Abstract
We exhibit a family of convex functionals with infinitely many, equal-energy C 1 stationary points that (i) occur in pairs v± satisfying det v± = 1 on the unit ball B in R 2 and (ii) obey the boundary condition v± = id on ∂B. When the parameter upon which the family of functionals depends exceeds √ 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±(x) and prove that they are proportional to (− 1//) ln |x| as x → 0 in B. The lowest-energy pairs v± are energy minimizers within the class of twist maps (see Taheri [30] or Sivaloganathan and Spector [22]), which, for each 0 ≤ r ≤ 1, take the circle {x ∈ B : |x| = r} to itself; a fortiori, all v± are stationary in the class of W 1,2 (B; R 2) maps w obeying w = id on ∂B and det w = 1 in B.