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

TL;DR

This paper introduces a new technique leveraging Hadamard-in-the-mean inequalities to identify unique minimizers in constrained variational problems related to incompressible elasticity.

Contribution

It develops a method to characterize global minimizers of Dirichlet energy under determinant constraints using mean coercivity and pressure tuning.

Findings

The technique applies to specific constrained minimization problems in nonlinear elasticity.

It establishes conditions for the uniqueness of minimizers in Sobolev spaces.

Explicit examples demonstrate the method's effectiveness in incompressible elasticity contexts.

Abstract

Let $\mathbb{D}(u)$ be the Dirichlet energy of a map $u$ belonging to the Sobolev space $H^1_{u_0}(\Omega;\mathbb{R}^2)$ and let $A$ be a subclass of $H^1_{u_0}(\Omega;\mathbb{R}^2)$ 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 $H^1_{u_0}(\Omega;\mathbb{R}^2)$. A key ingredient is the mean coercivity of $F(\varphi)$ on $H^1_0(\Omega;\mathbb{R}^2)$, 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.