Regularity of the free boundaries for the two-phase axisymmetric inviscid fluid

In the seminal paper (Alt, Caffarelli and Friedman, Trans. Amer. Math. Soc., 282, (1984).), the regularity of the free boundary of two-phase fluid in two dimensions via the so-called ACF energy functional was investigated. It was shown the $C^1$ regularity of the free boundaries and asserted that the two free boundaries coincide under some additional assumptions. Later on the standard technique of Harnack inequality could be applied to improve the regularity to $C^{1,\eta}$. A recent significant breakthrough in the regularity of two-phase fluid is due to De Philippis, Spolaor and Velichkov, who investigated the free boundary of the two-phase fluid with the two-phase functional (De Philippis, Spolaor and Velichkov, Invent. Math., 225, (2021).), and the $C^{1,\eta}$ regularity of the whole free boundaries was given in dimension two. Moreover, the free boundaries of the two-phase fluids do not coincide and the zero level set may process positive Lebesgue measure. In this paper, we consider the free boundaries for the two-phase axisymmetric fluid and show the free boundary is $C^{1,\eta}$ smooth. The Lebesgue measure of the zero level set of may also be positive, and the main difference lies in the degenerate elliptic operator and the free boundary conditions. More precisely, we use partial boundary Harnack inequalities and establish a linearized problem, whose regularity of the solutions implies the flatness decay of the two-phase free boundaries. Then the iteration argument gives the smoothness of the free boundaries.