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.
