Hadamard's inequality in the mean

Let $Q$ be a Lipschitz domain in ${\mathbb{R}}^n$ and let $f\in L^{\mathrm{\infty }}(Q)$. We investigate conditions under which the functional $$I(\phi )={\int }_Q|\mathrm{\nabla }\phi {|}^n+f(x)det\mathrm{\nabla }\phi dx$$ obeys $I(\phi )\ge 0$ for all $\phi \in W_0^{1,n}(Q,{\mathbb{R}}^n)$. We prove that there are $f$ such that $I\ge 0$ holds and is strictly stronger than the best possible inequality that can be derived using Hadamard's pointwise inequality $n^{n/2}|detA|\le |A{|}^n$ alone. Almost all of the $f$ we consider are piecewise continuous, and we specialize to the case $n=2$ in many occasions. We find that it is both the geometry of the `jump sets' as well as the sizes of the `jumps' themselves that determine whether $I\ge 0$ holds. We also outline connections to quasiconvexity at the boundary, sequential weak lower semicontinuity and to Agmon's conditions in elasticity. Theoretical results will be complemented with various numerical experiments. This is a joint work with J.~Bevan (Surrey) and J.~Valdman (Prague).