Dipartimento di Matematica

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1 Dipartimento di Matematica P. Bonicatto, L. Luardi Analyi of an integral equation ariing from a variational problem Rapporto interno N. 5, luglio 29 Politecnico di Torino Coro Duca degli Abruzzi, Torino-Italia

2 ANALYSIS OF AN INTEGRAL EQUATION ARISING FROM A VARIATIONAL PROBLEM PAOLO BONICATTO AND LUCA LUSSARDI Abtract. In thi paper we conider an integral equation which arie from a Γ-convergence problem. More preciely we invetigate the exitence of the olution and we collect ome elementary propertie of uch a olution. The reult extend and partially complete what already proved in [5]. Keyword : integral equation. 2 Mathematic Subject Claification: 45D5.. Introduction The aim of thi paper i the tudy of an integral equation which arie from a Γ-convergence problem in the theory of the variational approximation of free dicontinuity functional. A number of problem in Calculu of Variation recently propoed involve integral functional with free dicontinuitie according to a terminology introduced in [4]): the variable function u i required to be mooth only outide a urface K, depending on u, and both u and K enter the tructure of the functional. A typical form i give by F u, K) = φ u ) dx + ϑ u + u ) dh n \K where i an open ubet of R n, K i a n )-dimenional compact et, u + u i the jump of u acro K, while φ and ϑ are given poitive function and H n denote the n )- dimenional Haudorff meaure). The natural weak formulation i obtained looking at K a the et of dicontinuitie of u, thu working in pace of function allowing hyperurface of dicontinuitie, uch a the pace BV ) of function of bounded variation. The main difficulty in the actual minimization of F i the preence of the urface energy, and then a uitable approximation, leading to the convergence of minimum point, by mean of more manageable functional naturally arie. In the cae of functional defined in all BV ) the aforementioned weak form of F take the form.) F u) = φ u ) dx + ϑ u + u ) dh n + c D c u ), S u where Du = u dx + u + u ) dh n + D c u i the decompoition of the meaure derivative of u in it abolutely continuou, jump and Cantor part, repectively, and S u denote the et of dicontinuity point of u. Auming that φ i convex and ϑ i concave, with K φt) ϑt).2) lim = c = lim, t + t t + t it turn out that F i lower emicontinuou with repect to the trong L -topology. The main example i given by the well-known Mumford-Shah functional: u 2 dx + H n S u ). Many author have tudied approximation cheme for free dicontinuity functional by mean of a family F ε ) ε>, for intance the cae where the functional F ε are retricted to finite element pace on regular triangulation of ize ε [6], [9], [9]); or the implicit contraint on the gradient through the addition of a higher order penalization [], [3], [8]); or the tudy of non-local model,

3 2 P. BONICATTO AND L. LUSSARDI where the effect of a large gradient i pread onto a et of ize ε: thi i the method which wa firt applied to the Mumford-Shah functional by Braide & Dal Mao in [8] ee alo [7], [], [], [2], [3]). We alo have to mention the Ambroio & Tortorelli approximation ee [4] and [5]) of the Mumford-Shah functional via elliptic functional, where an additional variable, ay v, which approache the characteritic function of the complement of the dicontinuity et, i introduced. A variant of thi lat method wa tudied in [2], [22] and [2] for functional with linear growth in the gradient: the attempt i to unify the curve evolution method ued in Computer Viion to detect boundarie, and the pre-proceing of the image to provide an edge-trength function v, which indicate the likelihood of an object boundary being preent at any point of the domain compare with the additional variable v in the Ambroio-Tortorelli functional). In particular in [6],[7] and [5] the following variant of the family invetigated in [8] wa treated: F ε u) = ) f ε ε uy) dy dx ε B εx) where B ε x) denote the ball of centre x and radiu ε. In thi cae the urface energy denity can be explicitely computed a ωn.3) ϑ) = 2 f ) t 2 ) n dt ω n where ω k denote the volume of the k-dimenional ball in R k. In order to obtain an approximation reult for a precribed free dicontinuity functional of the form.) we will ee that we only need to olve equation.3). More preciely we want to find a continuou olution f : [, + ) R of the integral equation.3) atifying the following condition: i) f, ii) f i non-decreaing, iii) f i concave, ft) iv) lim = c. t + t The integral equation.3) ha already been briefly invetigated in [5]: indeed, after a uitable change of variable, equation.3) become an Abel integral equation, and an explicit olution i known for thi kind of equation ee for intance [2]). In [5] ome technical aumption are aumed on the function ϑ in order to prove the approximation Theorem. Thi paper extend and partially complete the reult obtained in [5]. Without lo of generality we will retrict our attention to the following integral equation..4) ϑ) = f t 2 ) n) dt, n N. Indeed if f i a olution of.4) then g) := ) 2 f ωn ω n olve equation.3) with n replaced by n). Firt of all in ection 2 we will derive ome neceary condition on ϑ; then we will formulate the main Theorem. Section 3 i devoted to the proof of the main Theorem of the paper and finally in ection 4 we will try to tudy neceary and ufficient condition to obtain the concavity of the olution, which repreent the main difficult of the work. In the lat ection we will briefly ketch an application to a variational approximation cheme for free dicontinuity problem. 2. Formulation of the problem Taking into account the origin of the problem, a explained in the Introduction, we have to olve thi quetion: given n N and given a concave function ϑ: [, + ) [, + ) with ϑ) lim = c, + ) +

4 ANALYSIS OF AN INTEGRAL EQUATION ARISING FROM A VARIATIONAL PROBLEM 3 find a continuou function f : [, + ) [, + ) which i non-decreaing, concave, with for ome c, + ) and uch that 2.) ϑ) = f) lim = c + f t 2 ) n) dt for any c in general will not equal to c ince equation 2.) i implified with repect to the equation.3)). Notice that in thi cae n = the complete olution of the problem i obviou: indeed in uch a cae equation 2.) become ϑ) = f) dt = f) and then f = ϑ i the olution; f i alo non-decreaing ince it i non-negative and concave. Then we will take into account only the cae n. The following Lemma ay that if we have a concave olution f of equation 2.) then necearily ϑ i differentiable; oberve that thi fact doe not hold in the cae n =. Lemma 2.. Take a concave function f : [, + ) [, + ) with and let Then ϑ C, + ). f) lim = c, + ) + ϑ) = f t 2 ) n) dt. Proof. By the change of variable x = t 2 ) equation 2.) become ϑ) = 2 /n Letting ψ) = 2ϑ n/2 ) and = fx n/2 ) we eaily get and then ψ) = ψ) = x dx = fx n/2 ) x dx. gy) dy,. gy) dy It i ufficient to how that ψ C, + ). Since f i concave, f i a locally lipchitz function, and then, by Rademacher Theorem, it i differentiable a.e. in, + ). Thu g i alo differentiable a.e. in, + ). We how that ψ i differentiable proving that for any >. Fix > and conider ψ + h) ψ) lim = h h yg y) dy gy + hy) gy) R y h) =. h Taking into account the chain rule for Lipchitz function we get lim R yh) = yg y) h a.e. y, + ). Moreover there exit a poitive contant L which may depend on ) uch that R y h) L y for any y [, ]. By the Dominated Convergence Theorem we get ψ + h) ψ) yg ) y) R lim y h) yg y) dy = lim dy =. h h y h y

5 4 P. BONICATTO AND L. LUSSARDI Thi how that ψ i differentiable in, + ) with ψ ) = yg y) dy. Since g y) = n 2 yn/2 f y n/2 ) a.e. y > and ince f x) c a.e. x > we deduce that yg y) n 2 c y n/2 n/2 a.e. y >. Applying again the Dominated Convergence Theorem we conclude that ψ i continuou, and then ψ C, + ). Now we are in poition to tate the final form of the problem P: Given ϑ C, + ) with ϑ, with ϑ concave and with ϑ) lim = c, + ) + find a function f : [, + ) R uch that P) f. P2) f i continuou. P3) f i non-decreaing. P4) f i concave. P5) It hold P6) It hold ϑ) = ft) lim = c, + ). t + t f t 2 ) n) dt,. The main Theorem of thi paper i the following reult. Theorem 2.2. For any n N, n, problem P)-P2)-P5)-P6) ha the unique olution given by 2 t /n ϑt) + ntϑ t)) dt > 2.2) f) = nπ t =. 3. Proof of the Theorem 2.2 Thi ection i devoted to the proof of Theorem 2.2. Firt in the following Lemma we invetigate the computation of the olution of a general Abel integral equation. Lemma 3.. Let φ C [, + )) with φ) =. Then the function g : [, + ) R given by φ x) dx > 3.) g) = π x = i the unique continuou olution of 3.2) φ) = x dx,. Proof. Firt we will prove that the function g given by 3.) i a continuou olution of equation 3.2). We can rewrite g a φ y) g) = dy π y which i continuou in, + ) ince φ C, + ). Moreover for mall we get g) π up φ dy [,] y

6 ANALYSIS OF AN INTEGRAL EQUATION ARISING FROM A VARIATIONAL PROBLEM 5 and the right-hand ide tend to a + ince φ i bounded in, ). Thu g i continuou in = too. It remain to how that g i a olution of 3.2). By Fubini Theorem we have dx = x φ ) τ) dτ dx = ) φ τ) dx dτ. x π x x τ π x x τ A traightforward computation how that Thu we conclude that τ dx = arcin 2x τ x x τ τ x dx = φ τ) dτ = φ). Converely let g : [, + ) R be a continuou olution of 3.2). Then for any t it hold t and then, by Fubini Theorem, A before we get and thu t t x dx d = t x t τ τ φ) t d = π. t φ) d dx = d. t ) x) t t x t ) x) d = π t dx = t φ) t d = π t π Differentiating in t and integrating by part we obtain, ince φ) =, t t gt) = 2π t φty) dy + t = π and thi end the proof of uniquene. π yφ ty) dy = φ ty) dy = π π φty) dy φ ty) ) y + yφ ty) dy y φ ) t d Now we can conclude the proof of Theorem 2.2. After a change of variable equation 2.) become 3.3) ϑ) = 2 /n fx n/2 ) x dx. Let φ) = 2 ϑ n/2 ) and = fx n/2 ); then 3.3) can be rewritten a 3.4) φ) = x dx. Obviouly φ) = and φ C, + ); ince lim + ϑ)/ = c and ince n we eaily get φ +) < +, hence φ C [, + )). Since φ x) = ϑxn/2 ) + nx n/2 ϑ x n/2 ) x applying Lemma 3. equation 3.4) ha the unique continuou olution given by ϑx n/2 ) + nx n/2 ϑ x n/2 ) dx > g) = π x x =.

7 6 P. BONICATTO AND L. LUSSARDI Then we get f) = g ) = ϑx n/2 ) + nx n/2 ϑ x n/2 ) dx > π x x =. Letting x n/2 = t we finally have f) = 2 t /n ϑt) + ntϑ t)) dt > nπ t = which i formula 2.2). Letting t = u in the formula 2.2) for any > we have 3.5) f) = 2 nπ Then for each > we get f) = 2 nπ u /n ϑu) + nuϑ u)) u u /n ϑu) u du + 2 π du. u /n ϑ u) u du. Since ϑ)/ and ϑ ) tend to c for + by the Dominated Convergence Theorem we get Then lim + lim + f) u /n ϑu) u = c > where du = lim + c = and thi end the proof of Theorem cn + ) nπ u /n ϑ u) du = c u /n du. u u u /n u du 4. Some conideration about the concavity of the olution To olve completely problem P) P6) it remain to how that the function f given by 2.2) i concave; indeed P3) follow combining P) and P4). Exploiting 3.5) we ee that a ufficient condition for the concavity of f i the concavity of the map ϑ) + nϑ ). The main difficult i to obtain neceary condition for the concavity of the olution; we have only partial reult. Let u uppoe that f i concave and aume that ϑ belong to C 3 [, + )) thi regularity condition can be removed by mean of an approximation argument). Then we follow the proof of the Theorem 2.2 with the ame notation: = fx n/2 ) i concave providing that n/2, then if n =, 2. Since φ) = 2 ϑ n/2 ) belong to C 3 [, + )) then one can eaily ee that g C 2 [, + )), and then we have g x) for any x. Iterating Lemma 3. it i eay to ee that φ g x) ) = dx x and then φ ) which can be written a In particular we get 2ϑ n/2 ) + n n/2 ϑ n/2 )) ϑ n/2 ) + n n/2 ϑ n/2 ). ϑ n/2 ) + n n/2 ϑ n/2 )) ϑn/2 ) + n n/2 ϑ n/2 )

8 ANALYSIS OF AN INTEGRAL EQUATION ARISING FROM A VARIATIONAL PROBLEM 7 which implie that the map ϑ n/2 ) + n n/2 ϑ n/2 ) i concave. Oberve that if n = then we obtain, a neceary condition for the concavity of f, the concavity of ϑ ) + ϑ ) which i, in general, a different aumption with repect to the concavity of ϑ) + ϑ ). In the cae n = 2 we get a complete olution of problem P) P6): indeed in uch a cae the neceary condition become the concavity of ϑ) + 2ϑ ) which coincide with the neceary condition. To find neceary condition for the cae n > 2 i an open problem. 5. Approximation of free dicontinuity functional In [5] the following fact i proved. Let R n be a bounded open et with Lipchitz boundary. Let c, + ) and let φ: [, + ) [, + ) be a convex function with φ) = and with φt) lim = c. t + t Let f ε ) ε> be a family of function atifying the following condition. A) For every ε >, f ε : [, + ) [, + ) i a non-decreaing continuou function with f ε ) = ; moreover, there exit a ε > uch that a ε a ε and f ε i concave in a ε, + ). f ε t) A2) lim ε,t),) εφ ) t =. ε A3) f ε converge uniformly on the compact ubet of [, + ); we denote the limit of f ε by f, which i a continuou, concave and non-decreaing function. A4) There exit L > uch that f ε ) f ε t) L t,, t >. Conider the family F ε ) ε> of functional L ) [, + ] of the form ) f ε ε uy) dy dx if u W, ) ε 5.) F ε u) = B εx) + otherwie. Theorem 5.. Let F ε ) ε> be a in 5.), with f ε atifying condition A)-A4). Then the family F ε ) ε> Γ-converge, in the L -topology, a ε, to F : L ) [, + ] given by φ u )dx + ϑ u + u )dh n + c D c u ) if u GBV ) S Fu) = u + otherwie where ωn 5.2) ϑ) = 2 f ) t 2 ) n dt. ω n If we are able to olve completely problem P) P6) for intance if ϑ) + ϑ ) i concave) then we obtain a concave olution of equation 5.2) and thu we can apply Theorem 5. in view to approximate a given free dicontinuity functional Fu) = φ u )dx + ϑ u + u )dh n + c D c u ) S u

9 8 P. BONICATTO AND L. LUSSARDI for u GBV ). Indeed it i ufficient to conider the family f ε ) ε> given by εφ ) t ε if t ε β 5.3) f ε t) = ft ε β ) + εφε β ) if t ε β being f the concave olution of equation 5.2). However the unique olution of equation 5.2) i given by f up to contant) o that one can tudy by hand the concavity of f for a given explicit function ϑ. Example 5.2. Conider R 2, and the functional with linear growth given, on GBV ), by u + u F u) = u dx + S u + u + u dhn + D c u ), propoed in [2] a a egmentation functional. In order to obtain the approximation of thi functional we can apply the previou argument. Indeed in thi cae ϑ) = t + and i concave. ϑ) + ϑ ) = ) 2 Reference [] R. Alicandro, A. Braide, and M.S. Gelli. Free-dicontinuity problem generated by ingular perturbation. Proc. Roy. Soc. Edinburgh Sect. A, 6:5 29, 998. [2] R. Alicandro, A. Braide, and J. Shah. Free-dicontinuity problem via functional involving the L -norm of the gradient and their approximation. Interface and free boundarie, :7 37, 999. [3] R. Alicandro and M.S. Gelli. Free dicontinuity problem generated by ingular perturbation: the n-dimenional cae. Proc. Roy. Soc. Edinburgh Sect. A, 33): , 2. [4] L. Ambroio and V.M. Tortorelli. Approximation of functional depending on jump by elliptic functional via Γ-convergence. Comm. Pure Appl. Math., XLIII:999 36, 99. [5] L. Ambroio and V.M. Tortorelli. On the approximation of free dicontinuity problem. Boll. Un. Mat. Ital. B 7), VI):5 23, 992. [6] B. Bourdin and A. Chambolle. Implementation of an adaptive finite-element approximation of the Mumford- Shah functional. Numer. Math., 854):69 646, 2. [7] A. Braide and A. Garroni. On the non-local approximation of free-dicontinuity problem. Comm. Partial Differential Equation, 235-6):87 829, 998. [8] A. Braide and G. Dal Mao. Non-local approximation of the Mumford-Shah functional. Calc. Var., 5): , 997. [9] A. Chambolle and G. Dal Mao. Dicrete approximation of the Mumford-Shah functional in dimenion two. M2AN Math. Model. Numer. Anal., 334):65 672, 999. [] G. Corteani. Sequence of non-local functional which approximate free-dicontinuity problem. Arch. Rational Mech.Anal., 44:357 42, 998. [] G. Corteani. A finite element approximation of an image egmentation problem. Math. Model Method Appl. Sci., 92): , 999. [2] G. Corteani and R. Toader. Finite element approximation of non-iotropic free-dicontinuity problem. Numer. Funct. Anal. Optim., 89-):92 94, 997. [3] G. Corteani and R. Toader. Nonlocal approximation of noniotropic free-dicontinuity problem. SIAM J. Appl. Math., 594):57 59, 999. [4] E. De Giorgi. Free dicontinuity problem in calculu of variation. In Robert Dautray, editor, Frontier in pure and applied mathematic. A collection of paper dedicated to Jacque-Loui Lion on the occaion of hi ixtieth birthday. June 6, Pari 988, page 55 62, Amterdam, 99. North-Holland Publihing Co. [5] L. Luardi. An approximation reult for free dicontinuity functional by mean of non-local energie. Math. Meth. Appl. Sci., 38): , 28. [6] L. Luardi and E. Vitali. Non-local approximation of free dicontinuity functional with linear growth: the one dimenional cae. Ann. Mat. Pura e Appl., 864): , 27. [7] L. Luardi and E. Vitali. Non-local approximation of free dicontinuity functional with linear growth. ESAIM Control Optim. Calc. Var., 3): 35-62, 27. [8] M. Morini. Sequence of ingularly perturbed functional generating free-dicontinuity problem. SIAM Journal on Mathematical Analyi, 353):759 85, 23.

10 ANALYSIS OF AN INTEGRAL EQUATION ARISING FROM A VARIATIONAL PROBLEM 9 [9] M. Negri. The aniotropy introduced by the meh in the finite element approximation of the Mumford-Shah functional. Numer. Funct. Anal. Optim., 29-): , 999. [2] A. D. Polyanin and A. V. Manzhirov. Handbook of integral equation, pp.787. Boca Raton [etc]: CRC, 998. [2] J. Shah. A common framework for curve evolution, egmentation and aniotropic diffuion. In IEEE conference on computer viion and pattern recognition, 996. [22] J. Shah. Ue of elliptic approximation in computer viion. In R. Serapioni and F. Tomarelli, editor, Progre in Nonlinear Differential Equation and Their Application, volume 25, 996. Paolo Bonicatto) via Fenoglio n.5, Robaomero, 7 Torino, Italy. addre, P. Bonicatto: paolo.bonicatto@gmail.com Luca Luardi) Dipartimento di Matematica, Politecnico di Torino, c.o Duca degli Abruzzi n.24, 29 Torino, Italy. addre, L. Luardi: luca.luardi@polito.it