  We prove that any distribution $q$ satisfying the equation $\nabla q=\div{\bf
f}$ for some tensor ${\bf f}=(f^i_j), f^i_j\in h^r(U)$ ($1\leq r<\infty$) -the
{\it local Hardy space}, $q$ is in $h^r$, and is locally represented by the sum
of singular integrals of $f^i_j$ with Calder\'on-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 ${\bf u}$ satisfying $|\nabla {\bf u}|^2,
|{\rm cof}\nabla{\bf u}|^2\in h^1$. We also derive the system of Euler-Lagrange
equations for incompressible local minimizers ${\bf u}$ that are in the space
$K^{1,3}_{\rm loc}$; partially resolving a long standing problem. For H\"older
continuous pressure $p$, we obtain partial regularity of area-preserving
minimizers.

Analysis of PDEs;Classical Analysis and ODEs