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 \(\mathcal {A}\) 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 \(\mathcal {A}\) as the unique global minimizer of the associated functional \(F(u):=\mathbb {D}(u)+ \int _{\Omega } f(x) \, \det \nabla u(x) \, \, \textrm{d}x\) in the free class \(H^1_{u_0}(\Omega ;\mathbb {R}^2)\) . A key ingredient is the mean coercivity of F 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 [1]. 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.
