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

2008/07/24 by Nirmalendu Chaudhuri, Chaudhuri, Nirmalendu, Aram L. Karakhanyan +1
Mathematics · #35J60 #42A40 #73C50 #73V25 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.AP #math.CA #msc:35J60 #msc:42A40 #msc:73C50 #msc:73V25

paper · pdf · doi:10.48550/arxiv.0807.3810

23 pages

arxiv created 2008/07/24 · arxiv updated 2009/12/01

Abstract

We prove that any distribution q satisfying the equation ∇ q=÷\bf f for some tensor \bf f=(fⁱ_j), fⁱ_j∈ h^r(U) (1≤ r<∞) -the \it local Hardy space, q is in h^r, and is locally represented by the sum of singular integrals of fⁱ_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 |∇ \bf u|², |\rm cof∇\bf u|²∈ h¹. 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.

Related

• On a class of special Euler-Lagrange equations
• Construction of solutions to the 3D Euler equations with initial data in H^β for β>0
• On the energy and helicity conservation of the incompressible Euler equations
• Existence and regularity of minimizers for nonlocal energy functionals
• On the definiteness of associated energy-momentum tensors to a class of general variation problems