Nonlocal Minimal Surface Theory and Curvature Flows
Summary
Nonlocal minimal surface theory extends the classical notion of minimal surfaces by incorporating interactions over finite or infinite distances through integral functionals. Rather than depending solely on local curvature and area, nonlocal perimeters account for contributions from points separated by a kernel that decays with distance. This leads to a richer geometric landscape in which surfaces minimise an energy balancing local and nonlocal effects. Fractional curvature flows describe the evolution of interfaces driven by nonlocal mean curvature, resulting in motion laws that interpolate between classical mean curvature flow and global diffusion processes. These flows have been shown to capture fine interface dynamics in materials science, phase transitions and image processing, where long-range interactions play a significant role. The interplay between nonlocal energies and curvature-driven motion gives rise to new regularity phenomena, pattern formation and stability criteria, as well as novel variational characterisations of evolving boundaries.
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Nonlocal Minimal Surface Theory and Curvature Flows publication trend
The graph below shows the total number of articles in nonlocal minimal surface theory and curvature flows across all publications each year (not limited to Nature Index journals).
Technical terms
Nonlocal minimal surface: A set whose boundary minimises an energy defined by an integral of pairwise interactions weighted by a decaying kernel, generalising the notion of perimeter to account for long-range forces.
Fractional perimeter: A nonlocal functional measuring the boundary size of a set via double integrals of characteristic functions against a singular kernel, parametrised by an exponent controlling the interaction range.
Fractional mean curvature: The first variation of the fractional perimeter, given by a singular integral expressing a weighted average of boundary deviations over nonlocal neighbourhoods.
Curvature flow: A geometric evolution law in which the normal velocity of a boundary is prescribed by a curvature quantity, local or nonlocal, driving the shape towards energy minimisation.
Gamma-convergence: A variational convergence notion ensuring that minimisers of a sequence of energies converge to minimisers of a limiting energy, often used to justify passing from nonlocal to local models.
References
- On nonlocal minimal graphs. Calculus of Variations and Partial Differential Equations (2021).
- Minimisers of a fractional seminorm and nonlocal minimal surfaces. Interfaces and Free Boundaries Mathematical Analysis Computation and Applications (2020).
- Nonlocal diffusion of smooth sets. Mathematics in Engineering (2021).
- On the shape of small liquid drops minimizing nonlocal energies. ESAIM Control Optimisation and Calculus of Variations (2023).
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