Connecting solutions in second order phase transition models

DATES: 27 - 31 January 2020 (5 sessions)
TIME: 10:00 - 12:00 (a total of 10 hours)
LOCATION: BCAM Seminar room

ABSTRACT:
This course focuses on the vector Allen-Cahn equation which models coexistence of three or more phases of a substance. For the scalar case, De Giorgi in 1978 suggested a striking analogy with minimal surface theory that led to significant developments in P.D.E. and the Calculus of Variations. In the vector case, the mathematical phenomena are considerably richer. Our main focus will be on entire solutions converging to the phases (i.e. the global minima of the potential), as the variable goes to infinity along certain directions. This course is based on our recently published book [1]. It is addressed to researchers, with an interest in any of the following: differential equations ordinary or partial; nonlinear analysis; the calculus of variations; or the applied mathematics of materials science.

PROGRAMME:
1. Introduction
We will give an overview of the main results obtained for the Allen-Cahn equation including the De Giorgi conjecture. Then, we will present some useful tools applying for elliptic gradient systems: the stress-energy tensor, the monotonicity formula, Hamiltonian identities, Liouville type theorems, and a maximum principle.

2. One-dimensional solutions, connecting orbits
In space dimension one, the vector Allen-Cahn equation reduces to a Hamiltonian system of O.D.E. Assuming that the zero level set of the potential is partitioned into two compact subsets, we shall establish the existence of heteroclinic, homoclinic and periodic orbits.

3. Higher dimensional solutions: the triple junction, and other symmetricstructures
The triple junction is a two-dimensional solution converging to each of the three global minima of the potential, as the variable goes to infinity and remains in an appropriate sector. It has initially been constructed in 1996 by Bronsard, Gui and Schatzman, for symmetric potentials. More generally, symmetric solutions can be obtained in the equivariant class, (a) for general point groups, and (b) for general discrete reflection groups, thus establishing the existence of previously unknown lattice solutions.

4. Heteroclinic orbits in Hilbert spaces and applications to P.D.E.: the double heteroclinic solution
When the boundary conditions are appropriate, the Allen-Cahn P.D.E. can be reduced to a O.D.E. problem in a Hilbert space. By applying this approach, we shall construct a double heteroclinic solution, connecting in the plane the two global minima of the potential along the horizontal direction, and two distinct heteroclinic orbits along the vertical direction. This construction has initially been performed by Schatzman in 2002. It is an important result, providing the first examples of nontrivial minimal solutions for the Allen-Cahn system.

5. The Painlevé phase transition model
The second Painlevé O.D.E. is known to play an important role in many areas of Mathematics and Physics. The P.D.E. version of this equation is obtained by multiplying by x1 the linear term of the scalar Allen-Cahn P.D.E. It involves a non autonomous potential H which is bistable for every fixed x1 < 0, and thus describes as the Allen-Cahn equation a phase transition model. We shall construct in the plane, a solution connecting along the vertical direction x2, the two branches of minima of H parametrized by x1. This solution plays a similar role that the heteroclinic orbit for the Allen-Cahn equation. It is the first to our knowledge solution of the Painlevé P.D.E. both relevant from the applications point of view (liquid crystals), and mathematically interesting.

REFERENCES:
[1] N. D. Alikakos, G. Fusco, P. Smyrnelis: Elliptic systems of phase transition type. Progress in Nonlinear Differential Equations and Their Applications Vol. 91, Springer-Birkhauser (2018)
[2] P. Antonopoulos, P. Smyrnelis: On minimizers of the Hamiltonian system $u=nabla W(U)$, and on the existence of heteroclinic, homoclinic and periodic orbits. Indiana Univ. Math. J. 65 (5), 1503-1524 (2016)
[3] L. Bronsard, C. Gui, M. Schatzman: A three-layered minimizer in $R^2$ for a variational problem with a symmetric three-well potential. Commun. Pure. Appl. Math. 49 (7), 677-715
(1996)
[4] M. G. Clerc, M. Kowalczyk, P. Smyrnelis: The connecting solution of the Painlevé phase transition model. To appear inAnn. Scuola Norm. Sci.
[5] M. Schatzman: Asymmetric heteroclinic double layers. Control Optim. Calc. Var. 8, 965-1005 (2002). A tribute to J. L. Lions (electronic)


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To sign-up go to https://forms.gle/87WMfxTGGno62yyo7 and fill the registration form.

Student grants are available. Please, let us know if you need support for travel and accommodation expenses in the previous form before December 15th.

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