![]() Institute of Experimental Epilepsy and Cognition Research (IEECR)Ī fundamental challenge for the brain is to extract relevant information from an ever changing external world. Consequently, far-reaching implications beyond the proposed objectives are expected, in the development of new methods and applications, in diverse fields of Analysis. ![]() In this regard, the proposed methodology gathers novel ideas oriented to overcome such paramount challenges. This set of problems comprises significant theoretical obstacles at the forefront of the calculus of variations and geometric measure theory. Lastly, Theme II conjectures a complementary result to the ground-breaking De Philippis–Rindler theorem, which asserts that the regular part of an A-free measure is essentially unconstrained. The goal is to prove that these measure-theoretic generalizations of surfaces possess an underlying BV-like structure. It will also investigate the structure of integral varifolds with bounded firstvariation. Via potential and measure theory methods, it will attempt to produce substantial advances towards solving the sigma-finiteness conjecture in BD spaces. Theme II investigates the fine properties of PDE-constrained measures from three different perspectives. Building upon results recently pioneered by the PI, the purpose of Theme I is to prove a novel interpretation of Bouchitte’s Vanishing mass conjecture and a novel compensated integrability result, with profound implications for the compensated compactness theory. Theme I examines the qualitative and quantitative nature of PDE-constrainedĬoncentrations. ConFine will investigate the nature of concentrations and fine geometries arising from longstanding conjectures and novel questions of the calculus of variations. This phenomenon, modeled by weak forms of convergence, entails the formation of oscillations, concentrations, and fine patterns ubiquitous in geometric, physical, and materials science models. ![]() ![]() The interaction between microscopic and macroscopic quantities lies at the heart of fascinating problems in the modern theory of nonlinear PDEs. ![]()
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