Relaxation and integral representation for functionals of linear growth on metric measure spaces

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincar\'e inequality. Such a functional is defined through relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem related to the functional, boundary values can be presented as a penalty term.


Introduction
Let f : R+ ! R+ be a convex, nondecreasing function that satises the linear growth condition with some constants < m ≤ M < ∞. Let Ω be an open set on a metric measure space (X, d, µ). Throughout the work we assume that the measure is doubling and that the space supports a Poincaré inequality. For u 2 L loc (Ω), we dene the functional of linear growth via relaxation by where gu i is the minimal 1-weak upper gradient of u i . For f (t) = t, this gives the denition of functions of bounded variation, or BV functions, on metric measure spaces, see [1], [3] and [24]. For f (t) = p + t , we get the generalized surface area functional, which has been considered previously in [17] and [18]. Our rst result shows that if F(u, Ω) < ∞, then F(u, ·) is a Borel regular outer measure on Ω. This result is a generalization of [24,Theorem 3.4]. For corresponding results in the Euclidean case with either the Lebesgue measure or more general measures, we refer to [2], [4], [8], [9], [10], [13], [14], and [15].
Our main goal is to study whether the relaxed functional F(u, ·) can be represented as an integral in terms of the variation measure kDuk, as can be done in the Euclidean setting, see e.g. [2,Section 5.5]. To this end, let u 2 L (Ω) with F(u, Ω) < ∞. Then the growth condition implies that u 2 BV(Ω). We denote the decomposition of the variation measure kDuk into the absolutely continuous and singular parts by dkDuk = a dµ + dkDuk s , where a 2 L (Ω). Similarly, we denote by F a (u, ·) and F s (u, ·) the absolutely continuous and singular parts of F(u, ·) with respect to µ. For the singular part, we obtain the integral representation where f∞ = lim t!∞ f (t)/t. This is analogous to the Euclidean case. However, for the absolutely continuous part we only get an integral representation up to a constant where C depends on the doubling constant of the measure and the constants in the Poincaré inequality. Furthermore, we give a counterexample which shows that the constant cannot be dismissed. We observe that working in the general metric context produces signicant challenges that are already visible in the Euclidean setting with a weighted Lebesgue measure. In overcoming these challenges, a key technical tool is an equi-integrability result for the discrete convolution of a measure. As a by-product of our analysis, we are able to show that a BV function is actually a Newton-Sobolev function in a set where the variation measure is absolutely continuous.
As an application of the integral representation, we consider a minimization problem related to functionals of linear growth. First we dene the concept of boundary values of BV functions, which is a delicate issue already in the Euclidean case. Let Ω b Ω * be bounded open sets in X, and assume that h 2 BV(Ω * ). We dene BV h (Ω) as the space of functions u 2 BV(Ω * ) such that u = h µ-almost everywhere in Ω * \ Ω. A function u 2 BV h (Ω) is a minimizer of the functional of linear growth with boundary values h, if where the inmum is taken over all v 2 BV h (Ω). It was shown in [17] that this problem always has a solution. By using the integral representation, we can express the boundary values as a penalty term. More precisely, under suitable conditions on the space and Ω, we establish equivalence between the above minimization problem and minimizing the functional F(u, Ω) + f∞ˆ∂ Ω |T Ω u − T X\Ω h|θ Ω dH over all u 2 BV(Ω). Here T Ω u and T X\Ω u are boundary traces and θ Ω is a strictly positive density function. This extends the Euclidean results in [14, p. 582] to metric measure spaces. A careful analysis of BV extension domains and boundary traces is needed in the argument.

Preliminaries
In this paper, (X, d, µ) is a complete metric measure space with a Borel regular outer measure µ. The measure µ is assumed to be doubling, meaning that there exists a constant c d > such that < µ(B(x, r)) ≤ c d µ(B(x, r)) < ∞ for every ball B(x, r) with center x 2 X and radius r > . For brevity, we will sometimes write λB for B(x, λr). On a metric space, a ball B does not necessarily have a unique center point and radius, but we assume every ball to come with a prescribed center and radius. The doubling condition implies that µ(B(y, r)) µ(B(x, R)) ≥ C ⇣ r R ⌘ Q (2.1) for every r ≤ R and y 2 B(x, R), and some Q > and C ≥ that only depend on c d . We recall that a complete metric space endowed with a doubling measure is proper, that is, closed and bounded sets are compact. Since X is proper, for any open set Ω ⇢ X we dene Lip loc (Ω) as the space of functions that are Lipschitz continuous in every Ω 0 b Ω (and other local spaces of functions are dened similarly).
Here Ω 0 b Ω means that Ω 0 is open and that Ω 0 is a compact subset of Ω. For any set A ⇢ X, the restricted spherical Hausdor content of codimension is dened as where < R < ∞. The Hausdor measure of codimension of a set A ⇢ X is The measure theoretic boundary ∂ * E is dened as the set of points x 2 X in which both E and its complement have positive density, i.e.
A curve is a rectiable continuous mapping from a compact interval to X. The length of a curve is denoted by` . We will assume every curve to be parametrized by arc-length, which can always be done (see e.g. [16,Theorem 3.2]).
A nonnegative Borel function g on X is an upper gradient of an extended real-valued function u on X if for all curves in X, we have whenever both u(x) and u(y) are nite, and´ g ds = ∞ otherwise. Here x and y are the end points of . If g is a nonnegative µ-measurable function on X and (2.2) holds for -almost every curve, then g is a -weak upper gradient of u. A property holds for -almost every curve if it fails only for a curve family with zero -modulus.
A family Γ of curves is of zero -modulus if there is a nonnegative Borel function ρ 2 L (X) such that for all curves 2 Γ, the curve integral´ ρ ds is innite. We consider the following norm where the inmum is taken over all upper gradients g of u. The Newtonian space is dened as where the equivalence relation ⇠ is given by u ⇠ v if and only if ku − vk N , (X) = . In the denition of upper gradients and Newtonian spaces, the whole space X can be replaced by any µ- measurable (typically open) set Ω ⇢ X. It is known that for any u 2 N , loc (Ω), there exists a minimal -weak upper gradient, which we always denote by gu, satisfying gu ≤ g µ-almost everywhere in Ω, for any -weak upper gradient g 2 L loc (Ω) of u [5,Theorem 2.25]. For more on Newtonian spaces, we refer to [26] and [5].
Next we recall the denition and basic properties of functions of bounded variation on metric spaces, see [1], [3] and [24]. For u 2 L loc (X), we dene the total variation of u as where gu i is the minimal -weak upper gradient of u i . We say that a function u 2 L (X) is of bounded variation, and write u 2 BV(X), if kDuk(X) < ∞. Moreover, a µ-measurable set E ⇢ X is said to be of nite perimeter if kDχ E k(X) < ∞. By replacing X with an open set Ω ⇢ X in the denition of the total variation, we can dene kDuk(Ω). For an arbitrary set A ⇢ X, we dene For an open set Ω ⇢ X and a set of locally nite perimeter E ⇢ X, we know that [1,Theorem 5.3] and [3,Theorem 4.6]. The constant c P is related to the Poincaré inequality, see below.
The jump set of a function u 2 BV loc (X) is dened as where u^and u _ are the lower and upper approximate limits of u dened as Outside the jump set, i.e. in X \ Su, H-almost every point is a Lebesgue point of u [20,Theorem 3.5], and we denote the Lebesgue limit at x by e u(x). We say that X supports a (, )-Poincaré inequality if there exist constants c P > and λ ≥ such that for all balls B(x, r), all locally integrable functions u, and all -weak upper gradients g of u, we havê If the space supports a (, )-Poincaré inequality, by an approximation argument we get for every u 2 L loc (X) where the constant c P and the dilation factor λ are the same as in the (, )-Poincaré inequality. When u = χ E for E ⇢ X, we get the relative isoperimetric inequality Throughout the work we assume, without further notice, that the measure µ is doubling and that the space supports a (, )-Poincaré inequality.

Functional and its measure property
In this section we dene the functional that is considered in this paper, and show that it denes a Radon measure. Let f be a convex nondecreasing function that is dened on [, ∞) and satises the linear growth condition for all t ≥ , with some constants < m ≤ M < ∞. This implies that f is Lipschitz continuous with constant L > . Furthermore, we dene where the second equality follows from the convexity of f . From the denition of f∞, we get the simple estimate for all t ≥ . This will be useful for us later. Now we give the denition of the functional. For an open set Ω and u 2 N , (Ω), we could dene it as where gu is the minimal 1-weak upper gradient of u. For u 2 BV(Ω), we need to use a relaxation procedure as given in the following denition.
where gu i is the minimal 1-weak upper gradient of u i .
Note that we could equally well require that gu i is any 1-weak upper gradient of u i . We dene F(u, A) for an arbitrary set A ⇢ X by for any A ⇢ X. This estimate follows directly from the denition of the functional, the denition of the variation measure, and (3.1). It is also easy to see that for any sets B ⇢ A ⇢ X.

Remark 3.2.
In this remainder of this section we do not, in fact, need the convexity of f , or the fact that the space supports a (, )-Poincaré inequality.
In order to show the measure property, we rst prove a few lemmas. The rst is the following technical gluing lemma that is similar to [2, Lemma 5.44].
, v 2 Lip loc (V) and ε > . Choose k 2 N such that if the above integral is nite -otherwise the desired estimate is trivially true. For i = , . . . , k, dene the sets so that H S k i= H i , and dene the Lipschitz functions see [5,Lemma 2.18]. By also using the estimate In the last inequality we used (3.5). Thus we can nd an index i such that the function w = w i satises the desired estimate.
In the following lemmas, we assume that u 2 L loc (A [ B). and for every i 2 N. In the above inequality, the last integral converges to zero as i ! ∞, since H b B and Exhausting A with sets B concludes the proof, since then F(u, A \ B ) ! by (3.4).
Proof. First we note that every Therefore, according to Lemma 3.4, it suces to show that for every i 2 N. By the properties of H, the last integral in the above inequality converges to zero as i ! ∞, and then

Lemma 3.6. Let A, B ⇢ X be open and let
and Then, since A and B are disjoint, Unauthenticated Download Date | 12/5/16 11:31 AM Now we are ready to prove the measure property of the functional.

Theorem 3.7.
Let Ω ⇢ X be an open set, and let u 2 L loc (Ω) with F(u, Ω) < ∞. Then F(u, ·) is a Borel regular outer measure on Ω.
Proof. First we show that F(u, ·) is an outer measure on Ω. Obviously F(u, ;) = . As mentioned earlier, and thus letting n ! ∞ and ε ! gives us For general sets A i ⇢ Ω, we can prove (3.6) by approximation with open sets. The next step is to prove that Dene the sets . Thus by Lemma 3.6, Now letting ε ! shows that F(u, ·) is a Borel outer measure by Carathéodory's criterion. The measure F(u, ·) is Borel regular by construction, since for every where V A is a Borel set. As a simple application of the measure property of the functional, we show the following approximation result. and On the other hand, for any relatively closed set F ⇢ Ω we have The last inequality follows from (3.7), since Ω \ F is open. By the measure property of the functional, we can subtract F(u, Ω \ F) from both sides to get According to a standard characterization of the weak* convergence of Radon measures, the above inequality and (3.7) together give the result [11, p. 54].

Integral representation
In this section we study an integral representation for the functional F(u, ·), in terms of the variation measure kDuk. First we show an estimate from below. Note that due to (3.4), F(u, Ω) < ∞ always implies kDuk(Ω) < ∞. Proof. Pick a sequence u i 2 Lip loc (Ω) such that u i ! u in L loc (Ω) and Using the linear growth condition for f , presented in (3.1), we estimate For a suitable subsequence, which we still denote by gu i , we have gu i dµ * * dν in Ω, where ν is a Radon measure with nite mass in Ω. Furthermore, by the denition of the variation measure, we necessarily have ν ≥ kDuk, which can be seen as follows. For any open set U ⇢ Ω and for any ε > , we can pick an open set U 0 b U such that kDuk(U) < kDuk(U 0 ) + ε; see e.g. Lemma 3.4. We obtain On the rst line we used the denition of the variation measure, and on the second line we used a property of the weak* convergence of Radon measures, see e.g. [ The following lower semicontinuity argument is from [2, p. 64-66]. First we note that as a nonnegative nondecreasing convex function, f can be presented as for some sequences d j , e j 2 R, with d j ≥ , j = , , . . ., and furthermore sup j d j = f∞ [2, Proposition 2.31, Lemma 2.33]. Given any pairwise disjoint open subsets of Ω, denoted by A , . . . , A k , k 2 N, and functions for every j = , . . . , k and i 2 N. Summing over j and letting i ! ∞, we get by the weak* convergence Since we had ν ≥ kDuk, this immediately implies We recall that dkDuk = a dµ + dkDuk s . It is known that the singular part kDuk s is concentrated on a Borel set D ⇢ Ω that satises µ(D) = and kDuk s (Ω\D) = , see e.g. [11, p. 42]. Dene the Radon measure σ = µ+kDuk s , and the Borel functions As mentioned above, we have sup j h j = h, and we can write the previous inequality as Since the functions ϕ j 2 Cc(A j ), ≤ ϕ j ≤ , were arbitrary, we get

Since this holds for any pairwise disjoint open subsets
However, by the denitions of h and σ, this is the same aŝ Combining this with (4.1), we get the desired estimate from below. It is worth noting that in the above argument, we only needed the weak* convergence of the sequence gu i dµ to a Radon measure that majorizes kDuk. Then we could use the fact that the functional for measures is lower semicontinuous with respect to weak* convergence of Radon measures. This lower semicontinuity is guaranteed by the fact that f is convex, but in order to have upper semicontinuity, we should have that f is also concave (and thus linear). Thus there is an important asymmetry in the setting, and for the estimate from above, we will need to use rather dierent methods where we prove weak or strong L -convergence for the sequence of upper gradients, instead of just weak* convergence of measures. To achieve this type of stronger convergence, we need to specically ensure that the sequence of upper gradients is equi-integrable. The price that is paid is that a constant C appears in the nal estimate related to the absolutely continuous parts. An example that we provide later shows that this constant cannot be discarded.
We recall that for a µ-measurable set H ⇢ X, the equi-integrability of a sequence of functions g i 2 L (H), i 2 N, is dened by two conditions. First, for any ε > there must exist a µ- We will need the following equi-integrability result that partially generalizes [12, Lemma 6]. For the construction of Whitney coverings that are needed in the result, see e.g. [ Then the sequence g i is equi-integrable in H. Moreover, a subsequence of g i converges weakly in L (H) to a functionǎ that satisesǎ ≤ co a µ-almost everywhere in H. Proof. To check the rst condition of equi-integrability, let ε > and take a ball B = B(x , R) with x 2 X and R > so large that ν(Ω \ B(x , R)) < ε/co. Then, by the bounded overlap property of the Whitney balls, we haveˆH To check the second condition, assume by contradiction that there is a sequence of µ-measurable sets We know that there exists δ > such that if A ⇢ Ω and µ(A) < δ, then´A a dµ < ε. Note that by the bounded overlap property of the Whitney balls, (4.2) Fix k 2 N. We can divide the above sum into two parts: let I consist of those indices j 2 N for which µ(A i \ B i j )/µ(B i j ) > /k, and let I consist of the remaining indices. We estimate when i is large enough. Now we can further estimate (4.2): for large enough i 2 N. By letting rst i ! ∞, then k ! ∞, and nally ε ! , we get a contradiction with  , r)).
By the Radon-Nikodym theorem, µ-almost every x 2 H satises By using these estimates as well as the previous one, we get for µ-almost every where the rst term on the right-hand side is co a by the Radon-Nikodym theorem, and the second term is zero. Thus we haveǎ ≤ co a µ-almost everywhere in H.
Now we are ready to prove the estimate from above.  Proof. Since the functional F(u, ·) is a Radon measure by Theorem 3.7, we can decompose it into the absolutely continuous and singular parts as F(u, ·) = F a (u, ·) + F s (u, ·). The singular parts kDuk s and F s (u, ·) are concentrated on a Borel set D ⇢ Ω that satises µ(D) = and see e.g. [11, p. 42].
First we prove the estimate for the singular part. Let ε > . Choose an open set G with D ⇢ G ⇢ Ω, such that µ(G) < ε and kDuk(G) < kDuk(D) + ε. Take a sequence u i 2 Lip loc (G) such that u i ! u in L loc (G) and Thus for some i 2 N large enough, we havê The latter inequality necessarily holds for large enough i by the denition of the functional F(u, ·). Now, using the two inequalities above and the estimate for f given in (3.2), we can estimate In the last inequality we used the properties of the set G given earlier. Letting ε ! , we get the estimate from above for the singular part, i.e. For every i 2 N, take a Whitney covering We know that u i ! u in L (G) as i ! ∞, and that each u i has an upper gradient Unauthenticated Download Date | 12/5/16 11:31 AM with C = C(c d , c P ), see e.g. the proof of [20,Proposition 4.1]. We can of course write the decomposition g i = g a i + g s i , where and .
By the bounded overlap property of the coverings, we can easily estimatê for every i 2 N, with e C = e C(c d , c P , λ). Furthermore, by Lemma 4.2 we know that the sequence g a i is equiintegrable and that a subsequence, which we still denote g a i , converges weakly in L (G) to a functionǎ ≤ Ca, with C = C(c d , λ). By Mazur's lemma we have for certain convex combinations, denoted by a hat, where d i,j ≥ and P N i j=i d i,j = for every i 2 N [25, Theorem 3.12]. We note that b u i 2 Lip loc (G) for every i 2 N (the hat always means that we take the same convex combinations), b u i ! u in L loc (G), and g b u i ≤ b g i µ-almost everywhere for every i 2 N (recall that gu always means the minimal -weak upper gradient of u). Using the denition of F(u, ·), the fact that f is L-Lipschitz, and (4.4), we get By letting ε ! we get the estimate from above for the absolutely continuous part, i.e. By combining this with (4.3), we get the desired estimate from above.

Remark 4.5. By using Theorems 4.1 and 4.4, as well as the denition of the functional for general sets given in (3.3), we can conclude that for any µ-measurable set
where F a (u, ·) and F s (u, ·) are again the absolutely continuous and singular parts of the measure given by the functional. We obtain, to some extent as a by-product of the latter part of the proof of the previous theorem, the following converse, which also answers a question posed in [20]. A later example will show that the constant C is necessary here as well.   with C = C(c d , c P , λ). Note also that the proof of [16, Lemma 7.8], which we used above, is also based on Mazur's lemma, so the techniques used above are very similar to those used in the proof of Theorem 4.4.
Finally we give the counterexample that shows that in general, we can have The latter inequality answers a question raised in [24] and later in [3].
The unweighted integral of g and each g i over X is . Next dene the function Unauthenticated Download Date | 12/5/16 11:31 AM Now u is in N , (X) and even in Lip(X), since g is bounded. In this -dimensional setting, it can be seen that every 1-weak upper gradient of u is in fact an upper gradient, and then it is easy to see that the minimal 1-weak upper gradient of u is g. Approximate u with the functions The functions u i are Lipschitz, and they converge to u in L (X) and even uniformly, which can be seen as follows. Given i 2 N, the set A i consists of i intervals of length α i / i . If I is one of these intervals, we have −i =ˆI g dL =ˆI g i+ dL , (4.5) and alsoˆX Hence u i+ = u at the end points of the intervals that make up A i , and elsewhere |u i+ − u| is at most −i by (4.5).
Clearly the minimal 1-weak upper gradient of u i is g i . However, we havê Thus the total variation is strictly smaller than the integral of the minimal 1-weak upper gradient, demonstrating the necessity of the constant C in Theorem 4.6. On the other hand, any approximating sequence v i 2 Lip(X) satisfying v i ! u in L (X) converges, up to a subsequence, to u also pointwise µ-and thus Lalmost everywhere, and then we necessarily have for some such sequence  B(x, r)) ≤ , and if x 2 X \ A, we clearly have that the derivative is . On the other hand, if the derivative were strictly smaller than in a subset of A of positive µ-measure, we would get kDuk(X) < , which is a contradiction with the fact that kDuk(X) = . Thus dkDuk = a dµ with a = χ A . ¹ 1 We can further show that g i dµ * * a dµ in X, but we do not have g i ! a weakly in L (X), demonstrating the subtle dierence between the two types of weak convergence.

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Download Date | 12/5/16 11:31 AM To show that we can have F a (u, X) >´X f (a) dµ -note that F a (u, X) = F(u, X) -assume that f is given by (We could equally well consider other nonlinear f that satisfy the earlier assumptions.) Since a = χ A , we havê On the other hand, for some sequence of Lipschitz functions v i ! u in L (X), we have (4.7) By considering a subsequence, if necessary, we may assume that v i ! u pointwise µ-and thus L -almost everywhere. By Proposition 3.8, we have for any closed set Applying these two equalities together with the inequality f (t) ≥ t − , we obtain The last term on the last line can be made arbitrarily small. Inserting this into (4.7), we get The last inequality follows from the pointwise convergence of v i to u L -almost everywhere. Roughly speaking, we note that the total variation kDuk(X) is found to be unexpectedly small because the growth of the approximating functions u i is concentrated outside the Cantor set A, where it is "cheaper" due to the smaller value of the weight function. However, when we calculate F(u, X), the same does not work, because now the nonlinear function f places "extra weight" on upper gradients that take values larger than .
where the inmum is taken over all v 2 BV h (Ω).
Note that if u 2 L loc (Ω * ) and u = h in Ω * \ Ω, then u 2 L (Ω * ). Furthermore, if F(u, Ω * ) < ∞, then kDuk(Ω * ) < ∞ by (3.4). Thus it makes sense to restrict u to the class BV(Ω * ) in the above denition. Observe that the minimizers do not depend on Ω * , but the value of the functional does. Note also that the minimization problem always has a solution and that the solution is not necessarily continuous, see [17].
Since u = v µ-almost everywhere in Ω * \ Ω 0 , the rst terms on both sides of the inequality cancel out, and we have F(u, Ω 00 ) ≤ F(v, Ω 00 ).
Now we wish to express the boundary values of the minimization problem as a penalty term involving an integral over the boundary. To this end, we need to discuss boundary traces and extensions of BV functions.
The word "strong" refers to the condition kD(Eu)k(∂Ω) = , which is not (necessarily) part of the conventional denition of a BV extension domain. It can be understood as an additional regularity condition for the domain. As an example of a BV extension domain that fails to satisfy this additional condition, consider X = C = R and the slit disk This is a BV extension domain according to [21,Theorem 1.1]. However, the function u(z) = Arg(z) 2 BV(Ω) clearly cannot be extended such that the condition kD(Eu)k(∂Ω) = would be satised. These are the two conditions we will impose in order to have satisfactory results on the boundary traces of BV functions. Based on results found in [7], we proved in [22] that every bounded uniform domain is a strong BV extension domain and satises the weak measure density condition. An open set Ω is A-uniform, with constant A ≥ , if for every x, y 2 Ω there is a curve in Ω connecting x and y such that` ≤ Ad(x, y), and for all t 2 [,` ], we have dist The standard assumption in the classical Euclidean theory of boundary traces is a bounded domain with a Lipschitz boundary, see e.g. [2,Theorem 3.87]. It can be checked that such a domain is always a uniform domain, and so the theory we develop here is a natural generalization of the classical theory to the metric setting.
Now we give the denition of boundary traces.
Denition 5.6. For a µ-measurable set Ω and a µ-measurable function u on Ω, a real-valued function T Ω u dened on ∂Ω is a boundary trace of u if for H-almost every x 2 ∂Ω, we have Often we will also call T Ω u(x) a boundary trace if the above condition is satised at the point x. If the trace exists at a point x 2 ∂Ω, we clearly have where ap lim denotes the approximate limit. Furthermore, we can show that the trace is always a Borel function.
Let us recall the following decomposition result for the variation measure of a BV function from [3, Theorem 5.3]. For any open set Ω ⇢ X, any u 2 BV(Ω), and any Borel set A ⇢ Ω that is σ-nite with respect to H, we have The function θ and the lower and upper approximate limits u^and u _ were dened in Section 2. In particular, by [3, Theorem 5.3] the jump set Su is known to be σ-nite with respect to H. The following is our main result on boundary traces.
Theorem 5.7. Assume that Ω is a strong BV extension domain that satises the weak measure density condition, and let u 2 BV(Ω). Then the boundary trace T Ω u exists, that is, Proof. Extend u to a function Eu 2 BV(X). By the fact that kD(Eu)k(∂Ω) = and the decomposition (5.1), we have H(S Eu \ ∂Ω) = -recall that the function θ is bounded away from zero. Here Due to the Lebesgue point theorem [20,Theorem 3.5], we have in fact lim sup for H-almost every x 2 ∂Ω, where Q > was given in (2.1). However, we will not need this stronger result. Let us list some general properties of boundary traces.
Proposition 5.8. Assume that Ω is a µ-measurable set and that u and v are µ-measurable functions on Ω. The boundary trace operator enjoys the following properties for any x 2 ∂Ω for which both T Ω u(x) and T Ω v(x) exist: (iv) Let h > and dene the truncation u h = min{h, max{u, −h}}.
(v) If Ω is a µ-measurable set such that both Ω and its complement satisfy the weak measure density condition, and w is a µ-measurable function on X, then for H-almost every x 2 ∂Ω for which both traces T Ω w(x) and Proof. Assertions (i) and (ii) are clear. Since minimum and maximum can be written as sums by using absolute values, property (iii) follows from (i) and the easily veried fact that T Ω |u|(x) = |T Ω u(x)|. Assertion (iv) follows from (iii). In proving assertion (v), due to the symmetry of the situation we can assume that T Ω w(x) ≥ T X\Ω w(x). By using the denition of traces and Chebyshev's inequality, we deduce that for every ε > , To determine the lower and upper approximate limits, we use these results to compute To obtain the result "2 (, )" above, we used the weak measure density conditions. We conclude that w _ (x) = T Ω w(x), and since "lim sup" can be replaced by "lim inf" in the above calculation, we also get w^(x) = T X\Ω w(x).
A minor point to be noted is that any function that is in the class BV(X), such as an extension Eu of u 2 BV(Ω), is also in the class BV(Ω), and thus T Ω Eu = T Ω u. Eventually we will also need to make an additional assumption on the space, as described in the following denition that is from [3, Denition 6.1]. The function θ E was introduced earlier in (2.4). Denition 5.9. We say that X is a local space if, given any two sets of locally nite perimeter E ⇢ E ⇢ X, See [3] and [22] for some examples of local spaces, and [23,Example 5.2] for an example of a space that is not local, despite having a doubling measure and a Poincaré inequality. The assumption E ⇢ E can, in fact, be removed as follows. Note that for a set of locally nite perimeter E, we have kDχ E k = kDχ X\E k, i.e. the two measures are equal [24,Proposition 4.7]. From this it follows that θ E (x) = θ X\E (x) for H-almost every x 2 ∂ * E. Now, if E and E are arbitrary sets of locally nite perimeter, we know that E \ E and E \ E are also sets of locally nite perimeter [24,Proposition 4.7]. For every x 2 ∂ * E \ ∂ * E we have either x 2 ∂ * (E \ E ) or x 2 ∂ * (E \ E ). Thus by the locality condition, we have for H-almost every In a local space the decomposition (5.1) takes a simpler form, as proved in the following lemma.
Lemma 5.10. If X is a local space, Ω is a set of locally nite perimeter, u 2 BV(X), and A ⇢ ∂ * Ω is a Borel set, then we haveˆAˆu Note that since Ω is a set of locally nite perimeter, A ⇢ ∂ * Ω is σ-nite with respect to H. On the third line we used Fubini's theorem. On the fourth line we used the fact that if u^(x) < t < u _ (x), then x 2 ∂ * {u > t}. This follows from the denitions of the lower and upper approximate limits. By the locality condition we see that the right-hand side above equalŝ In the above characterization, we implicitly assume that the integral is well-dened -in particular, this is the case if Ω and Ω * \ Ω are also strong BV extension domains, due to Theorem 5.7. Furthermore, if X is a local space, we then have
To prove one direction of the proposition, let us assume (5.2). In particular, we assume that T Ω u(x) and T Ω * \Ω v(x) exist for H-almost every x 2 ∂Ω. For h > , dene the truncated functions u h = min{h, max{u, −h}} and v h = min{h, max{v, −h}}. Then see e.g. [20,Proposition 4.2]. Based on the decomposition of the variation measure given in (5.1), for H-almost every x 2 ∂Ω. Using Proposition 5.8 (iv) again, for H-almost every x 2 ∂Ω we have By the lower semicontinuity of the total variation as well as (5.3), (5.4) and (5.5), we now get Thus w 2 BV(Ω * ).
To prove the converse, assume that w 2 BV(Ω * ). Here we can simply again write the decomposition of the variation measure where α = α(c d , c P ) > , and just as earlier, note that for H-almost every x 2 ∂Ω. This combined with the previous estimate gives the desired result. If X is a local space, we combine the decomposition of the variation measure (5.1), Lemma 5.10, and (5.6) to obtain the last claim.
Next we show that if a set A (which could be e.g. the boundary ∂Ω) is in a suitable sense of codimension one, traces of BV functions are indeed integrable on A. Let us rst recall the following fact from the theory of sets of nite perimeter. Given any set of nite perimeter E ⇢ X, for H-almost every x 2 ∂ * E we have for every x 2 A and r 2 (, R], where R 2 (, dist(A, X \ Ω * )) and c A > are constants. Then where C = C(c d , c P , λ, A, R, c A ).
Thus we have δ = δ(c, ), and consequently δ = δ(c d , c P , λ, A, R). By the denition of E , we can nd a number r 2 (, R/] that satises This can be done by repeatedly halving the radius R/ until the right-hand side of the above inequality does not hold, and picking the last radius for which it did hold. From the relative isoperimetric inequality (2.5) we conclude that µ(B(x, r/λ)) r/λ ≤ c d µ(E \ B(x, r/λ)) r/λ ≤ C P (E, B(x, r)). (5.10) Using the radii chosen this way, we get a covering {B(x, r(x))} x2A\E of the set A \ E . By the 5-covering lemma, we can select a countable family of disjoint balls {B(x i , r i )} ∞ i= such that the balls B(x i , r i ) cover A \ E . By using (5.8) and (5.10), we get where C = (c d , c P , λ, c A ).
Then we consider the function u. Assume that x 2 A \ Su and u^(x) + u _ (x) > t, with t > . By the denitions of the lower and upper approximate limits, we know that x 2 ∂ * {u > s} for all s 2 (u^(x), u _ (x)). By the coarea formula (2.3), the sets {u > s} are of nite perimeter in Ω * for every s 2 T, where T is a countable dense subset of R. Thus, outside a H-negligible set, (5.7) holds for every x 2 ∂ * {u > s} and s 2 T. Assuming that x is outside this H-negligible set, we can nd s 2 ((u^(x) + u _ (x))/, u _ (x)) \ T and estimate This gives the estimate for A \ Su. For A \ Su, we simply note that if x 2 A \ Su and u^(x) = u _ (x) > t, then the approximate limit of u at x is larger than t, which easily gives x 2 {u > t} , and then we can use Cavalieri's principle as above.
Finally we get the desired representation for the minimization problem.
Theorem 5.13. Assume that X is a local space, and let Ω b Ω * be bounded open sets such that Ω and Ω * \ Ω satisfy the weak measure density condition, Ω is a strong BV extension domain, and ∂Ω satises the assumptions of Proposition 5.12. Assume also that h 2 BV(Ω * ) and that the trace T X\Ω h(x) exists for H-almost every x 2 ∂Ω, which in particular is true if Ω * \ Ω is also a strong BV extension domain. Then the minimization problem given in Denition 5.2, with boundary values h, can be reformulated as the minimization of the functional F(u, Ω) + f∞ˆ∂ Ω |T Ω u − T X\Ω h|θ Ω dH (5.12) over all u 2 BV(Ω).
Note that this formulation contains no reference to Ω * .
Proof. First note that due to the conditions of Proposition 5. |T Ω u − T X\Ω h|θ Ω dH + F(h, Ω * \ Ω), (5.13) where the rst equality follows from the measure property of F(u, ·) as well as the fact that µ(∂Ω) = , the second equality follows from the integral representation of the functional (see Remark 4.5), the third equality follows from the decomposition (5.1) and Lemma 5.10, and the fourth equality follows from Proposition 5.8 (v). Now, the term F(h, Ω * \ Ω) does not depend on u, so in fact we need to minimize (5.12). Conversely, assume that u 2 BV(Ω). Then we can extend u to Eu 2 BV(Ω * ). By Proposition 5.8 (v) we have for H-almost every x 2 ∂Ω. By the proof of Theorem 5.7 we have that T Ω Eu(x) is the Lebesgue limit of Eu for H-almost every x 2 ∂Ω. By Proposition 5.12, we now get
Remark 5.14. Note that in the latter part of the above proof we showed that, under the assumptions on the space and on Ω, the spaces BV(Ω) and BV h (Ω) ⇢ BV(Ω * ) can be identied.