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On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two

Jean Dolbeault, Régis Monneau (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in 2 . We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

Given a bounded open set Ω in n (or in a Riemannian manifold) and a partition of Ω by k open sets D j , we consider the quantity 𝚖𝚊𝚡 j λ ( D j ) where λ ( D j ) is the ground state energy of the Dirichlet realization of the Laplacian in D j . If we denote by k ( Ω ) the infimum over all the k -partitions of 𝚖𝚊𝚡 j λ ( D j ) , a minimal k -partition is then a partition which realizes the infimum. When k = 2 , we find the two nodal domains of a second eigenfunction, but the analysis of higher k ’s is non trivial and quite interesting. In this paper, we give...

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