LOCAL SOLVABILITY OF A CONSTRAINED GRADIENT SYSTEM OF TOTAL VARIATION

A 1 − harmonic map ﬂow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in R N is formulated by use of subd-iﬀerentials of a singular energy - the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result a local-in-time solution of 1 − harmonic map ﬂow equation is constructed as a limit of the solutions of p − harmonic ( p > 1) map ﬂow equation, when the initial data is smooth with small total variation under periodic boundary condition.


Introduction
We consider a gradient system of total variation of mappings with constraint of their values. We are interested in the solvability of its initial value problem. To see the difficulty let us write the equation at least formally. For a mapping u : Ω → R N let E p (u) denote its energy where Ω is a domain in R n and p ≥ 1. The energy E 1 is the total variation of u. Let since π v (w) = w − w, v w, where ·, · denotes the standard inner product in R N . An explicit form for (p−H) is given for example in [24]. Our constrained gradient system of total variation of mapping is the 1−harmonic flow of the form (1−H), i.e., This equation has a strong singularity at ∇u = 0 so that the evolution speed is expected to be determined by a nonlocal quantity. Even if one considers the corresponding unconstrained problem the speed where u is constant is determined by a nonlocal quantity (like the length of spatial interval where u is a constant when n = 1) [19], [14], [11]. The equation (EQ u ) is a nonlocal diffusion equation so even the notion of a solution is a priori not clear. Fortunately for (EQ u ) a general nonlinear semigroup theory (initiated by Y. Kōmura [21]) applies under appropriate boundary conditions since the energy is convex. The theory yields the unique global solvability of the initial value problem for (EQ u ) under Dirichlet boundary condition; see e.g. [8], [6] and also [19], [14], [17]; for a recent L 1 −theory see [3], [1], [2], [7]. However, for (EQ c ) such a theory does not apply since it cannot view as a gradient system of a convex functional. For scalar function a more general form of (EQ c ) without gradient structure is studied when n = 1 by extending the notion of viscosity solution [12], [13]. However, such a theory does not apply since (EQ c ) has no pointwise order preserving structure. For other examples of singular diffusion equations with nonlocal effects the reader is referred to a recent review article [11]. Our goal is to give a suitable notion of a solution of (EQ c ) and to solve its initial value problem under suitable boundary condition. We formulate (EQ c ) with Dirichlet boundary condition and periodic boundary condition by using the subdifferential of energy, which is an extended notion of differentials for nonsmooth functional like E 1 . A similar formulation is given in a recent work of [15]. In fact, they constructed a global solution for any piecewise constant initial data when n = 1, N = 2 and M = S 1 under Dirichlet boundary condition. They also studied its behavior and provided a numerical simulation. However, their analysis is limited for one dimensional piecewise constant mappings although their formulation of the problem is general. Our formulation is close to theirs but slightly different since we use the subdifferential of space-time functional To solve (EQ c ) we prepare an abstract convergence result. Roughly speaking it asserts that if a sequence of approximate energy converges to our energy in the sense of Mosco, the corresponding sequence of the solutions of the approximate problem converges to our original problem. (For this purpose the interpretation of − div(∇u/|∇u|) by a subdifferential of T 0 E 1 (u)dt is convenient.) We use this abstract result by approximating E 1 by E p (1 < p < 2). Compared with the harmonic map flow equation less is known for (p−H) for p ∈ (1,2). M. Misawa [23] proved the global existence of weak solution of the initial value problem with a Dirichlet boundary condition when M = S N −1 . However, his existence result is not enough to apply our abstract theory since it is not clear that div (|∇u p | p−2 ∇u p ) is in L 2 (Ω × (0, T )) for his solution u p of (p−H). Our formulation unfortunately requires such a structure. Moreover, we need the condition that div (|∇u p | p−2 ∇u p ) is bounded in L 2 (Ω × (0, T )) as p ↓ 1 to apply our existence theorem. Recently, A. Fardoun and R. Regbaoui [9] constructed a unique global weak solution for a general target manifold when Ω is a compact manifold without boundary for smooth initial data of small E p energy. Since we need to establish a bound of div (|∇u p | p−2 ∇u p ) ∈ L 2 (Ω × (0, T )), we estimate the Lipschitz norm. Fortunately, we establish a uniform spatially Lipschitz bound for u p in a small time interval, we are able to prove the local solvability of (EQ c ) under a periodic boundary condition when initial data is smooth with small total variation. The constructed solution is spatially Lipschitz continuous. Of course, since results in [9] are for a general source manifold, our results easily extend to such a general manifold by interpreting the gradient in an appropriate way. If u has a jump, the dynamics given by (EQ c ) depends not on the metric of M but also the metric of ambient space R N outside M . This is a serious difference between 1−harmonic flow equation and (p−H) for p > 1. Fortunately, our solution does not depend on that quantity since it has no jumps. We note that notion of BV for mapping in M is not clear as pointed out by [10]. The problem (EQ c ) for the case n = 2 and M = S N −1 is proposed by [27] in image processing. If we let I(x, y, 0) : Ω → R N represent the color data whose components stand for the brightness of each color pixel's of the image at (x, y) ∈ Ω, then its pixel's chromaticity u(x, y, 0) : Ω → S N −1 is expressed by the normalized vector u(x, y, 0) := I(x, y, 0)/|I(x, y, 0)|. The system (EQ c ) for the scaled chromaticity u(x, y, t) describes the process to remove the noise from original u(x, y, 0) maintaining the unit norm constraint and preserving chroma discontinuities. See the book [25] for background of our problem (EQ c ) and other PDEs from image processing. This type of the constrained problems also naturally arise in the modeling of multi-grain boundaries [20] where u represents a direction of grains embedded in a larger crystal of fixed orientation in the two-dimensional frame. We will formulate (EQ c ) by using the notion of subdifferential in Chapter 2. In Chapter 3, we will state three main theorems, which are an Abstract theorem providing the framework of our convergence results, Convergence theorem obtained by applying Abstract theorem, and Local existence theorem following from Convergence theorem by applying the result of [9]. From Chapter 4 to Chapter 6, we will prove these main theorems. In addition, we will prove some properties of general convex functionals, which is used to show Convergence theorem, in Appendix.

Formulation of the problems
In this chapter we formulate the initial value problem with periodic boundary condition.
  where T n := n i=1 (R/ω i Z) for given ω i > 0 (i = 1, 2, · · · , n) and the given initial data u 0 is a map from T n to M . We also formulate the initial boundary value problem   where Ω denotes a bounded domain with a Lipschitz continuous boundary ∂Ω and the initial data u 0 :Ω → M is Lipschitz continuous. We formulate (EQ pe ) and (EQ D ) as evolution equations on L 2 -space. Since some notations are different from each case, we state the formulation of each problem individually. Let M denote a smoothly embedded compact manifold in R N and π v denote the orthogonal projection from R N to the tangent space

Subdifferential formulation of the problem with a periodic boundary condition
We formulate the initial value problem of constrained total variation flow equation with a periodic boundary condition (EQ pe ). First, we define the energy functional φ pe of total variation of each function u ∈ L 2 (T n , R N ) by where BV (T n , R N ) denotes the space of functions of bounded variation on T n with values in R N . It is easy to see that φ pe is a proper, convex, and lower semicontinuous functional on L 2 (T n , R N ) (see [16]). We also consider a functional Φ T pe on L 2 (0, T ; L 2 (T n , R N )) by Φ T pe (u) := T 0 φ pe (u(t))dt.

Proposition 2.1 The functional Φ T pe is proper, convex, and lower semicontinuous on
Proof. The functional Φ T pe is obviously proper and convex on L 2 (0, T ; L 2 (T n , R N )). We will show that Φ T pe is lower semicontinuous. Assume that u m → u strongly in L 2 (0, T ; L 2 (T n , R N )) and Φ T pe (u m ) ≤ λ for any m ∈ N. Since BV (T n , R N ) is compactly embedded in L 1 (T n , R N ) ( [16]), by taking some subsequence of {u m } +∞ m=1 , we have that Then, the lower semicontinuity of φ pe and Fatou's lemma yield, . This implies that Φ T pe is lower semicontinuous on L 2 (0, T ; L 2 (T n , R N )). Now let us formally calculate the variational derivative of this Φ T pe with respect to the metric of L 2 (0, T ; Therefore, the variational derivative δΦ T pe (u)/δu of Φ T pe in L 2 (0, T ; L 2 (T n , R N )) can be formally written as We need several other notations to complete the formulation of (EQ pe ).
Let L 2 (0, T ; L 2 (T n , M )) denote the set of all L 2 -mappings from [0, T ] to L 2 (T n , M ). For any g ∈ L 2 (0, T ; L 2 (T n , M )) we define a map P g (·) : L 2 (0, T ; L 2 (T n , R N )) → L 2 (0, T ; L 2 (T n , R N )) by for any f ∈ L 2 (0, T ; L 2 (T n , R N )). By these notations of the function space, (2.1), and (2.2), (EQ pe ) is formally of the form The initial value problem (EQ1 pe ) does not have a rigorous mathematical meaning since the energy functional Φ T pe is not always differentiable. We need the notion of subdifferential to handle the problem caused by this singularity of the gradient of our Φ T pe and to complete the mathematical formulation of (EQ1 pe ). Let us recall the definition.

Definition 2.2 (Subdifferential)
Let ψ be a proper, convex functional on a real Hilbert space H equipped with the inner product ·, · H . We define the subdifferential of ψ denoted by ∂ψ(u) as Using the subdifferential ∂Φ T pe of Φ T pe , we are now able to formulate (EQ1 pe ) as an evolution equation (EQ2 pe ) in L 2 (0, T ; L 2 (T n , R N )) of the form where u 0 ∈ L 2 (T n , M ) is a given initial data. The initial value problem (EQ2 pe ) can be regarded as a mathematical formulation of (EQ pe ). Our goal is to show the existence of a solution of (EQ pe ); the definition of a solution is given below.

Definition 2.3
We call a function u : ) and satisfies (EQ2 pe ).

Subdifferential formulation of the problem with a Dirichlet boundary condition
In this section we formulate the initial value problem of constrained total variation flow equation with a Dirichlet boundary condition (EQ D ). Let L 2 (Ω, M ) the closed subset of We always choose an initial data v 0 which is a Lipschitz continuous map from Ω to M . Let v 0 denote a Lipschitz extension of v 0 to R n . We define the energy functional φ D with a Dirichlet boundary condition on L 2 (Ω, R N ) as following.
The definition is independent of the way of extension. It is easy to check that φ D is a proper, convex, and lower semicontinuous functional on L 2 (Ω, R N ) (see [16]). Note that the energy φ D also measures the discrepancy of v from v 0 on the boundary ∂Ω.
Since the proof parallels that of Proposition 2.1, we do not respect it. For g ∈ L 2 (0, T ; L 2 (Ω, M )) we define a map P g (·) : Note that each solution of (EQ1 D ) moves satisfying the Dirichlet boundary condition in order to keep minimizing the energy due to the discrepancy on the boundary. The notion of subdifferential of Φ T D allows us to formulate the formal equation (EQ1 D ) as an ) and solves (EQ2 D ).

Convergence results
In this chapter we state three main theorems. The first theorem shows the validity of our scheme to construct a solution of the equations formulated in the previous chapter. For applications we state the theorem in a general setting.
Let H be a real Hilbert space and G be a nonvoid closed subset of H.
be an operator satisfying following properties: Theorem 3.1 (Abstract theorem) Let Ψ m (m = 1, 2, · · · ) and Ψ be proper, convex, lower semicontinuous functionals on L 2 (0, T ; H). Assume that ∂Ψ m converges to ∂Ψ in the sense of Graph (see Remark below). Assume that u m ∈ L 2 (0, T ; H) (m = 1, 2, · · · ) satisfies following conditions; In addition, assume that Applying Theorem 3.1 to our cases, we obtain more explicit statements. Before we give the second theorem, we define approximate energies Φ T pe,m and Φ T D,m (m = 1, 2, · · · ) for our original energies Φ T pe and Φ T D respectively.
Note that these energy functionals are equivalent to p−energy in p−harmonic map flow equation for p = 1 + 1/m. We again associate Φ T 's with φ's.
It is not difficult to see that these functionals Φ T pe,m and Φ T D,m are proper, convex, and lower semicontinuous. We are now in position to state the second theorem. (1) (the case with a periodic boundary condition) Assume that u m ∈ L 2 (0, T ; and u satisfies the energy equality This means that u is a solution of (EQ pe ) in the sense of Definition 2.3.
(2) (the case with a Dirichlet boundary condition) Assume that v m ∈ L 2 (0, T ; and v satisfies the energy equality This means that v is a solution of (EQ D ) in the sense of Definition 2.5.
In some situation our Theorem 3.3 actually yields a solution of our limit problem. Indeed, the solvability result of p-harmonic map flow equation in [9] (1 < p < 2) with Theorem 3.3 and a priori estimate yield local existence of a solution of (EQ pe ) in the sense of Definition 2.3.
Theorem 3.4 (Local Existence theorem) For any K > 0 there exists ε 0 > 0 depending only on T n , M, and K such that if the initial data u 0 : T n → M satisfies following conditions; Remark 3.5 It was proved in [23] that the global weak solution which solves the initial value problem of p−harmonic map flow equation (1 < p < 2) with a Dirichlet boundary condition for the case that the target manifold is S N −1 is an element of ). This regularity of the solution is not sufficient to be a solution of our approximate problem v t ∈ −P v (∂Φ T D,m (v)), since we are considering this evolution equation in L 2 (0, T ; L 2 (Ω, R N )). We need the regularity of the solution as much as all the terms of the equation v t = div(|∇v| 1/m−1 ∇v) + |∇v| 1/m+1 v are elements of L 2 (0, T ; L 2 (Ω, R N )) to be a solution of our approximate problem. Therefore, we are unable to apply our convergence theorem (Theorem 3.3) in this setting. So even local existence is unknown for the Dirichlet problem (EQ2 D ).

Proof of Abstract theorem
We need a notion of convergence of sets in a Hilbert space to carry out the proof. We give the definition of the convergence first. such that x k x weakly in H as k → +∞}.

Proof. By definition for any
We take any (f, g) ∈ A and fix it. Since A m k converges to A as Graph, we see that there exists a sequence (f k , g k ) ∈ A m k (k = 1, 2, · · · ) such that f k → f and g k → g strongly in H as k → +∞. (4.2) By the convergences (4.1), (4.2) and the fact that any weak convergent sequence is bounded in H, we see that Proof. Since ∂Ψ m and ∂Ψ are maximal monotone operators in H, this is a direct consequence of the previous proposition.
We shall show that by taking a suitable subsequence of . Indeed, if we choose some subsequence of {u m k } +∞ k=1 so that the condition (iii) for P (·)(·) holds, then we see that for any h ∈ L 2 (0, T ; H) Here we have used the convergences that P (u m k ) * (h) → P (u) * (h) strongly in L 2 (0, T ; H) by the condition (iii) and (4.5). Therefore, by sending k → +∞ in the both side of f k = P (u m k )(v k ), we have f = P (u)(v) by (4.4) and (4.7). Moreover, the inclusion (4.6) yields that f ∈ P (u)(∂Ψ(u)) holds. Then the desired inclusion has been proved. Now we are ready to prove Theorem 3.1.
Proof of Theorem 3.1. By the condition (ii), we see that {u m,t } +∞ m=1 is bounded. Thus, one can choose a subsequence {u m k ,t } +∞ k=1 ⊂ {u m,t } +∞ m=1 so that u m k ,t converges weakly to some u in L 2 (0, T ; H). Moreover, the convergence u m k → u in L 2 (0, T ; H) yields that u = u t and u m k ,t → u t weakly in L 2 (0, T ; H).
, the definition of sequentially weak upper limit and Proposition 4.5 assure that The properties u ∈ L 2 (0, T ; G) and u| t=0 = u 0 obviously follow from the assumptions. The proof is now complete.

Proof of Convergence theorem
In this chapter we prove Theorem 3.3 as an application of Theorem 3.1. We shall check that the situation of Theorem 3.3 satisfies the assumptions of Theorem 3.1. Some convergence results of the convex functionals assure that Theorem 3.1 is available for our problem. Especially, we show that the functionals φ pe,m , φ D,m , Φ T pe,m , and Φ T D,m defined in Chapter 3 converge to our original energy functionals in the sense of Mosco. The following lemma proved in [26] is the first step. We give its proof for the completeness only under Dirichlet boundary condition, since the proof under periodic boundary condition is easier.  Proof. We first show the condition (i) of Mosco convergence. Assume that u m u weakly in L 2 (Ω, R N ).
It is sufficient to show in the case that u m ∈ D(φ D,m ). Thus, we may assume that u m ∈ W 1,1+1/m (Ω, R N ). By Hölder's inequality we see that u m ∈ BV (Ω, R N ) and Thus, by the lower semicontinuity of φ D , we obtain This implies that (i) holds. Next we show the condition (ii) of Mosco convergence is satisfied. Take any u ∈ D(φ D ) and fix it. Sinceũ ∈ BV (Ω, R N ), by [ and the trace of u j on ∂Ω is equivalent to the trace of u. (5.1) The properties (5.1) yield that u j ∈ D(φ D ) and Moreover, we observe that u j ∈ D(φ D,m ) for any m ∈ N and We take ε i := 1 j , u i := u j for any i ∈ {i * j , · · · , i * j+1 } and any j ∈ N, and u i := u 1 for i ∈ {1, · · · , i * 1 − 1}, and observe thatû i ∈ D(φ D,i ) for any i ∈ N and u i → u in L 2 (Ω, R N ) as i → +∞.
This implies that the condition (ii) of Mosco convergence holds. It needs some technical arguments to prove this proposition. We will give the proof in a general setting in Appendix. The consequence follows from Proposition 5.1, Proposition A.2, and Proposition A.4 which will be proved in Appendix (also see [4], [5]). We can derive energy equalities which are necessary to prove Theorem 3.3 by applying Proposition A.1 also shown in Appendix later. Proof. We only prove (5.3). We can show (5.2) by the same argument as below.

Now we show Theorem 3.3.
Proof of Theorem 3.3. We present the proof only under Dirichlet boundary condition, since the proof is similar for periodic boundary value problem. First we note that Proposition 5.3 actually gives the assumption for the Graph convergence of the subdifferential of energy functionals in Theorem 3.1.
We shall check that our projection P · (·) satisfies the conditions of Theorem 3.1. Since it is easy to check that the conditions (i),(ii) hold, we only show that the condition (iii) holds.
Thus, by compactness [16,Theorem 1.19] this BV bound implies that the sequence We are now able to use Ascoli-Arzela's theorem (for C([0, T ], L 2 (Ω, R N )) ) and conclude that there exists a subsequence {v m(l) We now observe that all the assumptions of Theorem 3.1 are fulfilled. Thus, Theorem 3.1 yields the desired result.

Proof of Local Existence theorem
Since we have already established Convergence theorem, it is sufficient to find approximate solutions of p−harmonic map flow equation which satisfies the assumptions of Convergence theorem. First of all, let us calculate ∂Φ T pe,m to see that solutions of p−harmonic map flow equation solve our approximate problem in our notation with ∂Φ T pe,m .

Lemma 6.1 The subdifferential ∂Φ T pe,m is of the form
Proof. Let v ∈ ∂Φ T pe,m (u). Then by the definition of subdifferential, for any f ∈ C ∞ 0 (T n × [0, T ], R N ) and ε > 0 (The left side of (6.1)) Thus, we have By dividing the both side by ε, sending ε ↓ 0, and integrating by parts, we obtain that By taking negative ε < 0 and sending ε ↑ 0 in the same way, we also obtain This implies that v = − div |∇u| 1/m−1 ∇u . The proof is now complete.
We need to know the solvability result of p−harmonic map flow equation as an approximate problem for our problem. By Lemma 6.1 we safely transfer result of [9] into our setting. Proposition 6.2 (Global solvability of p−harmonic map flow equation [9]) For m ∈ N and K > 0 there exists ε 0 > 0 depending only on K, M, T n and m such that for the initial data u 0,m : T n → M satisfying conditions: Then, there uniquely exists a function u m :

and the energy inequality
for any T > 0. In addition,

Remark 6.3
In [9] this theorem was proved not only for our manifold T n but also for a general compact Riemannian manifold without boundary. The dependence of ε 0 with respect to m is not explicitly stated in [9]. However, if one examines the proof, one conclude that ε 0 can be chosen independently of m ≥ 3 as stated below. Proof. Let us follow the arguments in [9] briefly. In [9] the global solution was obtained as a limit of a function u δ,m : T n × [0, T δ ) → M , which is a solution of following regularized problem (6.5) as δ ↓ 0. (1 + 1/m) sup{1, 2 (1/m−1)/2 }2 1+n/2 . Now by calculation we can check that ε 1 := inf m≥3 ε 1 is still positive and there exists Using these ε 1 > 0 and C > 0, we define ε 0 > 0 by Then by the proof of [9, Theorem 1], one is able to prove that u 0 ∈ C 2+α (T n , R N ), φ m (u 0 ) ≤ ε 0 and ∇u 0 L ∞ (M ) ≤ K yield the consequences of Proposition 6.2.
Corollary 6.5 For any K > 0 there exists ε 0 > 0 depending only on T n , M, and K such that if the initial data u 0 : T n → M satisfies following conditions; Thus, Corollary 6.4 assures the existence of u m : T n × [0, ∞) → M with the desired properties.
We are now in position to prove local existence theorem.
Proof of Theorem 3.4. It is sufficient to show that there exist R > 0 and T > 0 such that for approximate solutions u m whose existence is assured by Corollary 6.5, the inclusion ∂Φ T pe,m (u m ) ⊂ B R holds for any m ≥ m 0 . (6.7) Then, all the assumptions of Theorem 3.3 are satisfied and Theorem 3.3 yields the existence of a solution of (EQ2 pe ) for this T > 0. We see that the approximate equation u m,t ∈ −P u m ∂Φ T pe,m (u m ) is equivalent to the following equation.
for any (x, t) ∈ T n × (0, +∞), (6.9) where C is a positive constant depending only on M . By the inequality (6.4) and the assumption (iii) of Theorem 3.3, we know that there exists R > 0 such that u m,t ∈ B R for any m ≥ m 0 . Thus, if we prove that there exists K > 0 and T > 0 such that then, by (6.8) and (6.9) we have that for some R > 0 independent of m. This inclusion implies that (6.7) holds. We shall show the inequality (6.10).
, by a standard argument for system of uniform parabolic equation (see [22]), we conclude that u m ∈ C ∞ (U ). We put w m (x, t) := |∇u m (x, t)| 2 and differentiate the both side in time. Noting the equality (6.8), we see that Moreover by calculation we obtain (the first term of (6.11)) = n i,j=1 (the second term of (6.11)) = (the third term of (6.11)) ≤ i are continuous functions in U and C is a positive constant depending only on M . More precisely, we see that a ij is written as where δ ij is Kronecker's delta. We can check that (a ij ) > 0 in U . Indeed, by Schwarz's inequality Substituting the (in)equalities (6.12) into (6.11) we obtain the inequality that i . Let f m (t) be a solution of the following initial value problem.
Then f m is of the form (6.14) Evidently, f m is strictly increasing and blows up when t = t m , where t m is given by t m := 2 Plug v m into (6.11) and we obtain that By taking λ sufficiently large we may assume that d( Thus, the standard maximum principle for parabolic equations assures that there exists a boundary point ( We obviously observe that at least one of the following properties holds. ( For each case it is easy to check that v m (x,t) ≤ 0. As the conclusion, the inequality ). Moreover, the definition of U and (6.16) yield that By (6.14) and (6.15) we obtain that if m 1 ≤ m 2 , then Then f blows up when t 0 = 2/ C max{1, K 2 } and we observe that t m < t 0 for any m ∈ N and t m t 0 as m → +∞. Now take any T ∈ (0, t 0 ) and fix it. Then there exists a natural number m T ≥ m 0 such that the blow up time t m T of f m T is larger than T . Noting (6.18) and (6.19), we see that In other word, |∇u m (x, t)| ≤ f m T (T ) for any (x, t) ∈ T n × [0, T ] and any m ≥ m T .
If we set K := max then we finally obtain (6.10). Thus, Theorem 3.3 yields the existence of a solution of (EQ2 pe ) in L 2 (0, T ; L 2 (T n , R N )). The energy equality (3.2) follows by the same argument as Proposition 5.4.
Therefore, it follows from (A.5) and (A.6) that Next we show one proposition for Mosco convergence of convex functional which assures the statement of Proposition 5.3. We follow the arguments in [4]. Let us set some notations used below in advance. Let H denote a real Hilbert space and φ m (m = 1, 2, · · · ) and φ be proper, convex, and lower semicontinuous functionals on H. Define functionals Φ m (m = 1, 2, · · · ) and Φ on L 2 (0, T ; H) by Φ m (u) := Remark A.3 This is generized to time dependent φ t m , φ t by N. Kenmochi [18] under suitable assumptions.
We recall a property for Mosco converging energy functional.
Proposition A.4 (See [4] or [5]) The following properties are equivalent. Remark A.5 Note that the convergence that ∂φ m → ∂φ in the sense of resolvent is equivalent to the convergence ∂φ m → ∂φ in the sense of Graph (see [4] or [5]).
The previous proposition means that to show the property (b) for Φ m and Φ is sufficient to attain our purpose. We prepare some lemmas to show the property (b).   On the other hand, the second convergence of (A.11) and the definition of Mosco convergence assure that lim inf k→+∞ φ m k (z k ) ≥ φ(x 0 ). (A.14) We have by (A.13) and (A.14) that φ(x 0 ) = −∞. This is a contradiction since we took x 0 ∈ D(φ). In addition, since z m ∈ (I + λφ m )(ξ m ), we have ξ m = J λ z m . Similarly we can get ξ = J λ z. Thus we see that φ m (J λ z m ) → φ(J λ z) as m → +∞, which implies that φ λ m (z m ) → φ λ (z) as m → +∞. Therefore we observe that