Sharp bounds for the p-torsion of convex planar domains
Lamboley, Jimmy; Gazzola, Filippo; Fragalà, Ilaria (2013), Sharp bounds for the p-torsion of convex planar domains, in Alvino, Angelo; Sakaguchi, Shigeru; Magnanini, Rolando, Geometric Properties for Parabolic and Elliptic PDE's, Springer : Berlin, p. 97-115. http://dx.doi.org/10.1007/978-88-470-2841-8_7
TypeCommunication / Conférence
External document linkhttp://hal.archives-ouvertes.fr/hal-00654228/fr/
Conference titleGeometric Properties for Parabolic and Elliptic PDE's
Book titleGeometric Properties for Parabolic and Elliptic PDE's
Book authorAlvino, Angelo; Sakaguchi, Shigeru; Magnanini, Rolando
Series titleSpringer INdAM
Number of pages292
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Abstract (EN)We obtain some sharp estimates for the $p$-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the boundary), and on the behaviour of the inner parallel sets of convex polygons. As an application of our isoperimetric inequalities, we consider the shape optimization problem which consists in maximizing the $p$-torsion among polygons having a given number of vertices and a given area. A long-standing conjecture by Pólya-Szegö states that the solution is the regular polygon. We show that such conjecture is true within the subclass of polygons for which a suitable notion of ''asymmetry measure'' exceeds a critical threshold.
Subjects / Keywordsconvex shapes; web functions; shape optimization; isoperimetric inequalities
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