We study optimal embeddings for the space of functions whose Laplacian Δu belongs to , where is a bounded domain. This function space turns out to be strictly larger than the Sobolev space in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when , we establish a sharp embedding inequality into the Zygmund space . On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem , 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 are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.
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}, pages = {73--93}, publisher = {Elsevier}, volume = {27}, number = {1}, year = {2010}, doi = {10.1016/j.anihpc.2009.07.006}, zbl = {1194.46048}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.07.006/} }
TY - JOUR AU - Cassani, Daniele AU - Ruf, Bernhard AU - Tarsi, Cristina TI - Best constants in a borderline case of second-order Moser type inequalities JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 73 EP - 93 VL - 27 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.07.006/ DO - 10.1016/j.anihpc.2009.07.006 LA - en ID - AIHPC_2010__27_1_73_0 ER -
%0 Journal Article %A Cassani, Daniele %A Ruf, Bernhard %A Tarsi, Cristina %T Best constants in a borderline case of second-order Moser type inequalities %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 73-93 %V 27 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.07.006/ %R 10.1016/j.anihpc.2009.07.006 %G en %F 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://archive.numdam.org/articles/10.1016/j.anihpc.2009.07.006/
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