Best constants in a borderline case of second-order Moser type inequalities
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, p. 73-93

We study optimal embeddings for the space of functions whose Laplacian Δu belongs to L 1 (Ω), where Ω N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W 2,1 (Ω) in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N=2, we establish a sharp embedding inequality into the Zygmund space L 𝑒𝑥𝑝 (Ω). On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem Δu=f(x)L 1 (Ω), u=0 on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N3 are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.

DOI : https://doi.org/10.1016/j.anihpc.2009.07.006
Classification:  46E35,  35B65
Keywords: Sobolev embeddings, Pohožaev, Strichartz and Trudinger–Moser inequalities, Best constants, Elliptic equations, Regularity estimates in L 1 , Brezis–Merle type results
@article{AIHPC_2010__27_1_73_0,
     author = {Cassani, Daniele and Ruf, Bernhard and Tarsi, Cristina},
     title = {Best constants in a borderline case of second-order Moser type inequalities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {1},
     year = {2010},
     pages = {73-93},
     doi = {10.1016/j.anihpc.2009.07.006},
     zbl = {1194.46048},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_1_73_0}
}
Cassani, Daniele; Ruf, Bernhard; Tarsi, Cristina. Best constants in a borderline case of second-order Moser type inequalities. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, pp. 73-93. doi : 10.1016/j.anihpc.2009.07.006. http://www.numdam.org/item/AIHPC_2010__27_1_73_0/

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