Regularity of optimal shapes. Applications.

这项建议包括两个部分。 [1]凸性约束下最优形状的正则性：艾萨克·牛顿考虑了由具有给定高度的凸函数描述的一类最小阻力物体的形状。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。我打算研究的规则性/奇异性的最佳形状下的凸约束在任何维度。 我的方法是解析的，所以这个问题被转化为函数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，定性性质的拉格朗日乘子，和一些精细的估计的第一和第二阶形状导数和偏微分方程（PDE）相关联。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的应用领域，如经济学，生物学或运营商理论的应用。 [2]人工心脏（AH）的最佳形状设计：描述AH的数学模型由一个非线性偏微分方程系统组成，该系统包括运动域中的三维Navier-Stokes方程（NSE）和一个模拟运动膜的双曲偏微分方程。我将研究这个系统的解的存在性、唯一性，并找到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AH室中血液的粘性应力、湍流和涡度。本研究计画对于求解具有周期性边界资料与移动边界之非结构化实体，以及相关之形状最佳化问题，贡献了原创性的成果。它有助于AH设计的改进，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最佳形状设计中具有应用。

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.