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On the existence of periodic solutions of an hyperbolic equation in a thin domain

Russell Johnson, Mikhail Kamenskii, Paolo Nistri (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

For a nonlinear hyperbolic equation defined in a thin domain we prove the existence of a periodic solution with respect to time both in the non-autonomous and autonomous cases. The methods employed are a combination of those developed by J. K. Hale and G. Raugel and the theory of the topological degree.

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou Lin, Marcello Lucia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider on a two-dimensional flat torus T defined by a rectangular periodic cell the following equation Δ u + ρ e u ∫ T e u - 1 | T | = 0 , ∫ T u = 0 . It is well-known that the associated energy functional admits a minimizer for each ρ ≤ 8 π . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting λ 1 ( T ) to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever ρ ≤ min { 8 π , λ 1 ( T ) | T | } . Our results hold more generally for solutions that are Steiner symmetric, up to a translation....

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