In this thesis, we will study a family of functionals, referred to as excess functionals, introduced by J

Bevan, M Kružìk and J Valdman [10] to derive so-called mean Hadamard inequalities, a functional

generalisation of the classical (pointwise) Hadamard inequality. In turn, these inequalities may be

used to construct examples of Jacobian constrained variational problems and minimisation problems

for functionals of a polyconvex integrand, that have explicit solutions. This is of particular relevance

in the field of elasticity [7], where these functionals describe the stored energy associated with the

deformation of a material and Jacobian constraints are a form of incompressibility condition.

The focus of the thesis will be primarily on developing explicit and constructive methods for deriving

examples and counter-examples of mean Hadamard inequalities. We will also describe how to use

these mean Hadamard inequalities to form Jacobian constrained classes in which the Dirichlet

energy has a global minimiser. In the cases where we can not show the existence of a minimiser,

we will show that the techniques used to bound the excess functional can also be used to bound the

Dirichlet energy on the constrained class. The results will mainly be for the case of a ball shaped

domain in two dimensions but can be generalised using conformal mappings.