论文标题
在接触约束下优化几何和PDE形状功能的表面的存在
Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint
论文作者
论文摘要
本文涉及在某些几何约束下将一般形状功能最小化的超曲面的存在最小化。我们认为是可允许的形状，可满足所谓的到达条件（也称为均匀的球特性），可确保c 1,1 hypersurface的规律性。在本文中，我们对[9，4，5]的结果进行了重新访问和概括。我们提供了更简单的框架和更简洁的证据，证明了这些参考文献中包含的某些结果，并将其扩展到涉及PDE的新问题。确实，通过使用Delfour和Zolesio引入的签名距离（例如，请参见[7]），我们避免了当地地图的密集和技术用途，就像上述参考文献中一样。我们最初开发的目的是解决[12]中的存在问题，可以很容易地扩展到涉及与域相关的不同数学对象的成本，例如超出表面上的椭圆方程解决方案。
This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as the uniform ball property, which ensures C 1,1 regularity of the hypersurface. In this paper, we revisit and generalise the results of [9, 4, 5]. We provide a simpler framework and more concise proofs of some of the results contained in these references and extend them to a new class of problems involving PDEs. Indeed, by using the signed distance introduced by Delfour and Zolesio (see for instance [7]), we avoid the intensive and technical use of local maps, as was the case in the above references. Our approach, originally developed to solve an existence problem in [12], can be easily extended to costs involving different mathematical objects associated with the domain, such as solutions of elliptic equations on the hypersurface.
