Special Mini-Course in Geometric PDE

Manifolds with geometric structure carry large and useful families of non-standard “subharmonic” functions. For example, any almost complex manifold with hermitian metric carries plurisubharmonic functions. Moreover, it also carries “Lagrangian subharmonic functions” whose restrictions to Lagrangian submanifolds are subharmonic. Similarly, a manifold with calibration $\varphi$ carries “$\varphi$-plurisubharmonic functions” which are subharmonic on all calibrated submanifolds. In all cases the extremals in these families, the “harmonic functions”, are interesting and often satisfy a basic non-linear second-order equation. I will discuss the Dirichlet Problem for such harmonic functions on bounded domains in a riemannian manifold. Existence and uniqueness will be established for quite general second-order equations. The result holds for all continuous boundary data subject to a geometric F-convexity of the boundary, defined entirely in terms of the equation F. Examples include all branches of the Monge-Amp`ere equation over {\bf R}, {\bf C} and {\bf H}, and all branches of the special lagrangian potential equation. The mini-course will be in two parts. The first will discuss the Dirichlet problem for domains in {\bf R}^n. The second will treat domains in general riemannian manifolds.