用多项式optimization证明一些函数不等式
Verification of some functional inequalities via polynomial optimization
摘要：
受李雅普诺夫方法应用于偏微分方程（PDE）的启发，我们研究了形式为$f(I_1(u),\ldots,I_k(u))\geq 0$的泛函不等式，其中$f$是一个多项式，$u$是满足规定约束的任意函数，而$I_1(u),\ldots,I_k(u)$是积分泛函，其被积函数在$u$及其导数和积分变量中是多项式.我们表明，这样的功能不等式可以加强到足够的多项式不等式，原则上可以检查通过半定规划使用标准技术多项式优化。这些充分条件也可以用来优化泛函与仿射依赖的可调参数，同时确保其非负性。我们的方法依赖于一个测量理论提升的原始功能的不平等，这扩展了最近的时刻放松策略PDE分析和对偶方法积分泛函不等式。
英文摘要：
Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities of the form $f(I_1(u),\ldots,I_k(u))\geq 0$ where $f$ is a polynomial, $u$ is any function satisfying prescribed constraints, and $I_1(u),\ldots,I_k(u)$ are integral functionals whose integrands are polynomial in $u$, its derivatives, and the integration variable. We show that such functional inequalities can be strengthened into sufficient polynomial inequalities, which in principle can be checked via semidefinite programming using standard techniques for polynomial optimization. These sufficient conditions can be used also to optimize functionals with affine dependence on tunable parameters whilst ensuring their nonnegativity. Our approach relies on a measure-theoretic lifting of the original functional inequality, which extends both a recent moment relaxation strategy for PDE analysis and a dual approach to inequalities for integral functionals.
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