Universitą degli studi di Modena e Reggio Emilia

Starting from fundamental works by Bernoulli, Euler, Lagrange, and Weierstrass, the Calculus of Variations constitutes a fundamental set of theories aimed at solving optimization problems where a functional (often of integral type) has to be minimized. Variational techniques specifically developed for the study of critical points of functionals (i.e., solutions to the associated Euler equations) play a key role in proving existence of solutions to boundary value problems for important classes of ordinary and partial differential equations. Many problems in applied sciences are formulated in terms of ordinary and partial differential equations with a variational structure. Traditionally, one of the characteristics of variational problems arising in applications is given by the presence of nonlinear terms with critical growth in classical function spaces, and new models are settled in limiting or borderline spaces which we aim to investigate. Many questions arising in applications lead to variational models with non-standard growth conditions. Relevant examples are in elasticity theory and material sciences to model strongly anisotropic materials, such as composite and biological materials. Local regularity of minimizers for functionals associated to these models has been widely studied in recent years, but several open questions still remain open.