Ph.D. Thesis

Inserted: 5 jan 2026
Last Updated: 5 jan 2026

Pages: 183
Year: 2025
Links: PhD Thesis_Ficola Eleonora

Abstract:

Object of this thesis is the study of convex first–order variational functionals F with linear growth in the gradient variable, coupled with a non–linear integral term with respect to a (possibly signed) Radon measure on given bounded domains of R^n. After achieving a generalized parametric lower semicontinuity result, we provide necessary and sufficient conditions for the existence of minimizers of F in the space of functions of bounded variation (BV), discussing (counter)examples and borderline cases for the assumptions. A first step in the existence theory consists of approaching semicontinuity of the (anisotropic) total variation problem with measures in continuation to the parametric outcome of 92, further exploiting lifting of functions to extend the result to a broader class of integrals. The transition from the standard total variation integral to possibly non–even anisotropies with measures is addressed by the choice of a suitable signed isoperimetric condition defined on sets of finite perimeter – also equivalently reformulated on multiple subclasses of BV functions. More importantly, we make use of coarea and layer-cake arguments to move from the parametric formulation with (anisotropic) perimeter to (anisotropic) variations with measures. This method represents the foundation of our global existence theory, and it is subsequently employed to achieve a more general existence result. In parallel, we determine the dual maximization problem corresponding to F set in the class of divergence–measure vector fields. The outcome of duality enables us to reformulate optimality relations for solutions of the two problems in terms of a refined version of Anzellotti’s pairing 7 between maximizing fields of assigned divergence measure and distributional derivatives of minimizers of F. By introducing a suitable notion of solution to the Euler-Lagrange equation associated to F, we then recover the usual meaning of necessary – and, under convexity, also sufficient – condition for minimizers of F, demonstrating that our BV formulation provides a natural extension of the Sobolev model in the sense of L1-relaxation.

Keywords: calculus of variations, functions of bounded variation, Sets of finite perimeter, convex analysis, Elliptic Partial Differential Equations, Euler-Lagrange equation