Regularity for Energy-Minimizing Area-Preserving Deformations

In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem inf(kappa) integral(Omega) vertical bar del v vertical bar(2), mu subset of R-2 as the Lagrange multiplier corresponding to the incompressibility constraint deta double dagger v=1 a.e. in Omega. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Omega (a <dagger)=u(Omega). This allows to construct a convex function psi, defined in the image domain, such that the measure of the normal mapping of psi controls the L (2) norm of the pressure. As a by-product we conclude that if the dual pressure (introduced in Karakhanyan, Manuscr. Math. 138:463, 2012) is nonnegative.