DSpace
 
  Repositório Institucional da Universidade de Aveiro > CIDMA - Centro de Investigação e Desenvolvimento em Matemática e Aplicações > CIDMA - Artigos >
 Problems of Minimal Resistance and the Kakeya Problem
Please use this identifier to cite or link to this item http://hdl.handle.net/10773/21409

title: Problems of Minimal Resistance and the Kakeya Problem
authors: Plakhov, Alexander
keywords: Newton’s problem of least resistance
Shape optimization
Kakeya problem
issue date: Aug-2015
publisher: SIAM
abstract: Here we solve the problem posed by Comte and Lachand-Robert in [8]. Take a bounded domain R2 and a piecewise smooth nonpositive function u : ¯ ! R vanishing on @ . Consider a flow of point particles falling vertically down and reflected elastically from the graph of u. It is assumed that each particle is reflected no more than once (no multiple reflections are allowed); then the resistance of the graph to the flow is expressed as R(u; ) = 1 | | R (1 + |ru(x)|2)−1dx. We need to find inf ,u R(u; ). One can easily see that |ru(x)| < 1 for all regular x 2 , and therefore one always has R(u; ) > 1/2. We prove that the infimum of R is exactly 1/2. This result is somewhat paradoxical, and the proof is inspired by, and partly similar to, the paradoxical solution given by Besicovitch to the Kakeya problem
URI: http://hdl.handle.net/10773/21409
ISSN: 1095-7200
publisher version/DOI: https://doi.org/10.1137/15M1012931
source: SIAM Review
appears in collectionsCIDMA - Artigos

files in this item

file description sizeformat
sirev.pdf346.01 kBAdobe PDFview/open
statistics

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.

 

Valid XHTML 1.0! RCAAP OpenAIRE DeGóis
ria-repositorio@ua.pt - Copyright ©   Universidade de Aveiro - RIA Statistics - Powered by MIT's DSpace software, Version 1.6.2