| Abstract: |
| In this talk, we present a diffuse interface formulation of the well-known Plateau`s problem (finding a minimal surface with a given boundary) with two parameters: thickness and volume. More precisely, we consider the family of problems
\begin{equation}
\inf \Big\{\int_{\Omega} \epsilon|\nabla u|^2 + \frac{1}{\epsilon}W(u) \, |\, \int_{\Omega} \mathcal{V}(u) =\delta \, ,\, \text{``$u$ spans a given wire frame``} \Big\}.
\end{equation}
Where $u$ represents the density distribution of the soap particles, the (unbounded) set $\Omega$ represents the exterior of a wire, $\delta$ the volume of the soap film, and $\epsilon$ its corresponding thickness. This point of view introduces the additional interfacial length scale to the Plateau`s problem with volume from King-Maggi-Stuvard, which is recovered in the limit of vanishing interfacial thickness.
We will discuss existence of solutions for the diffuse interface problem for small thickness, convergence to the sharp interface problem introduced in King-Maggi-Stuvard, and partial regularity of minimizers. Additionally, we will discuss how the optimal regularity of minimizers $u$ of the diffuse interface problem is closely related with the regularity of a new type of free boundary problem for semilinear elliptic equations. |
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