On double-covering stationary points of a constrained Dirichlet energy

The double-covering map udc : R 2 → R is given by udc(x) = 1 √ 2|x| ( x2 2 − x12 2x1x2 ) in cartesian coordinates. This paper examines the conjecture that udc is the global minimizer of the Dirichlet energy I(u) = ∫ B |∇u| dx among allW 1,2 mappings u of the unit ball B ⊂ R satisfying (i) u = udc on ∂B, and (ii) det∇u = 1 almost everywhere. Let the class of such admissible maps be A. The chief innovation is to express I(u) in terms of an auxiliary functional G(u−udc), using which we show that udc is a stationary point of I in A, and that udc is a global minimizer of the Dirichlet energy among members of A whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about udc in A using ODE techniques, we also show that udc is a local minimizer among variations whose tangent ψ to A at udc obeys G(ψ) > 0, where ψ is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint det∇u = 1 is identified by an analysis which exploits the well-known Fefferman-Stein duality. Abstract Le double-revêtement fonction udc : R 2 → R est donn par udc(x) = 1 √ 2|x| ( x2 2 − x12 2x1x2 )Le double-revêtement fonction udc : R 2 → R est donn par udc(x) = 1 √ 2|x| ( x2 2 − x12 2x1x2 ) en coordonnes cartesian. Cet article examine la conjecture que udc est le minimiseur global de l’énergie de Dirichlet I(u) = ∫ B |∇u| dx pour les fonctions satisfaisant (i) u ∈ W (B), où B est la boule unité de R , (ii) u = udc sur ∂B, et (iii) det∇u = 1 presque partout. Soit A la classe admissible de telles fonctions. La principale innovation est ici 2010 AMS Classification 35A15, 49J40, 49N60 Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK. tel: +44 (0)1483 682620. email: [email protected]