In this talk, I shall discuss a class of functional inequalities known as mean Hadamard inequalities. These inequalities generalise (in some sense) the standard Hadamard inequality. I will show how these inequalities relate to problems in elasticity, in particular, minimisation of the Dirichlet energy subject to a conservation of mass constraint. I will also take some time to discuss some various notions of convexity and how this motivates the search for new techniques to handle these minimisation problems.
Although this talk will mainly be an introduction to the problem of finding mean Hadamard inequalities, I will take some time to go over some results in this area. More specifically, I will show some sufficient conditions that can be derived analytically using Fourier decomposition and weighted Poincaré inequalities and necessary conditions that use novel constructions of solutions to partial differential inclusions and numerical integration.