Computing the best constant in the Sobolev inequality for a ball.
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Date
2017
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Abstract
Let B1 be the unit ball of R
N , N ≥ 2, and let p
? = N p/(N − p) if 1 < p < N and p
? = ∞ if p ≥ N. For
each q ∈ [1, p?
) let wq ∈ W1,p
0
(B1) be the positive function such that kwqkLq(B1) = 1 and
λq(B1) := min (
k∇uk
p
Lp(B1)
kuk
p
Lq(B1)
: 0 6≡ u ∈ W1,p
0
(B1)
)
= k∇wqk
p
Lp(B1)
.
In this paper we develop an iterative method for obtaining the pair (λq(B1), wq), starting from w1. Since w1
is explicitly known, the method is computationally practical, as our numerical tests show.
2010 Mathematics Subject Classification. 34L16; 35J25; 65N25
Keywords: Best Sobolev constant; extremal functions; inverse iteration method; p-Laplacian.
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ENCOLE, G.; ESPÍRITO SANTO, J. C do.; MARTINS, E. M. Computing the best constant in the Sobolev inequality for a ball. Applicable Analysis, v. 1, p. 1-17, 2018. Disponível em: <https://www.tandfonline.com/doi/full/10.1080/00036811.2017.1422723>. Acesso em: 16 jun. 2018.