Calculus of Variations and Geometric Measure Theory
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Incontri di Analisi Matematica tra Firenze, Pisa e Siena

Non-local characterizations of Sobolev spaces and bounded variation functions

Massimo Gobbino (Dip. Ing. Civ. e Ind. Univ. Pisa)

created by paolini on 30 Apr 2019

17 may 2019 -- 16:30   [open in google calendar]

Abstract.

We present two approximations of the p-norm of the gradient of a function u(x) through double integrals that do not involve derivatives. In the first one, the so called “horizontal approximation”, the gradient is replaced by finite differences. In the second one, the so called “vertical approximation”, the double integral measures some sort of interaction between the sublevels of u(x). In both cases, the integrand penalizes the pairs of points (x, y) such that x and y are close to each other, while the difference between u(x) and u(y) is large. We describe a common approach that in both cases leads to compute the pointwise limit and the Γ-limit of these functionals. The main steps are first reducing to dimension one, then localizing the result to an interval, and finally reducing the analysis to the asymptotic behavior of suitable multi-variable inequalities. (Based on some joint works with M.G. Mora (horizontal approach) and with C. Antonucci, M. Migliorini, and N. Picenni (vertical approach))

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