We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W 2,n/2(B n) into the exponential class, where B n is the unit ball of Rn. The corresponding sharp results for the spaces W0d,n/d(Ω) are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for d=1 and are due to Chang-Yang, Cianchi and Leckband. The proof of our result is based on a new integral representation formula for the "canonical" solution of the Poisson equation on the ball, that is, the unique solution of the equation δu=f which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function. © 2011 Elsevier Inc.

Fontana, L., Morpurgo, C. (2012). Sharp Moser-Trudinger inequalities for the Laplacian without boundary conditions. JOURNAL OF FUNCTIONAL ANALYSIS, 262, 2231-2271 [10.1016/j.jfa.2011.12.011].

Sharp Moser-Trudinger inequalities for the Laplacian without boundary conditions

FONTANA, LUIGI;
2012

Abstract

We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W 2,n/2(B n) into the exponential class, where B n is the unit ball of Rn. The corresponding sharp results for the spaces W0d,n/d(Ω) are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for d=1 and are due to Chang-Yang, Cianchi and Leckband. The proof of our result is based on a new integral representation formula for the "canonical" solution of the Poisson equation on the ball, that is, the unique solution of the equation δu=f which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function. © 2011 Elsevier Inc.
Articolo in rivista - Articolo scientifico
Moser–Trudinger; Sharp Sobolev inequalities; Exponential integrability
English
2012
262
2231
2271
none
Fontana, L., Morpurgo, C. (2012). Sharp Moser-Trudinger inequalities for the Laplacian without boundary conditions. JOURNAL OF FUNCTIONAL ANALYSIS, 262, 2231-2271 [10.1016/j.jfa.2011.12.011].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/53050
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