On Derivation of Euler-Lagrange Equations for incompressible energy-minimizers

We prove that any distribution q satisfying the grad-div system q=divf for some tensor f=(fji)fjihr(U)(1r) -the local Hardy space; q is in h r and q is locally represented by the sum of singular integrals of fji 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 u2cofu2h1 . We also derive the system of Euler–Lagrange equations for volume preserving local minimizers u that are in the space Kloc13 [defined in (1.2)]—partially resolving a long standing problem. In two dimensions we prove partial C 1,α regularity of weak solutions provided their gradient is in L 3 and p is Hölder continuous.