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Abstract: The isoperimetric-type problem is an archetypical problem in the geometric measure theory. It often corresponds to some kind of energy minimization and ties with calculus of variation. Not only many beautiful results are born, but also fundamental insights and powerful tools are introduced. In this talk, we study an open double minimization problem in unbounded domains proposed by Buttazzo, Carlier and Laborde. To model lipid bilayer membranes, the minimization problem consists of one perimeter term, which presents the interfacial energy, and p-Wasserstein distance, which takes into account the covalent bonding energy. They prove the existence in 2D and propose it holds in high dimensions. We provide a new approach to this double optimization problem in any dimension. Our method overcomes the lack of compactness in unbounded domain and does not rely on the geometric observation in 2D. Inspired by Almgren’s seminal paper for minimizing clusters problem, we reformulate the double optimization problem into an equivalent isoperimetric problem. We then provide a proof to the existence of minimizers to such an isoperimetric problem with the $p$-Wasserstein penalization in $\R^d$, for $\frac{1}{p}+\frac{2}{d}>1$ and the constricted volume is small. This is a joint work with my advisor Prof. Qinglan Xia.

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