Let ||•|| denote an arbitrary norm on |Rn and let ||•||_0 be its dual norm . Relative to this arbitrary norm, the perimeter of a set E of R^n is defined in a standard way that reduces to the usual classical perimeter in the case of the Euclidean norm. For u an arbitrary function defined on R^n let u^# be the convex rearrangement of u. The main result of this paper is the following: If u is a nonnegative function in the Sobolev space W^{1,p}(Rn), (1<p<∞), with the property that the set of critical points of |u^#| has measure zero and realize the equality in the Pòlya-Szegö inequality then u= u# a.e. in Rn (up to translations). The Euclidean version of this result was established by J. E. Brothers and the W.P. Ziemer and the original proof, even if based on a geometrical clear approach, the rigorous part justification of the argumants is accomplished after overcoming serious technical difficulties by means of results from geometric measure theory. In this paper is given a proof based on arguments from the classical theory of Sobolev spaces. The very worth of this method is that it is more flexible and can be easily adapted to more general problems. Indeed it has been used from other authors to prove quantitative version of this inequality.

Convex Symmetrization: The equality case in Pòlya-Szegö Inequality

FERONE, Adele;
2004

Abstract

Let ||•|| denote an arbitrary norm on |Rn and let ||•||_0 be its dual norm . Relative to this arbitrary norm, the perimeter of a set E of R^n is defined in a standard way that reduces to the usual classical perimeter in the case of the Euclidean norm. For u an arbitrary function defined on R^n let u^# be the convex rearrangement of u. The main result of this paper is the following: If u is a nonnegative function in the Sobolev space W^{1,p}(Rn), (1
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11591/189056
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