TY - JOUR

T1 - Best constants for two families of higher order critical Sobolev embeddings

AU - Shafrir, Itai

AU - Spector, Daniel Eli

PY - 2018/12/1

Y1 - 2018/12/1

N2 - In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into [Formula presented] and those that embed into slightly larger target spaces. Concerning the former, we show that for [Formula presented], [Formula presented] even, one has an optimal constant [Formula presented] such that [Formula presented]for all [Formula presented] (the case [Formula presented] was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams’ higher order inequality of J. Moser: For [Formula presented], [Formula presented] and [Formula presented], there exists [Formula presented] and optimal constant [Formula presented] such that [Formula presented]for all [Formula presented] such that [Formula presented], where [Formula presented] is the traditional semi-norm on the space [Formula presented].

AB - In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into [Formula presented] and those that embed into slightly larger target spaces. Concerning the former, we show that for [Formula presented], [Formula presented] even, one has an optimal constant [Formula presented] such that [Formula presented]for all [Formula presented] (the case [Formula presented] was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams’ higher order inequality of J. Moser: For [Formula presented], [Formula presented] and [Formula presented], there exists [Formula presented] and optimal constant [Formula presented] such that [Formula presented]for all [Formula presented] such that [Formula presented], where [Formula presented] is the traditional semi-norm on the space [Formula presented].

KW - Best constant

KW - Critical exponent

KW - Sobolev embedding

UR - http://www.scopus.com/inward/record.url?scp=85047193238&partnerID=8YFLogxK

U2 - 10.1016/j.na.2018.04.027

DO - 10.1016/j.na.2018.04.027

M3 - Article

AN - SCOPUS:85047193238

VL - 177

SP - 753

EP - 769

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -