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Regularity and singularities in elliptic PDE's: beyond monotonicity formulas

Objective

One of the oldest and most important questions in PDE theory is that of regularity. A classical example is Hilbert's XIXth problem (1900), solved by De Giorgi and Nash in 1956. During the second half of the XXth century, the regularity theory for elliptic and parabolic PDE's experienced a huge development, and many fundamental questions were answered by Caffarelli, Nirenberg, Krylov, Evans, Nadirashvili, Friedman, and many others. Still, there are problems of crucial importance that remain open.

The aim of this project is to go significantly beyond the state of the art in some of the most important open questions in this context. In particular, three key objectives of the project are the following. First, to introduce new techniques to obtain fine description of singularities in nonlinear elliptic PDE's. Aside from its intrinsic interest, a good regularity theory for singular points is likely to provide insightful applications in other contexts. A second aim of the project is to establish generic regularity results for free boundaries and other PDE problems. The development of methods which would allow one to prove generic regularity results may be viewed as one of the greatest challenges not only for free boundary problems, but for PDE problems in general. Finally, the third main objective is to achieve a complete regularity theory for nonlinear elliptic PDE's that does not rely on monotonicity formulas. These three objectives, while seemingly different, are in fact deeply interrelated.

Field of science

  • /natural sciences/mathematics/pure mathematics/mathematical analysis/differential equations/partial differential equations

Call for proposal

ERC-2018-STG
See other projects for this call

Funding Scheme

ERC-STG - Starting Grant

Host institution

UNIVERSITAT DE BARCELONA
Address
Gran Via De Les Corts Catalanes 585
08007 Barcelona
Spain
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 950 306,25

Beneficiaries (2)

UNIVERSITAT DE BARCELONA
Spain
EU contribution
€ 950 306,25
Address
Gran Via De Les Corts Catalanes 585
08007 Barcelona
Activity type
Higher or Secondary Education Establishments
UNIVERSITAT ZURICH

Participation ended

Switzerland
EU contribution
€ 384 943,75
Address
Ramistrasse 71
8006 Zurich
Activity type
Higher or Secondary Education Establishments