Addendum to ‘Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity’

Abstract

The paper ‘Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity’ published in Proc. R. Soc. A 2013, 469, 20130075 ( doi:10.1098/rspa.2013.0075 ) is clarified. Notably, much more general boundary conditions are given under which sharp lower bounds on the integrals of certain quadratic functions of the fields can be obtained. More precisely, if the quadratic form is Q *-convex then any solution of the Euler–Lagrange equations will necessarily minimize the integral. As a consequence, strict Q *-convexity is found to be an appropriate condition to ensure uniqueness of the solutions of a wide class of linear Euler–Lagrange equations in a given domain Ω with appropriate boundary conditions.