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Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems ( P ε ) : Δ u ε = u ε ( n + 2 ) / ( n 2 ) , u ε > 0 in A ε , u ε = 0 on A ε , where A ε is an annulus-shaped domain of n , n 3 , which becomes thinner as ε 0 . We show that for every solution u ε , the energy A ε | u | 2 as well as the Morse index tend to infinity as ε 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on n , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics

Sebastian Herr (2010)

Journées Équations aux dérivées partielles

This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds M with small initial data in H 1 ( M ) . The results include small data GWP for the quintic NLS in the case of the 3 d flat rational torus M = 𝕋 3 and small data GWP for the corresponding cubic NLS in the cases M = 2 × 𝕋 2 and M = 3 × 𝕋 . The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well...

Energy norm error estimates and convergence analysis for a stabilized Maxwell's equations in conductive media

Eric Lindström, Larisa Beilina (2024)

Applications of Mathematics

The aim of this article is to investigate the well-posedness, stability of solutions to the time-dependent Maxwell's equations for electric field in conductive media in continuous and discrete settings, and study convergence analysis of the employed numerical scheme. The situation we consider would represent a physical problem where a subdomain is emerged in a homogeneous medium, characterized by constant dielectric permittivity and conductivity functions. It is well known that in these homogeneous...

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