On the Second Boundary Value Problem for Monge–Ampère Type Equations and Geometric Optics
In this paper, we prove the existence of classical solutions to second boundary value problems 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.
Unable to display preview. Download preview PDF.
- 2.Fitzpatrick, P., Pejsachowicz, J.: Orientation and the Leray-Schauder Theory for Fully Nonlinear Elliptic Boundary Value Problems. Memoirs of the American Mathematical Society. vol. 10, No. 483, American Mathmatics Society, Providence (1993)Google Scholar
- 18.Trudinger, N.S.: On the prescribed Jacobian equation. Proceedings of Internationl Conference on for the 25th Anniversary of Viscosity Solutions, Gakuto International Series, Mathematical Sciences and Applications 20, 243–255, 2008Google Scholar
- 20.Trudinger, N.S.: On the local theory of prescribed Jacobian equations revisited (in preparation)Google Scholar
- 21.Trudinger, N.S., Wang, X.-J.: On convexity notions in optimal transportation. Preprint, 2008Google Scholar
- 23.Trudinger, N.S., Wang, X.-J.: On the second boundary value problem for Monge–Ampère type equations and optimal transportation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. VIII, 143–174 (2009)Google Scholar