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Advances in Nonlinear Analysis

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Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature

Serena Dipierro
  • Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA-6009 Perth, Australia; and Dipartimento di Matematica “Federigo Enriques”, Università degli studi di Milano, Via Saldini 50, 20133 Milano, Italy
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/ Andrea Pinamonti
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  • Dipartime nto di Matematica, Università di Trento, Via Sommarive 14, 38123 Povo, Trento, Italy
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/ Enrico Valdinoci
  • Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA-6009 Perth, Australia; Dipartimento di Matematica “Federigo Enriques”, Università degli studi di Milano, Via Saldini 50, 20133 Milano, Italy; School of Mathematics and Statistics, University of Melbourne, Grattan Street, Parkville, VIC-3010 Melbourne, Australia; and Istituto di Matematica Applicata e Tecnologie Informatiche, Via Ferrata 1, 27100 Pavia, Italy
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Published Online: 2018-06-07 | DOI: https://doi.org/10.1515/anona-2018-0013

Abstract

We present a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun, and we provide a classification result of stable solutions of linear elliptic problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature. The result obtained here is a refinement of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.

Keywords: Riemannian manifolds; elliptic problems; Robin condition

MSC 2010: 58J05; 58J32; 53C24

1 Introduction

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., [14]). 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 g=(gij). We denote by Δ the Laplace–Beltrami operator induced by g. Let ΩM be a compact orientable domain and ν be the outer normal vector of Ω lying in the tangent space TpM for any pΩ. We assume that Ω 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:

{Δu+f(u)=0inΩ,νu+h(u)=0onΩ,(1.1)

where f,hC1() and νu:=g(u,ν). Similar problems have been investigated in [11, 2, 3, 25].

As usual, we consider the volume term induced by g, that is, in local coordinates,

dV=|g|dx1dxm,

where {dx1,,dxm} is the basis of 1-forms dual to the vector basis {1,,m}, and |g|=det(gij)0. We also denote by dσ the volume measure on Ω induced by the embedding ΩM.

As customary, we say that u is a weak solution to (1.1) if uC1(Ω¯) and

Ωu,φ𝑑V+Ωh(u)φ𝑑σ=Ωf(u)φ𝑑Vfor any φC1(Ω).

Moreover, we say that a weak solution u is stable if

Ω|φ|2dV+Ωh(u)φ2dσ-Ωf(u)φ2dV0for any φC1(Ω).(1.2)

In order to state our result, we recall below some classical notions in Riemannian geometry. Given a vector field X, we set

|X|=X,X.

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

(Hϕ)ij=ij2ϕ-Γijkkϕ,

where Γijk are the Christoffel symbols, namely,

Γijk=12ghk(ighj+jgih-hgij).

Given a tensor A, we define its norm by |A|=AA*, where A* is the adjoint.

The above quantities are related to the Ricci tensor Ric via the Bochner–Weitzenböck formula (see, for instance, [4] and references therein):

12Δ|ϕ|2=|Hϕ|2+Δϕ,ϕ+Ric(ϕ,ϕ).(1.3)

Finally, we let 𝕀 and H denote the second fundamental tensor and the mean curvature of the embedding ΩΩ in the direction of the outward unit normal vector field ν, respectively.

We are now in position to state our main result.

Theorem 1.1.

Let uC3(Ω¯) be a stable solution to (1.1). Assume that the Ricci curvature is nonnegative in Ω, and that

  • (C1)

    for any pΩ , the quadratic form 𝕀-h(u)g~ on the tangent space Tp(Ω) is nonpositive definite.

If

Ω(h(u)f(u)+(m-1)(h(u))2H+h(u)(h(u))2)𝑑σ0,(1.4)

then u is constant in Ω.

Remark 1.2.

Theorem 1.1 has been proved in [2, Theorem 4.5] in the particular case where h(t):=αt for some α. 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.

Theorem 1.3.

Let u be a stable weak solution to (1.1). Then

Ω(Ric(u,u)+|Hu|2-||u||2)φ2dV-Ω(12|u|2,ν+h(u)|u|2)φ2dσΩ|u|2|φ|2dV(1.5)

for any φC(Ω).

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 L2-norm of any φC1(Ω) plus a boundary term by a weighted L2-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, [7] 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).

For related results in the framework of Markov Triples, see [12]. The next two sections are devoted to the proofs of Theorems 1.3 and 1.1, respectively.

2 Proof of Theorem 1.3

Applying (1.2) with φ replaced by |gu|φ, we get

Ωf(u)|u|2ϕ2𝑑VΩ(||u||2ϕ2+|u|2|ϕ|2+2ϕ|u|ϕ,|u|)𝑑V+Ωh(u)|u|2φ2𝑑σ=Ω(||u||2ϕ2+|u|2|ϕ|2+12ϕ2,|u|2)𝑑V+Ωh(u)|u|2φ2𝑑σ.

Therefore, integrating by parts, the third term in the last line, we get

Ωf(u)|u|2ϕ2𝑑VΩ(||u||2ϕ2+|u|2|ϕ|2-12ϕ2Δ|u|2)𝑑V+12Ωφ2|u|2,ν𝑑σ+Ωh(u)|u|2φ2𝑑σ.

Hence, recalling (1.3),

Ωf(u)|u|2ϕ2𝑑VΩ[||u||2ϕ2+|u|2|ϕ|2-(|Hu|2+Δu,u+Ric(u,u))φ2]𝑑V+12Ωφ2|u|2,ν𝑑σ+Ωh(u)|u|2φ2𝑑σ.(2.1)

Now, by differentiating the equation in (1.1), we see that

-Δu=f(u)u.

Plugging this information into (2.1), we conclude that

0Ω[||u||2ϕ2+|u|2|ϕ|2-(|Hu|2+Ric(u,u))ϕ2]𝑑V+12Ωφ2|u|2,ν𝑑σ+Ωh(u)|u|2φ2𝑑σ,

which completes the proof of Theorem 1.3.∎

3 Proof of Theorem 1.1

In this section we provide the proof of Theorem 1.1. We first state the following result that proves [2, Theorem 3.4] in the more general case in which h is any C1 function.

Theorem 3.1.

Let wC3(Ω¯) satisfy

νw+h(w)=0on Ω,(3.1)

for some hC1(R). Then

12ν|w|2=𝕀(~w,~w)-h(w)|~w|2-h(w)Hw(ν,ν)on Ω,

where ~w:=w-g(w,ν)ν is the tangential gradient with respect to Ω, and Hw is the Hessian matrix of the function w.

Proof.

We let {ei}, with i{1,,m}, be a Darboux frame along Ω, that is, such that em:=ν. In this setting, conditions (3.1) reads

wm=-h(w)on Ω.(3.2)

Also, for any i, j{1,,m-1}, we define

Hij:=g(𝕀(ei,ej),ν).

Then, reasoning as in the proof of [2, formula (3.32)], we obtain that, for any j{1,,m-1},

wjm=i=1m-1Hijwi-h(w)wjon Ω.

Therefore, multiplying both terms by wj, we get

wjmwj=i=1m-1Hijwiwj-h(w)wj2on Ω.(3.3)

On the other hand, for any i{1,,m},

12(|w|2)i=j=1mwjwji=i=1m-1wjwji+wmwmi=i=1m-1wjwji-h(w)wmi,

where we used (3.2) in the last passage.

As a consequence,

12ν|w|2=j=1m-1wjwjm-h(w)wmmon Ω.

From this and (3.3), we thus obtain

12ν|w|2=i,j=1m-1Hijwiwj-h(w)j=1m-1wj2-h(w)wmmon Ω,

which implies the desired result. ∎

Now we recall that Δ is the Laplace–Beltrami operator of the manifold (M,g), and we let Δ~ be the Laplace–Beltrami operator of the manifold Ω endowed with the induced metric by the embedding ΩM. We have that

Δw=Δ~w-(m-1)Hwν+Hw(ν,ν).(3.4)

With this, we can prove the following result.

Lemma 3.2.

Let uC3(Ω¯) be a stable solution of (1.1). Then

Ω(Ric(u,u)+|Hu|2-||u||2)φ2𝑑V-Ω(𝕀(~u,~u)-h(u)|~u|2+h(u)f(u)+(m-1)(h(u))2H+h(u)(h(u))2)φ2𝑑σΩ|u|2|φ|2dV-Ωh(u)~u,~φ2dσ(3.5)

for any φC(Ω).

Proof.

From Theorems 1.3 and 3.1, for every stable weak solution u to (1.1) and for any φC(Ω), we have that

Ω(Ric(u,u)+|Hu|2-||u||2)φ2𝑑V-Ω(𝕀(~u,~u)-h(u)|~u|2-h(u)Hu(ν,ν)+h(u)|u|2)φ2𝑑σΩ|u|2|φ|2dV.(3.6)

Now we use (3.4) to manipulate the integral on the boundary of Ω. In this way, we obtain from (3.6) that

Ω(Ric(u,u)+|Hu|2-||u||2)φ2𝑑V-Ω[𝕀(~u,~u)-h(u)|~u|2-h(u)(Δu-Δ~u+(m-1)Huν)+h(u)|u|2]φ2𝑑σΩ|u|2|φ|2dV.

Thus, recalling1 (1.1), we conclude that

Ω(Ric(u,u)+|Hu|2-||u||2)φ2𝑑V-Ω(𝕀(~u,~u)-h(u)|~u|2+h(u)f(u)+h(u)Δ~u+(m-1)(h(u))2H+h(u)|u|2)φ2𝑑σΩ|u|2|φ|2dV.(3.7)

Now we observe that

|u|2=|~u|2+|uν|2=|~u|2+(h(u))2on Ω.

Plugging this information into (3.7), we obtain that

Ω(Ric(u,u)+|Hu|2-||u||2)φ2𝑑V-Ω(𝕀(~u,~u)+h(u)f(u)+h(u)Δ~u+(m-1)(h(u))2H+h(u)(h(u))2)φ2𝑑σΩ|u|2|φ|2dV.

Now we notice that

Ωh(u)φ2Δ~u𝑑σ=-Ωh(u)|~u|2φ2𝑑σ-Ωh(u)~u,~φ2𝑑σ,

and therefore

Ω(Ric(u,u)+|Hu|2-||u||2)φ2𝑑V-Ω(𝕀(~u,~u)-h(u)|~u|2+h(u)f(u)+(m-1)(h(u))2H+h(u)(h(u))2)φ2𝑑σΩ|u|2|φ|2dV-Ωh(u)~u,~φ2dσ,

which proves the desired inequality. ∎

Before completing the proof of Theorem 1.1 we recall the following lemmata proved in [21, Lemma 5] and [21, Lemma 9], respectively.

Lemma 3.3.

For any smooth function ϕ:MR, we have that

|Hϕ|2||ϕ||2almost everywhere.

Lemma 3.4.

Suppose that the Ricci curvature of M is nonnegative and that Ric does not vanish identically. Let u be a solution of (1.1), with

Ric(u,u)(p)=0for any pM.

Then u is constant.

With this, we are able to finish the proof of Theorem 1.1.

Proof of Theorem 1.1.

Taking φ1 in (3.5), we see that

Ω(Ric(u,u)+|Hu|2-||u||2)𝑑VΩ(𝕀(~u,~u)-h(u)|~u|2+h(u)f(u)+(m-1)(h(u))2H+h(u)(h(u))2)𝑑σ.

Hence, using (C1) and (1.4), we obtain that

Ω(Ric(u,u)+|Hu|2-||u||2)𝑑V0.(3.8)

On the other hand, by Lemma 3.3,

|Hu|2-||u||20on Ω,

and so (3.8) gives

ΩRic(u,u)𝑑V0,

which implies that

Ric(u,u)=0in Ω.

Thus, the conclusion follows from Lemma 3.4. ∎

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Footnotes

  • 1

    Notice that, since u is regular enough, the equation holds true up to the boundary of Ω. 

About the article

Received: 2017-10-30

Accepted: 2018-01-16

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.

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