arXiv subjects

Luciano Sciaraffia

Publications and source records attributed to Luciano Sciaraffia.

4 recordsLinked to original sources

arXiv2026

A capillary problem, its dimension reduction, and its phase-field approximation

We study the behaviour of a given volume of liquid confined between two rough solid plates. When the separation between the plates is small relative to the liquid volume, capillary bridges are expected to form, which minimise Gauss's capillary energy locally. We derive a $Γ$-expansion for the energy as the plate separation approaches zero, yielding a dimensionally reduced problem in terms of the wetted regions on the plates. At leading order, the energy is determined by the area of the wetted regions, while the next order term is given by their perimeter, weighted by appropriate functions of the relative adhesion coefficients. This provides a framework for a subsequent phase-field approximation, which is employed in numerical simulations to study the evolution of the droplets under the normal movement of the plates. The theory also gives justification to models for capillarity that assume that liquids fill up the rough topography to the Kelvin radius.

math.AP↗

arXiv2024

Minimal networks on balls and spheres for almost standard metrics

We study the existence of minimal networks in the unit sphere $\mathbf{S}^d$ and the unit ball $\mathbf{B}^d$ of $\mathbf{R}^d$ endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of $θ$-networks in $\mathbf{S}^d$ and triods in $\mathbf{B}^d$, jointly with the Lusternik--Schnirelmann category.

math.DG↗

arXiv2023

Singularities of the network flow with symmetric initial data

We study the formation of singularities for the curvature flow of networks when the initial data is symmetric with respect to a pair of perpendicular axes and has two triple junctions. We show that, in this case, the set of singular times is finite.

math.AP↗

arXiv2019

Nontrivial solutions to Serrin's problem in annular domains

We construct nontrivial smooth bounded domains $Ω\subseteq \mathbb{R}^n$ of the form $Ω_0 \setminus \overlineΩ_1$, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value problem \[ -Δu = 1, \; u>0 \quad \text{in } Ω, \qquad u = 0 ,\; \partial_νu = \text{const} \quad \text{on } \partialΩ_0, \qquad u = \text{const} ,\; \partial_νu = \text{const} \quad \text{on } \partial Ω_1, \] where $ν$ stands for the inner unit normal to $\partialΩ$. From results by Reichel and later by Sirakov, it was known that the condition $\partial_νu \leq 0$ on $\partialΩ_1$ is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, the constructed domains are shown to be self-Cheeger.

math.AP↗