Equivalence between Poly\'a-Szeg\H{o} and relative capacity inequalities under rearrangement

The transformations of functions acting on sublevel sets that satisfy a P\'olya-Szeg\H{o} inequality are characterized as those being induced by transformations of sets that do not increase the associated capacity.Comment: 9 page

Desingularization of vortices for the Euler equation

We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation -\eps^2 \Delta u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.Comment: 40 page