Une théorie trois-dimensionnelle des ondes de surface de l'eau et le développement de Friedrichs. I. (A three dimensional theory of surface waves of water and the expansion of Friedrichs. I.) (Q1087068)

The author makes a rigorous analytical analysis expansion of \textit{K. O. Friedrichs} after a little parameter \(\delta\), used by him in Commun. Pure Appl. Math. 1, 1-87 (1948) for the Cauchy problem for three dimensional, irrotational surface waves. His study is a comprehensive one, divided into two parts, the second appearing in the following number of the present Journal. In this first part he makes a short historical sketch of the problem, followed by four chapters where he deduces the boundary problem of the movement, establishes some apriori estimates for the velocity potential, useful for the existence theorem which is proved locally in time. Giving the solution of the free boundary problem, the author shows that it is infinitely differentiable on the free boundary in the little parameter \(\delta\), after which K. O. Friedrichs developed the solution, giving so the mathematical base for this development.