Strong poyhedra approximation of simpe jump sets Tuomo Vakonen Abstract We prove a strong approximation resut for functions u W 1, (Ω \ J), where J is the union of finitey many Lipschitz graphs satisfying some further technica assumptions. We approximate J by a poyhedra set in such a manner that a reguarisation term η(div j u i ), (i = 0, 1, 2,...), is convergent. The boundedness of this reguarisation functiona itsef, introduced in [T. Vakonen: Transport equation and image interpoation with SBD veocity fieds, (2011)] ensures the convergence in tota variation of the jump part Div j u i of the distributiona divergence. Mathematics subject cassification: 26B30, 49Q20, 74R10. Keywords: bounded variation, approximation, discontinuities. 1. Introduction Let u SBV(Ω) be a specia function of bounded variation on the domain Ω R m. We woud ike to approximate u by a sequence of functions {u i } i=0 such that ui is reasonaby smooth in Ω \ Ĵui, (i = 0, 1, 2,...), and Ĵu i is a poyhedra (m 1)-dimensiona set, containing the jump set J ui. As the novety of our resuts, we woud ike convergence from a reguarisation term η(div j u i ), introduced in [11]. The boundedness of this term ensures that if Div j u i Div j u and Div j u i λ, then λ = Div j u. The notation Div j u here stands for the jump part of the distributiona divergence Div u, whie the absoutey continuous part wi be denoted by div u. Why do we want this kind of strong approximation property? In [11] we studied an extension of the transport equation invoving jump sources and sinks. With u = (1, b) the veocity fied and I the space-time data being transported, it can be stated as Div(Iu) I div u τ Div j u = 0 (1.1) for some τ defined on the jump set of u, modeing the sources and sinks. To show the stabiity of (1.1) with {I i } i=0 converging weaky in BV(Ω) and {ui } i=0 converging as in the SBV/SBD compactness theorems [3, 4], we needed to further assume that Div j u i (Ω) Div j u (Ω). To use (1.1) as a constraint in an optimisation probem (specificay, image interpoation), we thus had to introduce the reguarisation term η(div j u i ) ensuring this convergence. One possibiity for the definition is ) η(µ) := ( µ (Ω) 2 m µ(x + [0, 2 ] m ) dx, (µ M(Ω)). (1.2) R m =0 Roughy η(µ) < says that on average the differences 2 m ( µ (x + [0, 2 ] m ) µ(x + [0, 2 ] m ) ) go to zero as the scae 2 becomes smaer. Thus on sma sets µ is cose to µ. Institute for Mathematics and Scientific Computing, Kar-Franzens University of Graz, Austria. E-mai: tuomo.vakonen@iki.fi. This work has been financiay supported by the SFB research program Mathematica Optimization and Appications in Biomedica Sciences of the Austrian Science Fund (FWF). 1
The probem then becomes: can we, at east in principe, numericay sove probems invoving such reguarisation terms? That is, can we in particuar construct a sequence of discretisations of u such that η(div j u i ) η(div j u) aong with the standard convergences u i u and u i u in L 2, D j u i D j u weaky*, and H m 1 (J u i) H m 1 (J u )? In the present work, we intend to provide a partia answer. Specificay, we restrict our attention to functions u W 1, (Ω \ Ĵu), where Ĵu is the union of finitey many Lipschitz graphs with bounded variation gradient mapping, satisfying further technica conditions, given in Definition 5.1 beow. Assuming these conditions, we show that u can be approximated by functions u i W 1, (Ω \ Ĵui) with Ĵui poyhedra and satisfying Definition 5.1. Some of our proof techniques resembe those of the SBD approximation theorem of Chamboe [6, 7]. In SBV a counterpart approximation theorem is proved by quite different techniques by Cortesani and Toader [8]. Their resut provides argey simiar convergence properties as ours, but is missing the crucia convergence of η(div j u i ). Of course, the cass of functions that we are abe to study at the moment is significanty smaer. Finay, we aso study anisotropic approximation with Ĵu i restricted to ie on transations of the coordinate panes. We have organised this paper as foows. First, in Section 2, we introduce notation and some other we-known toos. In section 3 we study the functiona η, and estimates for bounding it. As a consequence we aso obtain some new SBV compactness resuts. In Section 4 we provide a series of further technica emmas of genera nature, needed to prove the approximation theorem. In the subsequent Section 5 we then introduce in detai the space where the approximated function u ies in, and provide further technica emmas regarding the covering of the boundary of the jump set by cubes. Our main approximation theorem is then stated and proved in Section 6. Finay, we study anisotropic approximation in Section 7. 2. Preiminaries 2.1. Sets and functions We denote the unit sphere in R m by S m 1, whie the open ba of radius ρ centred at x R m we denote by B(x, ρ). The boundary of a set A is denoted A, and the cosure by c A. For ν R m, the hyperpane orthogona to ν we denote by ν := {z R m ν, z = 0}. P ν denotes the projection onto the subspace spanned by ν, and P ν the projection onto ν. We denote by {e 1,..., e m } the standard basis of R m. The k-dimensiona Jacobian of a inear map L : R k R m, (k m), is defined as J k [L] := det(l L). A set Γ R m is a caed a Lipschitz d-graph (of Lipschitz factor L), if there exist a unit vector z Γ, an open set V Γ on a d-dimensiona subspace of z Γ, and a Lipschitz map g Γ : V Γ R m of Lipschitz factor at most L, such that Γ = {y R m g Γ (v) = y, v = P z Γ y V Γ }. We say that Γ is poyhedra if g Γ is piecewise affine and V Γ is a poyhedra set, i.e., consists of finitey many simpices. If g Γ is further affine, we say that Γ is affine. We define the boundary as Γ := g Γ ( V Γ ). Remark 2.1. Consider the situation d = m 1. If Γ is the graph of f : U R m 1 R, then g Γ (v) = (x, f(x)) for v = (x, 0) V Γ = U {0}. More generay, if V Γ zγ for some z Γ R m, and f : V Γ R is Lipschitz map, then g Γ (v) = v + z Γ f(v) defines a Lipschitz graph. Conversey, if Γ is a Lipschitz graph per the above definition, then defining f Γ (v) := g Γ (v), z Γ for v V Γ, we obtain the more conventiona description Γ = {v + f Γ (v)z Γ v V Γ }. For our purposes it is more convenient to work with the map g Γ, however. 2
2.2. Measures The space of (signed) Radon measures on an open set Ω is denoted M(Ω). If V is a vector space, then the space of V -vaued Radon measures on Ω is denoted M(Ω; V ). The k-dimensiona Hausdorff measure, on any given ambient space R m, (k m), is denoted by H k, whie L m denotes the Lebesgue measure on R m. For a measure µ and a measurabe set A, we denote by µ A the restriction measure defined by (µ A)(B) := µ(a B). The tota variation measure of µ is denoted µ. For a Bore map u : Ω R we denote µ(u) := Ω u dµ. A measure µ M(Ω) is said to be Ahfors-reguar (in dimension d), if there exists M (0, ) such that M 1 r d µ (B(x, r)) Mr d for a r > 0 and x supp µ. If ony the first or the second inequaity hods, then µ is said to be, respectivey, ower or upper Ahfors-reguar. We wi often refer to the foowing standard resut on weak* convergence. (See, e.g., [2, Proposition 1.62]). Proposition 2.1. Let µ i M(Ω), (i = 0, 1, 2,...), be such that µ i µ M(Ω), and µ i λ M(Ω). If E is a reativey compact µ-measurabe set such that λ( E) = 0, then µ i (E) µ(e). More generay, et u : Ω R be any compacty supported Bore function, and denote by E f the set of its discontinuity points. Then, if λ(e f ) = 0, we have Ω u dµi Ω u dµ. 2.3. Functions of bounded variation A function u : Ω R K on a bounded open set Ω R m, is said to be of bounded variation (see, e.g., [3] for a more thorough introduction), denoted u BV(Ω; R K ), if u L 1 (Ω; R K ), and the distributiona gradient Du is a Radon measure. We define the norm u BV(Ω;R K ) := u L 1 (Ω;R K ) + Du (Ω). Given a sequence {u i } i=1 BV(Ω; RK ), strong convergence to u BV(Ω; R K ) is defined as strong L 1 convergence u i u L 1 (Ω;R K ) 0 together with convergence of the tota variation u u i (Ω) 0. Weak convergence is defined as u i u strongy in L 1 (Ω; R K ) aong with Du i Du weaky* in M(Ω; R K m ). We denote by S u the approximate discontinuity set, i.e., the compement of the set where the Lebesgue imit ũ exists. The atter is, of course, defined by 1 im ρ 0 ρ m ũ(x) u(y) dy = 0. B(x,ρ) The distributiona gradient can be decomposed as Du = ul m + D j u + D c u, where the density u of the absoutey continuous part of Du equas (a.e.) the approximate differentia of u. The jump part D j u may be represented as D j u = (u + u ) ν Ju H m 1 J u, (2.1) where x is in the jump set J u S u of u if for some ν := ν Ju (x) there exist two distinct one-sided traces u ± (x) defined as satisfying 1 im ρ 0 ρ m u ± (x) u(y) dy = 0, (2.2) B ± (x,ρ,ν) 3
where B ± (x, ρ, ν) := {y B(x, ρ) ± y x, ν 0}. It turns out that J u is countaby H m 1 - rectifiabe, and ν is (a.e.) the norma to J u. Moreover, H m 1 (S u \ J u ) = 0. The remaining Cantor part D c u vanishes on any Bore set σ-finite with respect to H m 1. The space SBV(Ω; R K ) of specia functions of bounded variation is defined as those u BV(Ω; R K ) with D c u = 0. There we have the foowing compactness resut. Theorem 2.1 (SBV compactness [1]). Let Ω R m be open and bounded. Suppose ψ : [0, ) [0, ) is non-decreasing with im t ψ(t)/t =. Suppose {u i } i=0 SBV(Ω; RK ) with ( ) sup i u i L 1 + ψ( u i ) dx + D j u i (Ω) + H m 1 (J u i) <. Ω Then there exists u SBV(Ω; R K ) and a subsequence of {u i } i=0, unreabeed, such that D j u i u i u strongy in L 1 (Ω; R K ), u i u weaky in L 1 (Ω; R K m ), D j u weaky* in M(Ω; R K m ), and H m 1 (J u ) im inf i Hm 1 (J u i). We wi aso be working with functions that are of bounded variation on a subspace. That is, et z S m 1, and V z be open and bounded. We then denote u BV(V ; R K ) if u R z BV(Rz 1 V ; R K ), where R z R m (m 1) is an orthonorma basis matrix for z. We et u BV(V ;R K ) := u R z BV(R 1 z V ;R K ). We define the Soboev spaces W n,p (V ; R K ), (n 0, 1 p ), anaogousy. We are aso interested in the case when u has not just scaar or simpe vector vaues, but u = g L 1 (V ; R K z ). Then the definition becomes that u BV(V ; R K z ) if [x u(r z (x))r z ] BV(Rz 1 V ; R K (m 1) ) with u BV(V ;R K z ) := x u(r z (x))r z BV(R 1 z V ;R K (m 1) ). 2.4. Poincaré-type inequaities We wi ater need some Poincaré-type inequaities, which we study now. The foowing proposition can be found in, e.g., [12, Theorem 5.12.7]. Proposition 2.2. Let Ω R d be a connected domain with Lipschitz boundary, and µ a positive Radon measure on R d, that is upper Ahfors reguar with constant M in dimension d 1, and satisfies supp µ c Ω. Then there exists a constant C 1 = C 1 (Ω), such that for each u BV(Ω), we have M u µ(u)/µ(ω) L 1 (Ω) C 1 µ(c Ω) Du (Ω). Coroary 2.1. Suppose Ω = B(0, r) in Proposition 2.2. Then there exists a constant C 2 = C 2 (d), independent of r, such that u µ(u)/µ(ω) L 1 (Ω) r 2d 1 M C 2 Du (Ω), (u BV(Ω)). (2.3) µ(c Ω) Suppose, in particuar, that µ = L d Ω Ω with µ(u) = 0 and L d (Ω ) ρr d. Then, for a constant C 3 = C 3 (d), we have u L 1 (Ω) r d ρ (1 d)/d C 3 Du (Ω). (2.4) 4
Proof. We appy Proposition 2.2 on the domain B(0, 1) with u 1 (x) := u(rx) and µ 1 (A) := µ(ra), yieding A change of variabes gives and u 1 µ 1 (u 1 )/µ 1 (B(0, 1)) L 1 (B(0,1)) C 2 M µ1 µ(c B(0, 1)) Du 1 (B(0, 1)). Du 1 (B(0, 1)) = Du (B(0, r)), u 1 µ 1 (u 1 )/µ 1 (B(0, 1)) L 1 (B(0,1)) = r d u µ(u)/µ(b(0, r)) L 1 (B(0,r)) as µ 1 (u 1 ) = µ(u) and µ 1 (B(0, 1)) = µ(b(0, r)). Observing that the upper Ahfors constant M µ1 for µ 1 is at most Mr d 1, we get (2.3). As for the second resut, we just have to approximate M. Eementary manipuations give µ(b(x, s)) min{ω d s d, L d (Ω )} Ms d 1 for ω d the voume of the unit ba in R d, and M defined by Inserting this into (2.3) gives (2.4). M/L d (Ω ) = ( ω d /L d (Ω ) ) (d 1)/d (ρ 1 ω d ) (d 1)/d r 1 d. 3. Reguarisation of tota variation 3.1. Convergence of tota variation measures We now study a condition ensuring the convergence of the tota variation µ i (Ω) subject to the weak* convergence of the measures µ i, (i = 0, 1, 2,...). Improving a resut first presented in [11], we show in Theorem 3.1 beow that if {f } =0 is a normaised nested sequence of functions per Definition 3.1 beow, then it suffices to bound η(µ i ) := =0 η (µ i ), where η (µ i ) := µ i (Ω) µ i (τ x f ) dx. Here we empoy the notation τ x f(y) := f(y x). In the next subsection we wi then study an upper bound on η. Definition 3.1. Let f : R m R, ( = 0, 1, 2,...), be bounded Bore functions with compact support that are continuous in R m \ S f. (That is, the approximate discontinuity set is the discontinuity set.) Let aso {ν } =0 M(Rm ), ν (R m ) = 1. The sequence {(f, ν )} =0 is then said to form a nested sequence of functions if f (x) = f +1 (x y) dν (y) (a.e.). The sequence is said to be normaised if f 0 and f dx = 1. The sequence is said to be reguar, if it is normaised, and there exist constants α > 0 and β > 0, and a sequence h 0, im r 0 min{h, r} = 0, (3.1) =0 such that αh m χ B(0,βh ) f α 1 h m χ B(0,h ). Exampe 3.1. Exampes incude f = χ [ 1/2,1/2] m in R m, and f(t) = max{0, min{1 + t, 1 t}} in R (as we as simiar but more compicated shape functions in R m ). Reguarity hods in these cases, and in the more genera typica case f (x) := h m f(x/h ) for h 0 and some f αχ B(0,β) with compact support and f dx = 1. 5
Theorem 3.1. Let Ω R m be an open and bounded set, and {(f, ν )} =0 a normaised nested sequence of functions. Define η(µ) := =0 η (µ), where η (µ) := µ (Ω) µ(τ x f ) dx, (µ M(Ω)). (3.2) Suppose {µ i } i=0 M(Ω) weaky* converges to µ M(Ω) with sup i µ i (Ω)+η(µ i ) <. If aso µ i λ, then λ = µ. Moreover, each of the functionas η and η, ( = 0, 1, 2,...), is ower-semicontinuous with respect to the weak* convergence of {µ i } i=0. Provided that the weak* convergences hod in M(Rm ), then aso η (µ i ) η (µ), ( = 0, 1, 2,...). Proof. Let us suppose first that µ i µ and µ i λ weaky* in M(R m ) rather than just M(Ω). We denote by E f the discontinuity set of f, whie S f stands for the approximate discontinuity set. Fubini s theorem and the fact that S f is an L m -negigibe Bore set, impy that λ(s τxf ) dx = 0. This shows that λ(s τxf ) = 0 for a.e. x R m. Since, by assumption E f S f, it foows that λ(e τxf ) = 0, so that by Proposition 2.1 we have µ i (τ x f ) µ(τ x f ) for a.e. x R m. Likewise µ i (τ x f ) λ(τ x f ) for a.e. x R m. Since sup i µ i (Ω) <, and Ω is bounded, an appication of the dominated convergence theorem now yieds im µ i (τ x f ) dx = µ(τ x f ) dx. (3.3) i We stress that (3.3) hods because of the convergence µ i λ in M(R m ) and λ(e τxf ) = 0. If we can show that, as caimed, λ = µ, it foows immediatey from (3.3) and the definition (3.2) that η (µ i ) η (µ), showing that part of the caim of the emma. Moreover, since the tota variation µ i (Ω) is ower-semicontinuous with respect to weak* convergence, it foows from (3.3) that each η is ower-semicontinuous with respect to the simutaneous weak* convergence of {(µ i, µ i )} i=0. Consequenty aso η is ower-semicontinuous with respect to the simutaneous convergence (by Fatou s emma). However, assuming that { µ i } i=0 does not converge, et us take a subsequence {µin } n=0 such that η(µ in ) α := im inf i η(µ i ). Since sup i µ i (Ω) <, we may move to a further subsequence, unreabeed, such that aso µ in λ for some λ M(Ω). Since sti η(µ in ) α, we deduce from the ower semicontinuity with respect to the simutaneous weak* convergence that α η(µ). This competes the proof of the caim that η is ower-semicontinuous with respect to weak* convergence of {µ i } i=0 aone. Returning to the proof of λ = µ, observe that thanks to the fact that {(f, ν )} i=0 is a nested sequence of functions, {η (µ)} =0 forms a decreasing sequence (for any µ M(Ω)). Indeed, as f (x) = f+1 (x y) dν (y) and ν (R m ) = 1 with ν 0, we have µ(τ x f ) dx = µ(τ x+y f +1 ) dν (y) dx µ(τ x+y f +1 ) dν (y) dx = µ(τ x+y f +1 ) dx dν (y) = µ(τ x f +1 ) dx after a change of variabes in the ast step to eiminate y. Minding the definition (3.2), it foows from here that η (µ) η +1 (µ). To show λ = µ, that is µ i µ, we ony have to show µ i (Ω) µ (Ω). To see the atter, we choose an arbitrary ɛ > 0, and write µ (Ω) µ i (Ω) = η (µ) η (µ i ) + µ(τ x f ) µ i (τ x f ) dx. (3.4) Next we observe from the aready proved ower semi-continuity of η and the bound sup i η(µ i ) =: K < that η(µ) K as we. Therefore, recaing that {η (µ)} =1 and {η (µ i )} =1 for i = 0, 1,... are 6
decreasing sequences, as shown above, it foows that by taking j arge enough, we can ascertain that sup{η (µ), η (µ 1 ), η (µ 2 ),...} ɛ. (Note that η 0!) Empoying this observation in (3.4), we find that µ (Ω) µ i (Ω) 2ɛ + µ(τ x f ) µ i (τ x f ) dx for any arge enough j and a i. The integra term tends to zero as i by (3.3). Therefore, we have im µ i (Ω) µ (Ω) 3ɛ. i Since ɛ > 0 was arbitrary, this concudes the proof under the assumption that the weak* convergences are in M(R m ). If this assumption does not hod, we may sti switch to a subsequence for which µ i k µ and µ i k λ weaky* in M(R m ). Then the above reasoning shows that µ = λ. But, since Ω is open, necessariy µ Ω = µ and λ Ω = λ. This impies λ = µ. By the reasoning above, η (µ i k) η ( µ). Hence an appication of the triange inequaity gives η (µ) = η ( µ Ω) η ( µ) = im k η (µ i k ). As this bound hods for every subsequence, we deduce that each η, ( = 0, 1, 2,...), is owersemicontinuous, and consequenty η as we. This concudes the proof. Remark 3.1. Since, by assumption, f dx = 1, we may aternativey write η (µ) = R m µ (τ x f ) µ(τ x f ) dx. We wi occasionay refer to the foowing eementary properties that foow from the triange inequaity and the fact that supp f B(0, h ). Lemma 3.1. Let {(f, ν )} =0 be a reguar nested sequence of functions and A Rm a Bore set. (i) We have (ii) If {λ x } x R m M(Ω), then η (µ A) + η (µ R m \ A) η (µ) η (µ A) + 2 µ (R m \ A). A λ x (τ x f ) dx λ x (A + B(0, h )) (τ x f ). 3.2. A bound on geometrica compexity We now introduce a quantification of the geometrica compexity of a measure or set. It bears some resembance to definitions of uniform rectifiabiity, as studied by David and Semmes [9]. That notion, however, does not provide the reguarity we need, as it aows considerabe dense packing of the set, merey measuring ocay the deviation from a Lipschitz surface in a geometric sense. Our notion, by contrast, measures the deviation in the sense of measure. Definition 3.2. Let Ω R m open and bounded, and {(f, ν )} =0 a reguar nested sequence of functions per Definition 3.1. Let µ M(Ω) be a radon measure, d m 1 and L, M [0, ). We denote µ Sp d (Ω, L, M) if the foowing hod. 1. µ is upper Ahfors-reguar in dimension d with constant M. 2. There exist famiies G = {G } =0, G = {Γ x x Rm } of d-dimensiona Lipschitz graphs Γ x, of Lipschitz factor at most L, satisfying Sp(µ; G) := Sp (µ; G ) <, where Sp (µ; G ) := µ O x \ Γx (τ x f ) dx, (3.5) R m =0 with the notation O x := x + supp f. 7
(a) A simpe set with Sp(Γ) = (b) A compex set with Sp(Γ) < Figure 1: Exampes of sets satisfying and faiing the condition of Definition 3.2. Definition 3.3. We aso set Sp(µ) := inf G Sp(µ; G), and Sp (µ) := inf G Sp(µ; G ), where the infimum is taken over a famiies of the type specified above. Definition 3.4. For a bounded set E R m, we denote E Sp d (Ω, L, M) if H d E Sp d (Ω, L, M), and set Sp (E; G) = Sp (H d E; G), etc. Definition 3.5. For the Lipschitz graphs Γ x V x := V Γ x, g x := g Γ x, and zx := z Γ x. from Definition 3.2, we use the shorthand notations Remark 3.2. Even quite simpe sets may fai to satisfy this condition, as Exampe 3.2 beow demonstrates. This poses the question whether this is a reasonabe concept. As an eement of justification, in Exampe 3.3 we provide an exampe of a somewhat compex set that satisfies the condition. After that, in Proposition 3.1, we show that the condition impies rectifiabiity. Exampe 3.2. Let us choose h := 2 and f h (x) = h 2 χ Q (x/h) for Q := [ 1/2, 1/2] 2. We then set Γ 1 = [0, 1] {0} and Γ 2 = {(x, g(x)) x [0, 1]} for g(x) = e 1/x, and study µ := H 1 (Γ 1 Γ 2 ) on R 2. See Figure 1(a) for a sketch. Suppose h (0, 1) and et A h := (h /2, h /2) + {(x, y) x [0, 1 h], g(x + h) h, y [g(x + h) h, 0]}. Then, whenever (x, y) A h, both Consequenty, by the definition of f h, we find that If we set we then have H 1 (Γ i ((x, y) + hq)) h, (i = 1, 2). (H 1 Γ i )(τ (x,y) f h ) h 1, (i = 1, 2; (x, y) A h ). G i := {(Γ 1 Γ 2 \ Γ i ) ((x, y) + h Q) (x, y) R 2 }, h 1 L 2 (A h ) (H 1 Γ i )(τ (x,y) f h ) d(x, y) Sp (µ; G i ). A h We want to show that A h has too arge measure for condition (3.5) to be satisfied, that is h 1 L 2 (A h ) does not sum to a finite quantity (for any sequence h 0). For sma enough h, we have A h {(x, y) x 0, g(x + h) h/2, y [ h/2, 0]}. 8
Since g 1 (h) = 1/ og h, we thus have (for sma enough h) We observe g 1 h 1 L 2 (A h ) h 1 (h/2) h h/2 dx = ( 1/ og(h/2) h)/2. ( 1/ og(h /2) h ) = =0 Therefore =0 Sp (µ; G i ) =, (i = 1, 2). 0 (1/( + 1) 2 ) =. Finay, we observe that there do not exist famiies G, ( = 0, 1, 2,...), of Lipschitz graphs covering (Γ 1 Γ 2 ) ((x, y) + hq) with bounded constant, so ony Γ 1 or Γ 2 can be covered, as has been done above. To see this, one observes that for the Lipschitz constant to be bounded, there must exist α > 0 such that any Lipschitz graph Γ covering a part Γ 1 has z Γ, (1, 0) α. But then either z Γ is a tangent vector to Γ 2, or Γ 2 is occuded by Γ 1 when ooking in the direction of z. Thus µ fais (3.5). Exampe 3.3. Let r i := 2 i, and Γ i := {1 r i } [0, r i ], (i = 0, 1, 2,...). Set then R := i=0 Γ i, as sketched in Figure 1(b). We caim that R satisfies (3.5) with respect to f (x) = h 2 χ Q (x/h ), where Q := [ 1/2, 1/2] 2. Indeed, at every (x, y) R 2, et us choose Γ (x,y) as Γ i ((x, y) + h Q) for the smaest i such that 1 r i x h /2. A we then have to do is to cacuate Z i, := H 1 (Γ i \ Γ (x,y) )(τ (x,y) f ) d(x, y), (i = 0, 1, 2,...). (3.6) =0 The term H 1 (Γ i \ Γ (x,y) )(τ (x,y) f ) is non-zero ony when x + h /2 1 r i and x h /2 1 r i 1. Minding that r i 1 r i = r i, it foows that x is on an interva of ength h r i, and h r i. For fixed x we may thus cacuate that This gives the estimate y+h (H 1 Γ i )(τ (x,y) f ) dy = h 2 χ [0,ri ](t) dt dy r i /h. y Z i, { (h r i )r i /h, h r i, 0, otherwise, for the contribution (3.6) of Γ i, (i = 0, 1, 2,...), to (3.5). But h r i means i og 2 h, so summing the contributions of Γ i for i og 2 h, we obtain Sp (µ) Z i, i=0 i og 2 h (h r i )r i /h i og 2 h r i 2h. Thus (3.5) hods when =0 h <. Moreover, it is possibe to show that R is Ahfors-reguar in dimension 1, the maximum for the constant M for the upper bound being given at (1, 0). Proposition 3.1. Suppose Ω R m is open and bounded, and µ M(Ω) satisfies (3.5). Then µ is concentrated on a countaby d-rectifiabe set J. If µ Sp d (Ω, L, M), i.e., µ is aso upper Ahfors-reguar, then µ is d-rectifiabe, µ H d J. Proof. Let G be as in Definition 3.2. Let K be a compact set containing supp µ + B(0, h 0 ). To construct rectifiabe approximations of supp µ, we need a partiay discrete approximation of the Lebesgue integra over K. Denoting by α and β the reguarity constants for {f } =0 from Definition 3.1, we set A := B(0, βh ). With fixed for the moment, we then appy the Besicovitch covering theorem on 9
the famiy {x + A x K} to obtain an at most countabe (actuay finite) set G, such that for a dimensiona constant c m, we have χ K ξ G τ ξ χ A c m. It foows that L m c 1 m ξ G L m (ξ + A ). (3.7) Moreover, from the reguarity condition for f, there exists a constant C 4 > 0 dependent on α, β, and m aone, such that τ ξ f h m ατ ξ χ A h m αχ K C 4 /L m (A )χ K. (3.8) ξ G ξ G Now, with this preiminary setup taken care of, et us for any given y A set J y := x G +y Γx. Then J y is Hd -rectifiabe and we may, using (3.7) and (3.8), approximate Sp (µ; G ) = µ O x \ Γx (τ x f ) dx c 1 m µ O x \ Γx (τ x f ) dy c 1 m A x y+g A x y+g µ Ω \ J y (τ x f ) dy C 4 c m L m µ Ω \ J y (A ) A C 4 µ (Ω \ J y c m L m (A ) ) dy. A We thus deduce that there is a choice of y A with Setting J := j=0 J y Sp (µ; G )c m C 1 µ (Ω \ J y )., it foows from observing µ (Ω \ J y ) µ (Ω \ J) (τ y χ K ) dy and etting that µ (Ω \ J) = 0. Since J is H d -rectifiabe, this gives the first caim of the proposition. If µ is upper Ahfors-reguar in dimension d, we then have µ H d J. We concude that µ is rectifiabe. We finish this subsection by showing ower-semicontinuity of the functiona µ Sp(µ) + µ (Ω), and, consequenty, a cosure property of bounded sets in the space Sp d (Ω, L, M). Proposition 3.2. Let Ω R m be open and bounded. Suppose {µ i } i=0 Spd (Ω, L, M) with sup Sp(µ i ) + µ i (Ω) <. i=0,1,2,... Then any weak* imit µ of (a subsequence of) {µ i } i=0 satisfies µ Spd (Ω, L, M) and Sp(µ) + µ (Ω) im inf i Sp(µi ) + µ i (Ω). 10
Proof. Let ɛ > 0 be arbitrary. Let G i = {G i} =0, Gi = {Γx,i x R m }, be such that Then it suffices to show that Sp(µ i ; G i ) Sp(µ i ) + ɛ, (i = 0, 1, 2,...). Sp(µ; G) + µ (Ω) im inf i for some G = {G } =0, G = {Γ x x Rm }. We use the shorthand notation z x,i := z Γ x,i, and g x,i Sp(µi ; G i ) + µ i (Ω) := g Γ x,i. We may assume that V Γ x,i = P B(x, h z x,i ). This is because we may (see, e.g., [10]) extend g x,i increasing the Lipschitz constant. from V Γ x,i to the whoe space (z x,i ), without We may, moreover, assume that µ i µ M(Ω), and µ i λ M(Ω), where λ µ. The caim of the proposition now foows by an appication of Fatou s inequaity in (3.5), if we show for a = 0, 1, 2,... and amost a x R m that im inf µ i O x \ i Γx,i (τ x f ) µ O x \ Γx (τ x f ) (3.9) for some Lipschitz graph Γ x with constant at most L. Indeed, with = 0, 1, 2,... and x Rm fixed, we may for each i = 0, 1, 2,..., define a Lipschitz map g i : B(0, h ) R m 1 Γ x of constant at most L by g i (v) = g x,i (x + R z x,iv) with R z R m (m 1) the basis matrix of z. Then, since Lipschitz maps of bounded constant are compact in the topoogy of pointwise convergence, we define Γ x as the image of the pointwise imit g of a subsequence of {g i } i=0. Rotating the domain of g back on z with z a imit of a further subsequence of {z x,i } i=0 wi show that Γx is a Lipschitz graph. Let us then write µ i O x \ Γx,i (τ x f ) = µ i (τ x f ) µ i Γ x,i (τ x f ). (3.10) For amost a x R m, we have (as foows from, e.g., [2, Proposition 1.62]) Moreover, we have µ i (τ x f ) λ(τ x f ). (3.11) λ(τ x f ) = (λ O x \ Γx )(τ xf ) + (λ Γ x )(τ xf ) µ O x \ Γx (τ x f ) + (λ Γ x )(τ xf ). (3.12) On the other hand, any weak* imit λ of (a subsequence of) µ i Γ x,i satisfies λ λ Γ x. Moreover, for a.e. x R m, we have µ i Γ x,i (τ x f ) λ(τ x f ). Thus, minding (3.10) (3.12), we deduce im inf µ i O x \ ( µ i Γx,i i (τx f ) = im inf (τx f ) µ i Γ x,i ) i (τx f ) µ O x \ Γx (τ x f ) + (λ Γ x )(τ xf ) im sup µ i Γ x,i (τ x f ) i µ O x \ Γx,i (τ x f ) + (λ Γ x )(τ xf ) λ(τ x f ) µ O x \ Γx (τ x f ) for a.e. x R m. But this is (3.9). Since upper Ahfors reguarity ceary hods for µ with constant M by the ower semi-continuity of µ (B(x, r)) with respect to weak* convergence, we may concude the proof. 11
3.3. Bounds for η We now intend to derive bounds on η(µ) for measures µ Sp d (Ω, L, M). Throughout we assume that exacty the same reguar nested sequence of functions {(f, ν )} =0 is empoyed in the definition of Sp(µ; G) and η(µ). We begin with a technica definition. We need a concept of bounded variation on a famiy of Lipschitz surfaces. With this notion we can imit variations in the intensity of a rectifiabe measure µ, whie bounds on Sp(µ; G) imit variations in the geometry. Both bounds together then bound η(µ). Definition 3.6. Suppose θ is a Bore function on a countaby H d -rectifiabe set J R m, and G a famiy of Lipschitz d-graphs. We then set θ BV(G) := sup Γ i θ g Γi BV(VΓi ), where the supremum is taken over a finite disjoint sub-coections {Γ 1,..., Γ N } G, (N 1). We now state the bounding resut. We reca that α and {h } =0 denote reguarity constants for the maps {f } =0 from Definition 3.1. Condition (3.13) beow is required for uniform constants in Poincaré inequaities; it can triviay be satisfied by extending the domains V x of the Lipschitz graphs Γ x to the whoe space (z x), as can be done according to [10]. Proposition 3.3. Let Ω R m be open and bounded. Suppose µ = θh d J Sp d (Ω, L, M) with Sp(µ; G) < for the coections G = {G } =0, G = {Γ x x Rm }, of Lipschitz graphs of constant at most L. Suppose, moreover, that Then Γ x B(x, h ), and P z x Γx = P z x B(x, h ), ( = 0, 1, 2,... ; x R m ). (3.13) η (µ) C 5 h d θ BV(G ) + Sp (µ; G ) (3.14) for some constant C 5 = C 5 (L, m, d, α). In particuar, if =0 hd <, then ( η(µ) C 6 sup θ BV(G ) + Sp(µ; G) ) for C 6 = C 6 (L, m, d, α, h d ). =0,1,2,... Proof. Let {0, 1, 2,...} be fixed. By writing θ = θ + θ, where θ ± 0, we deduce η (µ) = µ (τ x f ) µ(τ x f ) dx { } = 2 min θ + τ x f dh d, θ τ x f dh d dx. J J (3.15) Writing J = (J Γ x ) (J \ Γx ), we get { } η (µ)/2 min θ + τ x f dh d, θ τ x f dh d dx + µ O x Γ x Γ x \ Γx (τ x f ) dx. (3.16) Since the minimum is non-zero ony if both θ + O x 0 and θ O x 0, ony points x in the set Z := {x R m 0 conv θ(γ x ), Γx B(x, h ) } 12
contribute to the first integra in (3.16). Appying (3.5), we thus obtain { } η (µ)/2 min θ + τ x f dh d, θ τ x f dh d dx + Sp (µ; G ) Z Γ x Γ x { } α 1 h m min θ + dh d, θ dh d dx + Sp (µ; G ). Z Γ x Γ x (3.17) In the fina step we have used the reguarity of {f } =0, i.e., f α 1 h m χ B(0,h ). Next we set B := B(0, (2L + 4)h ), and appy the Besicovitch covering theorem on the famiy {B + x x Z }. With c m a constant dependent on the dimension m aone, we thus find finite coections F 1,..., F cm Z satisfying x F i τ x χ B 1, (i = 1,..., c m ), and x F τ x χ B χ Z with F := c m i=1 F i. Appying the cover F + B of Z in (3.17), and denoting Γ x (θ) = Γ θ dh d, we x may write η (µ)/2 α 1 h m min{γ x (θ+ ), Γ x (θ )} dy + Sp (µ; G ) B x (F +y) Z C (3.18) 7 L m min{γ x (B ) (θ+ ), Γ x (θ )} dy + Sp (µ; G ) B x (F +y) Z for some constant C 7 = C 7 (α, m, L). By the definition of F as c m i=1 F i, it foows that to bound η (µ), it suffices to show that there exists C 8 = C 8 (d, L) such that min{γ x (θ+ ), Γ x (θ )} C 8 h d θ BV(G ) (3.19) x (F i +y) Z for L m -a.e. y B and a i {1,..., c m }. To begin the proof of (3.19), we observe that J d ( g x(v)) C 9 for some C 9 = C 9 (m, d, L). This is due to the continuity of J d and the bound g x (v) L. Thus the area formua yieds Γ x (θ± ) = θ ± dh d = (θ ± g x )J d( g x ) dv C 9 θ ± g x dv. (3.20) Γ x V x V x Let us momentariy fix x Z, and set V = V x, θ ± = θ ± g x, z = zx, and θ = θ g x. We intend to appy Coroary 2.1. Towards this end, we set µ (±) := L d (V \ supp θ ). Then µ (+) (V ) + µ ( ) (V ) L d (V ), so minding (3.13), we have max{µ (+) (V ), µ ( ) (V )} L d (V )/2 = L d (P z B(x, h ))/2 = h d Ld (B(0, 1))/2. Since µ (±) ( θ ± ) = 0, we may appy Coroary 2.1 to get either θ + L 1 (V ) h d C 10 θ + BV(V ) or θ L 1 (V ) h d C 10 θ BV(V ) for a constant C 10 = C 10 (d). As θ ± BV(V ) θ BV(V ), by the definition of θ ±, this gives min{ θ + L 1 (V ), θ L 1 (V )} h d C 10 θ BV(V ). That is min{ θ + g x L 1 (V x ), θ g x L 1 (V x ) } h d C 10 θ g x BV(V x ). (3.21) Next, we observe that with a {0, 1, 2,...}, i {1,..., c m }, and y B fixed, the graphs {Γ x x (y + F i Z )} are disjoint. This foows from the bas x + B, (x y + F i ), being disjoint 13
by construction, and from Γ x x + B = B(x, (2L + 4)h ). The atter hods due to assumption (3.13) and g x having Lipschitz factor at most L. Combining (3.21) with (3.20) thus finay yieds min{γ x (θ+ ), Γ x (θ )} C 9 C 10 h d θ g x BV(V x) x (F i+y) Z x (F i+y) Z (3.22) C 9 C 10 h d θ BV(G ). To concude the proof of the proposition, we ony have to observe that (3.22) yieds (3.19). Remark 3.3. Let {( f, ν )} =0 be another nested sequence of functions that satisfies f C f for some C > 0. Then in (3.16) we coud approximate µ O x \ Γx (τ x f ) dx C µ O x \ Γx (τ x f ) dx C Sp (µ; G ), where Sp denotes the functiona Sp obtained with the sequence {( f, ν )} =0. Thus it woud, at the expense of additiona technica compexity that we want to avoid, be possibe to express our resuts for different sequences of nested functions for the definitions of η and Sp. 3.4. Compactness in SBV(Ω; R K ) We finish this section by providing some compactness resuts in SBV(Ω; R K ) foowing immediatey from the resuts above. They can be usefu in appications for proving cosure properties. We need to work with vector-vaued measures µ M(Ω; R K m ). The resuts above on Sp(µ) can readiy be extended to this situation with no changes in proofs or definitions, but for concreteness we work through the foowing definition. Definition 3.7. For µ = (µ i,n ) [Sp m 1 (Ω, L, M)] K m, we denote Sp(µ) = K i=1 m n=1 Sp(µ i,j). Our main compactness resut is then as foows. The difference to the we-estabished Theorem 2.1 is that we repace H m 1 (J u i) by Sp(D j u i ). Theorem 3.2. Let Ω R m be open and bounded, and {u i } i=0 SBV(Ω; RK ). Suppose ψ : [0, ) [0, ) is non-decreasing with im t ψ(t)/t =. If each D j u i [Sp m 1 (Ω, L, M)] K m, (i = 0, 1, 2,...), and sup u i L 1 (Ω) + i ψ( u i (x)) dx + D j u i (Ω) + Sp(D j u i ) <, (3.23) there then exists u SBV(Ω; R K ) with D j u [Sp m 1 (Ω, L, M)] K m and a subsequence, unreabeed, such that u i u strongy in L 1 (Ω; R K ), (3.24) u i u weaky in L 1 (Ω; R K m ), (3.25) D j u i D j u weaky* in M(Ω; R K m ), and (3.26) Sp(D j u) im inf i Sp(Dj u i ). (3.27) Proof. Let us denote by K the supremum on the eft side of (3.23). We then deduce from (3.23) that sup u i L 1 (Ω) + Du i (Ω) <. i Moving to a subsequence, unreabeed, we may thus assume that u i u weaky in [BV(Ω)] k for some u BV(Ω; R K ). This gives (3.24). Moreover, because { u i } i=0 is an equi-integrabe famiy, 14
we have the existence of some w L 1 (Ω; R K m ), such that for a further unreabeed subsequence, u i w weaky in L 1 (Ω; R K m ). Sti, seecting another subsequence, we find from Proposition 3.2 that D j u i λ for some λ [Sp m 1 (Ω, L, M)] K m with Sp(λ) im inf i Sp(D j u i ). Minding that u i L m + D j u i = Du i and Du i Du by the weak convergence of {u i } i=0 in BV(Ω; RK ), we therefore have wl m + λ = Du = ul m + D j u + D c u. (3.28) Since λ [Sp m 1 (Ω, L, M)] K m, Proposition 3.1 shows that the measure λ is concentrated on a H m 1 rectifiabe set J. This gives w = u, showing (3.25). According to [2], the Cantor part D c u vanishes on any Bore set B that is σ-finite with respect to H m 1. In particuar D c u J = 0. Hence, by (3.28), λ = D j u and D c u = 0. This shows that u SBV(Ω; R K ) as we as (3.26) and (3.27), thus competing the proof. We now state a coroary that be used to prove the cosedness of equations ike (1.1). Specificay, we show stronger convergence for T D j u i with T : R K m R a bounded inear operator by bounding η(t D j u i ). When K = m, choosing T = Tr as the trace operator, we get the convergence in tota variation of the jump part Div j u i := Tr D j u i of the distributiona divergence, appearing in (1.1) and more precisey given by Div j u i (ϕ) = (Tr D j u i )(ϕ) = Here e 1,..., e m is the standard basis of R m. m e n, D j u i (ϕ)e n, (ϕ C c (Ω)). n=1 Coroary 3.1. Let Ω R m be open and bounded, and {u i } i=0 SBV(Ω; RK ). Suppose ψ : [0, ) [0, ) is non-decreasing with im t ψ(t)/t =, and T : R K m R a bounded inear operator. If each D j u i [Sp m 1 (Ω, L, M)] K m, (i = 0, 1, 2,...), and sup u i L 1 (Ω) + i ψ( u i (x)) dx + D j u i (Ω) + Sp(D j u i ) + η(t D j u i ) <, (3.29) then there exists u SBV(Ω; R K ) with D j u [Sp m 1 (Ω, L, M)] K m, and a subsequence, unreabeed, such that (3.24) (3.27) hod aong with T D j u i T D j u weaky* in M(Ω), and (3.30) T D j u i (Ω) T D j u (Ω). (3.31) Proof. Theorem 3.2 shows that (3.24) (3.27) hod. As an immediate consequence, we aso get (3.30). Now (3.31) foows from Theorem 3.1. 4. Technica resuts We now prove a coupe of genera technica resuts that we wi be needing in the proof of the approximation theorem. We begin with a resut on graph approximation, for which we need the foowing eementary emma. Lemma 4.1. Let Γ R m be a Lipschitz (m 1)-graph with norma fied ν Γ. Then for the inear operator A Γ defined by (ν Γ g Γ )(v) = A Γ g Γ (v)/ A Γ g Γ (v), (a.e. v V Γ ), A Γ G = (I H Γ G )z Γ, 15
with H Γ : z Γ Rm the injection operator and G : z Γ Rm an arbitrary inear operator. Moreover A Γ 1, (4.1) and the map defined by has Lipschitz factor Lip(F Γ ) = 1. F Γ (G) := A Γ G/ max{1, A Γ G } Proof. For some f Γ : zγ R we have g Γ(v) = H Γ v + f Γ (v)z Γ and g Γ (v) = H Γ + z Γ f Γ (v). We have HΓ z Γ = 0 and HΓ g Γ (v) = HΓH Γ + HΓz Γ f(v) = HΓH Γ = I, so that for any v zγ, v V Γ, we get (I H Γ ( g Γ (v)) )z Γ, g Γ (v)v = 0. Since the tangent cone T Γ (g Γ (v)) = g Γ (v)z Γ a.e., this says that ν Γ (g Γ (v)) = (I H Γ( g Γ (v)) )z Γ (I H Γ ( g Γ (v)) )z Γ = A Γ g Γ (v) A Γ g Γ (v), (a.e. v V Γ). (4.2) Thanks to H Γ z Γ = 0, we deduce that A Γ z Γ H Γ G z Γ = z Γ 2 + H Γ G z Γ 2 z Γ = 1, with G : z Γ Rm an arbitrary inear operator of norm G = 1. Finay, thanks to F Γ G A Γ G, we have F Γ G 1 F Γ G 2 A Γ G 1 A Γ G 2 = H Γ (G 1 G 2 ) z Γ G 1 G 2, so that F Γ is Lipschitz with factor Lip(F Γ ) = 1. Lemma 4.2. Let Γ R m be a Lipschitz (m 1)-graph with Γ int Ẑ and Hm 1 ( Ẑ Γ) = 0 for a cosed set Ẑ. Let {sk } k=0 (0, s) with sk 0, (k ). Suppose that g Γ BV(V Γ ; R m zγ ). Then we can find poyhedra Lipschitz graphs {Γ k } k=0 of factor at most L = L (Γ), satisfying Γ k Ẑ, z Γ k = z Γ, V Γ k V Γ, (k = 0, 1, 2,...), and We aso have the convergences Γ k Γ \ Ẑ + B(0, sk /2). (4.3) H m 1 Γ k H m 1 Γ \ Ẑ weaky* in M(Rm ), (k ), (4.4) ν Γ kh m 1 Γ k ν Γ H m 1 Γ \ Ẑ weaky* in M(Rm ; S m 1 ), (k ). (4.5) Regarding the maps {g Γ k} k=0, we have g Γ k BV(V Γ k; Rm z Γ ) with g Γ k g Γ L (V Γ k ;R m ) s k /2, (4.6) ν Γ k g Γ k ν Γ g Γ L 1 (V Γ k ;R m ) s k, and (4.7) ν Γ k g Γ k BV(VΓ k ;R m ) g Γ k BV(VΓ k ;R m zγ ) C ) 11( gγ L 1 (V Γ ;R m ) + g Γ BV(VΓ ;R m zγ ) (4.8) for some constant C 11 = C 11 (m). 16
Proof. Suppose we construct Γ k := g Γ k(ṽ k ) \ Ẑ for some g Γ k : z Γ Rm of Lipschitz factor at most L, and poyhedra Ṽ k V Γ with Γ g Γ (Ṽ k ) Ẑ. Then z Γ k = z Γ and V Γ k = g 1 (Γ k ) Γ Ṽ k with k Γ k Ẑ hoding. Moreover, (4.3) foows if we show (4.6). Since g Γ k(v) 1, (v Ṽ k ), we deduce from Lemma 4.1 that ν Γ k g Γ k = F Γ k g Γ k for the Lipschitz function F Γ k. Since g Γ k(x) 1 and F Γk (G) 1 for a x, G, we find that ν Γ k g Γ k L 1 (V Γ k ;R m ) = F Γ k g Γ k L 1 (V Γ k ;R m ) g Γ k L 1 (V Γ k ;R m ). If g Γ k BV(V Γ k; R m z Γ ), it thus foows from the BV chain rue and Lip(F Γ k) = 1 that ν Γ k g Γ k BV(V Γ k; R m ) with ν Γ k g Γ k BV(VΓ k ;R m ) = F Γ k g Γ k BV(VΓ k ;R m ) = F Γ k g Γ k R zγ BV(R 1 z Λ V Γ k ;R m ) x g Γ k(r zγ x)r zγ BV(R 1 z Λ V Γ k ;R m (m 1) ) = g Γ k BV(VΓ k ;R m z Γ ). From the Lipschitz property of F Γ k, we aso deduce that ν Γ k g Γ k ν Γ g Γ L 1 (V Γ k ;R m ) = F Γ k g Γ k F Γ g Γ L 1 (V Γ k ;R m ) Thus (4.7) and (4.8) foow from showing g Γ k g Γ L 1 (V Γ k ;R m z Γ ). g Γ k BV(VΓ k ;R m z Γ ) C 11( gγ L 1 (V Γ ;R m ) + g Γ BV(VΓ ;R m z Γ ) ), (4.9) and, respectivey, g Γ k g Γ L 1 (V Γ k ;R m z Γ ) sk. (4.10) Next we want to show that (4.4), (4.5) foow if we show (4.6) and (4.10). Indeed, et ϕ Cc (R m ) and define U := Rz 1 Γ Ṽ k, as we as g = g Γ R zγ and g k = g Γ k R zγ, where we reca that R z : R m 1 z is the basis matrix of z. Then the area formua gives g Γ k (Ṽ k ) ϕ dh m 1 ϕ dh m 1 g Γ (Ṽ k ) = U ϕ( g k (x))j m 1 ( g k (x)) dx ϕ( g(x))j m 1 ( g(x)) dx. U Empoying the fact that the map (x, y) xy is Lipschitz on bounded sets, it foows that g Γ k (Ṽ k ) ϕ dh m 1 ϕ dh m 1 ϕ( g k (x))j m 1 ( g k (x)) ϕ( g(x))j m 1 ( g(x)) dx g Γ (Ṽ k ) U ( C 12 ϕ( g k (x)) ϕ( g(x)) dx + J m 1 ( g k (x)) J m 1 ( g(x)) ) dx (4.11) U for some constant C 12 = C 12 (ϕ, L ). Minding (4.6), the first integra of (4.11) goes to zero because ϕ Cc (R m ) is uniformy continuous. For the second integra, we observe from (4.10) that g k converges to g in L 1, which we reca to impy amost uniform convergence for a subsequence. That is, after possiby switching to an unreabeed subsequence, for every ɛ > 0 there exists a measurabe subset E Ω with L m (Ω\E) < ɛ, and g k g uniformy on E. By the uniform Lipschitz continuity of {g k } k=0, the vaues of gk moreover ie in a bounded set. With these observations it now easiy U 17
foows that the second integra of (4.11) aso tends to zero. Thus the eft hand side of (4.11) tends to zero. We have therefore shown that H m 1 g Γ k(ṽ k ) H m 1 g Γ (V Γ ). By assumption H m 1 (Γ Ẑ) = 0, so that by Proposition 2.1 Minding the construction of Γ k, we have both H m 1 g Γ k(ṽ k ) \ Ẑ H m 1 g Γ (V Γ ) \ Ẑ. (4.12) H m 1 Γ k = H m 1 g Γ k(ṽ k ) \ Ẑ and Hm 1 g Γ (V Γ ) \ Ẑ = Hm 1 Γ \ Ẑ. (4.13) The convergence (4.4) now foows from (4.12) and (4.13). Since (4.5) can be shown in a simiar fashion with the hep of (4.7), we skip the detais. It remains to construct g Γ k and V Γ k such that (4.6), (4.9), and (4.10) hod. To begin with, et {T } =0, be a sequence of uniform trianguations of zγ, each a subdivision of the previous with edge ength approaching zero as. We then et Ṽ := {T T T V Γ }. For sufficienty arge, we have Γ \ Ẑ g Γ(Ṽ) and g Γ ( Ṽ) int Ẑ. Since g Γ BV(V Γ ; R m z Γ ), we may by moification approximate g Γ on Ṽ by smooth functions g ɛ, satisfying for sufficienty sma ɛ > 0 estimates of the type (4.6), (4.10) aong with g ɛ ( Ṽ k ) int Ẑ and g ɛ BV( Ṽ k ;R m z Γ ) g Γ BV(VΓ ;R m z Γ ). Moreover, the Lipschitz factor of g ɛ is bounded by that of g Γ. As a consequence of this approximation, we may assume that g Γ W 1, (V Γ ; R m ) W 2,1 (V Γ ; R m ). (4.14) For each = 0, 1, 2,..., we denote by {x,n } M n=1 the noda points of the trianguation T. Define ϕ,n such that it is affine on each T and supp ϕ,n K,n := T. We then define g k : Ṽ k R m as g k := M (k) n=1 T T :x,n T ϕ (k),n g(x (k),n ), (k = 0, 1, 2,...) for some (k) k. That is, g k is the Lagrange interpoation of g on T (k). Minding that we have without oss of generaity assumed (4.14), choosing (k) is sufficienty arge, we observe that g k satisfies for some constant C 13 = C 13 (m, T 1 ) the standard finite eement estimates (see, e.g., [5]) g k W 1, (Ṽ k ;R m ) C 13 g Γ W 1, (V Γ ;R m ), g k g Γ L (Ṽ k ;R m ) sk /2, and g k g Γ L 1 (Ṽ k ;R m m ) sk /4, (k = 0, 1, 2,...). In particuar, g k has Lipschitz factor at most L (Γ) = C 13 g Γ W 1, (V Γ ;R m ), and (4.6), (4.10) are satisfied. Finay, to show (4.9), we observe that g k BV(VΓ k ;R m z Γ ) C 14 g Γ W 2,1 (V Γ ;R m ), (k = 0, 1, 2,...), (4.15) 18
for some constant C 14 = C 14 (m, T 1 ). For piecewise affine shape functions, this does not foow from standard resuts due to insufficient reguarity. If we use smooth (or W 2,1 ) shape functions, we however get by standard resuts (see [5, Theorem 4.5.11]) that g k BV(VΓ k ;R m z Γ ) gk W 2,1 (V Γ k ;R m ) C 14 g Γ W 2,1 (V Γ ;R m ), (k = 0, 1, 2,...). Thus, to get (4.15), we can simpy approximate the piecewise affine shape functions by smooth shape functions on the same trianguation T k and pass to the imit. (To construct such smooth shape functions, for each ϕ = ϕ,n with support K = K,n, we may take a sequence of functions {ψ i } i=0 such that ψ i 1 on {x K dist( K, x) > 1/i}, and ψ i 0 on {x K dist( K, x) < 2/i}. As smooth approximations of ϕ supported on K, we take we take ϕ i := (ρ 1/(2i) Rz 1 Λ ) (ψ i ϕ), (i = 0, 1, 2,...). Here {ρ ɛ } ɛ>0 are the standard moifiers on R m 1 = Rz 1 Λ zγ.) Lemma 4.3. Let F be a finite coection of maps ψ C 1 (c Ω R m R m S m 1 ). Denote T ψ u := ψ(, u +, u, ν Ju )H m 1 J u, (ψ F). (4.16) Suppose that F incudes the functions ψ ν i : (x, u +, u, ν) ν i, and ψ ± i : (x, u +, u, ν) (u ± ) i for i {1,..., m}. Let {v, w 0, w 1, w 2,...} SBV(Ω; R K ) L M (Ω; RK ) satisfy sup H m 1 (J w k) <, (4.17) k sup η(t ψ w k ) <, (ψ F), (4.18) k ν Jw k Hm 1 J w k ν Jv H m 1 J v weaky* in M(Ω; S m 1 ), and, (4.19) (w k ) ± H m 1 J w k v ± H m 1 J v weaky* in M(Ω; R m ). (4.20) Then, after possiby moving to an unreabeed subsequence, we have T ψ w k T ψ v and T ψ w k T ψ v for a ψ F. Proof. Let ψ F. The function ψ is bounded on the compact set c Ω c B(0, M) c B(0, M) S m 1 ), so that, minding w k L (Ω;R m ) M, the sequence {T ψ w k } k=0 is aso bounded in M(Ω). Therefore, after possiby moving to a subsequence, we may assume the measures {T ψ w k } k=0 to converge weaky* to some ω ψ M(Ω), and the measures { T ψ w k } k=0 to converge weaky* to some λ ψ M(Ω). By (4.18) and Theorem 3.1 it foows that λ ψ = ω ψ. The question remains, whether ω ψ = T ψ v. Indeed, it foows from the weak* convergences (4.19) and (4.20) that ω ψ = T ψ v for ψ = ψi ν, ψ± i, (i = 1,..., m). In particuar for µ u := (u +, u, ν Ju )H m 1 J u M(Ω; R m R m S m 1 ). Minding that ν Ju (x) = 1, we may now write for f Cc ( ψ f (x, a, b, z) := f(x)ψ x, µ w k µ v and µ w k (Ω) µ v (Ω). (4.21) a z, (Ω) and b z, z z ) z that Ω f(x) dt ψ u(x) = = Ω Ω =: Ω f(x)ψ ( x, u + (x), u (x), ν Ju (x) ) dh m 1 J u f(x) ψ( x, u + (x), u (x), ν Ju (x) ) ( u + (x), u (x), ν Ju (x) ) d µ u (x). ( ) ψ f x, dµk d µ k d µ u (x). 19
The function ψ f is continuous, because ψ is C 1, ν Ju (x) = 1, and 1/ z(x) = (u + (x), u (x), ν Ju (x)) / ν Ju (x) = (u + (x), u (x), ν Ju (x)) 2M 2 + 1. It therefore foows from the Reshetnyak continuity theorem (see, e.g., [3, Theorem 2.39]) and (4.21) that T ψ w k T ψ v. Hence µ ψ = T ψ v. Next we prove a trace resut. Proposition 4.1. Let V R m 1 be an open and bounded, f : V R Lipschitz continuous of factor L, and ϱ > 0. Define Ω := {(x, s) V R s f(x) + ( ϱ, ϱ)}, and g(x) := (x, f(x)). Suppose u W 1, (Ω). Then u has a trace u Γ on Γ := g(v ), and u Γ g W 1, (V ) with u Γ g W 1, (V ) C 15 u W 1, (Ω) (4.22) for some constant C 15 = C 15 (L, m). Proof. The existence of a trace u Γ L 1 (Γ) foows from standard resuts. We just have show that u Γ g is Lipschitz on V. Let us set U := V ( ϱ, ϱ) and v(x, s) := u(x, f(x) + s) = u( g(x, s)) ((x, s) U), where g(x, s) := g(x) + (0, s). We have ( g(x) g(x, s) = 0 as we as so that ceary v W 1, (U) with the bound for some constant C 16 = C 16 (L, m). ) + ( ) 0 0, ((x, s) U), 0 1 v(x, s) = g(x, s) u( g(x, s)), v W 1, (U) C 16 u W 1, (Ω) (4.23) Since u is (Lipschitz) continuous, as is v, we observe that u Γ g = v 0 := v(, 0). But ceary, sti by continuity, Lipschitz continuity is preserved by traces on affine sets, in particuar on V {0}. We therefore obtain v 0 W 1, (V ) v W 1, (U). (4.24) Combining (4.23), (4.24) shows (4.22). Proposition 4.2. Let V R m 1 be an open and bounded, f : V R Lipschitz continuous of factor L, and ϱ > 0. Define Ω := {(x, s) V R s f(x) + ( ϱ, ϱ)}, Ω ± := {(x, s) V R s f(x) + (0, ±ϱ)}, and g(x) := (x, f(x)). Let Γ := g(v ). Suppose u W 1, (Ω\Γ) with H m 1 ({x Γ u + (x) u (x)}) = 0. Then there exist extensions v (±) W 1, (Ω) of u Ω ±, satisfying v (±) L (Ω) u L (Ω ± ) and v (±) W 1, (Ω) C 17 u W 1, (Ω ± ) (4.25) for some C 17 = C 17 (L, m, u). Moreover L m ({x Ω v (+) (x) = v ( ) (x)} = 0. (4.26) 20
Proof. From Proposition 4.1, we deduce that u ± g W 1, (V ) C 15 u W 1, (Ω) for C 15 = C 15 (L, m). Let q 0, q 1 : R + R + be the saw-tooth functions that osciate between the vaues 0 and 1 at sope q 0 = q 1 = 2 u L (Ω), with initia vaues q 0 (0) = 0 and q 1 (0) = 1. Let p(x) := g(p (0,1) (x)) be the projection of x on Γ (aong z Γ = (1, 0)). Then the functions u ± p are Lipschitz with factor at most L u L (Ω ± ;R m ). Consequenty, defining v (±) (x) = { u(x), x Ω ±, q 1 ( x p(x) )u ± (p(x)) + q 0 ( x p(x) )u (p(x)), x Ω, and minding that u ± and q 0, q 1 are bounded, we find that v ± are Lipschitz and (4.25) hods for some C 17 = C 17 (L, m, u). Moreover, we deduce (4.26) thanks to H m 1 ({x Γ u + (x) u (x)}) = 0 and L 1 ({s f(x) + ( δ, δ) v (+) (x, s) = v ( ) (x, s)}), (a.e. x V ). The atter foows from the fact that by construction the functions x q i ( x p(x) ), (i = 0, 1,), osciate faster than u on ines {y} R, (y V ). Remark 4.1. The property (4.26) together with preserving the L bound in (4.25) are the reason for not using standard Soboev or Lipschitz (cf. [10]) extension resuts. Remark 4.2. Both Proposition 4.1 and Proposition 4.2 can easiy by a rotation argument be extended to domains Ω = g Γ (V Γ ) + z Γ ( ϱ, ϱ) defined by a genera Lipschitz graph Γ. 5. The space and boundary covers We now introduce the space A(Ω; R K ) of functions admissibe for the approximation theorem stated in the next section. Definition 5.1. Given an open set Ω R m with Lipschitz boundary, we denote by A(Ω; R K ) the set of functions u : Ω R K that are in W 1, (Ω\J; R K ) for a (with respect to Ω) compact set J = Ĵu Ω satisfying the foowing: (i) H m 1 (J \ J u ) = 0. (ii) J = N i=1 Λ i, where Λ i is a Lipschitz (m 1)-graph of constant at most L. (iii) Λ i Λ n Λ i Λ n and Λ i Ω Λ i. with Λ i := g Λi ( V Λi ), (i, n = 1,..., N; i n), (iv) J Sp m 1 (Ω, L, M) for some M (0, ). (v) Each V Λi, (i = 1,..., N) has Lipschitz boundary. (vi) g Λi BV(V Λi ; R m zλ i ), (i = 1,..., N). We wi henceforth use the shorthand notation V i := V Λi, g i := g Λi, and z i := z Λi. Remark 5.1. Observe that if {u i } i=0 A(Ω; RK ) with the same constants L, M, i.e., Ĵ u i Sp m 1 (Ω, L, M), and if sup u i W i 1, (Ω\Ĵu i ) + Hm 1 (Ĵui) + Sp(Ĵui) <, then it foows from Theorem 3.2 and Proposition 3.2 that there exists u SBV(Ω; R K ) with Ĵ u Sp m 1 (Ω, L, M) such that the convergences (3.24) (3.27) hod for a subsequence. Simiar cosure properties for sets within the space A(Ω; R K ) itsef woud depend on further imiting the compexity and number of the graphs {Λ i } N i=1. 21
In the remainder of this section we provide a series of technica emmas studying the covering of N i=1 Λ i by cubes on a grid. We begin by definitions reated to the cover. Definition 5.2. We denote rq := [0, r] m and rq 0 := [0, r) m for r > 0. Definition 5.3. Suppose Z = X + rq for some set X y + rz m with r > 0 and y Q 0. We then say that E Z is a face of Z if for some ξ X the set E ξ is a face of rq, i.e., for some i = 1,..., m and θ {0, 1}, we have E = ξ + r{x Q x, e i = θ}. Definition 5.4. Suppose J = N i=1 Λ i is as in Definition 5.1. Denote J := N i=1 Λ i. Then for r > 0 and y Q 0, we et F r := {ξ rz m (ξ + 2rQ) J }, F y r := ry + F r, Z y r := F y r + rq. and The sets Z y r, (y Q 0 ), are the covers of the boundary we are interested in. We now show a bound on the size of the cover, and then an average density estimate for sets in the neighbourhood of this famiy of covers. Then we wi prove further emmas. Lemma 5.1. Let J be as in Definition 5.1. There then exists a constant C 18 = C 18 (J) such that for each r > 0 and i = 1,..., N there are K Cr 2 m open bas B 1,..., B K of diameter at most r with V Λi K k=1 B k. Proof. This is a consequence of the Lipschitz boundary property Definition 5.1(v). We take an open cover U 1,..., U M of V Λi such that V Λi U n is a Lipschitz graph (in the (m 1)-dimensiona space zλ i ) for each n = 1,..., M. Each of the sets V Λi U n, may, as a Lipschitz graph of dimension m 2, triviay be covered by C i,n r 2 m open bas of diameter at most r, for some C i,n = C i,n (J). Lemma 5.2. # F r C 19 r 2 m for C 19 = C 19 (J). Proof. One simpy considers the cover of V i by K Cr 2 m bas B 1,... B K of diameter r from Lemma 5.1. Since g i is Lipschitz of factor at most L, covering the images g i (B n ) by squares rq + ξ with ξ rz m produces the caim. Lemma 5.3. Let J be as in Definition 5.1 and J = N i=1 Λ i for Lipschitz (m 1)-graphs {Λ i }N i=1. Then there exists a constant C 20 = C 20 (J, N, m) such that for every r > 0 and h (0, r], we have the bound H m 1( J (Zr y + B(0, h)) \ Z y ) r dy C20 h. (5.1) Q 0 Proof. As χ F y r +rq(x) = ξ F r χ ξ+ry+rq (x) for L m -a.e. y Q 0, we begin by cacuating χ F y r +rq(x) dy = Q 0 Q 0 χ ξ+ry+rq (x) dy = r m ξ F r ξ F r rq 0 χ ξ+y+rq (x) dy. Using χ F y r +rq+b(0,h)(x) ξ F r χ ξ+ry+rq+b(0,h) (x), we simiary get the inequaity Q 0 χ F y r +rq+b(0,h)(x) dy r m ξ F r rq 0 χ ξ+y+rq+b(0,h) (x) dy. 22