TAILIEUCHUNG - Đề tài " The perimeter inequality under Steiner symmetrization: Cases of equality "

Steiner symmetrization is known not to increase perimeter of sets in Rn . The sets whose perimeter is preserved under this symmetrization are characterized in the present paper. 1. Introduction and main results Steiner symmetrization, one of the simplest and most powerful symmetrization processes ever introduced in analysis, is a classical and very well-known device, which has seen a number of remarkable applications to problems of geometric and functional nature. | Annals of Mathematics The perimeter inequality under Steiner symmetrization Cases of equality By Miroslav Chleb rik Andrea Cianchi and Nicola Fusco Annals of Mathematics 162 2005 525 555 The perimeter inequality under Steiner symmetrization Cases of equality By Miroslav Chlebík Andrea Cianchi and Nicola Fusco Abstract Steiner symmetrization is known not to increase perimeter of sets in Rn. The sets whose perimeter is preserved under this symmetrization are characterized in the present paper. 1. Introduction and main results Steiner symmetrization one of the simplest and most powerful symmetrization processes ever introduced in analysis is a classical and very well-known device which has seen a number of remarkable applications to problems of geometric and functional nature. Its importance stems from the fact that besides preserving Lebesgue measure it acts monotonically on several geometric and analytic quantities associated with subsets of Rn. Among these perimeter certainly holds a prominent position. Actually the proof of the isoperimetric property of the ball was the original motivation for Steiner to introduce his symmetrization in 18 . The main property of perimeter in connection with Steiner symmetrization is that if E is any set of finite perimeter P E in Rn n 2 and H is any hyperplane then also its Steiner symmetral Es about H is of finite perimeter and P Es P E . Recall that Es is a set enjoying the property that its intersection with any straight line L orthogonal to H is a segment symmetric about H whose length equals the 1-dimensional measure of L n E. More precisely let us label the points x x1 . xn E Rn as x x y where x x1 . xn-1 E Rn-1 and y xn assume without loss of generality that H x 0 x E Rn-1 and set Ex y E R x y E E for x E Rn 1 x íX Ex for x E Rn-1 526 MIROSLAV CHLEBÍK ANDREA CIANCHI AND NICOLA FUSCO and n E x e R 1 xr 0 where Lm denotes the outer Lebesgue measure in Rm. Then Es can be defined as Es X y e Rn x e n E y h x

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