We address the minimization of the Canham–Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds.

我们解决了存在多个阶段时Canham–Helfrich函数的最小化问题。该问题是由于异质生物膜的建模而引起的，该模型可能具有可变的弯曲刚度和自发曲率。关于先前的贡献，这里不假定最小化器的对称性。相应地，在定向曲率可变的较弱框架中重新提出并解决了该问题。我们提出了一个较低的半连续性结果，并证明了面积和封闭体积约束下单相和多相最小化器的存在。此外，我们讨论了最小化器的规律性并建立了直径的上限和下限。