Dirichlet duality and the nonlinear Dirichlet problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain Ω ⋐ ℝn. In our approach the equation is replaced by a subset F ⊂ Sym2(ℝn) of the symmetric n × n matrices with ∂F ⊆ {F = 0}. We establish the existence and uniquen...

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Veröffentlicht in:	Communications on pure and applied mathematics Jg. 62; H. 3; S. 396 - 443
Hauptverfasser:	Harvey, F. Reese, Lawson Jr, H. Blaine
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.03.2009 Wiley John Wiley and Sons, Limited
Schlagworte:	Convex and discrete geometry Differential equations Exact sciences and technology Geometry Mathematical analysis Mathematical problems Mathematics Nonlinear equations Partial differential equations Sciences and techniques of general use Several complex variables and analytic spaces Non linear equation Elliptic equation Lagrangian Existence of solution Symmetric matrix Dirichlet problem Geometric convexity Duality Nonlinear problems Riemann geometry Boundary condition Degenerate equation
ISSN:	0010-3640, 1097-0312
Online-Zugang:	Volltext
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Zusammenfassung:	We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain Ω ⋐ ℝn. In our approach the equation is replaced by a subset F ⊂ Sym2(ℝn) 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.
Bibliographie:	istex:3F86CC267F31EA293767555179A007981FA4D5F9 Institut des Hautes Études Scientifiques ark:/67375/WNG-37J6TX7G-W ArticleID:CPA20265 National Science Foundation SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14
ISSN:	0010-3640 1097-0312
DOI:	10.1002/cpa.20265