QUANTITATIVE ANALYSIS OF FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS

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1 QUANTITATIVE ANALYSIS OF FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS ANNIKA BACH, ANDREA BRAIDES, AND CATERINA IDA ZEPPIERI Abstract. Motivated by appications to image reconstruction, in this paper we anayse a finite-difference discretisation of the Ambrosio-Tortorei functiona. Denoted by the eipticapproximation parameter and by the discretisation step-size, we fuy describe the reative impact of and in terms of Γ-imits for the corresponding discrete functionas, in the three possibe scaing regimes. We show, in particuar, that when and are of the same order, the underying attice structure affects the Γ-imit which turns out to be an anisotropic freediscontinuity functiona. Keywords: finite-difference discretisation, Ambrosio-Tortorei functiona, Γ- convergence, eiptic approximation, free-discontinuity functionas. 000 Mathematics Subect Cassification: 49M5, 49J45, 68U0, 65M06.. Introduction The detection of obects and obect contours in images is a centra issue in Image Anaysis and Computer Vision. From a mathematica modeing standpoint, a grey-scae image can be described in terms of a scaar function g : [0, ] here, R n is a set parameterising the image domain, e.g., a rectange in the pane, which measures, at every point in, the brightness or grey-eve of the picture. After a mode introduced by Mumford and Shah [48], the reevant information from an input image g can be obtained from a restored image described by a function u which soves the minimisation probem { } min MSu + u g dx: u SBV,. where MSu = u dx + H n S u. is the so-caed Mumford-Shah functiona and SBV denotes the space of specia functions of bounded variation in [4], S u denotes the discontinuity set of u, and H n is the n - dimensiona Hausdorff measure. By soving. the discontinuous function g is repaced by a function u which is cose to g and at the same time is smooth outside its discontinuity set S u. The atter, moreover, having a minima n -dimensiona Hausdorff measure wi ony detect the reevant contours in the input image g. We note that a more compete Mumford- Shah functiona woud be of the form α u dx + βh n S u with α, β positive contrast parameters. In the anaysis carried out in the present paper it is not restrictive to set α = β =. Athough the reevant space dimension for Image Anaysis is n =, we define our probems in a n-dimensiona setting for the sake of generaity, and aso because in the case n = 3 the Mumford-Shah functiona has an important mechanica interpretation as it coincides with Griffith s fracture energy in the anti-pane case see [6]. Probem. is a weak formuation proposed by De Giorgi and Ambrosio of the origina minimisation probem proposed by Mumford and Shah, where the minimisation is performed on pairs u, K, with K piecewise-reguar cosed set and u smooth function outside K. In the weak formuation.-. the set K is repaced

2 A. BACH, A. BRAIDES, C.I. ZEPPIERI by the discontinuity set of u, and a soution of the origina probem is obtained by setting K = S u and proving reguarity properties of K see the recent review paper [4]. The existence of soutions to. foowing the direct methods of the Cacuus of Variations is by now cassica [6]. However, the numerica treatment of. presents maor difficuties which are mainy due to the presence of the surface term H n S u. A way to circumvent these difficuties is to repace the Mumford-Shah functiona in. with an eiptic approximation studied by Ambrosio-Tortorei [9, 0], which provides one of the reference approximation argument used in the iterature see e.g. [, 36, 37, 47, 49, 5]. Foowing the Ambrosio-Tortorei approximation argument, in pace of. one considers a famiy of scae-dependent probems { } min AT u, v + u g dx: u, v W,,.3 where AT u, v = v + η u dx + v + v dx..4 Formay, when the approximation parameter > 0 is sma, the first term in the second integra of.4 forces v to be cose to the vaue except on a sma set, which can be regarded as an approximation of S u. Additionay, the presence of the term v in the first integra aows u to have a arge gradient where v is cose to zero. Finay, the optimisation of the singuarperturbation term with v produces a transition ayer around S u giving exacty the surface term present in MS. The parameter η > 0 is used in the numerica simuations in order to have we-posed minimisation probems in.3; it is taken much smaer than, but does not intervene in the mathematica anaysis. It is interesting to note that the coefficient v + η can be aso interpreted as a damage parameter see e.g. [44], so that, within Fracture Theory, AT can be seen as an approximation of Griffith s Fracture by concentrated damage. More in genera the functionas AT are a prototype of phase-fied modes for free-discontinuity probems. Since the functionas in.3 are equi-coercive and AT converge to MS in the sense of Γ- convergence [0], soving.3 gives pairs u, v, where u approximates a soution u to. and v provides a diffuse approximation of the corresponding discontinuity set S u. Moreover, since the functionas AT are eiptic, the difficuties arising in the discretisation of the freediscontinuity set are prevented and finite-eements or finite-difference schemes for AT can be impemented. From the Γ-convergence of the numerica approximations of AT with mesh size at fixed and the Γ-convergence of AT to MS, a diagona argument shows that if the mesh size = is fine enough then the numerica approximations of AT with mesh size Γ-converge to MS. For finite-eements schemes Beettini and Coscia []; see aso Bourdin [5] for the numerica impementation showed more precisey that this hods if the mesh-size is chosen such that. In other words, this assumption on aows a separation-of-scae argument and to regard separatey the two imits as and tend to 0, respectivey. Conversey, note that if >> then the second integra in.4 diverges uness v is uniformy cose to, which impies that the domain of the Γ-imit of numerica approximations of AT with mesh size is with u W,, and hence the Γ-imit is not MS see aso the arguments of Section 6. In order to iustrate in genera the combined effect of and, in particuar when is of the same order of, we briefy reca some anayses which started from a different discrete approximation scheme for.. Chamboe in [33], considered a finite-difference approximation of MS based on an earier mode by Bake and Zissermann [3]: in the case of space-dimension

3 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 3 n =,, Chamboe studied the asymptotic behaviour of the discrete functionas given by F u = { n ui u min, },.5 i, Z n i = where the energies depend on finite differences through a truncated quadratic potentia with threshod energy /. If n = he showed that the functionas F Γ-converge to MS with respect to an appropriate discrete-to-continuum convergence of attice functions. In dimension n =, however, the Γ-imit of F turns out to be anisotropic and given by F u = u dx + ν u dh,.6 S u where ν u denotes the norma to S u and ν = ν + ν is the -norm of the vector ν, which appears in the imit due to the specific geometry of the underying attice Z. Using the attice energy.6 as a mode, some continuum approximations of the origina isotropic Mumford-Shah energy have been obtained. Notaby, a continuum finite-difference approximation was conectured by De Giorgi and proved by Gobbino [45], whie a non-oca version invoving averages of gradients in pace of finite differences was proved by Braides and Da Maso []. Various modifications of F have been studied, many of which in the direction of obtaining more genera surface terms in the imit energies. In [34] Chamboe introduced a variant of.5 where arbitrary finite differences and truncated energy densities with variabe threshod energies are considered. He showed that this new cass of functionas provides discrete approximations of image-segmentation functionas where the anisotropy is reduced with respect to.6 see aso the paper by Braides and Gei [4]. Braides and Piatnitski [6] examined random mixtures of truncated quadratic and simpy quadratic interactions producing surface energies whose anisotropy can be described through percoation resuts, whereas in the recent paper [50] Ruf shows that the anisotropy in the imit functiona can be prevented by considering discrete approximating functionas defined on statisticay isotropic attices. The form of the surface energy can be studied separatey by examining energies on attice spin functions see e.g. [3, 3] and [0] and the references therein; in particuar patterns of interactions corresponding to different threshod vaues in the truncated quadratic potentias satisfying design constraints and giving arbitrary surface energies has been recenty described by Braides and Kreutz [5]. As finitedifference schemes invoving energies as in.5 are concerned, in [35] Chamboe and Da Maso show that macroscopic anisotropy can be avoided by considering aternate finite-eements of suitabe oca approximations of the Mumford-Shah functiona. The finite-difference schemes described above suggest that in the numerica impementation of the Ambrosio-Tortorei approximation, for genera vaues of the mesh-size and the parameter the anisotropy of the surface term cannot be rued out as in the case considered in []. In terms of Γ-convergence, we may expect that for a genera dependence of on a discretisation of AT with mesh size sha not converge to the Mumford-Shah functiona but rather to some anisotropic functiona of the form Eu = u dx + ϕν u dh n,.7 S u where the surface integrand ϕ refects the geometry of the underying attice and may depend on the interaction between and. These considerations motivate the anaysis carried out in the present paper. In the spirit of a recent paper by Braides and Yip [3] in which the discretisation of the Modica-Mortoa functiona [46] is anaysed, here we propose and anayse a finite-difference

4 4 A. BACH, A. BRAIDES, C.I. ZEPPIERI discretisation of the Ambrosio-Tortorei functionas; i.e., we consider the functionas defined as E u, v = n v i u i u + n vi + n v i v.8 i, Z n i = i Z n i, Z n i = and we study their imit behaviour as and simutaneousy tend to zero. Since the discrete functionas in.8 are more expicit than the Beettini-Coscia finite-eements discretisation, we are in a position to perform a rather detaied Γ-convergence anaysis for E in a the three possibe scaing regimes; i.e., subcritica regime, critica regime, and supercritica regime. More precisey, if := im, in Theorem. we prove that for every [0, + ] and for n = the functionas E Γ-converge to E u = u dx + ϕ ν u dh, S u for some surface integrand ϕ : S [0, + ]. Furthermore, we show that in the subcritica regime ϕ 0 so that E 0 = MS, in the critica regime ϕ expicity depends on the norma ν see.0, beow, and finay in the supercritica regime ϕ +, so that E is finite ony on the Soboev Space W, and it coincides with the Dirichet functiona. It is worth mentioning that the convergence resuts in the extreme cases = 0 and = + actuay hod true in any space dimensions see Section 4 for the case = 0 and Section 6 for the case = +, whereas the convergence resut in the critica case is expicit ony for n =,. In fact, for 0, + the surface integrand ϕ can be expicity determined ony for n =, see Theorem 5.0 and Remark 5.6, whie for n > we can ony prove an abstract compactness and integra representation resut see Theorem 5.3 and Theorem 5.5 which, in particuar, does not aow us to excude that the surface energy density may aso depend on the ump opening. The main difference between the case n = and n = 3 and higher is reated to the probem of describing the structure of the sets of attice sites where the parameter v is cose to 0, which approximates the set ump S u. In principe, if that discrete set presents hoes the imit surface energy may depend on the vaues u ± of u on both sides of S u see [9]. In two dimensions this is rued out by showing that such attice sets can be ocay approximated by a continuous ine. In dimension n > deducing that such set is approximatey described by a hypersurface seems more compex and in this case the difficuties are simiar to those encountered in some attice spin probems e.g., when deaing with diute attice spin systems [7]. Beow we briefy outine the anaysis carried out in the present paper, in the three different scaing regimes. Subcritica regime: = 0. In this regime the Γ-imit of the finite-difference discretisation E is the Mumford-Shah functiona MS, as in the case of the finite-eements discretisation anaysed by Beettini and Coscia in []. Even if the scaing regime is the same as in [], the proof of the Γ-convergence resut for E is substantiay different. In particuar, the most deicate part in the proof of the Γ-convergence resut is to show that the ower-bound estimate hods true. Indeed, in our case the form and the non-convexity of the first term in E makes it impossibe to have an inequaity of the type E u, v AT ũ, ṽ + o, where ũ and ṽ denote suitabe continuous interpoations of u and v, respectivey. Then, to overcome this difficuty we first prove a non-optima asymptotic ower bound for E which aows us to show that the domain of the Γ-imit is GSBV see Proposition 3.4. Subsequenty, we combine this information with a carefu bow-up anaysis, which eventuay provides us with the

5 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 5 desired optima ower bound see Proposition 4.. Finay, the upper-bound inequaity foows by an expicit construction see Proposition 4.. Critica regime: 0,. When the two scaes and are comparabe, we appea to the so-caed direct methods of Γ-convergence to determine the Γ-imit of E. Namey, we show that E admits a Γ-convergent subsequence whose imit is an integra functiona of the form E u = u dx + φ [u], ν u dh n,.9 S u for some Bore function φ : R S n [0, +. Here in genera the surface integrand φ depends on the subsequence, on the ump-opening [u] = u + u, and on the norma ν u to the ump-set S u. The deicate part in the convergence resut as above is to show that the abstract Γ- imit satisfies the assumptions needed to represent it in an integra form as in.9. Specificay, a so-caed fundamenta estimate for the functionas E is needed see Proposition 5.. For n =, which is the most reevant case for the appications we have in mind, we are abe to expicity characterise the function φ. In particuar we prove that φ does not depend on the subsequence and on the ump-ampitude [u]. Specificay, we show that φ ϕ where ϕ ν = im T + T inf { i T Q ν Z v i + i, T Q ν Z i = v i v : channe C in T Q ν Z : v = 0 on C, v = otherwise near T Q ν}..0 In.0 a channe C see Definition 5.9 for a forma definition is a path on the square attice Z connecting two opposite sides of the square and can be interpreted as a discrete approximation of the discontinuity ine {x R : x, ν = 0}. When ν = e, e we show that the channe C in.0 is actuay fat and it coincides with the discrete interface {x R : x, ν = 0} Z. As a consequence, the minimisation probem defining ϕ turns out to be one-dimensiona see Remark 5.6. For n = the function φ is aso expicit and equa to a constant; the proof of this fact is a consequence of more eementary one-dimensiona arguments and is briefy discussed in Remark 5.6. Supercritica regime: = +. In this scaing regime discontinuities have a cost proportiona to / and are therefore forbidden. In fact the Γ-imit E turns out to be finite ony in W, see Proposition 3.4 and E u = u dx. In order to aow for the deveopment of discontinuities in the imit, in the spirit of Braides and Truskinovsky [30], in this case we aso anayse the asymptotic behaviour of a suitaby rescaed variant of E whose Γ-imit is sti a functiona of the form.7 with ϕν = ν see Theorem 6., so that in this supercritica regime we recover a crystaine surface energy. The paper is organised as foows. In Section we introduce a few notation and state the main Γ-convergence resut Theorem.. In Section 3 we determine the domain of the Γ-imit in the three scaing regimes, we prove an equicoercivity resut for a suitabe perturbation of the fuctionas E, and study the convergence of the associated minimisation probems. In Sections 4, 5, and 6 we prove the Γ-convergence resut Theorem., respectivey, in the subcritica, critica, and supercritica regime. In Section 6 we aso anayse the asymptotic behaviour of a sequence of functionas which is equivaent to E in the sense of Γ-convergence see [30]. Eventuay, in

6 6 A. BACH, A. BRAIDES, C.I. ZEPPIERI Section 7 we show that for n = and 0, +, the surface integrand ϕ interpoates the two extreme regimes = 0 and = +.. Setting of the probem and statement of the main resut Notation. Let n, we denote by R n an open bounded set of with Lipschitz boundary. We furthermore denote by A the famiy of a open subsets of and by A L A the famiy of a open subsets of with Lipschitz boundary. If A, A A are such that A A, we say that ϕ is a cut-off function between A and A if ϕ Cc A, 0 ϕ and ϕ on A. If t R we denote by t its integer part. If ν = ν,..., ν n R n we denote by ν the eucidian norm of ν. Moreover, we set ν := n k= ν k and ν := max k n ν k. We use the notation ν, ξ for the scaar product between ν, ξ R n. We set S n := {ν R n : ν = } and for every ν S n we denote by Π ν := {x R n : x, ν = 0} the hyperpane through 0 and orthogona to ν. We aso denote by Π + ν and Π ν the two haf spaces defined, respectivey, as Π + ν := {x R n : x, ν > 0} and Π ν := {x R n : x, ν 0}. For every ν S n we denote by Q ν R n a given cube centred at 0 with side ength and with one face orthogona to ν, and for a x 0 R n and ρ > 0 we set Q ν ρx 0 = x 0 + ρq ν. If {e,..., e n } denotes the standard basis in R n and ν = e k for some k n we choose Q = Q ν the standard coordinate cube and simpy write Q ρ x 0. By L n and H k we denote the Lebesgue measure and the k-dimensiona Hausdorff measure in R n, respectivey. For p [, + ] we use standard notation L p for the Lebesgue spaces and W,p for the Soboev spaces. We denote by SBV the space of specia functions of bounded variation in for the genera theory see e.g., [8, 7]. If u SBV we denote by u its approximate gradient, by S u the approximate discontinuity set of u, by ν u for the generaised outer norma to S u, and u + and u are the traces of u on both sides of S u. We aso set [u] := u + u. Moreover, we consider the arger space GSBV, which consists of a functions u L such that for each m N the truncation of u at eve m defined as u m := m u m beongs to SBV. Furthermore, we set and SBV := {u SBV : u L and H n S u < + } GSBV := {u GSBV : u L and H n S u < + }. It can be shown that SBV L = GSBV L. Let u, w be two measurabe functions on R n and et A R n be open, bounded and with Lipschitz boundary; by u = w near A we mean that there exists a neighbourhood U of A in R n such that u = w L n -a.e. in U A. Setting. Throughout the paper > 0 is a stricty positive parameter and = > 0 is a stricty increasing function of such that such that 0 decreasingy as 0 decreasingy. Set := im 0.. We now introduce the discrete functionas which wi be anaysed in this paper. To this end et R n be open, bounded, and with Lipschitz boundary. Let := Z n denote the portion of the square attice of mesh-size contained in and for every u : R set u i := ui, for i. It is customary to identify the discrete functions defined on the attice with their piecewise-constant counterparts beonging to the cass A := {u L : u constant on i + [0, n for a i Z n },

7 by simpy setting FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 7 ux := u i for every x i + [0, n and for every i.. If u is a sequence of functions defined on the attice and u L, by u u in L we mean that the piecewise-constant interpoation of u defined as in. converges to u in L. We define the discrete functionas E : L L [0, + ] as E u, v := i, i = n v i u i u + n vi + i i, i = n v i v if u, v A, 0 v, + otherwise in L L,.3 It is aso convenient to consider the functionas F, G given by F u, v := n v i u i u.4 and i, i = G v := n vi + i i, i = n v i v so that in more compact notation we may write { F u, v + G v if u, v A, 0 v, E u, v := + otherwise in L L.,.5 In what foows we wi aso make use of the foowing equivaent expressions for F and G : F u, v = n n v i u i u i+e k n + u i u i e k i and k= i+e k G v = n vi + i i n k= i+e k k= i e k n vi v i+e k, For U A we wi aso need to consider the ocaised versions of F and G ; i.e., for every U A we set U := U Z n,.6 and F u, v, U := n v i i U G v, U := i U n v i + n k= i±e k U n k= i+e k U ui u i±e k vi v i+e k,,

8 8 A. BACH, A. BRAIDES, C.I. ZEPPIERI so that finay E u, v, U := { F u, v, U + G v, U if u, v A, 0 v, + otherwise in L L..7 Sometimes it wi be usefu to distinguish between points i U such that a their nearest neighbours beong to U and points i U such that i ± e k U for some k n. Then, for a given U A we set U := {i U : U for every Z n s.t. i = }, and U := U \ U. With the identification above, we wi describe the Γ-imits of energies E with respect to the strong L L -topoogy, in the spirit of recent discrete-to-continuum anayses see e.g. [4, 8, 5, ] for some genera resuts in different imit functiona settings and [8, 9] for some introductory materia. In a that foows we use the standard notation for the Γ-iminf and Γ-imsup of the functionas E see [8] Section.; i.e., for every u, v L L and every U A we set E u, v, U := Γ- im inf 0 E u, v, U and E u, v, U := Γ- im sup E u, v, U..8 0 When U = we simpy write E u, v and E u, v in pace of E u, v, and E u, v,, respectivey. The foowing Γ-convergence theorem is the main resut of this paper. Theorem. Γ-convergence. Let be as in. and et E be as in.3. Then, E i Subcritica regime If = 0 the functionas E Γ-converge to E 0 defined as u dx + H n S u if u GSBV, v = a.e. in, E 0 u, v := + otherwise in L L ; ii Critica regime If 0, + there exists a subsequence such that the functionas Γ-converge to E defined as u dx + φ [u], ν u dh n if u GSBV, v = a.e. in, E u, v := S u + otherwise in L L, for some Bore function φ : R S n [0, + possiby depending on the subsequence. If moreover n = the function φ does not depend on the subsequence. Furthermore, for every t, ν R S n we have φ t, ν = ϕ ν where ϕ : S n [0, + is given by ϕ ν := im T + { T inf i T Q ν Z v i + i, T Q ν Z i = v i v : v A T Q ν, channe C in T Q ν Z : v = 0 on C, v = otherwise near T Q ν }, see Definition 5.9 for a precise definition of channe;

9 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 9 iii Supercritica regime If = + the functionas E Γ-converge to E defined as u dx if u W,, v = a.e. in, E u, v := + otherwise in L L. The proof of Theorem. wi be divided into a number of intermediate steps and carried out in Sections 4, 5, and Domain of the Γ-imit and compactness In this section we prove a compactness resut for the functionas E. This resut is first obtained for n = and then extended to the case n by means of a sicing-procedure see [8] Section 5. The main resut of this section is the foowing. Theorem 3. Domain of the Γ-imit. Let u, v L L be such that u, v u, v in L L and sup E u, v < + >0 and et be as in.. i Subcritica and critica regime If [0, + then u GSBV and v = a.e. in. ii Supercritica regime If = + then u W, and v = a.e. in. The proof of Theorem 3. wi be carried out in Proposition 3. and Proposition 3.4 beow. 3.. The one-dimensiona case. In this subsection we dea with the case n =. In what foows we ony consider the case = I := a, b with a, b R, a < b. The case of a genera open set can be treated by repeating the proof beow in each connected component of. Proposition 3.. Let u, v L I L I be such that u, v u, v in L I L I and sup E u, v < + >0 and et be as in.. i Subcritica and critica regime If [0, + then u SBV I and v = a.e. in I. Moreover, E u, v u dt + #S u. I ii Supercritica regime If = + then u W, I and v = a.e. in I. Moreover, E u, v u dt. Proof. The proof wi be divided into two steps. Step : proof of i; i.e., the case [0, +. Let u, v L I L I be as in the statement. We caim that E u, v can be bounded from beow by AT ũ, ṽ for suitabe functions ũ and ṽ with ũ, ṽ u, v in L I L I. Then the concusion foows appeaing to the cassica Ambrosio and Tortorei convergence resut [0, Theorem.]. I

10 0 A. BACH, A. BRAIDES, C.I. ZEPPIERI For our purposes it is convenient to rewrite E as foows E u, v = vi + v i+ u i u i+ + vi i I i I i+ I + i I i+ I v i v i+ We define moreover ũ, ṽ as the piecewise affine interpoations of u, v on I, respectivey; i.e., ũ t := u i + ui+ ṽ t := v i + vi+ We note that ũ, ṽ u, v in L I L I. Let η > 0 be fixed; for sufficienty sma we have a + η, b η therefore v i v i+ i I u i t i if t [i, i +, i, i + I, v i t i if t [i, i +, i, i + I. = i I i+ i i I [i, i +, ṽ dt b η a+η ṽ dt, 3. for sma. Moreover, in view of the definition of ṽ and the convexity of z z, for every i I we get i+ from which we get i ṽ dt = i+ vi = i t i i+ i v i + t i t i v i + v i+, i I v i b η a+η vi+ dt dt + vi+ i+ i t i dt ṽ dt, 3. for sma. Finay, the definition of ũ together with the convexity of z z yied i+ ṽ ũ dt = u i u i+ i+ i t i v i + t i i vi+ dt u i u i+ v i i+ t i i+ dt + v i+ t i i i dt = vi + v i+ u i u i+, for every i I, and thus i I v i + v i+ u i u i+ b η a+η ṽ ũ dt, 3.3.

11 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS for sma. Eventuay, gathering we deduce E u, v AT ũ, ṽ, a + η, b η, 3.4 where AT denotes the Ambrosio-Tortorei functiona; i.e., AT u, v := v u d x + v + v dx, for every u, v W, W, with 0 v. Hence the thesis foows first appeaing to [0, Theorem.] and then by etting η 0. Step : proof of ii; i.e., the case = +. Let u, v L I L I be as in the statement, then in particuar sup >0 vi < +. i I Hence there exists a constant c > 0 such that for every i I and for every > 0 v i c. Let η 0, be arbitrary; since by assumption / 0 there exists 0 = 0 η > 0 such that v i < η for every i I and for every 0, 0. Then, up to choosing sma enough, we have v i ui ui± b η η ũ dt. 3.5 i I a+η Since ũ u in L I, in view of the bound on the energy, from 3.5 we may deduce that ũ u in W, a + η, b η so that in particuar u W, a + η, b η. Moreover, 3.5 entais im inf 0 E u, v im inf 0 i v i I b η η u dt, a+η ui u i± η im inf 0 b η a+η ũ dt so that the desired ower bound foows by etting η 0. Remark 3.3. Let u, v L I L I be a sequence such that u, v u, v in L I L I and sup E u, v < + ; et moreover [0, +. In view of , arguing as in [0, Lemma.] we note that the two inequaities aso hod. im inf 0 F u, v b a u dt, im inf 0 G v #S u

12 A. BACH, A. BRAIDES, C.I. ZEPPIERI 3.. The n-dimensiona case. In this section we dea with the case n. The foowing proposition wi be obtained by combining the one-dimensiona resut in Proposition 3. and a sicing procedure in the coordinate directions. To this end it is convenient to introduce the foowing notation. For every k {,..., n} set Π k := {x R n : x k = 0} and et p k : R n Π k be the orthogona proection onto Π k. For a y Π k et k,y := {t R : y + te k } 3.6 and k := {y Π k : k,y }. 3.7 For every w : R, t k,y, and y k we set Proposition 3.4. Let u, v L L be such that w k,y t := wy + te k. 3.8 u, v u, v in L L and sup E u, v < + >0 and et be as in.. i Subcritica and critica regime If [0, + then u GSBV and v = a.e. in. Moreover, E u, v u dx + ν u dh n. 3.9 S u ii Supercritica regime If = + then u W, and v = a.e. in. Moreover, E u, v u dx. 3.0 Proof. The proof wi be divided into two steps. Step : proof of i; i.e., the case [0, +. Let u, v L L be a sequence converging to u, v in L L and such that sup E u, v < +. Note that v in L, so that v = a.e. in. We now show that u GSBV. To this end et k {,, n} be fixed and for every y k consider the two sequences of functions u k,y, v k,y defined on k,y as in 3.8 with w repaced by u and v, respectivey. Let η > 0 be fixed; set η := {x : distx, R n \ > η} and et η k,y be as in 3.6 with repaced by η. Moreover et η k be defined according to 3.7.

13 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 3 Set Π k := Πk Z n ; a direct computation yieds E u, v n = n η k Π k i v i i±e k F k u k,y i k, Z i±e k k, + u i u i±e k i k, Z v k, i u k, vk, i + + vi + i i u k, i ± e k i k, Z i+e k k,, v k,y, k,y + G k v k,y, k,y dh n y, v k, i i+e k v i v i+e k i v k, i + e k 3. with F k u k,y, v k,y, k,y := i k,y Z i±e k k,y v k,y i u k,y i u k,y i ± e k and G k v k,y, k,y := vk, i + i k, Z i k, Z i+e k k,y where u and v are identified with their piecewise-constant interpoations. Then invoking Fatou s Lemma gives im inf 0 E u, v η k im inf 0 F k u k,y v k,y i v k,y i + e k, v k,y, k,y + G k v k,y, k,y dh n y. Since for H n -a.e y k there hods u k,y u k,y in L k,y, Proposition 3.-i together with Remark 3.3 yied u SBV η k,y for Hn -a.e. y η k and im inf 0 E u, v η k η k im inf 0 η k,y F u k,y, v k,y, k,y + im inf 0 G v k,y, k,y dh n y u k,y dt + # S u k,y k,y η dh n y. 3. Since 3. hods for every k {,..., n}, appying [7, Theorem 4. and Remark 4.] we deduce that u GSBV η.,

14 4 A. BACH, A. BRAIDES, C.I. ZEPPIERI In order to prove the ower bound 3.9 we notice that for every k {,..., n} we aso have im inf E u, v 0 n = n = η η im inf 0 η,y Fu,y, v,y,,y dh n y + u,y t dt n u = dx + ν u, e k dh n = η x S u η = u dx + ν u, e k dh n. η S u η η k im inf 0 G k v k,y, k,y dh n y dh n y + # S u η k,y η k,y dh n y k Then taking the sup on k {,..., n} we get im inf E u, v u dx + ν u dh n. 0 η S u η Finay, by etting η 0 we both deduce that u GSBV and 3.9. Step : proof of ii; i.e., the case = +. Arguing as in Step and now appeaing to Proposition 3.-ii yied both u W, and the ower-bound estimate 3.0. Remark 3.5. From the proof of 3.9 in Proposition 3.4-i we get that if [0, + and u, v L L is such that u, v u, v in L L with sup E u, v < + then the two inequaities hod aso true. im inf 0 F u, v u dx, im inf 0 G v S u ν u dh n Remark 3.6. For ater use we notice that Proposition 3.4 can be ocaised in the foowing sense. Let [0, +, U A L and et E be as in.8. Then E u,, U u dx + ν u dh n for every u GSBV. U S u U 3.3. Convergence of minimisation probems. On account of the Γ-convergence resut Theorem. in this subsection we estabish a convergence resut for a cass of minimisation probems associated to E. Specificay, we consider a suitabe perturbation of E which wi aso satisfy the needed equi-coercivity property. To this end, having in mind appications to image-segmentation probems, for a given g L we define and consider the functionas g i := n i+[0, n gxdx E g u, v := E u, v + i n u i g i. 3.3 Moreover, we wi ony focus on the subcritica and critica regimes; i.e., [0, +, as these are the ony regimes giving rise to a nontrivia Γ-imit.

15 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 5 The foowing resut hods true. Proposition 3.7 Equicoercivity. Let [0, + and g L. Then the functionas E g defined as in 3.3 are equi-coercive with respect to the strong L L -topoogy. More precisey, for every sequence u, v L L satisfying sup E g u, v < + there exist a subsequence not reabeed and a function u GSBV such that u, v u, in L L. Proof. For n = the thesis directy foows by combining the estimate 3.4 with the equicoercivity of the perturbed Ambrosio-Tortorei functiona [0, Theorem.]. For n the proof is aso standard and the equi-coercivity of E g foows from the onedimensiona case invoking e.g. [, Theorem 6.6] see aso [, Section 3.8]. We are now ready to prove the foowing resut on the convergence of the associated minimisation probems. Coroary 3.8 Convergence of minimisation probems. For every fixed > 0 the minimisation probem m := min {E g u, v : u, v A A }. admits a soution û, ˆv. Let [0, + ; then, up to subsequences, the pair û, ˆv converges in L L to û, with û soution to m := min { E u, + moreover, m m as 0. u g dx : u SBV, u L g L } ; 3.4 Proof. The existence of a minimising pair û, ˆv foows by appying the direct methods. Indeed, et > 0 be fixed and et u k, v k be a minimising sequence for E g. Then there exists a constant c > 0 such that i n u i k gi c, for every k N. Since for every i it hods g i g L, we deduce that u i k c for every i and every k N where now the constant c possiby depends on. Hence, up to subsequences not reabeed, im k u i k = ûi, for some û i R. Since moreover, 0 v i k for every k N and every i, up to subsequences, we aso have im k v i k = ˆvi for some ˆv i [0, ]. Since for fixed > 0 the set { u i k, vi k : i } is finite, up to choosing a diagona sequence we can aways assume that im k + ui k, vi k = ûi, ˆv i for every i. Then, to deduce that û, ˆv is a minimising pair for E g it suffices to notice that m = im inf k + Eg u k, v k E g û, ˆv. Since E decreases by truncations in u, by definition of E g it is not restrictive to assume that û L g L. Moreover, invoking Proposition 3.7 gives the existence of a subsequence not reabeed û, ˆv and of a function û GSBV such that û, ˆv û, in L L ; further, the Dominated Convergence Theorem aso yieds û û in L. Since ceary û L g L, we actuay deduce that û SBV. Then it ony remains to show that û is a soution to 3.4. To this end, et g A be the piecewise-constant function defined by g x := g i for every x i + [0, n, for every i.

16 6 A. BACH, A. BRAIDES, C.I. ZEPPIERI Then g g a.e. in ; since moreover g L g L the Dominated Convergence Theorem guarantees that g g in L. Therefore Theorem. gives m E û, + û g dx im inf E g û, ˆv = im inf m On the other hand, for every w GSBV L with w L g L Theorem. provides us with a sequence u, v such that u, v w, in L L and im sup E u, v E w,. 0 Then if ū is the sequence obtained by truncating u at eve w L, we ceary have ū w in L and so that im sup 0 Hence by the arbitrariness of w we get Eventuay, gathering 3.5 and 3.6 yieds and thus the thesis. im sup E ū, v E w,, 0 m im sup E g ū, v E w, + w g dx. 0 im sup m m im m = m = E û, + 0 û g dx 4. Proof of the Γ-convergence resut in the subcritica regime = 0 In this section we study the Γ-convergence of E in the subcritica regime; i.e., when = 0. This regime corresponds to the case where the mesh-size is much smaer than the approximation parameter. We show that under such assumptions the discreteness of the probem does not pay a roe in the imit behaviour of the functionas E whose Γ-imit is actuay given by the Mumford-Shah functiona, as in the continuous case. We reca the foowing definition u dx + H n S u if u GSBV, v = a.e. in, E 0 u, v := MSu, v = + otherwise in L L. We dea with the ower-bound and upper-bound inequaities separatey. 4.. Lower-bound inequaity. In this subsection we estabish the iminf inequaity for E when = 0. The main resut of this subsection is as foows. Proposition 4. Lower-bound for = 0. Let E be as in.3 and = 0. Then for every u, v L L and every u, v L L with u, v u, v in L L we have im inf 0 E u, v E 0 u, v.

17 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 7 Proof. Up to subsequences we can aways assume that sup E u, v < +, otherwise there is nothing to prove. Then Proposition 3.4-i gives u GSBV and v = a.e. in. Thus, it remains to show that im inf E u, v u dx + H n S u 4. 0 hods true for every u GSBV. We first prove 4. for u SBV. To this end we use the Fonseca and Müer bow-up procedure [43]. For every > 0 we define the discrete Radon measure µ := i n n v i k= u i u i±e k where i denotes the Dirac deta in i. Since sup >0 + vi + n k= µ sup E u, v < + >0 v i v i+e k i, we deduce that, up to subsequences, µ µ weaky* in the sense of measures, for some positive finite Radon measure µ. Then, appeaing to the Radon-Nikodỳm Theorem we can write µ as the sum of three mutuay orthogona measures We caim that and µ = µ a L n + µ J H n S u + µ s. µ a x 0 ux 0 for L n -a.e. x 0 4. µ J x 0 for H n -a.e. x 0 S u. 4.3 Let us assume for the moment that we can prove 4. and 4.3. Then, by choosing an increasing sequence of cut-off functions ϕ k Cc such that 0 ϕ k and sup k ϕ k = we get im inf E u, v im inf µ im inf ϕ k dµ = ϕ k dµ ϕ k dµ a + ϕ k dµ J S u u ϕ k dx + ϕ k dh, hence the concusion foows by etting k + and appeaing to the Monotone Convergence Theorem. Let us now prove 4. and 4.3. Step : proof of 4.. By virtue of the Besicovitch Derivation Theorem and the Caderòn-Zygmund Lemma for L n -a.e. x 0 we have S u µ a x 0 = im ρ 0 + µq ρ x 0 L n Q ρ x 0 = im ρ 0 + µq ρ x 0 ρ n 4.4 and im ρ 0 + ρ n+ ux ux 0 ux 0, x x 0 dx = Q ρx 0 Let x 0 be fixed and such that both 4.4 and 4.5 hod true. Since µ is a finite Radon measure, there hods µq ρ x 0 = µq ρ x 0 except for a countabe famiy of ρ s. Moreover, for

18 8 A. BACH, A. BRAIDES, C.I. ZEPPIERI ρ sufficienty sma the upper semicontinuous function χ Qρ has compact support in. Thus, appeaing to [8, Proposition.6a] we deduce that for every ρ m 0 and every 0 we have µ a x 0 = im m + ρ n µq ρm x 0 = im m m + ρ n µq ρm x 0 = im m m + ρ n χ Qρm x 0 dµ m im im sup χ m + Qρm x 0 dµ im im sup µ Q ρm x 0. + m + + ρ n m We now want to estimate ρ n m µ Q ρm x 0. To this end, we notice that for every and for every m we can find x 0 Z n and ρ m, > 0 such that x 0 x 0, ρ m, ρ m, as + and so that µ Q ρm x 0 ρ n m = where n ρ n m i Z n Q ρm, x 0 ρ m, Z n Q ρ m v i Z n Q ρm, x 0 = Z n Q ρm x 0 n u i u i± e k k= n n v,m k= u e ρ k,m m, u±,m ρ n m + vi + n k= v i v i+ e k u,m := u x 0 +ρ m,, v,m := v x 0 +ρ m, for every Z n Q. 4.6 ρ m, Now we define a new sequence w,m on the attice ρ m, Z n by setting w,m := u,m ux 0 ρ m, for every ρ m, Z n Q. Since u,m ux 0 + ρ m in L Q as +, appeaing to 4.5 we deduce that by etting first + and then m + we get w,m w 0 in L Q, where w 0 x := ux 0, x for every x Q. Moreover by definition Hence ρ n m u,m u± ρ m, e k,m µ Q ρm x 0 ± e ρ k m,,m = w,m w for every Z n Q. ρ m, ρ m, Z ρ n Q m, n ρ m n v,m k= w,m e ρ k m, w±,m ρ m, = ρm, ρ m n F ρ m, w,m, v,m, Q,

19 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 9 then, a standard diagonaisation argument yieds the existence of a sequence m + as + such that σ := 0, w,m, v,m w 0, in L Q L Q, as +, and ρ m, µ a x 0 im inf F σ w,m, v,m, Q. + w 0 dx = ux 0, Q where the second inequaity foows by Remark 3.5 in view of the equiboundedness of the tota energy E σ w,m, v,m, Q. Step : proof of 4.3. and To prove 4.3 et x 0 S u be such that Q ν ρ x 0 µ J x 0 = im H µq ν ρx 0 n Q ν = im ρx 0 S u ρ 0 + ρ n 4.7 ρ 0 + µ im ρ 0 + ρ n ux u ± x 0 dx = 0, 4.8 Q ν ρ x 0 ± where ν := ν u x 0 and Q ν ρx 0 ± := {x Q ν ρx 0 : ± x x 0, ν > 0}. We notice that 4.7 and 4.8 hod true for H n -a.e. x 0 S u. Then, arguing as in Step we deduce that for every ρ m 0 and every 0 we have hence we are now ed to estimate m, by a change of variabes we have ρ n m µ J x 0 im im sup m + + ρ n µ Q ν ρ m x 0 where u,m and v,m are as in 4.6. Set u 0 x := ρ n m µ Q ν ρ m x 0, µ Q ν ρ m x 0 from beow. To this end, arguing as in Step ρm, ρ m n E ρ m, u,m, v,m, Q ν, { u + x 0 if x, ν 0 u x 0 if x, ν < 0; in view of 4.8 we deduce that u,m u 0 in L Q ν if we first et + and then m +. Then appeaing to a diagonaisation argument we may find a sequence m + as + such that σ := 0, u, v := u,m, v,m u 0, in L Q ν L Q ν, as +, and ρ m, µ J x 0 im inf + E σ u, v, Q ν. Therefore to prove 4.3 we now need to show that im inf E σ u, v, Q ν For the sake of carity in what foows we ony consider the case n = ; the proof for n > can be obtained by means of anaogous constructions and arguments. Upon possiby extracting a subsequence, we assume that the iminf in 4.9 is actuay a imit.

20 0 A. BACH, A. BRAIDES, C.I. ZEPPIERI We now define suitabe continuous counterparts of u and v. To this end, set τ := /ρ m, and consider the trianguation T of Q ν defined as foows: For every i τ Z Q ν set T + i := conv{i, i + τ e, i + τ e } and T i := conv{i, i τ e, i τ e }, and T := {T i +, T i : i τ Z Q ν }. Then we denote by ũ, ṽ the piecewise affine interpoations of u and v on T, respectivey. Moreover, we aso consider the piecewise-constant function ˆv x := v i if x We ceary have ũ u 0, ˆv, and ṽ in L Q ν. Let η > 0 be fixed; we now caim that E σ u, v, Q ν Q ηˆv ũ dx + ν for arge. For sufficienty arge there hods Q ν η σ ṽ dx Moreover, for every i τ Z Q ν there hods i τ Z Q ν k= T i + Ti. Q ν η σ τ ṽ σ + σ ṽ dx 4.0 v i vi+τ e k τ. 4. ṽ x = λ 0 xv i + λ xv i+τ he + λ xv i+τ he x T + i, for some λ 0 x, λ x, λ x [0, ] satisfying λ 0 x + λ x + λ x = for every x T i + λ k xdx = 3 L T i + = 6 τ, for every k = 0,,. T + i and Then, the convexity of z z yieds ṽ dx = λ 0 xv T + σ i σ T + i + λ xv i+τ e + λ xv i+τ e dx i v i λ 0 xdx + v i+τ e λ xdx + v i+τ e σ T + i = v i 6 τ + v i+τ e + v i+τ e, σ σ σ T + i T + i λ xdx and anaogousy for T i T + i +. Therefore summing up on a trianges T i, T i T yieds Q ν η ṽ σ dx i τ Z Q ν τ for sufficienty arge. Further, for every i τ Z Q there hods ˆv ũ dx = τ vi u i ui+τ e τ v i σ, 4. u i + ui+τ e τ

21 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS and anaogousy on Ti so that we may deduce Q ηˆv ũ dx τ v ν i i τ Z Q ν u i ui±τ e k τ for sufficienty arge. Finay, gathering entais 4.0. Now et Π ν := {x R : x, ν = 0}, y Q ν Π ν and set Ceary, ũ ν,y ũ ν,y k= t := ũ y + tν, ˆv ν,y t := ˆv y + tν. W, +η, η for H -a.e. y Q ν η Π ν, moreover there hods ũ ν,y h 4.3 uν,y 0 in L /, / with S u ν,y = {0}, for H -a.e. y Q ν Π ν. By Fubini s Theorem we have η ˆv ũ dx = ˆv y + tν ũ y + tν dt dh y Q ν η Q ν η Πν Q ν η Πν +η η +η ˆv ν,y ũ ν,y dt dh y, thus, from the bound on the energy we deduce the existence of a set N Π ν with H N = 0 such that sup η +η ˆv ν,y ũ ν,y dt < for every y Q ν η Π ν\n. Further, it is not restrictive to assume that ũ ν,y W, +η, η for every y Q ν η Π ν \ N. Therefore, in view of , appeaing to a cassica onedimensiona argument see e.g. the proof of [7, Theorem 3.5] for every y Q ν η Π ν \ N we can find a sequence s y +η, η satisfying ˆv ν,y s y 0 as Now et ṽ ν,y be the one-dimensiona sice of ṽ in the direction ν; i.e., ṽ ν,y t := ṽ y + tν. For every y Q ν η Π ν \ N et s y be as in 4.6 and consider ṽν,y s y. Let moreover d > 0 be fixed; we want to exhibit a set N d Π ν with H N d 0 as + with the foowing property: for every y Q ν η Π ν \ N N d there exists 0 := 0 d, y N satisfying To this end, for every i τ Z Q ν set set moreover ṽ ν,y s y d for every 0. M i := max{ v i v : τ Z Q ν, i = τ }; 4.7 I d := { i τ Z Q ν : M i d }. 4.8 From the energy bound we deduce the existence of a constant c > 0 such that c σ τ v i v τ σ M i #I d d 4 σ i I d i I d τ Z Q ν i =τ

22 A. BACH, A. BRAIDES, C.I. ZEPPIERI for every. Hence, there exists a constant cd > 0 such that #I d cd σ for every. 4.9 Let p ν : R n Π ν be the orthogona proection onto the hyperpane Π ν and set N d := i I d p ν T i + T i ; then in view of 4.9 we have H N d τ #I d cd τ 0 σ as +, 4.0 where the convergence to zero comes from the identity τ /σ = /. Now et N be arge, et y Q ν η Π ν \ N N d, and consider the corresponding s y as in 4.6. By definition of ˆv we deduce the existence of i 0 := i 0 y τ Z Q ν \ I d such that y + s y ν T i + 0 Ti 0 and v i 0 0 as +. Therefore for every d > 0 and every y Q ν η Π ν \ N N d there exists 0 := 0 d, y N such that v i 0 < d/ for every 0. Moreover, since i 0 τ Z Q ν \ Ih d we aso have v i 0±τ e k h < v i 0 + d < d, for k =, and for every 0. Therefore, since ṽ y + s y ν is a convex combination of either the tripe vi 0, v i 0+τ e, v i 0+τ e or of the tripe v i 0, v i 0 τ e, v i 0 τ e we finay get ṽ y + s y ν < d, for every 0. Since on the other hand up to a possibe extraction ṽ ν,y a.e., we can find r y, ry +η, η such that ry < sy < ry and ṽ ν,y r y > d, ṽν,y r y > d for sufficienty arge. Hence, for every fixed y Q ν η Π ν \ N N d using the so-caed Modica-Mortoa trick as foows we get that η +η ṽ ν,y σ + σ ṽ ν,y dt s y r y d d ṽ ν,y ṽ ν,y dt + r y s y ṽ ν,y ṽ ν,y dt zdz = d, 4. for every 0. Moreover, by 4.0 we deduce that up to subsequences χ Q ν η Π ν\n N d H -a.e. in Q ν η Π ν so that im inf + η +η ṽ ν,y σ + σ h ṽ ν,y dt χ Q ν η Π ν\n N d d 4.

23 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 3 for H -a.e. y in Q ν η Π ν. Thus, in view of 4.0, the Fatou s Lemma together with 4. give im E σ u, v, Q ν ṽ im inf + σ ṽ dx + + Q ν η Πν im inf + η +η Q ν η σ ṽ ν,y σ + σ ṽ ν,y dt d H Q ν η Π ν = d η, so that we deduce 4.3 by first etting d 0 and then η 0. Step 3: extension to the case u GSBV. χ Q ν η Π ν\n N d dh y Let u GSBV and for m N et u m be the truncation of u at eve m. Ceary, u m u in L as m +. Therefore, since E 0 = MS is ower semicontinuous with respect to the strong L -topoogy and E 0, v decreases by truncations, we deduce and hence the thesis. E 0u, im inf m + E 0u m, im inf m + MSum, MSu, = E 0 u,. 4.. Upper-bound inequaity. In this subsection we prove the imsup inequaity for E when = 0. To this end we start by recaing some we-known facts about the so-caed optima profie probem for the Ambrosio-Tortorei functiona. We define { + } m := min f + f dt : f W, oc 0, +, f0 = 0, im ft =, 4.3 t + 0 a straightforward computation shows that m = and that this minimum vaue is attained at the function ft = e t. For our purposes it is aso convenient to notice that = m = m where { } T m := inf T >0 inf 0 f + f dt: f C [0, T ], f0 = 0, ft =, f T = f T = 0 We are now ready to prove the foowing proposition Proposition 4. Upper-bound for = 0. Let E be as in.3 and = 0. Then for every u, v L L there exists u, v L L with u, v u, v in L L such that im sup E u, v E 0 u, v. 0 Proof. We can consider ony those target functions u GSBV and v = a.e. in, otherwise there is nothing to prove. By virtue of [38, Theorem 3.9 and Coroary 3.], using a standard density and diagonaisation argument it suffices to approximate those functions u which beong to the space W defined as the space of a the SBV -functions satisfying the foowing conditions: S u is essentiay cosed; i.e., H n S u \ S u = 0, S u is the intersection of with the union of a finite number of pairwise disoint cosed and convex sets each contained in an n dimensiona hyperpane, and whose reative boundaries are C, 3 u W, \ S u for a N.

24 4 A. BACH, A. BRAIDES, C.I. ZEPPIERI We prove the imsup inequaity ony in the case S u = K with K Π ν, K cosed and convex, the proof in the genera case being anaogous. Let p ν denote as usua the orthogona proection onto Π ν and for x R n set dx := distx, Π ν. For every h > 0 define K h := {x Π ν : distx, K h}. Let η > 0 be fixed; by 4.4 there exist T η > 0 and f η C [0, T η ] such that f η 0 = 0, f η T η =, f ηt η = f η T η = 0, and Tη 0 f η + f η dt + η. Ceary, up to setting f η t = for every t T η we can aways assume that f η C [0, +. Let T > T η and choose ξ > 0 such that ξ / 0 as 0. We set A := {x R n : p ν x K + n, dx ξ + n}, B := {x R n : p ν x K + n, dx ξ + n + T }, C := {x R n : p ν x K /, dx ξ /}, D := {x R n : p ν x K, dx ξ }, and according to.6 we denote by A,, B,, C,, D, the corresponding discretised sets. Let ϕ be a smooth cut-off function between C and D and set u x := ux ϕ x. Since u W, \ S u for every N, we can assume in particuar that u C \ D so that u C. We notice that u u in L by the Lebesgue Dominated Convergence Theorem. Moreover, choose a smooth cut-off function γ between K + n and K + n and define v x := γ p ν xh dx + γ p ν x, where h : [0, + R is given by 0 if t < ξ + n, h t := f t ξ n η if ξ + n t < ξ + n + T, if t ξ n + T. By construction v W, C 0 C \ A and v in L. We then define the recovery sequence ū, v A A by setting ū i := u i, v i := 0 v i for every i. We ceary have v in L as 0. Moreover, ū u L u i u x dx i+[0, n = i i i+[0, n \D i+[0, n ui ux dx + i i+[0, n D i+[0, n u i u x dx c u L + u L ξ, 4.6 where to estimate the first term in the second ine we have used the mean-vaue Theorem whie to estimate the second term we used the fact that #{i : i + [0, n ξ D } = O n. Hence, the convergence ū u in L foows from 4.6 and the fact that u u in L.

25 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 5 Thus it remains to prove that We ceary have im sup E ū, v 0 im sup 0 E ū, v im sup 0 u dx + H n S u. F ū, v + im sup G v We now estimate the two terms in the right-hand-side of 4.7 separatey. We start with F ū, v. Since v i = v i = 0 for a i A,, we have F ū, v = n n v i u i u i ± e k. 4.8 i \A k= i±e k Let i \ A. By construction i ± e k \ D for every k {,..., n} and since u = u on \ D, using Jensen s inequaity we deduce that u i ± e k u i = ui ± te k, e k dt ui ± te k, e k dt, 0 for every k {,..., n}. Therefore, thanks to the reguarity of u and to the fact that 0 v, the mean-vaue Theorem gives F ū, v n ui ± te k, e k dt dh n y so that i \A k= = n i \A k= i+[0, n i+[0, n uy ± te k, e k dt dh n y + O ux dx + O, 4.9 im sup F ū, v u dx We now turn to estimate the term G v. We have G v Gv = n v i n + i \B k= + n v i n + i B, \A k= v i n We start noticing that + i \B n i A, n v i k= + k= v i v i+e k v i v i+e k v i v i+e k. 4.3 n + v i v i+e k =

26 6 A. BACH, A. BRAIDES, C.I. ZEPPIERI Indeed, for i \ B we have di ξ + n + T. Then, since T > T η, for sufficienty sma we deduce that di + e k di ξ + n + T η for every k {,..., n} so that by definition of h we get v i = v i+e k =, hence 4.3. We now caim that i A, n v i + n v i v i+e k k= c ξ + H n K + n 0, 4.33 as 0. To prove the caim we observe that v i = v i+e k = 0 for every i A, and every k {,..., n} such that i + e k A,. Hence we have v i n + v i v i+e k i A, n = i A, n i A, n k= i+e k A, = #A, n c ξ + H n K + n Then, it ony remains to estimate the energy for those i A, and k {,..., n} such that i + e k B, \ A, ; i.e., to estimate the term n v i v i+e k. k= i+e k B, \A, To this end, we observe that in genera, for every i and every k {,..., n}, thanks to the reguarity of v, by Jensen s inequaity we have v i v i+e k v i v i + e k v i + te k, e k dt Since v x = h dx γ p ν xd ν px ± γ p ν xh dxν, where Dpx ν denotes the Jacobian of p ν evauated at x, using that h W, 0, + satisfies h L C, whie γ L C and Dν p L, from 4.35 we obtain v i v i+e k = O for every i, for every k {,..., n} 4.36 thus, consequenty i A, n n k= i+e k B, \A, v i v i+e k which together with 4.34 gives We finay come to estimate i B, \A, n v i 0 c# A, n c Hn K n +, n + v i v i+e k. k=

27 FINITE-DIFFERENCE APPROXIMATIONS OF FREE-DISCONTINUITY PROBLEMS 7 To this end, it is convenient to write B \ A as the union of the pairwise disoint sets M, V, W defined as: M := {x R n : p ν x K + n, ξ + n < dx ξ + n + T }, V := {x R n : p ν x K + n \ K + n, dx ξ + n}, W := {x R n : p ν x K + n \ K + n, ξ + n < dx ξ + n + T }. Further, we denote with M,, V,, W, their discrete counterparts as in.6. We now estimate the energy aong the recovery sequence in the three sets as above, separatey. To this end we start noticing that #V, = O ξ and #W n, = O. Thus, appeaing to 4.36 we n deduce that n vi n + v i v i+e k = Oξ i V, k= and i W, n vi + n v i v i+e k k= = O. Finay, again using 4.36 we get i M, n vi + n v i v i+e k k= i M, n vi + n v i v i+e k k= + c n # M,; moreover we notice that # M, c n H n K + n +. Let now i M, ; then i + [0, n M. Hence, by definition of v we have dx ξ n v x = f η and v x = ± f η dx ξ n ν in i + [0, n. Since f η C [0, T ], appeaing to the Mean-Vaue Theorem we deduce that for every x i + [0, n dx ξ n f η di ξ n f η f η L 0,T, whie for every y i + [0, n, every t 0,, and every k {,..., n} dy + f η tek ξ n di + f η tek ξ n ν, e k f η L 0,T.