Symmetry of steady periodic water waves with vorticity

We discuss certain a priori geometric properties of two-dimensional steady gravity water waves with vorticity. The main result states that for an arbitrary distribution of vorticity, any periodic wave of finite depth with a single trough (a minimum over one period) is symmetric about a single crest (a maximum over one period) and the wave profile decreases (strictly) monotonically from crest to trough if every streamline attains its minimum below the trough and the wave profile is monotone near the trough. The proof involves the method of moving planes as adapted to nonlinear elliptic boundary value problems. The mathematical existence theory for periodic waves with vorticity dates back to the construction by Dubreil-Jacotin (1934) of small amplitude waves of infinite depth, and it includes the works by Constantin & Strauss (2002) and Constantin & Strauss (2004) in the finite depth case and Hur (2006) in the infinite depth case on the global bifurcation of large amplitude waves. The construction by Constantin & Strauss (2002) and Constantin & Strauss (2004) assumes that the wave profiles are symmetric; our result establishes a priori their symmetry and monotonicity properties. In the irrotational setting, Garabedian (1965) considered the symmetry property of periodic waves of finite depth with a variational approach provided that each streamline has a single crest and a single trough per wavelength except for the flat bottom; a direct proof is due to Toland (2000), which combines with the divergence theorem and Dirichlet’s principle for harmonic functions. Further demonstration of symmetry appeared with the advent of the so-called method of moving planes. In particular, with its extension by Berestycki & Nirenberg (1988) to nonlinear elliptic problems, Craig & Sternberg (1988) proved the Phil. Trans. R. Soc. A (2007) 365, 2203–2214 doi:10.1098/rsta.2007.2002 Published online 13 March 2007