We study optimal embeddings for the space of functions whose Laplacian \Delta u belongs to L^1(\Omega), where \Omega\subset\R^N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W^{2,1}(\Omega) 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}(\Omega). On one hand, this result enables us to improve the Brezis--Merle \cite{BM} regularity estimate for the Dirichlet problem \Delta u=f(x)\in L^1(\Omega), u=0 on \partial\Omega; on the other hand, it represents a borderline case of D.R. Adams'' \cite{DRA} generalization of Trudinger-Moser type inequalities to the case of higher order derivatives. Extensions to dimension N\geq are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions

Best constants in a borderline case of second order Moser type inequalities / D. Cassani, B. Ruf, C. Tarsi. - In: ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE. - ISSN 0294-1449. - 27:1(2010 Jan), pp. 73-93. [10.1016/j.anihpc.2009.07.006]

Best constants in a borderline case of second order Moser type inequalities

D. Cassani
Primo
;
B. Ruf
Secondo
;
C. Tarsi
Ultimo
2010

Abstract

We study optimal embeddings for the space of functions whose Laplacian \Delta u belongs to L^1(\Omega), where \Omega\subset\R^N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W^{2,1}(\Omega) 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}(\Omega). On one hand, this result enables us to improve the Brezis--Merle \cite{BM} regularity estimate for the Dirichlet problem \Delta u=f(x)\in L^1(\Omega), u=0 on \partial\Omega; on the other hand, it represents a borderline case of D.R. Adams'' \cite{DRA} generalization of Trudinger-Moser type inequalities to the case of higher order derivatives. Extensions to dimension N\geq are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions
Best constants; Brezis-Merle type results; Elliptic equations; Pohožaev, Strichartz and Trudinger-Moser inequalities; Regularity estimates in L1; Sobolev embeddings
Settore MAT/05 - Analisi Matematica
gen-2010
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/143287
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