Regularity of optimal shapes. Applications.

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This proposal contains two parts. [1] Regularity of optimal shapes under convexity constraint: Isaac Newton considered the shape of the body of minimal resistance in the class of bodies described by a convex function with given height. Newton found one smooth 'solution', but on 1995 it was realized that Newton's solution was wrong. I intend to study the regularity/singularities of optimal shapes under the convexity constraint in any dimension. My approach is analytic, so the problem is transformed to a minimization problem in a functional setting. The key ingredients I will use are the optimality conditions, the characterization of the convexity constraint, qualitative properties of the Lagrange multiplier, and some fine estimates of first and second order shape derivatives and the partial differential equations (PDEs) associated. Regularity of optimal shapes is a challenging issue in Mathematics. The involved shape functionals I will consider have applications in several pure and applied areas, such as economics, biology or operator theory. [2] Optimal shape design of artificial hearts (AH): The mathematical model describing an AH consists of a nonlinear PDEs system comprising three dimensional Navier-Stokes equations (NSEs) in a moving domain and of a hyperbolic PDE modeling the moving membrane. I will study the existence, uniqueness of the solution to this system and I will find efficient numerical methods for solving it. Next, I will solve the optimal shape of the AH, which is the solution of a shape optimization problem. The shape functional involved measures the viscous stress, turbulence and the vorticity of the blood in the AH chamber. This research program contributes with original results to the solution of NSEs with periodic boundary data and moving boundary, and to the related shape optimization problem. It contributes to the improvement of AH design, and so brings hope to the patients suffering from heart diseases. The theory developed for this project has applications in the optimal shape design of different medical devices.