Some inequalities of Sobolev type on two-dimensional spheres. (English) Zbl 0652.26020

Talenti, Giorgio

General inequalities 5, 5th Int. Conf., Oberwolfach/FRG 1986, ISNM 80, 401-408 (1987).

[For the entire collection see Zbl 0621.00004.]
Let \(S^ 2\) be the two-dimensional sphere in \(R^ 3\) and let \(H^ 2(dx)\) be the usual measure on \(S^ 2\). The main results read as follows: The (exact) inequalities \[ \int_{S^ 2}[u(x)-H^ 2(S^ 2)^{-1}\int_{S^ 2}u(y)H^ 2(dy)]^ 2H^ 2(dx) \]
\[ \leq (4\pi)^{-1}(\int_{S^ 2}| \text{grad} u(x)| H^ 2(dx))^ 2 \] and \[ \max u-\min u\quad \leq \quad c(\int_{S^ 2}| \text{grad} u(x)|^ pH^ 2(dx))^{1/p} \] hold for any smooth real function u on \(S^ 2\), \(p>2\) and \[ c = (2\pi)^{-1/p}[\pi^{1/2}\Gamma (1-p/(2p- 2))/\Gamma (1-1/(2p-2))]^{1-1/p}. \]

Reviewer: J.Rákosník

MSC:

26D10

Inequalities involving derivatives and differential and integral operators

Keywords:

Sobolev type inequalities; two-dimensional sphere in \(R^ 3\)

Citations:

Zbl 0621.00004