Print Email Facebook Twitter The Poincare Inequality on Smooth and Bounded Domains in Rd Title The Poincare Inequality on Smooth and Bounded Domains in Rd Author Krylov, Ivan (TU Delft Electrical Engineering, Mathematics and Computer Science) Contributor Lorist, E. (mentor) van Gennip, Y. (graduation committee) Degree granting institution Delft University of Technology Corporate name Delft University of Technology Programme Applied Mathematics Date 2023-07-07 Abstract In this thesis we study for which domain types the Poincare inequality holds for all functions having continuous first derivative. We first consider the classical Poincare inequality, which we prove holds for a very large class of open sets in Rd. We then constructively prove that bounded, open, and connected domains in Rd, which also possess a smooth C1-boundary, must satisfy the Poincare-Wirtinger inequality. We do this in six successive steps.First, we show that an arbitrary open rectangle in Rd must satisfy the inequality.Second, we prove that a C1-diffeomorphism with a sufficient condition, between a set which satisfies the inequality and an open, bounded and connected set implies the open, bounded and connected set also satisfies the Poincare-Wirtinger inequality. Third, we show that there exists such a C1-diffeomorphism between a domain in the class of open rectangles with one face distorted by a C1-function and another domain in the class of arbitrary open rectangles in Rd. Fourth, we show the class of all open rectangles with one face distorted by a C1-function satisfies the Poincare-Wirtinger inequality. Fifth, we show the union of non-disjoint open sets which satisfy the inequality in turn also satisfies the Poincare-Wirtinger inequality. Lastly, we cover the open, bounded and connected domain with a C1-boundary by a collection of rectangles from the classes of open rectangles with one face distorted by a C1-function and arbitrary open rectangles to show that the domain satisfies thePoincare-Wirtinger inequality.Finally, we extend our function space to the first-order Sobolev space and show that we can directly extend our results to this function space. Subject Poincare InequalityPoincare-WirtingerSobolev SpacesSmooth Boundary To reference this document use: http://resolver.tudelft.nl/uuid:c112b0a6-250b-4918-a2b1-54df7bf76b68 Part of collection Student theses Document type bachelor thesis Rights © 2023 Ivan Krylov Files PDF Poincare_Inequalities_Iva ... Krylov.pdf 337.05 KB Close viewer /islandora/object/uuid:c112b0a6-250b-4918-a2b1-54df7bf76b68/datastream/OBJ/view