[论文解读] Hyperbolic polynomials and the Dirichlet problem

F. Reese Harvey, H. Blaine Lawson|arXiv (Cornell University)|Dec 29, 2009

Mathematical Dynamics and Fractals参考文献 13被引用 27

一句话总结

本文提供了Gårding关于双曲多项式理论的自完备阐述，建立了特征值函数的实解析排列，并证明了PDE理论中至关重要的单调性性质。它表明Gårding理论自然地为完全非线性PDE生成了子方程，确保在$(p,E)$-拟凸域上Dirichlet问题具有唯一连续解。

ABSTRACT

This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. In a second, independent part of the paper, the relationship of Garding's theory to the authors' recent work (arXiv:0710.3991) on the Dirichlet problem for fully nonlinear partial differential equations is investigated. Let p be a homogeneous polynomial of degree m on S^2(R^n) which is hyperbolic with respect to the all positive directions A \geq 0. Then p has an associated eigenvalue map lambda:S^2(R^n) o R^m, defined modulo the permutation group acting on R^m. Consequently, each closed symmetric set E of R^m induces a second-order p.d.e. by requiring, for a C^2-function u in n-variables, that (D^2 u)(x) lie in the boundary of E for all x. Assume that E + (R_+)^m is contained in E. A main result is that for smooth domains in R^n whose boundary is suitably (p,E)-pseudo-convex, the Dirichlet problem has a unique continuous solution for all continuous boundary data. This applies to a vast collection of examples the most basic of which are the m distinct branches of the equation p(D^2 u) =0. In the authors' recent extension of results from euclidean domains to domains in riemannian manifolds (arXiv:0907.1981), a new global ingredient, called a monotonicity subequation, was introduced. It is shown in this paper that for every polynomial $p$ as above, the associated Garding cone is a monotonicity cone for all branches of the the equation p(Hess u) = 0 where Hess u denotes the riemannian Hessian of u.

研究动机与目标

• 通过关于特征值解析性的新结果，提供Gårding双曲多项式理论的自完备、基础性的阐述。
• 在Gårding锥中元素相加下建立特征值函数的单调性，这对PDE应用至关重要。
• 通过子方程将Gårding理论与完全非线性二阶PDE的Dirichlet问题联系起来。
• 证明Gårding锥是黎曼流形上方程$p(\mathrm{Hess}\,u) = 0$的单调性锥。
• 通过方向导数的迭代应用和容量估计，将Gurvits不等式作为Gårding基本不等式的改进形式加以证明，并明确等号成立条件。

提出的方法

• 提出一个新定理（定理2.9），表明在Gårding锥中元素平移下，双曲多项式的特征值可实现为实解析参数化。
• 利用代数曲线的经典统一参数化定理，建立特征值函数的解析性与严格单调性。
• 应用Bauschke-Guler-Lewis-Sendov凸性结果，推导Gårding锥作为凸集的关键性质。
• 通过特征值映射$\lambda: \mathrm{Sym}^2(\mathbb{R}^n) \to \mathbb{R}^m$，从闭对称集$E \subset \mathbb{R}^m$构造子方程，从而导出形式为$\lambda(D^2u) \in \partial E$的PDE。
• 通过利用特征值的单调性，验证正性条件$F + P \subset F$（其中$P > 0$），从而确保子方程结构。
• 通过方向导数的迭代应用和容量估计，证明Gurvits不等式，从而对Gårding原始不等式进行改进。

实验结果

研究问题

• RQ1 在Gårding锥中元素平移下，双曲多项式的特征值函数能否被全局排列为实解析函数？
• RQ2 在正定矩阵相加下特征值的单调性是否可推广至整个Gårding锥，并支持非线性PDE子方程的构造？
• RQ3 在何种条件下，方程$p(D^2u) = 0$的Dirichlet问题在域$\Omega \subset \mathbb{R}^n$上存在唯一连续解？
• RQ4 Gårding理论如何为黎曼流形上方程$p(\mathrm{Hess}\,u) = 0$生成单调性锥？
• RQ5 Gurvits不等式能否通过方向导数的迭代应用和容量估计，作为Gårding不等式的改进形式推导得出？

主要发现

• 对于$b \in \Gamma$，特征值函数$\lambda_k(x + tb)$是从$\mathbb{R}$到$\mathbb{R}$的严格递增实解析映射，且具有实解析逆映射。
• Gårding锥$\Gamma$是凸集，且满足单调性性质$\lambda_k^\uparrow(x + b) > \lambda_k^\uparrow(x)$对所有$b \in \Gamma$成立，这对子方程理论至关重要。
• 对于任意闭对称集$E \subset \mathbb{R}^m$满足$E + \mathbb{R}_+^m \subset E$，由$\lambda(D^2u) \in \partial E$定义的Dirichlet问题在边界为$(p,E)$-拟凸的光滑域$\Omega$上存在唯一连续解。
• Gårding锥$\overline{\Gamma}$是黎曼流形上方程$p(D^2u) = 0$所有$m$个不同分支的单调性锥。
• Gurvits不等式通过表明$\frac{1}{m^m} \mathrm{Cap}(p) \leq \frac{1}{m!} p^{(m)}_{b_1,\dots,b_m}$（其中$b_1,\dots,b_m \in \Gamma(p)$）强化了Gårding不等式，且等号成立条件与Gårding结果相同。
• 容量$\mathrm{Cap}_{b_1,\dots,b_m}(p)$满足归纳不等式$\frac{1}{k^k} \mathrm{Cap}_{b_1,\dots,b_k}(p^{(m-k)}_{b_{k+1},\dots,b_m}) \leq \frac{1}{k(k-1)^{k-1}} \mathrm{Cap}_{b_1,\dots,b_{k-1}}(p^{(m-k+1)}_{b_k,\dots,b_m})$，从而提供容量估计的递归改进。