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Dirichlet duality and the nonlinear Dirichlet problem
Author(s) -
Harvey F. Reese,
Lawson H. Blaine
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20265
Subject(s) - mathematics , uniqueness , dirichlet problem , convexity , bounded function , dirichlet distribution , duality (order theory) , dirichlet boundary condition , pure mathematics , boundary (topology) , boundary value problem , degenerate energy levels , dirichlet's energy , mathematical analysis , dirichlet's principle , nonlinear system , physics , quantum mechanics , financial economics , economics
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F (Hess u ) = 0 on a smoothly bounded domain Ω ⋐ ℝ n . In our approach the equation is replaced by a subset F ⊂ Sym 2 (ℝ n ) of the symmetric n × n matrices with ∂ F ⊆ { F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “ F ‐convexity” assumption on the boundary ∂Ω. We also study the topological structure of F ‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F̃ that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F̃ is F , and in the analysis the roles of F and F̃ 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‐Ampère 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. © 2008 Wiley Periodicals, Inc.