Abstract. In this talk we present some recent results regarding compactly supported solutions of the 2D steady Euler equations. Under some assumptions on the support of the solution, we prove that the streamlines of the flow are circular. The proof uses that the corresponding stream function solves an elliptic semilinear problem $-\Delta \phi = f(\phi)$ with $\nabla \phi=0$ at the boundary. One of the main difficulties in our study is that $f$ can fail to be Lipschitz continuous near the boundary values.

If $f(\phi)$ vanishes at the boundary values we can apply a local symmetry result of F. Brock to conclude. Otherwise, we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of $f$. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, Hopf lemma and Serrin corner lemma for elliptic linear operators with singular coefficients.

Bio. David Ruiz obtained his degree in mathematics from the University of Granada in 1999. In 2002 he received his doctorate degree from the same university. As a post-doc, he made several research stays at the Scuola Internazionale Superiore di Studi Avanzati (SISSA), in Trieste, Italy. During the academic year 2006/07 he obtained a Ricercatore position at SISSA, and afterwards he became an associate professor at the University of Granada where he is a professor nowadays.

David Ruiz works in Nonlinear Elliptic Partial Differential Equations that appear in Mathematical Physics and Geometry. The main tools for this study are variational methods, topological methods, and perturbation arguments. Moreover, he has worked on Liouville-type equations on compact surfaces that appear in Chern-Simons theory and Geometrical Analysis. Another field of interest concerns semilinear elliptic problems under overdetermined boundary conditions.