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Attainability of Second Order Adams Inequalities with Navier Boundary Conditions

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Abstract

We prove the existence of extremal functions for second order Adams inequalities with Navier boundary conditions on balls in ℝn in any dimension n ≥ 4. The proof is based on a symmetrization argument and the ideas introduced by L. Carleson and S.-Y. A. Chang in (Bulletin des Sciences Mathématiques 110, 113–127 1986) to prove the existence of extremal functions in the first order case, i.e. extremal functions for the Trudinger-Moser inequality on balls.

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Acknowledgements

The author would like to thank the referee for the useful comments provided, and in particular for suggesting the interesting open questions about the symmetry of extremal functions, and a possible relation between Adams’ inequality and Hardy-Adams inequality mentioned at the end of the Introduction.

The author is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Appllicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica “F. Severi” (INdAM), whose support is acknowledged.

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  1. Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via Campi 213/A –, 41125, Modena, Italy

    Federica Sani

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  1. Federica Sani
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Sani, F. Attainability of Second Order Adams Inequalities with Navier Boundary Conditions. Potential Anal 60, 365–386 (2024). https://doi.org/10.1007/s11118-022-10053-6

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