## Dirichlet Duality and the Nonlinear Dirichlet Problem

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Harvey, F. Reese

Lawson, Jr, H. Blaine

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We study the Dirichlet problem for fully nonlinear, degenerate elliptic
equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In
our approach the equation is replaced by a subset F of the space of symmetric
nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence
and uniqueness of continuous solutions under an explicit geometric
``F-convexity'' assumption on the boundary bdy(F). The topological structure of
F-convex domains is also studied and a theorem of Andreotti-Frankel type is
proved for them. Two key ingredients in the analysis are the use of subaffine
functions and Dirichlet duality, both introduced here. Associated to F is a
Dirichlet dual set F* which gives a dual Dirichlet problem. This pairing is a
true duality in that the dual of F* is F and in the analysis the roles of F and
F* are interchangeable. The duality also clarifies many features of the problem
including the appropriate conditions on the boundary. Many interesting examples
are covered by these results including: All branches of the homogeneous
Monge-Ampere equation over R, C and H; equations appearing naturally in
calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and
all branches of the Special Lagrangian potential equation.

##### Keywords

Mathematics - Analysis of PDEs, Mathematics - Complex Variables, Mathematics - Differential Geometry, 35J25, 35J70, 32W20