This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme in the panel ERC Starting grant under grant agreement No 810867.

The ERC’s mission is to encourage the highest quality research in Europe through competitive funding and to support investigator-driven frontier research across all fields, on the basis of scientific excellence. ERC grants are awarded through open competition to projects headed by starting and established researchers, irrespective of their origins, who are working or moving to work in Europe. The sole criterion for selection is scientific excellence.

ERC has the aim to recognise the best ideas, and confer status and visibility on the best
brains in Europe, while also attracting talent from abroad.
In the long term ERC looks to substantially strengthen and shape the European research

In the long term ERC looks to substantially strengthen and shape the European research system.

For more information, please visit ERC’s website.


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.

Home Institution

The University of Barcelona (UB), established in 1450, is one of the top universities in Southern Europe, and it ranks first among Spanish universities according to the QS World University Rankings and the Shanghai Ranking. It has become a European benchmark for research activity, both in terms of the number of research programmes that it conducts and the excellent results achieved.

The Mathematics Department at UB has a long tradition and it has been an important center of research in mathematics for decades. Many of their professors are members of the IMUB, an institute that was created to foster and support research in all areas of Mathematics in 2000. The IMUB hosts conferences, workshops, seminars and advanced courses, and promotes interdisciplinary collaboration. In addition, the department is part of the Barcelona Graduate School of Mathematics (BGSMath), a collaborative initiative of mathematical research groups in Barcelona which was awarded a “Unit of Excellence Maria de Maeztu” distinction by the Spanish Government in 2015.

Xavier Ros-Oton

Principal investigator: Prof. Xavier Ros-Oton

Xavier Ros-Oton

Xavier is an ICREA Research Professor at the University of Barcelona since 2020. Previously, he has been Assistant Professor at Universität Zürich, as well as R. H. Bing Instructor at the University of Texas at Austin. He is a mathematician who works on Partial Differential Equations (PDEs). Specifically, he studies the regularity of solutions to elliptic and parabolic PDEs, and he is mostly known for his results on free boundary problems and integro-differential equations. He is the PI of an ERC Starting Grant „ElipticPDE” (2019-2024) and has received several awards for young mathematicians in Spain, as well as the Scientific Research Award from the Fundación Princesa de Girona in 2019.

His research interests are on Partial Differential Equations (PDE) and he works mainly on topics related to the regularity of solutions to nonlinear elliptic/parabolic PDE. This is one of the most basic and important question in PDE theory: to understand whether all solutions to a given PDE are smooth or if, instead, they may have singularities. Some of his main contributions have been in the context of free boundary problems. These are PDE problems that involve unknown/moving interfaces, such as ice melting to water (phase transitions). From the mathematical point of view, they give rise to extremely challenging questions, and their study is closely connected to geometric measure theory. In particular, the study of free boundary problems has a strong geometrical flavor.

For further information on Prof. Ros Oton’s previous publications and research career, please refer to his personal website.