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Minimal solutions to nonlinear systems of PDEs

Periodic Reporting for period 1 - MinSol-PDEs (Minimal solutions to nonlinear systems of PDEs)

Reporting period: 2019-12-01 to 2021-11-30

Many physical phenomena are ruled by the principle of least action, according to which the temporal evolution of a physical system tends to minimize its "energy". For instance, "the catenoid", the curve that a hanging chain or cable assumes under its own weight, minimizes its potential energy.

In my project I studied minimization problems from the mathematical viewpoint.

On the one hand, I developed the mathematical theory of these minimal solutions which are also related to the geometry of minimal surfaces.
The problem is to classify and describe the solutions (or the surfaces) whose local deformations result in an increase of their energy (or area).

On the other hand, my research has a direct application in the Physics of liquid crystals. The objective is to model the orientation of the molecules in a liquid crystal film illuminated by a laser light, and understand the structure of the vortices that are created. Manipulating light vortices has technological applications in the areas of quantum computing, telecommunications and astronomy (improvement of astronomical images, detection of exoplanets).
The research work carried out focuses on the construction of minimal solutions of elliptic PDEs (WP2), on some of their fundamental properties (WP3), and on the applications in the theory of liquid crystals (WP4).

Four articles have been produced :
D.2.1: P. Smyrnelis: Double layered solutions to the extended Fisher-Kolmogorov P.D.E. Nonlinear Differ. Equ. Appl., 28:48, 22 pages (2021)
The first article published (WP2) provides the first examples of minimal solutions for the Extended Fisher-Kolmogorov P.D.E. which is a fourth order phase transition model.

D.3.1: J. Jendrej, P. Smyrnelis: :Nondegeneracy of heteroclinic orbits for a class of potentials on the plane. Applied Mathematics Letters, 124 (2022) 107681
The second article published (WP3) establishes the nondegeneracy of heteroclinic orbits for a class of potentials on the plane.

D.3.2: P. Smyrnelis: A comparison principle for vector-valued minimizer of semilinear energy, with application to dead cores. Indiana Univ. Math. J., Vol. 70, No. 5, 1745--1768
The third article published (WP3) studies the dead core properties of minimizers of semilinear elliptic energy.

D.4.1: M. Kowalczyk, X. Lamy, P. Smyrnelis: Entire vortex solutions of negative degree for the anisotropic Ginzburg-Landau system. arXiv 2110.07651 to appear in Archive for Rational Mechanics and Analysis.
A fourth article (to appear), proves the existence of vortices of negative degree for the anisotropic Ginzburg-Landau equation, which is relevant in the theory of liquid crystals (WP4).

There is also a fifth article (in preparation) studying some asymptotic properties and Liouville type results for entire solutions to semilinear elliptic P.D.E. (cf. WP3). The aforementioned articles will be further exploited (cf. the next section below).

The researcher has been trained (cf. WP5) on the tensorial theory of liquid crystals at BCAM. By visiting foreign universities (University Sorbonne Paris Nord, Polish Academy of Sciences, University of Athens), and inviting at BCAM foreign experts, he also got involved in a wide range of new research areas (the nonlinear wave equation, the theory of anisotropic operators, PDE problems in manifolds, simulations and computational mathematics).

As far as dissemination and communication are concerned (cf. WP6), the researcher has given a mini course at BCAM. In addition, he has participated in two conferences: 1) Workshop on singularities in variational models, January 8–10, 2020, Toulouse, France, 2) SIAM Conference on Mathematical Aspects of Materials Science, May 17–28, 2021, virtually, and he has given as well invited talks in the seminars of various institutions. Finally, he has presented his habilitation at the Polish Academy of Sciences, and he has been awarded the title of Habilitated Doctor.
From a scientific point of view the main achievements are D.2.1) the construction of the first examples of two-dimensional minimal solutions for the extended Fisher-Kolmogorov PDE. These solutions play a crucial role in phase transition models, since they are closely related to the De Giorgi conjecture, D.3.1) the first explicit examples provided of vector potentials for which heteroclinic orbits are nondegenerate. This property is essential in a wide range of problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. D.3.2) the description of the dead core phenomena for vector minimizers of semilinear elliptic energy, D.4.1) the construction of anisotropic Ginzburg-Landau vortices, which are relevant in the theory of liquid crystals.

The aforementioned articles will be further exploited. On the one hand, D.3.1 is a preparation work for an ongoing project on the interaction of kinks in the context of the vector nonlinear wave equation. On the other hand, the results in D.4.1 will be extended in my future research to develop the anisotropic theory of vortices, which has a direct application in the Physics of liquid crystals (e.g. quantic computation, telecommunications, and astronomy).