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

TL;DR

This paper establishes the existence, representation, and regularity properties of pressure fields in incompressible energy-minimizing deformations, advancing the understanding of Euler-Lagrange equations in this context.

Contribution

It proves the local representation of pressure via singular integrals, derives Euler-Lagrange equations for incompressible minimizers, and addresses a longstanding open problem in the field.

Findings

Pressure q is in local Hardy space h^r and represented by singular integrals.

Existence and local representation of hydrostatic pressure p for elastic minimizers.

Partial regularity results for area-preserving minimizers with H"older continuous pressure.

Abstract

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.