Skip to Main content Skip to Navigation
Theses

Results of compactness and regularity in a non-local Ginzburg-Landau model resulting from micromagnetism. Poincaré lemma and domain regularity

Abstract : In this thesis, we study some boundary value problems involving micromagnetic models and differential forms. In the first part, we consider a nonlocal Ginzburg-Landau model arising in micromagnetics with an imposed Dirichlet boundary condition. The model typically involves S²-valued maps with an energy functional depending on several parameters, which represent physical quantities. A first question concerns the compactness of magnetizations having the energies of several Néel walls of finite length and topo- logical defects when these parameters converge to 0. Our method uses techniques developed for Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-valued vector fields by S¹-valued vector fields away from the vortex balls. We also carry out in detail the proofs of the C^infinite regularity in the interior and C(^1,alpha) regularity up to the boundary, for all alpha belong to (0, 1/2), of critical points of the model. In the second part, we study the Poincaré lemma, which states that on a simply connected domain every closed form is exact. We prove the Poincaré lemma on a domain with a Dirichlet boundary condition under a natural assumption on the regularity of the domain: a closed form ƒ in the Hölder space C(^r,alpha) is the differential of a C(^r+1,alpha) form, provided that the domain itself is C(^r+1,alpha). The proof is based on a construction by approximation, together with a duality argument. We also establish the corresponding statement in the setting of higher order Sobolev spaces.
Document type :
Theses
Complete list of metadata

https://tel.archives-ouvertes.fr/tel-03191333
Contributor : Abes Star :  Contact
Submitted on : Wednesday, April 7, 2021 - 9:23:10 AM
Last modification on : Thursday, April 8, 2021 - 3:26:07 AM

File

2019TOU30315a.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-03191333, version 1

Citation

Hoang Phuong Nguyen. Results of compactness and regularity in a non-local Ginzburg-Landau model resulting from micromagnetism. Poincaré lemma and domain regularity. Optimization and Control [math.OC]. Université Paul Sabatier - Toulouse III, 2019. English. ⟨NNT : 2019TOU30315⟩. ⟨tel-03191333⟩

Share

Metrics

Record views

35

Files downloads

2