W.P. Ziemer, Weakly Dierentiable Functions. Springer, New York, 1989. 30  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Z w 1 w 2 dx ; jw 1 j 0 : w 1 ; w 1 ) 1=2 0 (76) and jwj 1 : ess sup x2 jw(x)j. Now assume that (75) were not true. Then there exists a sequence fw n g H with kw n kH = 1, such that Z (rf T w n ) 2 0 jrw n j 2 dx 0 for n 1 : 77) Using the Poincar e inequality [62] Z (v v) 2 dx C( Z jrvj 2 dx 8v 2 H 1( 78) where v : 1 j j R v dx, it follows from (77) and (78) that jw n w n j 2 0 0; 79) with w n : 1 j j R w n dx. By means of jrf T wj 2 0 2 jf x 1 j 2 1 jw 1 j 2 0 2 jf x 2 j 2 1 jw 2 j 2 0 (80) and ....

W.P. Ziemer, Weakly Dierentiable Functions. Springer, New York, 1989. 30

....estimate holds also in L p , for 1 p 1. Sketch of proof. Let = jj dist(rv; SO(n)jj L 2 (Q) By scaling we may assume that Q is the unit cube and it suces to consider 1. We rst prove an interior estimate on a concentric subcube Q 0 . Step 1 (truncation) A truncation argument [15, 21, 5] shows that there exists a constant M such that if the result holds for v with jrvj M , then it is true in general. Step 2 (bound by p ) For a matrix F 2 M n n let cof F denote the matrix of (n 1) n 1) minors such that F T cof F = det F Id. Then for all v 2 W 1;1 we have div cof rv ....

W. Ziemer, Weakly Dierentiable Functions, Springer, 1989. 11

.... (x) This vector eld of normals can be also de ned (hence extended to all Q) as the Radon Nikodym derivative of the measure ru with respect to jruj, i.e. it formally satis es ru = jruj and, also, j j 1 a.e. For further information concerning functions of bounded variation we refer to [1, 17, 40]. Let us now introduce the function spaces for . Let Q be an open bounded subset of IR 2 with a Lipschitz boundary. We de ne W 1;p (div; Q) f 2 L p (Q) 2 : div( 2 L p (Q)g; 1 p 1; and M(div; Q) f 2 L 1 (Q) 2 : div( is a Radon measure in Qg: The Trace Theorem ....

W. P. Ziemer, Weakly Dierentiable Functions, GTM 120, Springer Verlag, 1989. 30

...., which will be denoted by D E , while = jD E j is its total variation; the perimeter P (E; B) of E in a Borel set B IR N 21 is de ned by jD E j(B) and we use the notation P (E) in the case B = IR N . For further information on sets of nite perimeter we refer to [2] 12] 14] 15] [49]. De nition 6 Let be an open subset of IR N . We say that a Borel function u : 1; 1] has weakly bounded variation in if P (fu tg; 1 for a.e. t 2 IR: The space of such functions will be denoted by WBV( We call total variation of u and denote by jDuj the measure de ned on ....

....jDuj the measure de ned on every Borel subset B as jDuj(B) Z 1 1 P (fu tg; B) dt: Remarks. i) It follows from the properties of the perimeter that jDuj is a additive measure on the Borel sets of Let BV( denote the space of functions of bounded variation in (see for instance [2, 12, 14, 15, 49]) Remark that by Lemma 1 in [3] BV( WBV( 1 Furthermore, if is bounded with Lipschitz boundary, u 2 WBV( and jDuj( 1 then, by [3] u 2 BV ( and, by the coarea formula, jDuj coincides with the total variation of u. ii) It must be emphasized that WBV is not a vector space. Take indeed ....

W.P. Ziemer, Weakly Dierentiable Functions, GTM 120, Springer Verlag, 1989. 37

....the notation P er(E) P (E; IR N ) If E has smooth boundary, by the classical GaussGreen formula, we have P (E; H N 1 ( E ; being H N 1 the Hausdor (N 1) dimensional measure in IR N . For further information concerning functions of bounded variation we refer to [2] 17] and [26]. Also several results from [7] see also [23] are needed. Following [7] let X( fz 2 L 1( IR N ) div(z) 2 L 1( g: If z 2 X( and w 2 BV( L 1( the functional (z; Dw) C 1 0( IR is de ned by the formula (z; Dw) Z w div(z) dx Z w z r dx: 2 Then (z; ....

....(4.6) give that jH N 1 ( M E 1 ) H N 1 ( M E 2 ) k 0 jEjj H N 1 ( M E : 4.9) Now, observe that H N 1 ( M E H N 1 ( M E cos 0 cos min(jEj; j n Ej) 4.10) for some constant 0 0. Indeed, by the relative isoperimetric inequality ( 17] [26]) we have that H N 1 ( M E Cmin(jEj; j n Ej) N 1 N ; for some constant C 0, and, since min(jEj; j n Ej) N 1 N min(jEj; j n Ej) j j 1 N ; we obtain H N 1 ( M E C j j 1 N min(jEj; j n Ej) This implies (4.10) Observe that (4.9) and (4.10) ....

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W.P. Ziemer, Weakly Dierentiable Functions, GTM 120, Springer Verlag, 1989. 22

....1. In all our arguments, and unless stated otherwise, C denotes a generic constant, the value of which may vary even within the same proof. 2 Some properties of BV functions For a detailed treatment of BV functions including the proofs of the following fundamental results, we refer the reader to [14] or [12] Although we shall not use it in the sequel, we rst recall the alternate (and equivalent) de nition of BV by nite di erences: if is an open set of R d , f 2 L 1( has bounded variation if and only if the quantity sup jhj 1 kf f( h)k L1( h ) jhj ; 2.1) is nite where h : ....

....jhj ; 2.1) is nite where h : fx 2 : x th 2 for t 2 [0; 1]g. Moreover for a xed this quantity is equivalent to the total variation jf jBV( We also recall that the space BV is (non compactly) embedded in L d ( with d = d d 1 and that we have the embedding inequality (see [14], p. 81) kfk L d C( kfkBV( 2.2) We shall use the possibility of approximating the functions of BV by smooth functions in the following sense (see e.g. 12] p.172 or [14] p.225) Theorem 2.1 Let f 2 BV . Then there exists a sequence ff k g k 0 in BV ( C 1( such that lim k 1 kf ....

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Ziemer, W., Weakly dierentiable functions, Springer Verlag, New York, 1989.

....1: 16) For suciently large , inequality (16) will become false and the cross over value is the screening length scr . By the Poincar e inequality on (0; 2 , inequality (16) is a consequence of Z (0; 2 jrU j 2 1: 5 Now R jrU j 2 is just the electrostatic capacity (see e.g. [21]) of the union of all particles within (0; 2 . Since R d, we have that if the number of particles inside the box is suciently small, the capacity of the union of particles is approximately the sum of the capacities of the individual particles. For radially symmetric particles of radius R ....

W. P. Ziemer. Weakly dierentiable functions. Springer, New York, 1989. 23

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W.P. Ziemer, Weakly Dierentiable functions, Springer, Berlin 1989.

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