# Holder Space Theory for the Rotation Problem of a Two-Phase Drop
## 2 Setting of a problem
The domains \(\Omega_{0}^{+}\) and \(\Omega_{0}\) are supposed to be close to equilibrium figures \(\mathcal{F}^{+}\) and \(\mathcal{F}\) of the same volumes:  
\[|\Omega_{0}^{+}|=|\mathcal{F}^{+}|,\quad|\Omega_{0}|=|\mathcal{F}|. \tag{2}\]  
In view of the incompressibility of the fluids, equalities (2) hold for arbitrary \(t>0\):  
\[|\Omega_{t}^{+}|=|\mathcal{F}^{+}|,\quad|\Omega_{t}|=|\mathcal{F}|. \tag{3}\]  
Mass conservation follows from the constancy of the densities. We set \(\mathcal{G}^{+}=\partial\mathcal{F}^{+}\), \(\mathcal{G}^{-}=\partial\mathcal{F}\) and \(\mathcal{F}^{-}=\mathcal{F}\setminus\overline{\mathcal{F}^{+}}\) (see Figure 1). The uniform rotation of a two-phase drop about the \(x_{3}\)-axis with constant angular velocity \(\omega=\beta/\mathcal{I}\) is governed by the homogeneous steady Navier-Stokes equations:  
\[\bar{\rho}(\boldsymbol{\mathcal{V}}\cdot\
abla)\boldsymbol{\mathcal{V}}-\bar{ \mu}\
abla^{2}\boldsymbol{\mathcal{V}}+\
abla\mathcal{P}=0,\qquad\
abla\cdot \boldsymbol{\mathcal{V}}=0\quad\text{in}\;\cup\mathcal{F}^{\pm}\]  
with the step-function of dynamical viscosity \(\bar{\mu}\equiv\mu^{+}\) in \(\mathcal{F}^{+}\) and \(\bar{\mu}\equiv\mu^{-}\) in \(\mathcal{F}^{-}\). The solution of this system is the couple of velocity vector field  
\[\boldsymbol{\mathcal{V}}(x)=\omega\boldsymbol{e}_{3}\times\boldsymbol{x}\equiv \omega\boldsymbol{\eta}_{3}\]  
and the function of pressure  
\[\mathcal{P}(x)=\bar{\rho}\frac{\omega^{2}}{2}|x^{\prime}|^{2}+p_{0}^{\pm},\]  
where \(|x^{\prime}|^{2}=x_{1}^{2}+x_{2}^{2}\) and \(\bar{\rho}\) and \(p_{0}^{\pm}\) are step-functions in \(\mathcal{F}^{\pm}\). In order to find the equations of the surfaces \(\mathcal{G}^{\pm}\) of the domains \(\mathcal{F}^{\pm}\), we substitute \(\boldsymbol{\mathcal{V}},\mathcal{P}\) into boundary conditions (7)  
\[ \begin{split}\sigma^{-}\mathcal{H}^{-}(x)+\rho^{-}\frac{\omega^{2 }}{2}|x^{\prime}|^{2}+\rho^{-}\varkappa\mathcal{U}+p_{0}^{-}=0,\quad x\in \mathcal{G}^{-},\\ \sigma^{+}\mathcal{H}^{+}(x)+[\bar{\mu}]\big{|}_{\mathcal{G}^{+} }\frac{\omega^{2}}{2}|x^{\prime}|^{2}+[\bar{\mu}]\big{|}_{\mathcal{G}^{+}} \varkappa\mathcal{U}+[p_{0}^{\pm}]\big{|}_{\mathcal{G}^{+}}=0,\quad x\in \mathcal{G}^{+},\end{split}  \tag{8}\]  
where \(\mathcal{H}^{-}\) and \(\mathcal{H}^{+}\) are twice the mean curvatures of \(\mathcal{G}^{-}\) and \(\mathcal{G}^{+}\), respectively, and  
\[\mathcal{U}(x)=\int_{\mathcal{F}}\frac{\bar{\rho}\,\mathrm{d}z}{|x-z|}.\]  
In [8], the existence of the boundaries \(\mathcal{G}^{\pm}\) satisfying Equation (8) was proved, provided that \(\beta\) is small enough, and \(\varkappa,\big{|}\bar{\rho}\big{|}_{\mathcal{G}^{+}}>0\) (Proposition 3.3). It was noted there that \(\mathcal{G}^{\pm}\) are flattened spheroids. If \(\varkappa=0\), the condition \(\big{[}\bar{\rho}\big{]}\big{|}_{\mathcal{G}^{+}}>0\) is not necessary. We cite this proposition. Proposition 1: _Let the angular momentum \(\beta\) be small enough, and \(\varkappa>0\) and \(\rho^{+}>\rho^{-}\). Then, for given volumes \(|\mathcal{F}^{+}|\) and \(|\mathcal{F}|\), there exists a unique equilibrium figure which is axially symmetric about the axis \(x_{3}\) and symmetric about the plane \(x_{3}=0\). The surfaces \(\mathcal{G}^{\pm}\) are close to the spheres \(S_{R_{\pm}}=\{|x|=R_{\pm}\}\), respectively, where \(R_{+}\), \(R_{-}\) are such that_  
\[\frac{4\pi}{3}R_{+}^{3}=|\mathcal{F}^{+}|,\quad\frac{4\pi}{3}R_{-}^{3}=| \mathcal{F}|\]  
_and_  
\[\mathcal{G}^{\pm}=\{x=z+\phi^{\pm}(z)\frac{z}{|z|},\quad z\in S_{R_{\pm}}\}.\]  
Thus, we admit the axial symmetry of \(\mathcal{F}^{\pm}\) and their symmetry about the plane \(x_{3}=0\). Then  
\[\int_{\mathcal{F}}\bar{\rho}x_{i}dx=0,\quad i=1,2,3, \tag{9}\]  
\[\int_{\mathcal{F}}\bar{\rho}x_{3}x_{j}dx=0,\quad j=1,2.\]  
Relation (9) corresponds to the first condition in (5). It means that mass center of the fluids is in the origin all the time.