Convex solutions to integral inequalities in two-dimensional domains

This paper presents a method to verify integral inequalities on two-dimensional domains. The integral expressions are given by line integrals on the boundaries and by surface integrals: both are quadratic on the dependent variables and their derivatives. The proposed approach can verify the inequalities for a set of the dependent variables defined by their boundary values. We apply the results to solve integral inequalities related to Lyapunov stability conditions for exponential stability of Partial Differential Equations.