Complete Shape Metric and Geodesic

We develop the framework for moving domain and geometry under minimal regularity (of moving boundaries). This question arose in shape control analysis and non cylindrical PDE analysis. We apply here this setting to the morphic measure between shape or images. We consider both regular and non smooth situations and we derive complete shape metric space with characterization of geodesic as being solution to Euler fluid-like equation. By the way, this paper also addresses the variational formulation for solution to the coupled Euler-transport system involving only condition on the convected terms. The analysis relies on compactness results which are the parabolic version to the Helly compactness results for the BV embedding in the linear space of integrable functions. This new compactness result is delicate but supplies to the lack of convexity in the convection terms so that the vector speed associated with the optimal tube (or moving domain), here the shape geodesic, should not be curl-free so that the Euler equation does not reduce to a classical Hamilton-Jacobi one. For topological optimization this geodesic construction is developed by level set description of the tube, and numerical algorithms are in the next paper of this book.