Computing the best constant in the Sobolev inequality for a ball

ABSTRACT

Let $B_1$ be the unit ball of ${\mathbb{R}}^N,$ $N\ge 2,$ and let $p^{\star }=Np/(N-p)$ if $1<p<N$ and $p^{\star }=\infty$ if $p\ge N.$ For each $q\in [1,p^{\star })$ let $w_q\in W_0^{1,p}(B_1)$ be the positive function such that ${∥w_q∥}_{L^q(B_1)}=1$ and $${\lambda }_q(B_1):=min\left\{\frac{{∥\mathrm{\nabla }u∥}_{L^p(B_1)}^p}{{∥u∥}_{L^q(B_1)}^p}:0\not\equiv u\in W_0^{1,p}(B_1)\right\}={∥\mathrm{\nabla }{w}_q∥}_{L^p(B_1)}^p.$$

In this paper, we develop an iterative method for obtaining the pair $({\lambda }_q(B_1),w_q)$, starting from $w_1$. Since $w_1$ is explicitly known, the method is computationally practical, as our numerical tests show.

KEYWORDS:

• Best Sobolev constant
• extremal functions
• inverse iteration method
• p-Laplacian

AMS SUBJECT CLASSIFICATIONS:

• 34L16
• 35J25
• 65N25

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author thanks the support of FAPEMIG/Brazil [grant number CEX-PPM-00165]; CNPq/Brazil [grant number 483970/2013-1], [grant number 306590/2014-0].