Isometric Immersions and Compensated Compactness

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. TComment: 25 pages, 6 figue

BRIEF NOTES A Flow Potential Function for Hyperbolic Sine Creep

Table 1 p|q, 2 The distribution of hydrostatic stress p along the v-axis of symmetry "f ihe nlastlc field, for elliptical holes of aspect ratio a/b > 1 of the plastic f dx ay __ d£ x H . d0 (4a) (46) On substituting (4) into the expressions for xp and yp, and carrying out an integration by parts, we obtain the following set of equations for the moving coordinates These equations can now be rewritten in a form suitable for numerical integration, with the aid of (3), as follows: In the present application of (6) the argument Z of the Bessel functions is complex; thus the modified Bessel functions h(Z) = -iJ\(iZ) and h(Z) = Jo(iZ) are used in the numerical quadrature of the line integrals. The Solution of the Elliptical-Hole Problem Equations . + sin 0 w the application of the present solution to the problem of the stapihty of incompressible plasticity in the presence of microvoids Reference