Xavier Cabré (Universidad Politécnica de Catalunya)

THURSDAY (June 19):  10:00-11:00

Phase transition layers, minimal surfaces, and ground states

Phase transitions or interfaces appear in physical problems when two different states coexist and there is a balance between two opposite tendencies: a diffusive effect that tends to mix the materials, and a mechanism that drives them into their pure state. Due to surface tension, interfaces often tend to minimize their area. Nonlinear elliptic PDEs modeling these phenomena, such us the scalar Ginzburg-Landau equation, have been extensively studied. Their solutions develop sharp transition layers, whose level sets correspond to the physical interfaces. The local study of the interfaces through a blow-up analysis leads to nonlinear elliptic problems in the whole or in a half space. We will discuss recent developments on such global semilinear elliptic problems. They establish rich relations among different qualitative properties of solutions: their stability, minimality, monotonicity in one variable, and their symmetry. For reactions in the interior, one of such problems was posed by E. De Giorgi in 1978, and progress has been made only recently. The main results state that layer solutions (that is, bounded solutions in the whole space which are monotone in one variable) are always local minimizers of the energy, and that in low space dimensions they are necessarily functions of only one Euclidean variable (1 d symmetry). We will also present new related results for ``ground states'' or, more generally, for radial solutions. They establish that, in low dimensions, radial solutions in the whole space are always unstable, and hence they are never local minimizers.

For phase transitions occurring only on the boundary, the blow-up analysis leads to harmonic functions in a half space subject to a nonlinear Neumann boundary condition. We will describe recent developments concerning layer solutions for this problem (that is, solutions which are monotone in one tangential variable), their existence, uniqueness, stability, minimality, and 2d symmetry properties.