GIOVANNI COMI AND MONICA TORRES

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1 ONE-SIDED APPROXIMATION OF SETS OF FINITE PERIMETER GIOVANNI COMI AND MONICA TORRES Abstract. In this note we present a new proof of a one-sided approximation of sets of finite perimeter introduced in [2], in order to fill a gap in the original proof. 1. Introduction It is a classical result in geometric measure theory that a set of finite perimeter E can be approximated with smooth sets E such that (1.1) L N (E ) L N (E) and P (E ) P (E), where P (E) is the perimeter of E and L N is the Lebesgue measure in R N. The approximating smooth sets (see for instance Ambrosio-Fusco-Pallara [1, Remar 3.42] and Maggi [6, Theorem 13.8]) are the superlevel sets of the convolutions of χ E, which can be chosen for a.e. t (0, 1). The one-sided approximation refines the classical result in the sense that it distinguishes between the superlevel sets for a.e. t ( 1 2, 1) from the ones corresponding to a.e. t (0, 1 2 ), thus providing an interior and an exterior approximation of the set respectively (see Theorem 3.1 and Theorem 4.1). Indeed, in the first case, the difference between the level sets and the measure theoretic interior is asymptotically vanishing with respect to the H N 1 -measure; in the latter, we obtain the same result for the measure theoretic exterior. The main aim of this note is to fill a gap in the original proof of the main approximation Theorem 4.1 (see [2, Theorem 4.10]). Thus, the results of this note are not only interesting by themselves, but they also validate the application of Theorem 4.1 to the construction of interior and exterior normal traces of essentially bounded divergence-measure fields. This construction was developed in [2] and it was motivated by the study of systems of hyperbolic conservation laws with Lax entropy condition, where these vector fields naturally appear. 2. preliminaries In what follows we will wor in R N. We introduce now a few basic definitions and results on the theory of functions of bounded variations and sets of finite perimeter, for which we refer mainly to [1], [4] and [6] (see also [5] and [7]). Definition 2.1. A function u L 1 (R N ) is called a function of bounded variation if Du is a finite R N -vector valued Radon measure on R N. A measurable set E R N is called a set of finite perimeter in R N (or a Caccioppoli set) if χ E BV (R N ). Consequently, Dχ E is an R N -vector valued Radon measure on R N whose total variation is denoted as Dχ E. By the polar decomposition of measures, we can write Dχ E = ν E Dχ E, where ν E is a Dχ E - measurable function such that ν E (x) = 1 for Dχ E -a.e. x R N. We define the perimeter of E as { } P (E) := sup div(ϕ) dx : ϕ Cc 1 (R N ; R N ), ϕ 1 E Date: June Key words and phrases. sets of finite perimeter, one-sided approximation, tangential properties of the reduced boundary. 1

2 2 GIOVANNI COMI AND MONICA TORRES and it can be proved that P (E) = Dχ E (R N ). The notion of perimeter generalizes the idea of H N 1 -measure of the boundary of the set E. It is a well-nown fact that the topological boundary of a set of finite perimeter can be very irregular, it can even have full Lebesgue measure. This suggests that for a set of finite perimeter is interesting to consider subsets of E instead. In [3], De Giorgi considered a set of finite H N 1 -measure on which Dχ E is concentrated, which he called reduced boundary. Definition 2.2. We say that x E, the reduced boundary of E, if (1) Dχ E (B(x, r)) > 0, r > 0; 1 (2) lim ν E (y) d Dχ E (y) = ν E (x); r 0 Dχ E (B(x, r)) B(x,r) (3) ν E (x) = 1. It can be shown that this definition implies a geometrical characterization of the reduced boundary, by using the blow-up of the set E around a point of E. Theorem 2.3. If x E, then and E x ε (R N \ E) x ε H + ν E (x) := {y R N : y ν E (x) 0} in L 1 loc(r N ) as ε 0 H ν E (x) := {y R N : y ν E (x) 0} in L 1 loc(r N ) as ε 0. The proof can be found in [4, Section 5.7.2, Theorem 1]. Formulated in another way, for ε > 0 small enough, E B(x, ε) is asymptotically close to the half ball H ν E (x) B(x, ε). Because of this result, we call ν E (x) measure theoretic unit interior normal to E at x E, since it is a generalization of the concept of unit interior normal. In addition, De Giorgi proved that Dχ E = H N 1 E, so that Dχ E = ν E H N 1 E and P (E) = H N 1 ( E) (see [4, Section 5.7.3, Theorem 2]). For every α [0, 1] we set E α := {x R N : D(E, x) = α}, where B(x, r) E D(E, x) := lim, r 0 B(x, r) and we give the following definitions: (1) E 1 is called the measure theoretic interior of E. (2) E 0 is called the measure theoretic exterior of E. We recall (see Maggi [6, Example 5.17]) that every Lebesgue measurable set is equivalent to the set of its points of density one; that is, (2.1) L N (E E 1 ) = L N ((R N \ E) E 0 ) = 0. It is also a well-now result due to Federer that there exists a set N with H N 1 (N ) = 0 such that R N = E 1 E E 0 N (see [1, Theorem 3.61]). The perimeter P (E) of E is invariant under modifications by a set of L N -measure zero, even though these modifications might largely increase the size of the topological boundary. In this paper we consider the following representative (2.2) E := E 1 E. Given a smooth nonnegative radially symmetric mollifier ρ Cc (B 1 (0)), we denote the mollification of χ E by u (x) := (χ E ρ ε )(x) for some positive sequence ε 0. We define, for t (0, 1), (2.3) A ;t := {u > t}.

3 ONE-SIDED APPROXIMATION OF SETS OF FINITE PERIMETER 3 By Sard s theorem 1, we now that, since u : R N R is C, L 1 -a.e. t (0, 1) is not the image of a critical point for u and so A ;t has a smooth boundary for these values of t. Thus, for each there exists a set Z (0, 1), with L 1 (Z ) = 0, which is the set of values of t for which A ;t has not a smooth boundary. If we set Z := + =1 Z, then L 1 (Z) = 0 and, for each t (0, 1) \ Z and for each, A ;t has a smooth boundary. It is a well-nown result from BV theory (see for instance [1, Corollary 3.80]) that every function of bounded variations u admits a representative which is the pointwise limit H N 1 -a.e. of any mollification of u and which coincides H N 1 -a.e. with the precise representative u : 1 lim u(y) dy if this limit exists u (x) := r 0 B(x, r) B(x,r). 0 otherwise For any set of finite perimeter E, we denote the precise representative of the function χ E by u E, which is given by 1, x E 1 u E (x) = 0, x E , x E Since H N 1 (R N \ (E 1 E E 0 )) = 0, the function u E is well defined H N 1 -a.e.. In order to prove Theorem 4.1, we need to use the classical coarea formula, for which we refer to [4, Section 3.4, Theorem 1]. Theorem 2.4. Let u : R N R be Lipschitz. Then, for any L N -measurable set A, we have (2.4) u dx = H N 1 (A u 1 (t)) dt. A R 3. The approximation of E with respect to any µ << H N 1 The one-sided approximation theorem allows to extend (1.1) to any Radon measure µ such that µ << H N 1. More precisely, for any bounded set of finite perimeter E, there exist smooth sets E ;i, E ;e, such that (3.1) µ(e ;i ) µ(e 1 ), P (E ;i ) P (E) and (3.2) µ(e ;e ) µ(e), P (E ;e ) P (E). The convergence of the perimeters in (3.1) and (3.2) follows as in the standard proof of (1.1). However, the convergence with respect to µ is a consequence of the following result. Theorem 3.1. Let µ be a Radon measure such that µ << H N 1 and E be a bounded set of finite perimeter in R N. Then: (a) µ (E 1 A ;t ) 0, for 1 2 < t < 1; (b) µ (E A ;t ) 0, for 0 < t < 1 2. Proof. We have (3.3) u (x) u E (x) for H N 1 -a.e. x. Since {0 < u u E 1} E δ := {x R N : dist(x, E) δ}, for any if δ > max ε, and E δ is bounded, then we can apply the dominated convergence theorem with respect to the measure µ, 1 For which we refer to [6, Lemma 13.15].

4 4 GIOVANNI COMI AND MONICA TORRES taing 1 as summable majorant since µ is a Radon measure. Hence, for any ε > 0, there exists large enough such that, if 1 2 < t < 1, we have ε u (x) u E (x) d µ R n u (x) u E (x) d µ + u E (x) u (x) d µ A ;t \E 1 E 1 \A ;t (t 1 2 ) µ (A ;t \ E 1 ) + (1 t) µ (E 1 \ A ;t ) min {t 12 }, 1 t µ (A ;t E 1 ). Thus, for large enough and 1 2 < t < 1, we obtain µ (A ;t E 1 ε ) min { t 1 2, 1 t}, which is (a). Analogously, for 0 < t < 1 2, we have ε u (x) u E (x) d µ R n u (x) u E (x) d µ + u E (x) u (x) d µ A ;t \E E\A ;t t µ (A ;t \ E) + ( 1 2 t) µ (E \ A ;t) min {t, 12 } t µ (A ;t E). Thus, for large and 0 < t < 1 2, which gives (b). µ (A ;t E) ε min { t, 1 2 t}, Remar 3.2. The convergence in (3.1) follows easily from Theorem 3.1: we have and it is clear that (a) implies µ(e 1 ) µ(a ;t ) = µ(e 1 \ A ;t ) µ(a ;t \ E 1 ) µ(e 1 \ A ;t ) µ (E 1 \ A ;t ) 0, µ(a ;t \ E 1 ) µ (A ;t \ E 1 ) 0. One can show (3.2) in a similar way using (b). We also notice that Theorem 3.1 has been proved for any t (0, 1 2 ) ( 1 2, 1). However, since the sets A ;t have smooth boundary only for almost every t, we shall consider only t / Z, where Z is the set of singular values defined in the introduction. Remar 3.3. With µ = H N 1 E, we obtain from Theorem 3.1: (a) H N 1 ( E A ;t ) 0 for 1 2 < t < 1; (b) H N 1 ( E (R N \ A ;t )) 0 for 0 < t < 1 2. Indeed, this is clear from the following identities E (E 1 A ;t ) = E [(E 1 \ A ;t ) (A ;t \ E 1 )] = E A ;t, E (E A ;t ) = E [(E \ A ;t ) (A ;t \ E)] = E (R N \ A ;t ).

5 ONE-SIDED APPROXIMATION OF SETS OF FINITE PERIMETER 5 Remar 3.4. Using Remar 3.3 we can also show that we have: (a) H N 1 ( E u 1 (t)) 0 for 1 2 < t < 1; (b) H N 1 ( E u 1 (t)) 0 for 0 < t < 1 2. Indeed, u 1 (t) A ;s for 1 2 < s < t < 1 and u 1 (t) RN \ A ;s for 0 < t s < 1 2. In addition, we observe that µ (u 1 (t)) = 0 for L1 -a.e. t, since µ is a Radon measure. It is in fact clear that u 1 (t) = A ;t, that A ;t A ;s A ;0 if 0 < s < t < 1, with A ;0 bounded, and that the sets A ;t are pairwise disjoint. Hence, since µ is finite on bounded sets and additive, the set {t (0, 1) : µ ( A ;t ) > ε} is finite for any ε > 0. This implies that the set {t (0, 1) : µ ( A ;t ) > 0} is at most countable (see also the observation at the end of Section 1.4 of [1]). Then we obtain also: (a) H N 1 ( E u 1 (t)) = 0 for a.e. 1 2 < t < 1; (b) H N 1 ( E u 1 (t)) = 0 for a.e. 0 < t < The main approximation result The following theorem, together with Theorem 3.1, shows that indeed we have an interior approximation of E for a.e. t ( 1 2, 1). Theorem 4.1. Let E be a set of finite perimeter in R N. There exists a sequence ε converging to 0 such that, if u := χ E ρ ε, we have (4.1) lim + HN 1 (u 1 (t) \ E1 ) = 0 for a.e. t ( 1 2, 1). Proof. We tae s > 1 2 and a sequence ε, with ε 0, and we consider the set A ;s := {u > s}. By the coarea formula (2.4), we have (4.2) A ;s \E 1 u dx = = s H N 1 (u 1 (t) (A ;s \ E 1 )) dt H N 1 (u 1 (t) \ E1 ) dt, since, for t s, u 1 (t) (A ;s \ E 1 ) =, while, for t > s, u 1 (t) (A ;s \ E 1 ) = u 1 (t) \ E1. We claim that (4.3) u L1 (A ;s \E 1 ) 0. In order to prove the claim, we observe that, for any x R N, u (x) = χ E (y) x ρ ε (x y) dy = χ E (y) y ρ ε (x y) dy R N R N = ρ ε (x y)ν E (y) d Dχ E (y) = (ρ ε Dχ E )(x). R N Hence, u = (Dχ E ρ ε ) = ( Dχ E ν E ρ ε ), which implies (4.4) u Dχ E ρ ε.

6 6 GIOVANNI COMI AND MONICA TORRES Recalling from (2.1) that L N (E E 1 ) = 0, (4.4) leads to u L 1 (A ;s \E 1 ) R = u χ A;s \E dx N ( Dχ E ρ ε ) χ A;s \E dx = (ρ ε χ A;s \E) d Dχ E = R N R N = (ρ ε χ A;s \E) dh N 1. E Thus, we need to investigate, for any x E, the behaviour of (ρ ε χ A;s \E)(x) as +. We have ( ) x y (ρ ε χ A;s \E)(x) = ε N ρ χ A;s (y) χ R ε (R N \E)(y) dy N = [y = x + ε z] = ρ(z) χ A;s (x + ε z) χ (R N \E)(x + ε z) dz. We observe that x + ε z R N \ E if and only if z (RN \E) x ε, hence χ (R N \E)(x + ε ) = χ (R N \E) x ( ) χ H νe (x) ( ) in L1 (B 1 (0)) as +. ε In particular, this means that the L 1 limit of χ (RN \E)(x + ε z) is not L N -a.e. zero only if z ν E (x) 0, so we can restrict the integration domain to B 1 (0) H ν E (x). On the other hand, x + ε z A ;s = {u > s} if and only if u (x + ε z) > s. We see that u (x + ε z) = ρ ε (x + ε z y) χ E (y) dy R N = [y = x + ε z + ε u] = ρ(u) χ E (x + ε (u + z)) du. Arguing as before, we obtain χ E (x+ε (z + )) χ H + νe (x) (z + ) in L1 (B 1 (0)) as +, for any x E and z B 1 (0). Now, we recall that z ν E (x) 0, and, since we have χ H + νe (x)(z + u) = 1 if and only if 0 (z + u) ν E (x), we conclude that 0 z ν E (x) u ν E (x) 1; that is, u belongs to the half ball B 1 (0) H ν + E (x). This implies that, for any x E and z B 1 (0) Hν E (x), (4.5) lim u (x + ε z) := v(x, z) = ρ(u) χ + H + νe (x) (z + u) du 1 2. (4.6) Therefore, these calculations yield (ρ ε χ A;s \E)(x) = ρ(z) χ A;s (x + ε z) χ (R N \E)(x + ε z) dz ρ(z) χ {v(x,z)>s} (z) χ H νe (x)(z) dz, for any x E. Equation (4.5) shows then that the limit in (4.6) is identically zero, since { z R N : v(x, z) > s > 1 } B 1 (0) Hν 2 E (x) =, for any x E. We can now apply the Lebesgue dominated convergence theorem with respect to the measure H N 1 E and the sequence of functions ρ ε χ A;s \E (since the constant 1 is clearly a summable majorant), thus obtaining (4.3).

7 ONE-SIDED APPROXIMATION OF SETS OF FINITE PERIMETER 7 Finally, up to passing to a subsequence (which we shall eep calling ε with a little abuse of notation), (4.2) and (4.3) yield (4.1), for a.e. t > s. Since s > 1 2 is fixed arbitrarily, we can conclude that (4.1) is valid for a.e. t > Remar. An analogous result holds for the measure theoretic exterior; namely, there exists a sequence ε converging to 0 such that, if u := χ E ρ ε, we have (4.7) lim + HN 1 (u 1 (t) \ E0 ) = 0 for a.e. t (0, 1 2 ) Remar. It is not difficult to see that Theorem 3.1 and Theorem 4.1 are also valid if we wor in an open set Ω and we consider a set of finite perimeter E Ω (which is the framewor of [2]). Indeed, the extension to 0 in R N \ Ω of the function χ E is a function of bounded variations in R N ; that is, E can be seen as a set of finite perimeter in R N. In this way, the previous results follow. Acnowledgments The authors would lie to than Prof. Luigi Ambrosio for useful discussions. References [1] 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 Yor, [2] G.-Q. Chen, M. Torres and W. P. Ziemer. Gauss-Green Theorem for Wealy Differentiable Vector Fields, Sets of Finite Perimeter, and Balance Laws. Communications on Pure and Applied Mathematics, 62(2): , [3] E. De Giorgi. Nuovi teoremi relativi alle misure (r 1)-dimensionali in uno spazio a r dimensioni. Ricerche Mat., 4 (1955), [4] L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press: Boca Raton, FL, [5] E. Giusti. Minimal Surfaces and Functions of Bounded Variation. Birhäuser Verlag, Basel, [6] F. Maggi. Sets of finite perimeter and geometric variational problems. Cambridge, [7] W. P. Ziemer. Wealy Differentiable Functions. Springer-Verlag, New Yor, [G. Comi] Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy address: giovanni.comi@sns.it [M. Torres]Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN , USA. torres address: torres@math.purdue.edu