基于面积差弹性模型，用弛豫法探寻满足开口膜泡边界条件的欧拉-拉格朗日方程组的新解，得到了双开口的哑铃形分支解，并结合以前得到的单开口哑铃形及闭合哑铃形，对它们之间的相变进行了深入的研究。为了探究实验上是否可能发现这些形状，与以往实验上观察到的较小约化弛豫面积差的杯形、管形、烟囱形开口形状的能量进行了比较，发现这些新形状在较大的约化弛豫面积差值时，在某些线张力区间比以往发现的形状能量更低。另外为了对比，本文对于实验上已知的杯形、管形、烟囱形及球形之间的相变行进行了探讨，并对两者之间的不同特点进行了对比。%Due to the discovery and study of opening-up lipid vesicles, the theoretical analysis and numerical calculation have aroused increasing interests of researchers. In the previous study, Suezaki and Umeda gave the opening-up vesicles near the spherical vesicles, such as the dish and cup shapes with one hole, and the tube and funnel shapes with two holes. These shapes are found at relatively low values of reduced, relaxed area difference ∆a0. However, what are the stable shapes for high values of ∆a0 is not known. Kang et al. found solutions of opening up dumbbell shapes with one hole. Whether or not there exist dumbbell shapes with two holes, and the phase transformation behavior between them remains unknown. The purpose of this paper is to explore a new kind of two-hole dumbbell shaped lipid vesicles and phase transformations between this kind of vesicle and previously found vesicles. Based on the area-difference-elasticity model, this paper tries to explore new solutions of the Euler-Lagrange equations of the opening-up membrane vesicles which meet the boundary conditions by using the relaxation method. A new branch of solution of dumbbell shapes with two holes is found. The phase transformations of closed dumbbell shapes and opening-up dumbbell shapes with one hole and two holes are studied in detail. To explore whether these shapes could be found in experiments, the energy of the cup, tube, and funnel shaped vesicles are also compared with the opening-up dumbbell shapes. It is found that at high values of ∆a0, all the cup, tube, and funnel shapes will transform into closed spherical vesicles. So the energy of new opening-up dumbbell vesicles can be compared to that of closed spherical vesicles and closed dumbbell vesicles. It is found that the dumbbell shapes with one hole and two holes all have stable regions, implying that it is possible for these open dumbbells to be observed. Since the distance in the functional space is too far between the open dumbbell shapes and spherical vesicles, experimental test is needed to verify whether the dumbbell shapes with two holes will evolve continuously to the closed dumbbell shapes or to the closed spherical vesicles. It has been noticed that for relatively small values of ∆a0, two holes vesicles may exhibit symmetrical tube shapes and asymmetric funnel shapes between which the phase transformation is continuous, because the funnel solutions bifurcate from the tube solutions. In order to check whether there exist asymmetric opening-up dumbbell shapes with two holes and the similar bifurcation behavior, a thorough search is made in the parameter space. So far no asymmetric dumbbell shape with two holes is found.
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