Dirichlet duality and the nonlinear Dirichlet problem
Dirichlet duality and the nonlinear Dirichlet problem
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DOI：
10.1002/cpa.20265
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发表时间：
2007-10
影响因子：
3
通讯作者：
F. R. Harvey;H. Lawson
中科院分区：
文献类型：
--
作者：
F. R. Harvey;H. Lawson
研究了光滑有界区域Ω n上F（Hess u）= 0形式的完全非线性退化椭圆型方程的Dirichlet问题.在我们的方法中，方程被替换为对称n × n矩阵的子集F <$Sym2（<$n），其中<$F <${F = 0}。我们在边界条件<$Ω上的一个明确的几何“F-凸性”假设下建立了连续解的存在唯一性。研究了F-凸域的拓扑结构，证明了Andreotti-Frankel型定理.分析中的两个关键要素是使用“次仿射函数”和“狄利克雷对偶”。与F相关联的是一个Dirichlet对偶集F，它给出一个对偶Dirichlet问题。这个配对是一个真正的对偶，因为F的对偶是F，并且在分析中F和F的角色是可以互换的。对偶性也阐明了问题的许多特征，包括边界上的适当条件。这些结果涵盖了许多有趣的例子，包括：齐次Monge-Ampère方程在π、π和π上的所有分支;在校准几何、拉格朗日几何和p凸黎曼几何中自然出现的方程;以及特殊拉格朗日势方程的所有分支。© 2008 Wiley Periodicals，Inc.
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.