A Mini-course on EllipticâHyperbolic Equations - ICMS
A Mini-course on EllipticâHyperbolic Equations - ICMS
A Mini-course on EllipticâHyperbolic Equations - ICMS
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The isometric embedding problem: When can a 2-dimensi<strong>on</strong>alRiemannian manifold be “seen” via a local realizati<strong>on</strong> in R 3 ? Thisproblem is associated with the existence of a local soluti<strong>on</strong> to thefully n<strong>on</strong>linear Darboux equati<strong>on</strong>. But that equati<strong>on</strong> has beensuccessfully approached via Nash–Moser methods involving thelinearizati<strong>on</strong>(aKu x ) x+ bu yy + cKu x + du y ,where a, b > 0 and K is the Gaussian curvature of the surface.[Han, Commun. Pure Appl. Math. 58 (2005); Han and Khuri,Commun. Anal. Geom. 18 (2010)]