Preprint Article Version 6 Preserved in Portico This version is not peer-reviewed

Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

Version 1 : Received: 24 September 2022 / Approved: 8 October 2022 / Online: 8 October 2022 (03:50:31 CEST)
Version 2 : Received: 11 October 2022 / Approved: 12 October 2022 / Online: 12 October 2022 (03:54:18 CEST)
Version 3 : Received: 24 October 2022 / Approved: 24 October 2022 / Online: 24 October 2022 (05:39:11 CEST)
Version 4 : Received: 24 November 2022 / Approved: 24 November 2022 / Online: 24 November 2022 (06:54:28 CET)
Version 5 : Received: 4 December 2022 / Approved: 5 December 2022 / Online: 5 December 2022 (02:57:53 CET)
Version 6 : Received: 7 December 2022 / Approved: 7 December 2022 / Online: 7 December 2022 (02:47:54 CET)
Version 7 : Received: 14 December 2022 / Approved: 14 December 2022 / Online: 14 December 2022 (06:25:21 CET)
Version 8 : Received: 16 December 2022 / Approved: 16 December 2022 / Online: 16 December 2022 (07:34:56 CET)
Version 9 : Received: 19 December 2022 / Approved: 20 December 2022 / Online: 20 December 2022 (02:32:43 CET)
Version 10 : Received: 4 January 2023 / Approved: 4 January 2023 / Online: 4 January 2023 (03:45:53 CET)
Version 11 : Received: 23 January 2023 / Approved: 25 January 2023 / Online: 25 January 2023 (04:21:30 CET)
Version 12 : Received: 29 January 2023 / Approved: 29 January 2023 / Online: 29 January 2023 (09:58:00 CET)

A peer-reviewed article of this Preprint also exists.

Botelho, F.S. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations. Mathematics 2022, 11, 63, doi:10.3390/math11010063. Botelho, F.S. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations. Mathematics 2022, 11, 63, doi:10.3390/math11010063.

Abstract

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. In the first sections, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section we present some numerical results concerning the generalized method of lines applied to a Ginzburg-Landau type equation.

Keywords

Duality principles; Generalized method of lines; Ginzburg-Landau type equations

Subject

Computer Science and Mathematics, Applied Mathematics

Comments (1)

Comment 1
Received: 7 December 2022
Commenter: Fabio Botelho
Commenter's Conflict of Interests: Author
Comment: Dear Sir Editor

We have made some corrections in the section 6 of the previous version v.5.

Indeed, we have re-written some parts of such a concerning section.

The related duality principle now has a local nature.
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