Computing the best constant in the Sobolev inequality for a ball. Issue 7 (19th May 2019) - Ercole, Grey; Espírito Santo, Júlio César do; Martins, Eder Marinho
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.