Hadamard's inequality in the mean

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.

Figures (14)

• Figure 1: Distribution of $f$ yielding $I(\varphi) \geq 0$ for any $\varphi \in W^{1,2}_0(Q;\mathbb{R}^2)$ .
• Figure 4.2: Distribution of rectangles.
• Figure 4.3: This explains the conversion of the minimization problem posed on the central domain $R_1 \cup R_2$ into two separate problems defined on the unit square $Q$. The minimizer of $\mathbb{D}(F_2,R_2)$ corresponds to the 'left-hand half' of the minimizer of $\mathbb{D}(\cdot,Q)$ under the boundary conditions stated on $Q$, suitably shifted, as seen in the problem of $\ominus-$type. The minimizer of $\mathbb{D}(F_1,R_1)$ corresponds to the 'right-hand half' of the minimizer of $\mathbb{D}(\cdot,Q)$ under the boundary conditions stated on $Q$, suitably shifted, as seen in the problem of $\oplus-$type.
• Figure 4.4: Distribution of $Q_1,\ldots,Q_4$.
• Figure 5.5: Distribution of $f$ (left), components $\varphi_1$ (middle), and $\varphi_2$ (right) of the corresponding minimizer $\varphi=(\varphi_1, \varphi_2)$ providing $I(\varphi)<0$.

Theorems & Definitions (36)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3
• Proposition 3.4