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A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs
Nonlinear Differential Equations and Applications (NoDEA) ( IF 1.1 ) Pub Date : 2020-12-28 , DOI: 10.1007/s00030-020-00667-3
Jonathan J. Bevan , Jonathan H. B. Deane

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.



中文翻译:

连续扰动的Dirichlet能量,具有保留区域的固定点,这些固定点“屈曲”并以等能量对形式出现

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\({\ mathbb {R}} ^ 2 \)和(ii)服从边界条件\(v _ {\下午} = \文本{ID} \)\(\部分B \)。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\(X \ RIGHTARROW 0 \)。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\(v _ {\ pm} \)\(W ^ {1,2}(B; {\ mathbb {R}} ^ 2)\)类中都是固定的地图瓦特服从\(W = \文本{ID} \)\(\局部乙\)\(\ DET \ nabla瓦特= 1 \)

更新日期:2020-12-28
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