Regularity of optimal shapes. Applications.

本提案包括两部分凸性约束下最优形状的规则性：艾萨克·牛顿考虑了用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。牛顿找到了一个光滑的“解”，但在1995年人们意识到牛顿的解是错误的。我打算研究在任何维度的凸性约束下最优形状的正则性/奇异性。我的方法是分析的，所以这个问题被转化为一个功能设置中的最小化问题。我将使用的关键成分是最优性条件、凸性约束的表征、拉格朗日乘子的定性性质、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（PDEs）。最优形状的正则性是数学中一个具有挑战性的问题。所涉及的形状泛函我将考虑在几个纯粹和应用领域的应用，如经济学，生物学或算子理论人工心脏（AH）的最佳形状设计：描述人工心脏的数学模型包括一个非线性偏微分方程系统，该系统由运动域的三维Navier-Stokes方程（NSEs）和运动膜的双曲偏微分方程组成。我将研究这个方程组解的存在唯一性，并找到求解它的有效数值方法。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。形状功能涉及测量粘性应力，湍流和涡度的血液在AH室。本研究项目对具有周期边界数据和运动边界的nse问题的求解，以及相关的形状优化问题的解决，都有独到的研究成果。它有助于改进AH设计，为心脏病患者带来希望。本课题提出的理论可应用于不同医疗器械的最佳形状设计。

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.