The isometric immersion of two-dimensional Riemannian manifolds with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss--Codazzi equations for the first and second fundamental forms. The large $L^\infty$ solution is obtained, which leads to a $C^{1,1}$ isometric immersion. The approximate solutions are constructed by the Lax--Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The $H^{-1}$ compactness is also derived. A compensated compactness framework is applied to obtain the existence of a large $L^\infty$ solution to the Gauss--Codazzi equations for surfaces that are more general than those in the literature.
['Wentao Cao', 'Feimin Huang', 'Dehua Wang']
Isometric Immersion of Surface with Negative Gauss Curvature and the Lax-Friedrichs Scheme