An obstacle problem for the p-elastic energy

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler--Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.

Obstacle problems have been attracting attention from a lot of researchers.Moreover, the p-elastica functional has recently raised a lot of interest, see e.g.[1, 16, 19-22, 33, 34] for the study of critical points and see e.g.[2,[26][27][28]31] for the study of gradient flows.In this article we study the regularity of minimizers.In cases where the absence of obstacle implies smoothness of minimizers, i.e., any solution of the Euler-Lagrange equation for the corresponding energy functional is smooth, one can generally observe that the presence of obstacle induces a loss of regularity of minimizers.The collapse of the regularity occurs at the coincidence set between minimizers and obstacle.On the other hand, if the Euler-Lagrange equation for the corresponding energy has a degeneracy, it is significant to ask the following mathematical question: Problem 1. Which is the main cause of the loss of regularity of minimizers, the presence of obstacle or the degeneracy of the Euler-Lagrange equation?
The critical points of the elastic energy E 2 pγq " ˆγ κpsq 2 ds with the length constraint are called elastica and have been studied well in the mathematical literature.Here, γ is a planar curve, and κ and s denote the curvature and the arc length parameter of γ, respectively.Since the curvature of any critical point of E 2 with no constraint satisfies the Euler-Lagrange equation 2κ ss `κ3 " 0, it is well known that any critical point of E 2 is analytic, cf.[13].On the other hand, the Euler-Lagrange equation for the p-elastic energy E p pγq " ˆγ |κ| p ds is formally written as (1.4) pp|κ| p´2 κq ss `pp ´1q|κ| p κ " 0.
The degeneracy of the Euler-Lagrange equation (1.4) induces a loss of regularity of critical points of E p .One of the reasons for the loss of regularity is the emergence of flat core solutions, which are critical points of E p and contain flat parts, i.e., parts with κ " 0 (see [19,34]).In [34], it was proved that the flat core solutions do not belong to C 3 .Recently, a complete classification of solutions of (1.4) has been obtained, as well as the optimal regularity (see [19]).
One of the purposes of this paper is to give an answer to Problem 1 for the obstacle problem (1.2).Following this motivation, we consider the more general obstacle problem (1.5) inf Assumption 2 (Assumptions on G).We assume that G P C 8 pRq.For the derivatives of G we use the notation 9 G, : G, . ... We require (1) G is odd, i.e.Gp´sq " ´Gpsq, G ą 0, ps Þ Ñ s 9 Gpsqq P L 1 pRq.
In particular the case p " 2 and G " EU 2 resembles the classical elastic energy of graphpuq.Generalized functionals like in (1.6) have applications in image restoration, cf.[5].
To study the optimal regularity of minimizers of problem (1.5), we introduce a notion of critical points of problem (1.5).Inspired by the work [8, p. 248] of Euler on the elastic energy, we consider first the following substitution.Definition 1.1.Let u P W 2,p p0, 1q.Then the quantity w P L p p´1 p0, 1q given by (1.8) wpxq :" w u pxq :" ´p 9 Gpu 1 pxqq p´1 |u 2 pxq| p´2 u 2 pxq a.e.x P p0, 1q is called Euler's substitution.
Then critical points for problem (1.5) are defined as follows: Definition 1.2.We say that u P M pψq is a critical point of E with obstacle constraint if there exists a nonnegative Radon measure µ supported on tu " ψu such that Euler's substitution w " w u satisfies (1.9) ´ˆ1 0 wpxq d dx ´9 Gpu 1 pxqqϕ 1 pxq ¯dx " ˆ1 0 ϕpxq dµpxq for all ϕ P W 2,p p0, 1q X W 1,p 0 p0, 1q.Our first main result Theorem 1.3 is concerned with a qualitative property and the optimal regularity of critical points.
One of the novelties of this paper is the generalized Euler's substitution (1.8) for the Euler-Lagrange equation.For each critical point u, the substitution allows us to apply the maximum principle to the (second order) elliptic PDE satisfied by w u and, in turn, to prove several qualitative properties as well as the optimal regularity of w u .We prove Theorem 1.3 by translating the qualitative properties and the optimal regularity on w u into those of u.
We prove that any minimizer of problem (1.5) is a critical point in the sense of Definition 1.2.Thus any minimizer of problem (1.5) satisfies the property (1.10) which implies that any minimizer of problem (1.5) is concave and has no flat core part.Then we observe that the collapse of regularity of minimizers may occur only at the boundary and the coincidence set.Moreover, we infer from the optimal regularity (i) and (ii) in Theorem 1.3 that for p P p2, 8q the degeneracy of the Euler-Lagrange equation plays a more important role on the loss of regularity, whereas for p P p1, 2s it is the effect of obstacle constraint.
Remark 1.4.On the existence of minimizers of problem (1.5), we prove the existence for small obstacles and non-existence for large obstacles (see Theorem 4.3 and Proposition 5.1, respectively).Moreover, with the aid of a nonlinear Talenti inequality, we prove existence of a symmetric minimizer for suitably small symmetric obstacles ψ (see Proposition 4.6 and Corollary 4.8).
The second purpose of this paper is to give the complete classification of solvability of problem (1.2) in terms of the 'size' of obstacle ψ, where ψ denotes a specified obstacle, the so-called cone obstacle.Recently, obstacle problems for the elastic energy have been attracting attention and have been studied well in the mathematical literature, e.g., problem (1.2) with p " 2 ( [3,6,18,23,35]), the obstacle problem for E 2 with the clamped boundary condition ( [10]), and gradient flows corresponding to problem (1.2) with p " 2 ( [24,25,29]).Following our second purpose, we focus on the case of symmetric cone obstacle, given by functions ψ P C 0 pr0, 1sq such that ψ| r0,1{2s and ψ| r1{2,1s are affine linear, and ψpxq " ψp1 ´xq for x P r0, 1s.More precisely, we consider the obstacle problem (1.11) inf vPMsympψq E p pvq with (1.12) M sym pψq :" tv P M pψq : vpxq " vp1 ´xq for x P r0, 1su, where ψ is a symmetric cone satisfying Assumption 1.For problem (1.11) with p " 2 it was proved by [3,18,23,35] that there exists a threshold h ą 0 satisfying the following: if ψp 1 2 q ă h, then there exists a unique minimizer u P M sym pψq of E 2 in M sym pψq; if ψp 1 2 q ě h, then there is no minimizer of E 2 in M pψq.Theorem 1.5 gives an extension of the result to the case p ‰ 2: Theorem 1.5.Let p ą 1.There exists a constant h ˚ą 0 satisfying the following : Let ψ be a symmetric cone and satisfy Assumption 1.
(i) If ψp 1 2 q ă h ˚, then there is a unique minimizer u P M sym pψq of E p in M sym pψq.(ii) If ψp 1  2 q ě h ˚, then there is no minimizer of E p in M pψq.Here we note that the value h ˚can be explicitly given (see (6.41) below).To prove Theorem 1.5, we employ a strategy, given by [18], making use of the polar tangential angle.To this aim, using a substitution ω :" |κ| p´2 κ, first we derive an explicit formula of a solution of the Euler-Lagrange equation (1.4).The explicit formula gives us a special curve, the so called p-rectangular.By way of the prectangular we extend the polar tangential strategy into the case p ‰ 2. Both the substitution ω and the Euler's substitution w u reduce the second-order ODE in terms of κ to a second-order ODE in terms of u.However, the reduced equations are different, and each reduced equation independently provides properties of critical points (more precisely see Remark 6.3

below).
This paper is organized as follows: In Section 2, using the Euler's substitution, we motivate the definition of critical points of E in M pψq.In Section 3, we study the optimal regularity and qualitative properties of critical points which are introduced in Section 2.More precisely, first we prove the regularity of the Euler's substitution in Section 3.1.By way of the results in Section 3.1 we show the following for critical points of E in M pψq: Nondegeneracy (Section 3.2); properties of the coincidence set (Section 3.3); optimal regularity (Section 3.4).In Section 4 we prove the existence of minimizers of E in M pψq (Section 4.1) and symmetry of minimizers of E in M pψq under a smallness condition on symmetric obstacles (Section 4.2).The nonexistence of minimizers for E in M pψq for large cone obstacles ψ is proved in Section 5. Finally we prove Theorem 1.5 in Section 6.
Euler's substitution to our generalized p-elastica functional with a refined substitution.In the following we always consider p P p1, 8q as fixed if not stated otherwise.
To this end, let p P p1, 8q and notice that E : W 2,p p0, 1q Ñ R is Fréchet differentiable.Its Fréchet derivative reads (for each ϕ P W 2,p p0, 1q and using Assumption 2) One can now observe by the product rule that the expression in parentheses is nothing but d dx p 9 Gpu 1 qϕ 1 q and thus one obtains In particular, the equation DEpuq " 0 is a very weak form of an elliptic equation of second order for w, namely of Notice that this equation is elliptic since 9 G ą 0 by Assumption 2. It becomes obvious that the quantity w plays an important role in our analysis.

2.1.
A measure-valued Euler-Lagrange equation.The basis of our measurevalued approach is given by the study of variational inequalities, which is a theory of critical points for functionals defined on a convex subset of a Banach space, cf.[11] for details.For obstacle problems it is usual that a notion of critical points can be established by means of a variational inequality.Such inequality can be reformulated as a measure-valued Euler-Lagrange equation.The measure that appears is always supported on the coincidence set tu " ψu.Since this measurevalued equation is (second-order) elliptic in terms of w u , we can proceed our analysis using methods for measure-valued elliptic PDEs.We refer to [30] for a very detailed introduction to these types of PDEs.
We will next derive that the critical point equation in Defintion 1.2 is a necessary criterion for a minimizer.Proposition 2.1 (Euler-Lagrange equation).Let u P M pψq be a minimizer.Then there exists a finite positive Radon measure µ supported on tu " ψu such that with w " w u being Euler's substitution, see (1.8).We call µ coincidence measure.
Proof.Let u P M pψq be a minimizer.Since M pψq (see (1.3)) is a convex subset of W 2,p p0, 1q X W 1,p 0 p0, 1q, for each v P M pψq one has that u `tpv ´uq P M pψq for all t P r0, 1s.In particular, minimality of u implies Epu `tpv ´uqq " DEpuqpv ´uq.
In the sequel we will study (optimal) regularity properties of critical points.Doing so we will discover certain useful geometric properties that will help to characterize existence and nonexistence of minimizers.

Optimal Regularity of critical points
Now that a convenient critical point equation is established, elliptic regularity can be applied to examine its solutions u P M pψq.A problem is that this elliptic regularity discussion first only applies to Euler's substitution w u and not to u itself.In order to translate regularity results for w u into regularity results for u, a degeneracy phenomenon must be taken into account.Such phenomenon happens -if p ą 2 -exactly on the set tu 2 " 0u, as we shall see.The elliptic maximum principle will enable us to overcome this problem.Indeed, one can show that tu 2 " 0u " t0, 1u.In particular, the mentioned degeneracy phenomenon can only occur at the boundary Bp0, 1q.

3.1.
Regularity for Euler's substitution.In the sequel we denote by L 1 the one-dimensional Lebesgue measure.With this notation fixed we can use some results for measure-valued PDEs to obtain regularity of Euler's substitution w.The following proposition shows W 1,8 -regularity of w and, as an immediate consequenence of (1.8), also (non-optimal) C 2 -regularity of u.Proposition 3.1 (Regularity and boundary conditions for w).Let u P M pψq be a critical point in the sense of Definition 1.2 and w be Euler's substitution.
Proof.By (1.9) computing d dx p 9 Gpu 1 qϕ 1 q with the chain rule we infer for all ϕ P W 2,p p0, 1q X W 1,p 0 p0, 1q.Since u P W 2,p ãÑ C 1 pr0, 1sq, w P L p p´1 and G P C 8 , the function f :" : Gpu 1 qu 2 w belongs to L 1 p0, 1q.With the introduced notation we infer for all ϕ P W 2,p p0, 1q X W 1,p 0 p0, 1q.We can now use Fubini's theorem to rewrite the measure term in a convenient way.For ϕ P W 2,p p0, 1q X W 1,p 0 p0, 1q one has pf `g `Cqξ dx, whereupon one concludes that w 9 Gpu 1 q P W 1,1 p0, 1q and pw 9 Gpu 1 qq 1 " f `g `C.By the fact that in one dimension W 1,1 Ă C 0 we find that w 9 Gpu 1 q P C 0 pr0, 1sq.In particular, since 1   9   G˝u 1 is continuous (as 9 G ą 0) one has that w is continuous.From (1.8) and the fact that z Þ Ñ |z| p´2 z has a continuous inverse function it follows that also u 2 is continuous, i.e. u P C 2 pr0, 1sq.Recalling the definition of f we find that f P C 0 pr0, 1sq.With this knowledge we infer from pw 9 Gpu 1 qq 1 " f `g `C that w 9 Gpu 1 q P W 1,8 p0, 1q.Using that 1 9 G˝u 1 P W 1,8 (as 9 G ą 0 and u 2 P L 8 ) we infer from the Banach algebra property of W 1,8 that w P W 1,8 .This information given, we may again look at (3.4) for arbitrary ϕ P W 2,p p0, 1q X W 1,p 0 p0, 1q and integrate by parts to find ´"w 9 Gpu Using again pw 9 Gpu 1 qq 1 " f `g `C we find ´"w 9 Gpu 1 qϕ 1 ı 1 0 " ˆ1 0 Cϕ 1 dx " 0 @ϕ P W 2,p p0, 1q X W 1,p 0 p0, 1q.
By the fundamental lemma of calculus of variations we infer that there exists a constant C P R such that (3.5) apxqw 1 pxq ´µppx, 1qq " C a.e. in p0, 1q.
Next we define mpxq :" µppx, 1qq `C for x P p0, 1q.This is clearly bounded since µ is a finite measure and also nonincreasing.Moreover, since µptu ą ψuq " 0, it is also locally constant on tu ą ψu.Rearranging (3.5) implies that aw 1 " m almost everywhere.□ Remark 3.2.Equation (3.2) implies that w 1 " 1 a m P BV p0, 1q, since m is BV as monotone bounded function and 1 a " 1 9 G˝u 1 P C 1 pr0, 1sq.From these regularity properties of w one can already infer some basic properties of critical points.Proof.Since sgnpu 2 q " ´sgnpwq it suffices to show that w ě 0 on p0, 1q with w " w u being Euler's substitution.Let a, m be as in the previous proposition.Then apxqw 1 pxq " mpxq a.e. in p0, 1q.
Since a ą 0 one has sgnpmq " sgnpw 1 q a.e.. Since m is nonincreasing there must exist a (not necessarily unique) c P r0, 1s such that w 1 ě 0 a.e. on p0, cs and w 1 ď 0 a.e. on pc, 1q.In particular w is nondecreasing on r0, cs and nonincreasing on rc, 1s.This monotonicity behavior and wp0q " wp1q " 0 imply immediately that w ě 0. □ Remark 3.4.An important consequence of the concavity of u is that Euler's substitution w can be rewritten as Proof.By the previous corollary and the chain rule a P C 1 pr0, 1sq and a ą 0. Moreover Proposition 3.1 yields that w 1 " m a where m is locally constant on tu ą ψu.In particular, the chain rule then implies that w 1 P C 1 ptu ą ψuq and hence w P C 2 ptu ą ψuq.Equation (3.7) follows directly from (3.1) and the fact that sptpµq X tu ą ψu " H. □ We have now already inferred an optimal(!)regularity result for w, which is w P W 1,8 p0, 1q.(Optimality can for example be seen with [18,Remark 3.7] for p " 2).The rest of this section is devoted to the optimal regularity of u.Notice that the regularity of u and the regularity of w are two different questions.Indeed, u 2 and w are related via (3.6), which means that a p ´1'st root has to be taken in order to retrieve u 2 from w.This p ´1'st root imposes regularity issues on tu 2 " 0u, at least when p ą 2 (since then the p ´1'st root is not locally Lipschitz in R).This explains that in order to find the optimal regularity, we have to pay special attention to the set tu 2 " 0u (a flat core), where a degeneracy phenomenon might happen.
3.2.Nondegeneracy in the interior.An obstruction to the regularity of the problem is imposed by the phenomenon of flat cores.This means that the solution may contain line segments.
While flat cores will generally occur for critical points with fixed length, cf.[34], we will argue in this section that they will not occur for our critical points without fixed length assumption.Understanding (non)degeneracy of minimizers with fixed length will be subject to future research.
The basis of our argument is again the critical point equation satisfied by Euler's substitution.
Proof.Let u P M pψq be a critical point and w be its Euler's substitution.It is sufficient to prove that tw " 0u " t0, 1u.Let m : p0, 1q Ñ R be the function from Proposition 3.1.Arguing as in the proof of Corollary 3.3 we see that there must exist c P r0, 1s such that w 1 ě 0 a.e. on r0, cs and w 1 ď 0 a.e. on rc, 1s.In particular, w can be decomposed into two monotone parts.Since wp0q " wp1q " 0 we conclude from this that tw " 0u " r0, α 1 s Y rα 2 , 1s for some α 1 ě 0 and α 2 ď 1.This is due to the elementary fact that monotone functions may attain a certain value only at a point or an interval.It remains to show that α 1 " 0 and α 2 " 1.For a contradiction assume that α 1 ą 0. Then in particular w " 0 on p0, α 1 q and therefore one also has w 1 " 0 on p0, α 1 q.Notice that then mpxq " apxqw 1 pxq " 0 in p0, α 1 q.
Since m is nonincreasing on p0, 1q one infers from this that m ď 0 on p0, 1q.In particular one has w 1 pxq " mpxq apxq ď 0 a.e.x P p0, 1q, since a " 9 G ˝u1 ą 0. Hence w is nonincreasing.This and wp0q " wp1q " 0 however imply that w " 0 on r0, 1s.As a consequence one has u 2 " 0 on r0, 1s.A contradiction to Assumption 1 since this and up0q " up1q " 0 imply u " 0. We infer that α 1 " 0. Analogously one can show that α 2 " 1 (the only difference is that in the case of α 2 ă 1, w turns out to be nondecreasing instead of nonincreasing).□ This nondegeneracy will allow us to take the mentioned p ´1'st root of w in p0, 1q without losing any differentiablity.This can be used to prove many regularity statements and also further structural results, e.g.results about the coincidence set.
An immediate consequence is e.g. the following local regularity result away from the obstacle.Corollary 3.7 (Interior regularity away from the obstacle).Let u P M pψq be a critical point in the sense of Definition 1.2.Then u P C 8 pp0, 1q X tu ą ψuq.
Proof.By Corollary 3.5 we have (cf.(3.6)) that w " p 9 Gpu 1 q p´1 p´u 2 q p´1 P C 2 ptu ą ψuq.Using that by Proposition 3.6 ´u2 ą 0 on p0, 1q and taking the p ´1'st root we find 9 Gpu 1 qu 2 " ´`1 p w ˘1 p´1 P C 2 pp0, 1q X tu ą ψuq.We conclude d dx pG˝u 1 q P C 2 pp0, 1qXtu ą ψuq and therefore G˝u 1 P C 3 pp0, 1qXtu ą ψuq.Since G P C 8 and 9 G ą 0 we infer that G is a local C 3 -diffeomorphism and thus u 1 P C 3 pp0, 1q X tu ą ψuq.From this follows immediately that u P C 4 pp0, 1q X tu ą ψuq.This (and the nondegeneracy) at hand allow us to bootstrap further.For each x P tu ą ψu one has (cf.Proposition 3.1) that p 9 G˝u 1 qw 1 " const. in a neighborhood of x.We use this information to derive an equation for the fourth order derivatives of u that can be written as where α, β P R and F : R 3 Ñ R is a smooth function.This formula and the fact that 9 G ą 0 and ´u2 ą 0 (cf.Proposition 3.6) make a bootstrapping argument possible, whereupon we conclude that u P C 8 pp0, 1q X tu ą ψuq.□ 3.3.The coincidence set.In this section we discuss some properties of the coincidence set tu " ψu.Understanding the coincidence set is important for the regularity discussion, since it is a set where regularity can potentially be lost.In light of the interior nondegeneracy result of the previous section, coincidence with ψ is actually the only phenomenon that can obstruct interior regularity.
Proposition 3.8 (Nonempty coincidence set).Let u P M pψq be a critical point in the sense of Definition 1.2.Then tu " ψu ‰ H.
Another consequence of the nondegeneracy and the C 2 -regularity is that obstacles can only be touched at concave points, in the following sense.Proposition 3.10 (Noncoincidence at convex points).Let u P M pψq be a critical point in the sense of Definition 1.2.Let x 0 P p0, 1q be such that ψ is twice continuously differentiable in a neighborhood of x 0 and ψ 2 px 0 q ě 0. Then x 0 R tu " ψu.

3.4.
Optimal regularity for critical points.In this section we study the optimal (global) regularity of critical points.Recall that we have two major obstructions to the regularity.The first one is the obstacle constraint -hitting the obstacle will affect the regularity.The second one is the degeneracy of the equation when u 2 " 0. While the latter phenomenon can (by the previous section) only occur at the boundary Bp0, 1q, the first phenomenon can only occur in the interior (as ψp0q, ψp1q ă 0).In this section we investigate which of the two phenomena imposes a harsher restriction on the regularity.The answer to this question will depend on p. Indeed, for p P p2, 8q the degeneracy plays a more important role, whereas for p P p1, 2s it is the obstacle constraint.We remark that in both cases we will experience a substantial loss of regularity compared to the C 8 -regularity that holds where none of the two obstructions are present, cf.Corollary 3.7, that shows regularity away from the obstacle and away from the boundary (where the degeneracy occurs).Proposition 3.12 (Case 1: p ď 2).Let u P M pψq be a critical point in the sense of Definition 1.2 and p ď 2. Then u P W 3,8 p0, 1q and u 3 P BV p0, 1q.This regularity is optimal in the sense that there exist obstacles ψ ˚satisfying Assumption 1 such that critical points (and even minimizers) in M pψ ˚q can not lie in C 3 p0, 1q.
Step 2. We show the BV -regularity of u 3 .Proposition 3.1 and the chain rule imply for a.e.x P p0, 1q with c p a constant depending only on p. Rearranging and using that w ą 0 we find for a.e.x P p0, 1q (3.9) ´: We conclude from the previous formula and the chain rule in BV that u 3 P BV p0, 1q if and only if m ¨w 2´p p´1 P BV p0, 1q.To prove the latter, we show that there exists M ą 0 such that ˇˇˇˆ1 For p " 2 this is trivial by the BV -property of m (as decreasing bounded function).For p P p1, 2q fix ϕ P C 8 0 p0, 1q.Since w ą 0 on p0, 1q and w P W 1,8 we find ˇˇˇˆ1 Since mpxq " µppx, 1qq `C (see (3.5)) with a finite measure µ, proceeding as in (3.3) in the first integral, we obtain where we used again w 1 " m 9 Gpu 1 q from Proposition 3.1.By boundedness of m, µp0, 1q ă 8 and w P L 8 p0, 1q one infers for some finite M ą 0. It remains to show that the integral on the right hand side is finite.If p ď 3 2 this is trivial.For other values of p we let δ ą 0 be chosen such that tu " ψu X r0, δq Y p1 ´δ, 1s " H. Since by Proposition 3.6 w ą 0 on p0, 1q one has that |w| 3´2p p´1 P L 8 ppδ, 1 ´δqq.Therefore it only remains to show integrability on p0, δq and on p1 ´δ, 1q.Next we show the integrability on p0, δq.By the local constancy of m on tu ą ψu (cf.Proposition 3.1) there exists a constant B 0 P R such that m " B 0 on p0, δq.We claim that B 0 ‰ 0. Indeed, B 0 " 0 would imply w 1 " 0 on p0, δq, which in turn would yield wpxq " wp0q " 0 for all x P p0, δq, contradicting the nondegeneracy statement in Proposition 3.6.We infer that | 9 | is uniformly bounded above and below one finds c, C ą 0 such that c ď |w 1 | ď C on p0, δq.This, wp0q " 0 and w| p0,1q ą 0 implies that (3.11) cx ď wpxq ď Cx @x P p0, δq.
p´1 is integrable on p0, δq (as 3´2p p´1 " ´2 `1 p´1 ą ´1) we obtain that |w| 3´2p p´1 is also integrable on p0, δq.Integrability on p1 ´δ, 1q follows the same lines.We conclude that the integral on the right hand side of (3.10) is finite.The claimed BV -regularity follows.
Step 3. We construct an obstacle ψ ˚as in the statement.We actually show that for cone obstacles (cf.Remark 3.11) each critical point (and therefore each minimizer) does not lie in C 3 p0, 1q.To this end let ψ ˚be an arbitrary cone obstacle satisfying Assumption 1, say ψ ˚|r0,θs and ψ ˚|rθ,1s are affine linear for some θ P p0, 1q.By Remark 3.11, each critical point u ˚P M pψ ˚q must have the tip of the cone as point of coincidence, i.e. tu ˚" ψ ˚u " tθu.We fix now any critical point u ånd for a contradiction assume now that u ˚P C 3 p0, 1q.Let w ˚:" w u˚b e Euler's substitution, see Definition 1.1.By the nondegeneracy in p0, 1q, cf.Proposition 3.6, and Proposition 3.1, it follows that w ˚P C 1 p0, 1q.Moreover, by Proposition 3.1 there exists m ˚: p0, 1q Ñ R nonincreasing and locally constant on tu ˚ą ψ ˚u and a ˚P C 1 p0, 1q, a ˚ą 0 such that a ˚pxqw 1 ˚pxq " m ˚pxq for a.e.x P p0, 1q.Since tu ˚ą ψ ˚u " p0, 1qztθu, the only point of nonconstancy of m ˚can be at θ and thus one has for some real constants c 1 ě c 2 .However, since a ˚, w 1 ˚P C 0 p0, 1q one infers that c 1 " c 2 and therefore there exists some c " c 1 " c 2 P R such that a ˚pxqw 1 ˚pxq " c @x P p0, 1q.In particular one has d dx pa ˚pxqw 1 ˚pxqq " 0 for all x P p0, 1q.Once this is established one can follow the lines of the proof of Proposition 3.8 (after (3.8)) to infer the contradiction u " 0. □ Remark 3.13.The previous proof reveals that all critical points whose coincidence set is a singleton may not lie in C 3 p0, 1q.Actually, the same argument would reveal that whenever the coincidence set of a critical point is discrete then it cannot lie in C 3 p0, 1q.If, on the contrary, the coincidence set has a nondiscrete part, more regularity might be possible.Indeed, by (3.1) it becomes obvious that the regularity depends strongly on the coincidence measure µ.If, say, µ is absolutely continuous with respect to the Lebesgue measure and has a smooth density, then standard elliptic regularity theory implies higher regularity.Notice however that a necessary condition for such absolute continuity is L 1 psptpµqq ą 0, which implies L 1 ptu " ψuq ě L 1 psptpµqq ą 0, i.e. coincidence on a very large set.
Remark 3.14.Finally, we clarify that in the case of p ď 2, the major reason for the loss of regularity is the obstacle.Indeed, for these values of p the degeneracy phenomenon at the boundary does not affect the regularity since the p ´1'st root is locally Lipschitz on r0, 8q when p ď 2. This is why the Sobolev regularity of u 2 and w must coincide, cf.Proposition 3.12.That the major regularity problem is imposed by the obstacle becomes now obvious when recalling from Proposition 3.1 that apxqw 1 pxq " mpxq for a.e.x P p0, 1q.Since m is locally constant on tu ą ψu (and therefore smooth on tu ą ψu) and a P C 1 pr0, 1sq, w 1 (and hence also u 3 ) may only lose its C 0 -regularity on tu " ψu, i.e. because of coincidence with the obstacle.
We have seen in Proposition 3.12 that this will actually happen.
We have now seen that touching the obstacle can potentially shrink regularity to W 3,8 .In the case of p ą 2 we will actually lose even more regularity due to the degeneracy at the boundary.Proposition 3.15 (Case 2: p ą 2).Let u P M pψq be a critical point in the sense of Definition 1.2 and p ą 2. Then u P W 3,q p0, 1q for all q P r1, p´1 p´2 q but u R W 3, p´1 p´2 p0, 1q .
Step 1.We show that u R W 3, p´1 p´2 p0, 1q.Reasoning as in the proof of Proposition 3.12 (below (3.10)) there exists a constant B 0 P R, B 0 ‰ 0 and δ ą 0 s.t.m " B 0 on p0, δq.In particular, from (3.9) for a.e.x P p0, δq we have ´: Gpu 1 pxqqu 2 pxq 2 with B 0 ‰ 0. We infer that on p0, δq, |u 3 | has the same integrability as w 2´p p´1 .Indeed, the prefactor 1 9 G˝u 1 can safely be disregarded since it is bounded from above and below.The summand p : G ˝u1 qu 22 lies in Cpr0, 1sq Ă L 8 p0, 1q, cf.Proposition 3.1, and hence can also be disregarded for the integrability.To finish the proof we investigate the integrability of w 2´p p´1 on p0, δq.Similar to the discussion in (3.11) in Proposition 3.12 we can derive that there exists c, C ą 0 such that cx ď wpxq ď Cx for all x P p0, δq.This in particular yields that w 2´p p´1 has the same integrability as x 2´p p´1 .This on contrary implies by the discussion above that |u 3 | has the same integrability as x 2´p p´1 on p0, δq.In particular, Step 2. We show that u 3 P L q p0, 1q for all q P r1, p´1 p´2 q.Note that (3.9) and 9 G ˝u1 ě a 0 on p0, 1q imply that for a.e.x P p0, 1q one has This and w ą 0 implies again that for δ ą 0 as in Step 1 one has w P L 8 pδ, 1 ´δq.Hence it only remains to show L q -integrability on p0, δq Y p1 ´δ, 1q.Concerning p0, δq, let c, C ą 0 be chosen as in Step 1. Then one has for x P p0, δq The last summand is finite by Proposition 3.1 and the first summand lies clearly in L q p0, δq for all q P r1, p´1 p´2 q.Along the same lines, one can show the same integrability on p1 ´δ, 1q.From the above discussion we conclude that □ u 3 P L 8 pδ, 1 ´δq X L q pp0, δq Y p1 ´δ, 1qq Ă L q p0, 1q.
Finally we combine the results in this section to obtain Theorem 1.3.
Proof of Theorem 1.3.The theorem follows combining the claims of Proposition 3.1 and of Propositions 3.6, 3.12 and 3.15.□ Remark 3.16.We remark that in the case of p ą 2 the BV -property of u 3 can not be true.This is due to the fact that in one dimension BV p0, 1q Ă L 8 p0, 1q.However a close examination of the proof of Proposition 3.12 (Step 2, in particular (3.10)) still yields that u 3 P BV loc p0, 1q.This is due to the fact that |w| 3´2p p´1 is integrable on pδ, 1 ´δq by nondegeneracy.

Existence and symmetry of minimizers
4.1.Existence of minimizers.In this section we will show an existence result for small obstacles for any p P p1, 8q.The smallness condition on the obstacle is expressed in terms of an upper bound on inf uPM pψq Epuq.In Section 5 we prove a non-existence result for large obstacles, i.e. smallness requirements on ψ are really necessary.The bounds that we formulate in this section are likely not optimal.However in Section 6 we will obtain a sharp existence result when minimizing among symmetric functions only.Notice that it is delicate to show symmetry of minimizers for a higher order problem due to the lack of a maximum principle.Nevertheless we will present a symmetry result for small obstacles in Section 4.2.
In order to quantify a sufficient upper bound on the infimum we will define the constant We also define some carefully chosen comparison functions, inspired by [3].For c P p0, c p pGqq we set (4.2) Clearly, one has u c P C 8 pr0, 1sq and This implies that (4.4) We also define (4.5) U G 0 pxq :" lim cÑcppGq u c pxq.
Remark 4.1.The map c Þ Ñ u c pxq is actually increasing for each x P r0, 1s, which is why the limit in (4.5) is a pointwise supremum.Indeed, one can calculate with (4.2) and (4.3) for fixed x P r0, Using the concavity of u c (cf. (4.3)) and x P r0, 1  2 s one infers that B c u c pxq is nonnegative as a sum of nonnegative terms.For x P r 1  2 , 1s one obtains the same result by symmetry.
It is easy to see (using that ps Þ Ñ s ¨9 Gpsqq P L 1 by Assumption 2) that U G 0 P C 8 p0, 1q X Cpr0, 1sq, U G 0 p0q " U G 0 p1q " 0 and pU G 0 q 1 pxq " G ´1 ˆcp pGq 2 ´cp pGqx ˙@x P p0, 1q.This is why we sometimes write by abuse of notation U G 0 " u cppGq .Next we give a universal upper bound on inf uPM pψq Epuq independent of the obstacle.We expect this bound to be sharp for large obstacles since it is sharp for the minimization among symmetric functions, as we shall see in Section 6.
In particular we conclude that Gpu 1 c,δ q 1 " ´c 1´2δ χ pδ,1´δq .Hence a straightforward computation implies Epu c,δ q " p1 ´2δq ˆc 1 ´2δ ˙p ď c p pGq p p1 ´2δq p´1 .Since δ ą 0 can be chosen arbitrarily small we obtain the claim.□ Finally we can formulate an existence theorem for small obstacles under a more restrictive energy bound.Then there exists u ˚P M pψq such that Epu ˚q " inf uPM pψq Epuq.Moreover u ˚is a critical point in the sense of Definition 1.2 (and hence satisfies all the regularity properties discussed in Section 3).
Proof.Let pu n q nPN Ă M pψq be a minimizing sequence.We first claim that pu n q nPN is bounded in W 1,8 p0, 1q.Indeed, since u n p0q " u n p1q " 0 there must exist ξ n P p0, 1q such that u 1 n pξ n q " 0. Further let x n P r0, 1s be such that |u 1 n px n q| " ||u 1 n || L 8 .Now there exists δ ą 0 such that for n large enough one has by Jensen's inequality From this it directly follows that there exists C ą 0 such that ||u 1 n || L 8 ď C. Therefore we obtain n || L 8 by the zero boundary conditions we infer that pu n q nPN is bounded in W 2,p p0, 1q, implying that (up to a subsequence) u n á u ˚in W 2,p p0, 1q.One readily checks (since then u n Ñ u ˚in Cpr0, 1sq) that u ˚P M pψq.It only remains to show that E is lower semicontinuous in u ˚.To see this we use that u q such that ψpxq ď u c pxq for all x P p0, 1q then (4.6) holds true.This becomes obvious when using u c as a test function and looking at (4.4).
Remark 4.5.We will show in the coming sections that without the energy bound (4.6) nonexistence may occur -even though the energy infimum is always finite (cf.Proposition 4.2).Numerics for p " 2 and G " EU 2 (cf.[3]) suggest that minimizers develop vertical slope at x " 0, 1 once the obstacles reach a certain height.Vertical slopes are however not allowed in our class M pψq.This makes it natural to look at a relaxation of the problem.In the case of p P p1, 8q and G " EU p one would want to find a relaxation that maintains the geometric meaning of the functional.This suggests looking at the p-elastic energy of curves in the larger class of pseudographs, i.e. curves that can consist of graph parts and vertical parts, cf.[23].In [23] it is observed that for p " 2 the elastic energy possesses a pseudograph minimizer, regardless of the height of the obstacle.We are confident that the analysis carries over to the situation of arbitrary p.
In the coming section we will look at symmetry of minimizers.One should also point out another geometric property which we already derived now -the strict concavity, i.e. u 2 ă 0, which we have already shown for every critical point in Proposition 3.6.4.2.Symmetry.In this section we use a nonlinear Talenti inequality to prove symmetry of minimizers u P M pψq for suitably small symmetric obstacles ψ.While this symmetry property is also expected to hold for general symmetric obstacles, it is somewhat delicate to prove, since the Euler-Lagrange equation is fourth order and hence lacks a maximum principle.Notice also that curvature functionals generally exihibit nonsymmetric critical points despite symmetric boundary data, cf.[17].The smallness assumption is needed since Talenti's inequality (see [32]) isoriginally -a tool to examine linear equations.Thus a control of the nonlinearity is needed.Proposition 4.6 (Symmetry under smallness condition).Suppose that ψ is symmetric and ψ| r0, 1  2 s is increasing.Suppose further that there exists some minimizer u P M pψq that satisfies ||u 1 || L 8 ď C 0 , where C 0 ą 0 is such that 2 9 Gpxq `x : Gpxq ą 0 for all x P r0, C 0 s.Then one can find a minimizer v P M pψq which is additionally symmetric.
Proof.Let u P M pψq be as in the statement.Define f :" Gpu 1 q 1 .Then f P L p p0, 1q.Let f ˚P L p p0, 1q be the symmetric decreasing rearrangement of f , cf.
[25, Section 7.1].Note in particular that ||f ˚|| L p p0,1q " ||f || L p p0,1q .Intermediate claim.There exists a solution v P W 2,p p0, 1q of # Gpv 1 q 1 " f ˚in p0, 1q, vp0q " vp1q " 0, which additionally satisfies where u ˚is the symmetric decreasing rearrangement of u.Notice that such v lies in M pψq since v ě u ˚ě ψ ˚" ψ.Here we have used that the symmetric decreasing rearrangement is order-preserving (cf.[15, Section 3.3]) and ψ ˚" ψ since ψ is symmetric and radially decreasing (due to the fact that ψ| r0, 1 2 s is increasing).Now Step 2. We show convexity of 1 G ´1 on r0, Gp||u 1 || L 8 qs.Notice that due to the assumption this is a subset of r0, GpC 0 qs.We compute This expression is positive on r0, GpC 0 qs iff p2 9 G `x ¨: GqpG ´1psqq ą 0 for all s P r0, GpC 0 qs.Monotonicity of G, G ´1 implies the claim.□ Remark 4.7.We remark that for G " EU p one has 2 9 Gpxq `x : Gpxq ą 0 for all Corollary 4.8.Suppose that ψ is symmetric and ψ| r0, 1 2 s is increasing.Moreover, suppose that ψ ď u c for some c P p0, 1  2 c p pGqq which is such that 2 9 Gpxq `x : Gpxq ą 0 on r0, p a G ´1pc p qs. Then there exists a symmetric minimizer in M pψq.
Proof.It follows from Remark 4.4 that there exists some minimizer u P M pψq.
In order to show that there exists also a symmetric minimizer, we apply Proposition 4.6.In order to do so it remains to ensure that ||u 1 || L 8 ď C 0 :" p a G ´1pc p q, where c is as in the statement.Let E ˚:" Epuq.The fact that u is minimizing and ψ ď u c yields that E ˚ď c p .Now Jensen's inequality yields (4.7) c p ě E ˚" Epuq ě rGpu 1 p1qq ´Gpu  This symmetry result shows that under suitable conditions on G and ψ minimization in M pψq and minimization in the smaller class M sym pψq yield the same result.In Section 6 we will study minimization in M sym pψq in detail and show further properties, such as uniqueness and explicit parametrization of minimizers, if ψ is a cone obstacle.

Nonexistence for large cone obstacles
In this section we will use the properties of minimizers that we studied to obtain a nonexistence result for suitably large symmetric cone obstacles (see Remark 3.11).For this nonexistence result we do not have to require symmetry of minimizers, only the obstacle ψ is required to satisfy ψ " ψp1 ´¨q.Hence the tip of the cone is at x " 1  2 and θ in Remark 3.11 is equal to 1 2 .We will derive a sharp bound on the L 8 norm of minimizers for general G, provided that the supremum in the following lemma is finite.In particular, if ||ψ `|| L 8 is larger than the right hand side in (5.1), then there does not exist a minimizer u P M pψq.
Proof.Let x 0 P p0, 1q be a point where u 1 px 0 q " 0. Such x 0 exists due to the boundary conditions up0q " up1q " 0. Without loss of generality we may assume x 0 P p0, 1 2 s, otherwise we may consider up1 ´¨q, which is also a minimizer.By concavity (cf.Corollary 3.3) one concludes that u 1 p 1 2 q ď 0. Recall from Propositions 3.12 and 3.15 that u P W 3,q p0, 1q for some q ą 1 and that since u ą ψ on r0, 1szt 1 2 u 9 Gpu 1 pxqqw 1 pxq " const.on p0, 1 2 q, where w is Euler's substitution.This in particular yields some B P R such that w 1 pxq " B 9 Gpu 1 pxqq @x P p0, 1 2 q.
Note that u 1 pxq ă u 1 p0q for all x ą 0 since by Proposition 3.6 one has u 2 pxq ă 0 for all x P p0, 1q.This fact also implies that B ą 0. Rewriting now in terms of u we find pp ´1qp´u 2 pxqq p 9 Gpu 1 pxqq p " Bpu 1 p0q ´u1 pxqq.
Taking the p-th root we infer ´u2 pxq 9 Gpu 1 pxqq pu 1 p0q ´u1 pxqq and hence (5.2) Let now F : p´8, u 1 p0qs Ñ R be given by Then F is invertible as F 1 ă 0 on p´8, u 1 p0qq.Further, F pu 1 p0qq " 0 since Moreover, u 1 pxq " F ´1pxq for all x P r0, The bound in (5.1) is only nontrivial provided that the supremum in (5.1) is finite.This is the content in the following lemma whose proof we postpone to the appendix.HpAq " 0, lim
Remark 5.3.We remark that for each p P p1, 8q the p-elastica functional G " EU p satisfies (5.3) for any ϵ P p0, 1 ´1 p q. Hence this nonexistence result applies to the p-elastica functional.
Remark 5.4.Under additional assumptions, H can be shown to be (eventually) increasing, e.g. in the case of p " 2 and G " EU 2 , cf. [25,Lemma 7.6].If this monotonicity can be shown, then the previous lemma yields actually an explicit bound for nonexistence, namely 1 2 Hp8q " 1 cppGq ||x ¨9 G|| L 1 p0,8q , cf. (5.1) and Lemma 5.2.In the case of G " EU p , the same sharp bounds can also be obtained using the geometry of p-elastic curves, which we pursue in Section 6.

Uniqueness of symmetric minimizers
In this section we consider problem (1.2), i.e., the minimization problem inf vPMsympψq E p pvq, where the functional E p and the admissible set M sym pψq are respectively defined by (1.1) and (1.12), and ψ denotes a cone obstacle.Here we recall that the p-elastic energy E p is given by the functional E with G " EU p , which is defined by (1.7).
6.1.Existence of minimizers.Following the strategy in Section 4.1 we first prove the existence of minimizers of problem (1.2).Taking G " EU p in (4.1), we define the constant c p by (6.1) c p :" c p pEU p q " 2 ˆ8 0 1 p1 `t2 q Then there exists a minimizer u P M sym pψq of E p in M sym pψq.
Proof.Let tu k u kPN Ă M sym pψq be a minimizing sequence for E p .By assumption (6.2) we may assume that (6.3) E p pu k q ď pc p ´2δq p for all k P N for some δ ą 0. Since EU p is an odd function, we have EU p pu 1 k pxqq " EU p p´u 1 k p1 ´xqq " ´EU p pu 1 k p1 ´xqq for all x P r0, 1s and k P N, and then for all k P N.This together with (6.3) yields a uniform estimate ´δq ă 8 for all k P N.Then, along the same lines as in the proof of Theorem 4.3, we complete the proof of Theorem 6.1.□ Let u P M sym pψq be a minimizer of E p in M sym pψq, which is obtained by Theorem 6.1.Similar to (2.4), we see that DE p puqpv ´uq ě 0 for any v P M sym pψq.(6.4)While the admissible set is restrictive, employing the same argument as in [35,Lemma 2.1], we improve (6.4) to DEpuqpv ´uq ě 0 for any v P M pψq.(6.5) (For a proof of (6.5), see Appendix A.2.) Recalling that E p is given by E with G " EU p , we observe from (6.5) and Theorem 4.3 that u is a critical point of E p in M pψq in the sense of Definition 1.2.Then, Corollary 3.3 implies the concavity of u, and Propositions 3.12 and 3.15 give us the optimal regularity of u.Moreover, we notice that the (measure-valued) Euler-Lagrange equation (1.9) becomes As we mentioned in Section 1, if γ is a critical point of E p and parameterized by arc length, then the curvature κ satisfies (1.4) in the sense of distributions.Thus it is quite natural that we have: Lemma 6.2 (Proof in Appendix A).Let u P W 3,q p0, 1q for some q ą 1 and κu be the curvature of graphpuq parameterized by arc length, i.e., κu psq :" κ u pxpsqq, where xpsq is the inverse of spxq " ´x 0 a 1 `u1 pξq 2 dξ.If κ u satisfies on E :" px 1 , x 2 q Ă p0, 1q (6.6) for all ϕ P C 8 0 px 1 , x 2 q, then κu satisfies (6.7) for all φ P C 8 0 pspx 1 q, spx 2 qq.6.2.Explicit formulae of free p-elastica.Let u be a minimizer of E p in M sym pψq.Since Assumption 1 implies u ą ψ near x " 0, we infer from Lemma 6.2 that u near x " 0 can be characterized as a C 2 -planar curve whose curvature κ P Cpr0, Lsq vanishes at x " 0 and satisfies p `|κ| p´2 κ ˘2 `pp ´1q|κ| p κ " 0 in the sense of distributions.
Remark 6.3.Euler's substitution w u (see Definition 1.1) and ω introduced in (6.9) are different from each other and play different roles.In fact, with the choice of G " EU p in (1.8), one notices that w u pxq " ´p|κ u pxq| p´2 κ u pxq `1 `u1 pxq 2 ˘p´1 2p .
Notice that in this sense ω and w u have in common that they involve the term |u 2 | p´2 u 2 as the term of highest order.Their roles are as follows: Euler's substitution allows w u to satisfy a second order differential equation without zeroth-order term; the transformation ω changes the quasilinear Euler-Lagrange equation into a semilinear equation.
The proof of Lemma 6.4 is similar to that of [3, Proof of Proposition 3.2], and given in the Appendix.
First the function sin q,r x is defined on r0, π q,r {2s, and then it is extended symmetrically to the interval r0, π q,r s, more precisely sin q,r x :" sin q,r pπ q,r ´xq for x P rπ q,r {2, π q,r s.Further it can be extended as a C 1 -function on R first extending it to r´π q,r , π q,r s as an odd function and then to all of R as a 2π q,r -periodic function.
Notice that sin q,r x is strictly increasing on r0, π q,r {2s.The function cos q,r : R Ñ R is defined by (6.13) cos q,r x :" d dx sin q,r x, x P R.
6.3.Proof of Theorem 1.5.In order to prove the uniqueness of minimizers of problem (1.2), in what follows we suppose that ψ is a symmetric cone obstacle (see Section 5).Then thanks to the properties of Γ λ such as the scale invariance and the monotonicity of the curvature, the strategy developed in [18] can work.First let us introduce the notion of the polar tangential angle (e.g.see [18, Definition 2.1]).Definition 6.11.Let an arc length parameterized curve γ P C 2 pr0, Ls; R 2 q satisfy γp0q " p0, 0q and γpsq ‰ p0, 0q for s P p0, Ls.For γ, the polar tangential angle function ϖ : p0, Ls Ñ R is defined as a continuous function such that R ϖ γpsq |γpsq| " γ s psq for all s P p0, Ls, where R θ stands for the counterclockwise rotation matrix through angle θ P R.
From now on, ϖ λ : p0, 2L λ s Ñ R denotes the polar tangential angle function for λ psq for all s P p0, 2L λ s. (6.28) Remark 6.12.Since k λ P Cpr0, 2L λ sq, the curve Γ λ belongs to C 2 pr0, 2L λ s; R 2 q.In [18], the polar tangential angle function is defined for curves of class C 8 to ensure the differentiability of ϖ.However, we notice that for each C 2 -curve one can still obtain ϖ P C 1 .Thus the methods developed in [18] are applicable to C 2 -curves, including Γ λ .Lemma 6.13 (Monotonicity and scale invariance of ϖ λ ).For each λ ą 0, the polar tangential angle ϖ λ is strictly monotonically decreasing in p0, L λ q and satisfies the following scaling property : (6.29) tan `ϖλ psq ˘" tan `ϖ1 pλ 1 p sq ˘for all s P p0, L λ s.
Moreover, it follows that for each λ ą 0 (6.30) lim sÓ0 tan ϖ λ psq " 0, tan ϖ λ pL λ q " ´Y1 pL 1 q X 1 pL 1 q , < l a t e x i t s h a 1 _ b a s e 6 4 = " X Q G + 7 d q w i K h i 5 < l a t e x i t s h a 1 _ b a s e 6 4 = " s y  < l a t e x i t s h a 1 _ b a s e 6 4 = " X Q G + 7 d q w i K h i 5 x < l a t e x i t s h a 1 _ b a s e 6 4 = " R C I q G S i e e f c b k n 6 Z y v u R G b z B j e 8 = " > A w 3 / o A L P 0 F c K g j i w t s 0 I C r q H W b m z J l 7 7 p y Z 0 W x D u h 7 R Q 0 j p 6 O z q 7 u n t C / c P D A 5 F o s M j 2 6 5 V c 3 S R 1 S 3 D c v K a 6 g p D m i L r S c 8 Q e d s R a l U z R E 6 r r L X 2 c 3 X h u N I y t 7 y G L X a q a t m U e 1 J X P a Z K 0 d H Z Y l 1 1 b F k q G i z a V a d j 7 k w p G q c k + R h m i b D t C q a u G K K n 7 6 5 3 9 U l M 4 r m 6 Z m 9 6 B L X b q S s 3 U q 7 q m e E z l y x W 5 E k 9 S i v x I / A R y A J I I I m P F r 7 C N X a n H 5 8 M W X s H T u 5 j 5 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a t N B 2 + r D J C j s G k p N 3 g S 4 r a K i w l E = " > A A A C h 3 i c h V E 9 S y t B F D 3 u 8 y P G r / h s B J t g U L Q w 3 h X x B S u f N p Z + v K h g N M y u E x 3 c 7 C 6 7 m 4 B v S S / + g V d Y K Y i I Y K u 9 j X / A w p 8 g r 1 S w s f B m s y A q 6 h 1 m 5 s y Z e + 6 c m T F c S / k B 0 V 2 T 9 q O 5 p b U t 0 Z 7 s 6 O z q 7 k n 1 / l z x n Y p n y r z p W I 6 3 Z g h f W s q W + U A F l l x z P S n K h i V X j d 2 5 + v 5 q V X q + c u w / w Z 4 r N 8 p i 2 1 Y l Z Y q A q W J q c K x Q F Z 6 r i v p I w W L Z l t g M C y V P m K F e C 9 1 a L e 2 P F l M Z y l I U 6 Y 9 a Y f q U 0 + x e L u s T K N I b q l M 3 q g G z q n e 3 r + t F Y Y 1 a h 7 2 e P Z a G i l W + w 5 6 F 9 + + l Z V 5 j n A z q v q S 8 8 B S s h F X h V 7 d y O m f g u z o a / + / f e w P L 0 0 F A 7 T M f 1 n / 0 d 0 R 9 d 8 A 7 v 6 a J 4 s y q V D J P k D 9 P f P / R G s T G T 1 q S w t T m Z m Z u O v S G A A g x j h 9 / 6 F G c x j A X k + d x 8 X u M S V 1 q 6 N a 1 N a r p G q N c W a P r w J 7 f c L h r i W v A = = < / l a t e x i t > $ 1 ( 1 p s) Figure 3. Polar tangential angle in Definition 6.11 (left).Scaling behavior of the polar tangential angle, cf.Lemma 6.13 (right).and (6.31) Proof.To begin with, we show the monotonicity of ϖ λ psq.Lemma 6.9-(iv) asserts that the curvature k λ of Γ λ satisfies k λ psqk 1 λ psq ą 0 for s P p0, L λ q, and hence by applying [18,Corollary 2.7] to Γ λ , we deduce that ϖ λ is monotonically decreasing in p0, L λ q.
Step 1.We show the existence of minimizers.Suppose that ψp 1 2 q ă h ˚.Then we have ψpxq ă U 0 pxq for all x P r0, 1s.Since u c , defined by (6.40), uniformly converges to U 0 as c Ò c p , there is c ˚P p0, c p q such that u c˚ě ψ in r0, 1s.Moreover, in view of (4.4) we notice that u c˚P M sym pψq, E p pu c˚q " pc ˚qp ă pc p q p .Thus we observe from Theorem 6.1 that there exists a minimizer u of E p in M sym pψq.
Step 3. Finally we prove the nonexistence of minimizers of E p in M pψq with ψ satisfying ψp 1 2 q ě h ˚.Suppose that there is a minimizer u P M pψq of E p in M pψq.Since u : r0, 1s Ñ R is continuous and up0q " up1q " 0, u attains its maximum u max ą 0 at x max P p0, 1q.Without loss of generality we may assume that x max P p0, 1  2 s.Since ψ is a symmetric cone, we infer from Remark 3.11 that u ą ψ in p0, x max q.Moreover, recalling that u is also a critical point of E p in M pψq in the sense of Definition 1.2, we see that u satisfies ı dx " 0 for all ϕ P C 8 0 p0, x max q, up0q " 0, u 2 p0q " 0, upx max q " u max , u 1 px max q " 0, and u 2 pxq ă 0 for all x P p0, x max q.Thus, using Lemma 6.14 with x 0 " x max , we obtain 2h ˚ď 2ψp 1 2 q ď ψp 1 2 q x max ď upx max q x max " tan `´ϖ λu pspx max qq ˘.
Here we used that upx max q ě up 1 2 q ě ψp 1 2 q.On the other hand, by Lemma 6.13 and (6.37), for the right-hand side of the above we see that tan `´ϖ λu pspx max qq ˘" tan `´ϖ 1 pλ 1{p u spx max qq ą tan `´ϖ 1 pL 1 q ˘" Y 1 pL 1 q X 1 pL 1 q " 2h ˚.
This is a contradiction.Therefore Theorem 1.5 follows.□   Gpsq ds c p pGq , whereupon the claim follows.Boundedness of H follows from the continuity, the explained asymptotics and the fact that ps Þ Ñ s 9 Gpsqq P L 1 p0, 8q.□ A.2. Technical proofs in Section 6.We prove (6.5) for general functional E which is defined by (1.6).

Corollary 3 . 3 (
Concavity).Let u P M pψq be a critical point in the sense of Definition 1.2.Then u is concave.

(3. 6 ) wpxq " p 9 Corollary 3 . 5 (
Gpu 1 pxqq p´1 p´u 2 pxqq p´1 .Regularity away from the obstacle).Let u P M pψq be a critical point in the sense of Definition 1.2 and let a " 9 G ˝u1 as in Proposition 3.1.Then w P C 2 ptu ą ψuq and there holds (in the classical sense) 0 @x P tu ą ψu.
r d t p E B w m N t B w J z L S A / 3 J U q I s b b A W Z I z B L P 7 P O 7 y a j t i L V 6 X a 3 q h W u N T D O 4 u K / s x S I 9 0 T a / 0 Q D f 0 T B + / 1 g r C G m U v B z y r F a 1 0 s h 3 H v W v v / 6 p M n n 3 s f a n + 9 O w j j 6 n Q q 8 7 e n Z A p 3 0 K r 6 I u H p 6 9 r 06 u D w R B d 0 A v 7 P 6 c n u u c b W M U 3 7 X J F r p 4 h x h + Q + v n c 1 W B j L J m a S N L K e G J m L v q K J v R h A M P 8 3 p O Y w S K W k e Z z T 3 C L O 9 w r P c q 0 M q v M V 1 K V m k j T j W + h L H 0 C m Y S b j Q = = < / l a t e x i t > (s) | (s)| < l a t e x i t s h a 1 _ b a s e 6 4 = " W t X L R o O z o t P p s x o y s y c 2 7 k 8 0 v z Q = " > A A A C d 3 i c h V H L L g R B F D 3 T 3 u M x g 4 3 E g p h g b C a 3 R R A r Y c F y B o P E y K S 6 F T r T r 3 T 3 T D D x A 3 7 A Q i I h E c R n 2 P g B C 5 8 g l i Q i s X C n p x N B c C t V d er U P b d O V W m u a f g B 0 U N M a W h s a m 5 p b Y u 3 d 3 R 2 J Z L d P a u + U / Z 0 m d c d 0 / H W N e F L 0 7 B l P j A C U 6 6 7 n h S W Z s o 1 r T 1 e b U S s z e t a T T 9 U 6 3 y K y d 1 j 5 S C G 6 Z 6 u 6 Z n u 6 I Y e 6 f 3 X W t W w R s 3 L P s 9 a X S v d Y u K o b / n 1 X 5 X F c 4 D d T 9 W f n g N s Y z r 0 a r B 3 N 2 R q t 9 D r + s r B 8 f P y z N J w d Y T O 6 Y n 9 n 9 E D 3 f I N 7 M q L f p G T S y e I 8 w e o 3 5 / 7 J 1 g d z 6 i T G c p N p G b n o q 9 o R T + G k O b 3 n s I s F p F F n s / d w y k u c R V 7 U w a U E S V d T 1 V i k a Y X X 0 J R P w D P M 5 C m < / l a t e x i t > 0 (s) H 7 C V I B i C O I D S t 6 i S J 2 Y U F H D V U I m P A Y G 1 D h c i s g B Y L N 3 A 6 a z D m M p L 8 v c I g w a 2 u c J T h D Z b b C Y 5 l X h Y A 1 e d 2 q 6 f p q n U 8 x u D u s j C F B 9 3 R N z 3 R H N / R I 7 7 / W a v o 1 W l 4 a P G t t r b B L k a O x z O u / q i r P H v Y / V X 9 6 9 r C H J d + r Z O + 2 z 7 R u o b f 1 9 Y P j 5 8 x y O t G c p H N 6 Y v 9 n 9 E C 3 f A O z / q J f b I r 0 C c L 8 A a n v z / 0 T b M 8 l U w t J 2 p y P r 6 w G X 9 G L c U x g m t 9 7 E S t Y x w a y f G 4 D p 7 j C d e h N i S l T y k w 7 V Q k F m l F 8 C W X u A 8 e h k R U = < / l a t e x i t > $ (s)< l a t e x i t s h a 1 _ b a s e 6 4 = "

1 <
e f a w 9 6 n 6 0 7 O H K l Z 8 r z p 7 t 3 2 m c w u t q 2 8 e t l / y q 7 n Z 1 h x d 0 D P 7P 6 c H u u U b m M 1 X 7 T I r c m e I 8 A f I 3 5 / 7 J y g u p O S l F G U X k + m 1 4 C v C m M Y M 5 v m 9 l 5 H G B j I o 8 L k 1 n O A U 7 d C j F J U m p M l u q h Q K N O P 4 E l L i A y P h i o E = < / l a t e x i t> X l a t e x i t s h a 1 _ b a s e 6 4 = " L S f o v z N c 1 d U F B S r 5 a w C y I + m J a d 4 = "

1 <
m 9 3 m s 8 m o z Y E 1 e t 2 u 6 v l r j U w z u D i s T S N E 9 X d M L 3 d E N P d H 7 r 7 W a f o 2 2 l w O e 1 Y 5 W 2 J X Y 8 U T h 7 V 9 V j W c P e 5 + q P z 1 7 2 M W i 7 1 V n 7 7 b P t G + h d f S N w 9 Z L Y S m f a k 7 T B T 2 z / 3 N 6 o F u + g d l 4 1 S 5 z I n + G C H + A / P 2 5 f 4 L S b F q e T 1 N u L p l Z D r 4 i j E l M Y Y b f e w E Z r C K L I p 9 b x Q l O 0 Q o 9 S l F p T B r v p E q h Q D O K L y E l P g A l 4 4 q C < / l a t e x i t > Y l a t e x i t s h a 1 _ b a s e 6 4 = " H Y / H h N u L l l U 0 p 1 S A K D M 8 ), we obtain (6.38) κu psq " |ξ u psq| 2´p p´1 ξ u psq " |ω λu psq| 2´p p´1 ω λu psq " k λu psq.
dt for c P p0, c p q, where c p is defined by (6.1).Similarly to (4.5) we also define U 0 pxq :" lim cÒcp u c pxq.