Refined Gauss–Green formulas and evolution problems for Radon measures

In this PhD thesis, we present some developments in the theory of sets of finite perimeter, weak integration by parts formulas and systems of coupled evolution equations for nonnegative Radon measures. First, we introduce a characterization of the perimeter of a measurable set in ℝⁿ via a family of functionals originating from a BMO-type seminorm. This result comes from a joint work with Luigi Ambrosio and is based on a previous paper by Ambrosio, Bourgain, Brezis and Figalli. In this paper, the authors considered functionals depending on a BMO-type seminorm and disjoint coverings of cubes with arbitrary orientations, and proved the convergence to a multiple of the perimeter. We modify their approach by using, instead of cubes, covering families made by translations of a given open connected bounded set with Lipschitz boundary. We show that the new functionals converge to an anisotropic surface measure, which is indeed a multiple of the perimeter if we allow for isotropic coverings (e.g. balls). This result underlines that the particular geometry of the covering sets is not essential. We then present the proof of a one-sided interior approximation for sets of finite perimeter, which was introduced in a paper of Chen, Torres and Ziemer. The original proof contained a gap, which was corrected in a joint work with Monica Torres. Given a set of finite perimeter E, the key idea for the approximation consists in taking the superlevel sets above 1/2 (respectively, below) of the mollification of the characteristic function of E. Then, we have that, asymptotically, the boundary of the approximating sets has negligible intersection with the measure theoretic interior (respectively, exterior) of E with respect to the (n − 1)-dimensional Hausdorff measure. The main motivation for the study of this finer type of approximation was the aim to establish Gauss–Green formulas for sets of finite perimeter and divergence-measure fields; that is, Lp-summable vector fields whose divergence is a Radon measure. Exploiting an alternative approach, we lay out a direct proof of generalized versions of the Gauss–Green formulas, which relies solely on the Leibniz rule for essentially bounded divergence-measure fields and scalar essentially bounded BV functions. In addition, we show some recent refinements. In particular, we provide a new Leibniz rule for Lp-summable divergence-measure fields and scalar essentially bounded Sobolev functions with gradient in Lp0 and we derive Green’s identities on sets of finite perimeter. This part is based on joint works with Kevin R. Payne and with Gui-Qiang Chen and Monica Torres. Due to the robustness of the Euclidean theory of divergence-measure fields, we can extend it to some non-Euclidean context. In particular, based on a joint work with Valentino Magnani, we develop a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Thanks to some nontrivial approximation arguments, we prove a Leibniz rule for essentially bounded horizontal divergence-measure fields and essentially bounded scalar function of bounded h-variation. As a consequence, we achieve the existence of normal traces and the related Gauss–Green theorem on sets of finite h-perimeter. Despite the fact that the Euclidean theory of normal traces relies heavily on De Giorgi’s blow-up theorem, which does not hold in general stratified groups, we are able to provide alternative proofs for the locality of the normal traces and other related results. Finally, we present a work in progress concerning the study of dislocations in crystals and their connection with evolution equations for signed measures, based on a current research project with Luigi Ambrosio, Mark A. Peletier and Oliver Tse. Starting from previous works of Ambrosio, Mainini and Serfaty, we consider couples of nonnegative measures instead of signed measures. Then, we employ techniques from the theory of optimal transport in order to represent the evolution equations as the gradient flows of a given energy with respect to a suitable distance among couples of nonnegative measures. To this purpose, we study a version of a Hellinger-Kantorovich distance introduced by Liero, Mielke and Savaré. In particular, we prove the existence of (weakly) continuous minimizing curves of measures which realize this distance and investigate its alternative representation as infimum of some action functional. Future research shall go in the direction of analyzing further properties of this Hellinger-Kantorovich distance, such as its dual representation, with the final aim to apply the classical methods of minimizing movements to prove the existence of solutions satisfying a certain type of energy dissipation equality.