*Phd Thesis*

**Inserted:** 2 aug 2016

**Last Updated:** 2 aug 2016

**Year:** 2016

**Abstract:**

This thesis is dedicated to the study of isoperimetric inequalities in some Carnot-Carathéodory spaces, related to the Heisenberg geometry.

The thesis is organized as follows. Chapter 1 is concerned with some preliminaries: we consider Carnot-Caratheodory spaces, and we define Grushin spaces and H-type groups. Then, we introduce the notion of X-perimeter, showing the validity of a non-sharp isoperimetric inequality. In Chapter 2, we study the sharp isoperimetric inequality in H-type groups and Grushin spaces. Several techniques are needed here, such as representation formulas for the X-perimeter, a concentration-compactness type argument, and non-classical rearrangements. In Chapter 3, we prove quantitative isoperimetric inequalities in the Heisenberg group and in some Grushin spaces. To this purpose, we use a technique, known in the Calculus of Variations as subcalibration, in a suitable class of sets of finite X-perimeter. Finally, in Chapter 4, we address the problem of studying quantitative isoperimetric inequalities in the Grushin plane in a class of symmetric sets, starting from Euclidean techniques. Crucial differences arise from the lack of invariance under translation of the X-perimeter and lead us to study a variational problem, which has connections with the study of soap bubbles in the Grushin plane.

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