The study of partial differential equations on manifolds has a long tradition in analysis and geometry, see, e.g., [28, 32, 33, 26, 1]. The interest for such topic may come from different perspectives. On the one hand, at a local level, classical equations with variable coefficients can be efficiently comprised into the manifold setting, allowing more general and elegant treatments. In addition, at a global level, the geometry of the manifold can produce new interesting phenomena and interplay with the structure of the solutions, thus creating a novel scenario for the problems into consideration.
Of course, given the complexity of the topic, the different solutions of a given partial differential equation on a manifold can give rise to a rather wild “zoology” and it is important to try to group the solutions into suitable “classes” and possibly to classify all the solutions belonging to a class.
In this spirit, very natural classes of solutions in a variational setting arise from energy considerations. The simplest class in this framework is probably that of “minimal solutions”, namely the class of solutions which minimize (or, more generally, local minimize) the energy functional.
On the other hand, it is often useful to look at a more general class than minimal solutions, that is, the class of solutions at which the second derivative of the energy functional is nonnegative. These solutions are called “stable” (see, e.g., ). Of course, the class of stable solutions contains that of minimal solutions, but the notion of stability is often in concrete situations more treatable than that of minimality. For instance, it is typically very difficult to establish whether or not a given solution is minimal, since one, in principle, should compare its energy with that of all the possible competitors, while a stability check could be more manageable, relying only on a single, and sometimes sufficiently explicit, second derivative bound.
The goal of this paper is to study the case of a linear elliptic equation on a domain of a Riemannian manifold with nonnegative Ricci curvature, endowed with nonlinear boundary data. We will consider stable solutions in this setting and provide sufficient conditions to ensure that they are necessarily constant.
The framework in which we work is the following. Let M be a connected m-dimensional Riemannian manifold endowed with a smooth Riemannian metric . We denote by Δ the Laplace–Beltrami operator induced by g. Let be a compact orientable domain and ν be the outer normal vector of lying in the tangent space for any . We assume that is orientable for the outer normal to be well defined and continuous.
In this paper we study the solutions to the following boundary value problem:
As usual, we consider the volume term induced by g, that is, in local coordinates,
where is the basis of 1-forms dual to the vector basis , and . We also denote by the volume measure on induced by the embedding .
As customary, we say that u is a weak solution to (1.1) if and
Moreover, we say that a weak solution u is stable if
In order to state our result, we recall below some classical notions in Riemannian geometry. Given a vector field X, we set
Also (see, for instance, [26, Definition 3.3.5]), it is customary to define the Hessian of a smooth function ϕ as the symmetric 2-tensor given in a local patch by
where are the Christoffel symbols, namely,
Given a tensor A, we define its norm by , where is the adjoint.
The above quantities are related to the Ricci tensor via the Bochner–Weitzenböck formula (see, for instance,  and references therein):
Finally, we let and H denote the second fundamental tensor and the mean curvature of the embedding in the direction of the outward unit normal vector field ν, respectively.
We are now in position to state our main result.
Let be a stable solution to (1.1). Assume that the Ricci curvature is nonnegative in Ω, and that
for any , the quadratic form on the tangent space is nonpositive definite.
then u is constant in Ω.
Theorem 1.1 has been proved in [2, Theorem 4.5] in the particular case where for some . We point out that with this particular choice of h, Theorem 1.1 here weakens the assumptions on the sign of α of [2, Theorem 4.5].
The proof of Theorem 1.1 is based on a geometric Poincaré-type inequality, which we state in this setting as follows.
Let u be a stable weak solution to (1.1). Then
for any .
We notice that formula (1.5) relates the stability condition of the solution with the principal curvatures and the tangential gradient of the corresponding level set. Since this formula bounds a weighted -norm of any plus a boundary term by a weighted -norm of its gradient, we may consider this formula as a weighted Poincaré type inequality.
The idea of using weighted Poincaré inequalities to deduce quantitative and qualitative information on the solutions of a partial differential equation has been originally introduced by Sternberg and Zumbrun in [30, 31] in the context of the Allen–Cahn equation, and it has been extensively exploited to prove symmetry and rigidity results, see, e.g., [15, 18, 19]. See also [16, 21, 20, 23, 24, 29] for applications to Riemannian and sub-Riemannian manifolds,  for problems involving the Ornstein–Uhlenbeck operator, [6, 17] for semilinear equations with unbounded drift and [22, 8, 9, 10] for systems of equations.
Recently, in [13, 11], the cases of Neumann conditions for boundary reaction-diffusion equations and of Robin conditions for linear and quasilinear equations have been studied, using a Poincaré inequality that involves also suitable boundary terms.
We point out that Theorem 1.1 comprises the classical case of the Laplacian in the Euclidean space with homogeneous Neumann data, which was studied in the celebrated papers [5, 27]. In this spirit, our Theorem 1.1 can be seen as a nonlinear version of the results of [5, 27] on Riemannian manifolds (and, with respect to [5, 27], we perform a technically different proof, based on Theorem 1.3).
2 Proof of Theorem 1.3
Applying (1.2) with φ replaced by , we get
Therefore, integrating by parts, the third term in the last line, we get
Hence, recalling (1.3),
Now, by differentiating the equation in (1.1), we see that
Plugging this information into (2.1), we conclude that
which completes the proof of Theorem 1.3.∎
3 Proof of Theorem 1.1
for some . Then
where is the tangential gradient with respect to , and is the Hessian matrix of the function w.
We let , with , be a Darboux frame along , that is, such that . In this setting, conditions (3.1) reads
Also, for any i, , we define
Then, reasoning as in the proof of [2, formula (3.32)], we obtain that, for any ,
Therefore, multiplying both terms by , we get
On the other hand, for any ,
where we used (3.2) in the last passage.
As a consequence,
From this and (3.3), we thus obtain
which implies the desired result. ∎
Now we recall that Δ is the Laplace–Beltrami operator of the manifold , and we let be the Laplace–Beltrami operator of the manifold endowed with the induced metric by the embedding . We have that
With this, we can prove the following result.
Let be a stable solution of (1.1). Then
for any .
Now we observe that
Plugging this information into (3.7), we obtain that
Now we notice that
which proves the desired inequality. ∎
For any smooth function , we have that
Suppose that the Ricci curvature of M is nonnegative and that does not vanish identically. Let u be a solution of (1.1), with
Then u is constant.
With this, we are able to finish the proof of Theorem 1.1.
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About the article
Published Online: 2018-06-07
Funding Source: Australian Research Council
Award identifier / Grant number: DP170104880
Part of this work was done while Andrea Pinamonti was visiting the Dipartimento di Matematica “Federigo Enriques” of the University of Milan. The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Supported by the Australian Research Council Discovery Project Grant “N.E.W. Nonlocal Equations at Work”.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1035–1042, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0013.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0