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We consider linear homogeneous differential equations of the form ẋ=A(t)x where A(t) is a square matrix of C1, real and T-periodic functions, with T>0. We give several criteria on the matrix A(t) to prove the asymptotic stability of the trivial solution to equation ẋ=A(t)x. These criteria allow us to show that any finite configuration of cycles in Rn can be realized as hyperbolic limit cycles...
We prove that if there exists a positive nonconstant function u which is p-harmonic (1<p⩽n) in a punctured domain Ω∖{0}⊂Rn and such that both u and ∂u∂ν are constant on ∂Ω, then u is radial and ∂Ω is a round sphere. The proof is based on a combination of integral identities, maximum principles and the isoperimetric inequality.
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