Abstract
Let \(Q\) be a Lipschitz domain in \(\mathbb{R}^n\) and let \(f \in L^{\infty}(Q)\). We investigate conditions under which the functional $$I_n(\varphi)=\int_Q |\nabla \varphi|^n+ f(x)\,\mathrm{det} \nabla \varphi\, \mathrm{d}x $$ obeys \(I_n \geq 0\) for all \(\varphi \in W_0^{1,n}(Q,\mathbb{R}^n)\), an inequality that we refer to as Hadamard-in-the-mean, or (HIM). We prove that there are piecewise constant \(f\) such that (HIM) holds and is strictly stronger than the best possible inequality that can be derived using the Hadamard inequality \(n^{\frac{n}{2}}|\det A|\leq |A|^n\) alone. When \(f\) takes just two values, we find that (HIM) holds if and only if the variation of \(f\) in \(Q\) is at most \(2n^{\frac{n}{2}}\). For more general \(f\), we show that (i) it is both the geometry of the `jump sets' as well as the sizes of the `jumps' that determine whether (HIM) holds and (ii) the variation of \(f\) can be made to exceed \(2n^{\frac{n}{2}}\), provided \(f\) is suitably chosen. Specifically, in the planar case \(n=2\) we divide \(Q\) into three regions \(\{f=0\}\) and \(\{f=\pm c\}\), and prove that as long as \(\{f=0\}\) `insulates' \(\{f= c\}\) from \(\{f= -c\}\) sufficiently, there is \(c>2\) such that (HIM) holds. Perhaps surprisingly, (HIM) can hold even when the insulation region \(\{f=0\}\) enables the sets \(\{f=\pm c\}\) to meet in a point. As part of our analysis, and in the spirit of the work of Mielke and Sprenger (1998), we give new examples of functions that are quasiconvex at the boundary.