We prove the smoothness of weak solutions to an elliptic complex Monge–Ampère equation, using the smoothing property of the corresponding parabolic flow.

We consider the cubic Szegő equation $i\partial tu=\Pi \left(|u{|}^{2}u\right)$ in the Hardy space $L_{+}^2\left(ℝ\right)$ on the upper half-plane, where $\Pi$ is the Szegő projector. It was first introduced by Gérard and Grellier as a toy model for totally nondispersive evolution equations. We show that the only traveling waves are of the form $C∕\left(x-p\right)$, where $p\in ℂ$ with $Imp<0$. Moreover, they are shown to be orbitally stable, in contrast to the situation on the unit disk where some traveling waves were shown to be unstable.

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein–Gordon equation, in the spirit of Kenig and Merle for the $H^1$ critical wave and Schrödinger equations. Our result includes the $H^1$ critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger–Moser inequality. The main difficulty is the lack of scaling invariance in both the linear and the nonlinear terms.

Rayleigh-type surface waves correspond to the characteristic variety, in the elliptic boundary region, of the displacement-to-traction map. In this paper, surface quasimodes are constructed for the reduced elastic wave equation, anisotropic in general, with traction-free boundary. Assuming a global variant of a condition of Barnett and Lothe, the construction is reduced to an eigenvalue problem for a selfadjoint scalar first order pseudodifferential operator on the boundary. The principal and the subprincipal symbol of this operator are computed. The formula for the subprincipal symbol seems to be new even in the isotropic case.