We prove the existence and regularity of minimisers for the Canham–Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the 2-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to proving the main results we establish the lower semicontinuity of the Canham–Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions.

中文翻译：

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最小化 Canham-Helfrich 能量的球体的存在性和规律性

我们证明了 Canham-Helfrich 能量在 2 球体的弱（可能是分支和气泡）浸入类中的最小化子的存在和规律性。这解决了（球形情况）由 Helfrich 在 1973 年提出的最小化问题，模拟脂质双层膜。在证明主要结果的过程中，我们在弱浸入（可能是分支和冒泡的）弱收敛下建立了 Canham-Helfrich 能量的下半连续性。