J. J. Bevan

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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; the minimizer u is C and is such that det∇u vanishes at one point. Mathematics Subject Classification. — Please, give AMS classification codes —. Received October(More)
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 ≥ 2. The systems are singular in the sense that they arise as the EulerLagrange equations of the functionals I(u) = ∫ B W (x,∇u(x)) dx,(More)
Abstract. A family of integral functionals F which model in a simplified way 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(More)
In this note we formulate a sufficient condition for the quasiconvexity at x 7→ 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)
After introducing the topics that will be covered in this work we review important concepts from the calculus of variations in elasticity theory. Subsequently the following three topics are discussed: The first originates from the work of Post and Sivaloganathan [Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, 127(03):595–614, 1997](More)
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)
We prove that any C1 weak local minimizer of a certain class of elastic stored-energy functionals I(u) = ∫ Ω f(∇u) dx subject to a linear boundary displacement u0(x) = ξx on a star-shaped domain Ω with C1 boundary is necessarily affine provided f is strictly quasiconvex at ξ. This is done without assuming that the local minimizer satisfies the(More)