STRICT INTERIOR APPROXIMATION OF SETS OF FINITE PERIMETER AND FUNCTIONS OF BOUNDED VARIATION

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1 STRICT INTERIOR APPROXIMATION OF SETS OF FINITE PERIMETER AND FUNCTIONS OF BOUNDED VARIATION THOMAS SCHMIDT Abstract. It is well nown that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate an open set of finite perimeter in R n strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the n 1)-dimensional Hausdorff measure of the topological boundary equals the perimeter of. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation of BV-functions from a prescribed Dirichlet class. 1. Introduction and statement of the results For arbitrary n N, the perimeter P) of a measurable set in R n is defined as { } P) := sup divϕdl n : ϕ C cpt Rn, R n ), sup ϕ 1 [0, ] R n with the n-dimensional Lebesgue measure L n ). As a consequence of the divergence theorem, this definition generalizes the more classical notions of perimeter, and in particular, for sets with smooth boundary, one has the identity P) = H n 1 ) with the n 1)-dimensional Hausdorff measure H n 1. Even for non-smooth sets, the above notion of perimeter is very reasonable and useful, but in general one only has the inequality see [1, Proposition 3.62]) 1.1) P) H n 1 ). Though a set with finite perimeter P) < in R n is in several regards well-behaved, one often needs to approximate with smooth sets in such a way that also the perimeters converge, and indeed it is very well nown see [1, Theorem 3.42]) that this is always possible: one can find open sets ε with smooth boundaries in R n such that ε converges to in measure and 1.2) H n 1 ε ) = P ε ) converges to P) for ε ց 0. In this paper we are concerned with strict interior approximation: that means, we investigate whether, for an open, one can additionally tae 1.3) ε. For bounded Lipschitz domains, this requirement can be achieved by the constructions described in [18, 17, 9]. For more general, in contrast, some related problems have been considered in [20, 23, 24, 19, 21, 22], but it seems that approximations with 1.2) and 1.3) have not been found. The main obstacle in this regard is that the analogous approximation is impossible for arbitrary bounded open sets with finite perimeter in R n : indeed, when one Date: November 5, Mathematics Subject Classification. Primary 28A75, 26B30, 41A63, 41A30; Secondary 28A78, 26B15. The research leading to these results has received funding from the European Research Council under the European Community s Seventh Framewor Programme FP7/ ) / ERC Grant agreement GeMeThnES n

2 2 THOMAS SCHMIDT considers = 0,1) n 1 [ 0,1) \ { 1 2 }], then it follows using the lower semicontinuity [1, Proposition 3.38b)] of the perimeter on both halves of ) that for all approximations ε one necessarily has liminf εց0 P ε ) 2n+2, but H n 1 ) = 2n+1 and P) = 2n are strictly smaller. Here, we will show that these and similar examples are ruled out and strict interior approximation is again possible if one only imposes the mild extra assumption 1.4) below. This is actually the first main result of the present paper, which will be proved in Section 3: Theorem 1.1 strict interior approximation of the perimeter). Suppose that is a bounded open set in R n whose topological boundary is well-behaved in the sense of 1.4) H n 1 ) = P). Then, for every ε > 0, there exists an open set ε with smooth boundary in R n and with 1.5) ε, \ ε N ε ) N ε ε ), P ε ) P)+ε, where we have used the notation N ε ) for ε-neighborhoods of sets in R n. Several comments on this approximation theorem follow. First of all, we remar that boundedness of is not essential, but is just imposed in order to simplify our statements and proofs. In fact, the extension of Theorem 1.1 and of Theorem 1.2 stated below) to unbounded is discussed in [5, Section 3.3]. Furthermore, by De Giorgi s structure theorem [1, Theorem 3.59] combined with [1, Proposition 3.6]), whenever the perimeter P) is finite, then one has equality 1.6) P) = H n 1 F) with the measure of the reduced boundary F of. In this case, we can thus rephrase the hypothesis 1.4) also taing into account that F holds by definition as the requirement 1.7) H n 1 \F) = 0. In this light, 1.4) yields a measure-theoretic control on the topological boundary, and it seems plausible that such a condition may be relevant in order to incorporate the topological condition 1.3) into the measure-theoretic theory of the perimeter. A more detailed discussion of 1.4), including the proof of a restricted optimality property, is postponed to Section 5. Turning from the hypothesis of Theorem 1.1 to its conclusion, we observe that 1.5) implies ε>0 ε = and lim εց0 1 ε 1 L 1 R n ) = lim εց0 L n \ ε ) = 0. Via the lower semicontinuity of the perimeter and the smoothness of ε ), we thus infer that 1.2) is valid for the approximations of the theorem. Furthermore, whenever P) is finite, it equals the total variation D1 R n ) of the gradient measure of 1, so that the approximating sequence 1 1/ ) N converges strictly in BVR n ) to 1 in the sense of [1, Definition 3.14]). In addition, we also record that the first two assertions in 1.5) imply the bounds 1.8) d H ε,) ε and d H ε, ) ε in the Hausdorff distance 1 d H. Last but not least, let us emphasize that the precise form of the last condition in 1.5) is significant; indeed, if one requires only the weaer bound 1.9) P ε ) CH n 1 ) with a certain dimensional constant C 1, ), then the existence of interior approximations ε with 1.9) can be concluded from a standard argument, which is based on a covering of 1 For non-empty, bounded subsets A and B of R n, the Hausdorff distance is defined by d H A,B) := max { sup a A dista,b), sup b B dista,b) } [0, ).

3 STRICT INTERIOR APPROXIMATION OF BV SETS AND FUNCTIONS 3 withsuitableballsandobtainsthe ε byremovingtheseballsfromcomparewith[1, Proofof Proposition 3.62]). The same argument has been used in connection with [7, Proposition 8.1], where a statement similar to 1.5) has been claimed, but as an inspection of the proof reveals only a statement of the type 1.9) has been established. Indeed, it seems that 1.5) cannot be obtained via a covering of with balls, but that rather coverings with suitably flat objects are needed. In the proof of Theorem 1.1 we will see that such refined coverings can be constructed, when one involves ideas from the proof of De Giorgi s structure theorem in order to first decompose into a negligible set and countably many locally flat-looing pieces. Next, for arbitrary n,n N, we turn to approximation results for R N -valued functions of bounded variation on an open subset of R n. We fix u 0 W 1,1 R n, R N ), for every u BV, R N ) we set { ux) for x ux) := u 0 x) for x R n \, and we define BV u0, R N ) := {u BV, R N ) : u BVR n, R N )}. We observethat BV ) u0, R N contains the Dirichlet classwu 1,1 0, R N ) := u 0 +W 1,1 0, RN ) and isclosed inthe wea- topologyofbv, R N ). Hence, BV ) u0, R N maybe seen asanatural replacement for Wu 1,1 0, R N ) in many regards, for instance when dealing with minimization problems for variational integrals in Dirichlet classes compare [5, Section 2.2]). If has sharp external cusps, then BV ) u0, R N is strictly smaller than BV, R N ), but whenever has a bounded Lipschitz boundary, then [1, Corollary3.89] yields BV ) u0, R N = BV, R N ); thus, in general, functions in BV ) u0, R N do not attain the boundary values of u 0 in any reasonable sense, but rather their derivative comprises some information about the deviation from these boundary values. We now focus onthe strictapproximationoffunctions u BV ) u0, R N from Wu 1,1 0, R N ). Whenever this approximation is possible for all functions u in the class, then BV ) u0, R N is in fact the sequential closure of Wu 1,1 0, R N ) in both the wea- and the strict topology of BV, R N ) andthus oftenthe smallestpossiblereplacement. Beforestatingourcorresponding result, we mention a variety of closely related approximation theorems in the literature: first, the simpler case of functions u which do attain 2 the boundary values u 0 on is treated in [12, Remar 2.12], [6, Lemma B.1], [14, Lemma 1] and does not depend on the regularity of ; generalized boundary conditions are considered in [3, Lemma 5.1] and [13, Lemma A.2] under a C 2 -assumption on, and in [16, Theorem 3.1] for u 0 0 and Lipschitz boundaries; finally, [2, Fact 3.3] deals with arbitrary generalized boundary data on bounded Lipschitz domains, but it seems that a complete proof in this generality has only been published in [6, Lemma B.2]. Our second main result, which will be established in Section 4, extends all these statements to cases with possibly irregular boundaries as follows. Theorem 1.2 strict approximation of BV-functions from a given Dirichlet class). Suppose that is a bounded open set in R n with H n 1 ) = P) <. Then, for u 0 W 1,1 R n, R N ), u BV u0, R N ), and every ε > 0 there exists a with 1.10) u ε u 0 +C cpt, RN ) u ε u dl n ε and L n,du ε ) ) L n,du) ) +ε. 2 In our terminology, attainment of the boundary values means Du ) = 0; compare also [15, Theorem 1] for the construction of certain piecewise affine approximations in this situation.

4 4 THOMAS SCHMIDT Here, L n,du) denotes the R Nn+1 -valued measure on R n whose first component is L n and whose other components are given by the R Nn -valued gradient measure Du, and L n,du ε ) is understood analogously. Writing Du = u)l n +D s u for the Lebesgue decomposition of Du with respect to L n, the second inequality in 1.10) can thus be restated as 1+ uε 2 dl n 1+ u 2 dl n + D s u ) +ε. Finally, we stress that via the Reshetnya continuity theorem [1, Theorem 2.39] we can also achieve variants of 1.10), as for instance Du ε ) Du ) +ε. Acnowledgement. The author would lie to than Lisa Bec for several comments on a preliminary version of the manuscript. 2. Some notation and terminology We have tried to eep the notation of this paper as close as possible to the standard, and most of our terminology follows the one in the monographs [1, 12]. Nonetheless, we comment on some specific conventions: Elementary geometry. First of all, B x 0 ) is the open ball with center x and radius in R n, and we set ω n := L n B 1 0)) and ω 0 := 1). For a vector subspace T of R n and y R n, we write Ty) for the orthogonal projection of y on T, and we set T y) := y Ty). With this notation, the cone with axis T and opening parameter ε > 0 is defined as 2.1) C ε T) := {y R n : T y) ε Ty) }. Reduced boundaries. Following [1, Definition 3.54], we write F for the reduced boundary of any L n -measurable subset of R n, taen in the largest open set in which has locally finite perimeter. By ν x) we denote the generalized) exterior unit normal to at x F, and we set B ± x) := {y B x) : ±y ν x) 0} where the relevant is indicated by the context). Finally, Tan n 1 F,x) is the approximate tangent space at x F, that is an n 1 F x n 1)-dimensional subspace T of R n such that, for ց 0, the measures H locally wea- converge to H n 1 T on R n ; compare [1, Definition 2.79]. By [1, Theorem 3.59], Tan n 1 F,x) exists and equals the orthogonal complement of ν x) for x F. 3. Strict interior approximation of the perimeter We start this section with a lemma on the decomposition of F into flat pieces. Lemma 3.1. Consider an L n -measurable set in R n. For every ε > 0, the reduced boundary F can be decomposed into countably many disjoint Borel subsets R 1,R 2,R 3,... such that L n B + x)) ε 2 n for all 0, i] 1, x R i = R i B 1/i x) x+c ε Tan n 1 F,x)). Proof. For j N, we set inductively { H n 1 F B x)) 1 2 A j := x F : ωn 1 n 1 } j 1 L n B + x)) for all 1 ε 2 n j \ A m. Evidently the A j are disjoint, by Fubini s theorem they are Borel sets, and by De Giorgi s structure theorem [1, Theorem 3.59] their union is all of F. By a well-nown argument, taen from the proof of the structure theorem, one can moreover show that for every x A j there exists some r x > 0 depending also on ε) with 3.1) A j B rx x) x+c ε Tan n 1 F,x)). m=1

5 STRICT INTERIOR APPROXIMATION OF BV SETS AND FUNCTIONS 5 The required argument for 3.1) is described in the proof of [1, Theorem 2.83ii)], but for the sae of clarity we setch it in the following. Indeed, if 3.1) failed for arbitrarily small radii, then, abbreviating T := Tan n 1 F,x), we could find a sequence x l ) l N in A j \ [x+c ε T)] such that l := x l x converges to 0 for l. By a straightforward computation, we would get B χ lx l ) [x+c ε/2 T)] = for χ := ε/2+2ε), and in combination with the choice of A j, this would in turn give H n 1 F)B 1+χ) lx) \ [x+c ε/2 T)]) H n 1 F B χ lx l )) 1 2 ω n 1χ l) n 1 for l 1. Hence, for the rescalings l := x)/ l, we would have H n 1 F l )B 1+χ 0) \ C ε/2 T)) 1 2 ω n 1χ n 1 for l 1, and every local wea- limit of the measures H n 1 F l would have positive mass in B 1+χ 0)\C ε/2 T), but, by [1, Theorem 3.59b)], the limit measure is H n 1 T, which does not have this property. This contradiction completes the derivation of 3.1). Now the r x can moreover be taen 3 as Borel functions of x, and when we choose an enumeration j1),1)),j2),2)), j3),3)),... of N N with ji) i and i) i for all i N, then it is easy to see that the sets R i := {x A ji) : r x i) 1 }\R 1 R 2... R i 1 ) have the required properties. The following proof of Theorem 1.1 is essentially a special case of the argument used for Proposition 4.1 below. Nevertheless, we will first carry out the details in the more geometric and slightly less technical situation of the theorem. Proof of Theorem 1.1. We first observe that it suffices to construct arbitrary measurable ε with 1.5), not necessarily with smooth boundaries. Indeed, as soon as we achieve this, we can mollify the characteristic functions of these sets and employ the coarea formula in order to select smooth ε as superlevel sets of the mollifications. This reasoning is detailed in the proof of the classical approximation property [1, Theorem 3.42], and we omit further details. Furthermore, we claim that it suffices to establish the claim only with the weaer requirement 3.2) \ ε N ε ) instead of the second condition in 1.5). To see this, we assume that the weaer form of 1.5) holds for sets 1/ with N where 1/ replaces ε), and we show by an elementary contradiction argument that, for every given ε > 0, we can choose the desired ε as one of the 1/ with N. Indeed, if this were not possible, we would necessarily have \ 1/ N ε 1/ ) for all ε 1. Then, we could choose x \ 1/ ) \ N ε 1/ ), and a subsequenceofthe x wouldconvergetosomex with distx, 1/ ) > distx, 1/ ) ε/2 = distx, 1/ ) ε/2 ε/2 for 1 where the last two steps exploit x / 1/ and x / N ε 1/ ), respectively). Consequently, we could also find a y with disty, 1/ ) ε/2 for 1, but this would result in a contradiction, as we started with \ 1/ N 1/ ) and would thus have y 1/ for > disty, ) 1. In view of the preceding reduction steps, the case P) = can be concluded by the simple choice ε := {x : distx, ) > ε/2}, and in the following we assume P) <. We further wor with a given ε 1 2 and the corresponding sets R i of Lemma 3.1. By definition of 3 For instance, fixing a dense subset { m : m N} of 0, ), one can choose r x = sup m N r m x, where rm x equals m in the case A j B mx) x+c εtan n 1 F,x)) and equals 0 otherwise.

6 6 THOMAS SCHMIDT the Hausdorff measure 4, there exist x i,j R n and r i,j 0, ) with 3r i,j min { 1 i,ε}, R i B ri,j x i,j ), and j=1 j=1 ω n 1 r n 1 i,j H n 1 R i )+2 i ε. Evidently, we can assume R i B ri,j x i,j ) so that this intersection contains at least one point x i,j, for all i,j N. Abbreviating T i,j := Tan n 1 F, x i,j ) and taing 3r i,j 1 i into account, the assertions of Lemma 3.1 imply 3.3) L n B + 3r i,j x i,j )) 3 n ε 2 r n i,j, R i B ri,j x i,j ) R i B 1/i x i,j ) x i,j +C ε T i,j ). Recalling the notations introduced around 2.1), we infer that R i B ri,j x i,j ) is also contained in the quite flat) cylinder {y R n : T i,j y x i,j ) < r i,j and T i,jy x i,j ) < 2εr i,j }. We will slightly enlarge this cylinder in order to get control on a part of its boundary. To this end we tae some h i,j [2εr i,j,3εr i,j ] with H n 1 {y : T i,j y x i,j ) < r i,j and T i,jy x i,j ) = h i,j ν x i,j )} ) 3 n εr n 1 i,j. Such a choice is possible by 3.3) and a Fubini type argument, as in view of ε 1 2 relevant sets are indeed contained in B + 3r i,j x i,j ). We now introduce all the C i,j := {y R n : T i,j y x i,j ) < r i,j and T i,j y x i,j) < h i,j } B ε x i,j ), and record that top and bottom of C i,j have each H n 1 -measure ω n 1 r n 1 i,j, while the side has measure 2h i,j n 1)ω n 1 ri,j n 2. Involving also the preceding choice of h i,j, we get 3.4) H n 1 C i,j ) [ ω n 1 +3 n ε+6εn 1)ω n 1 ] r n 1 i,j. Now we exploit the reformulation 1.7) of 1.4), and we cover also \F by countably many balls B 1y 1 ),B 2y 2 ),B 3y 3 ),... with radii ε/2 such that B y ) and =1 n 1 ε. Since the C i,j cover F, the C i,j and B y ) together form an open cover of. Exploiting boundedness of, we choose a finite sub-cover S := C i1,j 1 C i2,j 2... C im,j M B 1 y 1 ) B 2 y 2 )... B N y N ) of, and we tae ε := \S. Then, ε is an open set with ε and \ ε S N ε ). Thus, the first condition in 1.5) and the weaer form 3.2) of the second condition are valid. In order to verify the third condition in 1.5), we exploit in turn 1.1), the preceding definition of ε, 3.4), the choices of 4 At this point, we wor with the spherical Hausdorff measure, but we also use results from [1], formulated with the standard Hausdorff measure. Nevertheless, our argument is consistent, as we only evaluate these measures on H n 1 -rectifiable sets, where they coincide by [11, Theorem ]. Notice in addition that the theory of the perimeter can also be developed using only the spherical measure from the very beginning.

7 STRICT INTERIOR APPROXIMATION OF BV SETS AND FUNCTIONS 7 the C i,j and B y ), the fact that the R i are disjoint in F, and 1.6). In this way we deduce P ε ) H n 1 ε ) H n 1 C i,j )+ H n 1 B y )) i,j=1 1+A n ε) 1+A n ε) i,j=1 =1 ω n 1 r n 1 i,j +A n =1 n 1 [ H n 1 R i )+2 i ε ] +A n ε i=1 = 1+A n ε)[h n 1 F)+ε]+A n ε P)+1+2A n +A n P))ε with the dimensional constant A n := max{3 n /ω n 1 +6n 1),nω n }. The same reasoning applies with the smaller quantity ε/1+2a n +A n P)) in place of ε. Thus, we obtain the last assertion of 1.5), and the proof is complete. 4. Strict approximation of BV-functions from a given Dirichlet class For a BV-function w, defined near F, we write w + F for the exterior trace and w F for the interior trace on F with orientation given by the exterior normal ν ); see [1, Theorem 3.77]. With this notation, Theorem 1.1 corresponds essentially to the special case of a constant w in the next proposition. Proposition 4.1. Suppose that is a bounded open set in R n with H n 1 ) = P) <. Then, for w BVR n, R N ) L R n, R N ) and every ε > 0, there exists an open set ε with finite perimeter in R n such that ε, L n \ ε ) ε, w F ε dh n 1 w F dh n 1 +ε. F ε Proof. We assume ε 1 2 and tae L := max{1, w L R n,r N )}. Woring with the sets R i from Lemma 3.1, we pass on to the sets R i of all x R i such that, for some r x > 0 and all 0,r x ], we have 4.1) 4.2) 4.3) R i B x) F H n 1 R i B x)) 1 1+ε ωn 1 n 1, w w F x) dl n ε 2 n, B x) w F w F x) dh n 1 εh n 1 R i B x)). With this terminology we have H n 1 R i \ Ri ) = 0 for the following reasons: the almosteverywhere validity of the first property is guaranteed by one implication of) the Besicovitch- Marstrand-Mattila theorem [1, Theorem 2.63, Theorem 2.83i)]; similarly, for the second property we rely on the defining property of traces [1, Theorem 3.77], and in connection with the thirdoneweexploitthefactthat H n 1 -a.e. pointofr i isalebesguepointofthel -function w F with respect to the finite Radon measure H n 1 R i. The family { B x) : x Ri, 0 < 2 min{r x, 1 i }, H n 1 R i B x)) = 0 } is a fine cover of R i, and by the Vitali covering theorem see [1, Theorem 2.19]) for the Radon measure H n 1 R i we can find a countable subfamily { B ri,j x i,j ) : j N }, consisting of

8 8 THOMAS SCHMIDT disjoint balls with x i,j Ri and 2r i,j min{r xi,j, 1 i }, such that ) 4.4) H R n 1 i \ B ri,j x i,j ) = 0 j=1 holds. As a consequence of Lemma 3.1, we have 4.5) L n B + 2r i,j x i,j )) 2 n ε 2 ri,j n, 4.6) R i B ri,jx i,j) x i,j +C ε T i,j) with the abbreviation T i,j := Tan n 1 F,x i,j ). Next we introduce the cylinders C i,j := {y R n : T i,j y x i,j ) < r i,j and T i,j y x i,j) < h i,j }, where h i,j εr i,j,2εr i,j ] will now be chosen such that we get good estimates for parts of C i,j. Indeed, via 4.5), 4.2), the observation that C i,j B 2ri,j x i,j ) holds whenever h i,j 2εr i,j ), and Fubini s theorem we can fix h i,j εr i,j,2εr i,j ] such that, for we have 4.7) + C i,j := {y R n : T i,j y x i,j ) < r i,j and T i,jy x i,j ) = h i,j ν x i,j )}, such that H n 1 -a.e. point in H n 1 + C i,j ) 2 n+1 εr n 1 i,j, C i,j := {y R n : T i,j y x i,j ) < r i,j and T i,jy x i,j ) = h i,j ν x i,j )} is a Lebesgue point of w, and such that we have 4.8) w w F x i,j) dh n 1 2 n+1 εri,j n 1. C i,j From 4.4), 4.6), and h i,j > εr i,j we infer 4.9) H n 1 R i \ ) C i,j = 0, j=1 and via the boundedness of w, via the fact that the side FC i,j \ + C i,j C i,j ) of C i,j has H n 1 -measure 2h i,j n 1)ω n 1 r n 2 i,j, and via 4.8), 4.7), 4.1), 4.3) we arrive at 4.10) w + FC i,j dh n 1 FC i,j w dh n 1 +LH n 1 + C i,j )+2h i,j n 1)Lω n 1 r n 2 i,j C i,j w F x i,j) H n 1 C i,j )+ [ 2 n+1 +2 n+1 ] L+4n 1)Lω n 1 εr n 1 i,j w F x i,j) ω n 1 r n 1 ) i,j L 2 εωn 1 r n 1 i,j w F x i,j) 1+ε)H n 1 R i B ri,j x i,j ))+ L 2 ) εh n 1 R i B ri,j x i,j )) 1+ε) w F dh n 1 + LεH n 1 R i B ri,j x i,j )), R i B ri,j x i,j) where we abbreviated L := 2 n+3 /ω n 1 +8n)L. Via 1.7), Lemma 3.1, and 4.9), we get ) H \ n 1 C i,j = 0. i,j=1

9 STRICT INTERIOR APPROXIMATION OF BV SETS AND FUNCTIONS 9 Then, as in the proof of Theorem 1.1, we cover \ i,j=1 C i,j by countably many balls B 1y 1 ),B 2y 2 ),B 3y 3 ),... with radii 1 such that =1 n 1 ε. The C i,j and B y ) form an open cover of the compactum, and we can choose a finite sub-cover S := C i1,j 1 C i2,j 2... C im,j M B 1 y 1 ) B 2 y 2 )... B N y N ). Then ε := \S is an open set with finite perimeter in R n this follows, for instance, from the fact that the multiplication of BV L -functions is still in BV L ). Next we employ 4.10), where we control the right-hand side of this estimate via the fact that B ri,j x i,j )) j N is a disjoint family of balls for fixed i N). Exploiting that also the R i are disjoint, we deduce w F ε dh n 1 w + FC i,j dh n 1 +Lnω n 1 n F ε i,j=1 FC i,j =1 [ ] 1+ε) w F dh n 1 + LεH n 1 R i ) +Lnω n ε i=1 R i 1+ε) w F dh n 1 + LεH n 1 F)+Lnω n ε. F Finally, relying on r i,j 1 and 1, on 4.1), and on the fact that all R i B ri,j x i,j ) are disjoint, we have L n \ ε ) L n C i,j )+ L n B y )) i,j=1 4ε i,j=1 4ε1+ε) =1 ω n 1 ri,j n +ω n =1 n H n 1 R i B ri,j x i,j ))+ω n i,j=1 [6H n 1 F)+ω n ] ε. Possibly decreasing ε suitably, the last two estimates yield the claims. Next we apply Proposition 4.1 with suitable truncations of u u 0 in place of w. Involving also a mollification procedure, this leads to a Proof of Theorem 1.2. We fix ε > 0, for L 0 we set { { u if u L u L := L u u if u > L and ul 0 := u0 if u 0 L L u0 u 0 if u 0 > L, and we start with the observation that u L equals u L 0 outside and satisfies 4.11) L n,du L ) ) L n,du) ). =1 n 1 Now we fix also L depending on ε) so large that we have u L u dl n ε and u L 0 u 0 + u L 0 u 0 ) dl n ε. Then we apply Proposition 4.1 to the bounded function u L u L 0, finding open sets ε with P ε ) <, L n \ ε ) ε,

10 10 THOMAS SCHMIDT and 4.12) u L ) F ε u L 0 dh n 1 F ε F u L ) F ul 0 dh n 1 +ε = Du L F)+ε where we have just written u L 0, as all one- or both-sided sided traces of the W1,1 -function u L 0 coincide). Now we use that, by the absolute continuity of the Lebesgue integral, there exists an increasing η: [0, ) [0, ) with lim δց0 ηδ) = 0 and u + u0 + u 0 ) dl n ηl n A)) for all measurable subsets A of. A We also introduce the notation M ε for a smoothing operator, based on mollification with a radius r ε < dist ε, R n \) which is taen small enough that M ε 1 ε u L ) 1 ε u L dl n ε, Mε 1 ε u L 0 ) 1 ε u L 0 + M ε1 ε u L 0 ) 1 ε u L 0 ) dl n ε hold. Consequently, we can estimate M ε 1 ε u L ) u dl n Mε 1 ε u L ) 1 ε u L + u L u ) dl n + u dl n 4.13) \ ε 2ε+ηε) and, very similarly, 4.14) M ε 1 ε u L 0 ) u 0 dl n + M ε 1 ε u L 0 ) u 0 dl n 2ε+ηε). At this stage, we define u ε as the restriction of u 0 + M ε 1 ε u L u L 0 )) to. Then, by the choice of r ε, we have u ε u 0 +C cpt, R N ), and 4.13) and 4.14) imply u ε u dl n M ε 1 ε u L ) u dl n + u 0 M ε 1 ε u L 0) dl n 4ε+2ηε). With the help of [1, Theorem 3.84], we compute, L n -a.e. on, u ε = u 0 +M ε D1 ε u L u L 0)) = u 0 M ε 1 ε u L 0 )+M εdu L ε ) M ε [u L ) F ε u L 0 ] ν ε H n 1 F ε ) where we understand, as usual, the mollification of a measure as a function). Starting from this formula, we now use 4.14) and the inequality M ε µ) L 1 R n,r m ) µ R n ) which holds for every R m -valued Radon measure on R n ) to deduce M ε 1 )L n,du ε ) ) u 0 M ε 1 ε u L 0 ) dl n + M ε L n,du L ε ) L1,R Nn+1 ) + M ε [u L ) F ε u L 0 ] ν ε H n 1 F ε ) L 1,R Nn ) 2ε+ηε)+ L n,du L ) )+ u L ) F ε u L 0 F ε dh n 1 3ε+ηε)+ L n,du L ) F) 3ε+ηε)+ L n,du) ), where we also involved 4.12) and 4.11) in the last steps. Combining the preceding estimate with L n M ε 1 )L n ) = 1 M ε 1 ) dl n L n \ ε ) ε, we conclude L n,du ε ) ) L n,du) )+4ε+ηε).

11 STRICT INTERIOR APPROXIMATION OF BV SETS AND FUNCTIONS 11 Decreasing ε suitably, we have established 1.10), and the proof is complete. 5. On the hypothesis H n 1 ) = P) In this section, we are concerned with the basic assumption H n 1 ) = P), which we have imposed in Theorems 1.1 and 1.2. We emphasize that this hypothesis is actually not completely new, but has already occurred in a related context, namely in the trace theorems [4, Teoremi 5, 7, 10]. In the following, we prove some optimality of the assumption in the 2-dimensional case, and we discuss a refined example for its necessity. Theorem 5.1 optimality property in R 2 ). Consider a bounded open set in R 2 such that has finitely many connected components, while has at most countably many path-connected components. If, for every ε > 0, there exists an open set ε with smooth boundary in R 2 and with 1.5), then we already have H 1 ) = P). Proof. For, we denote the number of connected components of by M N, and we write 1, 2,..., M for these components. We record that the open and connected sets i are automatically path-connected, and we fix one point p i in each i. For sufficiently large and only those we will consider), 1.5) implies that all p i are contained in 1/, and we can define i as the path-connected component of 1/ which contains p i. For the open set := M i=1 i we will now show 5.1) lim d H, ) = 0. Indeed, from 1/ and the observations 1.8) we deduce sup x distx, ) sup distx, ) d H 1/, ) 0. x 1/ Hence, the claim 5.1) could only fail if we had γ := limsup sup distx, ) > 0 x and thus lim l distx l, l ) = γ for some subsequence l ) l N and some sequence x l ) l N in which converges to x. Now, on one hand, we would have distx, l ) γ/2 for l 1, while on the other hand we could find an x with x x < γ/2 and a path from x to an p i in. For sufficiently large l, the whole path would be contained in 1/l, therefore we would have x i l l and distx, l ) x x < γ/2. The latter estimate contradicts the previously observed lower bound for distx, l ), and thus 5.1) is proved. Next we write R j ) j N for the family of path-connected components of where, in case of finitely many components, we understand R j = for large j). For the moment we will wor with R 1, R 2,..., R N, wheren N is arbitrary, but fixed. We define E j asthe path-connected component of R n \ which contains Rj, we set and we claim: 5.2) 5.3) lim inf d M H i=1j=1 R i,j := i E j, R i,j is path-connected, N ) R i,j,x N = 0 for some closed X N with N R j X N. j=1

12 12 THOMAS SCHMIDT In order to prove 5.2) we tae arbitrary y,ỹ R i,j. As 1/ is smooth and bounded, we can write the path-connected component of R i,j which contains y as the image of a smooth closed curve l. Moreover, by the path-connectedness of i and Ej we can connect y and ỹ by curves c int : [0,1] R 2 and c ext : [0,1] R 2 with c int 0) = c ext 0) = y and c int 1) = c ext 1) = ỹ such that c int remains inside and c ext remains outside apart from the endpoints). Now, asiml locally separates and Rn \, this means that c intt) and c ext t) lie on different sides of the loop Iml for 0 < t 1. As neither c int nor c ext can cross Iml, we infer that the common endpoint ỹ is contained in Iml. But then a part of l connects y and ỹ in, and we have shown that Ri,j is path-connected. R i,j Turning to 5.3), we first observethat the validity ofthe convergencefor some closed set X N is immediate by the Blasche selection theorem see [1, Theorem 6.1] or [10, Theorem 3.16]), so it only remains to justify the claimed inclusion of N j=1 Rj in X N. To this end, we fix j 0 {1,2,...,N} and consideran arbitraryx R j0. By the compactness of, we can find x with x x = distx, ); thenthe linesegmentfromx toxdoesnotintersect, hence thissegmentiscontainedin Ej0, andwehavex E j0. In viewof = M i=1 i we can also fix an i 0 {1,2,...,M} such that x i0 and therefore x R i0,j0 hold for infinitely many N. Taing into account x x = distx, ) d H, ) and 5.1), the x converge to x, and hence x is contained in the Hausdorff limit X N of M N i=1 j=1 Ri,j. We have thus shown R j0 X N, and the derivation of 5.3) is complete. In view of 5.2), the number of connected components of M i=1 N j=1 Ri,j is bounded by M N, and then [8, Corollary 35.15] that is essentially Golab s theorem; compare also [10, Theorem 3.18]) guarantees lower semicontinuity of H 1 along the convergence in 5.3). We can thus deduce M H 1 X N ) limsup H 1 N i=1 j=1 ) R i,j. Using the inclusion N j=1 Rj X N from 5.3) on the left-hand side and R i,j 1/ on the right-hand side, we infer H 1 N j=1 ) R j limsup H 1 1/ ). Finally, wetaeintoaccountthat H 1 1/ ) = P 1/ ) P)+1/ holdsbythesmoothness of 1/ and 1.5) so that we end up with N ) H 1 R j P). j=1 As N is arbitrary and j=1 Rj equals, this shows H 1 ) P). As observed in 1.1), the opposite inequality is always valid, and thus the proof is complete. In the introduction we have given a concrete example for the failure of the boundary hypothesis H n 1 ) = P), but in that case the problematic set is in some sense quite negligible. Indeed, in the mentioned example and in many similar situations, one may pass to an open set which satisfies H n 1 ) = P ), allows strict interior approximation, and is equivalent in the sense of L n ) = 0. To some extent, this concept of equivalence seems reasonable, as it leaves the perimeter and the related measure-theoretic notions invariant, but evidently it changes the topological notions of interior approximation and boundary.

13 STRICT INTERIOR APPROXIMATION OF BV SETS AND FUNCTIONS 13 Anyhow, every equivalence class of open sets contains a maximal with respect to set inclusion) representative := {x R n : L n B x) \ ) = 0 for some > 0}, which moreover has minimal topological boundary in the class. Whenever the boundary hypothesis holds or strict interior approximation is possible for an open set, then this is a fortiori the case for. Nonetheless, in the following we show that these assertions may fail even for. Example 5.2. For n 2, we tae := 0,1) n \ B i yi, 2) 1 R n, i=1 where the i > 0 and y i R n 1 are chosen such that the n 1)-dimensional balls B iy i ) are disjoint with dense union in [0,1] n 1 and such that we have i=1 ωn 1 n 1 i < 1. Then, the Cantor type set A := [0,1] n 1 \ B iy i ) R n 1 i=1 has positive L n 1 -measure, equals, and we have P) H n 1 ) = 2n+nω n i=1 n 1 i + L n 1 A). In addition, the lower semicontinuity of the perimeter, applied on ± := {x : ±x n 1 2 ) > 0}, shows that interior approximations ε necessarily fulfill lim inf P ε) 2n+nω n 1 n i +2L n 1 A). εց0 Hence, strict interior approximation of, in the sense of 1.5), is impossible. To close this section, we remar that the situation may change for non-open sets. Looing at the representative 1 := {x R n : lim ց0 n L n B x) \) = 0} which is even larger than, has even smaller boundary, but is not necessarily open), for instance, one may still hope to find strict approximations ε 1. However, we stress that such ε may potentially touch and even 1 ; thus, this type of approximation seems vaguely related to ideas of [23, 24, 19], but quite different from the considerations of the present paper. i=1 References 1. L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, Oxford, G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Am. Math. Soc ), G. Anzellotti, BV solutions of quasilinear PDEs in divergence form, Commun. Partial Differ. Equations ), G. Anzellotti and M. Giaquinta, Funzioni BV e tracce, Rend. Sem. Mat. Univ. Padova ), L. Bec and T. Schmidt, Convex duality and uniqueness for BV-minimizers, Preprint 2013), 32 pages. 6. M. Bildhauer, Convex variational problems. Linear, nearly linear and anisotropic growth conditions, Springer Lecture Notes in Mathematics 1818, Berlin, G.-Q. Chen, M. Torres, and W.P. Ziemer, Gauss-Green theorem for wealy differentiable vector fields, sets of finite perimeter, and balance laws, Commun. Pure Appl. Math ), G. David, Singular sets of minimizers for the Mumford-Shah functional, Birhäuser, Basel, P. Dotor, Approximation of domains with Lipschitzian boundary, Čas. Pěst. Mat ), K.J. Falconer, The geometry of fractal sets, Cambridge University Press, Cambridge, H. Federer, Geometric Measure Theory, Springer, Berlin, E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birhäuser, Basel, G. Gregori, Generalized solutions for a class of non-uniformly elliptic equations in divergence form, Commun. Partial Differ. Equations ),

14 14 THOMAS SCHMIDT 14. J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in W 1,1 and BV, Arch. Ration. Mech. Anal ), ; correction in Arch. Ration. Mech. Anal ), J. Kristensen and F. Rindler, Piecewise affine approximations for functions of bounded variation, Preprint 2013), 14 pages. 16. S. Littig and F. Schuricht, Convergence of the eigenvalues of the p-laplace operator as p goes to 1, Calc. Var. Partial Differ. Equ ), U. Massari and L. Pepe, Sull approssimazione degli aperti lipschitziani di R n con varietà differenziabili, Boll. Unione Mat. Ital., IV. Ser ), J. Nečas, Sur les domaines du type N, Czech. Math. J ), Russian, French summary). 19. W.F. Pfeffer, The Gauss-Green theorem, Adv. Math ), T. Quentin de Gromard, Approximation forte dans BV), C. R. Acad. Sci., Paris, Sér. I ), T. Quentin de Gromard, Strong approximation of sets in BV), Proc. R. Soc. Edinb., Sect. A, Math ), T. Quentin de Gromard, Approximation forte des ensembles dans BV) par des ensembles à frontière C 1, C. R., Math., Acad. Sci. Paris ), I. Tamanini and C. Giacomelli, Approximation of Caccioppoli sets, with applications to problems in image segmentation, Ann. Univ. Ferrara, Nuova Ser., Sez. VII ), I. Tamanini and C. Giacomelli, Un tipo di approssimazione dall interno degli insiemi di perimetro finito, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl ), Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, Pisa, Italy address: thomas.schmidt@sns.it Institut für Mathemati, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland address: thomas.schmidt@math.uzh.ch