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Minicourse on BV Functions Minicourse on BV functions University of Oxford Panu Lahti June 16, 2016 University of Oxford Panu Lahti Minicourse on BV functions Literature L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. viii+268 pp. L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983. vii+272 pp. σ-algebras and positive measures Let X be a topological space. A collection E of subsets of X is a σ-algebra if ; 2 E, X n E 2 E whenever E 2 E, and for any sequence (Eh) ⊂ E, we have S E 2 E. h2N h We say that µ : E! [0; 1] is a positive measure if µ(;) = 0 and for any sequence (Eh) of pairwise disjoint elements of E, ! [ X µ Eh = µ(Eh): h2N h2N We say that M ⊂ X is µ-negligible if there exists E 2 E such that M ⊂ E and µ(E) = 0. The expression "almost everywhere", or "a.e.", means outside a negligible set. The measure µ extends to the collection of µ-measurable sets, i.e. those that can be presented as E [ M with E 2 E and M µ-negligible. Vector measures N Let N 2 N. We say that µ: E! R is a vector measure if µ(;) = 0 and for any sequence (Eh) of pairwise disjoint elements of E ! [ X µ Eh = µ(Eh): h2N h2N If µ is a vector measure on (X ; E), for any given E 2 E we define the total variation measure jµj(E) as ( ) X [ sup jµ(Eh)j : Eh 2 E pairwise disjoint; E = Eh : h2N h2N We can show that jµj is then a finite positive measure on (X ; E), that is, jµj(X ) < 1. Borel and Radon measures We denote by B(X ) the σ-algebra of Borel subsets of X , i.e. the smallest σ-algebra containing the open subsets of X . A positive measure on (X ; B(X )) is called a Borel measure. If it is finite on compact sets, it is called a positive Radon measure. N A vector Radon measure is an R -valued set function that is a vector measure on (K; B(K)) for every compact set K ⊂ X . We say that it is a finite Radon measure if it is a vector measure on (X ; B(X )). Lebesgue and Hausdorff measures n n We will denote by L the n-dimensional Lebesgue measure in R . n n Sometimes we write jAj instead of L (A) for A ⊂ R . k We denote by !k the volume of the unit ball in R . Definition n Let k 2 [0; 1) and let A ⊂ R . The k-dimensional Hausdorff measure of A is given by k k H (A) := lim Hδ (A); δ!0 where for any 0 < δ ≤ 1, ( ) !k X [ Hk (A) := inf [diam(E )]k : diam(E ) < δ; A ⊂ E δ 2k i i i i2N i2N with the convention diam(;) = 0. Besicovitch differentiation theorem Theorem Let µ be a positive Radon measure in an open set Ω, and let ν be N an R -valued Radon measure in Ω. Then for µ-a.e. x 2 Ω the limit ν(B(x; r)) f (x) := lim r!0 µ(B(x; r)) N exists in R and ν can be presented by the Lebesgue-Radon-Nikodym decomposition ν = f µ + νs , where νs = ν E for some E ⊂ Ω with µ(E) = 0. Definition of BV functions n The symbol Ω will always denote an open set in R . Definition Let u 2 L1(Ω). We say that u is a function of bounded variation in Ω if the distributional derivative of u is representable by a finite Radon measure in Ω, i.e. Z Z @ 1 u dx = − dDi u 8 2 Cc (Ω); i = 1;:::; n Ω @xi Ω n for some R -valued Radon measure Du = (D1u;:::; Dnu) in Ω. The vector space of all functions of bounded variation is denoted by BV(Ω). We can always write Du = σjDuj, where jDuj is a positive Radon measure and σ = (σ1; : : : ; σn) with jσ(x)j = 1 for jDuj-a.e. x 2 Ω. Examples of BV functions W 1;1(Ω) ⊂ BV(Ω), since for u 2 W 1;1(Ω) we have Du = ru Ln. On the other hand, for the Heaviside function 1;1 χ(0;1) 2 BVloc(R) we have Du = δ0, and u 2= Wloc (R). 0 0 We say that u 2 BVloc(Ω) if u 2 BV(Ω ) for every Ω b Ω, i.e. every open Ω0 with Ω0 compact and contained in Ω. Mollification, part I 1 n n Let ρ 2 Cc (R ) with ρ(x) ≥ 0 and ρ(−x) = ρ(x) for all x 2 R , supp ρ ⊂ B(0; 1), and Z ρ(x) dx = 1: Rn −n Choose " > 0. Let ρ"(x) := " ρ(x="), and Ω" := fx 2 Ω : dist(x;@Ω) > "g: 1 Then for any u 2 L (Ω), we define for any x 2 Ω" Z Z −n x − y u ∗ ρ"(x) := u(y)ρ"(x − y) dy = " u(y)ρ dy: Ω Ω " Mollification, part II Similarly, for any vector Radon measure µ = (µ1; : : : ; µN ) on Ω, we define Z µ ∗ ρ"(x) = ρ"(x − y) dµ(y); x 2 Ω": Ω 1 We can show that µ ∗ ρ" 2 C (Ω") and r(µ ∗ ρ") = µ ∗ rρ": If v 2 Liploc(Ω), then r(v ∗ ρ") = rv ∗ ρ". Also, given any v 2 L1(Ω) and a vector Radon measure µ on Ω, from Fubini's theorem it follows easily that Z Z n (µ ∗ ρ")v dL = v ∗ ρ" dµ Ω Ω n if either µ is concentrated in Ω" or v = 0 L -a.e. outside Ω". Mollification of BV functions If u 2 BV(Ω), we have r(u ∗ ρ") = Du ∗ ρ" in Ω": 1 To see this, let 2 Cc (Ω) and " 2 (0; dist(supp ; @Ω)). Then Z Z Z n n n (u ∗ ρ")r dL = u(ρ" ∗ r ) dL = ur(ρ" ∗ ) dL Ω Ω Ω Z Z n = − ρ" ∗ dDu = − Du ∗ ρ" dL : Ω Ω n If u 2 BV(R ) and Du = 0, u is constant, which we can see 1 n as follows. For any ", u ∗ ρ" 2 C (R ) and r(u ∗ ρ") = Du ∗ ρ" = 0. Thus u ∗ ρ" is constant for every 1 n " > 0, and since u ∗ ρ" ! u in L (R ) as " ! 0, we must have that u is constant. Lipschitz test functions Let u 2 BV(Ω) and 2 Lipc (Ω). Then for small enough " > 0 we 1 have ∗ ρ" 2 Cc (Ω), and thus Z Z @( ∗ ρ") n u dL = − ∗ ρ" dDi u; i = 1;:::; n: Ω @xi Ω As " ! 0, we have ∗ ρ" ! uniformly and @( ∗ ρ") @ @ = ∗ ρ" ! @xi @xi @xi almost everywhere, so by the Lebesgue dominated convergence theorem we get Z Z @ n u dL = − dDi u; i = 1;:::; n: Ω @xi Ω That is, we can also use Lipschitz functions as test functions in the definition of BV. A Leibniz rule Lemma If φ 2 Liploc(Ω) and u 2 BVloc(Ω), we have uφ 2 BVloc(Ω) with D(uφ) = φ Du + urφ Ln. Proof. 1 1 Clearly uφ 2 Lloc(Ω). We have for any 2 Cc (Ω) and i = 1;:::; n Z @ Z @(φψ) Z @φ u φ dx = u dx − u dx Ω @xi Ω @xi Ω @xi Z Z @φ = − φ dDi u − u dx; Ω Ω @xi since φψ 2 Lipc (Ω). The variation, part I Definition 1 Let u 2 Lloc(Ω). We define the variation of u in Ω by Z n 1 n V (u; Ω) := sup u div dL : 2 [Cc (Ω)] ; j j ≤ 1 : Ω Theorem Let u 2 L1(Ω). Then u 2 BV(Ω) if and only if V (u; Ω) < 1. In addition, V (u; Ω) = jDuj(Ω). 1 n Proof. Let u 2 BV(Ω). Then for any 2 [Cc (Ω)] with j j ≤ 1, Z n Z n Z n X X u div dL = − i dDi u = − i σi djDuj Ω i=1 Ω i=1 Ω with jσj = 1 jDuj-almost everywhere, so that V (u; Ω) ≤ jDuj(Ω). The variation, part II Assume then that V (u; Ω) < 1. By homogeneity we have Z n 1 u div dL ≤ V (u; Ω)k kL1(Ω) 8 2 Cc (Ω): Ω 1 Since Cc (Ω) is dense in Cc (Ω) and thus in C0(Ω) (which is just the closure of Cc (Ω), in the k · kL1(Ω)-norm) we can find a continuous linear functional L on C0(Ω) coinciding with Z n 7! u div dL Ω 1 on Cc (Ω) and satisfying kLk ≤ V (u; Ω). Then the Riesz n representation theorem says that there exists an R -valued Radon measure µ = (µ1; : : : ; µn) with jµj(Ω) = kLk and n Z X n L( ) = i dµi 8 2 [C0(Ω)] : i=1 Ω The variation, part III Hence we have Z n Z n X 1 n u div dL = i dµi 8 2 [Cc (Ω)] ; Ω i=1 Ω so that u 2 BV(Ω), Du = −µ, and jDuj(Ω) = jµj(Ω) = kLk ≤ V (u; Ω): Lower semicontinuity Lemma 1 If uh ! u in Lloc(Ω), then V (u; Ω) ≤ lim infh!1 V (uh; Ω).
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