Convex integration, Staircase Laminates, and Applications
to March 20, 2025
When: march 17th , 18th , 19th and 20th , 2025, from 15:00 to 16:30
Where: room iA, Faculty of Math and Comp Science, historical building of Universitat de Barcelona (Gran Via 585)
Free registration through SING IN
Introduction
The method of convex integration, originated in the celebrated construcction of isometric inmersions by Nash in the 60s, has become anindespensable tool to understand irregular solutions in partial differential equations, leading to celebrated results as the solution of Onsaguer conjecture in the context of turbulent hidrodynamics. In the course I will explain a variant of the method, based on the so called Staircase Laminates and its relation with Singular Integral Operators, quasiconformal maps, elliptic equations and electromagnetism.
Contents
The course will start with an introduction to the theory of differential inclusions, with an special focus on the L^p theory. We will explain how problems like regularity of solutions to elliptic equations, boundedness of singular integral operators or problems in hidrodynamics can be brought to this framework.
The second part of the course will review the sharp L^p theory of quasilinear elliptic equations in the plane and the conenction with the theory of quasiconformal mappings. I will explain also recent results on the autonomous Beltrami equation and announce the theorem that we will be able to prove at the end of the course.
The third part of the course will introduce the concepts of gradient distributions, laminates, and the basic building block. I will explain what convex integration is, and prove a baby convex integration theorem.
The fourth part of the course will introduce the theory of staircase laminates and construct various examples, concerning the non Ornstein inequalities, the L^p theory of linear and nonlinear elliptic equations, and patological homeomorphisms.
Lecturer
Daniel Faraco
Institute of Mathematical Sciences (ICMAT)
I conducted my PhD thesis at the University of Helsinki, under the direction of Prof. Kari Astala, on quasiconformal functions, calculus of variations and their application to material science. Motivated by this last aspect I joined S. Müller’s group at the Max-Planck Institute for Applied Mathematics in Leipzig. After my postdoctoral stay I obtained a Ramón y Cajal scholarship and joined the Universidad Autónoma de Madrid (UAM), where I am full professor of applied mathematics since 2020.
In 2012 I was granted with an ERC CONSOLIDATOR “Geometric function theory, Inverse Problems and fluid dynamics”, ended in 2018. Currently I lead the UAM Node of Quamap (ERC Advanced Grant), the UAM Node of Agapi 2018-2021 and the group of inverse problems and mechanics of the UAM. I have also been involved in the four “Severo Ochoa” accreditations awarded to the Institute of Mathematical Sciences (ICMAT), being currently guarantor and currently a member of its executive committee. Recently, I have been teaching courses in the Hausdorff center in Bonn, Oberwolfach and the Bedlewo Summer School in Nonlinear Analysis and PDEs from which this advance course has been nurtured.
Although my scientific interest has changed over time, it has always lied at the intersection between mathematical analysis and applied mathematics. Applied problems with a strong theoretical component or theoretical problems with a potential use in applications. Quasiconformal mappings are particularly dear to me, which on the one hand investigate how many of the beautiful ideas of complex analysis can be extended to less rigid functions and, on the other hand, constitute a sufficiently regular class of elastic deformations to be handled with mathematical technical analysis.
Although quasiconformal mappings emerge in practically every aspect of my research, there are three topics that interest me fundamentally: calculation of vector variations and non-linear elasticity, the inverse problems and the study of turbulence in fluid mechanics.
References
On background
Astala, K., Iwaniec; T. and Martin, G., Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009. ISBN 9780691137773.
Iwaniec, T. ; Sbordone, C. Quasiharmonic fields. Annales de l’I.H.P. Analyse non linéaire, Vol. 18, no. 5, pp. 519-572 (2001).
Kirchheim, B. (2001). Geometry and rigidity of microstructures. Tesis Doctoral.
On Elliptic Staircase Laminates
Faraco, D. Milton’s conjecture on the regularity of solutions to isotropic equations, Annales de l’Institut Henri Poincaré C, Analyse non linéaire. 2003, vol. 20, Issue 5: p. 889-909, ISSN 0294-1449
Astala, K., Clop, A., Faraco, D., Jäskelainen, J., Koski, A., Martin. G, Laminates and the Autonomous equation. 2024.
On L^1, L^2 Staircase Laminates
Faraco, D., Wild Mappings built on unbounded Laminates. Proceedings Benevento 2005.
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