Appendix A Total Variation, Compactness, etc. I hate T.V. I hate it as much as peanuts. But I can’t stop eating peanuts. Orson Welles, The New York Herald Tribune (1956) A key concept in the theory of conservation laws is the notion of total varia- tion,T.V. (u), of a function u of one variable. We define T.V. (u):=sup u (xi) u (xi 1) . (A.1) | − − | i X We will also use the notation u := T.V. (u). The supremum in (A.1)is | |BV taken over all finite partitions xi such that xi 1 <xi. The set of all func- tions with finite total variation{ on}I we denote by− BV (I). Clearly, functions in BV (I) are bounded. We shall omit explicit mention of the interval I if (we think that) this is not important, or if it is clear which interval we are referring to. For any finite partition x we can write { i} u (x ) u (x ) = max (u (x ) u (x ) , 0) | i+1 − i | i+1 − i i i X X min (u (x ) u (x ) , 0) − i+1 − i i X =: p + n. Then the total variation of u can be written T.V. (u)=P + N := sup p +supn. (A.2) We call P the positive, and N the negative variation, of u. If for the moment we consider the finite interval I =[a, x], and partitions with a = x < < 1 ··· xn = x, we have that px nx = u(x) u(a), a − a − 451 452 A Total Variation, Compactness, etc. x x where we write pa and na to indicate which interval we are considering. Hence px N x + u(x) u(a). a a − Taking the supremum on the left-hand side we obtain P x N x u(x) u(a). a − a − Similarly, we have that N x P x u(a) u(x), and consequently a − a − u(x)=P x N x + u(a). (A.3) a − a In other words, any function u(x)inBV can be written as a di↵erence be- tween two increasing functions,1 u(x)=u+(x) u (x), (A.4) − − x x where u+(x)=u(a)+Pa and u (x)=Na . Let ⇠j denote the points where u is discontinuous. Then we have− that u(⇠ +) u(⇠ ) T.V. (u) < , | j − j− | 1 j X and hence we see that there can be at most a countable set of points where u(⇠+) = u(⇠ ). Observe6 that− functions with finite total variation are bounded, as u(x) u(a) + u(a) u(x) u(a) +T.V. (u) . | || | | − || | Equation (A.3) has the very useful consequence that if a function u in BV is also di↵erentiable, then u0(x) dx =T.V. (u) . (A.5) | | Z This equation holds, since d x d x u0(x) dx = P + N dx = P + N =T.V. (u) . | | dx a dx a Z Z ⇣ ⌘ We can also relate the total variation with the shifted L1-norm. Define λ(u, ")= u(x + ") u(x) dx. (A.6) | − | Z If λ(u, ") is a (nonnegative) continuous function in " with λ(u, 0) = 0, we say that it is a modulus of continuity for u. More generally, we will use the 1 This decomposition is often called the Jordan decomposition of u. ATotalVariation,Compactness,etc. 453 name modulus of continuity for any continuous function λ(u, ") vanishing at 2 p " =0 such that λ(u, ") u( + ") u p,where p is the L -norm. We will need a convenient characterizationk · − ofk total variationk·k (in one variable), which is described in the following lemma. Lemma A.1. Let u be a function in L1(R).Ifλ(u, ")/ " is bounded as a function of ", then u is in BV and | | λ(u, ") T.V. (u)=lim . (A.7) " 0 " ! | | Conversely, if u is in BV , then λ(u, ")/ " is bounded, and thus (A.7) holds. In particular, we shall frequently use | | λ(u, ") " T.V. (u) (A.8) | | if u is in BV . Proof. Assume first that u is a smooth function. Let xi be a partition of the interval in question. Then { } xi xi u(x + ") u(x) u (xi) u (xi 1) = u0(x) dx lim − dx. − " 0 " | − | xi 1 ! xi 1 Z − Z − Summing this over i we get λ(u, ") T.V. (u) lim inf " 0 " ! | | for di↵erentiable functions u(x). Let u be an arbitrary bounded function in 1 L , and uk be a sequence of smooth functions such that uk(x) u(x) for almost all x, and u u 0. The triangle inequality shows that! k k − k1 ! λ (u ,") λ (u, ") 2 u u 1 0. | k − | k k − kL ! Let x be a partition of the interval. We can now choose u such that { i} k uk (xi)=u (xi) for all i.Then λ(uk,") u (xi) u (xi 1) lim inf . − " 0 | − | ! " X | | Therefore, λ(u, ") T.V. (u) lim inf . " 0 " ! | | Furthermore, we have 2 This is not an exponent, but a footnote! Clearly, λ(u, ")isamodulusofcontinuity if and only if λ(u, ")=o (1) as " 0. ! 454 A Total Variation, Compactness, etc. j" u(x + ") u(x) dx = u(x + ") u(x) dx | − | (j 1)" | − | Z j Z − X " = u(x + j") u(x +(j 1)") dx | − − | j Z0 X" = u(x + j") u(x +(j 1)") dx | − − | Z0 j " X T.V. (u) Z0 = " T.V. (u) . | | Thus we have proved the inequalities λ(u, ") λ(u, ") λ(u, ") T.V. (u) lim inf lim sup T.V. (u) , (A.9) " " 0 " " 0 " | | ! | | ! | | which imply the lemma. ut Observe that we trivially have λ˜(u, "):= sup λ(u, σ) " T.V. (u) . (A.10) σ " | | | || | For functions in Lp care has to be taken as to which points are used in the supremum, since these functions in general are not defined pointwise. The right choice here is to consider only points xi that are points of approximate continuity 3 of u. Lemma A.1 remains valid. We measure the variation in the case of a function u of two variables u = u(x, y) as follows T.V.x,y (u)= T.V.x (u)(y) dy + T.V.y (u)(x) dx. (A.11) Z Z The extension to functions of n variables is obvious. We include a useful characterization of total variation. Definition A.2. Let ⌦ Rn be an open subset. We define the set of all functions with finite total✓ variation with respect to ⌦ as follows: 1 BV (⌦)= u Lloc(⌦) sup u(x)divφ(x) dx < . { 2 | 1 n 1} φ C0 (⌦;R ), φ 1 ⌦ 2 k k1 Z 3 Afunctionu is said to be approximately continuous at x if there exists a measurable set A such that limr 0 [x r, x + r] A / [x r, x + r] =1(here B denotes the measure of the set B!), and| u−is continuous\ | at| x−relative to| A.(EveryLebesguepoint| | is a point of approximate continuity.) The supremum (A.1)isthencalledtheessential variation of the function. However, in the theory of conservation laws it is customary to use the name total variation in this case, too, and we will follow this custom here. ATotalVariation,Compactness,etc. 455 For u BV (⌦)wewrite 2 Du =sup u(x)divφ(x) dx, k k 1 n φ C0 (⌦;R ), φ 1 ⌦ 2 k k1 Z and u BV (⌦) L1(⌦)wedefine 2 \ u = u 1 + Du . k kBV k kL (⌦) k k Remark A.3. If u is integrable with weak derivatives that are integrable func- tions, we clearly have Du = u(x) dx. k k |r | Z In one space dimension there is a simple relation between Du and T.V. (u) as the next theorem shows. k k Theorem A.4. Let u be a function in L1(I) where I is an interval. Then T.V. (u)= Du . (A.12) k k Proof. Assume that u has finite total variation on I. Let ! be a nonnegative function bounded by unity with support in [ 1, 1] and unit integral. Define − 1 x ! (x)= ! , " " " ⇣ ⌘ and u" = ! u. (A.13) " ⇤ Consider points x <x < <x in I.Then 1 2 ··· n " " u (xi) u (xi 1) | − − | i X " !"(x) u(xi x) u(xi 1 x) dx | − − − − | " i Z− X T.V. (u) . (A.14) Using (A.5) and (A.14) we obtain " " (u )0(x) dx =T.V. (u ) | | Z " " =sup u (xi) u (xi 1) | − − | i X T.V. (u) . Let φ C1 with φ 1. Then 2 0 | | 456 A Total Variation, Compactness, etc. " " u (x)φ0(x) dx = (u )0(x)φ(x) dx − Z Z " (u )0(x) dx | | Z T.V. (u) , which proves the first part of the theorem. Now let u be such that Du := sup u(x)φx(x) dx < . k k φ C1 1 0 Z φ2 1 | | First we infer that " " (u )0(x)φ(x) dx = u (x)φ0(x) dx − Z Z = (! u)(x)φ0(x) dx − " ⇤ Z = u(x)(! φ)0(x) dx − " ⇤ Z Du . k k Using that (see Exercise A.1) f L1(I) =sup f(x)φ(x) dx, k k φ C1(I), 0 Z 2φ 1 | | we conclude that " (u )0(x) dx Du . (A.15) | | k k Z Next we show that u L1. Choose a sequence uj BV C1 such that (see, e.g., [62, p. 172]) 2 2 \ u u a.e., u u 0,j , (A.16) j ! k j − kL1 ! !1 and u0 (x) dx Du ,j .