On Derivation of Euler-lagrange Equations for Incompressible Energy-minimizers

We prove that any distribution q satisfying the equation ∇q = div f for some tensor f = (f i j), f i j ∈ h (U) (1 ≤ r < ∞) -the local Hardy space, q is in h, and is locally represented by the sum of singular integrals of f i j with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incompressible elastic energy-minimizing deformation u satisfying |∇u|, |cof∇u| ∈ h. We also derive the system of EulerLagrange equations for incompressible local minimizers u that are in the space K loc (defined in (1.2)); partially resolving a long standing problem. For Hölder continuous pressure p, we obtain partial regularity of area-preserving minimizers.