A biharmonic analogue of the Alt-Caffarelli problem

We study a natural biharmonic analogue of the classical Alt-Caffarelli problem, both under Dirichlet and under Navier boundary conditions. We show existence, basic properties and C 1 ,α -regularity of minimisers. For the Navier problem we also obtain a symmetry result in case that the boundary data are radial. We find this remarkable because the problem under investigation is of higher order. Computing radial minimisers explicitly we find that the obtained regularity is optimal.


Introduction
For a given bounded and suitably smooth domain Ω ⊂ R n and a given boundary datum φ ∈ H 2 (Ω), Dipierro, Karakhanyan, and Valdinoci studied in [4] existence, regularity and further qualitative properties of minimisers of the following functional for The set N models the so-called Navier boundary conditions where only the height of u is prescribed on ∂Ω and not also the slope.The functional J in (1) resembles a lot the Alt-Caffarelli-functional in [1] where |∆u| 2 has to be replaced by |∇u| 2 .However, since Alt and Caffarelli considered only positive boundary data, thanks to the maximum principle their original problem remains the same when replacing 1 {u>0} (x) by 1 {u̸ =0} (x).In fact, for large enough domains Ω minimisers in [1] exhibit large flat regions with u(x) = 0.These flat regions have a physical meaning.They describe e.g. the shadow zone or wake which a jet of a fluid leaves after hitting an obstacle, cf.[21].Moreover, this problem is related to optimisation problems involving capacities, cf.[7,Chapter 14].
As for the functional J( .) in (1), introduced by Dipierro, Karakhanyan, and Valdinoci the situation is completely different.The second author proved in [14] that minimisers of J have no flat parts at all (if n = 2), since for each minimiser ∇u ̸ = 0 is satisfied on {u = 0}.
In order to retrieve minimisers with flat regions we propose to look at a modified functional, namely where the set models the so-called Dirichlet boundary conditions, where the height of u as well as its slope are prescribed on ∂Ω.
We think that the functional F exhibits minimisers which resemble more those obtained by Alt and Caffarelli, as flatness is rewarded.Example 1 and the examples from Section 4 give evidence to this statement.
Nevertheless one should have in mind that the variational problem is of second order, i.e. that any admissible function and in particular any minimiser has to be in H 2 which means that on the boundary of the "flat set" {x ∈ Ω : u(x) = 0} (or, more precisely, its interior), not only the values of the function but also of its derivatives have to match somehow.Outside this flat set, minimisers turn out to be biharmonic, and for biharmonic functions no strong maximum principle and, in general, not even a weak form of a comparison principle is available.In the variational context this is reflected by the fact that for u ∈ H 2 , in general u + ̸ ∈ H 2 , i.e. the "Stampacchia trick" (comparing the energy of u and u + ) does not apply.For features like positivity of superbiharmonic functions in balls, sign change in general domains, but also dominance of positivity in general domains, one may give a look at [8, Chapters 1, 2.6, 5 and 6].Having these features in mind one may expect that in the variational problem under investigation, minimisers for positive boundary data may look "almost positive" but may at the same time exhibit also oscillations, in particular close to the "flat set".Although a detailed investigation of these qualitative properties of minimisers would be in our opinion very interesting, this is beyond the scope of the present work and may be addressed in future research, in particular in combination with numerical approximations.
In the present paper we study existence, some basic qualitative properties and (optimal) regularity of minimisers in D and N .Next we summarise the main findings.
We show further that minimisers u ∈ N with radial boundary data in Ω = B 1 (0) are always smooth in a neighborhood of the boundary and radially symmetric.

II. Boundary regularity (for radial data).
The proof of the boundary regularity result is given in Section 3.2 and relies on a "Kelvin transform"-like reflection trick applied to ∆u.It is by no means obvious how to extend this proof and how to keep the free boundary of {x ∈ Ω : u(x) = 0} apart from the boundary ∂Ω, when general smooth domains and general smooth boundary data are considered.We have to leave this as a challenging open problem.Only in dimensions n ∈ {1, 2, 3} such a reasoning is obvious thanks to Sobolev embedding.

III. Radial symmetry.
The proof of the radial symmetry result is based upon a modification of Talenti's symmetrisation method ( [20]) in annular shaped domains, applied to |∆u|, cf.Section 5 and Appendix B. We find this result remarkable, because for higher order problems symmetry results are in general rather involved and their validity is often limited due to the abovementioned lack of maximum and comparison principles, see e.g.[8,Section 7.1].At the same time we have to leave the question open whether the same radiality result holds for minimisers in D.
We finally obtain that minimisers in N do in general not lie in C 2 (Ω), which shows that the obtained C 1,α -regularity result is optimal.According to the abovementioned symmetry result, Navier minimisers u in balls for constant boundary data φ(x) ≡ const = u 0 are always radial.In B 1 (0) ⊂ R 2 we compute these minimisers explicitly.We see that for u 0 > 0 small enough they exhibit a flat region {x ∈ B 1 (0) : u(x) = 0} ̸ = ∅ and we infer that they do not lie in C 2 (Ω), cf.Section 4. See Corollary 2 in Section 5.
One should remark that the regularity results I and IV leave open whether C 1,1 -regularity can generally be obtained.Due to the absence of a PDE for the minimizer, this regularity discussion goes beyond the scope of this article.Notice however that all the radial minimisers we study in Section 4 are C 1,1 -regular.In [3], Dipierro, Karakhanyan and Valdinoci study a singular perturbation problem for the functional J in (1).It is observed that the C 1,1 -norm of the constructed approximate solutions may degenerate, cf.[3,Theorem 1.7].This is a remarkable phenomenon.Notice however that it does not imply that solutions are generally not C 1,1 -regular.Indeed, the second author proves that (if n = 2) minimisers of J even have higher C 2,1 -regularity, cf.[15,Theorem 5.2].We think that the discussion of C 1,1 -regularity is an interesting, yet difficult, question.

Existence, basic properties, and regularity of minimisers
In this section we prove the existence and basic properties of minimisers in D and N .We formulate the regularity result, which will be proved in Section 3.1 and whose optimality is shown in Corollary 2 at the end of Section 5.
In one dimension minimisers can be computed explicitly.Doing so, we see that the flat set {u = 0} may become arbitrarily large, but it may also become void.Moreover we study the value of the infimum of the energy.
Theorem 1 (Existence of minimisers).Assume that Ω ⊂ R n is a bounded C 2 -smooth domain and that φ ∈ H 2 (Ω).The functional F attains its infimum on N as well as on D which are defined in (2) and (4).
Proof.We give the proof only on N , because on D the proof is similar and as for the H 2boundedness even simpler.Since Ω is bounded the infimum of F on N is finite, i.e.
We consider a minimising sequence In particular, (∥∆v k ∥ L 2 (Ω) ) k∈N is bounded.Thanks to elliptic H 2 -theory [10, Chapter 9] and the C 2 -smoothness and boundedness of Ω we see that also (∥v k ∥ H 2 (Ω) ) k∈N is bounded.Hence we find a u ∈ H 2 (Ω) so that after passing to a subsequence we have This yields immediately that also u − φ ∈ H 1 0 (Ω), hence u ∈ N .(When working on D, one exploits that φ + H 2 0 (Ω) is convex and closed and so, weakly sequentially closed in H 2 (Ω).This shows that u ∈ φ + H 2 0 (Ω).)As it is well known, the H 2 -norm is weakly sequentially lower semicontinuous in H 2 , in particular almost everywhere in Ω. Fatou's lemma then yields All in all we see that This proves that u minimises F in N .
Remark 1.In case that Ω = B R (0) is a ball of radius R and the boundary datum φ is radially symmetric, one may also consider the smaller sets N rad := {v ∈ N : v is radially symmetric} and D rad := {v ∈ D : v is radially symmetric}.Similar to Theorem 1 one shows that F attains its minimum also on N rad and D rad , respectively.An interesting question in the context of biharmonic problems is whether we have symmetry, i.e. whether the minimisers on the radial and the general sets of admissible functions coincide.We show in Section 5 that minimisers in N rad coincide indeed with those in N .In D, i.e. under Dirichlet boundary conditions, we have to leave this question open.
Example 1.In the special one-dimensional case the problem may be solved explicitly.We consider the interval Ω = (−R, R) of length 2R under Navier boundary conditions According to Theorem 6-or more elementary, according to the remark at the end of this example-any minimiser is even.Since minimisers in this case have to be at least C 1 -smooth and biharmonic on Ω \ {x ∈ Ω : u(x) = u ′ (x) = 0}, candidates for a minimiser are either the function (biharmonic in the whole interval and empty "free part") or the following functions for ρ ∈ (0, R): (with [−ρ, ρ] as "free part" and biharmonic on (ρ, R)).
As for the energies of these candidates we calculate Admitting (formally) ρ ∈ [0, R) we see that the optimal choice ρ min of ρ and the corresponding energy are: Comparing this with u 0 we see that the minimiser/s u min of our functional and its energy is/are given by These calculations may be carried out separately for the "left" and the "right" part of the minimisers.From this one can directly infer that they must be symmetric about 0.
Remark 2. One thing that becomes visible is that minimisers may be not unique.In the previous example this is the case for This means that minimisers do not depend continuously on the domain.
Another thing that becomes visible is that the flat set {u = 0} in the previous example cannot be arbitrarily small, if one excludes cases where {u min = 0} = ∅.Indeed, notice that if attains precisely the values in [ 1  4 , 1].Next we state our main regularity result for minimisers.
We prove this result at the end of Section 3.1.We will moreover see in Section 4 and Section 5 that minimisers do in general not lie in C 2 (Ω).
Remark 3. The regularity result only studies interior regularity.However, one would also like to conclude boundary regularity, provided that the assumptions keep the flat region {x ∈ Ω : u(x) = ∇u(x) = 0} away from the boundary.To this end we assume that Ω has smooth boundary and that φ is smooth and strictly positive on Ω.Then it is possible to conclude further in the special situation of Theorem 5 and in general in dimensions n = 1, 2, 3 where H 2 (Ω) → C 0 (Ω).Indeed, since φ| ∂Ω > 0 and each minimiser u lies in C 0 (Ω) one has that {x ∈ Ω : u(x) > 0} is a neighbourhood of ∂Ω.In particular there exists also a smooth tubular neighbourhood Ω ′ ⊂ Ω of ∂Ω such that u is biharmonic and smooth on Ω ′ \ ∂Ω.The smoothness of its boundary values φ on ∂Ω yield that then also u ∈ C ∞ (Ω ′ ).As a consequence, the Dirichlet or Navier boundary conditions are attained in a classical sense.In the case of Navier boundary conditions this means that u| ∂Ω = φ| ∂Ω and ∆u| ∂Ω = 0.
In the case of minimisation in N rad or D rad (cf.Remark 1) the boundary regularity can also be obtained for arbitrary dimensions, since by radial Sobolev embedding N rad , D rad ⊂ C 0 (Ω \ {0}).
In contrast to this, a minimiser u ∈ N solves a mixed Dirichlet-Navier problem This means that in both cases our problem does not decouple into a second order system for u and ∆u.

The energy infimum
Another noteworthy thing is that in the case of the Navier problem, the energy infimum is bounded above independently of the boundary datum, while in the case of the Dirichlet problem this is not the case.
Lemma 1 (The energy infimum).Let Ω ⊂ R n be a domain with C 2 -smooth boundary and and Here H n−1 denotes the n − 1-dimensional Hausdorff measure.
Using that ∇ψ = ∇φ on ∂Ω (in the sense of Sobolev traces) we obtain the claim.
For k large and ε small we have produced in the C 0 -sense small boundary data with large optimal energies.Remark 6.For the minimisation in N we actually obtain a dichotomy result.If inf ψ∈N F (ψ) = |Ω| then one immediately can obtain a harmonic minimiser with empty free boundary, as the proof of Lemma 1 reveals.If inf ψ∈N F (ψ) < |Ω| we infer that |{u = 0}| > 0, i.e. the free boundary is nonempty.

Nonemptiness of the "flat set" and free boundary
We show that when the set Ω is large enough compared to the boundary conditions, then any minimiser u ∈ N or u ∈ D exhibits a nonempty flat set {x ∈ Ω : u(x) = 0}.For brevity we discuss only the case of Dirichlet conditions; Navier conditions can be treated analogously.More precisely we have the following result which holds irrespective of the shape of domains. where For the flat set of any minimiser u ∈ D R we conclude: 3 BM O-estimates and regularity and φ ∈ H 2 (Ω).We consider any minimiser u of F on N or on D, respectively.Then for any R 0 ∈ (0, 1 3 r Ω ) there exists a constant C = C(u, R 0 ) such that for all x 0 ∈ Ω with dist(x 0 , ∂Ω) ≥ 3R 0 and all r ∈ (0, R 0 ) one has the following estimate of bounded mean oscillation type ˆBr where (∆u) x 0 ,r := denotes the mean value of ∆u in B r (x 0 ).In particular, ∆u ∈ BM O loc (Ω).
For the existence of such minimisers we refer to Theorem 1.
Proof.This is pretty much along the lines of [4, Theorem 1.1].However, for the reader's convenience we give a very detailed elaboration in Appendix A.
With this regularity we are finally able to prove Theorem 2.
Proof of Theorem 2. Let u be as in the statement.That u ∈ C 1,α (Ω) follows from the previous remark.We also notice that {x ∈ Ω : u(x) = ∇u(x) = 0} is closed in Ω by the fact that u ∈ C 1 (Ω).Hence the set Ω \ {u = ∇u = 0} is actually open.The proof of the biharmonicity on Ω \ {u = ∇u = 0} is divided into two steps.
Step 2. Biharmonicity on {u = 0, ∇u ̸ = 0}.Let x 0 ∈ Ω be such that u(x 0 ) = 0 and ∇u(x 0 ) ̸ = 0. We will show that there exists a neighbourhood of x 0 on which u is biharmonic.To this end first notice that by C 1 -regularity there exists r > 0 such that ∇u ̸ = 0 on B r (x 0 ) and B r (x 0 ) ⊂⊂ Ω.In particular this nonvanishing gradient implies that {u ).This minimiser w additionally satisfies ∆ 2 w = 0 weakly in B r (x 0 ).Next define v : Ω → R to be (a.e.) Since the Sobolev traces of u and w and of their first derivatives match at ∂B r (x 0 ) and B r (x 0 ) ⊂⊂ Ω we further obtain that v ∈ H 2 (Ω) and has the same boundary values as u (in both D and N ).Therefore v is admissible for our minimisation problem.In particular Moreover as we have discussed above one has |{u ̸ = 0} ∩ B r (x 0 )| = |B r (x 0 )| and hence also The previous two inequalities together with (9) yield 0 ≤ F (v) − F (u) ≤ 0 and hence F (u) = F (v).This in particular implies that (10) must hold with equality.By the uniqueness in the minimisation problem for w one concludes that w = u| Br(x 0 ) , implying biharmonicity of u on B r (x 0 ).Since x 0 ∈ {u = 0, ∇u ̸ = 0} was arbitrary we infer biharmonicity in (an open neighbourhood of) {u = 0, ∇u ̸ = 0}.

A global BM O-estimate for the Laplacian of Navier minimisers in balls under constant boundary conditions
In this section we prove a BM O-estimate up to the boundary for the Laplacian of minimisers in N in the (unit) ball Ω = B 1 (0) ⊂ R n , with n ∈ N arbitrary, in the special case of constant Navier boundary conditions φ = u 0 ≡ const > 0. As is shown in Corollary 1 to Theorem 5 below this suffices to see that u ∈ W 2,q (B 1 (0)) for every q ∈ [1, ∞).
In the proof of Theorem 5 below we need to find a weak solution of the biharmonic equation under mixed Dirichlet-Navier boundary conditions.The key issue for this is the following definition of a suitable function space.
together with the usual H 2 -norm or, equivalently, and the corresponding scalar products is a closed subspace of the Hilbert space H 1 0 ∩ H 2 (B 2 ) and hence a Hilbert space itself.
Let u ∈ H 2 (B 2 ) be arbitrary.Minimising v → ´B2 (∆v) 2 dx on the affine space u + H yields a weak solution h of the mixed Dirichlet-Navier boundary value problem Proof.The Riesz-Fischer theorem yields the closedness of H .That v → ´B2 (∆v) 2 dx has a minimum h on the affine space u + H follows by adapting Dirichlet's classical principle.As a necessary condition we obtain the following Euler-Lagrange-equation: This is the weak (variational) formulation of (11).We emphasise that the "weak" attainment of the Navier boundary data is encoded in the space H of admissible testing functions.
Theorem 5. Let Ω = B 1 (0) ⊂ R n be the (unit) ball and φ = u 0 ≡ const > 0. We consider any minimiser u of F on N .We consider Then for any R 0 ∈ (0, 1  8 ) there exists a constant C = C(u, R 0 ) such that for all x 0 ∈ Ω and all r ∈ (0, R 0 ) one has the following estimate of bounded mean oscillation type ˆBr(x0) where denotes the mean value of U in B r (x 0 ).
We have to leave the question open whether (12) holds for every x 0 ∈ R n .For our purposes, however, the previous result is strong enough.This will be shown in Corollary 1 below.
For the existence of such minimisers we refer again to Theorem 1.
We keep x 0 and R 0 fixed in what follows.In the following argument we mainly have the case in mind that B 2R (x 0 ) ∩ ∂Ω ̸ = ∅.If on contrary B 2R (x 0 ) ⊂ Ω, the reasoning below still applies, but the modifications are void and it runs along the lines of [4,Theorem 1.1].
For regularity purposes we first take a function We apply Lemma 2 with B 1 = B 2R (x 0 ) and B 2 = Ω = B 1 (0).We recall the Hilbert space from there and find a minimum h ∈ ũ+H of v → ´B2R (x 0 )∩Ω (∆v) 2 dx on ũ+H .This minimiser weakly solves in the sense of smooth functions.We define h ∈ N via By minimality of u we have that F (u) ≤ F (h) which implies that This yields with a universal constant C > 0. Since h| B 2R (x 0 )∩Ω = h ∈ ũ+H minimises v → ´B2R (x 0 )∩Ω (∆v) 2 dx on ũ + H and u − h ∈ H is an admissible testing function we have This gives With this, we conclude from (13) that Since ∆u and ∆h vanish on ∂Ω at least in a variational sense, it is a natural idea to introduce the following odd "Kelvin transformed" extensions U, H ∈ L 2 loc (R n ) One should observe that (thanks to using ũ instead of u for introducing h) we have in a classical sense This yields that we have also for the extended function For this one should have in mind: ) is a subset of the union of Ω ∩ B R (x 0 ) and its inversion.In order to see this we take x ∈ B R (x 0 ) with |x| > 1 and need to show that x/|x| 2 ∈ B R (x 0 ).This follows in turn from the following inequalities: ) and harmonic in B R (x 0 ) \ ∂Ω.Hence, it is weakly and consequently classically harmonic in B R (x 0 ).
Since the extension operator, defined in (15) By means of Hölder's inequality We deduce now a Campanato type inequality for H, which is harmonic in B R (x 0 ).We fix an arbitrary α ∈ (0, 1).For any harmonic function f on B R (x 0 ) local elliptic estimates yield (with [f ] β,G denoting the Hölder seminorm in C 0,β (G)) If we assume further that f (x 0 ) = 0 we conclude that ∀r ∈ (0, R 4 ): For r ∈ ( R 4 , R), r R is bounded from below and we directly see that All in all we have ∀r ∈ (0, R) : with a constant C = C(n, α).We apply this to the harmonic function (making use of its mean value property in B R (x 0 )) and find with a constant Putting all the estimates together we find the following with constants C = C(n, α): 18),( 19) Introducing the notation this estimate rewrites as In order to proceed we need to introduce the following increasing variant of Φ: which obeys the same relation as Φ: Lemma 2.1 from [9, Chapter III] yields that then with C = C(n, α) and in particular that (Recall again the local boundedness of the extension operator in (15).)Since we may e.g.fix α = 1/2 we end up with with a constant C = C(n, R 0 , F (u)).This proves (12).
Proof.According to [19, Chapter IV, Remark 1.1.1]the definition of BM O does not depend on whether one is working with balls or with cubes.Hence the previous Theorem 5 shows that ∆u ∈ BM O(B 1 (0)) in the sense of Jones [11, p. 41].According to the remarks between Theorem 2 and Theorem 3 in Jones' work, his Theorem 1 (where for the sufficiency part he gives strong credits to Reimann [18]) applies in particular to the ball B 1 (0).This yields that ∆u has an extension Ũ ∈ BM O(R n ).Applying [19, Corollary in Chapter IV. 1.3] shows then that Ũ ∈ L q loc (R n ) for any q ∈ (1, ∞).Hence ∆u ∈ L q (B 1 (0)) and elliptic regularity yields that u ∈ W 2,q (B 1 (0)).By Sobolev embedding the Hölder regularity follows.
Remark 8. We know from Remark 3 that in dimensions n ∈ {1, 2, 3} the previous boundary regularity result can be extended to general sufficiently smooth strictly positive boundary data, to general sufficiently smooth domains, and to the case of Dirichlet boundary conditions.
We have to leave the question open whether such generalisations are available in dimensions n ≥ 4.

Radial minimisers
In Remark 1 we have seen that minimisers in D rad and N rad can be found.In the sequel we want to compute these minimisers explicitly and study their properties on balls Ω = B R (0) with φ ≡ const =: u 0 ∈ (0, ∞).
Recall that each radial biharmonic function is given by Hence each radial minimiser must be either of the form u ≡ u 0 or We remark that C 1 , C 2 , C 3 , C 4 can be uniquely determined by ρ via the boundary conditions.
Figure 1: Plots of f 1 (0.05, .), f 1 (0.075, .), and f 1 (0.1, .) (left to right).The straight line marks the threshold energy level π.This shows that for small enough u 0 there is a "flat" set, while for large u 0 there is none.
Remark 9. We have seen that for radial minimisers in N rad , n = 2 one has ∆u ̸ ∈ C 0 (B 1 (0)) unless u ≡ const.Indeed, one infers from the previous example that if the infiumum in (25) is smaller than λπ one has a minimiser u ∈ N rad with ∆u where ρ min ∈ (0, 1) is some value where the infimum in (25) is attained.This discontinuity phenomenon also occurs in arbitrary dimension.Indeed, here we show the following Claim.Suppose that u ∈ N rad is a minimiser with |{u = 0}| > 0. Then ∆u ̸ ∈ C 0 (B 1 (0)).In particular u ̸ ∈ C 2 (B 1 (0)).
Remark 10.The expression in ( 25) is strictly increasing in u 0 until it reaches the value λπ.
From there on it is constant.Also this is true in any arbitrary dimension, as we shall prove here in the case of λ = 1. Claim.
Proof of the claim.Let 0 < u 1 < u 0 be arbitrary constant boundary data.Then there exists α ∈ (0, 1) such that u 1 = αu 0 .Let u ∈ N rad (u 0 ) be a minimiser.The rescaled function u α := αu lies in N rad (u 1 ) and satisfies inf ψ∈N rad (u 1 ) The inequality in the first estimate (28) is strict iff ´Ω(∆u) 2 dx ̸ = 0. Notice that this integal vanishes iff u is harmonic in Ω. Uniqueness of solutions for the harmonic Dirichlet problem yields that in this case u ≡ u 0 ≡ const.Hence inf This and (28) imply that where the inequality is strict, when u 0 admits a nonconstant minimiser.
The assertion follows.
Example 3. We minimise F in N rad again in dimension n = 2, but this time with Ω = B R (0) for any arbitrary R ∈ (0, ∞) and constant boundary value u 0 .We write F (• | B R (0)) to distinguish between the minimisation problems for different values of R. One readily checks that for each u ∈ N rad one can define u R : B 1 (0) → R via u R (x) := u(Rx) and obtains This yields that for λ = R 4 one has where F λ is as in Example 2. Using this and the fact that This can again be studied computer assistedly.
Example 4. We now minimise F defined in D rad for n = 2 on Ω = B 1 (0).In particular we prescribe u − u 0 ∈ H 2 0 (Ω) for some constant function u 0 > 0. Again either u ≡ u 0 or (21).The conditions we have to ensure are u = u 0 , ∂ r u = 0 on ∂B 1 (0) and u = ∂ r u = 0 on ∂B ρ (0).We compute on [ρ, 1]: Evaluated at r = 1 and r = ρ this amounts to Multiplying the first equation with ρ and subtracting the second one we find which yields Reinserting this into (29) yields Next we use that u ≡ u 0 on ∂B 1 (0) yields This yields an explicit fomula for C 1 , namely Therefore one has also Next we use that for Using that C 1 = C 1 (u 0 , ρ) and C 3 = C 3 (u 0 , ρ) one can therefore determine the F (u) only in terms of u 0 , ρ.An explicit formula for the infimum can therefore be found.Finally, we can compute the energy of each admissible u = u(C 1 , C 2 , C 3 , C 4 , ρ) in terms of u 0 , ρ and may minimise computer-assistedly. where we infer where In particular inf Figure 2: Plots of g(0.03, .), g(0.045, .), and g(0.05, .) (left to right).The straight line marks the threshold energy level π.
Remark 11.The nonconstant minimisers in the previous example display some qualitative properties, which we can show in any dimension.Let u ∈ D rad be a nonconstant radial minimiser on B 1 (0) ⊂ R n , which exists for sufficiently small boundary data u 0 ≡ const > 0. [The existence can be seen as follows: Take Notice that these limits are well defined as ∆u| B 1 (0)\Bρ(0) is a radial harmonic function.We prove the following Claim.
2. Sign-change of the Laplacian.It is immediate from (32) that ∆u changes sign on B 1 (0) \ B ρ (0).This behaviour is in contrast to the situation of Navier boundary conditions, cf.Example 2.

Radiality of Navier minimisers
In this section we investigate radiality of Navier minimisers in N in the case that Ω = B 1 (0) and φ = u 0 ≡ const.Showing radiality of minimisers means that minimisers in N rad coincide with minimisers in N .This means in particular that the observations we have made for radial Navier minimisers (e.g. that their Laplace is never continuous on B 1 (0), cf.Remark 9) are also observations about minimisers in N .
Remark 12.We want to investigate radiality of minimisers in the case of Ω = B 1 (0) and u 0 ≡ const > 0. In Lemma 1 we have already shown that inf In the case that equality holds one already obtains trivially a radial minimiser, namely the constant solution u(x) ≡ u 0 .Indeed, if equality holds then one computes In this case it is an obvious guess that any minimiser should be radially symmetric, although according to Example 1 there may also be nonconstant minimisers.However, one may wonder whether also in the generic case, where inf ψ∈N (u 0 ) F (ψ) < e n , minimisers were always radial.
The following theorem answers this question in the affirmative.
The following proof builds on the Talenti symmetrisation technique, introduced by G. Talenti in [20].This technique uses the symmetric decreasing rearrangement of the function ∆u, where u is a minimiser.Our rearrangement works slightly differently, we want to rearrange functions so that they are defined on an annular region.The reason for this is that the classical symmetric decreasing rearrangement would relocate the set {∆u = 0} to the boundary of B 1 (0).However for minimisers this set is to be expected in the middle of B 1 (0), because it must contain (up to a null set) the flat set {u = 0}.
Since we define different to [20] the rearrangement in annular regions, we have to argue in what way Talenti's symmetrisation results carry over.We therefore collect some properties of the used rearrangement procedure in Appendix B. The proofs of these properties do not defer very much from the classical case.Nevertheless, we present them for the reader's convenience.
If Claims 1, 2 and 3 are shown one can finish the proof.Indeed, employing the (strict) monotonicity properties proven in Remark 10 we find: This shows that v minimises F in N rad (w(1)).Since v is not a constant, the inequality sign ( * ) is even strict, again according to Remark 10.This contradiction shows that each minimiser must be radial.
Proof of Claim 2. We define for t ≥ 0 the increasing function For short we also use the notation µ(t) = |{u < t}|.Assume now that t ∈ (0, u 0 ) is a regular value of u.In view of the smoothness u ∈ C ∞ (B 1 (0) \ Ω 0 ) this is true by Sard's theorem for a.e.
We infer from this, the Cauchy Schwarz inequality and the isoperimetric inequality that the following holds for a.e.t ∈ (0, u 0 ): Now we estimate with Lemma 4, the definition of Ω 0 = {u = 0} and Lemma 3(b) By Example 5 one has that en ) 1/n .Using this and layer cake integration in (42) we find ˆ{u<t} |∆u| dx ≤ ˆr(t) Plugging this into (41) we infer that As a consequence we have for a.e.t ∈ (0, u 0 ) that Since h is in view of (39) strictly positive and continuous on (e n r n 0 , ∞) it possesses an increasing primitive on (e n r n 0 , ∞), which we call H. Integrating (43) on (ε, u 0 ) we find according to [16, Kap.VIII, §2, Satz 5] that Notice that the second inequality sign is due to the fact that H •µ is in general merely increasing and not necessarily absolutely continuous.Using that µ(u 0 ) ≤ e n , µ(ε) ≥ e n r n 0 + |{u < 0}| for all ε > 0 and letting ε → 0+ we find In view of the strict positivity of h this inequality is strict in case that |{u < 0}| = 0 (45) is violated.Employing the substitution τ = e n σ n , using (43) and observing that for all r ∈ [r 0 , 1] one has (by (36)) that d dr (r n−1 w ′ (r)) = r n−1 f (r) and w(r 0 ) = w ′ (r 0 ) = 0 we infer Here J z ⊂ R is the open maximal interval of existence and contains the point 0. The existence of such a solution follows from Peano's theorem and Zorn's lemma, see e.g.[17,Theorem 3.22].Outside of B 1 (0) we have to choose a Hölder continuous extension of ∇u |∇u| .The chosen extension will, however, not be relevant for our argument as we will only look at values s such that x z (s) ∈ B 1 (0).Notice that for all s ≤ 0 one has that x z (s) ∈ B 1 (0) since B 1 (0) = {u < u 0 } and d ds u(x z (s)) = |∇u|(x z (s)) > 0 for all s ∈ J z , s ≤ 0. From now on let Jz = J z ∩ (−∞, 0].We compute for s ∈ Jz that Notice that for all s 1 , s 2 ∈ Jz one has where we used in the last step that |∇ϕ( By virtue of the triangle inequality this implies that x z | Jz is a straight line and is moreover parameterised with unit speed.Therefore there exists v ∈ R n such that |v| = 1 and x z (t) = z + tv for all t ∈ Jz .Looking at (46) at s = 0 we infer also that Hence x z (s) = (1+s)z and for all s ∈ Jz and (since |∇u| > 0 on {u ̸ = 0}) maximality of Jz yields Jz = {s ∈ (−1, 0) : u((1 + s)z) ̸ = 0}.For each z ∈ ∂B 1 (0) we define f z : Jz → R, s → u((1 + s)z).
To prove radiality it suffices to show that f z does not depend on z.For all s ∈ Jz one has Therefore for any z, f z is a solution of As η is not Lipschitz continuous, no general uniqueness result is applicable to show that f z is independent of z.However, we can still obtain uniqueness by separation of variables.Indeed, If η is a primitive of 1 η we obtain that Using that η is strictly increasing and hence invertible we obtain that We infer that f z (and also Jz , since Jz = {f z > 0}), are independent of z.Hence z → u((1 + s)z) is independent of z on the annulus {u > 0}.We infer that u is radial and {u > 0} is an annulus with centre zero.The radiality is shown.This completes the proof of Claim 3 and also the proof of the theorem.
As a consequence we obtain the optimality of the C 1,α -regularity of minimisers, which we already mentioned in the introduction.This implies that for any minimiser u ∈ N the flat set Ω 0 = {x ∈ Ω : u(x) = 0} is not empty.The previous theorem yields that each minimiser is radial.Since nonconstant minimisers in N rad are not in C 2 by Remark 9 we infer that no minimiser can lie in C 2 .
Remark 13.This regularity behaviour is fundamentally different from the Alt-Caffarelli-type problem in [4], where C 2 -regularity can be expected and has already been proved in the case of n = 2 in [14].

A Proof of the local BM O-estimate
Proof of Theorem 4. We choose some R 0 ∈ (0, To this end we observe that for all x ∈ B 1 (0) \ B r 0 (0) one has In order to prove Theorem 6 we need a rearrangement inequality for annular symmetric decreasing rearrangements.The proof is completely anlogous to the classical case of rearrangements in balls, cf.[13,Theorem 3.4].We present it for the sake of the reader's convenience.
Proof.(a) This claim follows from the equimeasurability of the level sets of f and f * and Fubini's theorem:

C Stampacchia's lemma for second derivatives
In this appendix we prove an analogue to Stampacchia's lemma, cf.[12, Chapter II, Lemma A.4] for second derivatives, which will prove helpful for the argument.

3. 1 A
local BM O-estimate for the Laplacian of minimisers Theorem 4. Let Ω ⊂ R n be a bounded C 2 -smooth domain with maximal inscribed radius r Ω := sup x∈Ω d(x, ∂Ω)
t. Then {u < t} ⊂⊂ B 1 (0) is a C 1 -smooth domain with boundary {u = t} and outward unit Satz 4], µ is differentiable in almost all t because µ is increasing.For such t we infer from the coarea formula (see e.g.[5, Section 3.3.4,Proposition 3]) that