John E. Hutchinson

Convergence of Phase Interfaces in the van der Waals-Cahn-Hilliard theory

Abstract. We study the general asymptotic behavior of critical points, including those of non-minimal energy type, of the functional for the van der Waals-Cahn-Hilliard theory of phase transitions. We prove that the interface is close to a hypersurface with mean curvature zero when no Lagrange multiplier is present, and with locally constant mean curvature in general. The energy density of the limiting measure has integer multiplicity almost everywhere modulo division by a surface energy constant.

C1, ? Partial regularity of functions minimising quasiconvex integrals

We prove C1, α almost everywhere regularity for minimisers of functionals of the form ∫F(x,u,Du), where F is uniformly strictly quasiconvex. This extends a recent result of Evans in which F is allowed only to depend on Du.

Partial regularity for minimisers of certain functionals having nonquadratic growth

SummaryWe prove partial regularity results for local minimisers of

The discrete plateau problem: algorithm and numerics

\begin{array}{*{20}c} {\int\limits_\Omega {(G^{\alpha \beta } (x,u)g_{ij} (x,u)D_\alpha u^i D_\beta u^j )^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} ,} } & {where p \geqslant 2.} \\ \end{array}

The discrete plateau problem: convergence results

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On the approximation of unstable parametric minimal surfaces

Lower semicontinuity for polyconvex functionals of the form ∫Ωg(detDu)dx with respect to sequences of functions fromW1,n (Ω;ℝn) which converge inL1 (Ωℝn) and are uniformly bounded inW1,n−1 (Ω;ℝn), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations of these results.

Partial regularity and everywhere continuity for a model problem from non-linear elasticity

We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unstable, surfaces. In this paper we introduce the general framework and some preliminary estimates, develop the algorithm and give the numerical results. In a subsequent paper we prove the convergence estimate. The algorithmic procedure is to find stationary points for the Dirichlet energy within the class of discrete harmonic maps from the discrete unit disc such that the boundary nodes are constrained to lie on a prescribed boundary curve. An integral normalisation condition is imposed, corresponding to the usual three point condition. Optimal convergence results are demonstrated numerically and theoretically for non-degenerate minimal surfaces, and the necessity for non-degeneracy is shown numerically.

Finite element approximations to surfaces of prescribed variable mean curvature

Regularity is proven almost everywhere for minimisers of problems motivated by nonlinear elasticity. Model problems treated include \[ \int_\Omega {| {Du} |^2 + | {\det Du} |^2 } ,\] where