Abstract
Some special inequalities are helpful in the study of the p-Laplace operator.
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Notes
- 1.
By conjugation (III) follows from (I). To see this, let \(1<p<2\) and write \(q=p/(p-1) >2.\) By (I)
$$ 2^{2-q}|B-A|^q \le \langle |B|^{q-2}B-|A|^{q-2}A, B-A\rangle . $$Use \(a=|A|^{q-2}A,\, A = |a|^{p-2}a\) and the same for B to obtain
$$2^{2-q}\left| |b|^{p-2}b-|a|^{p-2}a\right| ^q\,\le \,\langle |b|^{p-2}b-|a|^{p-2}a, b-a\rangle \,\le |b-a|\left| |b|^{p-2}b-|a|^{p-2}a\right| .$$It follows that
$$2^{2-q}\left| |b|^{p-2}b-|a|^{p-2}a\right| ^{q-1}\,\le |b-a|.$$Thus, since \((p-1)(q-1)=1,\)
$$\begin{aligned} \boxed {\left| |b|^{p-2}b-|a|^{p-2}a\right| \,\le \, 2^{2-p}|b-a|^{p-1},\qquad 1< p <2.} \end{aligned}$$This directly implies (III) with \(\gamma (p) = 2^{2-p}.\)
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Lindqvist, P. (2019). Inequalities for Vectors. In: Notes on the Stationary p-Laplace Equation. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-14501-9_12
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DOI: https://doi.org/10.1007/978-3-030-14501-9_12
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