F{u). If sup f{x) i=sup fix) xe(o,i) xe[i,2) set u{y) = 0 and apply Case 4 .

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= z<^qz + qf-f = ^q + ^v^ {n)j < (/)^ usy; q-Â’z, = vfj -(n)^ < {f)j fijMsp udyx q-Z^'ofi (sÂ‘i] q-z ] \ = Â‘(tÂ‘ 0) 3 o J puv Â‘{{^)f-z) uira Â‘(^)/ (iÂ‘o) uim q V P7 -daoqv a?/^ Suififsi^vs (gÂ‘o) uo snonuiiuoo sq / p/ og '(SÂ‘rJ uo z> {X)f puv (rÂ‘o) UO 0^ (x); f ssbq n = / 0? f 9 SVJ fijddv puv 2 = n 79/ Â‘z = // '{n)j < (/)t7Â»?/7 99s 9 ffi -g ^ /? (gÂ‘T] 3 ^ 2 1 I = {fi)n Â‘(lÂ‘ 0 ) 3 /i (x)/ (^Â‘o) 9 ^jui j 9 UJ/ 9 P (t Â‘0) 3 X IP jof 0 < (x); f I -q Â•(Â™)^ < {f)j f^ixvvp dm udijx Pt '(l Â‘o) 3 x diuos jo/ q = (x)/ fj ts z < {x)I 7077 yons (gÂ‘x] 3 X uv spixd 9^977 puv (x Â‘o) 3 x gv jo/ 0 < (x)/ : Â£ asBQ < U)d os 'On = n gyv; uoy^ 'Z < {x)I 7Â»?/7 yons {z'\] 3 x puv q > [x)/ py; yons (x Â‘o) 9 x pixd dJdyx Z 9sbq Â•n = / o'l f dsvQ dfddv puv 0 = n 79s 0 = (x)/ // '{n)j < {/)j davy dm uivBy -g ^ (x)/ /i (sÂ‘l]3/i (x)/ 'I } = Â‘(tÂ‘0) 3/1 0 J 797 udyjp -Q 9 *x dmos jo/ q = (*x)/ udy^ '(x Â‘o) 9 x diuos jo/ q < {x)/ /j -q 9

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7 Thus we conclude that if f is any continuous function on (0,2) then there exists a function u with a jump discontinuity such that F{f) > F{u). Thus by uniqueness, no f ^ C{fl) can be a minimizer of F. argument f and hence f can not be a minimizer for F. As an improvement on the TV functional, Strong and Chan [23, 9] introduced the weighted TV functional for spatially adaptive (selective) image restoration. The function a is chosen so that a is larger away from possible edges and smaller near a likely edge. Hence we allow for greater smoothing away from edges and less smoothing at the edges. Certain choices of a{x) were given by Strong and Chan [23, 9], and their numerical results were very promising. However, many theoretical questions such as the existence and uniqueness for the minimizer of the weighted TV norm with some penalized term, and for the related evolution problems when a is a function on VL (not only piecewise constant) remain. The question whether the solution of the evolution equation converges to the minimizer or not as t Â— > oo also remains open. The goal of this chapter is to investigate these problems. Here we would like to point out in the case of constant a, results were obtained by Chambolle and Lions [8] for the minimization problem Now if f E PT^Â’^(f2), then by the Sobolev embedding theorem f is a.e. equal to an absolutely continuous function f and hence F{f) = F{f). Thus by the above minimize with where cr^ is known and A is a continuous linear operator on IF{Tl), and by Acar and Vogel [3] for the problem

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8 where A is a linear operator on L^(fi) and Jn(u) = [ ^\Vu\Â‘ + D Jn is defined on BV space. Results were also obtained, for instance, by Vese [29] for the functional and its corresponding flow. Here a > 0 is constant and (/? : R Â— ) R+ is a convex, even function nondecreasing in R+ with linear growth, and K ; is a linear, continuous, injective operator. However, the results for the flow are only in the dimensions one and two, due to the methods employed there, that is, using general results on maximal monotone operators and evolution operators in Hilbert spaces. In addition, Hardt and Zhou [18] consider the flow related to with Dirichlet boundary data u = g on dVt for a bounded domain fl,u = Uq on flx{0}, and any convex linear growth functional cp. In their analysis, they approximated the above solution by the flow associated with = rje * v{p) where is the usual mollifier on RÂ”. In this work we shall extend the results of Chambolle and Lions, Acar and Vogel, as well as those of Hardt and Zhou, to the adaptive TV scheme. In particular, we shall develop mathematical theories for the problem of that is = divxP>p{Vu), {(fp = V(/?) minf (Â«) = TV^ + f 11Â“ Â“ollii(n) ( 2 . 2 )

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9 over BV{Vl), and its corresponding evolution equation, dvi Â— = diw:,{a{x)ipp{Vu)) P{u-uq) du Â— = 0 on dO, X RtÂ’, on u{0) = uq on 0 , on n X Up, (2.3) (2.4) (2.5) where 0 is an open, bounded Lipschitz domain, = [0, T], ,5 > 0 is a parameter, (p{p) = \p\ on RÂ”, (pp{p) = Vp{p) = p/|p| on R", and a{x) is a parameter (smooth) function used for edge detection and to control the speed of smoothing. In image processing problems a is often chosen as a{x) 1 1 + k\WG(j * Uq\^ ( 2 . 6 ) k being a parameter, and being the Gaussian filter with parameter a. The definition of a weak solution to (2.3)(2.5) is similar to that in Zhou [30] or Hardt and Zhou [18]. However, our method of proving existence differs from Hardt and Zhou in two ways, the first being the approximation equation. Second, and most importantly, is the way most of the necessary estimates of the approximate solution are obtained. Finally, our use of an approximate PDF to the corresponding flow is in contrast to Vese. Also, our result holds for all n. 2.2 Preliminaries and Definitions We start with the definition of / a\Vf \ for functions in / G L^(fi). Jn Definition 2.2.1 Let LI be a bounded open subset of R". Let f be a real valued function on such that f G L^{L1). Also let a{x) > 0 be a continuous real valued function on O. Then we define the a-total variation of f or a-TV of f to be / a\Vf\= sup < / f div{(/))dx : \(f{x)\ < a{x) \/x e Ll\ Jn 0 eci(n,R") Un J where 4> is a vector-valued function

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10 Definition 2.2.2 We define f G to be in a-BV if sup W / div{(j))dx : \4>{x)\ < a{x) Vx G Q > < oo. 0eci(f2,R") 1 7a J Definition 2.2.3 If f & a-BV we define the a-BV seminorm by / a|V/| = sup iff div{(t))dx : \4>{x)\ < a{x) Vx G 7a 1 7a and the a-BV norm to be a-BV = f o|V/| 7a + z-i(a)In the sequel we will write the above norm as Remark 2.2.4 It is easy to show that if f E [ a\Vf\ = [ a\Vf\dx. 7a 7a Remark 2.2.5 Note that if f E BV{Q) and functions a and fi both satisfy the conditions of Definition 2.2.1 where a{x) < fi{x) for every x E fl and f E fi-BV, then we have f E a-BV and [ a|V/|< f l3\Vf\. 7a 7a This follows directly from the Definition 2.2.3 since |^/>(x)| < a{x) implies |

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11 Now take the supremum over (j) to get / ci|V/| < liminf [ a\Vfj\. Jn Jn Next is an important approximation result for functions in BV. Theorem 2.2.7 Let f Â€ BV where a{x) > 0 is continuous on fl. Then there is a sequence {fj} of functions from CÂ°Â°{Ll) such that lim [ \ fj f\dx = Jn and lim f o;|V/,|dx = f q:|V/|. j^co Proof: We essentially apply the argument of Giusti [13] with an important modification. Given e > 0 we construct the covering {Ai} of where Ai A\ = O2, with Ojt = G : dist{x, dQ) > = 0, 1, 2, and where m is large enough such that Next we construct the sequence {/J so that (2.7) i=l where rj^. is the usual mollifier on R" and {f>i} is a partition of unity subordinate to {Ai}. We then choose the efs such that the following four conditions hold simultaneously for each ef 1. Ci < e

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12 ix < e2' 2[ IVci * f^i\da JQ 3[ \Ve.*{fV(^>^)-fV(|>,\dx / in L^(Q), and by Theorem 2.2.6, / q;| V/ l < lim inf [ o;|V/e|dx. Jn ^-*0 Jn Now let g G Co(Q,R") be such that |^(x)| < a{x) Vx G Then n OO n 00 n / /ediv(i;)da; = V / (^e^ * (/0i))div(i/)dx = V] / /idiv(r/,. * an an an (2.8) so / /ediv(Â£/)dx = / /div(0i?7,, * + Y] / /div((;^i77,; * g)dx (2.9) an an an OO ^ X] / f^(l>i))dx. Z=1 Denote the three terms on the right side of (2.9) by I, II, and III respectively. Note III< ellallioo by our choice of the e^s. By uniform continuity of a, there exists an increasing function ca such that uj{r) Â— > 0 as r 0 and |o:(2) Â— Oi{z')\ < cu(r) for all z, z' such that \z Â— z'\ < r. Consequently a{z) < u>{ei) + a{x) for all |a; Â— < e,. Now write g = ag' where g' = Q

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13 if q; = 0 and \g'\ < 1 Then for i > 1 and any x e \(i>i{x){r]i* g){x)\ = \(j)i{x){qi * ag'){x)\ < / (j)i{x)gi{x z)a{z)\g' {z)\dz Ju < / 4>i{x)gi{x z){a{x) uj{ti))\g' {z)\dz Jn = (j)i{x)a{x) / gi{x z)\g' {z)\dz Jn +(j)i{x)uj{ei) / gi{x z)\g' {z)\dz Jn < a{x)+uj{e). So we get, for z = 1, 1= f * g)dx < f a\Vf\+u{ei) / |V/| Jn Jn Jn and also 11 = ^ / fdiv{(j)ig,. * g)dx < 3 / a\Vf\ + 3oj{e) [ \Vf\ i =2 ^ JnÂ—no Jn < 3e + 3u{e) [ |V/|, Jn with the last inequality following from (2.7). Therefore [ fcdW{g)< [ a|V/|+a;(ei) [ \V f\ + 3e + 3oo{e) [ |V/| + Jn Jn Jn q\a\Loc for every g Â€ with | 5 '(x)| < a{x) Vx G fl. Hence taking the supremum over g and then taking the limsup as e ^ 0 we get limsup / a|V/(:|da; < f a|V/|. 0 Jn Jn From (2.8) and (2.10) one finally has J|m f a\Vf,\dx = j a|V/|. (2.10)

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Remark 2.2.8 If f E BV{n) n a G C(fi), and dQ Lipschitz, then there exists a sequence of functions {/Â«} C such that ( 2 . 11 ) And if f E LÂ°Â°{Q), we also have ll/n||LÂ°Â°(n) < C'(^)||/|UÂ°Â°(n)( 2 . 12 ) In fact, in the proof of Theorem 2.2.7 we choose the eÂ’s to satisfy l-f and in addition \ve, * if (pi) f(pi\^dx < e2 \ Then we can take fj E CÂ°Â°(fl) n VK^Â’^(Jl) n L^(fi), Since CÂ°Â°(Sl) is dense in n L^(f2) there exists for each fj a sequence {fj,k} Â€ CÂ°Â°{Sl) such that as k ^ 0 Then (2.11) follows from (2.12>), (2.1A) and a standard diagonal argument applied to {fj,k} to obtain the desired sequence {fn}By the construction of {fj} and [fj^k] if in addition f E we obtain (2.12). Theorem 2.2.7 now allows us to prove a compactness theorem. Theorem 2.2.9 Let {fj} be a bounded sequence in a-BV where a E C(Q) and in addition a{x) > 5 > 0 ^x E Q. Also assume that C RÂ” is such that 09. is Lipschitz. Then there is a subsequence of {fj}, also denoted by {fj}, and an f E IT{9) such that fj f strongly in Lp{ 9) where 1 < p < and weakly in Jn such that (2.13) \\fj,k ~ fj\\L^{n) -> 0 and \\fj^k ~ /j||wci(n) -> 0. (2.14) LÂ—^{9). Proof: Since 0 < 5 < a[x) \/x E 9 and by remark 2.2.5 we have

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15 Therefore f |V*| < C. Jn Thus fj is bounded in BV norm and the theorem follows from the compactness result in Giusti [13]. 2.3 Minimization Problem We now consider the minimization problem min [ alVuj f (u Â— Uo)^dx. (2-15) BV{n)nLHa) Jn 2 Here uq is the initial noisy image, /3 is a positive parameter, and is a bounded open subset of R" with Lipschitz boundary. In the sequel we will always assume that a is a smooth function satisfying the conditions of Theorem 2.2.9. In practice we can take a{x) in (2.6) for some cr > 0 and A; > 0. Assuming uq G we see 1 Â“ 1 + C'||uo|licx>(n) The constant 5 in Theorem 2.2.9 can therefore be chosen as 1 1 + C'llÂ“o|lioo(n) To conclude this section we verify that (2.15) does have a unique minimizer. Lemma 2.3.1 The functional in problem (2.\b) has a unique solution in BV{Q.) n L2(Q). Proof: Clearly, the functional is convex, coercive in BV (fl) nL^(fi) and by Theorem 2.2.6 is lower semicontinuous. So by standard results, (2.15) has a solution in BV{Q,)C\ The uniqueness follows from strict convexity of the functional in (2.15). 2.4 Flow Related to the Minimization Problem To motivate the definition of a weak solution to (2.3)-(2.5) where (p{p) = p, P

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16 smooth to justify the following calculations and that (p is as mentioned above. For arbitrary v E L^([0, T]; we multiply the equation in (2.3)-(2.5) by u Â— u to get, after integrating by parts and using the formula p(p) Â— p(g) > pp(g) (p Â— g) (due to the convexity of p), / u(v Â— u)dx + / ap(Vv)dx > / ap{Vu)dxÂ— / P{u Â— uq){v Â— u)dx. (2.16) Jci Jo. Jn Jn Then integrate with respect to t to get n u{v Â— u)dxdt + / ap{V ! Jo Jn v)dxdt > n apiyu)dxdt (2-17) ! Â— I I /3{u Â— Uq){v Â— u)dxdt. Jo Jn On the other hand if (2.16) holds, by selecting v = u + X4> for (f) E C^(Q) we get that u is a solution to (2.3) in the sense of distributions. We are thus led to the following definition of a weak solution to (2.3)-(2.5): Definition 2.4.1 A function u E L^([0,T]; BV(fi)) is called a weak solution of (2.3)(2.5) if ii = dfU E LÂ‘^{Cl x Rr), u(0) = uq, and u satisfies (2.17) for every v E L\[0,T]-BV{Q)), a.e. sG [0,T]. Before we continue we list some properties of the smooth approximating function of p, p^{p) = defined on RÂ”, which will be used in the subsequent discussion: 1. p^^{p) is convex in p, 2. p],{p) p>0 Vp, 3. p^ ^ p uniformly with respect to p as e Â— ) 0. In fact 0 < |

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17 Consider the following approximation problem of (2.3)-(2.5): du m eAu + div(o;(pp(Vu)) Â— /3(u Â— Uq) on x R1Â—1 oo du dn 0 on dTl X Rp, (2.19) u(0) = Uq on n. (2.20) where Uq G CÂ°Â°(0) with Uq uq in (2.21) ||'f^olUÂ°Â° Â— C'(fI)||uo||LÂ°Â°) (2.22) 1 a(f{V ul) < C {fl) f a(f{'Vuo). Jn Jn (2.23) The existence of Uq is from Remark (2.2.8) if uq G BV{Tt) n LÂ°Â°(fl). The idea is to prove an existence result for the above quasilinear uniformly parabolic PDE, obtain bounds for the solution independent of e and 6, and pass the limit as e Â— > 0 and 5^0. Indeed, the lemma below provides an existence and uniqueness result for (2.18)-(2.20). Lemma 2.4.2 The approximation problem (2.\%)-(2.2Q) admits a unique weak solution u^'^ where G LÂ°Â°([0, T]; i7^(Q)), G L^([0, T]; and f f {u^Â’^)Â‘^dxdt f ^\Vu^Â’^{t)\Â‘^dx-\[ ap{Vu^'^{t))dx Jo Ju Jn 2 Jn < / + o;(p(Vuo)d2; + e Jn 2 for a.e t E [0, T], Proof: By using the Galerkin method, the fact that pp is a monotone operator ([6] and [20]) we have a weak solution to (2.18)-(2.20) such that G L^([0, Tj; L^(fl)), ^e,s ^ LÂ°Â°([o,T];/7i(fi)), and ^ J (u'Â’Â‘)Â‘Â‘dxdt + j + aip'(Vu'^Â’\t)) + ^(uÂ‘* -ulf^dx I Jn -|VuqP + o(p^(VMo)dx.

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18 Since (p{p) < ip^{p) < (p{p) + e for any p we arrive at f f {u^Â’^)'^dxdt+ f + f ap{Vu^'^{t))dx Jo Jn Jn 2 Jq Â— / Jq 2 We also have the following LÂ°Â°(n) bound for the solution to (2.18)-(2.20) obtained above: Theorem 2.4.3 Suppose Uq 6 LÂ°Â°{Q.)f]BV{fl) andu^Â’^ is a solution of initial boundary value problem (2.lS)-(2.20). Then we have ||^^Â’Â‘^||LÂ°Â°(nxRT) ^ C'(^)||^o||LÂ°Â°(n)Proof: Let M = ||uolUÂ°Â°(n)For any A > 0, multiply (2.18) by Â— M)+, where g At^Â£,(5 __ g At^Â£,5 _ > Q 0 otherwise. and integrate over fl to get du^Â’^ r du^>^ r J Â—e-^\e-^*u^Â’^-M)+dx + eJ Vu^Â’^ e~^^ e~^^Vu^Â’^dx + f app{Vu^'^)e~^* e~^^Vu^Â’^dx + P f {u^Â’^ uo)e~^\e~^^u"Â’^ M)pdx = 0. Jq Jq Then since the last three integrals are non-negative we see du^Â’^ Let Then f ^ Â— e Â— M)+dx < 0. Jq nt) = ljHe-V-MM^dx. e,6 2 Jo dt Fin Q dt )dx I JQ -At Â£.5 du dt e ;rÂ— (e M)pdx f -Xe M)+dx < Q. Jq

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19 Therefore I{t) > 0 is decreasing in t with /(O) = 0. Hence [ |(e-^VÂ’-^-M)+|"dx = 0 Jn Vt, and then C Â— a.e on Q, VA > 0 and Vt > 0. Letting A ^ 0 we obtain u^Â’\t) < M = ||4l|L~(n)Similarly, > Â—M = Â— ||uo||Â£)-(2.2D) have uniformly bounded LÂ°Â°{Q x Rj^) and L^{Q x R^) norms in e for and respectively. Then there is a subsequence of still denoted by such that as e Â— )Â• 0 1. ^ h weakly in L^(f 2 x R 7 -) for some h 2. weakly in LÂ‘^{Q, x RjÂ’) for some where h = ii^ and u'^(O) = Ug. For the proof see Zhou [30] or Temam [25]. Theorem 2.4.5 Suppose uq Â£ LÂ°Â°{Q) n BV{Q.), then there exists a unique u Â€ L^{[0,T]] BV{fl) n LÂ°Â°(f2)), ii G x Rt), and u{0) = uq such that u satisfies (2.17) for a.e. s G [0,T] and every v G L^([0,T]; RH(fl)). Proof: Let be the solution to (2.18)-(2.20). By Lemma 2.4.2 and Theorem 2.4.3 it satisfies ||'t^^Â’Â‘^||LÂ°Â°(nxR 7 Â’) ^ ||^olUÂ°Â°(n) < C'(f2)||Mo||LÂ°Â°(n) (2.24)

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20 and I \ii^Â’^\^dxdt -f / a^piy u^'^)dx J UxRtÂ’ Ju < 1 [ \Vul\Â‘^dx + C 'n < 1 [ \v4\'^dx + c 'u (2.25) / a(p{Vul)d. Ju X + e X + e. By (2.25) / \u nxRx + ^{Vu^'^))dxdt < ||u^Â’'||By(nxR^) < C|h-' with C Â— C(q;, Q,T). So for fixed 5 > 0, there exists a subsequence of such that as e ^ 0, > vr strongly in L^(fi x R^) and a.e in f] x R-b and u ^ if weakly in L^[Q, x R^). (2.26) Notice that by letting e ^ 0 in (2.24) with fixed 5 we have ||'wÂ‘^||LÂ°Â°(nxRT) ^ C'(^)lko|UÂ°Â°(n)) (2.27) since from (2.26) a.e. in fi x R^. By (2.26) we can also extract a subsequence, still denoted by such that u ^.<5 ^ u" strongly in L^{Q) for a.e. t E [0, T]. Also notice that as well since Â— > u'^ in L^(Q X R^) as e Â— >Â• 0 (2.28) (2.29) / \u^'^ Â— HxRtÂ’ dxdt < C(n)||uo||LÂ°Â°(u) f \u^Â’^ Â— u^\ dxdt t/s7 X Rt by (2.24) and (2.27).

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21 As is also a weak solution to (2.18)-(2.20) we have as in the motivation of Definition 2.4.1 n u'^Â’^{v Â— u^Â’^)dxdt + ^ f f \Vv\^dx+ f f aip'^{Vv)dxdt ^ Jo Jn Jo Jn > / /* \Vu^Â’^\Â‘^dxdt + f f aip^{Vu^'^)dxdt ^ Jo Jn Jo Jn Â— f f {u^Â’^ Â— Uq){v Â— u^Â’^)dxdt Jo Jn > j I aip^ (Vu^Â’^)dxdt Â— ^ f [ ~ ul){v Â— u^'^)dxdt Jo Jn Jo Jn for all V Â€ L^([0,T]; Now let e -> 0 in the above inequality to arrive at n ii^{v Â— u^)dxdt + f f a(p{Vv) ! Jo Jn >liminf^y J a(p^(Vu^Â’^)dxdt ~ (3 J J Â— Uq){v Â— u^Â’^)dxdt^ > f f aip{Vu^)dt Â— P f [ {u^ Â— Uq){v Â— u^)dxdt. Jo Jn Jo Jn for all V G L^([0, T]; and hence also for all v G L^([0, T]; 5V(D)) by Proposition 2.2.8. Here we used (2.26), (2.28), (2.29), Theorem 2.2.6, the fact that (p{p) for all p, and uniform convergence of to p. This shows that is a weak solution of (2.3)-(2.5) with initial data Un. Additionally from (2.28) and Theorem 2.2.6 it follows that / p{Vu^) < liminf f p{Vu^Â’^)dx for a.e. t G [0,T]. J Q, t/ (2.30) Thus letting e Â— )Â• 0 in (2.25), lii^pdxdt + 0 X Rj* f 0 u u u u in L^(D X Rr), hence in LP{VL x R 7 -) from (2.27), Â— > w in L^{Q) for a.e. t G [0,f], and ^ Â« in L^(0 X Rr). (2.31)

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22 Finally pass to the limit as 0 in the inequality n ii\v u^)dxdt + f f a(p{Vv) > f f aip{'Vu^)dt ! Jo Jn Jo Jn n {u^ Â— uo)(v Â— u^)dxdt ; to get n ii{v-u)dxdt+ f f a(p{Vv) > f f a(p{'Vu)dt ! Jo Jn Jo Jn ~P I / {u Â— Uq){v Â— u)dxdt Jo Jn for all V e L2([0,r]; W(f2)). Therefore we have the existence of a weak solution u to (2.3)-(2.5). Using (2.27), (2.30), and (2.31) we see as before that u E LÂ°Â°([0,T];BV(n)nLÂ°Â°(Q)). To prove uniqueness, consider two weak solutions Ui, U 2 to (2.3)-(2.5) with Ui(0) = Â«2(0) = Uq. We have the two inequalities n u\{u2 Ui)dxdt + / / aip{Vu2)dt > a(f{Vui)dt ! Jo Jn Jo Jn Â— / P{ui Â— Uq){u 2 Â— Ui)dxdt Jo Jn and n U2{ui U2)dxdt + f f a(p{Wui)dt > f f aip{Vu2)dt ' Jo Jn Jo Jn / /3{u 2 ~ Uo){ui U2)dxdt. Jo Jn Adding the above inequalities and combining we get n {u2 Â— ui){ui Â— U2)dxdt > / / /3{ui Â— U2)^dxdt. 1 Jo Jn And hence giving [ ^ / (^1 U2fdxdt < [ [ !J{ui U2fÂ‘dxdt < 0 Jo dt Jq Jq ||ui(-, s) U 2 {-, s)||l 2 ( 0 ) = 0 for Â£-a.e. s. Therefore Ui = U 2 D

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2.5 Stability and Asymptotic Behavior Lemma 2.5.1 If Ui and U 2 are two weak solutions of (2.3) with initial data uio and U 20 , then for C-a.e. tE [0,T], ll'^l ~ U2\\L<^(n) < ll^^io Â— 'i^2o||LÂ«>(n)Proof: Let M = ||uio Â— U 2 o||LÂ°Â°(n)For Â£-a.e. t we have / Ui{v Â— Ui)dx ~\/ aip{Vv)> / aip{Vui)Â— / ^{u\ Â— uiq){v Â— Ui)dx (2.32) Jri Jn Jn Jn and / U 2 {v Â— U 2 )dx + / acp{Vv)> / aip{'Vu 2 ) Â— / I3{u Â— U 2 o){v Â— U 2 )dx. (2.33) Jn Jn Jn Jn Define = u{ Â— {u\ Â— ^2 Â— M)_|_ and = U2 + {u\ Â— U2 Â— M)+ where u\ and are the approximation functions from Theorem 2.2.7. Inserting and w^ into (2.32) and (2.33) respectively we obtain / ui{v^ Â— ui)dx+ / aifCVv^) > / aip(Vui) Â— / ^{ui Â— uiq){v^ Â— ui)dx Jn Jn Jn Jn and / U2{w^ Â— U2)dx + / ai^[Vw^)> / a^p{Vu2)Â— / ^{u2 Â— U2 q){w^ Â— U2)dx. Jn Jn Jn Jn Now add the above two inequalities to get / ui{v^ Â— ui)dx+ / U2{w^ Â— U2)dx > / a^p{Vui)+ / a(p(Vu2) Jn Jn Jn Jn

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24 Observing that / aip{S/w^) + / aip{Vv^) = / a(f{Vu\) + / o;(^(V'Ui) >/n Jo. Jn Jn with ^ V and ^ to in L?{Q) where V = U\ Â— {Ui Â— U2 Â— M)^ and (2.34) (2.35) W = U2 + {Ui U2~ M) + we see after letting e Â— > 0 that I Ui{v Â— Ui)dx + / U 2 {w Â— U 2 )dx > Â— I P{ui Â— Uio){v Â— Ui)dx Jn Jn Jn / P{U2 ~ U2q){w U2)dx. Jn But the right hand side of the above inequality satisfies p / (ui tt 2 + U 20 Uio){ui -U 2 M)+dx > 0. Jn Thus / ui{v Â— U\)dx + / U2{w Â— U2)dx > 0 . Jn Jn Hence, using equalities (2.34) and (2.35) for v and to in the above and combining, / (ui Â— U 2 ){ui Â— U 2 Â— M)+dx < 0 Jn which implies / |(tii Â— U2 Â— M)+pdx < 0 Jn A dt and therefore / |(tti Â— ti2 Â— M)+|^dx < / \{uio Â— U 20 Â— M)^\^dx = 0 Jn Jn by the choice of M. Similarly U\ Â— U 2 > Â—M. To conclude, we investigate the asymptotic behavior of the weak solution to (2.3) by showing that the solution converges weakly in L^(fi) and strongly in L^(0) to a minimizer of (2.15).

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25 Remark 2.5.2 It is straightforward to show that if inequality (2.17) is satisfied for all V G LÂ‘^{[0,T]\ BV{il)), then (2.1Q) holds for all v G LÂ‘^{[0,T]; a.e. t. Then by using YoungÂ’s inequality, inequality (2.1Q) implies )dx [ f / (^ Jn ^ Jn uo)Â‘^dx > for all V G BV{fl), a.e. t. Finally we prove the following theorem concerning the asymptotic convergence of our solution u to (2.3) as t Â— ) oo: Theorem 2.5.3 The solution u to (2.3) weakly converges in L?Â‘{VL) to a minimizer Uoo of (2.13). First, let F be defined on BV{Q.) n L^(n) by F{u) = f q;|Vu| f{u Â— Uo)^dx. Jn 2 Jq Second, we recall the definition of the subdifferential of a proper convex functional G : H (Â— 00 , 00 ] where H is a Hilbert space with inner product < Â•, Â• >: the subdifferential of G at u, written dG[u], is defined as dG[u] = [w G H\G{v) > G{u)+ < w,v Â— u > Vu G H}. We also let D{dG) C H he the set of all u with dG[u] 0. Noting that the above F is defined on a subspace of F^(0), let the operator dF{u) be the subdifferential of F at u so that F{v) > F{u) + Jqw(v Â— u)dx Yw G dF{u), \/v G BV{Q) riL^(f2). By Brezis [6], dF is a maximal monotone operator and by the above remark our solution u to (2.3) satisfies for a.e. t Using the above, we can prove Theorem 2.5.3 by the following lemma ([7]): Lemma 2.5.4 Let (/?://Â—) (Â— 00 , + 00 ] be a proper lower semicontinuous convex function which assumes a minimum in H . Then for any Xq G D{dp), there exists a

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26 unique function x : [0, oo) Â— > H which is absolutely continuous on [5, oo) for all 5 > 0 and which satisfies x{t) G D{d(p) for all t > 0, x{t) G Â—dcp{x{t)) a.e., x(0) = Xq, and iw-limt^oo exists and is a minimum point of ip. To prove Theorem 2.5.3 we take H = p = F where, r F(u) \iueBV{Q) F{u) = { ( oo ue LÂ‘^{n)\BV{n) X Â— u, and Xq = uq. As in the proof of Theorem (2.2) ([18]), the function u : [0,oo) -> L^(fl) is absolutely continuous for all nonnegative t. So by Lemma 2.5.4 we have u{t) Â— ^ Uqo weakly in L^(fl) as t Â— > oo and Uqo a minimizer of F in Z/^(f]). By uniqueness, Uoo Â€ 1/^(0) fl BV{Q). Since u{t) is uniformly bounded in BV{^), we may also conclude that any sequence {u(tÂ„)} has a convergent subsequence still denoted by {u(tÂ„)} converging to u^o strongly in L^(0). Hence u{t) Â— >Â• Uoo strongly in L^(fl). The theorem is now proved. Note in fact that the minimizer u = Uqo from problem (2.15) is actually in To see this, note that the inequality in Lemma 2.4.2 and the proof of Theorem 2.4.5 imply that ||u(t)||Bv(n) < ^'(Q, (5)||uo||By{n) and ||w(t)||LÂ°Â°(n) < C'(f^)||^to||LÂ°Â°{n) for a.e t G [0,Tj. Hence by compactness, we can extract a subsequence {-{/(tn)} such that u{tn) Â— > u in L^{fl) and u(tÂ„) Â— ) u a.e. on fl. Thus u G LÂ°Â°(fi) with ll'a||LÂ°Â°(n) < C'(fl)||?^o|lLÂ°Â°(n)-

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27 2.6 Numerical Results For the numerical experiments we approximate (2.3) with du = div(o;(^p(Vu)) Â— P{u Â— Ug) in x R^, = 0 on do, X Rt-, u(0) = Uq on Q X {0}. dt du dn We write the above as du dt = adiv Vu \A + |v + Va Vti u\ Ve+|V p{u ul). Using forward time differences and the Neumann boundary condition we compute n = 1,2, . . . , N , N = number of iterations, by = ul + Ai6W,) where uF = uo{xi,yj), and Q{u) = a div Vu The term y/e + I Vu| Va Â• Va Â• Vu Vu i/e-b |Vup p{u-ul). V^e+TV u\ is discretized using the following scheme developed by Osher and Sethian [21] to permit the development of discontinuities at object boundaries. Here we let Aj. Uij Ax Uij Ay UiJ AyUiJ 2 ^ '^hj ^ij + 1 ~ 2

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28 Figure 2.1: Original 256 x256 image Then we use (Va Â• = max(Aj;Q;ij, 0) Uij + min(Aa;Q;i,j, 0)A+Ujj + max(Aj,Q;ij, 0)A~Uij + min(Aj/0;jj, 0)A+Ujj as in Osher and Sethian [21], while the term + [Vup is computed using the central differences AxUij and AyUij. For the term \^e+\Vu\\ we use the scheme from Rudin, Osher, and Fatemi [22] with central differences, that IS div Vu y/e-h |Vu|^ +a: These figures demonstrate the implementation of the above numerical scheme using 256 by 256 gray level images, with pixel values 0 to 255. Figure 2.1 shows

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29 hMiMmsiMms Pm%Wâ€¢ Figure 2.2: Noisy image with SNR=1:1 Figure 2.3: Restored image using ATV with A:=0.001, cr=0.5, e=0.01, /3=0.001, 300 iterations

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30 Figure 2.4: Restored image using TV with ^ Â— 0.001, 300 iterations Figure 2.5: Noisy image at j/=200

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31 Figure 2.6: Dashed line is restored image from Figure 1 using ATV at y=200, dotted line is plot of original image at y=200 Figure 2.7: Dashed line is restored image using TV, /3=0.001, 300 iterations at y=200, dotted line is plot of original image at y=200

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32 Figure 2.8: Original 256 x 256 image Figure 2.9: Noisy image, SNR=1:1.5

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33 Figure 2.10: Restored image using ATV with A;=0.005, (7=0.7, e=0.01, ^=0.002, 300 iterations the original image; figure 2.2 shows the original image with added noise; figure 2.3 the restored image using the above ATV scheme; and figure 2.4 the restored image using the above scheme with a = 1 (restoration using TV). The next three figures examine a cross section of the images from figures 2.1 through 2.4. Figure 2.5 is a cross section of the noisy image which includes the bottom tip of the thin ellipse. Figures 2.6 and 2.7 demonstrate the effectiveness of the ATV scheme in preserving the tip of the ellipse. By plotting the cross sections of the restored images and the original image, we see that the ATV scheme is better able to preserve the tip of the ellipse. Figures 2.8 through 2.10 also demonstrate the effectiveness of the method to preserve thin lines while reducing noise. For better display, we threshold the restored images to be between the gray levels 0 and 255. 2.7 Updated Selective Smoothing In this section we will investigate problem (2.15) with our choice of a being 1 a{x) = l + k\V{G^*u)\^' (2.36)

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34 So in the time dependent case, the selective smoothing functional now uses a continuously updated version of the image u. Fix a and denote by G. Assuming sufficient smoothness we can formally derive the Euler-Lagrange equation for (2.15) with a as above to be Â‘2^kGi,{y x)wi{y)\V u\dy^ + div (q;(^p(Vu)) /3(u uq) = 0 with Wi Gi * u 1% 2r^ n a Gi^*U, ^ = 0 an {l + k\V{G*u)\^Y and where Gl^ is the partial derivative of G with respect to its zth argument. The first integral on the left hand side was obtained by using the definition of convolution and then reversing the order of integration. In order to extend the solution to the space BV{Q.) n we multiply the above hy v Â— u, use convexity of cp, and integrate to get Â“^kGi^iy x)wi{y)\Vu\^ dx^ {v u) + j ap{Vv) > / a(/?(Vu) -h / P{u Â— Uo){v Â— u)dx. Jn Jn Notice that in the first integral on the left hand side the coefficient function of | Vu| is not necessarily nonnegative as in Definition 2.2.1. However, the definition of / Of|Vu| Jo. for arbitrary a G C(D) can be extended to be / dlVul = / d'^'lVul Â— a |Vu| Jn Jn Jn where dc^{x) Â— max(d(a:), 0) and a (x) = Â— min(o;(x), 0). One can easily verify that if u G then / a\Vu\dx= / oi^\Vu\dxÂ— / a~\Vu\dx. Jn Jn Jn Note that if a is not assumed to a nonnegative function then we no longer have lower semicontinuity of [ d|V Jn u\. From this definition combined with Remark 2.2.5 we have the simple result below.

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35 Theorem 2.7.1 If a ^ C(fi) n LÂ°Â°(r2) and u G BV{Q), then / q;|Vm| < oo and Jn [ Â«|V Jn u\ < ||q;| LÂ°Â°{n) f |VÂ«|. Jn Instead of studying the complicated flow corresponding to the above EulerLagrange equation, we may instead consider a simplifled version of this flow as follows: du dt du dn u = diVi(o;(/Jp(Vu)) Â— P{u Â— Uq) on x Rp = 0 on 50 X Rp = uo on 0 X {0} with a as in (2.36). Note that a{x, t) now depends on u{x, t). Although the deflnition of a weak solution of the above PDE is the same as Deflnition 2.4.1, the dependence of Of on t greatly complicates an existence argument. So here we will only prove the existence of a minimizer of (2.15) with the above choice of a. Uniqueness is complicated by the observation that F is no longer a convex function. We thus only prove an existence result for the stationary problem. Theorem 2.7.2 There exists a solution belonging to BV (Tl) C\ (il) to (2.1b) where a is given by 1 l + k\V{G*u)\^' Proof: Let F{u) = I Q!(a:)| V(u)| + f {u Â— uq)^ dx Jn Jn and let {Â«Â„} be a minimizing sequence for F. Then ||uÂ„||p2(n) < M, M depending only on D. Now for any u G L^(Q) we have where a{x) 1 1 + |VG*u|2 < <

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36 with C depending only on 0, a. Denote 1 1 + |V(j * Un\^ by an{x). Then since ||iin||z,2(n) ^ M hy the above we have o;Â„ > <5 > 0 for some 5 depending only on 0, a. Thus F{Un)> / (5|VuÂ„l+ / {Un~ Uofdx > 5 / iVuÂ„ 7n JQ. Jn Therefore / |VuÂ„| < M and by compactness there exists a subsequence which we Jn still denote by {uÂ„} and a u in BV{Q.) fl i/(D), for any 1 < p < 2, such that Un u in UÂ’{Q). Let a{x) Since Â— > u in LJ{Q) we have 1 1 + / G^.{x y)un{y)dy ^ y)u{y)dy Jn Jn for every x E Q. Hence an{x) o;(x) and |o;Â„(a:)| < 1 for every x E G,. So o;Â„ Â— >Â• o; in I/^(D) for any 1 < p < oo by LebesgueÂ’s Dominated Convergence Theorem. Now computing dan dxi 'iUo,,(x~v)uMdy )(/. Gx,xA^ y)un{y)dy {l + \VG*Un\^)^ ^ Un {x y)un{y)dy )(/. Gx 2X\ {x y)un{y)dy 2\2 (1 + |VG*uÂ„|^) gives dan dxi da dxi pointwise on Q, and similarily for dan dx2 ' dan dxi < M' Since

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37 for i = 1, 2 and M' depending only on G we again have by LebesgueÂ’s Dominated Convergence Theorem, > a in for any 1 < g < oo. By definition, / anlVunl = sup < UÂ„div(0)dx : |0i(x)| < an{x) V(a;) Jn <^>ec^(n)" Un e 0 = sup < / Un{Van{x) Â• (j)) + an{x)div{(f)))dx : \(f)i{x)\ < 1 V(a;) Â€ D (j>ec^{n)^ I in We choose p, q with 1 < p < 2 such that + = 1. Then P Q Un ^ u in L^{Q) and ^ ct in Thus for any (j) G with |0j(a;)| < 1 V(x) G D we have / u{V a 4> + adiv {(/))) dx = liminf / ttn(Vo!Â„ Â• 0 + o;Â„div(0))dx in in = liminf / uÂ„div(o;Â„)dx n->oo < liminf / 0 !Â„|VuÂ„|. rn.oo Hence taking the supremum over all (f) of the left hand side of the above we have / q;|Vu| < in Finally using this result, infF(uÂ„) = liminf ( / o;Â„|VuÂ„| + {un-Uo)^dx Vin in y > liminf / aÂ„| VuÂ„| + lim inf / {un Â— Uofdx n^oo n-)-oo > / q;|Vu| + (u Â— Uo)^dx. in in liminf / o;Â„|Vu^ n-Fcx) In The last inequality follows by convexity of the second integral and by the fact that we can also choose {Â«Â„} to converge weakly to u in L^(D). Thus F has a minimizer in L2(Q) nBC(D).

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38 2.8 Other Selective Smoothing Models We mention here some other proposed selective smoothing models used for image restoration. One such model was proposed by Chambolle and Lions [8] which uses a combination of TV and isotropic diffusion. Here edge preservation is achieved using total variation where the image gradient is above a certain threshold and smoothed isotropically where the gradient is below that threshold. The model is the minimization problem min ^ f iVuPdx-f f (iVnl Â— |) -b ^ f (u Â— lYdx. BV{n)nL^{u) 2e J\yu\<( J\vu\>e 2 ^ Jn with given corrupted image I Â€ BV (fl) nL^(n). This model will be discussed further in chapter 3. A model proposed by Chen, Levine, and Stanich [10] is min f (bix.Du) + Â—{u Â— iV BV(n)nL2(Q) 2 where (j){x,z) = < where ,5 > 0 and 1 < a < p{x) < 2. For example we can use ^ ^ l + k\VG^*I{x)\' This model uses intermediate values of the exponent of of |Vn| for anisotropic diffusion, that is 1 < p < 2 as well as TV-based diffusion (p = 1) along edges and isotropic diffusion in homogeneous regions (p = 2). The anisotropic diffusion is used for where the difference between noise and edges is unclear. Another model proposed is by Blomgren, Chan, Mulet, and Wong [4] which avoids the difficulty of choosing the threshold e or ^5 as used in the above models.

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39 The problem is min [ Jn where lim|vÂ«|-).oP = 2 and lim|vu|->.ooP = 1This model also uses a combination of edge preserving TV-based diffusion as well as anisotropic and isotropic smoothing.

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CHAPTER 3 SELECTIVE SMOOTHING AND PARTIAL REGULARITY 3.1 Introduction In the first two sections of this chapter, we focus on the problem (3.1) where is the following convex function defined on R" 0 C R" is a bounded domain with Lipschitz boundary, and I G LÂ°Â°(D) Cl BV{fl) is given. Such a functional for image restoration was considered in Chambolle and Lions [8]. Here the restored image is taken to be the minimizer of a combination of the total variation and the squared L^(Q) norm of the gradient. That is we minimize edge preserving via TV-based diffusion where edges are more pronounced (|Vu| > e). Without loss of generality we take e = 1 as in (3.1). In order to define for u G BV (D) we note that since the gradient of u is a measure Du, we can decompose Du into its absolutely continuous and singular parts with respect to Lebesgue measure, that is See Evans and Gariepy ([12]) for a complete discussion. Then we define ([15]) Thus we expect to have isotropic diffusion where the image is uniform (|Vu| < e), and Du = Vu dx -tD^u. 40

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41 with [ \D^u\ = [ d\D^u\ = |Il"u|(Q). Ja Jn It is important to note ([30] or [15]) that the functional J can also be defined by J{u)= sup \~ [ ( dx : \(f){x)\ < 1 yx E Q. ec^(n,R") I JO \2 / Using this definition, we see that the functional J is lower semicontinuous with respect to convergence in The proof is similar to that of the proof of lower semicontinuity of the BV seminorm. Since we have established lower semicontinuity of J, is now straightforward to show that there is a unique solution to (3.1) by standard methods. We now state the two main regularity results. Theorem 3.1.1 If u is the solution to (3.1), then for any given 0 < /r < 1 there exist positive constants cq o,nd kq depending only on n and /j, such that if ^ [ \Du-l\< eo ^r\ J Brio.) \Br\ JBr{a) holds for some Br{a) CC and for some I G RÂ”, with rC (l -I|]/]|L(n)) < kq and |/| < 1 2/r, for some constant C depending only on n and Q then, D^u\{Br/ 2 {a)) = 0 and |Vu| < 1 Â— p, on Br/ 2 {o) and u solves Â—Au = I Â— u on Br/ 2 {a). Hence u Â€ C^Â’Â°Â‘{Br /2 (a)) for any a < 1. Theorem 3.1.2 Let u be as in Theorem (3.1.1/ 7/Â£"({|Vu| < 1}) > 0, then there exists a nonempty open region E on which u is |Vr/| < 1 and u solves Â—Au = I Â— u on E.

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42 In addition we have |Vu| > 1 a.e. onH\E. It is actually straightforward to show that Theorem 3.1.2 is a direct consequence of Theorem 3.1.1 using standard results from measure theory and analysis. Thus from Theorem 3.1.2, we do indeed have smoothing where |Vu| < 1. Here we should point out that regularity results were obtained in Anzellotti and Giaquinta [1] for minimizers in BV (fl) of functionals of the form f {F{x, Du) -f G{x, u)) Jn where F{x,p) is a convex function in p with Ci|p| < F{x,p) < C 2 (l + |p|) for all p e R" and G{x, z) satisfies certain continuity conditions in both x and z. In our case, G{x,z) = \j2[z Â— I{x))Â‘^ with only the stated assumption on I. The proof of the above theorems follows the ideas of Tonegawa [26], where the above theorems were proved for the minimizer to the plasticity functional min < / (fiDu) Â— / fudx > uÂ€BV{n)nL^Q) j for p as above and / E LÂ°Â° . 3.2 Proof of Theorem 3.1.1 and Theorem 3.1.2 First we will show that the solution u to (3.1) is in LÂ°Â°{D). To prove this we could consider the time evolution problem corresponding to (3.1), as in the case of the previous selective smoothing problem with parameter function a, prove an LÂ°Â° bound for the time dependent solution u(x, t), and then consider the time asymptotic limit u. Thus we would conclude as in that case that u E LÂ°Â°(Q). The next lemma however provides a proof of this without having to consider the time evolution of (3.1). Lemma 3.2.1 If u is the solution to then u E LÂ°Â°{D). In fact, we have ||Â«||LÂ°Â°(n) < ll^||LÂ°Â°(n)-

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43 Proof: Let be defined on RÂ” by ^e{p) IpI if bl > 1, and consider the minimization problem min I [ (Pf(Vu) + [ (u Â— I)^dx\. uew^^{u)nLHQ) [J^ 2 Ja j By standard methods, there is a unique solution to the this problem. We follow a standard truncation argument where we fix e and f > 0 and let v = Noting that v G fl with { Vue if Ue < ^ 0 if > t, we have [ ^ f {u^-lfdx< [ (pe(Vu) + i [ {v-lfdx, Jn ^ Jn Jn ^ Jq and thus after subtracting / PeC^u^)dx+ / {u^ Â— I)Â‘^dx< / {t Â— I^dx. J {Ue>t} J {Uc>t} J {Ue>t} (3.2) Hence f (Ue Â— I)^dx < [ {t Â— ly J {uc>t) J {ue>i\ dx. Hue>t} J {Ue>t) But setting t = ||/||LÂ°Â°(n) we see that if ess sup u^> t then ^dx which contradicts the above, hence ess sup < ||/||LÂ°Â°(n)Applying a similar argument to u = max(Uf, Â— t) for t = ||/||LÂ°Â°{n) we get ess inf > Â— ||/||Loo(n) and thus |b6||LÂ°Â°(n) < ||L|UÂ°Â°{n)Furthermore, letting u = 0 in (3.2) we see that is bounded in n L'^{Q) C BV{VL) fl L?(Vt) independent of e. Thus there is a

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44 u G BV{Q) n and a subsequence of {ue}, still denoted by {ue}, such that ^ u strongly in L^(Q), ^ u weakly in and ^ u a.e in Q. Letting e Â— > 0 in (3.2), noting that (/?Â£->Â• / v?(Vu) and ^ u in Jn Jn L^(Q), and since v G LÂ‘^{Q) from the construction of ([15]) we can also take > n in Z/^(fl). Therefore we see that the above holds for all v G BV (Q) fl L^(fi) as well. Hence u solves (3.1). By uniqueness, u = u. By the uniform LÂ°Â° bound for and the convergence of Uj to u a.e. in Q we have u G LÂ°Â°{Q) with ||n||Loo(n) < ][/[|LÂ°Â°(n)D We note here that in the above proof, we could have chosen instead, for instance, a regularization of (3.1) using an appropriate smoothing of I with ||-fe||LÂ«>(n) < PI|LÂ«>(n) and If I in T^(fl), instead of We also mention that in passing to the regularized problem, we avoid having to consider the singular measures in the truncation argument. Throughout the rest of this section, we fix ^ > 0 and let ci, C 2 , . . . represent constants that depend only on n, /x, u, f2, (/? and possibly I. Unnumbered constants will be clearly labeled on what they depend. From Tonegawa [26] we have Lemma 3.2.2 Let u G BV{Br{a)) and h G C^{Br{a)) with sup |V/i| < 1 Â— /X, Br{a)

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45 then / ip{Du) Â— / (p{Vh)dx >11 \D^u\ + V{u Â— h) Vhdx J Br{a) J Br{a) J Br{a) J Br{a) + [ D^u-Vh + ^ [ \Vu\dx + l [ \\/{u-h)\^ J Br{a) 2 yBr(a)n{|Vu|>l} 2 ys^(a)n{| Vu| 0 and 0 < /3 < 1 to be chosen later. Also let v be defined by v{x) Â— v[x) Â— I Â• x. Let Tje be the usual mollifier on R" and denote = rjrp*v and = r]r^*v. We also have the estimates ([24]) sup \Vvp Â— ^1 = sup iVn^l < , Br(a) Br{a) sup \vp Â— v\ = sup \vp Â— v\