Regularity of optimal shapes. Applications.

这项提议包括两个部分。 [1]凸性约束下最优形状的正则性：艾萨克·牛顿考虑了由给定高度的凸函数描述的一类物体中阻力最小的物体的形状。牛顿找到了一个平稳的“解”，但在1995年，人们意识到牛顿的解是错误的。研究任意维凸约束下最优形状的正则性/奇异性。 我的方法是解析的，所以问题转化为函数环境下的最小化问题。我将使用的关键成分是最优性条件，凸性约束的特征，拉格朗日乘子的定性性质，以及一阶和二阶形状导数的一些精细估计以及相关的偏微分方程(PDE)。最优形状的正则性是数学中的一个具有挑战性的问题。我将考虑的形状泛函在几个纯粹的和应用的领域都有应用，例如经济学、生物学或算子理论。 [2]人工心脏的最优形状设计：描述人工心脏形状的数学模型由一个由运动区域中的三维N-S方程组成的非线性偏微分方程组和一个模拟运动膜的双曲偏微分方程组组成。我将研究这个系统解的存在唯一性，并寻找有效的数值方法来求解它。接下来，我将求解AH的最优形状，这是一个形状优化问题的解。所涉及的形状泛函测量了AH室中血液的粘性应力、湍流和涡度。该研究程序为求解具有周期边界数据和移动边界的网格子空间问题，以及相关的形状优化问题做出了贡献。它有助于改进AH的设计，从而给心脏病患者带来希望。为该项目开发的理论在不同医疗器械的形状优化设计中具有应用价值。

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.