  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.
