Description. Ellipticity is a crucial aspect of Partial Differential Equations (PDEs) in relation to the regularity of solutions. The most studied class of elliptic equations is that of uniformly elliptic PDEs, characterized by the uniform boundedness of the ellipticity ratio, i.e., the ratio between the the highest and the lowest eigenvalue of the governing elliptic matrix. The smoothing effect on energy solutions of a uniformly bounded ellipticity ratio is classical after Ural'tseva-Uhlenbeck foundational work [Ura68,Uhl77]. However, such a regularization process may be inhibited by plugging in ingredients, like forcing or transport terms, or space-depending coefficients, or by weakening uniform ellipticity condition. This is at the core of fundamental open problems in elliptic and parabolic regularity theory, revolving around various nonclassical, degenerate or nonuniform ellipticity types. Longstanding questions include degenerate Calderón-Zygmund theory, singular set estimates in vectorial problems, maximal regularity for monotone, possibly nonlocal operators and nonuniformly parabolic flows. In this respect, the ultimate goal of project NEW is to achieve a transformative nonlinear regularity theory yielding sharp results subject to nonclassical ellipticity and rough ingredients.

Funding

Funding Agency: European Research Council

Funding Scheme: Starting Grant

Call year: 2025    Panel: PE1    Project number: 101220121

Reference: ERC-2025-StG 101220121 NEW