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

Chaudhuri, Nirmalendu; Karakhanyan, Aram L.

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

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Description

We prove that any distribution $q$ satisfying the equation $\nabla q=÷{\bf f}$ 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 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 ${\bf u}$ 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 Hölder continuous pressure $p$, we obtain partial regularity of area-preserving minimizers.
23 pages