Verification of some functional inequalities via polynomial optimization

Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities form f(I1(u), . ., Ik(u)) ? 0 where f is a polynomial, u any function satisfying prescribed constraints, and I1(u), .,Ik(u) are integral functionals whose integrands polynomial in u, its derivatives, integration variable. We show that such can be strengthened into sufficient inequalities, which principle checked via semidefinite programming using standard techniques for optimization. These conditions used also optimize with affine dependence on tunable parameters whilst ensuring their nonnegativity. Our approach relies measure-theoretic lifting original inequality, extends both recent moment relaxation strategy PDE analysis dual functionals.