Linear elliptic PDEs with measurable coefficients naturally arise in the framework of Sobolev spaces. The qualitative behaviour of their solutions strongly depends on the regularity of the data, giving rise to a few typical scenarios. Interestingly, similar patterns of behaviour appear in broader contexts, including nonlinear systems, parabolic equations, and X-elliptic equations.
Gromov’s work on the convergence of metric structures and the results by Lott, Sturm and Villani opened the door to the study of nonsmooth spaces. In a loose sense, these objects play a role similar to that of Sobolev functions in the analysis of partial differential equations: they allow for weak notions of curvature, even in the absence of differentiability. Since recent developments in the theory of general relativity suggest that the universe may not be a smooth object, it is natural to ask whether ideas from metric geometry could help us understand the geometry of Lorentzian manifolds.
The analysis of heart rate variability (HRV) is a method to investigate the complex interaction between brain and heart. The study of parasympathetic, sympathetic, and Baevsky stress indexes allows a better understanding of Cardiac Autonomic Modulations under physical of phychological stress.