The talk will review the motivations, state of the art, recent results, and open questions on four very related PDE models related to phase transitions: Allen-Cahn, Peierls-Nabarro, Minimal surfaces, and Nonlocal Minimal surfaces. We will focus on the study of stable solutions (critical points of the corresponding energy functionals with nonnegative second variation). We will discuss new nonlocal results on stable phase transitions, explaining why the stability assumption gives stronger information in presence of nonlocal interactions. We will also comment on the open problems and obstructions in trying to make the nonlocal estimates robust as the long-range (or nonlocal) interactions become short-range (or local).[-]

(Work in collaboration with C. Bardos and I. Moyano). Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $\sigma \to +\infty$ and $1 − \alpha ∼ C/\sigma$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of
a kinetic model which is based on the harmonic extension definition of $\sqrt{−\Delta}$. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (“heavy tails”) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273–280].[-]

The importance of ultrasound is well established in the imaging of human tissue. In order to enhance image quality by exploiting nonlinear effects, recently techniques such as harmonic imaging and nonlinearity parameter tomography have been put forward. These lead to a coefficient identification problem for a quasilinear wave equation. Another characteristic property of ultrasound propagating in human tissue is frequency power law attenuation leading to fractional derivative damping models in time domain. In this talk we will first of all dwell on modeling of nonlinearity on one hand and fractional damping on the other hand. Then we will discuss the linear inverse problem of photoacoustic tomography with fractional damping. Finally some first results on nonlinearity parameter imaging are shown.[-]