On the second boundary value problem for Monge-Ampere type equations and geometric optics

In this paper, we prove the existence of classical solutions to second boundary value prob- lems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics. Our results depend in particular on a priori second derivative estimates recently established by the authors under weak co-dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables, (A3w). We also avoid domain deformations by using the convexity theory of generating functions to construct unique initial solutions for our homotopy family, thereby enabling application of the degree theory for nonlinear oblique boundary value problems.Comment: Final version to appear in Archive for Rational Mechanics and Analysi

Oblique boundary value problems for augmented Hessian equations I

In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge-Amp`ere type operators in optimal transportation and geometric optics, the general theory here embraces prescribed mean curvature problems in conformal geometry as well as oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations.Comment: Revised version containing minor clarification

Material removal investigation in bonnet polishing of CoCr alloy

The manufacture of orthopaedic joint bearings surfaces requires exceptionally high levels of control of not only the surface finish but also the surface form. In the case of hip joints, the form of femoral head should be controlled to within ± 50ìm from a given diameter. It has been shown that a better form control of bearing component could enhance clearances creating the correct volume of lubrication to fill the bearing surface gap and reduce wear particle generation. This element is especially critical for the new generation non-spherical head designs. Bonnet polishing which is used successfully in the area of optics is potentially an excellent finishing process to control the form and finish of artificial joints. In the process of form control polishing an “influence function” which defines the material removal rate is of vital importance in developing a corrective polishing procedure. However, the effects of polishing parameters (such as precess angle, head speed, tool pressure and tool offset) on influence function are not very clear for CoCr alloys. These elements must be assessed if a deterministic polishing process is to be developed. Therefore, it is of paramount importance to understand the contribution of each polishing factors to influence function and consequent part polishing. This study has investigated the effects of polishing parameters on influence function, including geometric size and volumetric material removal rate (MRR). The experimental results indicate that the polishing parameter of precess angle and tool offset affect the geometric size of influence function significantly; the polishing parameter of head speed and tool pressure affect the geometric size of influence function to a lesser degree; the polishing parameter of precess angle, head speed and tool offset affect MRR greatly