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 \(0\le r\le 1\), take the circle \(\{x\in B: \ |x|=r\}\) 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.

我们展示了一个凸函数族，它们具有无限等能量\（C ^ 1 \）固定点，这些定点（i）成对满足（（det \ nabla v _ {\ pm}）对（\ v _ {\ pm} \）= 1 \）单位球上的乙在\（{\ mathbb {R}} ^ 2 \）和（ii）服从边界条件\（v _ {\下午} = \文本{ID} \）上\（\部分B \）。当功能族依赖的参数\（\ epsilon \）超过\（\ sqrt {2} \）时，固定点似乎在B中心附近“屈曲”，并且其能量随屈曲量而单调增加。到哪个B受。我们还发现与地图相关联的拉格朗日乘子\（V _ {\点}（X）\） ，并证明他们是成正比\（（\小量-1 / \小量）\ LN | X | \）作为\（X \ RIGHTARROW 0 \）在乙。最低能量对\（v _ {\ pm} \）是扭曲图类中的能量最小化器（请参见2009年Topol Methods Nonlinear Anal 33（1）：179–204中的Taheri或Arch Ration Mech Anal中的Sivaloganathan和Spector。 196：363–394，2010），对于每个\（0 \ le r \ le 1 \），将圆圈\（\ {x \ in B：\ | x | = r \} \）移到自身；一个fortiori，所有\（v _ {\ pm} \）在\（W ^ {1,2}（B; {\ mathbb {R}} ^ 2）\）类中都是固定的地图瓦特服从\（W = \文本{ID} \）上\（\局部乙\）和\（\ DET \ nabla瓦特= 1 \）在乙。