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

Abstract

We prove that any distribution $q$ satisfying the equation $\nabla q=÷\mathbf{f}$ for some tensor $\mathbf{f}=(f_j^i),f_j^i\in h^r(U)$ ($1\le r<\infty$) -the {\it local Hardy space}, $q$ is in $h^r$, and is locally represented by the sum of singular integrals of $f_j^i$ 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 $\mathbf{u}$ satisfying $|\nabla \mathbf{u}{|}^2,|\mathrm{cof}\nabla \mathbf{u}{|}^2\in h^1$. We also derive the system of Euler-Lagrange equations for incompressible local minimizers $\mathbf{u}$ that are in the space $K_{\mathrm{loc}}^{1,3}$; partially resolving a long standing problem. For Hölder continuous pressure $p$, we obtain partial regularity of area-preserving minimizers.

23 pages