Detalles de Evento
Título: Minimal boundary cones
Conferenciante: Edoardo Cavallotto (Université Paris-Scaly, Francia)
Resumen: The Plateau problem arises from physics, and in particular from soap bubbles and soap films. Solving the Plateau problem means to find a surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists in giving a suitable sense to the notions of "surface", "area" and "boundary". Given $0 < d < n$ we will consider a setting, due to Almgren, in which the considered objects are sets with locally finite d-dimensional Hausdorff measure, the functional we will try to minimize is the Hausdorff area $H^d$, and the boundary condition is given in terms of a one-parameter family of deformations. Almgren minimizers turn out to have nice regularity properties, in particular an Almgren minimizer is a $C^{1,\alpha}$ embedded submanifold of $\mathbb{R}^n$ up to a negligible set, and the tangent cone to any point of such a minimizer is a minimal cone. Therefore in order to give a complete characterisation of these object we need to know how minimal cones look like. The complete list of minimal cones of $\mathbb{R}^2$ and $\mathbb{R}^3$ has been well known long time ago while in higher dimensions the list is far from being complete and we only know few examples. My talk will focus to a small variation of this setting which we call "sliding boundary" and to minimal cones that arise in this frame.
Fecha: 10 de abril de 2018, 11:30 - 12:30
Lugar: Seminario 1, IEMath-GR
Conferenciante: Edoardo Cavallotto (Université Paris-Scaly, Francia)
Resumen: The Plateau problem arises from physics, and in particular from soap bubbles and soap films. Solving the Plateau problem means to find a surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists in giving a suitable sense to the notions of "surface", "area" and "boundary". Given $0 < d < n$ we will consider a setting, due to Almgren, in which the considered objects are sets with locally finite d-dimensional Hausdorff measure, the functional we will try to minimize is the Hausdorff area $H^d$, and the boundary condition is given in terms of a one-parameter family of deformations. Almgren minimizers turn out to have nice regularity properties, in particular an Almgren minimizer is a $C^{1,\alpha}$ embedded submanifold of $\mathbb{R}^n$ up to a negligible set, and the tangent cone to any point of such a minimizer is a minimal cone. Therefore in order to give a complete characterisation of these object we need to know how minimal cones look like. The complete list of minimal cones of $\mathbb{R}^2$ and $\mathbb{R}^3$ has been well known long time ago while in higher dimensions the list is far from being complete and we only know few examples. My talk will focus to a small variation of this setting which we call "sliding boundary" and to minimal cones that arise in this frame.
Fecha: 10 de abril de 2018, 11:30 - 12:30
Lugar: Seminario 1, IEMath-GR
