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We exhibit a family of convex functionals with infinitely many equal-energy    $$C^1$$  C 1 stationary points that (i) occur in pairs    $$v_{\pm }$$  v ± satisfying    $$\det \nabla v_{\pm }=1$$  det ∇ v ± = 1 on the unit ball  B  in    $${\mathbb {R}}^2$$  R 2 and (ii) obey the boundary condition    $$v_{\pm }=\text {id}$$  v ± = id on    $$ \partial B$$  ∂ B . When the parameter    $$\epsilon $$  ϵ upon which the family of functionals depends exceeds    $$\sqrt{2}$$  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    $$v_{\pm }(x)$$  v ± ( x ) and prove that they are proportional to    $$(\epsilon -1/\epsilon )\ln |x|$$  ( ϵ - 1 / ϵ ) ln | x | as    $$x \rightarrow 0$$  x → 0 in  B . The lowest-energy pairs    $$v_{\pm }$$  v ± 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    $$0\le r\le 1$$  0 ≤ r ≤ 1 , take the circle    $$\{x\in B: \ |x|=r\}$$  { x ∈ B : | x | = r } to itself;  a fortiori , all    $$v_{\pm }$$  v ± are stationary in the class of    $$W^{1,2}(B;{\mathbb {R}}^2)$$  W 1 , 2 ( B ; R 2 ) maps  w  obeying    $$w=\text {id}$$  w = id on    $$\partial B$$  ∂ B and    $$\det \nabla w=1$$  det ∇ w = 1 in  B .

A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs