OVERDETERMINED ELLIPTIC PROBLEMS IN NONTRIVIAL CONTRACTIBLE DOMAINS OF THE SPHERE DAVID RUIZ PIERALBERTO SICBALDI AND JING WU

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OVERDETERMINED ELLIPTIC PROBLEMS IN NONTRIVIAL

CONTRACTIBLE DOMAINS OF THE SPHERE

DAVID RUIZ, PIERALBERTO SICBALDI, AND JING WU

Abstract In this paper, we prove the existence of nontrivial contractible domains

Ω⊂Sd,d≥2, such that the overdetermined elliptic problem











−ε∆gu+u−up= 0 in Ω,

u > 0in Ω,

u= 0 on ∂Ω,

∂νu=constant on ∂Ω,

admits a positive solution. Here ∆gis the Laplace-Beltrami operator in the unit

sphere Sdwith respect to the canonical round metric g,ε > 0is a small real param-

eter and 1< p < d+2

d−2(p > 1if d= 2). These domains are perturbations of Sd\D,

where Dis a small geodesic ball. This shows in particular that Serrin’s theorem

for overdetermined problems in the Euclidean space cannot be generalized to the

sphere even for contractible domains.

Keywords : Overdetermined boundary conditions; semilinear elliptic problems;

bifurcation theory.

Mathematics Subject Classiﬁcation (2020). 35J61, 35N25

1. INTRODUCTION

Semilinear overdetermined elliptic problems in the form











∆u+f(u)=0 in Ω,

u > 0in Ω,

u= 0 on ∂Ω,

∂νu=constant on ∂Ω,

(1.1)

have received much attention in the last decades. Typically Ωis a regular domain

in Rd,fis a Lipschitz function and ∂νuis the derivative of uin the direction of

the outward normal unit vector νon the boundary ∂Ω. These problems are called

“overdetermined” because of the two boundary conditions, and appear quite nat-

urally in many different phenomena in Physics, see [38,40] for more details.

If Ω⊂Rdis bounded the problem has been completely solved by J. Serrin,

who proved in 1971 that if (1.1) is solvable, then Ωmust be a ball (see the original

work [35] and also [28] for more details). Serrin’s proof is based on the moving

D. R. has been supported by the FEDER-MINECO Grant PGC2018-096422-B-I00 and by J. An-

dalucia (FQM-116). P. S has been supported by the FEDER-MINECO Grant PID2020-117868GB-I00

and by J. Andalucia Grant P18-FR-4049. J. W. has been supported by the China Scholarship Council

(CSC201906290013) and by J. Andalucia (FQM-116). D. R. and P. S. also acknowledge ﬁnancial support

from the Spanish Ministry of Science and Innovation (MICINN), through the IMAG-Maria de Maeztu

Excellence Grant CEX2020-001105-M/AEI/10.13039/501100011033.

arXiv:2210.10826v2 [math.AP] 7 Jun 2023

plane method, introduced in 1956 by A. D. Alexandrov in [1] to prove that the only

compact, connected, embedded hypersurfaces in Rdwith constant mean curvature

are the spheres. This proof showed an analogy between overdetermined elliptic

problems and constant mean curvature surfaces. From that moment, the moving

plane method has become a very important tool in Analysis to obtain symmetry

results for solutions of semilinear elliptic equations.

Starting from Serrin’s result, two natural lines of research have been considered:

ﬁrst, the case of unbounded domains Ωin the Euclidean space, and second, the

case of domains Ω⊂M, where Mis a Riemannian manifold.

The ﬁrst topic is motivated from the fact that overdetermined elliptic problems

arise naturally in free boundary problems, when the variational structure imposes

suitable conditions on the separation interface (see, for example, [2]). As is well

known, several methods for studying the regularity of the interface are based on

blowup techniques that lead to the study of an overdetermined elliptic problem

in an unbounded domain. In this framework, H. Berestycki, L. Caffarelli and L.

Nirenberg [6] stated the following conjecture:

BCN Conjecture (1997). Assume that Ωis a smooth domain with connected

complement, then the existence of a bounded solution to problem (1.1) for some

Lipschitz function fimplies that Ωis either a ball, a half-space, a generalized cylin-

der Bk×Rd−k(Bkis a ball in Rk), or the complement of one of them.

Many works have shown that under some assumptions on the function for on

the domain Ωthe conjecture is true, see [17,21,22,29,30,32,44]. Nevertheless, the

conjecture is false in its generality and was disproved for d≥3in [36], where the

second author found a periodic perturbation of the straight cylinder Bd−1×Rthat

supports a periodic solution to the problem (1.1) with f(u) = λu, λ > 0. After such

ﬁrst construction, other examples of nontrivial solutions have been obtained, see

for instance [9,14,25,33,34]. In all these examples the boundary of the domain has

a shape that looks like an unbounded constant mean curvature surface, showing

again an important analogy with those surfaces. Another type of counterexample

to the conjecture, of particular interest for this paper, has been given in [31]. In that

work it is shown that (1.1) admits a solution for some nonradial exterior domains

(i.e. the complement of a compact region in Rdthat is not a closed ball), for a

suitable function f(u). In dimension 2, this represents this ﬁrst construction of a

counterexample to the BCN conjecture, that turns to be false in any dimension. It is

worth pointing out that such result breaks the analogy with the theory of constant

mean curvature surfaces.

Overdetermined elliptic problems (1.1) posed in complete Riemannian mani-

folds instead of the Euclidean setting have also been a natural ﬁeld of research. In

this framework, we need to replace in (1.1) the classical Laplacian by the Laplace-

Beltrami operator ∆gassociated to the metric gof the manifold M:











∆gu+f(u) = 0 in Ω,

u > 0in Ω,

u= 0 on ∂Ω,

∂νu=constant on ∂Ω,

(1.2)

where Ωis a domain of M, and νis the normal outward unit vector about ∂Ω

with respect to g. For general Riemannian manifolds, solutions of overdetermined

elliptic problems of the form (1.2) are obtained in [10,11,16,26,27,37].

It is clear that any symmetry result on the solutions of (1.2) tightly depends on

the symmetry of the ambient manifold. In a given arbitrary manifold, geodesic

balls are not domains where (1.2) can be solved. As shown in [10], for small vol-

umes it is possible to construct solutions of (1.2) with f(u) = λ u in perturbations

of geodesic balls centered at speciﬁc points of the manifold, but such domains in

general are not geodesic balls. In fact, one can expect to obtain a Serrin-type result

only for manifolds that are symmetric in a suitable sense. More precisely (see also

the introduction of [11]), the key ingredient for the moving plane method is the

use of the reﬂexion principle in any point and any direction. For this we need that

for any p∈Mand any two vectors v, w ∈TpMthere exists an isometry of Mleav-

ing pﬁxed and transporting vinto w(i.e. Mis isotropic) and that such isometry

is induced by the reﬂection with respect to a hypersurface. Such last hypersur-

face must be totally geodesic, being the set of ﬁxed points of an isometry. But an

isotropic manifold admitting totally geodesic hypersurfaces must have constant

sectional curvature (see for example [7], p. 295) and the only isotropic Riemannian

manifolds of constant sectional curvature are the Euclidean space Rd, the round

sphere Sd, the hyperbolic space Hdand the real projective space RPd(see [42]).

Now, domains in RPdarise naturally to domains in its universal covering Sd, so

we are left to consider our problem in Rd,Sdand Hd.

Being the problem in Rdcompletely understood by Serrin, the framework of

the other two space form manifolds has been treated in 1998 in the paper by S.

Kumaresan and J. Prajapat [24]. In the case of Hdthey obtained a complete coun-

terpart of the Serrin’s theorem: namely, by using the moving plane method, they

show that if Ωis a bounded domain of Hdand (1.2) admits a solution, then Ωmust

be a geodesic ball. The case of Sdis different. In fact, even if the reﬂexion principle

is valid in any point, one needs to have a totally geodesic hypersurface that does

not intersect the domain in order to start the moving plane. This is not a problem

in Rdor Hd, but it is in Sd. Since the totally geodesic hypersurfaces are the equa-

tors, one can start the moving plane method if and only if the domain is contained

on a hemisphere. And this is exactly the case considered in [24]: if Ωis contained

in a hemisphere and (1.2) admits a solution, then Ωmust be a geodesic ball.

Other natural domains of Sdwhere (1.2) has solutions are symmetric neigh-

borhoods of any equator. Such symmetric annuli are not contractible and their

existence comes from the geometry of Sdin the same way as they exist in a cylin-

der or in a torus. Moreover, perturbations of such domains in Sdwhere (1.2) still

admits a solution have been built in [15] in the same way as this has been done for

the same kind of domains in cylinders or in tori [36].

Taking these facts in account, the following question arises naturally: is it true

that if Ω⊂Sdis contractible and (1.2) can be solved, then Ωmust be a geodesic

ball? In [12], J.M. Espinar and L. Mazet give an afﬁrmative answer to this question

if d= 2 but under some extra assumptions on the nonlinear term f(u). The proof

of such result shows again an analogy between overdetermined elliptic problems

and constant mean curvature surfaces, because it is highly inspired by the proof of

the Hopf’s Theorem that states that the only immersed constant mean curvature

surfaces of genus zero in R3are the spheres.

In this paper we show that the answer to the previous question is negative:

there exist contractible domains Ω⊂Sd, different from geodesic balls, where (1.2)

can be solved for some nonlinearities f. This construction works for any dimen-

sion d≥2. In view of [24], such domains cannot be contained in any hemisphere.

Our main result can be stated as follows.

Theorem 1.1. Let d∈N, d ≥2and 1< p < d+2

d−2(p > 1if d= 2).Then there exist

domains D, which are perturbations of a small geodesic ball, such that the problem











−ε∆gu+u−up= 0 in Sd\D ,

u > 0in Sd\D ,

u= 0 on ∂D ,

∂νu=constant on ∂D ,

(1.3)

admits a solution for some ε > 0.

FIGURE 1. The domain Sd\D

A more precise formulation of the above result will be given in Section 2.

The main idea of the proof is the following. First, one uses a dilation to pass

to a problem posed in Sd(k), the sphere of radius 1/k, where kwill be a small

parameter. We take a geodesic ball of radius 1 in Sd(k), and consider now Ωkthe

complement of such ball in Sd(k). The main idea is that, as k→0, the domain Ωk

converges (in a certain sense) to the exterior domain Rd\B(0,1). Thanks to the

result in [31], we have the existence of nontrivial solutions of (1.1) for a suitable

choice of f, bifurcating from a family of regular solutions uλof the problem:











−λ∆u+u−up= 0 in Rd\B(0,1) ,

u > 0in Rd\B(0,1) ,

u= 0 on ∂B(0,1) ,

∂νu=constant on ∂B(0,1).

(1.4)

In this paper we ﬁrst show that for ksufﬁciently small there exists a similar family

of solutions posed in Ωk. This is accomplished by making use of a (quantitative)

Implicit Function Theorem. Then we study the behavior of the linearized operator

by using a perturbation argument, and taking into account the case studied in

[31]. In such way, we can use the Krasnoselskii bifurcation theorem to show the

existence of a branch of nontrivial solutions to (1.2).

In our arguments, we rely on the study of the linearized operator given in [31].

At a certain point the assumption p < d+2

d−2is needed in [31], and hence our result

is also restricted to that case. Moreover, such assumption is needed also in order

to get L∞uniform estimates on the solutions.

The rest of the paper is organized as follows. In Section 2we set the notations

and we give a more precise statement of Theorem 1.1. The existence of a radial

family of solutions in Ωkis shown in Section 3. In Section 4we construct the non-

linear Dirichlet-to-Neumann operator and compute its linearization under certain

nondegeneracy assumptions. These hypotheses will be veriﬁed in Section 5, and

that section is also devoted to studying the properties of the linearized operator

computed in Section 4. With all those ingredients, we can use a local bifurcation

argument to prove our main result; this is done in Section 6.

Acknowledgment: The authors wish to express their sincere gratitude to the

referee for his/her careful reading and the many comments, which have deﬁni-

tively improved the quality of this work.

2. NOTATIONS AND STATEMENT OF THE MAIN RESULT

If k > 0, let Sd(k)be the d-dimensional sphere of radius 1

knaturally embedded

in Rd+1 (d≥2). We consider Sd(k)as a Riemannian manifold with the metric gk

endowed by its embedding in Rd+1. The sectional curvature of such manifold is

equal to k2. When k= 1 we write directly Sdas usual. We ﬁx two opposite points

S, N ∈Sd(k)(let’s say respectively the south and the north pole) and we use the

exponential map of Sd(k)centered at S,

expS:B0,π

k→Sd(k)\ {N},

where B0,π

k⊂Rdis the Euclidean ball of radius π

kcentered at the origin. Given

any continuous function v:Sd−1→0,π

k, we deﬁne the domain

Bv=expSx∈Rd: 0 ≤ |x|< v x

|x|⊂Sd(k).

The precise statement of our result is the following:

Theorem 2.1. Let d∈N,d≥2, let 1< p < d+2

d−2(p > 1if d= 2). Then, there exists

a real number k0>0, such that for any 0< k < k0the following holds true: there exist

a sequence of real parameters λm=λm(k)converging to some λ∗(k)>0, a sequence

of nonconstant functions vm=vm(k)∈C2,α(Sd−1)converging to 0in C2,α, and a

sequence of positive functions um∈C2,α(Sd(k)\B1+vm), such that the equation











−λm∆um+um−up

m= 0 in Sd(k)\B1+vm,

um= 0 on ∂B1+vm,

∂νum=constant on ∂B1+vm,

(2.1)

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OVERDETERMINEDELLIPTICPROBLEMSINNONTRIVIALCONTRACTIBLEDOMAINSOFTHESPHEREDAVIDRUIZ,PIERALBERTOSICBALDI,ANDJINGWUAbstractInthispaper,weprovetheexistenceofnontrivialcontractibledomainsΩ⊂Sd,d≥2,suchthattheoverdeterminedellipticproblem−ε∆gu+u−up=0inΩ,u>0inΩ,u=0on∂Ω,∂νu=constanton∂Ω,admitsapositi...

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