A continuously perturbed Dirichlet energy with area-preserving stationary points that \u2018buckle\u2019 and occur in equal-energy pairs

Abstract

We exhibit a family of convex functionals with infinitely many equal-energy C^1 stationary points that (i) occur in pairs v_{pm } satisfying det nabla v_{pm }=1 on the unit ball B in {mathbb {R}}^2 and (ii) obey the boundary condition v_{pm }=text {id} on partial B. When the parameter epsilon upon which the family of functionals depends exceeds sqrt{2}, the stationary points appear to ‘buckle’ near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps v_{pm }(x) and prove that they are proportional to (epsilon -1/epsilon )ln |x| as x rightarrow 0 in B. The lowest-energy pairs v_{pm } are energy minimizers within the class of twist maps (see Taheri in Topol Methods Nonlinear Anal 33(1):179–204, 2009 or Sivaloganathan and Spector in Arch Ration Mech Anal 196:363–394, 2010), which, for each 0le rle 1, take the circle {xin B: |x|=r} to itself; a fortiori, all v_{pm } are stationary in the class of W^{1,2}(B;{mathbb {R}}^2) maps w obeying w=text {id} on partial B and det nabla w=1 in B.