| Contents |
| This talk will focus on the stability of the manifold of equilibria for the moving boundary problem
$$\begin{cases}
-\Delta u=\lambda\, ,&\text{ in }\Omega(t)\text{ for }t>0\,,\\
u=0\, ,&\text{on }\partial\Omega(t)\text{ for }t>0\,,\\
\int _{\Omega(t)}u(x)\, dx=V_0\, ,&\text{for }t>0\\
V=F(|Du|)\, ,&\text{on }\partial\Omega(t)\text{ for }t>0\,,\\
\Omega(0)=\Omega _0\, .
\end{cases}$$
satisfied by a viscous droplet sitting on a homogeneous surface and driven by contact angle.
The unknown function $u$ describes the height of the droplet, $\Omega(t)$ the wetted region at time $t$, $V_0$ the droplet volume, $V$ the front velocity in direction normal to $\partial\Omega(t)$, and $\lambda$ is a Lagrange multiplier.
After the system is recast into a nonlinear nonlocal evolution for the boundary of the moving domain, the analysis proceeds with the explicit computation of the linearization in the equilibria and the introduction of suitable coordinates for $\partial\Omega(t)$ in the space of shapes. In these coordinates the equation takes on a particularly convenient form which simplifies the stability analysis. |
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