Liouville theorem for Beltrami flow

We prove that the Beltrami flow of ideal fluid in $R^3$ of a finite energy is zero.Comment: To appear in GAF

On stationary solutions of two-dimensional Euler Equation

We study the geometry of streamlines and stability properties for steady state solutions of the Euler equations for ideal fluid

Sets of unique continuation for heat equation

We study nodal lines of solutions to the heat equations. We are interested in the global geometry of nodal sets, in the whole domain of definition of the solution. The local structure of nodal sets is a well understander subject, while the global geometry of nodal lines is much less clear. We give a detailed analysis of a simple component of a nodal set of a solution of the heat equation

Conformal Spectrum and Harmonic maps

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that the map associating a finite number of eigenvectors of the maximizing $\lambda_1$ into the sphere is harmonic, establishing a link between conformal spectrum and harmonic maps

Maximization of higher order eigenvalues and applications

The present paper is a follow up of our paper \cite{nS}. We investigate here the maximization of higher order eigenvalues in a conformal class on a smooth compact boundaryless Riemannian surface. Contrary to the case of the first nontrivial eigenvalue as shown in \cite{nS}, bubbling phenomena appear