On Some Rigidity Properties in PDEs

Abstract

This thesis is dedicated to the study of three rigidity properties arising in different partial differential equations: (1) the backward uniqueness property of the heat equation in two-dimensional conical domains, (2) the weak and strong unique continuation principles for fractional Schrödinger equations with rough potentials and (3) the rigidity and non-rigidity of exactly stress-free configurations of a differential inclusion describing the cubic-to-orthorhombic phase transition in the geometrically linearized theory of elasticity.