Nonuniform Ellipticity Widespread

forthcoming

Project NEW tackles key challenges in nonlinear elliptic and parabolic regularity theory. The outcome answers to fundamental open questions. Achievements include sharp analysis data-solutions in nonuniformly elliptic or parabolic PDEs, singular set estimates in vectorial problems, and maximal regularity in degenerate (possibly nonlocal) monotone operators lacking strong monotonicity properties. These results are supported by innovative hybrid techniques, combining deep insights from harmonic analysis, nonlinear potential theory, and variational analysis, specifically devised to be implemented directly at the PDE level. Beyond technical and theoretical advances, the accomplishments will significantly impact Non-Newtonian fluid mechanics, homogenization, and geometric analysis, transforming techniques and perspectives developed through half a century of progress in nonlinear regularity theory.