In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.

The first aim of this PhD Thesis is devoted to the study of geodesic distances defined on a subdomain of a Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. Then one shows that the uniform convergence, on compact sets, of these distances can be equivalently characterized in terms of Gamma-convergence of several kinds of variational problems. Moreover, it investigates the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. The second purpose is to obtain the integral representation of some classes of local functionals, depending on a family of vector fields, that satisfy a weak structure assumption. These functionals are defined on degenerate Sobolev spaces and they are assumed to be not translations-invariant. Then one proves some Gamma-compactness results with respect to both the strong topology of L^p and the strong topology of degenerate Sobolev spaces.

The aim of this thesis is to propose a systematic exposition of some analytic and geometric problems arising from the study of sub-Riemannian geometry, Carnot-Carathéodory spaces and, more broadly, anisotropic metric and differential structures. We deal with four main topics. 1 Calculus of variations for local functionals depending on vector fields 2 PDEs over Carnot-Carathéodory structures. 3 Regularity theory for almost perimeter minimizers in Carnot groups. 4 Geometry of hypersurfaces in Heisenberg groups.

Very few standards exist for tting products to people. Footwear fit is a noteworthy example for consumer consideration when purchasing shoes. As a result, footwear manufacturing industry for achieving commercial success encountered the problem of developing right footwear which is fulfills consumer's requirement better than it's competeries. Mass customization starts with understanding individual customer's requirement and it finishes with fulllment process of satisfying the target customer with near mass production efficiency. Unlike any other consumer product, personalized footwear or the matching of footwear to feet is not easy if delivery of discomfort is predominantly caused by pressure induced by a shoe that has a design unsuitable for that particular shape of foot. Footwear fitter have been using manual measurement for a long time, but the combination of 3D scanning systems with mathematical technique makes possible the development of systems, which can help in the selection of good footwear for a given customer. This thesis, provides new approach for addressing the computerize footwear fit customization in industry problem. The design of new shoes starts with the design of the new shoe last. A shoe last is a wooden or metal model of human foot on which shoes are shaped. Despite the steady increase in accuracy, most available scanning techniques cause some deficiencies in the point cloud and a set of holes in the triangle meshes. Moreover, data resulting from 3D scanning are given in an arbitrary position and orientation in a 3D space. To apply sophisticated modeling operations on these data sets, substantial post-processing is usually required. We described a robust algorithm for filling holes in triangle mesh. First, the advance front mesh technique is used to generate a new triangular mesh to cover the hole. Next, the triangles in initial patch mesh is modified by estimating desirable normals instead of relocating them directly. Finally, the Poisson equation is applied to optimize the new mesh. After obtaining complete 3D model, the result data must be generated and aligned before taking this models for shape analysis such as measuring similarity between foot and shoe last data base for evaluating footwear it. Principle Component Analysis (PCA), aligns a model by considering its center of mass as the coordinate system origin, and its principle axes as the coordinate axes. The purpose of the PCA applied to a 3D model is to make the resulting shape independent to translation and rotation asmuch as possible. In analysis, we applied "weighted" PCA instead of applying the PCA in a classical way (sets of 3D point-clouds) for alignment of 3D models. This approach is based on establishing weights associated to center of gravity of triangles. When all of the models are aligned, an efficient algorithm to cut the model to several sections toward the heel and toe for extracting counters is used. Then the area of each contour is calculated and compared with equal sections in shoe last data base for finding best footwear fit within the shoe last data base.