Stable solutions of semilinear elliptic equations in unbounded domains
This paper establishes some properties for stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions in unbounded domains. A seminal result of Casten, Holland  and Matano  states that, in convex bounded domains, such solutions must be constant. This paper examines if this property extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain further satisfies , when . If the domain is a straight cylinder, a natural additional assumption is needed. We also derive some symmetry properties. Our results can be seen as an extension to more general domains of some results on the De Giorgi’s conjecture.
Semilinear elliptic equations StabilitySymmetryNeumann boundary conditionsDe Giorgi’s conjectureLiouville propertyConvex domainsUnbounded domainsGeneralized principal eigenvalue
AMS Class. No: 35B53, 35B06, 35B35, 35J15, 35J61, 35J60.
The author is deeply thankful to Professor Henri Berestycki for proposing the subject and for all the very instructive discussions.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi -ReactionDiffusion Equations, Propagation and Modelling held by Henri Berestycki.
1.1 General Framework
Consider the following semilinear elliptic equation, with homogeneous Neumann boundary conditions
where denotes the outer normal derivative, is a function and is a uniformly domain. In two seminal papers, Casten, Holland  and Matano  established the following: if the domain is bounded and convex, then
|any stable solution of () in is constant.||()|
In this work, we investigate if property () extends to unbounded convex domains. We also state some symmetry results, namely that stable solutions of () inherit symmetries from the domain’s invariances by translations or planar rotations.
Note that we only consider bounded solutions and that our results do not hold for unbounded solutions. For example is a nonconstant solution of in which is stable non-degenerate. Note also that we need no other assumptions on but smoothness. We choose to stick to this general context even if additional assumptions could lead to some stronger results. For example, Casten and Holland proved in  that if is convex or concave, then holds in any bounded domain, possibly not convex.
We point out that the question we address, in the case , is strongly related to the De Giorgi’s conjecture, to which an extensive literature is devoted (see [2, 3, 21, 25, 26] and references therein). This conjecture claims that any solution of the Allen-Cahn equation in which is monotonic in one variable must be planar. As monotonicity implies stability (see e.g Corollary 4.3 of ), the question of classifying stable solutions in is crucial in this context and has been considered by many authors. To that extent, our results in the particular case of the whole space are already contained in several papers, see for example [17, 19, 21]. See also  for radial solutions. Moreover, the method we use is inspired by this literature, in particular, Theorem 1.7 from , of which a refined version is presented in Lemma. This work can thus be seen as an extension to more general domains of some results on the De Giorgi’s conjecture.
The paper is organized as follows. We introduce the results in section 2 and give a brief discussion on the available counterexamples. In section 3, we recall the classical proof of the Casten, Holland and Matano result and prove Theorem. In section 4 we state some general properties of the generalized principal eigenvalue. The proofs of Theorem and Theorem are done in section 5 and the proof of Theorem in section 6. In the appendix, the reader can find more details on the generalized principal eigenvalue and on the notion of stability.
1.2 Definition of stability
We define stability through a linearization at equilibrium, as follows.
In the sequel, we often omit to mention the dependance on and simply write or . It is important to note that, if the domain is bounded, coincides with the classical principal eigenvalue of the linearized operator with Neumann boundary conditions. It formally corresponds to the lowest eigenvalue of the second variation of the energy associated with (), thus implies that the solution is a local minimum of the energy. The properties of in unbounded domains have been studied in a general context (see  and references therein). In the appendix, we give a brief discussion on the link between this definition of stability and the usual dynamical point of view.
2 The results
2.1 Convex domains
Now, let us focus on the classification of stable possibly degenerate solutions. We need a further assumption on the size of the domain at infinity, namely
This condition comes from Lemma, introduced below. Note that under this assumption, we allow for instance convex domains that are subdomains of , or of the form with bounded and , or of the form
where for all , with when ( can be empty for some ranges of values of ).
The case of a convex straight cylinder, namely with convex, turns out to be a specific case in our results. Note that it corresponds to domains for which convexity is degenerate in one direction. Indeed, some nonconstant stable solutions (consisting of planar waves) may exist in such domains. For example, the Allen-Cahn equation in
admits the explicit solution which is stable (degenerate). Note that, in this example, the nonlinearity is balanced, that is, . This leads up to the following assumption:
is the set of the stable zeros of . Formally, the sign of corresponds to that of the speed of a possible traveling wave connecting and . Thus, assumption (3) prevents the existence of a stationary wave (with speed ) connecting two stable states. Note that, if the domain is not a straight cylinder, planar waves do not exist anyway and this assumption is not needed.
Let be a convex domain which satisfies (1) and be a stable solution of ().
If is not a straight cylinder, then is constant.
If is a straight cylinder, then is either constant, or a planar monotonic stationary wave connecting two stable roots ,such that . As a consequence, if we further assume (3) then is constant.
2.2 Symmetry properties
The following result deals with straight cylinders, possibly not convex.
If is stable non-degenerate, then does not depend on .
If is stable degenerate and , then is monotonic with respect to .
If is stable degenerate and , the dependance of with respect to is only through a single scalar variable . Moreover, is monotonic with respect to .
We are now interested in cylinders which are invariant with respect to a planar rotation.
A domain is said to be -invariant if , where in some cylindrical coordinates .
When considering a -invariant domain , we further assume that the “radial section" is uniformly bounded:
Theorem and Theorem deal with domains which are invariant with respect to a translation or a planar rotation. More generally, one can consider domains which are invariant with respect to a vector field and ask whether stable solutions inherit the same symmetry, that is to say is zero. Based on the following observation, it is reasonable to think that the answer is negative in general. To fix ideas, let and let be a stable solution of (). Our method requires to satisfy the linearized equation
This essentially means that and commute, which implies and . Thus, the line integrals of are parallel lines or concentric circles, i.e the domain is invariant with respect to a translation or a planar rotation, which is already covered by Theorem and Theorem.
The latter case amounts to the existence of a counterexample to Lemma. This problem, which is related to the De Giorgi’s conjecture, turns out to be intricate. In , it is proved that Lemma does not hold in , . For this reason, it is reasonable to think that Theorem does not hold in , either. However, this seems to be an open question in low dimensions (See  for , or  for ). See section 4.2 for more details
Regarding counterexamples in bounded nonconvex domains, Matano constructs in  a “dumbbell domain” (consisting in two balls connected by a narrow passage) that admits a nonconstant stable solution, for a general class of bistable nonlinearities. Similar constructions can be found in [9, 7].
Theorem along with a blow-up argument leads to the following remarkable corollary. We further assume
Let be a smooth domain. We assume without loss that . For we define the dilated domain .
A complete proof of Corollary can be adapted from the proof of Theorem 5 in . We only give a sketch of the proof. By contradiction, assume there exists a sequence of nonconstant stable solutions of () in , for . From assumption (6), we infer a uniform bound for , then a uniform bound by elliptic estimates. Thus we can extract a subsequence that converges to some , which is a solution of () in the whole space . Using Corollary (below), we can prove that is stable. Now, from assumption (6) and the fact that is not constant, we can show that is not constant, which contradicts Theorem.
This result may be put in perspective with the aforementioned Matano’s counterexample. Corollary somehow states that such a counterexample could not be achieved in a domain with no “narrow passage”, regardless of its nonconvexity, at least for . In the same spirit, it is proved in  that considering a generalized traveling front, there is a complete invasion if is a cylinder containing a sufficiently wide straight cylinder whose section is star-shaped. This geometrical assumption somehow formulates the fact that the domain has no narrow passage.
3.1 The classical case of bounded convex domains
To give a grasp of the method and the difficulties arising when the domain is unbounded, we recall the proof of () for bounded convex domains.
Let be a convex bounded domain, a stable solution of () and set , for all .
From an integration by part we have
with from Definition. On the other hand, as is stable, we have and
Step 2. When the domain is convex, the above integral turns out to be nonpositive, as stated in the following key lemma. This is where the convexity of the domain comes into play. It can be found in [16, 23], but a simple proof is presented at the end of the section for completeness. Note that the lemma still holds when the domain is unbounded.
We then conclude that for all , we have thus minimizes . Note that, at this step, if we assume that is not constant, i.e for some , then we deduce , i.e is stable degenerate.
Note also that, if is unbounded, the computations are not licit and need to be adapted. This is done in section 3.2.
Step 3. Owing to the above conclusion, we deduce as a classical fact that for all , is a multiple of the principal eigenvalue associated to (7), which is denoted and is positive in .
Note that if is unbounded, this step also needs to be adapted. This is done in section 4.
Step 4. From on the closed surface , we deduce that vanishes on some point of the boundary. But as is colinear to , we conclude , which completes the proof.
Note that if is a straight cylinder, the above conclusion fails and may be a nonzero multiple of .
Before proving Lemma, we need the following definition.
Let . A “representation of the boundary” is a pair where is a function defined on a neighborhood of such that
where is the outer normal unit vector of at .
It is classical that such a representation of the boundary always exists for domains, see e.g section 6.2 of .
Proof (of Lemma)
Let us consider a representation of the boundary for . Equation (8) becomes
As is tangential to , we can differentiate the above equality with respect to the vector field . It gives, on ,
From this, we infer
Since is convex, for all , we have that is a nonnegative quadratic form in the tangent space of at . As is tangential to , we deduce from the above equation that is nonpositive.
In , the authors give the following remarkable geometrical interpretation of the above lemma. Consider a bounded convex domain . As satisfies Neumann boundary conditions, its level set cross the border orthogonally. Since the domain is convex, these level sets go apart one from each other as we move outward . As corresponds to the inverse of the distance of two level sets, it implies that decreases as we move outward , hence the result.
3.2 Non-degenerate stable solutions - proof of Theorem
The following proof is adapted from the first two steps of section 3.1. The method is inspired from . As is unbounded, the computations which lead to "" are not licit. We shall instead perform the computations on a truncated function . For , we set
for a smooth nonnegative function such that
Let be bounded and satisfy
If is stable non-degenerate, then .
Proof (of Lemma)
Now, let us show
If (13) holds, then , which contradicts the fact that is stable non-degenerate and thereby completes the proof. By contradiction, let us assume We have . Iterating, we find, for large enough
where positive constants are generically denoted . In addition, is bounded, hence . We have
Fixing large enough, we reach a contradiction as goes to . Thereby, we have proved (13) and the proof is complete.
We denote for . As is convex, Lemma implies
From a differentiation of (), we find that all the satisfy (7). Moreover, since is bounded, classical global Schauder estimates (see e.g Theorem 6.30 in ) ensure that all the are bounded. Then, Lemma implies in particular for all , i.e all the terms of the sum in (14) are nonnegative. As the sum is nonpositive, all the terms must be zero. From Lemma, we find for all , i.e is constant, which completes the proof.
4 Properties of
4.1 Existence of a positive eigenfunction
If the domain is bounded, it is classical that there exists a positive eigenfunction associated to . This property extends to unbounded domains, as stated in the following proposition.
This statement is adapted from the results of . However, presenting a full proof would be too technical and slightly off topic. See the appendix for more details.
As a corollary, we can prove that is equivalent to the existence of a positive supersolution. This result is essentially classical in the theory of linear elliptic equations. The proof is postponed to the appendix.
In the sequel, could be replaced by any satisfying (16) for our purposes.
4.2 A Liouville result, or the simplicity of
When the domain is bounded, it is classical that the only minimizers of are multiples of . The following lemma claims that, if the minimizer is a bounded function, the above conclusion extends to unbounded domains which satisfy (1). It is a refinement of Theorem 1.7 from .
Proof (of Lemma)
From (15) and , we obtain
Multiplying by (defined in (10)), integrating on and using the divergence theorem, we find
As on , the boundary term reads as , which is nonpositive from (12). Using the Cauchy-Schwarz inequality, we deduce
The cornerstone of the proof is that implies , where . The litterature refers to this property as the Liouville property. Originally introduced in , it has been extensively discussed (see [4, 6, 20, 21, 24]) and used to derive numerous results (e.g [10, 15, 17, 19]), in particular to prove the De Giorgi’s conjecture in low dimension (see [2, 3, 14, 21]). Lemma is a refinement of this property for domains with a boundary, instead of . This is why we need a boundary condition (12).
This is the only step where (1) is needed, it is thus a natural question to ask if this assumption can be relaxed. In the proof, (1) is used to derive (18), thus the choice of seems crucial. However, in , the authors consider the optimal by taking a solution of the minimization problem
That, in fact, does not allow to substantially relax condition (1). In , Barlow uses a probabilistic approach to establish that the aforementioned Liouville property (and consequently Lemma) does not hold in , . It is thus reasonable to think that condition (1) cannot be relaxed, yet this is an open question. We also cite , in which (1) is proved to be sharp, however we point out that, there, the condition is not satisfied.
Note also that, in this work, we only apply Lemma to functions which are derivatives of , which is a stronger condition than (11). In this context, not much is known about whether (1) could be relaxed. Indeed, up to the author’s knowledge, the only available counterexamples are for , , as a consequence of  in which the authors construct counterexamples to the De Giorgi’s conjecture for .
However, we can sometimes relax (1) under further assumptions, either on , , or . From a remark in , if , we can relax (1) to . Note also that, if , then (1) can be replaced by , or , see [8, 15]. In addition, we can show that Lemma holds for a large class of domains satisfying
More precisely, let be of the form
where , is bounded and
Then, to show (18) we use the cut-off
5 Proof of the symmetry properties
5.1 Proof of Theorem
We set for , which is bounded and satisfies (11). Since is straight in the directions , we have on , therefore satisfies (12). Thus, the first assertion follows from Lemma. Next, the second assertion follows from Lemma and the fact that .
To show the second assertion (we assume ), we follow an idea from [8, 21] and apply Lemma to , for being any unit vector of . We infer that there exists a constant such that . Since depends continuously on , it must vanish for some when moves on the sphere from direction to . Using a change of coordinates, we may assume . Hence and depends on one coordinate only. The monotonicty is deduced from the second assertion.
We set . Note that in a Cartesian system of coordinates
If is -invariant and is a smooth function satisfying
Proof (of Lemma)
Therefore , which completes the proof of Theorem.
6 Proof of Theorem
For all , for some constant .
Proof (of Lemma)
Differentiating (), we find that satisfies (11). Moreover, since is bounded, classical global Schauder’s estimates guarantee that all the are bounded.
On the other hand, since on , Lemma implies, in particular, that for all , i.e all the terms of the above sum are nonnegative. As the sum is nonpositive, all the terms must be zero.
Then, we apply Lemma to conclude.
6.1 First case: is not a straight cylinder
We have to prove that , for all . We proceed by induction: given , we show for all . If , the claim is trivial. Let us assume the claim for a fixed . We denote . As is not straight in any direction, there exists such that . Up to an isometric transformation of the orthonormal basis , we can assume without loss that . From and the induction assumption, we find . Since on , we have , which ends the proof.
6.2 Second case: is a straight cylinder
If is a straight cylinder, since it is convex and satisfies (1), it is either of the form , or , with a bounded domain. From the previous case and Theorem, we infer that depends on only one variable. Thus, we can assume without loss of generality that . Moreover, as , it is of constant sign, hence is monotic. Since is bounded, it has a limit when . Setting and using classical elliptic estimates, we can extract a subsequence that converges in to a stable solution of () (note that is invariant under translation in the direction). From , we deduce that must be a stable root of . Identically, has a constant limit as .
If , then is constant. Let us assume , and fix . Integrating (7) on gives
where is a positive constant. As (indeed, is integrable and is bounded), when goes to we obtain . The proof is thereby complete.
Note that if a stationary traveling wave exists, it is always a stable (degenerate) solution of (). This can be shown using Corollary with .
Appendix A Generalized principal eigenvalue
Given an elliptic operator and a smooth bounded domain along with some proper boundary conditions, the classical Krein-Rutman theory provides a minimal eigenvalue, refered as the principal eigenvalue. This notion has been extended to non smooth and non bounded domains, under Dirichlet boundary conditions, see [13, 11]. Considering smooth unbounded domains, the approach of  can be adapted to Neumann boundary conditions, by substituting the functional space with . Indeed, in  the authors define
and prove the existence of an associated positive eigenfunction, namely
Proposition (Theorem 3.1 and Proposition 1 in )
There exists , , which is positive on and satisfies
In fact, the quantities and (from Definition) are equal, as stated in the following lemma. Note however that the definition of relies on the fact that the operator is self-adjoint, whereas can be defined for more general operators.
In the definition of , it is equivalent to take the infimum on compactly supported smooth test functions, namely
where is the space of continuously differentiable functions with compact support in . As a consequence, .
Proof (of Corollary)
If is stable, the existence of is a direct consequence of Proposition.
Appendix B Stability
When considering stability from a dynamical point of vue, one can come up with the two following definitions.
where is the solution of (28).
The following proposition clarifies the hierarchy of the different definitions of stability.
Let be a solution of () and from Definition. The following implications hold
The first implication is trivial. Let us show the second implication by contradiction: assume and that is dynamically stable. From the dominated convergence theorem, there exists a bounded domain such that
Note that could be replace by a larger subdomain of , therefore we can assume without loss that and do not intersect tangentially. From technical but classical arguments (see, e.g. Theorem 3.1 in ) we know that there exists a positive function , such that
We choose the normalization