An estimate for the Morse index of a Stokes wave

Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler-Lagrange equation of a certain functional. This allows one to define the Morse index of a Stokes wave. It is well known that if the Morse indices of the elements of a set of non-singular Stokes waves are bounded, then none of them is close to a singular one. The paper presents a quantitative variant of this result.Comment: This version contains an additional reference and some minor change

Positivity properties of phase‐plane distribution functions

The aim of this paper is to compare the members of Cohen's class of phase-plane distributions with respect to positivity properties. It is known that certain averages (which are in a sense compatible with Heisenberg's uncertainty principle) of the Wigner distribution over the phase-plane yield non-negative values for all states. It is shown in this paper that the Wigner distribution is unique in this respect among the members of Cohen's class that have correct marginals or that satisfy Moyal's formula for all states. The subset of members of Cohen's class (not necessarily satisfying one of these two conditions) with positivity properties comparable with those for the Wigner distribution is shown to be rather small

Regularity at the free boundary for Dirac-harmonic maps from surfaces

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior ϵ-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions

Intertwining operators between some spaces of differential operators on a manifold

The Lie algebra of vector fields Vect(M) of a smooth manifold M acts by Lie derivatives on the space D^p_{\lambda,\mu} of differential operators of order less than or equal to p that map fields of densities of weight -\lambda on fields of densities of weight -\mu. In this paper, we determine all the intertwining operators between these modules, when the dimension of the manifold is greater than or equal to 2