Dottorato in MODELLI MATEMATICI PER L'INGEGNERIA, ELETTROMAGNETISMO E
NANOSCIENZE

Titolo della tesi: Regularizing effects on solutions to problems deriving from functionals with asymptotically linear growth

In this thesis, we discuss the regularizing effects that the gradient term present on the left side of the equation and the singular term on the right side of the equation have on the solutions of elliptic differential equations with homogeneous Dirichlet boundary condition, which involve $1$-Laplacian type operators with minimally regular data, specifically in $L^1$. Our main objective is to establish the conditions for the solutions to be of finite energy, that is, functions of bounded variation, and to be non-trivial. Furthermore, we also verify whether they have additional properties, such as being bounded and zero jump part of the derivative.