# Holder Space Theory for the Rotation Problem of a Two-Phase Drop
## 2 Setting of a problem
By definition, \(f\in C^{(\gamma,1+\alpha)}(Q_{T})\) if  
\[\begin{split}&\int_{\Omega_{t}}\rho^{\pm}y_{j}\,\mathrm{d}y=0, \quad j=1,2,3,\quad\text{(centroid conservation)},\\ &\int_{\Omega_{t}}\rho^{\pm}\tilde{\mathbf{v}}(y,t)\,\mathrm{d}y=0 \quad\text{(impulse conservation)},\\ &\int_{\Omega_{t}}\rho^{\pm}\tilde{\mathbf{v}}(y,t)\cdot\mathbf{\eta}_{j} (y)\,\mathrm{d}y+\omega\int_{\Omega_{t}}\rho^{\pm}\mathbf{\eta}_{3}\cdot\mathbf{\eta}_{ i}(y)\,\mathrm{d}y=\omega\int_{\mathcal{F}}\tilde{p}\mathbf{\eta}_{3}\cdot\mathbf{\eta}_{i}(y)\, \mathrm{d}y=\beta\delta_{i}^{3}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\where \[|f|_{Q_{T}}=\sup_{t\in(0,T)}\sup_{x\in\Omega}|f(x,t)|,\qquad\langle f\rangle_{Q_{T} }^{(\alpha,\alpha/2)}=\langle f\rangle_{x,Q_{T}}^{(\alpha)}+\langle f\rangle_{ t,Q_{T}}^{(\alpha/2)},\]3  
Finally, if a function \(f\) has finite norm  
\[\begin{split}&\int_{\Omega_{t}}\rho^{\pm}y_{j}\,\mathrm{d}y=0, \quad j=1,2,3,\quad\text{(centroid conservation)},\\ &\int_{\Omega_{t}}\rho^{\pm}\tilde{\mathbf{v}}(y,t)\,\mathrm{d}y=0 \quad\text{(impulse conservation)},\\ &\int_{\Omega_{t}}\rho^{\pm}\tilde{\mathbf{v}}(y,t)\cdot\mathbf{\eta}_{j} (y)\,\mathrm{d}y+\omega\int_{\Omega_{t}}\rho^{\pm}\mathbf{\eta}_{3}\cdot\mathbf{\eta}_{ i}(y)\,\mathrm{d}y=\omega\int_{\mathcal{F}}\tilde{p}\mathbf{\eta}_{3}\cdot\mathbf{\eta}_{i}(y)\, \mathrm{d}y=\beta\delta_{i}^{3}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\where \[|f|_{Q_{T}}=\sup_{t\in(0,T)}\sup_{x\in\Omega}|f(x,t)|,\qquad\langle f\rangle_{Q_{T} }^{(\alpha,\alpha/2)}=\langle f\rangle_{x,Q_{T}}^{(\alpha)}+\langle f\rangle_{ t,Q_{T}}^{(\alpha/2)},\]4  
where  
\[|f|_{t,Q_{T}}^{(\mu)}= \begin{cases}|f|_{Q_{T}}+\langle f\rangle_{t,Q_{T}}^{( \mu)}&\text{if $\mu>0$,}\\ |f|_{Q_{T}}&\text{if $\mu=0$,}\end{cases} \]  
we consider that \(f\in C^{\gamma,\mu}(Q_{T})\). We consider that a vector-valued function belongs to a Holder space if all of its components belong to this space and define its norm as the maximum of component norms. The same applies to a tensor-valued function. We set  
\[|f|_{D_{T}}^{(k+\alpha,\frac{k+\alpha}{2})}\equiv|f|_{Q_{T}^{T}}^{(k+\alpha,\frac {k+\alpha}{2})}+|f|_{Q_{T}^{T}}^{(k+\alpha,\frac{k+\alpha}{2})},\qquad D_{T} \equiv Q_{T}^{\pm}\cup Q_{T}^{-},\]  
\[|f|_{\cup\Omega^{\pm}}^{(k+\alpha)}\equiv|f|_{\Omega^{-}}^{(k+\alpha)}+|f|_{ \Omega^{+}}^{(k+\alpha)}.\]  
**3. A Linearized Problem**  
We suppose the surfaces \(\tilde{\Gamma}_{t}^{\pm}\) to be given by the relations  
\[\tilde{\Gamma}_{t}^{\pm}=\{y=z+\mathbf{N}(z)r(z,t),\quad z\in\mathcal{G}^{\pm}\}, \tag{14}\]  
where \(\mathbf{N}\) is the outward normal to \(\mathcal{G}^{-}\cup\mathcal{G}^{+}\). Let us map \(\tilde{\Omega}_{t}^{\pm}\) into \(\mathcal{F}^{\pm}\) by the inverse transformation to the Hanzawa transform  
\[y=z+\mathbf{N}^{*}(z)r^{*}(z,t)\equiv e_{r}(z,t), \tag{15}\]  
where \(\mathbf{N}^{*}\), \(r^{*}\) are some extensions of \(\mathbf{N}\) and \(r\) into \(\mathcal{F}\), respectively. In order to analyze problem (11) with initial data close to the regime of rotation as a rigid body (see Figure 1), we linearize it. To this end, we calculate the first variation with respect to \(r\) of the differences \(H(y)-\mathcal{H}(z)\), \(|y^{\prime}|^{2}-|z^{\prime}|^{2}\), \(U(y,t)-\mathcal{U}(z)\).