Regularity for Energy-Minimizing Area-Preserving Deformations

Abstract

In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem inf KˆΩ |∇v| 2 , Ω ⊂ R 2 as the Lagrange multiplier corresponding to the incompressibility constraint det ∇v = 1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ⋆ = u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L 2 norm of the pressure. As a by-product we conclude that u ∈ C 1 2 loc (Ω) if the dual pressure (introduced in [6]) is nonnegative.

Sufficient conditions for regularity of area-preserving deformations

“…The advantage of div {∇u − pcof∇u} = 0 is that it is "almost" linear unlike the second equation det ∇u = 1 in (1.4). Hence some of the basic tools from linear theory of elliptic systems would work under suitable conditions on p. A different approach, utilizing the scale invariance of (1.4), will appear in the forthcoming paper [10].…”

A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

“…Let the energy functional E(u) be given by (1), and let f (x, ξ) be given by (2), where M ∈ L ∞ (B, R 16 ) is symmetric and satisfies (3) for some ν > 0. Let u be a stationary point of E in the sense of (7) and assume that the corresponding pressure λ satisfies (9) ∇λ…”

Energy Minimisers with Prescribed Jacobian

“…Establishing regularity of energy minimisers for (1.2) is a challenging task, even in the incompressible case f = 1. There is an extensive literature on the topic, and we refer the reader to [7,10,16,27,45,46], as well as the references therein, for further information.…”