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Strict inequalities for integrals of decreasingly rearranged functions. (English) Zbl 0601.49012

The authors are interested in finding sufficient conditions to have equality in the well known inequality \[ (i)\quad \int_{R^ n}\int_{R^ n}h(x-y)f(x)g(y)dx dy\leq \int_{R^ n}\int_{R^ n}h^*(x-y)f^*(x)g^*(y)dx dy, \] where f, g, h are nonnegative functions and \(f^*\), \(g^*\), \(h^*\) are their symmetrically decreasing rearrangements. The same problem is observed for the inequality \[ (ii)\quad (\int | \nabla u^*|^ p)^{1/p}\leq (\int | \nabla u|^ p)^{1/p},\quad 1\leq p\leq \infty, \] where \(u^*\) is a spherically decreasing rearrangement of the function u. So, assuming that \(h=h^*\) and f and g are nonconstant, equality holds in (i) if and only if \(f=f^*\), and \(g=g^*\), while equality holds in (ii) (under some additional assumptions) if and only if \(u=u^*\). Some applications to variational problems of these results are sketched.
Reviewer: C.Zălinescu

MSC:

49K10 Optimality conditions for free problems in two or more independent variables
26D15 Inequalities for sums, series and integrals
42C20 Other transformations of harmonic type
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References:

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