Computing the best constant in the Sobolev inequality for a ball. Issue 7 (19th May 2019)

• Record Type: Journal Article
• Authors: Ercole, Grey; Espírito Santo, Júlio César do; Martins, Eder Marinho
• Is Part Of: Applicable analysis. Volume 98:Issue 7(2019)
• Page Start: 1307
• Page End: 1323
• Publication Date: 2019-05-19
• Subjects: Best Sobolev constant -- extremal functions -- inverse iteration method -- p-Laplacian; 34L16 -- 35J25 -- 65N25; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2017.1422723
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: LetB 1 be the unit ball ofR N, N ≥ 2, and letp ⋆ = N p / ( N - p ) if1 < p < N andp ⋆ = ∞ ifp ≥ N . For eachq ∈ [ 1, p ⋆ ) letw q ∈ W 0 1, p ( B 1 ) be the positive function such thatw q L q ( B 1 ) = 1 andλ q ( B 1 ) : = min ∇ u L p ( B 1 ) p u L q ( B 1 ) p : 0 ≢ u ∈ W 0 1, p ( B 1 ) = ∇ w q L p ( B 1 ) p . In this paper, we develop an iterative method for obtaining the pair( λ q ( B 1 ), w q ), starting fromw 1 . Sincew 1 is explicitly known, the method is computationally practical, as our numerical tests show.