On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

Abstract

This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of π/2.

Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

“…Large amplitude 2D periodic solutions, including those with angle 2 =3 satisfying the Stokes conjecture, were constructed later by Krasovskiȋ [61], Keady-Norbury [60], Toland [92], Amick-Toland [17], Amick-Fraenkel-Toland [16], Plotnikov [80], and McLeod [69]. For more recent work on Stokes waves, see Plotnikov-Toland [81] and Gravina-Leoni [41,42] and the references therein. Solitary nonperiodic solutions in 2D were constructed by Beale [20].…”

Tangential contact between free and fixed boundaries for variational solutions to variable coefficient Bernoulli type Free boundary problems

“…Very recently, works of Indrei [12], [13] study interactions of free boundaries and fixed boundaries for fully non-linear obstacle problems. We refer to [10] where authors shed more light into angle of contact between fixed boundary and free boundary for one phase Bernoulli problem.…”