A comparison theorem for super- and subsolutions of $\nabla^2u+f(u)=0$ and its application to water waves with vorticity

A comparison theorem is proved for a pair of solutions that satisfy opposite nonlinear differential inequalities in a weak sense. The nonlinearity is of the form $f(u)$ with $f$ belonging to the class $L^p_\mathrm{loc}$ and the solutions are assumed to have nonvanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary free-surface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.