The elastic flow with obstacles: small obstacle results

We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined.

Since we deal with obstacle problems, our admissible functions are required to lie above a suitable obstacle function ψ : p0, 1q Ñ R which we will specify later. The boundary conditions we want to impose are 'fixed ends', i.e. up0q " up1q " 0, so that the admissible set can be chosen as (1.1) C ψ :" tu P W 2,2 p0, 1q X W 1,2 0 p0, 1q : u ě ψ a.e.u. Existence (and nonexistence) of minimizers of E in C ψ has been studied in [7] and [18], minimization with slightly different frameworks has also been examined in [15], [16], [17] and [8].
The articles [7], [18] and [17] reveal that under certain smallness conditions on ψ minimizers do exist whereas they do not exist in general if the obstacle is too large.
A useful necessary criterion for minimizers is the variational inequality. More precisely -if u P C ψ is a minimizer, then u solves (1.2) DEpuqpv´uq ě 0 @v P C ψ , where DE denotes the Frechét derivative of E : W 2,2 p0, 1q X W 1,2 0 p0, 1q Ñ R. In the following we will also call solutions of (1.2) constrained critical points.
Once minimizers are found, an object of interest is the coincidence set Γ :" tu " ψu, which forms the so-called free boundary of the problem. For higher order variational problems like this one, a description of this free boundary is particularly challenging because of the lack of a maximum principle.
In this article we do not want to study minimizers but rather approximation of critical points by a certain type of L 2 -gradient flow, called parabolic obstacle problem in the literature.
Parabolic obstacle problems are time-dependent evolutions that flow towards solutions of the variational inequality. Such evolutions are driven by the so-called flow variational inequality, for short F V I. In our situation this reads p 9 uptq, v´uptqq L 2`DE puptqqpv´uptqq ě 0 @v P C ψ .
Parabolic obstacle problems form a large class of time-dependent free boundary problems, sometimes also called moving boundary problems. Here the moving boundary is given by Γ t :" tuptq " ψu.
In more beneficial frameworks parabolic obstacle problems can also be seen as gradient flows in the metric space pC ψ , d L 2 q, which immediately implies that evolutions dissipate energy in a direction that is steepest possible, cf. [3], [2].
Many authors have studied moving boundary problems driven by second order operators but recently fourth order problems have also raised a lot of interest, cf. [20], [21], [19], [9], [22]. The energies in [20], [21] are (semi-)convex which implies that the evolution can easily be regarded as a metric gradient flow in the sense of [3], [2]. We emphasize that the general framework in [3], [2] really relies on convexity assumptions, which E does not satisfy.
In [19] the lack of convexity is circumvented by looking at the gradient flow in a different flow metric, namely in the metric space pC ψ , d W 2,2 XW 1,2 0 q. We remark that in this metric space, E is locally semiconvex. Our given energy is neither L 2semiconvex nor do we want to use any other flow metric than the L 2 -metric. For this we have to pay a price.
Firstly, we must require that the obstacle is appropriately small to stay in a region where the elastic flow and the biharmonic heat flow show similar behavior. Most of our arguments will work by comparision to the biharmonic heat flow, controlling the nonlinearities with the various smallness requirements.
Secondly, we are unable to fit the flow into the framework of metric gradient flows. Properties like energy dissipation are thus not immediate consequences and have to be examined seperately. Nevertheless the flow follows now dynamics that are analytically very accessible, which makes the aforementioned comparision to the biharmonic heat flow possible. This is the reason why we study this particular dynamics.
The techniques used to construct the flow mainly rely on De Giorgi's minimizing movement scheme, a 'variational time discretization' for the problem. We remark that the evolution was constructed independently in [22], where the authors use the same scheme but carry out a different approach when passing to the limit.
After the construction of our flow is finished we examine further properties such as well-posedness, size of the moving boundary, regularity and convergence behavior. A byproduct of this study is that we show reflection symmetry of minimizers of E in C ψ for some obstacles ψ using symmetric decreasing rearrangements in a setting of nonlinear higher order equations.
Definition 2.1 (Elastic energy). We define the elastic energy E : W 2,2 p0, 1q X W 1,2 0 p0, 1q Ñ R to be The function G is important for many quantities that we consider, hence we will fix G as in (2.1) for the rest of the article.
We also require some conditions on the obstacle for the entire article, which we state here.
Assumption 1 (Assumptions on the obstacle). We always assume that ψ P Cpr0, 1sq is such that ψp0q, ψp1q ă 0 and there exists x 0 P p0, 1q such that ψpx 0 q ą 0. The admissible set C ψ is then defined as in (1.1). We define also I ψ :" inf uPC ψ
In particular, I ψ can become arbitrarily small for small obstacles, cf. Figure 1.
In the following theorems we will always fix an initial value u 0 P C ψ that satisfies a certain energy bound. For such an initial value to exist one usually needs a condition on I ψ which we will not write explicitly.
Remark 2.4. The existence and uniqueness of the required Cpr0, 8q, L 2 p0, 1qqrepresentative follows from the Aubin-Lions lemma from which also follows that the solution lies in Cpr0, 8q, C 1 pr0, 1sqq. Whenever we address the flow from now on we will always identify it with its Cpr0, 8q, C 1 pr0, 1sqq-representative unless explicitly stated otherwise. This means in particular that evaluations at fixed times t are well-defined -at least in C 1 pr0, 1sq. Next we state our main existence theorem.
The energy threshold of c 2 0 4 is necessary for our approach since below this threshold one can obtain a control of the nonlinearities in the Euler-Lagrange equation. The same threshold (and the same control) is used in [22], where an existence result is obtained independently. If Epu 0 q ă c 2 0 4 one can also show that the F V I gradient flow starting at u 0 is unique, cf. [22,Section 3].
We also discuss regularity in time: As we shall see in Section 5 almost every point t P p0, 8q is a point of continuity of u in the W 2,2 p0, 1q-topology.
Another interesting question is whether the flow touches the obstacle in finite time. This is in particular of interest because in case that the flow does not touch the obstacle, each F V I-Gradient flow just coincides with a regular L 2 gradient flow. If this were the case, it could have been constructed with much less effort. However for small initial data there holds Proposition 2.7 (Coincidence in finite time). For each u 0 P C ψ such that Epu 0 q ă Gp b 2 3 q 2 there exists a sequence t n Ñ 8 such that upt n q touches ψ. Next, we are interested in the asymptotic behavior of the flow. For this we first examine closely what candidates for limits are available. Definition 2.8 (Critical point). We say that u P C ψ is a (constrained) critical point in C ψ if it is a solution of the variational inequality (2.4) DEpuqpv´uq ě 0 @v P C ψ .
In our classification we seek to understand only symmetric critical points, i.e. critical points u that satisfy u " up1´¨q. The reason why those critical points are more important than the others is that under some conditions on the obstacle, the minimizers of E in C ψ can be shown to be symmetric. The condition on the obstacle are precisely that ψ itself is symmetric, "small" and radially decreasing i.e. ψ is a decreasing function of |x´1 2 |. An important special case are symmetric cone obstacles, i.e. ψ is symmetric and ψ| r0, 1 2 s is affine linear. The main technique used here is a nonlinear version of Talenti's inequality, a classical symmetrization procedure, cf. [23]. Once symmetry is shown one can obtain uniqueness of critical points by the following Theorem 2.9 (Uniqueness and minimality of symmetric critical points). Let ψ be a symmetric and radially descreasing obstacle and u 0 P C ψ such that Epu 0 q ď Gp2q 2 . Then there exists a symmetric minimizer of E in C ψ . Moreover, if ψ is a symmetric cone obstacle, this minimizer is the unique symmetric critical point in C ψ .
We remark that the symmetric critical points (and their uniqueness) have been studied independently also in [25] via the shooting method. Here we present a slightly different (but more or less equivalent) approach involving hypergeometric functions.
In Section 7 we show a subconvergence result. Here we first specify what we mean by subconvergence. Definition 2.10 (Subconvergence). Let A Ă r0, 8q be an unbounded set, X be a Banach space and u : A Ñ X. Let M Ă X be a set. We say that u is Xsubconvergent to points in M if each sequence pθ n q Ă A such that θ n Ñ 8 possesses a subsequence θ kn such that upθ kn q converges in X to an element of M . If A " r0, 8q we say u is fully X-subconvergent to points in M . Theorem 2.11 (Subconvergence to critical points). Let u 0 P C ψ be such that Epu 0 q ă c 2 0 4 . Let u be an F V I gradient flow starting at u 0 . Then u : r0, 8q Ñ C 1 pr0, 1sq is fully C 1 pr0, 1sq-subconvergent to points in Moreover, for each ą 0 there exists a set B Ă r0, 8q with |B| ă such that u | r0,8qzB : r0, 8qzB Ñ W 2,2 p0, 1q is W 2,2 p0, 1q-subconvergent to points in M crit .
Here we note that smallness requirements on the obstacle are really necessary for such a subconvergence result: For large obstacles ψ, it is shown in [19,Corollary 5.22] that no critical points exist in C ψ . This shows that the energy requirement in Theorem 2.11 may not be omitted.
Subconvergence improves to convergence as soon as there is only one element that is still in the limit candidate set. This is the case in the situation of Theorem 2.9. The following theorem summarizes many of the findings above. Figure 2. A useful byproduct of our approach is that we can find explicit formulas for minimizers if ψ is a small cone obstacle. In the situation of the first plot, Theorem 2.9 yields that u is a minimizer and the only symmetric critical point. In the second plot, u is the only symmetric critical point but the obstacle is too big to conclude with Theorem 2.9 that u is a minimizer. We strongly suspect that it is a minimizer anyway. Theorem 2.12 (Convergence for cone obstacles). Suppose that ψ is a symmetric cone obstacle. Let u 0 P C ψ be symmetric and Epu 0 q ă mintGp2q 2 , c 2 0 4 u. Then there exists a unique FVI gradient flow t Þ Ñ uptq in C ψ that converges to the unique symmetric minimizer of E in C ψ .
In particular we have shown that small obstacles and small initial data lead to convergent evolutions that respect the obstacle condition.

Preliminaries
3.1. Basic properties of the energy. Here we discuss basic estimates and properties of E that will be useful in the following. Most energy estimates will be expressed in terms of the function G : R Ñ p´c 0 2 , c0 2 q, where G and c 0 are defined as in (2.1), (2.2). Proof. If u P C ψ then up0q " up1q " 0. This implies that we can choose ξ P p0, 1q such that u 1 pξq " 0. Also, since u 1 P Cpr0, 1sq we can find η P p0, 1q such that u 1 pηq " ||u 1 || 8 . Note that |ξ´η| ď 1. By the Cauchy Schwarz inequality we have Remark 3.2. Note that each v P C ψ such that Epvq ă implies existence of a global minimizer of E in C ψ . It is also worth noting that each v P C ψ with Epvq ă These observations are the crucial reason for the energy bounds in the statement of Theorem 2.6 and Theorem 2.11.
In the following proposition we discuss first properties of the Frechét derivative DE. Most of those computations have already been made in [7], [18]. dx.
If additionally u P W 3,1 p0, 1q and u 2 p0q " u 2 p1q " 0 then Proof. Equation (3.1) can be found in [7,Equation 1.5]. If u is now as in the second part of the claim we can perform an integration by parts in the first summand to get DEpuqpφq " Since the boundary terms vanish by assumption we only end up with the last two integrals, whereupon (3.2) can easily be verified using the product rule.
Remark 3.4. The astoundingly compact formula (3.2) was already known to Euler, see [11], and will be of great use for us. The notation A u :" will also be used throughout the article.
Remark 3.5. Another useful consequence of (3.1) is that for all u, φ P W 2,2 p0, 1q X W 1,2 0 p0, 1q one has (3.3) |DEpuqpφq| ď p2||u|| W 2,2`5||u|| 2 W 2,2 q||φ|| W 2,2 . Here we have used that ||φ 1 || L 8 ď ||φ 2 || L 2 for all φ P W 2,2 X W 1,2 0 . 3.2. Basic properties of FVI gradient flows. In this section we will briefly discuss why the F V I gradient flow generalizes the concept of an L 2 -gradient flow. Furthermore we will discuss some basic regularity properties that follow immediately by the definition. Recall that for a Hilbert space H that is dense in L 2 , a Frechét differentiable functional F : H Ñ R is said to have an L 2 -gradient at u P H if DFpuq P H˚possesses an extension to a linear continuous functional in pL 2 q˚. We denote by ∇ L 2 Fpuq P L 2 the representing element of this functional in L 2 .  uptq "´∇ L 2 Epuptqq.
Looking first at a positive and then at a negative value of and dividing both times by we find DEpuptqqpφq " p´9 uptq, φq @φ P W 2,2 p0, 1q X W 1,2 0 p0, 1q. Since W 2,2 X W 1,2 0 is dense in L 2 we find that DEpuptqq can be extended to an element of L 2 p0, 1q˚, represented by´9 u. By the very definition of L 2 -gradient follows that´9 uptq " ∇ L 2 Epuptqq and hence the claim.
The F V I gradient flow starts with significantly less regularity in time than the one in [19]. However we can extract some immediate regularity properties: Here we expose a basic feature that will be very important for our analysis: Uniform C 0, 1 2 pp0, 8q; L 2 p0, 1qq-estimates. Proof. Let u 0 , u, s, t be as in the statement.

Existence Theory
In this section we construct the F V I gradient flow by a variational discretization scheme. We first show existence of so-called discrete flow trajectories, which we will define. The discrete stepwidth will always be denoted by τ ą 0. Once the discrete trajectories are defined we can look at their asymptotics as τ Ñ 0. To get desirable limit trajectories we need a suitable compactness, which we will achieve by discussing additional regularity properties of the discrete trajectories in parabolic function spaces.
That the minimum problem in (4.2) has a solution for all τ ą 0 and k P N is due to Lemma 4.1. To ensure that the Lemma is applicable for all k P N we have to check that Epu kτ q ă c 2 0 4 for all k P N and τ ą 0. This follows by induction since for all k P N it holds that Epu pk`1qτ q ď Epu kτ q. The latter inequality is an immediate consequence of the observation that Φ u kτ τ pu pk`1qτ q ď Φ u kτ τ pu kτ q by (4.2). Another noteworthy implication of this inequality is that for all τ ą 0 the map r0, 8q Q t Þ Ñ Epu τ ptqq P R is nonincreasing and coincides with Epu 0 q at t " 0. Monotonicity of the energy is not immediate for the piecewise linear interpolations, which reveals an important advantage of the pievewise constant interpolation.
Remark 4.5. Note that the piecewise linear interpolation is weakly differentiable in t and satisfieŝ Hence we have a uniform estimate for || 9 u τ || L 2 pp0,8q,L 2 p0,1qq independently of τ . Moreover, for all T ą 0 u τ lies in W 1,2 pp0, T q, L 2 p0, 1qq with norm bounded by where C T is a constant that depends only on T and not on τ .
Notice that both u τ andū τ play a role in this variational inequality.
Remark 4.7. Another fact we will make use of is that the L 2 distance of both defined interpolations can be uniformly controlled in time, more precisely This is an immediate consequence of (4.3). . Then pu τ q τ ą0 is bounded in L 8 pp0, 8q, W 2,2 p0, 1qq With the next lemma, we can obtain a global limit trajectory of u τn for a carefully chosen sequence τ n Ñ 0. The convergence will unfortunately not be good enough to show that this limit trajectory is an F V I gradient flow. To this end we have to obtain more regularity first and work with bothū τn and u τn .
Here we fix the right subsequence to consider for the additional regularity. A property that we will use very often in the arguments to come is that weak topologies have the Urysohn property, i.e. a sequence converges weakly if and only if each subsequence has a weakly convergent subsequence and the limit of all those subsequences coincide.
It remains to prove that u lies in L 8 pp0, 8q, W 2,2 p0, 1qq X D 1,2 pp0, 8q, L 2 p0, 1qq and u τ ptq converges weakly to uptq in W 2,2 p0, 1q for each t ą 0. We start with the latter assertion. We know that for each t ą 0, pu τ n n ptqq converges to uptq in C 1 pr0, 1sq. Since also ||u τ n n ptq|| W 2,2 is uniformly bounded by Lemma 4.8, we obtain that each subsequence of u τ n n has a weakly W 2,2 -convergent subsequence to some r u P W 2,2 p0, 1q that may depend on the choice of the subsequence. However, by uniqueness of limits in C 1 pr0, 1sq, we get that r u " uptq for each possible choice of a subsequence. This being shown, the Urysohn property of weak convergence implies that u τ n n ptq á uptq in W 2,2 p0, 1q. Note that uptq P C ψ as C ψ is weakly closed in W 2,2 p0, 1q. The uniform boundedness of pu τ n n q 8 n"1 in L 8 pp0, 8q, W 2,2 p0, 1qq implies then that u P L 8 pp0, 8q, W 2,2 p0, 1qq. We now show that u P D 1,2 pp0, 8q, L 2 p0, 1qq. For weak differentiability of u on p0, 8q fix φ P C 8 0 pp0, 8q; Rq. Observe that there exists T ą 0 such that supppφq Ă p0, T q. Since u τ n n á u in W 1,2 pp0, T q, L 2 p0, 1qq we obtain where wlim denotes the weak limit in L 2 p0, 1q. It only remains to show that 9 u P L 2 pp0, 8q, L 2 p0, 1qq. To this end, note that for each N P N one has by Remark 4 which is a bound that is independent of N . Letting N Ñ 8 the monotone convergence theorem implies the claim.
In the next lemma we show an estimate that will later account for L 2 pp0, T q, W 3,8 p0, 1qqbound of pu τ q τ ą0 . This will be the needed extra regularity to pass to the limit in the energy space. Another useful byproduct are the Navier boundary conditions that are a natural consequence of the underlying variational inequalities (cf. (4.5)). . Then there exist C, D ą 0 such that for each τ ą 0 and each t ą 0, u τ ptq P W 3,8 p0, 1q, pu τ ptqq 2 p0q " pu τ ptqq 2 p1q " 0. Moreover (4.6) ||u τ ptq 3 || L 8 ď C`D|| 9 u τ ptq|| L 2 p0,1q @t ą 0 for constants C, D that may depend on u 0 and the obstacle but not on τ .
Proof. Let T ą 0 be fixed. That pu τ q τ ą0 is uniformly bounded in W 1,2 pp0, T q, L 2 q has already been shown in Remark 4.5. Fix t, τ ą 0 and fix k P N 0 such that t P pkτ, pk`1qτ s. By (4.6) one has where C, D are chosen as in (4.6). For the next computation we set for convenience of notation u´1 τ :" u 0 . We can use the above estimate and (4.3) to find We infer that pu τ q τ ą0 is bounded in L 2 pp0, T q, W 3,8 q X W 1,2 pp0, T q, L 2 q, which embeds by the Aubin-Lions-Lemma compactly into L 2 pp0, T q, C 2 pr0, 1sq. Let τ n be the sequence constructed in Lemma 4.9. Then by the L 2 pp0, T q, W 3,8 q X W 1,2 pp0, T q, L 2 q-bound each subsequence of τ n must have another subsequence along which pu τn q 8 n"1 converges weakly in L 2 pp0, T q, W 3,2 p0, 1qq and strongly in L 2 pp0, T q, C 2 pr0, 1sqq. Because of uniqueness of weak limits in L 2 pp0, T q, W 2,2 q we deduce that all those subsequences must converge to the same u as constructed in Lemma 4.9. By the Urysohn property pu τn q 8 n"1 converges to u strongly in L 2 pp0, T q, C 2 pr0, 1sqq and weakly in L 2 pp0, T q, W 3,2 p0, 1qq. Convergence in the claimed spaces follows as T ą 0 was arbitrary. Choosing a further subsequence of pτ n q 8 n"1 we may also assume that u τn Ñ u pointwise almost everywhere in C 2 pr0, 1sq as L 2 -convergence of Banach-space valued functions implies the existence of a pointwise almost everywhere convergent subsequence. That uptq 2 p0q " uptq 2 p1q " 0 for almost every t ą 0 is then an immediate consequence of this fact.
So far we have shown convergence of the piecewise linear interpolations. As mentioned in Remark 4.3 we also need results on the behavior of the piecewise constant interpolations to control the energy. . Let u 0 be as before. Then pu τ q τ Pp0,1q is precompact in L 2 pp0, T q, C 2 pr0, 1sqq for each T ą 0.
Proof. The proof relies on a discrete version of the Aubin-Lions lemma -the socalled discrete Aubin-Lions-Dubinskii lemma, see [10,Theorem 1]. To apply this we just need to show that for all T ą 0 the expression is uniformly bounded in τ . The claim follows then since the embedding W 3,8 p0, 1q ãÑ C 2 pr0, 1sq is compact. That the second summand is uniformly bounded in τ follow from (4.6) and (4.4). For the first summand let N τ P N be such that pN τ´1 qτ ď T ď N τ τ and calculate using (4.3) Hence [10, Theorem 1] is applicable and the claim follows. Proof. The fact that uptq P C ψ for almost every t ą 0 follows from the fact that C ψ is weakly closed in W 2,2 p0, 1q. For the proof of the F V I inequality we choose v P C ψ . Let pu τn q 8 n"1 be a sequence chosen as in Lemma 4.11. Further let 0 ă a ă b be arbitrary. Integrating (4.5) we find that b a p 9 u τn ptq, v´u τn ptqq dt`ˆb a DEpu τn ptqqpv´u τn ptqq dt ě 0.
Since u τn converges to u uniformly in L 2 p0, 1q we can infer from Remark 4.7 that for each t ą 0 u τn ptq converges to uptq in L 2 p0, 1q. We also infer from Lemma 4.12 that -after choosing an approprate subsequence of pτ n q 8 n"1 again with a straightforward diagonal argument -we can ensure that for almost every t ą 0 the sequence pu τn ptqq 8 n"1 converges to uptq in C 2 pr0, 1sq. From this one immediately concludes that DEpu τn ptqqpv´u τn ptqq Ñ DEpuptqqpv´uptqq a.e. t ą 0.
This shows (2.3). It remains to show that t Þ Ñ Epuptqq coincides almost everywhere with a nonincreasing function f that satisfies f p0q " Epu 0 q. By Remark 4.3 t Þ Ñ Epu τn ptqq is nonincreasing for each τ ą 0. By Helly's theorem (cf. [3, Lemma 3.3.3]) this sequence of functions has a pointwise limit, which is a nonincreasing function, call it f . We have already shown in Lemma 4.12 that u τn ptq converges to uptq in C 2 pr0, 1sq for almost every t ą 0 so that Epu τn ptqq converges to Epuptqq pointwise almost everywhere. Hence t Þ Ñ Epuptqq coincides almost everywhere with f . Remark 4.14. A useful byproduct of this approach is that also u τn ptq Ñ uptq in L 2 p0, 1q for all t ą 0 (and not just almost everywhere). More can be said: Boundedness of pu τn ptqq 8 n"1 in W 2,2 p0, 1q (cf. (A.3)) implies that u τn ptq á uptq weakly in W 2,2 p0, 1q for all t ą 0.
Finally, we have constructed an F V I gradient flow. Before we can prove Theorem 2.6 we need to discuss some further properties of the constructed flow.

4.2.
Space regularity and Navier boundary conditions. The minimizing movement construction in the first part of this section is a highly nonunique concept. In general Theorem 2.6 asserts however some additional regularity properties that hold true for every possible choice of a F V I gradient flow starting at u 0 . To show this, we will not use the above construction and work directly with the definition instead. and let u be a F V I-Gradient Flow. Then uptq P C ψ for all t ą 0 and for each sequence t n Ñ t one has upt n q Ñ uptq weakly in W 2,2 p0, 1q. In particular t Þ Ñ uptq is a bounded curve in W 2,2 p0, 1q.
Proceeding similar to the proof of Lemma 4.10 we can derive the claimed regularity and the Navier boundary conditions.
Proof of Theorem 2.6. Existence of u follows from Corollary 4.13. That t Þ Ñ uptq P W 2,2 p0, 1q is everywhere defined and bounded follows from the last sentence of Lemma 4.15. The additional space regularity and the Navier boundary conditions follow from Lemma 4.16.

Energy dissipation.
In the rest of this section we will prove an energy dissipation inequality. This shows that energy is dissipated in p0, T q is comparable to || 9 u|| 2 L 2 pp0,T q,L 2 p0,1qq , which is what one would expect for a gradient flow. The speed of energy dissipation we obtain might however be worse than in the usual formulation of metric gradient flows. The expected dissipation speed can be described by De Giorgi's energy dissipation identity, cf. [ Proof. Let pu τn q 8 n"1 be the sequence from Lemma 4.11. For n P N we define k n P N 0 to be the unique integer such that k n τ n ď T ď pk n`1 qτ n . By weak convergence of u τn in W 1,2 pp0, T q, L 2 p0, 1qq (cf. Lemma 4.9) and weak W 2,2´c onvergence of u τn pT q to upT q (cf. Remark 4.14) we obtain with (4.3) pu pkn`1qτn q`k n ÿ l"0 pEpu l q´Epu l`1 qq¸" Epu 0 q.

4.4.
Uniqueness and preservation of symmetry. Now that we have shown existence of F V I gradient flows one can ask whether they are unique. This uniqueness has been obtained in [22,Section 3]. It has an important consequence for our later studies of the asymptotics -namely that evolutions are symmetry preserving, as we shall show.  Proof. Let u be as in the statement. We show that r u : t Þ Ñ uptqp1´¨q is an F V I gradient flow. As u and r u have the same initial datum, they must coincide by uniqueness. From the symmetry of ψ follows that r uptq P C ψ for almost every t ą 0. The regularity requirements are also easily to be checked. Moreover, by symmetry of E, E˝r u " E˝u coincides almost everywhere with a nonincreasing function that takes the value Epu 0 q at t " 0.
For arbitrary v P C ψ we infer by symmetry properties of the L 2 scalar product that p 9 r uptq, v´r uptqq`DEpr uptqqpv´r uptqq " p 9 uptqp1´¨q, v´uptqp1´¨qq`DEpuptqp1´¨qqpv´uptqp1´¨qq for a.e. t ą 0, because u is an F V I Gradient Flow and vp1´¨q P C ψ because of the symmetry of the obstacle.

Qualitative Behavior
Describing the qualitative behavior of higher order PDEs is in general a challenging task as there is no maximum priciple available that would allow a comparision of solutions. In the field of parabolic obstacle problems one is however interested in several qualitative aspects, in particular the description of the coincidence set tuptq " ψu that forms the now time-dependent free boundary of the problem.

The coincidence set.
Here we prove that the obstacle is touched in finite time, provided that the initial energy is suitably small. Not much more can be said about the size of the coincedence set as there exist critical points for which the coincedence set is only a singleton (cf. [ From Remark 3.2 also follows that inf uPC ψ Epuq " min uPC ψ Epuq ą 0. We here prove the slightly stronger statement that each time interval of length larger than must contain a time t such that uptq touches ψ. Suppose that pa, bq is an interval of length exceeding L 0 such that tuptq " ψu " H on pa, bq. Note that then uptq ą ψ and Proposition 3.6 yields that (5.1) p 9 uptq, φq`DEpuptqqpφq " 0 @φ P W 2,2 p0, 1q X W 1,2 0 p0, 1q for almost every t P pa, bq. We use again the Lions-Magenes-Lemma to compute in the sense of distributions we have d dtˆt 0 uptq 2 dx " 2p 9 uptq, uptqq.
Since t Þ Ñ uptq is absolutely continuous with values in L 2 , so is t Þ Ñ ||uptq|| L 2 with values in R. By the product rule for Sobolev functions and the fact that t Þ Ñ ||uptq|| L 2 is uniformly bounded in t, t Þ Ñ ||uptq|| 2 L 2 lies in W 1,1 p0, 1q. Hence the above inequality holds also pointwise almost everywhere and the fundamental theorem of calculus can be applied. By Theorem 2.6, uptq P C ψ X W 3,2 p0, 1q and uptq 2 p0q " uptq 2 p1q " 0 for almost every t. For those t we can define A t :" uptq 2 p1`uptq 12 q 5 4 and use (5.1) and (3.2) to find which is negative by the assumptions. By the fundamental theorem of calculus (whose applicability we have discussed above) and Remark 3.2 we find which results in a contradiction as the expression on the left hand side must be nonnegative.

Time regularity.
Since the constructed evolution is not driven by an equation but rather by an inequality one can not immediately obtain time regularity from space regularity. In general, time regularity for parabolic obstacle problems is an important problem. A technique that has been applied in previous works, e.g. [20], is to consider the flow as singular limit of perturbed evolutions without obstacle. We refer to [4] for a discussion of this technique. We remark that this approach heavily relies on uniqueness which is not the focus of this article. This is why we present a different approach. Proof. Since u is an F V I Gradient Flow, there exists a nonincreasing function φ : r0, 8q Ñ R and a set r B Ă r0, 8q such that |r0, 8qz r B| " 0 and E˝u " φ on r B. Also, let B be all the points of continuity of φ that lie in r B. Since φ is monotone, r BzB is at most countable and we find that |r0, 8qzB| " 0. Moreover define E :" ts : s is not a Lebesgue point of || 9 u|| L 2 , or F V I does not hold true at su.
Since |E| " 0 it suffices to show that each point s P BzE is a point of W 2,2continuity. We fix therefore s P BzE and let first t ą 0 be arbitrary. All we know then is that uptq P C ψ by Lemma 4.15. Now we compute, again using that is Lipschitz continuous and Remark 3. where D ą 0 is an appropriately chosen constant that does not depend on t. We find that there exists C 0 ą 0 such that (5.2) ||uptq´upsq|| W 2,2 ď C 0 pEpuptqq´φpsqq`pD`|| 9 upsq|| L 2 q||upsq 1´u ptq 1 || L 8 q for all arbitrary t ą 0. Now let ą 0 be arbitrary. Since s is a point of continuity of φ there exists δ 1 ą 0 such that sup tPB δ 1 psq |φptq´φpsq| ă 2C0 . Moreover || 9 upsq|| L 2 ă 8 as s is a Lebesgue point of 9 u and therefore there exists δ 2 ą 0 such that sup tPB δ 2 psq ||uptq 1´u psq 1 || L 8 ă 2C0pD`|| 9 upsq||q . Now choose δ :" mint δ1 2 , δ 2 u. Let t P p0, 8q be such that |t´s| ă δ. Then there exists a sequence t n Ñ t such that pt n q 8 n"1 Ă B as |p0, 8qzB| " 0. We can assume without loss of generality that for all n P N one has |t n´t | ă δ1 2 , which implies |t n´s | ă δ 1 for all n P N. Now note that by weak lower semicontinuity of E (cf.

Critical Points
In the next section we want to examine the critical points of E in C ψ . One question that could be asked is how many critical points exist. A partial answer is given in [19,Corollary 5.22] and [17], where it is shown that there exist no critical points above an obstacle of a certain height. This is also why our convergence results may only hold true for small obstacles.
Once existence is ensured, another question one can look at is symmetry of critical points, which is to expect since the equation has a symmetry: If u P C ψ solves (2.4) and ψ " ψp1´¨q then also up1´¨q P C ψ is a solution of (2.4), as follows directly from (4.10). Lemma 6.1 (Regularity and concavity of critical points). Let u P C be a critical point. Then u P W 3,8 p0, 1q, u 2 p0q " u 2 p1q " 0 and u is concave. Moreover, if u ą ψ on some interval pa, bq then u | ra,bs P C 8 pra, bsq and " const. on pa, bq.
Proof. For the regularity, (6.1) and the fact that u 2 p0q " u 2 p1q " 0, we refer to [7, Corollary 3.2 and Theorem 5.1]. For the concavity observe that by (3.2) By density we obtain that the same holds true for all φ P W 1,2 0 p0, 1q such that φ ě 0. Plugging in φ " maxtA u , 0u, which is admissible as by the previous regularity A u P W 1,8 p0, 1q and A u p0q " u 2 p0q " 0 and A u p1q " u 2 p1q " 0. We obtain that This implies that maxtA u , 0u " 0 a.e. and hence A u ď 0 a.e.. In particular we can conclude that u 2 ď 0 a.e. which implies the concavity of u.
Remark 6.2. Note that concavity of critical points implies in particular that those are nonnegative, i.e. u ě 0. 6.1. Symmetry of minimizers. Critical points of special importance are minimizers, which exist by Remark 3.2 whenever inf uPC ψ Epuq ă c 2 0 4 . Here we investigate symmetry of those. The main method used will use is a nonlinear version of Talenti's inequality for which we need some additional notation. For f P L 1 p0, 1q we denote by µ f ptq :" |tf ą tu| and by f˚pxq :" inftt ą 0 : µ f ptq ă xu. Moreover we define f˚pxq :" f˚p2|x´1 2 |q and call f˚the symmetric decreasing rearrangement of f . Note that for each decreasing function g : p0, 1q Ñ p0, 1q one has that rpxq :" gp2|x´1 2 |q satisfies r˚" r. Another important fact is that ||f || L p " ||f˚|| L p " ||f˚|| L p for each p P r1, 8s, cf. [14,Section 3.3].
The proof of the next result can be regarded as a special case of [23, Theorem 1] in one dimension. Since the assumptions in this article differ however slightly from our situation we give a self-contained proof, which however follows the lines of the proof in [23].
Observe that then for each h ą 0 Hence for almost every t ą 0 we have by the previous computation and by (6.6) Note that W is increasing. With the mean value theorem for integrals it can be shown that W is differentiable at t at all points of differentiability of µ u and at all such points (6.7) yields W 1 ptq ě 1. By [6,Proposition 4.7] t "ˆt 0 1 ds ďˆt 0 W 1 psq ds ď W ptq´W p0q.
Note that W p0q " 0 as µ u p0q " 1. This is so since u is by (6.3) concave and nonnegative and therefore tu " 0u " t0, 1u or u " 0 where we excluded the last case in the beginning of the proof. Hence tu " 0u is a Lebesgue null set and therefore |tu ą 0u| " 1. We obtain that t ď W ptq, i.e. Finally u˚pxq " u˚p2 |x´1{2|q ďˆ1 where we used (6.5) in the last step.
We will now use Lemma 6.3 to deduce that v ě puq˚ě ψ˚" ψ. To apply Lemma 6.3 we have to check that 1 2 ||f || L 2 ă ||G|| 8 and that 1 G´1 is convex on r0, Gp||u 1 || 8 qs. For the L 2 -bound we can look at For the convexity of 1 G´1 we can compute for arbitrary s P p0, ||G|| 8 q that 1
6.2. Uniqueness of symmetric critical points. We show now uniqueness of critical points for symmetric cone obstacles. This will follow from a more general uniqueness result for solutions to ODEs that we will prove in the appendix.
Lemma 6.5 (Uniqueness of strictly concave solutions, Proof in Appendix C). Let x 0 ą 0 and let J : r0, x 0 s Ñ R be nonnegative and decreasing such that J ą 0 on p0, x 0 q Further assume that J is locally Lipschitz continuous on p0, x 0 q and Jpx 0 q " 0. Then there exists at most one solution f P C 2 pr0, 1 2 sq to $ ' & ' % f 1 prq " Jpf prqq r P r0, 1 2 s, f p 1 2 q " x 0 , f p0q " 0, f is strictly concave on p0, 1 2 s. Here we call a C 2 -function strictly concave on a set A if f 2 ă 0 on A. The previous lemma is inspired by the following observation: A primary example for nonuniqueness of solutions to initial value problems is # 9 xptq " 2 a |xptq|, xp0q " 0.
It possesses infinitely many solutions but only one of them, namely t Þ Ñ t 2 , is strictly convex in p0, 8q, cf. Figure 3.
The following analysis of critical points has been obtained independently in [25].
Lemma 6.6 (Symmetric cone obstacles). Let ψ P Cpr0, 1sq be a symmetric cone obstacle, i.e. ψ is symmetric and affine linear on r0, 1 2 s. Then there exists at most one symmetric constrained critical point of E in C.
Proof. Let u be a symmetric critical point. We will in the following derive an explicit formula for u that characterizes it uniquely. We claim first that tu " ψu " t 1 2 u. In case that upaq " ψpaq for some a P p0, 1 2 q one gets u 1 paq " ψ 1 paq and by concavity (cf. Lemma 6.1) one has for all x P p0, 1 2 q upxq ď upaq`u 1 paqpx´aq " ψpaq`ψ 1 paqpx´aq " ψpxq, a contradiction to the nonnegativity of u. Hence u cannot touch ψ on p0, 1 2 q and similarly one shows that u cannot touch ψ on p 1 2 , 1q. Morover, we assert that u has to touch ψ, for if not then one obtains by Proposition 3.3 that u P C 8 pr0, 1sq and " const. on p0, 1q.
But since A u p0q " A u p1q " 0 one can find a point ξ P p0, 1q such that A 1 u pξq " 0. Therefore A 1 u p1`u 12 q 5 4 " 0 and hence A u " const. This yields u 2 " 0 but then the boundary conditions imply u " 0, a contradiction to Assumption 1. Hence tu " ψu " t 1 2 u. In particular by basic properties of the variational inequality there exists C P R such that As a further intermediate claim we assert that C ‰ 0. Indeed, if C " 0 then A 1 u " 0 which implies together with A u p0q " 0 that A u " 0. Then however u 2 " 0 on p0, 1 2 q which implies that u 1 " const on r0, 1 2 s. But u is symmetric and therefore u 1 p 1 2 q " 0 resulting in u 1 " 0. As a result u " 0 which yields again a contradiction to u P C ψ . Hence C ‰ 0.
If there were now x 0 P p0, 1 2 s such that u 2 px 0 q " 0, the above equation would imply that u 1 px 0 q " u 1 p0q and because of monotonicity of u 1 one has that u 1 " u 1 p0q on p0, x 0 q. Another look at (6.9) implies that then u 2 " 0 on p0, x 0 q. This implies that A 1 u " 0 on p0, x 0 q. This however is a contradiction to C " 0 when looking at (6.8). Since u is strictly concave on p0, 1 2 s we find that for all x ą 0 one has u 1 pxq ă u 1 p0q and now u 2 pxq "´a2|C| a u 1 p0q´u 1 pxqp1`u 1 pxq 2 q 5 4 @x P p0, 1 {2q.
Next we fix ą 0 and integrate from to some arbitary x P p0, 1 2 q to get after a substitution of s " u 1 pxq ds " a 2|C|px´ q We can pass to the limit as Ó 0 using the monotone convergence theorem on the left hand side to obtain As u is symmetric one has u 1 p 1 2 q " 0 which implieŝ Note that this means dz . (6.10) Note that this already yields an equation for u with only one free parameter, namely u 1 p0q. We show next that u 1 p0q is uniquely determined by ψp 1 2 q. To this end we compute (6.11) dz ": Hpu 1 p0qq, where (6.12) We show in Appendix B that H is a strictly monotone and smooth function of A.
Hence u solves on p0, 1 2 q (6.13) which is well defined because of the positivity of the integrand. The only problem that remains is that maximal solutions to (6.13) are not necessarily unique as J is not locally Lipschitz around ψp 1 2 q. It follows however by Lemma 6.5 that it does have a unique solution that is strictly concave in p0, 1 2 s. As we have shown strict concavity of each critical point, u is uniquely determined by x P p0, 1 2 q, up 1 2 q " ψp 1 2 q, up0q " 0, u strictly concave on p0, 1 2 s. Hence there can exist at most one such u as in (6.13).
Proof of Theorem 2.9. Follows now immediately from the previous Lemma and Corollary 6.4. 6.3. Compactness of the critical set. In the rest of this section we discuss compactness of the set of critical points. This will be of high importance later when examining the convergence. Here we do not impose any further assumption on ψ anymore, except for Assumption 1. Then M crit pAq is compact in W 2,2 p0, 1q.
By Lemma 6.1 we can integrate by parts without boundary terms and obtaiňˇˇˇˆ1 Together with this (6.17) and (6.16) we obtain Note that δ is chosen independently of w. This also implies that for all x P p0, 1q one has Aq we obtain with the explicit formula for A w that |w 2 | ď p1`G´1p ?
Finally note that Combining this with (6.18) and (6.19) we get ?

Convergence Behavior
In this section we want to examine whether the flow converges in the energy space W 2,2 p0, 1q. For large obstacles the absence of critical points already shows nonconvergence, cf. [19,Corollary 5.22]. For small obstacles and small initial energies however, convergence is true. Lemma 7.1 (W 2,2 -subconvergence disregarding small sets). Let u 0 P C ψ be such that Epu 0 q ă c 2 0 4 . Then for each ą 0 there exists a set B Ă r0, 8q with |B| ă such that u | r0,8qzB : r0, 8qzB Ñ W 2,2 p0, 1q is W 2,2 p0, 1q-subconvergent (in the sense of Definition 2.10) to points in M crit , where M crit is defined as in (2.5).
Proof. Let ą 0 and let φ be a nonincreasing function that coincides with E˝u almost everywhere. Let M be as in the statement. Define for each n P N the set Q n :" " t ą 0 : Epuptqq ‰ φptq or FVI is not true at t or || 9 uptq|| L 2 ą 1 n * .
Note that by Chebyshov's inequality, Q n has finite measure for all n and therefore there exists some k n P N such that |Q n X rk n , 8q| ă 2 n . Without loss of generality we can also achieve that k n`1 ą k n for all n P N. Therefore | Ť 8 n"1 pQ n Xrk n , 8qq| ă . Define B :" Ť 8 n"1 pQ n X rk n , 8qq. Suppose now that pθ n q 8 n"1 is an arbitrary sequence satisfying pθ n q Ă r0, 8qzB and θ n Ñ 8.
We conclude with this and (7.1) that u 8 P M crit .
So far we have proved a W 2,2 -subconvergence result for F V I gradient flows with an exceptional set B of artbitrary small measure. The next step is now to use the uniform Hölder continuity of F V I gradient flows in L 2 p0, 1q to get rid of the exceptional set. The topology however changes for the worse but can be improved again in the rest of the section. Proof. Let u 0 ,u be as in the statement. We start by showing full L 2 p0, 1q-subconvergence. For this let t n Ñ 8. Now set 1 :" 1 ą 0 and apply Lemma 7.1 with " 1 . This yields a set B " Bp 1 q such that |Bp 1 q| ă 1 and u | r0,8qzBp 1 q W 2,2 -subconverges to points in M crit . Note that for all n P N there exists s 1 n P pt n´ 1 , t n` 1 q such that s 1 n P r0, 8qzBp 1 q, since the contrapositive of this statement contradicts |Bp 1 q| ă 1 . By Lemma 7.1 there exists a subsequence pl 1 n q 8 n"1 Ă N such that ups 1 Hence there exists some n 1 P N such that for all n ě n 1 ||upt l 1 n q´u 1 8 || L 2 p0,1q ď 2D ? 1 . We start an iterartive procedure by repeating the process starting with the sequence pt l 1 n q něn1 and for 2 :" 1 2 , more precisely: We again choose a measurable set Bp 2 q of measure smaller than 2 such that u r0,8qzBp 2q W 2,2 -subconverges to points in M crit . We again observe that for all n ą n 1 there exists s 2 n P pt l 1 n´ 2 , t l 1 n` 2 q X r0, 8qzBp 2 q. Therefore we can find a subsequence of ps 2 n q něn1 along which u converges to some u 2 8 P M crit . As above this yields now a subsequence pt l 2 n q něn1 of pt l 1 n q něn1 such that lim sup nÑ8 ||upt l 2 n q´u 2 8 || L 2 p0,1q ă D In particular we can choose n 2 ě n 1 such that for all n ě n 2 ||upt l 2 n q´u 2 8 || L 2 p0,1q ă 2D ? 2 . Keeping going, we can find for all k P N nested subsequences pt l k n q nąn k Ă pt l k´1 n q něn k´1 Ă ... Ă pt n q ně1 and tu 1 8 , ..., u k 8 u Ă M crit such that for all q P t1, ..., ku and n ą n q one has (7.2) ||upt l q n q´u q 8 || L 2 p0,1q ă 2D Because of the compactness of M crit by Lemma 6.7 we obtain that pu q 8 q 8 q"1 has a W 2,2 -covergent subsequence, denoted by pu qm 8 q 8 m"1 . We denote the limit of this sequence simply by u 8 P M crit . The subsequence we consider now is pt l qm nq m q mPN . For the sake of simplicity of notation we define a m :" t l qm nq m . Now we observe by (7.2) Now both terms on the right hand side of this inequality tend to zero as m Ñ 8 and thus upa m q Ñ u 8 in L 2 p0, 1q. As pa m q 8 m"1 was a subsequence of pt n q 8 n"1 the claim follows.
Proof of Theorem 2.11. Let pt n q 8 n"1 be an arbitrary sequence that diverges to infinity. By Lemma 7.2 there exists a subsequence which we call again t n and some u 8 P M crit such that upt n q Ñ u 8 in L 2 p0, 1q. By Theorem 2.6, upt n q is bounded in W 2,2 p0, 1q hence we can choose a further subsequence which we do not relabel such that upt n q á u 8 weakly in W 2,2 p0, 1q. By compact embedding we obtain upt n q Ñ u 8 in C 1 pr0, 1sq and hence the claim follows.
Proof of Theorem 2.12. Let u 0 ,u be as in the statement. By Corollary 4.19 one has uptqp1´¨q " uptq for all t ą 0. Let now w P C ψ be the unique symmetric critical point in C ψ (cf. Theorem 2.9). By Theorem 2.9, w is a minimizer of E in C ψ . Now let t n Ñ 8 be a sequence. Observe that by Theorem 2.11 there exists a subsequence t ln Ñ 8 such that upt ln q converges in W 1,8 p0, 1q to some critical point u 8 P C ψ . Now since upt ln qp1´¨q " upt ln q for all n P N one obtains by the L 2 -convergence that u 8 p1´¨q " u 8 . From this follows that u 8 " w by Theorem 2.9. By the Urysohn property of L 2 -convergence we obtain that upt n q Ñ w in L 2 p0, 1q. As the sequence pt n q 8 n"1 was arbitrary we obtain that uptq Ñ w as t Ñ 8.

Open problems and perspectives
In this final section we summarize some problems that could be interesting for future research. We also discuss some ways to approach them.
Open Problem 8.1 (Optimal energy dissipation). The article shows that energydissipating F V I gradient flows can also be constructed even if the energy E is not L 2 -semiconvex. It is however unclear whether the energy dissipation rate is optimal. For L 2 -semiconvex functionals the dissipation rate will be optimal -in the sense of EDI-gradient flows in optimal transport theory, cf. [2,Definition 4.3]. Even more general -if E can be written as a sum of a convex and a Frechét differentiable functional then each F V I gradient flow is an EDI-gradient flow. This can be shown following the lines of [19,Section 2.3]. It is vital for the theory to understand what role the convexity assumption plays for the energy dissipation.
Open Problem 8.2 (Energy threshold and geometry). We have shown existence of the flow only below the energy threshold Epu 0 q ă c 2 0 4 . The reason for this threshold is that below one can obtain uniform control of ||B x upt,¨q|| L 8 and hence one has control of the nonlinearities. While this is helpful for our analysis, the control is lost for large obstacles, cf. [18], [19,Section 5]. The reason is that ||B x u|| L 8 is a quantity that disregards the nature of E as a geometric energy of curves, namely Epuq "ˆg raphpuq κ 2 ds.
More precisely: If ||B x u|| L 8 becomes large, graphpuq is not necessarily ill-behaved as a curve. If one wants to go beyond the threshold of c 2 0 4 one needs to work with curves and formulate a geometric minimizing movement scheme. While this causes additional difficulties, there has recently been progress, eg. in [5], for gradient flows of the (p-)elastic energy without an obstacle constraint.
Open Problem 8.3 (Symmetry breaking or not?). In the article we have seen that symmetric evolutions with Epu 0 q ă c 2 0 4 approach the unique symmetric critical point from Lemma 6.6. (Note that we have only shown uniqueness of this critical point, but its existence follows from symmetry-preserving and subconvergenceor alternatively from [25]).
We actually want to show convergence to a global minimizer. For this we have to show that a symmetric minimizer can be found. We have done so in Corollary 6.4 -but again only below an energy threshold of Gp2q 2 , which is even smaller than c 2 0 4 . The value of Gp2q 2 corresponds to a loss of convexity of 1 G´1 and hence poses a limitation to the nonlinear Talenti symmetrization. We expect that there exist symmetric minimizers also above this threshold, but a proof will require further techniques. Presumably one needs to find a more geometric approach to the symmetry problem, which will be subject to our future research. Therefore we can choose a minimizing sequence pu n q 8 n"1 Ă C ψ such that Φ f τ pu n q ď Epf q for all n P N. Hence Epu n q ď Φ f τ pu n q ď Epf q @n P N By Remark 3.2 we obtain that ||u 1 n || L 8 ď G´1p a Epf qq and pu n q 8 n"1 is uniformly bounded in W 2,2 p0, 1q. After choosing an appropriate subsequence (which we do not relabel), we can assume that u n á w in W 2,2 p0, 1q for some w P W 2,2 p0, 1q. Note that w P C ψ since C ψ is weakly closed as closed convex subset of W 2,2 p0, 1q. By Sobolev embedding u n Ñ w in C 1 pr0, 1sq and hence in particular in L 2 p0, 1q. Just like in the proof of [7, Lemma 2.5] we obtain now that (A.1) Epwq ď lim inf nÑ8 Epu n q.
Since w P C ψ is admissible we also have Φ f τ pwq ě I, which implies that w is a minimizer. Equation 4.1 now follows easily from the fact that for all v P C ψ one has 0 ď d dt |t"0 φ f τ pw`tpv´wqq which is due to the fact that w`tpv´wq is admissible for all t P r0, 1s.
Proof of Lemma 4.10. Fix t, τ ą 0. For the sake of simplicity of notation we define u :" u τ ptq. First we expand (4.5) to find that for each φ P W 2,2 p0, 1q X W 1,2 0 p0, 1q such that φ ě 0 Equation (A.4) implies that µ is supported on tu " ψu. Because of the assumptions on the obstacle µ is a Radon measure with support compactly contained in p0, 1q, hence also a finite measure. Now we want to bound µpp0, 1qq independently of τ .