Narrow Results

We discuss basic properties (uniqueness and regularity) of viscosity solutions to fully nonlinear elliptic equations of the form F(x, D²u) = 0, which includes also linear elliptic equations of nondivergent form. In the linear case we consider equations with discontinuous coefficients.

We review the out-of-equilibrium properties of a self-gravitating gas of particles in the presence of a strong friction and a random force (canonical gas). We assume a bare diffusion coefficient of the form D(ρ) = Tρ^1/n, where ρ is the local particle density, so that the equation of state is P(ρ) = D(ρ)ρ. Depending on the spatial dimension d, the index n, the temperature T, and whether the system is confined to a finite box or not, the system can reach an equilibrium state, collapse or evaporate. This article focuses on the latter cases, presenting a complete dynamical phase diagram of the system.