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Optimal principal eigenfunction for elliptic operators with large drift Rossi, Luca


In a series of papers, F. Hamel, N. Nadirashvili and E. Russ deal with the isoperimetric problem for eigenvalues of second order elliptic operators. With respect to the classical Faber-Krahn inequality, they consider an additional drift term, which acts as a control under L infinity constraint. They solve the problem and derive the precise asymptotic of the principal eigenvalue as the norm of the drift tends to infinity. As a preliminary step, they consider the optimization problem in a fixed domain. This leads to a nonlinear eigenvalue problem. They conjecture that, in such case, the maximal points of the optimal principal eigenfunction approach the points with maximal distance from the boundary, and that its gradient aligns with the gradient of the distance function. We present a proof of the first conjecture and give a partial result concerning the second one. These results have been obtained in collaboration with F. Hamel and E. Russ.

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