Geometric analysis: regularity theory for subelliptic PDEs and incompatible elasticity

This thesis is divided in two parts, which share a common theme of analysis in non-Euclidean spaces. The first one focuses on regularity of weak solutions of the $p$-Laplace equation in the Heisenberg group. In particular, we give a proof of the fact that, for $p>4$, solutions assumed to be in the horizontal Sobolev space $HW^{1,p}$ (consisting of $L^p$ functions whose horizontal gradient is in $L^p$), possess H\"older continuous horizontal derivatives. The argument is based on approximation via solutions of regularized problems: estimates independent of a non degeneracy parameter are obtained and passed to the limit. In particular, we show that the horizontal derivatives belong to a weighted De Giorgi space and then employ an alternative argument, not unlike the Euclidean case. The second part deals with non-Euclidean elasticity. We study incompatibly prestrained thin plates characterized by a prescribed Riemannian metric on their reference configuration. We analyze scaling of the elastic energy $E^h$ of order higher than $2$ in plate's thickness $h$, i.e. $\inf h^{-\beta}E^h$ for $\beta>2$. We find that, within this range, the only possible non trivial scaling is $\beta=4$. In this case we identify and study the $\Gamma$-limit functional, which consists of a von K\'arm\'an-like energy, given in terms of the first order infinitesimal isometries and of the admissible strains on the surface isometrically immersing the prestrain metric on the midplate in $\mathbb{R}^3$