Sobolev Inequalities with Remainder Terms

HaïM Brezis and Elliott H. Lieb
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HaïM Brezis: Université Paris VI, Département de Mathématiques
Elliott H. Lieb: Princeton University, Departments of Mathematics and Physics

A chapter in Inequalities, 2002, pp 581-594 from Springer

Abstract: Abstract The usual Sobolev inequality in ℝn, n≥3, asserts that 1 $$\left\| {\nabla f} \right\|_2^2 \geqslant {S_n}\left\| f \right\|_{2*}^2$$ , with S n being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ℝn. Two kinds of inequalities are established: (i) If f = 0 on ∂ Ω, then 2 $$\left\| {\nabla f} \right\|_2^2 \geqslant {S_n}\left\| f \right\|_{2*}^2 + C(\Omega )\left\| f \right\|_{p,w}^2$$ with p=2*/2. Some further results and open problems in this area are also presented.

Keywords: Sobolev Inequality; Fractional Integral; Remainder Term; Nonlinear Elliptic Equation; Sharp Constant (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-3-642-55925-9_46

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