ABSTRACT
Study of shape optimization began in the 1970s, and the first results concerned the shape derivatives, the shape gradient and the associated shape differential equation, which furnishes existence results for asymptotic evolution of a shape-gradient flow (see, for example, [9] and the historical introduction of [27]). Shape analysis became a branch of geometry in the 1990s and several results concerning topology, compactness and intrinsic analysis (mainly through the oriented distance function bΩ), and density perimeter are found in [10], [25], and the associated literature. Recently moving domains appeared in several settings such as fluid dynamics, optimal control theory, large deformations, fluid-structure coupling, moving front, images (traveling), dynamical antennas, and modelings of many industrial devices. A popular version of the shape differential equation is the level set method, which emphasizes a specific parameterization of a moving domain Ωt, its boundary being the level set zero of a one-parameter function φ(t, .). It is classical ([8], [25]) that a speed vector (whose flow mapping carries that moving domain) is
V φ(t, x) = − ∂ ∂t φ(t, x)
∇xφ(t, x) ||∇xφ(t, x)||2 (1.1)
Any other field building this tube is in the formW = V φ+Z with 〈Z(t), nt〉 = 0 on Γt.
