Teoria geométrica da medida e aplicações

Abstract

This master thesis aims to study some of the work of the Italian mathematician Ennio De Giorgi which refer to the existence and regularity of minimal surfaces but they do not fully contextualized within the framework of differential geometry but rather focused on a field of mathematics that implemented a few decades Geometric Measure Theory. According to the definitions of Ennio De Giorgi will study surfaces, which gave to the same as certain boundarys of sets, which are denoted sets Caccioppoli, this honor given by De Giorgi the Italian mathematician Renato Caccioppoli. These sets have many interesting geometric properties, such as adimits canonical tangent plan almost everywhere, and have "perimeter" finite. The above results will see that even the size 7 all the solutions to the problem of Plateau are regular and in general their regular class is C1,α. Finally, the results of this study are mostly based on the work: Minimal Surface and Function of Bounded Variation of the author Enrico Giusti, which summarizes the techniques of Geometric Measure Theory relating to the work of Ennio De Girogi on a regularity theory of minimal surfaces Theorem (De Giorgi- Federer- Massari - Miranda) Let Ω contains in R^n, n>1 an open set and E contains in R^n a Caccioppoli set satisties to α in (0, 1) ψ(E, Bρ(x)) < cρ^(n-1+2α) for every x in Ω and every ρ in (0, R), with c and R positive constants. Then the reduced boundary is a hipersurface analitic C1,α in Ω, e H^s((∂ E - ∂E) Ω) = 0 for every s > n - 8. Furthemore, suppose that Ej is a sequence of minimal sets in B1 locally converging at minimal set C. Let x in ∂ C and xj in Ej , xj converging at x. Then, if j is enough large , xj is a regular point for ∂Ej and νEj(xj) converges to ν(x), where ν(x) is a normal vector relative at Ej, ∂ E denots the boundary reduced of E and H^s denots the Hausdorff measure.