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- J. J. Bevan
- 2008

We extend the result [11] of Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to W… (More)

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of S 1 ; the minimizer u is C 1 and is such that det ∇u vanishes at one point.

- J. J. Bevan
- SIAM J. Math. Analysis
- 2011

A family of integral functionals F which model in a simplified way material mi-crostructure occupying a two-dimensional domain Ω and which take account of surface energy and a variable well depth is studied. It is shown that there is a critical well depth, whose scaling with the surface energy density and domain dimensions is given, below which the state u… (More)

We prove that any C 1 weak local minimizer of a certain class of elastic stored-energy functionals I(u) = Ω f (∇u) dx subject to a linear boundary displacement u 0 (x) = ξx on a star-shaped domain Ω with C 1 boundary is necessarily affine provided f is strictly quasiconvex at ξ. This is done without assuming that the local minimizer satisfies the… (More)

- J. Bevan
- 2015

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u(R, θ) = Rg(θ), where (R, θ) are plane polar coordinates and g takes values in R m , m ≥ 2. The systems are singular in the sense that they arise as the Euler

- J J Bevan
- 2008

A family of integral functionals F which, in a simplified way, model material microstructure occupying a two-dimensional domain Ω and which take account of surface energy and a variable well-depth is studied. It is shown that there is a critical well-depth, whose scaling with the surface energy density and domain dimensions is given, below which the state u… (More)

In this note we formulate a sufficient condition for the quasiconvexity at x → λx of certain functionals I(u) which model the stored-energy of elastic materials subject to a deformation u. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to Müller and Spector, on admissible deformations. Deformations… (More)

- J. Bevan
- 2014

We study the integral functional I(w) :=

- Jonathan Bevan
- 2013

The double-covering map u dc : R 2 → R 2 is given by u dc (x) = 1 √ 2|x| x2 2 − x1 2 2x1x2 in cartesian coordinates. This paper examines the conjecture that u dc is the global minimizer of the Dirichlet energy I(u) = B |∇u| 2 dx among all W 1,2 mappings u of the unit ball B ⊂ R 2 satisfying (i) u = u dc on ∂B, and (ii) det ∇u = 1 almost everywhere. Let the… (More)

We prove the local Hölder continuity of strong local minimizers of the stored energy functional E(u) = Ω λ|∇u| 2 + h(det ∇u) dx subject to a condition of 'positive twist'. The latter turns out to be equivalent to requiring that u maps circles to suitably star-shaped sets. The convex function h(s) grows logarithmically as s → 0+, linearly as s → +∞, and… (More)