This paper is concerned with the stability of periodic travelling-wave solutions of the Korteweg-de Vries equation $$ u_t+uu_x+u_{xxx}=0. $$ Here, $u$ is a real-valued function of the two variables $x,t\in{\mathbb R}$ and subscripts connote partial differentiation. These special solutions were termed cnoidal waves by Korteweg and de Vries. They also appear in earlier work of Boussinesq. It is shown that these solutions are stable to small, periodic perturbations in the context of the initial-value problem. The approach is that of the modern theory of stability of solitary waves, but adapted to the periodic context. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

We consider the problem $\Delta u+V(x)u=f'(u)$ in $\mathbb R^N$. Here the nonlinearity has a double power behavior and $V$ is invariant under an orthogonal involution, with $V(\infty)=0$. An existence theorem of one pair of solutions which changes sign exactly once is given.

We study the pointwise continuity of functions satisfying a certain inequality, deducible from parabolic obstacle problems and valid also for parabolic Q-minima. Employing techniques going back to De Giorgi, we give conditions on the obstacles sufficient to imply continuity of the solution. These conditions are formulated as inequalities for the capacities of sub- and super-level sets of the obstacles, and thus also thin obstacles are considered.