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Explore the mathematical analysis of elastic membrane models through the lens of variational calculus and free boundary problems in this 30-minute conference lecture. Delve into the minimization of the Canham-Helfrich energy for closed surfaces with prescribed area constraints within bounded containers, examining how these surfaces behave when trapped by geometric obstacles. Learn about the existence theory for minimizers within the framework of immersed bubble trees, a sophisticated mathematical structure that captures the complex topology of optimal configurations. Discover the derivation of Euler-Lagrange equations that govern these systems, paying particular attention to the measure terms that concentrate along free boundaries where the membrane contacts the container. Understand how these equations can be reformulated as conservation laws with Jacobian structure, enabling the determination of optimal regularity properties for solutions. Gain insights into the rigorous mathematical treatment of problems arising from physical models of biological membranes and soap films, with applications to understanding cellular structures and minimal surface theory.
