for the second boundary value problem we show that such a flow exists for all times and converges eventually to a solution of the prescribed Gauß curvature equation. We also discuss oblique boundary value problems and flows for Hessian equations.  2003 Éditions scientifiques et médicales Elsevier SAS MSC: primary 53C44; secondary 35K20, 53C42 RÉSUMÉ.-Nous considérons le flot d'une hypersurface strictement convexe piloté par la courbure de Gauß. Pour le problème aux limites de Neumann et pour le deuxième problème aux limites nous montrons qu'un tel flot existe pour tout temps et converge vers une solution de l'équation de courbure de Gauß prescrite. Nous étudions aussi des problèmes aux limites oblique et les flots pour des équations hessiennes.  2003 Éditions scientifiques et médicales Elsevier SAS

Translating solutions for Gaußcurvature flows with Neumann boundary conditions

“…Here, we used that 0 ∈ Ω implying x, ν < 0. Using these preparations, we prove a priori C 2 -estimates similarly to [6] and [7]. For the reader's convenience, we repeat the arguments incorporating the necessary modifications to the parabolic case.…”

“…The following argument is a modification of the proofs in [4,6] and [7]. We use the translating solution u ∞ , especially, its speed v ∞ , to construct an auxiliary barrier function.…”

“…As mentioned in [5], methods of [6] can be adapted to the non-parametric logarithmic Gauß curvature flow subject to the second boundary condition. For the Neumann boundary value problem, however, the lack of monotonicity requires a new proof to uniformly bound the oscillation of a solution.…”

“…The paper is organized as follows: As explained in [6], standard linear parabolic theory and the implicit function theorem imply short-time existence. We show uniform first-order estimates in Sections 2 and 3.…”

On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I

“…Using the parabolic methods, O.C. Schnürer and K. Smoczyk [10] also obtained the existence of solutions to (1.13) for τ = 0. As far as τ = π 2…”