On derivation of Euler–Lagrange equations for incompressible energy-minimizers

Abstract

We prove that any distribution q satisfying the grad-div system ∇q = div f for some tensor-the local Hardy space; q is in h r and q 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 u satisfying |∇u| 2 , |cof ∇u| 2 ∈ h 1 . We also derive the system of Euler-Lagrange equations for volume preserving local minimizers u that are in the space K 1,3 loc [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.

Sufficient conditions for regularity of area-preserving deformations

“…The following properties of the elements of A ( 0 ) are worth recording (see [14], see also Remark 3.3 [4]):…”

“…It is worth to point out that Theorem 1 is valid for a more general class of stored energy functions subject to suitable structural conditions [4].…”

“…This article is the sequel of [4], where under very weak assumptions we were able to derive the Euler-Lagrange equation of the minimization problem discussed below. Our purpose here is to further investigate the weak equations and to point out some geometric methods that lead to regularity of local minimizers in Hölder spaces.…”

On the Jacobian equation and the Hardy space H^1(C)

“…[Bal77], [LO81], [BOP92], [Le Dre85], [CK09], [Kar12] and the references contained therein. The setting of [CK09] and [Kar12] is mathematically particularly close to ours.…”

“…The idea behind the proof of the theorem dates back at least to [LO81] and is used under hypotheses more similar to ours in [CK09]. Some technicalities are, however, needed in order to establish the result in our setting, and we therefore present a proof.…”