New applications of Hadamard-in-the-mean inequalities to incompressible variational problems

Abstract

Let $\mathbb{D}(u)$ be the Dirichlet energy of a map $u$ belonging to the Sobolev space $H_{u_0}^1(\Omega ;{\mathbb{R}}^2)$ and let $A$ be a subclass of $H_{u_0}^1(\Omega ;{\mathbb{R}}^2)$ whose members are subject to the constraint $\det \nabla u=g$ a.e. for a given $g$, together with some boundary data $u_0$. We develop a technique that, when applicable, enables us to characterize the global minimizer of $\mathbb{D}(u)$ in $A$ as the unique global minimizer of the associated functional $F(u):=\mathbb{D}(u)+{\int }_{\Omega }f(x)\det \nabla u(x)dx$ in the free class $H_{u_0}^1(\Omega ;{\mathbb{R}}^2)$. A key ingredient is the mean coercivity of $F(\phi )$ on $H_0^1(\Omega ;{\mathbb{R}}^2)$, which condition holds provided the `pressure' $f\in L^{\infty }(\Omega )$ is `tuned' according to the procedure set out in \cite{BKV23}. The explicit examples to which our technique applies can be interpreted as solving the sort of constrained minimization problem that typically arises in incompressible nonlinear elasticity theory.