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Convex symmetrization and Pólya–Szegö inequality. (English) Zbl 1038.26014

Given a nonnegative convex function \(H\) on \({\mathbb R^n}\) which is homogeneous of degree one, the following inequality holds [cf. A. Alvino, V. Ferone G. Trombetti and P.-L. Lions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire {14}, 275–293 (1997; Zbl 0877.35040)]: \[ J(u):=\int_{\mathbb R^n}H(\nabla u)^2\,dx\geq \int_{\mathbb R^n}H(\nabla u^0)^2\,dx, \] where \(u^0\) denotes the “convex symmetrization” of \(u\), that is, a function whose level sets have the same measure as those of \(u\) and are homothetic to the polar of the level set \(\{x\in {\mathbb R^n}\mid H(x)<1\}\). In the particular case \(H(x)=| x| \) the inequality obtained is known as the Pólya-Szegö inequality. The main results of this paper are devoted to the equality problem in the above inequality (and in even slightly more general inequalities) stating that if \(H\) is strictly convex then \(J(u)=J(u^0)\) for some compactly supported function \(u\) without flat zones if and only if \(u\) agrees almost everywhere with a translate of \(u^0\). The proof utilizes a generalized version of the classical isoperimetric inequality.

MSC:

26D15 Inequalities for sums, series and integrals

Citations:

Zbl 0877.35040
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References:

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