Complete Rigidity at infinity and Existence of the Levinson Cavity

Abstract

We present a potential theoretic approach reducing the analysis of the asymptotic shape of free surfaces to the analysis of a precise ordinary differential equation resulting from the reduction process. Although the approach relies mainly on the principal part of the PDE operator to allow for a representation formula and is thus not restricted to problems of elliptic type, we present it at the clean-cut example of three-dimensional axially symmetric steady incompressible cavity flows, which are Neumann-type Bernoulli free boundary problems and for which frequency formulas are unknown and, if they do exist, insufficient to yield the very precise asymptotic behavior we prove here. In 1946 Norman Levinson derived by a power-law ansatz with a slowly varying correction a precise formula for the asymptotic shape of such cavities. However his result requires very strong assumptions such that it has remained an open problem for 80 years whether the cavity solutions we know to exist by a result by Garabedian-Lewy-Schiffer [12] actually share this asymptotic behavior, or whether at least one solution possessing the Levinson asymptotics exists. Here we answer both questions affirmatively, and we obtain complete rigidity at infinity of the Levinson solution in the class of axially symmetric solutions, that is, any solution satisfying mild and natural assumptions at the fixed boundary and infinity converges asymptotically to the Levinson profile $(\log r)^{-1/4}\sqrt{r}$.