Sobolev kuvaukset ja energia integraalit geometrisessa funktio teoriassa
Päärahoittaja
Rahoittajan antama koodi/diaarinumero: 334014
Päärahoittajan myöntämä tuki (€)
- 545 779,00
Rahoitusohjelma
Hankkeen aikataulu
Hankkeen aloituspäivämäärä: 01.09.2020
Hankkeen päättymispäivämäärä: 31.08.2024
Tiivistelmä
The proposal is a part of PI’s research program to develop variational techniques in Geometric Function Theory (GFT) with applied fields such as Nonlinear Elasticity (NE) in mind. The proposed problems of geometric nature originated from the Riemann Mapping Theorem; conformal mappings being univalent solutions of the Cauchy-Riemann system. Moving to the second order variational equations and their homeomorphic solutions offers new challenges. The goal is to characterize energy-functionals whose minimizers exist and resemble conformal maps. In these extremal problems homeomorphisms are free on the boundary, known as frictionless problems in NE. It is a common struggle in mathematical models of NE to establish the existence of energy-minimal deformations which comply the principle of no interpenetration of matter. To build a viable theory we have to adopt monotone and cellular Sobolev mappings as legitimate deformations in the 2D and 3D-elasticity. Minimizing among Sobolev homeomorphisms the basic questions
such as existence, uniqueness and regularity of energy-minimal mappings challenge the available methods in analysis and topology.
GFT is currently a field of enormous activity where the general framework of NE is extremely fruitful and significant. The proposed activity contributes to this interplay. It will open up a new direction in understanding properties of several important minima in GFT and NE. The tools and concepts (e.g. Monotone Sobolev Mappings and Free Lagrangians) that the PI have developed and continue to develop can be applied and will be valuable to solve the proposed projects here. These developments have already solved some old conjectures such as the Nitsche conjecture, the Ball-Evans problem and the Eells-Helein-Lemaire-Sealey problem.
As a broader impact, the proposed activity will encourage interchange and cooperation between pure and applied mathematicians to work together. The proposed problems are easy to state and offer a perfect mentoring opportunities to bring graduate students and postdoctoral scholars to Geometric Analysis. The PI is currently advising a Ph. D. student and working with postdoctoral scholars. The PI will supervise a Ph.D. student and mentor postdoctoral scholars working on problems closely related to this proposal. Finally, the PI with Tadeusz Iwaniec will provide educational materials, through the book entitled, "Sobolev Mappings and Variational Integrals in Geometric Function Theory".
such as existence, uniqueness and regularity of energy-minimal mappings challenge the available methods in analysis and topology.
GFT is currently a field of enormous activity where the general framework of NE is extremely fruitful and significant. The proposed activity contributes to this interplay. It will open up a new direction in understanding properties of several important minima in GFT and NE. The tools and concepts (e.g. Monotone Sobolev Mappings and Free Lagrangians) that the PI have developed and continue to develop can be applied and will be valuable to solve the proposed projects here. These developments have already solved some old conjectures such as the Nitsche conjecture, the Ball-Evans problem and the Eells-Helein-Lemaire-Sealey problem.
As a broader impact, the proposed activity will encourage interchange and cooperation between pure and applied mathematicians to work together. The proposed problems are easy to state and offer a perfect mentoring opportunities to bring graduate students and postdoctoral scholars to Geometric Analysis. The PI is currently advising a Ph. D. student and working with postdoctoral scholars. The PI will supervise a Ph.D. student and mentor postdoctoral scholars working on problems closely related to this proposal. Finally, the PI with Tadeusz Iwaniec will provide educational materials, through the book entitled, "Sobolev Mappings and Variational Integrals in Geometric Function Theory".
