Maximum principles for elliptic operators in unbounded Riemannian domains

Abstract

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

1 Introduction

In this work we address the validity of the maximum principle for bounded solutions to the problem

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta u\ge c u &{} \text {in}\ \Omega \\ u \le 0 &{} \text {on}\ \partial \Omega \end{array} \right. \end{aligned}$$

where $\Omega$ is an unbounded domain (i.e. an open and connected subset) inside the Riemannian manifold (M, g). We shall present two kinds of results where the common root is the assumption that $\Omega$ is “small” from the viewpoint of the operator. The first result requires that the underlying manifold has the special structure of a warped product cylinder and the smallness of the domain is encoded in its (Dirichlet) parabolicity. The second result has a more abstract flavour as it holds in any Riemannian manifold provided that the domain is small in a spectral sense.

In the Euclidean setting a classical Maximum Principle for unbounded domains contained in the complement of a cone states as follows (for a reference, see [3])

Theorem 1.1

Consider a possibly unbounded domain $\Omega \subset \mathbb {R}^n$, $n\ge 2$, whose closure is contained in the complement of a non-degenerate solid cone $\mathcal {C}\subset \mathbb {R}^n$.

If $u\in C^0(\overline{\Omega })\cap W^{1,2}_{loc}(\Omega )$ is a distributional solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + c\ u \le 0 &{} in\ \Omega \\ u\le 0 &{} on\ \partial \Omega \\ \sup _\Omega u <+\infty , \end{array} \right. \end{aligned}$$

where $0\le c\in C^0(\Omega )$, then $u\le 0$ in $\Omega$.

The proof is essentially based on the fact that the Euclidean space is a model manifold, that is, the manifold obtained by quotienting the warped product $([0,+\infty )\times \mathbb {S}^{n-1}, \text {d}r\otimes \text {d}r + r^2\,g^{\mathbb {S}^{n-1}})$ with respect to the relation that identifies $\{0\}\times \mathbb {S}^{n-1}$ with a point o, called pole, and then extending smoothly the metric in o.

Influenced by the model structure of $\mathbb {R}^n$, in Sect. 2 we obtain a transposition of the previous theorem to warped product manifolds satisfying certain (radial) curvature conditions and replacing the notion of cone with the notion of strip. The assumptions on the geometry of M and on $\Omega$ are needed to construct a suitable barrier function, crucial for the validity of the result. We stress that the main theorem of Sect. 2 will be first stated in the context of (Dirichlet-)parabolic manifolds and then reinterpreted in the language of maximum principles as follows.

Theorem A

(Unbounded maximum principle) Let $M=\mathbb {R}_{\ge 0}\times _\sigma N$ be a warped product manifold of dimension $\text {dim}(M)=m\ge 2$, where $\sigma :\mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{>0}$ is a smooth function and N is a closed manifold. Consider $\Omega \subset M$ an unbounded domain whose closure is contained in the strip $\mathbb {R}_{\ge 0}\times \Lambda$, where $\Lambda \subset N$ is a non-empty, smooth and connected open subset of N such that $\overline{\Lambda }\ne N$, and denote by $\lambda _1$ the first Dirichlet eigenvalue of the Laplace-Beltrami operator of $\Lambda$. Moreover, suppose that either one of the following conditions is eventually satisfies as $r\rightarrow +\infty$

• 1. $\sigma \sigma '\le \frac{\lambda _1}{(m-1)}r$;

• 2. $\sigma ''\le 0$.

If $u\in C^0(\overline{\Omega })\cap W^{1,2}_{loc}(\Omega )$ is a bounded above distributional solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + c\ u \le 0 &{} in\ \Omega \\ u\le 0 &{} on\ \partial \Omega , \end{array} \right. \ \\ \end{aligned}$$

where $0\le c\in C^0(\Omega )$, then $u\le 0$ in $\Omega$.

On the other hand, if we want to recover a maximum principle without requiring any assumption on the structure of the manifold (and of the domain), then we have to consider some additional hypotheses on the differential operator and on its spectrum. These kinds of assumptions are natural if one compares with the compact case.

Theorem 1.2

Let (M, g) a Riemannian manifold, $\Omega \subseteq M$ a bounded domain and $\mathcal {L}$ a linear elliptic operator with (sufficiently) regular coefficients. Then, the Maximum Principle holds for $\mathcal {L}$ in $\Omega$ with Dirichlet boundary conditions if and only if the first Dirichlet eigenvalue $\lambda _1^{-\mathcal {L}}(\Omega )$ of $\mathcal {L}$ on $\Omega$ is positive.

Inspired by this fact, one might wonder if this property can be generalized to unbounded domains. This is true in the Euclidean space according to the very interesting work [13] by Samuel Nordmann. In Sect. 3 we shall extend Nordmann result to Riemannian domains, dealing with operators of the form

$$\begin{aligned} \mathcal {L} u(x):=\text {div}\left( A(x)\cdot \nabla u(x) \right) +g(B(x),\nabla u(x))+ c(x) u(x), \end{aligned}$$

(1)

where $A\in \text {End}(TM)$ is a positive definite, smooth and symmetric endomorphism of the tangent bundle so that there exist two positive constant $c_0$ and $C_0$ satisfying

$$\begin{aligned} c_0\ g(\xi ,\xi ) \le g(A(x)\cdot \xi ,\xi )\le C_0\ g(\xi ,\xi ) \ \ \ \ \ \ \ \ \forall x\in M,\quad \forall \xi \in T_x M, \end{aligned}$$

(2)

$B\in C^\infty (M;TM)$ is a smooth vector field satisfying

$$\begin{aligned} B=A\cdot \nabla \eta \end{aligned}$$

(3)

for a smooth function $\eta :\Omega \rightarrow \mathbb {R}$ and $c\in C^0(M)$ is a continuous function. Moreover, we assume that the endomorphism A is bounded in the $C^1$-norm by a constant $a\in \mathbb {R}_{>0}$

$$\begin{aligned} \left| \left| A \right| \right| _{C^0(M,TM)}+\left| \left| dA \right| \right| _{C^0(TM,TM)}\le a. \end{aligned}$$

(4)

Denoting by $\lambda _1^{-\mathcal {L}}(\Omega )$ the bottom of the spectrum of the operator $\mathcal {L}$ acting on $\Omega$, the second maximum principle we present states as follows.

Theorem B

(Unbounded maximum principle) Let (M, g) be a complete Riemannian manifold, $\Omega \subset M$ a (possibly unbounded) smooth domain and $\mathcal {L}$ an operator of the form (1) and satisfying (2)–(4). Moreover, assume that $\lambda _1^{-\mathcal {L}}(\Omega )>0$.

If $u\in C^{2}(\overline{\Omega })$ is a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -\mathcal {L}u \le 0 &{} in\ \Omega \\ u \le 0 &{} on\ \partial \Omega \end{array} \right. \end{aligned}$$

and there exists $m\in \mathbb {N}$ so that

$$\begin{aligned} \int _{\Omega \cap B_R(x_0)} e^{\eta } (u^+)^2 \ \text {dv}= O(R^m)\quad as\ R\rightarrow +\infty \end{aligned}$$

(5)

for a fixed (and hence any) $x_0\in M$, then $u\le 0$ in $\Omega$.

To this end, we first obtain an ABP-like inequality for the differential operator $\mathcal {L}$ acting on bounded smooth domains. Next, we will use it to construct a couple of generalized eigenelements $(\lambda _1, \varphi )$ for $\mathcal {L}$ on possibly nonsmooth bounded domains and, using an exhaustion argument, on unbounded smooth domains. Following the proof obtained by Nordmann, in Theorem B we get a maximum principle for the operator $\mathcal {L}$ acting on an unbounded smooth domain $\Omega$ of a general Riemannian manifold (M, g) under the assumption that $\lambda _1^{-\mathcal {L}}(\Omega )>0$.

Remark 1.3

It is worth highlighting how the integral condition (5) intertwines both the volume growth of the domain $\Omega$ and the asymptotic behaviours of the functions $\eta$ and u. In particular, if the volume of $\Omega \cap B_R$ grows at most polynomially in R and the function $\eta$ is bounded from above, then any subsolution u whose positive part has a $L^2$-norm that grows at most polynomially on $\Omega \cap B_R$ (for instance, if u is bounded from above) satisfies condition (5), recovering also the Euclidean result obtained in [13]. On the opposite, if the volumes grow more than any polynomial, then one can still obtain a maximum principle on the family of subsolutions whose positive part decays fast enough at infinity.

In the last section we will apply Theorem B to generalize some of the results obtained in [6] by the author together with Stefano Pigola.

2 Maximum principle for unbounded domains in the complement of a strip

The already cited Theorem 1.1 is a milestone in the Euclidean analysis of PDEs. A possible proof makes use of the next classical lemma (see [3, Lemma 2.1]), which is based on the existence of a suitable positive $(-\Delta +c)$-subharmonic function. We state this result in a more general setting.

Lemma 2.1

Let (M, g) be a complete manifold. Given a (possibly unbounded) domain $\Omega \subset M$, suppose $u\in W^{1,2}_{_{loc}}(\Omega )\cap C^0(\overline{\Omega })$ is a distributional solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+c\ u\le 0 &{} in\ \Omega \\ u\le 0 &{} on\ \partial \Omega \\ \sup _\Omega u<+\infty , \end{array} \right. \end{aligned}$$

where $0\le c\in C^0(\Omega )$. If there exists a function $\phi \in C^2(\Omega )\cap C^0 (\overline{\Omega })$ (possibly depending on u) satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta \phi + c\ \phi \ge 0 &{} in\ \Omega \\ \phi >0 &{} in\ \overline{\Omega } \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \underset{p\in \Omega }{\limsup _{d^M (p,p_0)\rightarrow +\infty ,}} \ \frac{u(p)}{\phi (p)}\le 0 \end{aligned}$$

for any fixed $p_0\in \Omega$ (where $d^M$ is the intrinsic distance on M), then $u\le 0$ in $\Omega$.

Proof

Let $w:=\frac{u}{\phi }\in W^{1,2}_{loc}(\Omega )\cap C^0(\overline{\Omega })$. We have

$$\begin{aligned} \Delta w + 2 g\left( \nabla w, \frac{\nabla \phi }{\phi } \right) + w \frac{\Delta \phi }{\phi }= \frac{\Delta u}{\phi }\ge c\ \frac{u}{\phi }=c\ w \ \ \ \ \ \ \ \ \text {in}\ \mathcal {D}' \end{aligned}$$

i.e.

$$\begin{aligned} \mathcal {L}w:= -\Delta w-2g\left( \nabla w, \frac{\nabla \phi }{\phi } \right) + w \frac{-\Delta \phi + c\ \phi }{\phi }\le 0 \ \ \ \ \ \ \ \ \text {in}\ \mathcal {D}'. \end{aligned}$$

By assumption, for any $\epsilon >0$ and any fixed $p_0\in M$ there exists $0<R_\epsilon \xrightarrow {\epsilon \rightarrow 0} \infty$ so that $w(p)\le \epsilon$ for every $p\in \Omega$ satisfying $d^M (p,p_0)\ge R_\epsilon$. Hence, for $\Omega _\epsilon :=B^{M}_{R_\epsilon }(p_0)\cap \Omega$ we get

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}w\le 0 &{} \text {in any connected component of}\ \Omega _\epsilon \\ w\le \epsilon &{} \text {on the boundary of any connected component of}\ \Omega _\epsilon . \end{array} \right. \end{aligned}$$

Since $\frac{-\Delta \phi + c \phi }{\phi }\ge 0$, by the standard maximum principle $w\le \epsilon$ in any connected component of $\Omega _\epsilon$. Letting $\epsilon \rightarrow 0$ we get $w\le 0$ in $\Omega$, i.e. $u\le 0$ in $\Omega$. $\square$

As said above, the previous lemma is the key ingredient to obtain the unbounded maximum principle contained in Theorem 1.1. Indeed, for any bounded above subsolution u we only have to find a barrier function $\phi$ satisfying the assumptions of Lemma 2.1. Observe that, since in Theorem 1.1u is assumed to be bounded above, the dependence of $\phi$ on u may be bypassed just requiring that $\phi \xrightarrow []{|x|\rightarrow +\infty }+\infty$.

It is precisely the presence of the cone $\mathcal {C}$ in the complement of $\Omega$ that allows us to easily construct $\phi$.

Proof of Theorem 1.1

Consider the spherical coordinates $(r,\theta )$ on $\mathbb {R}^n$ and set $\Lambda =\mathbb {S}^{n-1} \setminus \mathcal {C}$. We define $\phi$ as the restriction to $\Omega$ of the function $\phi :(0,+\infty )\times \Lambda \rightarrow \mathbb {R}_{\ge 0}$ given by

$$\begin{aligned} \phi (r,\theta )=\left\{ \begin{array}{ll} \ln (r)+C_0 &{} \text {if}\ n=2\\ r^\alpha \psi (\theta ) &{} \text {if}\ n\ge 3, \end{array} \right. \end{aligned}$$

where $\psi$ is the first Dirichlet eigenfunction of $\Delta ^{\mathbb {S}^{n-1}}\Big |_{\Lambda }$ with associated first eigenvalue $\lambda _1>0$ and $\alpha \in \mathbb {R}$ satisfies the identity

$$\begin{aligned} \alpha (\alpha +n-2)-\lambda _1 =0. \end{aligned}$$

By the nodal domain theorem, it follows that $\phi >0$ in $\Omega$ and thus $(-\Delta +c)\phi \ge 0$. Moreover, by construction, $\phi$ diverges as $|x|\rightarrow +\infty$. By Lemma 2.1, the claim follows. $\square$

Using a different point of view, we can interpret Theorem 1.1 in terms of a the Dirichlet-parabolicity of the domain $\Omega$.

Definition 2.2

Given a Riemannian manifold (M, g) without boundary, we say that a domain $\Omega \subseteq M$ is Dirichlet parabolic ($\mathcal {D}$-parabolic) if the unique bounded solution $u\in C^0(\overline{\Omega })\cap C^\infty (\Omega )$ to the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=0 &{} in\ \Omega \\ u=0 &{} on\ \partial \Omega \end{array} \right. \end{aligned}$$

is the constant null function.

Remark 2.3

Note that in the definition of the $\mathcal {D}\text {-parabolic}$ity the boundary of the manifold (domain) at hand does not necessarily have to be smooth. For an interesting work about Dirichlet parabolicity, containing a detailed overview about the topic, we suggest [15].

As an application of what we have done so far, we get that any domain $\Omega \subset \mathbb {R}^n$ contained in the complement of a cone is $\mathcal {D}\text {-parabolic}$.

Corollary 2.4

If $\Omega \subset \mathbb {R}^n$, $n\ge 2$, is a (possibly unbounded) domain whose closure is contained in the complement of a non-degenerate solid cone $\mathcal {C}\subset \mathbb {R}^n$, then $\Omega$ is $\mathcal {D}$-parabolic.

Proof

Fixed any bounded function $u\in C^0(\overline{\Omega })\cap C^\infty (\Omega )$ satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=0 &{} \text {in}\ \Omega \\ u=0 &{} \text {on}\ \partial \Omega , \end{array} \right. \end{aligned}$$

by Theorem 1.1 we get $u\le 0$. Applying the same argument to $v=-u$, it also follows that $u\ge 0$, obtaining $u\equiv 0$. $\square$

2.1 From Euclidean space to warped products

Clearly, the previous construction is strongly based on the fact that the Euclidean space is a model manifold. Using this viewpoint, a natural question could be the following

$$\begin{aligned} \begin{array}{c} Can\ we\ retrace\ what\ we\ have\ done\ so\ far\ to\ obtain\ a\ suitable\ barrier\ \phi \\ on\ any\ warped\ product\ manifold\ M=I\times _\sigma N? \end{array} \end{aligned}$$

Remark 2.5

When we consider $\mathbb {R}^n$ as a warped product manifold, the cone $\mathcal {C}$ (whose vertex coincides with the pole o) can be seen as a strip that extends along the “radial” direction.

If we want to retrace the same construction step by step, we need the existence (and the positiveness) of the first eigenfunction $\phi$ of $\Delta ^N\big |_{\Lambda }$. This surely follows if the manifold N is compact. Whence, assuming that $\phi$ takes the form $\phi (r,\xi )=h(r) \psi (\xi )$ with $\psi$ the nonnegative first Dirichlet eigenfunction on a fixed subdomain $\Lambda \subset N$, by the structure of the Laplace-Belatrami operator acting on warped product manifolds, the inequality $(-\Delta +c)\phi \ge 0$ reduces to

$$\begin{aligned} \partial _r^2 h + (n-1) \frac{\sigma '}{\sigma } \partial _r h - \left( \frac{\lambda _1}{\sigma ^2}+c\right) h \le 0 \end{aligned}$$

(6)

and, in general, it is not easy to prove the existence of a positive solution to (6) that satisfies the asymptotic condition $h\xrightarrow []{r\rightarrow +\infty }+\infty$. This means that we are able to generalize Theorem 1.1 only requiring strong assumptions on the manifold at hand.

2.2 $\mathcal {D}$-Parabolicity and maximum principle for unbounded domains of warped product manifolds with compact leaves

Let $M=\mathbb {R}_{\ge 0}\times _\sigma N$ be a warped product manifold, with $\sigma :\mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{>0}$ a positive smooth function and N a closed manifold. Observe that, up to double M, we can equivalently assume $I=\mathbb {R}$ (and thus that the manifold is complete). In what follows, we consider $\Omega$ an unbounded domain whose closure is contained in the strip $(0,+\infty ) \times \Lambda$, where $\Lambda \subset N$ is a non-empty, connected open subset of N (with smooth boundary $\partial \Lambda$) such that $\overline{\Lambda }\ne N$.

While at the beginning of this section we explained how to prove $\mathcal {D}$-parabolicity using Lemma 2.1, for more general warped product manifolds we will apply the following Dirichlet–Khas’minskii test (see [15, Lemma 14]) to subdomains of the ambient manifold.

Lemma 2.6

($\mathcal {D}$-Khas’minskii test) Given a Riemannian manifold (M, g) with boundary $\partial M\ne \emptyset$, if there exists a compact set $K\subset M$ and a function $0\le \phi \in C^0(M{\setminus } \text {int}\ K)\cap W^{1,2}_{loc}(\text {int}\ M\ {\setminus } K)$ such that $\phi (x)\rightarrow \infty$ as $d^M(x,x_0)\rightarrow \infty$ for some (any) $x_0\in M$, and

$$\begin{aligned}&-\int _{\text {int}\ M\ \setminus K} g(\nabla \phi , \nabla \rho ) \le 0 \\&\quad \forall 0\le \rho \in C^0(M\setminus \text {int}\ K) \cap W^{1,2}_{loc}(\text {int}\ M\ \setminus K), \end{aligned}$$

then M is $\mathcal {D}\text {-parabolic}$.

As an application of the above $\mathcal {D}$-Khas’minskii test, we manage to prove the following theorem, that is the main result of this section.

Theorem 2.7

Let $M=\mathbb {R}_{\ge 0}\times _\sigma N$ be a warped product manifold of dimension $\text {dim}(M)=m\ge 2$, where $\sigma :\mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{>0}$ is a smooth function and N is a closed manifold. Consider $\Omega \subset M$ an unbounded domain whose closure is contained in the strip $\mathbb {R}_{\ge 0}\times \Lambda$, where $\Lambda \subset N$ is a non-empty, smooth and connected open subset of N such that $\overline{\Lambda }\ne N$, and denote by $\lambda _1$ the first Dirichlet eigenvalue of the Laplace–Beltrami operator of $\Lambda$. Moreover, suppose that either one of the following conditions is eventually satisfies as $r\rightarrow +\infty$

• 1. $\sigma \sigma '\le \frac{\lambda _1}{(m-1)}r$;

• 2. $\sigma ''\le 0$.

Then $\overline{\Omega }$ is $\mathcal {D}\text {-parabolic}$.

Remark 2.8

Observe that

$$\begin{aligned} {\left\{ \begin{array}{ll} \sigma >0\\ \sigma ''\le 0 \ \text {eventually} \end{array}\right. }\quad \Rightarrow \quad E\ge \sigma '\ge 0\ \text {eventually} \end{aligned}$$

for a positive constant E. Indeed, if A is large enough so that $\sigma ''\le 0$ in $[A,+\infty )$, fixed any $y>x>A$ and defined $z_t=\frac{y-tx}{1-t}$ for $t\in (0,1)$, thanks to the concavity and to the positivity of $\sigma$, one has

$$\begin{aligned} \sigma (y)=\sigma (tx+(1-t)z_t)\ge t\sigma (x)+(1-t)\sigma (z_t)> t\sigma (x) \end{aligned}$$

that implies, as $t\rightarrow 1$, $\sigma (y)\ge \sigma (x)$. Hence, $\sigma '\ge 0$. Moreover,

$$\begin{aligned} \sigma ''\le 0\ \text {in}\ [A,+\infty ) \quad&\Rightarrow \quad \sigma '\ \text {non-increasing}\ \text {in}\ [A,+\infty )\\&\Rightarrow \quad E:=\sigma '(A)\ge \sigma '\ \text {in}\ [A,+\infty ), \end{aligned}$$

and so $E\ge \sigma '\ge 0$ in $[A,+\infty )$.

Proof

We recall that $\Omega$ is $\mathcal {D}\text {-parabolic}$ if every $u\in C^\infty (\Omega )\cap C^0(\overline{\Omega })\cap L^\infty (\Omega )$ satisfying the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=0 &{} \text {in}\ \Omega \\ u=0 &{} \text {on}\ \partial \Omega \end{array} \right. \end{aligned}$$

(7)

vanishes everywhere. Observing that, by Pessoa et al. [15, Corollary 11], the $\mathcal {D}\text {-parabolic}$ity is invariant by removing compact subsets, it is enough to prove that there exists an appropriate $K\subset \Omega$ compact such that the resulting subdomain $U:=\Omega \setminus K$ is $\mathcal {D}\text {-parabolic}$. To this end, in turn, following the philosophy of Khas’minskii test, we only have to find a nonnegative function $\phi \in C^0(\overline{U})\cap W^{1,2}_{_{loc}}(U)$ satisfying the conditions

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta \phi \ge 0 \\ \underset{\underset{x\in U}{d^M(p_0,x)\rightarrow +\infty ,}}{\lim } \phi (x) = +\infty \end{array} \right. \end{aligned}$$

for any fixed $p_0 \in M$. Indeed, in this case given any solution $u\in C^\infty (U)\cap C^0(\overline{U})\cap L^\infty (U)$ of (7), suppose by contradiction that $\sup _U u >0$. Then there exists $x_0, x_1 \in U$ such that $\sup _U u \ge u(x_1)> u(x_0)=:u_0>0$. Define $v:=u-u_0-\epsilon \phi$, for $\epsilon$ small enough so that $v(x_1)>0$, and set $W:=\{x\in U\,\ v(x)>0\}$. Then $x_1 \in W$ and W is bounded since $\phi \rightarrow +\infty$ as $d^M(p_0,x)\rightarrow \infty$. By the fact that $\Delta v \ge 0$ weakly in W and $v\le 0$ on $\partial W$, using the strong maximum principle we get $v\le 0$ on W, thus obtaining a contradiction. It follows that $u\le 0$. By applying the same argument to the function $-u$, we conclude $u\equiv 0$, as desired.

It remains to prove the existence of the function $\phi$ and the corresponding compact set K. Thanks to the structure of the warped product manifold, we can assume $\phi$ to be of the form $\phi (r,\xi )=h(r)\psi (\xi )$. So, let $\psi$ be the positive first Dirichlet eigenfunction of the Laplacian on $\Lambda$

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _\Lambda \psi =\lambda _1 \psi \ge 0 &{} \text {in}\ \Lambda \\ \psi =0 &{} \text {on}\ \partial \Lambda . \end{array} \right. \end{aligned}$$

With this choice the differential inequality $-\Delta \phi \ge 0$ is equivalent to the second order ODE

$$\begin{aligned} h''+(m-1)\frac{\sigma '}{\sigma } h'-\frac{1}{\sigma ^2} \lambda _1 h\le 0. \end{aligned}$$

(8)

Whence, we are reduced to find a solution h to (8) so that $h\xrightarrow []{r\rightarrow +\infty }+\infty$. This is obtained via a case by case analysis:

• Case 1: let $A>0$ so that for every $r\ge A$

  $$\begin{aligned} \sigma (r) \sigma '(r)\le \frac{\lambda _1}{(m-1)}r \end{aligned}$$

  and define $h(r):=r$. It follows that

  $$\begin{aligned} h''+ (m-1)\frac{\sigma '}{\sigma } h'- \frac{1}{\sigma ^2} \lambda _1 h&= \frac{1}{\sigma ^2}\left( (m-1) \sigma \sigma ' - \lambda _1 r \right) \\&\le \frac{1}{\sigma ^2} \left( \lambda _1 r - \lambda _1 r \right) = 0 \end{aligned}$$

  for every $r\ge A$.

• Case 2: thanks to Remark 2.8 there exist $E>0$ and $A>0$ so that

  $$\begin{aligned} E\ge \sigma '\ge 0 \quad \text {in}\ [A,+\infty ). \end{aligned}$$

  In particular, it follows that

  $$\begin{aligned} \exists \lim _{r\rightarrow +\infty } \sigma (r)=l \in (0,+\infty ]. \end{aligned}$$

  If $l\in (0,+\infty )$, then the claim follows by the case 1. Hence, we only have to consider the case $l=+\infty$. Fix $\beta >0$ and define $h:=\sigma ^\beta$. We have

  $$\begin{aligned}&h''+ (m-1)\frac{\sigma '}{\sigma }h'-\frac{1}{\sigma ^2}\lambda _1 h\\&\quad = \sigma ^{\beta -2} \left[ (\sigma ')^2 \beta (\beta +m-2)-\lambda _1\right] + \underbrace{\beta \sigma ^{\beta -1} \sigma ''}_{\le 0} \end{aligned}$$

  for every $r\ge A$ and, by the boundedness of $\sigma '$, we can take a $\beta$ small enough so that

  $$\begin{aligned} (\sigma ')^2 \beta (\beta +m-2)-\lambda _1\le 0, \end{aligned}$$

  obtaining

  $$\begin{aligned} h''+&(m-1)\frac{\sigma '}{\sigma }h'-\frac{1}{\sigma ^2}\lambda _1 h\le 0 \quad \text {in}\ [A,+\infty ). \end{aligned}$$

As a consequence of the above analysis, by the previous argument follows that the domain $U:=\Omega \cap \left( [A,+\infty )\times N \right)$ is $\mathcal {D}\text {-parabolic}$.

Since $\Omega {\setminus } U = \left( [0,A]\times N\right) \cap \Omega$ is compact in $\Omega$ and U is $\mathcal {D}\text {-parabolic}$, by Pessoa et al. [15, Corollary 11] the domain $\Omega$ is itself $\mathcal {D}\text {-parabolic}$, thus completing the proof. $\square$

As a direct application of Theorem 2.7, we get the already stated Theorem A. Its proof is based on a characterization of the $\mathcal {D}\text {-parabolic}$ity contained in [15, Proposition 10], which asserts what follows.

Proposition 2.9

Let (M, g) be a Riemannian manifold with nonempty boundary $\partial M\ne \emptyset$. Then, M is $\mathcal {D}\text {-parabolic}$ if and only if the following maximum principle holds

$$\begin{aligned} \left\{ \begin{array}{l} u\in C^0(M)\cap W^{1,2}_{loc}(\text {int}\ M)\\ -\Delta u\le 0\ in\ M\\ \sup _M u <+\infty \end{array} \right. \quad \Rightarrow \quad \sup _M u = \sup _{\partial M} u. \end{aligned}$$

Proof of Theorem A

Consider $u\in C^0(\overline{\Omega })\cap W^{1,2}_{loc}(\Omega )$ a bounded above distributional solution to the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + c\ u \le 0 &{} \text {in}\ \Omega \\ u\le 0 &{} \text {on}\ \partial \Omega . \end{array} \right. \end{aligned}$$

If $u^+:=\max \{u,0\}$, by Kato’s inequality (see [17, Proposition 4.1]) we get

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u^+ \le -c u^+ \le 0 &{} \text {in}\ \Omega \\ u^+ = 0 &{} \text {on}\ \partial \Omega . \end{array} \right. \end{aligned}$$

Using Theorem 2.7, Proposition 2.9 and [17, Theorem 3.1], it follows that $u^+=0$ in $\Omega$, implying $u\le 0$ in $\Omega$. $\square$

Remark 2.10

Given a smooth function $\sigma :\mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{\ge 0}$ satisfying

• (i) $\sigma (r)>0$ for every $r>0$

• (ii) $\sigma ^{(2k)}(0)=0$ for every $k\in \mathbb {N}$ and $\sigma '(0)=1$,

the model manifold of warping function $\sigma$ is the Riemannian manifold obtained by quotienting the product manifold $[0,+\infty )\times \mathbb {S}^{n-1}$ with respect to the relation that identifies $\{0\}\times \mathbb {S}^{n-1}$ with a single point o and then endowed with the Riemannian metric obtained by smoothly extending the metric $dr^2+\sigma ^2 g^{\mathbb {S}^{n-1}}$ of the warped product $(0,+\infty )\times _\sigma \mathbb {S}^{n-1}$ at o.

Since the Dirichlet-parabolicity is a property that involves only the asymptotic behaviour of the manifold, we stress that both Theorems 2.7 and A can be adapted, without any change, to the case where M is a model manifolds.

3 A maximum principle for general unbounded domains in complete manifolds

In the present section we aim to prove a Maximum Principle for second order elliptic operators acting on unbounded domains of more general Riemannian manifolds. We stress that in the main theorem of this section, i.e. Theorem B, we only require the positivity (in the spectral sense) of the operator, with no further assumptions neither on the geometry or on the structure of the ambient manifold.

The result is obtained readapting the work made in the Euclidean case by Nordmann [13]. Most of the effort consists in generalizing in a Riemannian setting some classical Euclidean tools. In particular, it will be crucial the achievement of an Alexandroff–Bakelman–Pucci estimate, which will allow us to construct a (generalized) first eigenfunction in unbounded domains. The Maximum Principle will be a straightforward consequence of the existence of such eigenfunction.

3.1 ABP inequality

In the very interesting article [7], Cabré proved a Riemannian version of the Alexandroff-Bakelman-Pucci estimate for elliptic operators in nondivergent form acting on manifolds with nonnegative sectional curvature. In his work, he used the assumption on the sectional curvature to ensure two fundamental tools: the (global) volume doubling property for the Riemannian measure $\text {dv}$ and the classical Hessian comparison principle by Rauch. In particular, since these two tools (with different curvature bounds) are available in every relatively compact domain $\Omega \subset M$ regardless of any assumption on the sectional curvature of M, it is reasonable to expect that we can locally recover the results by Cabré up to multiply by appropriate constants depending on $\Omega$ and on the lower bound of its sectional curvature.

Among its various applications, the ABP inequality is one of the main ingredients used by Berestycki et al. [4] to prove the existence of the generalized principal eigenfunction of a second order differential operator $\mathcal {L}$ on Euclidean domains, that is, a generalization of the notion of eigenfunction to operators acting on possibly nonsmooth or unbounded domains. In this paper we will see how to transplant the construction of the generalized principal eigenfunction into general bounded (and into smooth unbounded) Riemannian domains: this will allow us to prove a maximum principle for uniformly elliptic second order differential operators acting on smooth unbounded domains.

Following the proof in [7], we get a version of the ABP inequality for uniformly elliptic operators of the form

$$\begin{aligned} \mathcal {M}u(x) := \text {div}\left( A(x)\cdot \nabla u(x) \right) +g(B(x),\nabla u(x)), \end{aligned}$$

(9)

acting on a bounded Riemannian domain $\Omega \subset M$. In the present subsection we are supposing that $B\in C^\infty (M;TM)$ is a general smooth vector field and that $A\in \text {End}(TM)$ is a positive definite, smooth and symmetric endomorphism of the tangent bundle satisfying conditions (2) and (4).

The strategy we adopt to achieve the ABP inequality is strongly based on the existence of a suitable atlas composed by harmonic charts. To this aim, let’s start by introducing the following definition.

Definition 3.1

Given an n-dimensional Riemannian manifold (M, g), the $C^1$-harmonic radius of M at $x\in M$, denoted by $r_h(x)$, is the supremum among all $R>0$ so that there exists a coordinate chart $\phi :B_R(x)\rightarrow \mathbb {R}^n$ with the following properties

• 1. $2^{-1} g^{\mathbb {R}^n}\le g \le 2 g^{\mathbb {R}^n}$ in the local chart $(B_R(x),\phi )$;

• 2. $||\partial _k g_{ij}||_{C^0(B_R(x))}\le \frac{1}{R}$ for every $k=1,...,n$;

• 3. $\phi$ is an harmonic map.

Defining $r_h(M):=\inf _{x\in M} r_h(x)$, if we suppose that

$$\begin{aligned} |\text {Ric}|\le K \quad \text {and}\quad \text {inj}_{(M,g)}\ge i \end{aligned}$$

(10)

for some constants $K,i\in \mathbb {R}_{>0}$, by Hebey and Herzlich [11, Corollary] it follows that there exists a constant $r_0=r_0(n,K,i)>0$ so that $r_h(M)\ge r_0$. In particular, under the assumptions (10) we can choose a cover of harmonic charts (with fixed positive radius) providing a uniform $C^1$-control on the metric and on its derivatives. We will use the existence of a positive harmonic radius in the proof of the next Alexandroff–Bakelman–Pucci inequality, one of the main results of the present section.

Theorem 3.2

Let (M, g) be a complete Riemannian manifold of dimension $\text {dim}(M)=n$, $\Omega \Subset M$ a bounded smooth domain and $\mathcal {M}$ a differential operator of the form (9) satisfying (2) and (4). Fix $b>0$ so that $|B|\le b$ in an open neighbourhood of $\Omega$.

Then, there exists a positive constant C such that for every $u\in C^2(\Omega )$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} -\mathcal {M}u\le f\ in\ \Omega \\ \limsup _{x\rightarrow \partial \Omega } u(x) \le 0 \end{array} \right. \end{aligned}$$

with $f\in L^n(\Omega )$, it holds

$$\begin{aligned} \sup _\Omega u\le C\ \text {diam}(\Omega )\left| \left| f \right| \right| _{L^n(\Omega )}. \end{aligned}$$

(11)

Remark 3.3

The explicit expression of the constant C in (11) is the following

$$\begin{aligned} C=\frac{t^{1/p} 2^{n/p}\left[ t\left( 2^{n(p+1)/p}C_{\mathbb {R}^n} C_1 \right) ^t +1 \right] }{\left( \frac{|\widehat{\Omega }\setminus \Omega |}{|\widehat{\Omega }|} \right) ^{1/p}} \end{aligned}$$

where, denoting $r:=r_h(\overline{\Omega })$,

• $p=p(n,r,a,b,c_0,C_0)$ and $C_1=C_1(n,r,a,b,c_0,C_0)$ are the constants given in the following Theorem 3.4;

• $C_{\mathbb {R}^n}$ is the Euclidean doubling constant;

• $t\le \frac{|\Omega _r| 2^{n/2}}{|\mathbb {B}_{r/8}|}$, where $\Omega _r:=\{x\in M\,\ d(x,\Omega )< r\}$ and $\mathbb {B}_{r/8}$ is the Euclidean ball of radius $\frac{r}{8}$;

• $\widehat{\Omega }$ is a neighbourhood of $\Omega$ given by the union of t geodesic balls of radius $\frac{r}{2}$.

The key result we need to prove Theorem 3.2 is the following Euclidean integral Harnack inequality, whose proof can be found in [8, Theorem 9.22]

Theorem 3.4

Let $\mathcal {L}:= a^{ij} \partial _{i}\partial _{j}+b^i \partial _i+c$ be an uniformly elliptic differential operator acting on a bounded domain $U\subset \mathbb {R}^n$ with

$$\begin{aligned} c_0\le [a^{ij}]\le C_0 \quad and \quad |b^i\partial _i|,|c|\le b, \end{aligned}$$

for some positive constants $c_0,C_0$ and b, and let $f\in L^n(U)$. If $u\in W^{2,n}(U)$ satisfies $\mathcal {L}u\le f$ and is nonnegative in a ball $B_{2R}(z)\subset U$, then

where p and $C_1$ are positive constants depending on $n,\ bR,\ c_0$ and $C_0$.

Remark 3.5

If $b=0$, i.e. if $B=b^i\partial _i$ is the null vector field and $c\equiv 0$, then the constants p and $C_1$ in previous theorem do not depend on the radius R.

Remark 3.6

For later purpose, let us remark that if $\Omega$ is a bounded smooth domain and $u\in C^2(\Omega )\cap C^1(\overline{\Omega })$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}u\le f &{} \text {in}\ \Omega \\ u\equiv C &{} \text {on}\ \partial \Omega \\ \frac{\partial u}{\partial A\cdot \nu } \le 0 &{} \text {on}\ \partial \Omega , \end{array} \right. \end{aligned}$$

where $\nu$ is the outward pointing unit vector field normal to $\partial \Omega$, then we can consider a larger bounded smooth domain $\Lambda \Supset \Omega$ and we can extend u and f to $\Lambda$ by imposing $u\equiv C$ and $f\equiv 0$ in $\Lambda {\setminus } \overline{\Omega }$. In this way we get a function $u\in C^0(\Lambda )\cap W^{2,n}(\Lambda )$ satisfying $\mathcal {M}u\le f$ weakly in $\Lambda$, i.e. so that

$$\begin{aligned} \int _{\Lambda } \left[ -g(A\cdot \nabla u,\nabla \phi ) + g(B, \nabla u) \phi \right] \ \text {dv}\le \int _\Lambda f \phi \ \text {dv}\quad \quad \forall 0\le \phi \in C^\infty _c(\Lambda ). \end{aligned}$$

Remark 3.7

We stress that if $\Omega$ is a bounded smooth domain, $u\in C^2(\Omega )\cap C^1(\overline{\Omega })$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}u\le 0 &{} \text {in}\ \Omega \\ u\equiv C &{} \text {on}\ \partial \Omega \end{array} \right. \end{aligned}$$

and $x_0\in \partial \Omega$ is a global minimum for u in $\overline{\Omega }$, then

$$\begin{aligned} \frac{\partial u}{\partial A\cdot \nu }(x_0)\le 0. \end{aligned}$$

Indeed, by decomposing $A\cdot \nu = (A\cdot \nu )^\top +(A\cdot \nu )^\bot$, where $(A\cdot \nu )^\top$ and $(A\cdot \nu )^\bot$ are tangential and normal to $\partial \Omega$ respectively, one can check that

$$\begin{aligned} \frac{\partial u}{\partial A\cdot \nu }(x_0)=(A(x_0)\cdot \nu (x_0))^\bot \frac{\partial u}{\partial \nu }(x_0)=\underbrace{g\Big (A(x_0)\cdot \nu (x_0), \nu (x_0)\Big )}_{>0} \frac{\partial u}{\partial \nu }(x_0) \end{aligned}$$

where the first equality follows from the fact that $x_0\in \partial \Omega$ is a minimum for $u|_{\partial \Omega }$, implying that the tangential component (to $\partial \Omega$) of $\nabla u$ vanishes at $x_0$. Hence $\frac{\partial u}{\partial A\cdot \nu }(x_0)$ and $\frac{\partial u}{\partial \nu }(x_0)$ have the same sign. By standard Hopf’s Lemma it follows that $\frac{\partial u}{\partial A\cdot \nu }(x_0)\le 0$.

Remark 3.8

Using the local expression of the differential operator $\mathcal {M}$, we can estimate the constant of Theorem 3.4 in every local chart in terms of the coefficients A and B and of the fist order derivatives of the metric, i.e. in terms of the harmonic radius of M thanks to condition 2 of Definition 3.1. Indeed, if X is a vector field, in local coordinates

$$\begin{aligned} \text {div}\left( X \right) = \frac{\partial X^k}{\partial x^k}+ X^t \Gamma ^k_{kt} \end{aligned}$$

obtaining

$$\begin{aligned} \text {div}(A\cdot \nabla u) =\frac{\partial }{\partial x^j} \left( a_i^j g^{hi} \frac{\partial u}{\partial x^h} \right) +a_i^t g^{hi} \frac{\partial u}{\partial x^h} \Gamma ^k_{kt}. \end{aligned}$$

Hence the differential operator $\mathcal {M}$ writes as

$$\begin{aligned} \mathcal {M}u&=\text {div}\left( A\cdot \nabla u \right) + g(B,\nabla u)\\&= \frac{\partial }{\partial x^j} \left( a_i^j g^{hi} \frac{\partial u}{\partial x^h} \right) +a_i^t g^{hi} \frac{\partial u}{\partial x^h} \Gamma ^k_{kt} +B^k \frac{\partial u}{\partial x^k}\\&=a_i^j g^{hi} \frac{\partial ^2 u}{\partial x^j\partial x^h}+ \left( \frac{\partial }{\partial x^j}\left( a_i^j g^{ki} \right) +a_i^t g^{ki} \Gamma ^h_{ht}+B^k \right) \frac{\partial u}{\partial x^k}. \end{aligned}$$

As a consequence, fixed any bounded domain $\Omega \subset M$, if we consider $b>0$ so that $|B|\le b$ in an open neighbourhood U of $\Omega$, then under the assumptions (10) the coefficients of $\mathcal {M}$ have the same bounds in every harmonic chart contained in U. In particular, in Theorem 3.4 we can chose the same constants $p=p(n,r_h(M),a,b,c_0,C_0)$ and $C=C(n,r_h(M),a,b,c_0,C_0)$ for every harmonic chart, avoiding any dependence on the local chart.

Proof of Theorem 3.2

We start by supposing that u is smooth up to the boundary of $\Omega$. Consider the solution w of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}w = -F := - (\mathcal {M}u)^- \le 0 &{} \text {in}\ \Omega \\ w=0 &{} \text {on}\ \partial \Omega . \end{array} \right. \end{aligned}$$

By assumption, $u\in C^\infty (\overline{\Omega })$ and so $F=(\mathcal {M}u)^-$ is Lipschitz in $\overline{\Omega }$, implying that $w\in C^{2,\alpha }(\overline{\Omega })$ for any $\alpha \in (0,1)$. Moreover, by the standard maximum principle, we have $w\ge 0$. Now consider the function $w-u$: by definition

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}(w-u)\le 0 &{} \text {in}\ \Omega \\ w-u\ge 0 &{} \text {on}\ \partial \Omega \end{array} \right. \end{aligned}$$

and, again by standard maximum principle,

$$\begin{aligned} w\ge u \ \ \ \ \text {in}\ \Omega . \end{aligned}$$

Take $z_0\in \Omega$ so that $S=w(z_0)=\sup _{\Omega } w>0$ and consider the function $v:=S-w\ge 0$. Let $r:=r_h(\overline{\Omega })$ and consider the r-neighbourhood $\Omega _{r}$ of $\Omega$

$$\begin{aligned} \Omega _{r}:=\{x\in M\ :\ d(x,\Omega )<r\}. \end{aligned}$$

Since $v|_{\partial \Omega }\equiv S$, by Remark 3.7, we can extend v and F to $\Omega _r$ as done in Remark 3.6.

Observe that, without loss of generality, we can suppose $\text {diam}(\Omega )\ge r$. Otherwise, $\Omega$ is contained in an harmonic local chart and the theorem follows by the standard Euclidean ABP inequality.

Consider an open cover $\mathcal {W}$ of $\overline{\Omega }$ given by

$$\begin{aligned} \mathcal {W}:=\{(W_1:=B_{r/4}(x_1),\phi _1),...,(W_t:=B_{r/4}(x_t),\phi _t)\} \end{aligned}$$

satisfying the following assumptions

• $x_i\in \overline{\Omega }$ for every $i=1,...,t$;

• $d(x_i,x_j)\ge \frac{r}{8}$ for every $i\ne j$;

• $\phi _i$ is harmonic for every $i=1,...,t$;

• $\mathcal {W}$ is maximal (by inclusion).

For a reference see [10, Lemma 1.1]. Moreover, observe that by construction

$$\begin{aligned} \bigcup _{i\le t} W_i\subset \Omega _r. \end{aligned}$$

Since every chart of $\mathcal {W}$ is an harmonic chart, then

$$\begin{aligned} |\Omega _r|\ge \left| \cup _{1\le i \le t} B_{r/8}(x_i) \right| = \sum _{i \le t} |B_{r/8}(x_i)| \ge t 2^{-n/2} |\mathbb {B}_{r/8}| \end{aligned}$$

implying that

$$\begin{aligned} t\le \frac{|\Omega _r|2^{n/2}}{|\mathbb {B}_{r/8}|} \end{aligned}$$

(12)

where $\mathbb {B}_s$ denotes the Euclidean ball of radius s. Now let $\mathcal {U}$ and $\mathcal {V}$ the dilated covers obtained from $\mathcal {W}$

$$\begin{aligned}&\mathcal {U}:=\{(U_1:=B_{r}(x_1),\phi _1),...,(U_t:=B_{r}(x_t),\phi _t)\}\\&\mathcal {V}:=\{(V_1:=B_{r/2}(x_1),\phi _1),...,(V_t:=B_{r/2}(x_t),\phi _t)\}. \end{aligned}$$

Observe that

$$\begin{aligned} W_i\cap W_j \ne \emptyset \quad \Rightarrow \quad \exists x_{ij}\ :\ B_{r/4}(x_{ij})\subseteq V_i \cap V_j \end{aligned}$$

which implies, by property (1) in Definition 3.1,

$$\begin{aligned} \frac{|V_j|}{|V_i\cap V_j|}{} & {} = \frac{|B_{r/2}(x_j)|}{|V_i\cap V_j|} \le \frac{|B_{r/2}(x_j)|}{|B_{r/4}(x_{ij})|} \nonumber \\{} & {} \overset{\text {1}}{\le } \frac{2^{n/2}|\mathbb {B}_{r/2}|}{2^{-n/2}|\mathbb {B}_{r/4}|} = \frac{2^n|\mathbb {B}_{r/2}|}{|\mathbb {B}_{r/4}|} \le 2^n C_{\mathbb {R}^n} \end{aligned}$$

(13)

whenever $W_i\cap W_j \ne \emptyset$, where $C_{\mathbb {R}^n}=2^n$ is the Euclidean doubling constant. It follows that if $W_i\cap W_j \ne \emptyset$

(14)

where $C_D:=4^{n}$.

In any local chart $U_i$ we can apply Theorem 3.4, obtaining

(15)

that implies

(16)

Summing up over $i=1,...,t$, on the left side of (15) we have

(17)

where

$$\begin{aligned} \widehat{\Omega }:=\bigcup _{1\le i\le t} V_i \subseteq \Omega _r. \end{aligned}$$

Now let $j\in \{1,...,t\}$ be so that

$$\begin{aligned} \left( \inf _{V_j} v + r\left| \left| F \right| \right| _{L^n(U_j)} \right) =\max _{i\le t} \left( \inf _{V_i} v + r\left| \left| F \right| \right| _{L^n(U_i)} \right) . \end{aligned}$$

and let $\mathcal {S}:=\{W_{i_1},...,W_{i_m}\}\subseteq \mathcal {W}$ be a sequence of coordinate neighbourhoods joining $W_j=:W_{i_1}$ and $z_0\in W_{i_m}$ and such that

$$\begin{aligned}&W_{i_q}\ne W_{i_s} \ \ \ \ \forall q\ne s,\\&W_{i_q}\cap W_{i_{q+1}}\ne \emptyset \ \ \ \ \forall q=1,...,m-1. \end{aligned}$$

We get

where

$$\begin{aligned} \widetilde{\Omega }=\bigcup _{1\le i\le t} U_i. \end{aligned}$$

As $v(z_0)=0$, iterating

$$\begin{aligned} \inf _{V_j} v&\le (C_D \widetilde{C}_1)^m \left( \inf _{V_{i_m}} v + m\ r\left| \left| F \right| \right| _{L^n(\widetilde{\Omega })}\right) \\&=(C_D \widetilde{C}_1)^m \left( m\ r\left| \left| F \right| \right| _{L^n(\widetilde{\Omega })}\right) \\&\le (C_D \widetilde{C}_1)^t \left( t\ \text {diam}(\Omega ) \left| \left| F \right| \right| _{L^n(\widetilde{\Omega })}\right) \\&=C_2\ \text {diam}(\Omega ) \left| \left| F \right| \right| _{L^n(\widetilde{\Omega })} \end{aligned}$$

where, using (12), $C_2:=t(C_D \widetilde{C}_1)^t$ can be bounded from above by

$$\begin{aligned} C_2 \le \frac{|\Omega _r|2^{n/2}}{|\mathbb {B}_{r/8}|} (C_D \widetilde{C}_1)^{\frac{|\Omega _r|2^{n/2}}{|\mathbb {B}_{r/8}|}}. \end{aligned}$$

Observe that, without loss of generality, $C_D\widetilde{C}_1 \ge 1$. In this way we obtain

$$\begin{aligned} \sum _{i\le t} \widetilde{C}_1^p \left( \inf _{V_i} + r \left| \left| F \right| \right| _{L^n(U_i)}\right) ^p\le & {} t \widetilde{C}_1^p \left( \inf _{V_j} v + \text {diam}(\Omega ) \left| \left| F \right| \right| _{L^n(\widetilde{\Omega })}\right) ^p\nonumber \\\le & {} \widetilde{C}_2^p \left( \text {diam}(\Omega ) \left| \left| F \right| \right| _{L^n(\widetilde{\Omega })}\right) ^p \end{aligned}$$

(18)

where $\widetilde{C}_2:=t^{1/p}\widetilde{C}_1(C_2+1)$. Using (16)–(18), it follows

i.e.

(19)

Recalling that $v\equiv S$ in $\widehat{\Omega }\setminus \Omega$, we get

and, since $|F|\le |f| \chi _{\Omega }$, by (19)

Whence

$$\begin{aligned} \sup _\Omega w =S\le C\ \text {diam}(\Omega ) \left| \left| f \right| \right| _{L^n(\Omega )} \end{aligned}$$

(20)

where $C=\frac{\widetilde{C}_2}{\theta ^{1/p}}$. In particular, previous inequality implies that if $f\in L^\infty (\Omega )$

$$\begin{aligned} \sup _\Omega w\le C\ \text {diam}(\Omega )\ |\Omega |^{1/n} \left| \left| f \right| \right| _{L^\infty (\Omega )}. \end{aligned}$$

For the general case, i.e. removing the smoothness assumption on u up to the boundary, we can proceed by an exhaustion of $\Omega$ by smooth, relatively compact subdomains, as done in [7, Theorem 2.3]. Indeed, let $\{U_\epsilon \}_{\epsilon >0}$ be a family of relatively compact subdomain of $\Omega$ with smooth boundary so that $u\le \epsilon$ in $\Omega \setminus U_\epsilon$ (recall that $\limsup _{x\rightarrow \partial \Omega } u(x)\le 0$) and satisfying $\bigcup _\epsilon U_\epsilon =\Omega$ and define $u_\epsilon =u-\epsilon \in C^2(\overline{U_\epsilon })$. If we consider a sequence $\{u_{k}\}_k\subset C^\infty (\overline{U_\epsilon })$ approximating uniformly u and its derivatives up to order 2 then, defining $u_{k,\epsilon }:=u_k-\epsilon$ and $F_{k,\epsilon }:=\bigg (\text {div}\left( A\cdot \nabla u_{k,\epsilon } \right) +g(B,\nabla u_{k,\epsilon }) \bigg )^-$, by (20) in the previous step we get

$$\begin{aligned} \sup _{U_\epsilon } u_{k,\epsilon } \le C\ \text {diam}(\Omega ) \left| \left| F_{k,\epsilon } \right| \right| _{L^n(U_\epsilon )}. \end{aligned}$$

Thanks to the properties of the defined sequences, we get

$$\begin{aligned} \sup _{U_\epsilon } u_{k,\epsilon } \xrightarrow []{k} \sup _{U_\epsilon } u_\epsilon \end{aligned}$$

and

$$\begin{aligned} F_{k,\epsilon }\xrightarrow []{k} F \ \ \ \ \ \ \ \ \text {in}\ L^n(U_\epsilon ) \end{aligned}$$

that, together with previous inequality, imply

$$\begin{aligned} \sup _{U_\epsilon } u_\epsilon \le C\ \text {diam}(\Omega ) \left| \left| F \right| \right| _{L^n(U_\epsilon )}, \end{aligned}$$

i.e.

$$\begin{aligned} \sup _{U_\epsilon } u \le C\ \text {diam}(\Omega ) \left| \left| f \right| \right| _{L^n(U_\epsilon )}+\epsilon . \end{aligned}$$

Letting $\epsilon \rightarrow 0$, thanks to the fact that $\limsup _{x\rightarrow \partial \Omega } u\le 0$ and $U_\epsilon \rightarrow \Omega$, we finally get

$$\begin{aligned} \sup _\Omega u \le C\ \text {diam}(\Omega ) \left| \left| f \right| \right| _{L^n(\Omega )}. \end{aligned}$$

$\square$

Remark 3.9

Observe that the constant C in the previous theorem depends on $n,\ a,\ b,\ c_0,\ C_0$ and on the family of harmonic neighbourhoods $\mathcal {W}$ that $\Omega$ intersects. In particular, by construction if $\Omega$ and $\Omega '$ are covered by the same family of harmonic neighbourhoods $\mathcal {W}$, $|\Omega |>|\Omega '|$ and C and $C'$ are the constants given by Theorem 3.2 on $\Omega$ and $\Omega '$ respectively, then

$$\begin{aligned} C>C'. \end{aligned}$$

As a consequence, the constant C is monotone (increasing) with respect to the inclusion and so we can use the same $C=C(\Omega )$ for every subdomain $\Omega '\subseteq \Omega$.

Remark 3.10

Observe that in the Euclidean case we have $r_h=+\infty$, implying that if $\Omega \subset \mathbb {R}^n$ is a fixed bounded domain, then we can choose the radius $R=(8\ \text {diam}(\Omega ))$ in order to get $\Omega \subset \mathbb {B}_{R/8}$. By Remark 3.9, we can use the ABP constant of the domain $\mathbb {B}_{R/8}$ also for the domain $\Omega$. In particular, thanks to the Euclidean (global) doubling property, the constants t and $\theta$ of the domain $B_{R/8}$ do not depend neither on $\mathbb {B}_{R/8}$ nor $\Omega$, while the constants p and $C_1$ depend on $n,\ R$ (and hence on $\text {diam}(\Omega )$), $b,c_0$ and $C_0$. This means that in case $M=\mathbb {R}^n$ the constant in Theorem 3.2 depends on the domain $\Omega$ only through its diameter. Moreover, by Remark 3.5, this last dependence on the diameter of $\Omega$ is avoided in case $b=0$ (for instance for the Euclidean Laplacian).

3.2 Generalized principal eigenfunction in general bounded domains

As already claimed, the aim of this section is to prove a maximum principle for smooth unbounded domains in general Riemannian manifolds. While in the bounded case the validity of the maximum principle is strictly related to the positivity of the first Dirichlet eigenvalue, in unbounded domains the existence of classical principal eigenelements is not even guaranteed. In this direction, following what was done by Nordman in [13], we will consider a generalization of the notion of principal eigenvalue (and related eigenfunction) in order to extend this relation to unbounded smooth domains.

Definition 3.11

The generalized principal Dirichlet eigenvalue of the operator $\mathcal {L}$ acting on a (possibly nonsmooth) domain $\Omega \subset M$ is defined as

$$\begin{aligned} \lambda _1^{-\mathcal {L}}(\Omega ) := \sup \{\lambda \in \mathbb {R}\ :\ -(\mathcal {L}+\lambda )\ admits\ a\ positive\ supersolution\} \end{aligned}$$

where u is said to be a supersolution for the operator $-(\mathcal {L}+\lambda )$ if $u\in C^{2}(\overline{\Omega })$ and it satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} -(\mathcal {L}+\lambda ) u \ge 0 &{} \text {in}\ \Omega \\ u\ge 0 &{} \text {on}\ \partial \Omega . \end{array} \right. \end{aligned}$$

Clearly, the previous definition makes sense both in bounded and unbounded domains and in the former case it coincides with the classical notion of principal eigenvalue. Moreover, if $A^{-1}\cdot B=\nabla \eta$ for a smooth function $\eta$ (for instance, if $B\equiv 0)$, then $\mathcal {L}$ is symmetric on $L^2(\Omega , \text {dv}_\eta )$, where $\text {dv}_\eta =e^\eta \ \text {dv}$, and we have a variational characterization of $\lambda _1$ through the Rayleigh quotient

$$\begin{aligned} \lambda _1^{-\mathcal {L}}(\Omega )= \underset{\left| \left| \psi \right| \right| _{L^2(\Omega , \text {dv}_\eta )}=1}{\inf _{\psi \in H^1_0(\Omega , \text {dv}_\eta )}} \left( \int _\Omega g(A\cdot \nabla \psi , \nabla \psi ) \ \text {dv}_\eta -\int _\Omega c \psi ^2 \ \text {dv}_\eta \right) . \end{aligned}$$

The next step consists in proving the existence of a couple of generalized eigenelements. The first result we need is a boundary Harnack inequality, obtained adapting [2, Theorem 1.4] to the Riemannian setting.

Theorem 3.12

(Krylov–Safonov boundary Harnack inequality) Let (M, g) be a complete Riemannian manifold, $\Omega \subset M$ a bounded domain with possibly nonsmooth boundary and $\mathcal {L}$ a differential operator of the form (1). Let $b>0$ so that $|B|,|c|\le b$ in an open neighbourhood of $\Omega$. Fix $x_0\in \Omega$ and consider $G\subset \Omega \cup \Sigma$ compact, where $\Sigma$ is a smooth open subset of $\partial \Omega$. Then, there exists a positive constant C, depending on $x_0,\ \Omega ,\ \Sigma ,\ G,\ a,\ b,\ c_0$ and $C_0$, so that for every nonnegative function $u\in W^{2,p}_{loc}(\Omega \cup \Sigma )$, $p>n$, satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}u=0 &{} a.e.\ in\ \Omega \\ u>0 &{} in\ \Omega \\ u=0 &{} on\ \Sigma \end{array} \right. \end{aligned}$$

it holds

$$\begin{aligned} u(x)\le C u(x_0) \ \ \ \ \forall x \in G. \end{aligned}$$

Proof

Let $\mathcal {U}:=\{U_1,...,U_m\}$ be a family of local charts of M intersecting and covering $\partial \Omega$. Fix $\epsilon >0$ small enough so that $d^M(x_0,\partial \Omega )>2\epsilon$,

$$\begin{aligned} \emptyset \ne \{x\in \Omega \ :\ d(x,\partial \Omega )\in (\epsilon ,2\epsilon )\}\subseteq \bigcup _{1\le i\le m} U_i \end{aligned}$$

and

$$\begin{aligned} \{x\in \Omega \ :\ d(x,\partial \Omega )>2\epsilon \} \ne \emptyset . \end{aligned}$$

Let $\Omega _{\epsilon }$ a smooth subdomain of $\Omega$ satisfying

$$\begin{aligned} \{x\in \Omega \ :\ d(x,\partial \Omega )>2\epsilon \} \subseteq \Omega _{\epsilon } \subseteq \{x\in \Omega \ :\ d(x,\partial \Omega )>\epsilon \}. \end{aligned}$$

Clearly, $\partial \Omega _{\epsilon } \subset \bigcup _{1\le i\le m} U_i$. Now complete $\mathcal {U}$ to a cover of $\Omega$ by coordinate neighbourhoods of M

$$\begin{aligned} \mathcal {V}=\mathcal {U}\cup \mathcal {U}'= \mathcal {U} \cup \{U_{m+1},...,U_h\} \end{aligned}$$

so that

$$\begin{aligned} \overline{\Omega }_{\epsilon } \subset \bigcup _{m+1\le i\le h} U_i \ \ \ \ \ \ \ \ \text {and} \ \ \ \ \ \ \ \ \partial \Omega \cap \left( \bigcup _{m+1\le i\le h} U_i \right) = \emptyset . \end{aligned}$$

Up to considering a larger family $\mathcal {U}'$, we can suppose that for every $i=m+1,...,h$ there exists $W_i\Subset U_i$ open subset such that

$$\begin{aligned} \overline{\Omega }_{\epsilon } \subset \bigcup _{m+1\le i\le h} W_i, \ \ \ \ \ \ \ \ \partial \Omega \cap \left( \bigcup _{m+1\le i\le h} W_i \right) = \emptyset \end{aligned}$$

and

$$\begin{aligned} W_i \cap W_j \ne \emptyset \ \ \ \ \Leftrightarrow \ \ \ \ U_i \cap U_j \ne \emptyset . \end{aligned}$$

Lastly, up to considering a larger family $\mathcal {U}$ and a smaller $\epsilon$, we can suppose that for every $i\in \{1,...,m\}$ there exists a compact subset $E_i\subset \left( U_i \cap \overline{\Omega } \right)$ so that

$$\begin{aligned} \overline{\Omega }\setminus \Omega _\epsilon \subset \bigcup _{1\le i\le m} E_i \end{aligned}$$

and every $E_i$ intersects at least one $W_j$.

For every $i=m+1,...,h$ we can apply the Euclidean version of Krylov–Safonov Harnack inequality, Gilbarg and Trudinger [8, Corollary 8.21], to the couple $W_i \Subset U_i$. Let $C_i=C_i(n,U_i, b, c_0, C_0, W_i)>0$ be the corresponding constant and define

$$\begin{aligned} K:=\max _{m+1\le i\le h} C_i\ge 1. \end{aligned}$$

If $x\in G$, we have two possible cases:

• 1. $\underline{x\in G \cap \Omega _\epsilon }$: we can consider a sequence of distinct neighbourhoods $U_{i_1},..,U_{i_t}\in \mathcal {U}'$ so that

  $$\begin{aligned} x\in W_{i_1}, \ \ \ \ \ \ \ \ x_0 \in W_{i_t} \ \ \ \ \ \ \ \ \text {and}\\ W_{i_j} \cap W_{i_{j+1}} \ne \emptyset \ \ \forall j=1,...,t-1 \end{aligned}$$

  and by (Euclidean) Krylov–Safonov Harnack inequality, we get

  $$\begin{aligned} u(x)&\le \sup _{W_{i_1}} u \le K \inf _{W_{i_1}} u \le K \inf _{W_{i_1}\cap W_{i_2}} u \\&\le K \sup _{W_{i_2}}u \le ... \le K^t \inf _{W_{i_t}} u \le K^t u(x_0). \end{aligned}$$

  Since the sequence of neighbourhoods can be chosen with at most $h-m$ different elements, it follows that

  $$\begin{aligned} u(x)\le \widetilde{K}\ u(x_0) \end{aligned}$$

  where $\widetilde{K}:=K^{k-m}$ does not depend on the choice of $x\in G\cap \Omega _{\epsilon }$.

• 2. $\underline{x\in G\setminus \Omega _{\epsilon }}$: without loss of generality, we can suppose $x \in U_1$. By Theorem 1.4 in [2] applied to $U_1$ and $E_1$, we get

  $$\begin{aligned} u(x)\le B_1\ u(z(x)) \end{aligned}$$

  where $B_1=B_1 (n,a, b, c_0, C_0,U_1,E_1)>1$ and $z(x)\in U_1\cap W_j$ for some $j\ge m+1$, up to enlarge slightly $W_j$ and $E_1$. Retracing what done in the previous point, we obtain that

  $$\begin{aligned} u(x)\le B_1\ u(z(x)) \le B_1 \sup _{W_j} u \le B_1\ \widetilde{K}\ u(x_0). \end{aligned}$$

Choosing $B:=\max _{1\le i\le m} B_i$ and defining $C:=B \widetilde{K}\ge \widetilde{K}$, we get

$$\begin{aligned} u(x)\le C\ u(x_0) \end{aligned}$$

for every $x\in G$, obtaining the claim. $\square$

Remark 3.13

Observe that C actually depends only on the neighbourhoods that G intersects and not really on G, i.e. C is “stable” under small perturbations.

Next stage consists in the construction of a function $u_0$ which vanishes at those points of $\partial \Omega$ that admit a barrier. It will be needed to show that the generalized principal eigenfunction vanishes at smooth portions of $\partial \Omega$.

Definition 3.14

We say that $y\in \partial \Omega$ admits a strong barrier if there exists $r>0$ and $h\in W^{2,n}_{loc}(\Omega \cap B_r(y))$ which can be extended continuously to y by setting $h(y)=0$ and so that

$$\begin{aligned} \mathcal {M}h \le -1. \end{aligned}$$

Remark 3.15

As proved by Miller in [12], the strong barrier condition at $y\in \partial \Omega$ is implied by the exterior cone condition in any local chart, i.e. by the fact that in every local chart around y there exists an exterior truncated cone $C_y$ with vertex at y and lying outside $\overline{\Omega }$. In particular, on every smooth sector $\Sigma$ of $\partial \Omega$ every point $y\in \Sigma$ satisfies the (local) exterior cone condition, and thus the strong barrier condition.

Theorem 3.16

Let (M, g) be a complete Riemannian manifold. Given a (possibly nonsmooth) bounded domain $\Omega \subset M$, there exists $g_0\in \mathbb {R}_{>0}$ and $u_0$ a positive solution to $\mathcal {M}u_0=-g_0$ in $\Omega$ that can be extended as a continuous function at every point $y\in \partial \Omega$ admitting a strong barrier by setting $u_0(y)=0$.

Proof

Consider $\Lambda \subset M$ a bounded, open and smooth domain containing $\overline{\Omega }$ properly and let $\mathcal {G}$ be the positive Dirichlet Green function on $\overline{\Lambda }$ associated to the differential operator $\mathcal {M}-1$. Fixed $x_0\in \Lambda {\setminus } \overline{\Omega }$, let $G(\cdot ):=\mathcal {G}(x_0,\cdot )$ so to have

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}G=G &{} \text {in}\ \Omega \\ G>0 &{} \text {in}\ \overline{\Omega } \end{array} \right. \end{aligned}$$

and define

$$\begin{aligned} g_0=\min _{\overline{\Omega }} G \ \ \ \ \ \ \ \ \text {and} \ \ \ \ \ \ \ \ G_0=\max _{\overline{\Omega }} G. \end{aligned}$$

Consider an exhaustion $\{H_j\}_j$ of $\Omega$ by smooth nested subdomains satisfying $\overline{H}_j\subset H_{j+1}$ and let $u_j$ be the solutions to

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}u_j=-g_0 &{} \text {in}\ H_j\\ u_j=0 &{} \text {on}\ \partial H_j. \end{array} \right. \end{aligned}$$

In particular, by standard Schauder estimates up to the boundary (see [8]), $u_j\in W^{2,p}(H_j)$ for every $p>n$ and, by the standard maximum principle, $\{u_j\}_j$ is an increasing sequence of positive functions. Moreover

$$\begin{aligned} \mathcal {M}(u_j+G)=-g_0+G\ge 0 \end{aligned}$$

so, again by maximum principle, it follows that

$$\begin{aligned} u_j+G\le \max _{\partial H_j} G\le G_0, \end{aligned}$$

i.e. $u_j\le G_0-G\le G_0$ for every j. Hence there exists a function $u_0$ so that

$$\begin{aligned}&u_j \rightharpoonup u_0 \ \ \ \ \text {in}\ W^{2,p}(E)\\&u_j \rightarrow u_0 \ \ \ \ \text {in}\ C^1(E) \end{aligned}$$

for every $p>n$ and every $E\subset \Omega$ compact. Moreover, $\mathcal {M}u_0=-g_0$ and $0<u_0\le G_0$ by construction.

The next step consists in proving that $u_0$ can be extended continuously to 0 at every $y\in \partial \Omega$ admitting a strong barrier. Fix such a $y\in \partial \Omega$ admitting a strong barrier, i.e. so that for some $B_r(y)$ there exists in $U=B_r(y)\cap \Omega$ a positive function $h\in W^{2,n}_{loc}(U)$ satisfying $\mathcal {M}h\le -1$ which can be extended continuously to y by imposing $h(y)=0$. Without loss of generality, we can suppose $r<\text {inj}(y)$. Let h be the strong barrier associated to y and choose j big enough so that $V=H_j \cap B_{r/2}(y)\ne \emptyset$: choosing $\epsilon >0$ small so that

$$\begin{aligned} \epsilon \mathcal {M}\left( d(x,y)^2\right) \le \frac{1}{2} \ \ \ \ \text {in}\ U \end{aligned}$$

the function $\widetilde{h}=h+\epsilon d(x,y)^2$ satisfies

$$\begin{aligned} \mathcal {M}\widetilde{h}\le -\frac{1}{2} \ \ \ \ \text {in}\ U. \end{aligned}$$

Moreover, if $d(x,y)=\frac{r}{2}$ and $x\in \overline{H}_j$, then

$$\begin{aligned} \widetilde{h}(x)\ge \epsilon \frac{r^2}{4}=:\delta \end{aligned}$$

and, up to decrease $\epsilon$, we can suppose $\delta \le \frac{1}{2}$ and that the function $w=G_0 \frac{\widetilde{h}}{\delta }-u_j$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}w \le 0 &{} \text {in}\ V \\ w\ge 0 &{} \text {on}\ \partial V. \end{array} \right. \end{aligned}$$

By the Maximum Principle, it follows $w\ge 0$ in V, i.e.

$$\begin{aligned} u_j(x)\le G_0 \frac{\widetilde{h}(x)}{\delta } \ \ \ \ \text {in}\ V. \end{aligned}$$

Fixing $x\in H_j \cap B_{r/2}(y)$ and letting $j\rightarrow +\infty$, it follows

$$\begin{aligned} u_0(x)\le G_0 \frac{\widetilde{h}(x)}{\delta }. \end{aligned}$$

Since the previous inequality holds for every $x\in H_j\cap B_{r/2}(y)$ and for every j big enough, by the continuity of $\widetilde{h}$ in y the claim follows. $\square$

Remark 3.17

Theorem 3.16 has been obtained thanks to an adaptation of the argument presented in [4, Sect. 3]. Unless small details, the structure of the proof remained unchanged with respect to the one by Berestycki, Nirenberg and Varadhan.

Finally, we can prove the existence of a generalized principal eigenfunction in any bounded Riemannian domain

Theorem 3.18

Let (M, g) be a complete Riemannian manifold of dimension $\dim (M)=n$ and consider a (possibly nonsmooth) bounded domain $\Omega \subset M$. Let $b>0$ so that $|B|,|c|\le b$ in an open neighbourhood of $\Omega$. Then,

• 1. there exists a principal eigenfunction $\phi$ of $\mathcal {L}$

  $$\begin{aligned} \mathcal {L}\phi =-\lambda _1 \phi \end{aligned}$$

  so that $\phi \in W^{2,p}_{loc}(\Omega )$ for every $p<+\infty$;

• 2. normalizing $\phi$ to have $\phi (x_0)=1$ for a fixed $x_0\in \Omega$, there exists a positive constant C, depending only on $x_0,\ \Omega ,\ a,\ b,\ c_0$ and $C_0$, so that $\phi \le C$;

• 3. there exists a positive constant $E>0$ so that $\phi \le Eu_0$, where $u_0$ is the function obtained in Theorem 3.16.

Remark 3.19

The proof proceeds along the lines of Berestycki et al. [4, Theorem 2.1]. We present it for completeness.

Proof

Fix $x_0 \in \Omega$ and consider a compact subset $F\subset \Omega$ so that $x_0 \in \text {int}\ F$ and $|\Omega {\setminus } F|=\delta$, where $\delta >0$ is a constant (small enough) to be chosen. Let $\{\Omega _j\}_j$ be a sequence of relatively compact smooth subdomains of $\Omega$ with $F\subset \Omega _1$ and satisfying

$$\begin{aligned} \overline{\Omega }_i\subset \Omega _{i+1} \ \ \ \ \forall i \ \ \ \ \ \ \ \ \text {and} \ \ \ \ \ \ \ \ \bigcup _i \Omega _i =\Omega . \end{aligned}$$

By the smoothness of $\Omega _j$, for every j there exists a couple of principal eigenelements $(\mu _j,\phi _j)$ for $\mathcal {L}$ so that

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}\phi _j = -\mu _j \phi _j &{} \text {in}\ \Omega _j \\ \phi _j >0 &{} \text {in}\ \Omega _j\\ \phi _j=0 &{} \text {on}\ \partial \Omega _j \end{array} \right. \end{aligned}$$

rescaled so that $\phi _j(x_0)=1$ and with $\phi _j \in W^{1,p}(\Omega _j)$ for every $p<+\infty$. Moreover, since $\phi _k>0$ in $\overline{\Omega }_j$ for $k>j$, by the standard maximum principle (see for instance Theorem 10 in Sect. 5 of [18]) it follows that $\mu _j>\mu _{j+1}>\lambda _1:=\lambda _1^{-\mathcal {L}}(\Omega )$ for every j. In particular, by monotonicity $\{\mu _j\}_j$ converges to a certain $\mu \ge \lambda _1$.

By the standard Harnack inequality applied in $\Omega _1$ it follows that there exists a positive constant $C=C(n,a,b,c_0,C_0,x_0,\Omega _1,F)$ so that

$$\begin{aligned} \max _{F} \phi _j \le C\ \phi _j (x_0)=C \end{aligned}$$

(21)

for every $j\ge 1$.

Now consider $U_j:=\Omega _j {\setminus } F$ and $v=v_j=\phi _j-C$: we have

$$\begin{aligned} \mathcal {M}v = - c \phi _j - \mu _j \phi _j \ge -b \phi _j -\mu _j \phi _j \end{aligned}$$

and

$$\begin{aligned} \limsup _{x\rightarrow \partial U_j} v \le 0. \end{aligned}$$

Let $\Lambda$ be a smooth, bounded domain containing $\overline{\Omega }$ and let $C_{\Lambda }$ be the constant given by Theorem 3.2 on $\Lambda$. Without loss of generality, we can suppose $|B|,|c|\le b$ in $\Lambda$. Observing that $\overline{U}_j \subset \Lambda$ for every j, by Theorem 3.2 and Remark 3.9 it follows that

$$\begin{aligned} \max _{\overline{U}_j} \phi _j - C= & {} \max _{\overline{U}_j} v \nonumber \\\le & {} C_{\Lambda }\ \text {diam}(\Lambda )\ \left| \left| (b+\mu _j)\phi _j \right| \right| _{L^n(U_j)} \nonumber \\\le & {} C_{\Lambda }\ \text {diam}(\Lambda )\ (b+\mu _j)\ \max _{\overline{U}_j} \phi _j\ \delta ^\frac{1}{n}. \end{aligned}$$

(22)

Let $B_r$ be a ball completely contained in F: by [14, Lemma 6.3] there exists a positive constant K, depending only on $\text {dim}(M)$ and on the coefficients of $\mathcal {L}$, so that

$$\begin{aligned} \mu _j\le \frac{K}{r^2}. \end{aligned}$$

Using the previous inequality in (22), we get

$$\begin{aligned} \max _{\overline{U}_j} \phi _j - C&\le C_{\Lambda }\ \text {diam}(\Lambda )\ \left( b+\frac{K}{r^2}\right) \ \max _{\overline{U}_j} \phi _j\ \delta ^\frac{1}{n} \end{aligned}$$

and choosing $\delta$ small enough so that

$$\begin{aligned} C_{\Lambda }\ \text {diam}(\Lambda )\ \left( b+\frac{K}{r^2}\right) \ \delta ^\frac{1}{n}\le \frac{1}{2} \end{aligned}$$

we obtain

$$\begin{aligned} \max _{\overline{U}_j} \phi _j \le 2C \end{aligned}$$

that, together with (21), implies

$$\begin{aligned} \max _{\overline{\Omega }_j} \phi _j \le 2C=:C. \end{aligned}$$

By interior $W^{2,p}$ estimates [8, Theorem 6.2], it follows that

$$\begin{aligned} \left| \left| \phi _k \right| \right| _{W^{2,p}(\Omega _j)}\le C_j \ \ \ \ \ \ \ \ \forall k\ge j+1 \end{aligned}$$

implying the existence of a function $\phi$, positive in $\Omega$, so that

$$\begin{aligned}&\phi _j \rightharpoonup \phi \ \ \ \ \ \ \ \ \text {in}\ W^{2,p}_{loc}(\Omega )\\&\phi _j \rightarrow \phi \ \ \ \ \ \ \ \ \text {in}\ W^{1,\infty }_{loc}(\Omega ). \end{aligned}$$

By construction, $\phi$ solves

$$\begin{aligned} \mathcal {L}\phi =-\mu \phi \ \ \ \ \text {in}\ \Omega \end{aligned}$$

with $\phi (x_0)=1$ and $\phi \le C$. Moreover, by definition of $\lambda _1$ and by the fact that $\mu \ge \lambda _1$, it follows that $\mu =\lambda _1$, obtaining the claims 1 and 2.

Lastly, observing that

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}\phi _j=-(\mu _j+c)\phi _j\ge -(\mu _j+b)\phi _j &{} \text {in}\ \Omega _j\\ \phi _j=0 &{} \text {on}\ \partial \Omega _j \end{array} \right. \end{aligned}$$

and recalling that

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}u_0=-g_0 &{} \text {in}\ \Omega \\ u_0>0 &{} \text {in}\ \Omega \end{array} \right. \end{aligned}$$

we get

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {M}\left( \phi _j- \frac{C}{g_0} (\mu _j^+ + b)u_0 \right) \ge -(\mu _j+b) C + (\mu _j^+ + b) C\ge 0 &{} \text {in}\ \Omega _j \\ \phi _j- \frac{C}{g_0} (\mu _j^+ + b)u_0<0 &{} \text {on}\ \partial \Omega _j \end{array} \right. \end{aligned}$$

and, by standard maximum principle,

$$\begin{aligned} \phi _j \le \frac{C}{g_0}(\mu _j^+ + b)u_0 \ \ \ \ \text {in}\ \Omega _j. \end{aligned}$$

Letting $j\rightarrow \infty$, it follows

$$\begin{aligned} \phi \le \frac{C}{g_0} (\lambda _1^+ + b)u_0=Eu_0. \end{aligned}$$

$\square$

Remark 3.20

Using Remark 3.15, Theorem 3.16 and the third point of the previous theorem, we can see that the function $\phi$ vanishes on every smooth portion of $\partial \Omega$. As a consequence, if we consider a smooth domain $\Omega$ and $x_0\in \partial \Omega$, then for every $R>0$ there exists a couple of eigenelements $(\varphi ^R, \lambda _1^{-\mathcal {L}})$ of the following Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}\varphi ^R=-\lambda _1^R \varphi ^R &{} \text {in}\ \Omega \cap B_R(x_0)\\ \varphi ^R=0 &{} \text {on}\ \text {smooth portions of}\ \partial (\Omega \cap B_R(x_0)). \end{array} \right. \end{aligned}$$

3.3 Generalized principal eigenfunction in smooth unbounded domains

As a consequence of the previous construction, we get the analogue of Theorem 1.4 in [5]. The Euclidean proof can be retraced step by step thanks to Theorems 3.12 and 3.18. We propose it for completeness

Theorem 3.21

Given an unbounded smooth domain $\Omega \subset M$, for any $R>0$ consider the truncated eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}\varphi ^R=-\lambda _1^R \varphi ^R &{} \text {in}\ \Omega \cap B_R\\ \varphi ^R=0 &{} \text {on}\ \partial (\Omega \cap B_R), \end{array} \right. \end{aligned}$$

where $B_R=B_R(x_0)$ for a fixed $x_0\in \partial \Omega$. Then:

• 1. for almost every $R>0$ there exists and is well defined the couple of eigenelemnts $(\lambda _1^R,\varphi ^R)$, with $\varphi ^R$ positive in $\Omega \cap B_R$;

• 2. $\lambda _1^R \searrow \lambda _1$ as $R\rightarrow +\infty$;

• 3. $\varphi ^R$ converges in $C^{2,\alpha }_{loc}$ to some principal eigenfunction $\varphi$ of $\Omega$.

Proof

By the smoothness of $\Omega$, for any $i\in \mathbb {N}$ there exists $r(i)\ge i$ so that $\Omega \cap B_i$ is contained in a single connected component $\Omega _i$ of $\Omega \cap B_{r(i)}$. Moreover, we can suppose $\Omega _i\subset \Omega _{i+1}$ for every i. By Agmon [1], it follows that

$$\begin{aligned} \lim _{i\rightarrow \infty } \lambda _1^{-\mathcal {L}}(\Omega _i)=\lambda _1^{-\mathcal {L}}(\Omega ). \end{aligned}$$

Now fix $x_1\in \Omega _1$ and let $\varphi ^i$ the generalized principal eigenfunction of $-\mathcal {L}$ in $\Omega _i$, obtained by Theorem 3.18, normalized so that $\varphi ^i(x_1)=1$. Fixed $i>j\in \mathbb {N}$, since $\varphi ^i \in W^{2,p}(\Omega \cap B_j)$ for every $p<+\infty$ and vanishes on $\partial \Omega \cap B_j$, by Theorem 3.12 with $\Omega =\Omega _{j+1}$, $\Sigma =\partial \Omega \cap B_{j+1}$ and $G=\overline{\Omega \cap B_j}$, it follows that there exists a positive constant $C_j$ so that

$$\begin{aligned} \sup _{\Omega \cap B_j} \varphi ^i \le C_j\ \varphi ^i(x_1)=C_j \ \ \ \ \ \ \ \ \forall i> j. \end{aligned}$$

By Gilbarg and Trudinger [8, Theorem 9.13] it follows that $\{\varphi ^i\}_{i>j}$ are uniformly bounded in $W^{2,p}(\Omega \cap B_{j-1/2})$ for every $p<+\infty$. Thus, up to a subsequence

$$\begin{aligned} \varphi ^i \overset{i}{\rightharpoonup }\ \phi _j \ \ \ \ \ \ \ \ \text {in}\ W^{2,p}(\Omega \cap B_{j-1/2}) \ \ \forall p<+\infty \end{aligned}$$

and, by Gilbarg and Trudinger [8, Theorem 7.26],

$$\begin{aligned} \varphi ^i \overset{i}{\rightarrow }\ \phi _j \ \ \ \ \ \ \ \ \text {in}\ C^1(\overline{\Omega }\cap B_{j-1}) \end{aligned}$$

to a nonnegative function $\phi _j$ that solves

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}\phi _j=-\lambda _1^{-\mathcal {L}}(\Omega ) \phi _j &{} \text {a.e.}\ \text {in}\ \Omega \cap B_{j-1} \\ \phi _j=0 &{} \text {on}\ \partial \Omega \cap B_{j-1}. \end{array} \right. \end{aligned}$$

By construction, $\phi _j(x_1)=1$ and so $\phi _j$ is positive in $\Omega \cap B_{j-1}$ by the strong maximum principle. Using a diagonal argument, we can extract a subsequence $\{\varphi ^{i_k}\}_{i_k}$ converging to a positive function $\varphi$ that is a solution of the above problem for all $j>1$. $\square$

3.4 Maximum principle in smooth unbounded domains

Once that the existence of the couple of (generalized) principal eigenelements in smooth unbounded domains has been proved, we can proceed to show the validity of the maximum principle under the assumption that the generalized principal eigenvalue is positive.

In what follows we consider an operator $\mathcal {L}$ of the form (1) and satisfying (2)–(4). Before proving the main result of this section, we introduce two technical lemmas

Lemma 3.22

Let (M, g) be a Riemannian manifold, $\Omega \subset M$ a (possibly unbounded) smooth domain and $(\lambda _1,\varphi )$ the generalized Dirichlet principal eigenelements of $\mathcal {L}$ on $\Omega$. Consider $v\in C^2(\overline{\Omega })$ a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -\mathcal {L}v\le 0 &{} in\ \Omega \\ v \le 0 &{} on\ \partial \Omega \end{array} \right. \end{aligned}$$

and define $\sigma :=\frac{v}{\varphi }$.

Then,

$$\begin{aligned} \text {div}\left( \varphi ^2 e^\eta A\cdot \nabla \sigma \right) \ge \lambda _1 e^\eta \sigma \varphi ^2 \ \ \ \ in\ \Omega \end{aligned}$$

(23)

and

$$\begin{aligned} \sigma ^+ \varphi ^2 g(\nu , A\cdot \nabla \sigma )=0 \ \ \ \ on\ \partial \Omega \end{aligned}$$

(24)

where $\sigma ^+=\max (0,\sigma )$. Since $\varphi =0$ at $\partial \Omega$, condition (24) must be understood as the limit when approaching the boundary with respect to the direction $A\cdot \nu$, where $\nu$ is the outward pointing unit vector field normal to $\partial \Omega$.

Proof

By the assumptions, it clearly follows

$$\begin{aligned} \text {div}\left( e^\eta A\cdot \nabla v \right) =e^\eta [\text {div}\left( A\cdot \nabla v \right) +g(B,\nabla v)] \end{aligned}$$

that, together with the fact that v is a subsolution, implies

$$\begin{aligned} \text {div}\left( e^\eta A\cdot \nabla v \right) +e^\eta c v =e^\eta \mathcal {L}v \ge 0. \end{aligned}$$

Moreover, since $\varphi$ is a principal eigenfunction, we get

$$\begin{aligned} \text {div}\left( e^\eta A\cdot \nabla \varphi \right) +c\ e^\eta \varphi =-\lambda _1 e^\eta \varphi , \end{aligned}$$

that, using previous inequality, implies

$$\begin{aligned} \text {div}\left( \varphi ^2 e^\eta A\cdot \nabla \sigma \right)&\ge \underbrace{e^\eta \left[ g(\nabla \varphi , A\cdot \nabla v)-g(\nabla v, A\cdot \nabla \varphi )\right] }_{=0\ \text {by the symmetry of}\ A}+v\lambda _1 e^\eta \varphi \end{aligned}$$

obtaining (23).

Now let $x_0\in \partial \Omega$ and set $x_\epsilon :=\exp _{x_0}(-\epsilon A(x_0)\cdot \nu (x_0))$ for $\epsilon >0$ small enough, where $\nu$ is the outward pointing unit vector field normal to $\partial \Omega$. Recalling that $v\le 0$ at $\partial \Omega$, we have two possible cases:

• 1. ${\sigma (x_\epsilon )\le 0\, \textrm{as}\, \epsilon \, \mathrm{becomes\, small}}$: then, $\sigma ^+(x_\epsilon )=0$ and thus (24) trivially holds in the sense of the limit for x approaching the boundary of $\Omega$ along the direction $A(x_0)\cdot \nu (x_0)$.

• 2. ${v(x_0)=0\,\textrm{and}\, v(x_{\epsilon _n})>0\, \mathrm{for\, a\, sequence}\, \epsilon _n\xrightarrow []{n}0}$: in this case

  $$\begin{aligned} g(A(x_0)\cdot \nu (x_0),\nabla v(x_0))\le 0 \end{aligned}$$

  and, by the standard Hopf’s lemma, recalling that $\varphi \Big |_{\partial \Omega }=0$ and hence its gradients at $x_0$ has no tangential component to $\partial \Omega$,

  $$\begin{aligned} g(A(x_0)&\cdot \nu (x_0),\nabla \varphi (x_0))\\ {}&=g(A(x_0)\cdot \nu (x_0), \nu (x_0)) \ g(\nu (x_0), \nabla \varphi (x_0))<0, \end{aligned}$$

  obtaining that the limit $\lim _{\epsilon \rightarrow 0} \sigma (x_\epsilon )$ exists and

  $$\begin{aligned} \lim _{\epsilon \rightarrow 0}\sigma (x_\epsilon )=\frac{g(A(x_0)\cdot \nu (x_0),\nabla v(x_0))}{g(A(x_0)\cdot \nu (x_0), \nabla \varphi (x_0))}. \end{aligned}$$

  From the definition of $\sigma$ and the fact that $v(x_0)\le 0$, it follows that

  $$\begin{aligned}&\varphi ^2 (x_\epsilon ) \sigma ^+(x_\epsilon ) g\left( \nu (x_0),A(x_0)\cdot \nabla \sigma (x_\epsilon )\right) \\&\quad =[g\left( A(x_0)\cdot \nu (x_0),\nabla v(x_\epsilon )\right) \\&\qquad -\sigma (x_\epsilon ) g\left( A(x_0)\cdot \nu (x_0),\nabla \varphi (x_\epsilon )\right) ]\underbrace{v^+(x_\epsilon )}_{\xrightarrow []{\epsilon \rightarrow 0}0} \\&\quad \xrightarrow []{\epsilon \rightarrow 0}0 \end{aligned}$$

  implying the claim.

$\square$

Now consider the sequence of cut-off functions $\{\rho _k\}_k\subset C^\infty _c(M)$ satisfying

$$\begin{aligned} \rho _k = {\left\{ \begin{array}{ll} 1 \quad \text {in}\ B_k(x_0)\\ 0 \quad \text {in}\ B_{2k}(x_0), \end{array}\right. } \quad 0\le \rho _k\le 1 \quad \text {and} \quad \left| \left| \nabla \rho _k \right| \right| _{L^\infty (M)} \le \frac{E}{k}, \end{aligned}$$

(25)

where $x_0\in M$ is fixed and $E>0$ is a constant not depending on k. Without loss of generality we can suppose $\{\rho _k \ne 0\}\cap \partial \Omega \ne \emptyset$ for every k.

Lemma 3.23

Let (M, g) be a Riemannian manifold, $\Omega \subset M$ a (possibly unbounded) smooth domain and $\{\rho _k\}_k\subset C^\infty _c(M)$ a sequence of cut-off functions satisfying (25). Consider $v\in C^2(\overline{\Omega })$ a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -\mathcal {L}v\le 0 &{} in\ \Omega \\ v \le 0 &{} on\ \partial \Omega . \end{array} \right. \end{aligned}$$

If $\lambda _1:=\lambda _1^{-\mathcal {L}}(\Omega )\ge 0$, then

$$\begin{aligned} \lambda _1 \int _\Omega \rho _k^2 e^\eta (v^+)^2 \ \text {dv}\le \int _\Omega g(\nabla \rho _k, A\cdot \nabla \rho _k) e^\eta (v^+)^2\ \text {dv}\end{aligned}$$

for every k big enough.

Proof

Fix $k\in \mathbb {N}$ so that $\{\rho _k \ne 0\}\cap \partial \Omega \ne \emptyset$ and let $U_k\subset \subset M$ be an open domain so that

• $\text {supp}(\rho _k)\subset U_k$;

• $\Sigma _k:=U_k \cap \partial \Omega$ is smooth (possibly not connected).

Let $\nu$ be the outward pointing unit vector field normal to $\partial \Omega$ and, for $t>0$ small enough, define

$$\begin{aligned} S_{k,t}:=\left\{ y\in U_k \cap \Omega \ :\ y=\text {exp}_x\left( -t A(x)\cdot \nu (x) \right) \ \text {for}\ x\in \partial \Omega \right\} . \end{aligned}$$

Next step consists in proving that there exists $\epsilon _k>0$ so that $S_{k,t}$ is a (possibly not connected) smooth hypersurface of $\Omega$ for every $0\le t \le \epsilon _k$. To this aim, let $p\in M$ and define $O_p \subset T_p M$ as the set of vectors $X_p$ such that the length $l_{X_p}$ of the geodesic whose initial data is $(p,X_p)$ is greater than 1. Observe that if $\alpha \in \mathbb {R}_{>0}$, then $l_{\alpha X_p}=\alpha ^{-1} l_{X_p}$ and hence

$$\begin{aligned} X_p\in O_p \quad \Rightarrow \quad tX_p\in O_p\quad \forall t \in (0,1]. \end{aligned}$$

Set $O:=\cup _{p\in M} O_p$ and observe that the exponential map is smooth on O [16, Lemma 5.2.3].

Now fix $p\in \partial \Omega$. Since A(p) is nonsingular and linear, the differential of the map $\exp _p \circ A(p):O_p\cap N_p \partial \Omega \rightarrow M$ evaluated in $0_p\in O_p$ is nonsingular and it is given by

$$\begin{aligned} d_{0_p}(\exp _p \circ A(p))=\underbrace{d_{0_p} \exp _p}_{=Id} \circ \ d_{0_p} A(p)=A(p). \end{aligned}$$

Retracing the proofs of Proposition 5.5.1 and Corollary 5.5.3 in [16], we obtain that there exists an open neighbourhood W of the zero section in $N \partial \Omega$ (the normal bundle of $\partial \Omega$) on which $\exp \circ A$ is a diffeomorphism onto its image. In particular, there exists a continuous function $\epsilon :\partial \Omega \rightarrow \mathbb {R}_{>0}$ so that

$$\begin{aligned} (p,-t\nu (p))\in W \quad \forall t \in [0,\epsilon (p)] \end{aligned}$$

(see the proof of Petersen [16, Corollary 5.5.2]). Now consider a neighbourhood $V_k\subset \subset M$ of $U_k$ that intersects $\partial \Omega$ smoothly, so that for

$$\begin{aligned} \epsilon _k:= \min _{p\in \overline{V_k}\cap \partial \Omega } \epsilon (p) \end{aligned}$$

we have

$$\begin{aligned} Z_{k,t}:=\left\{ (p,-t \nu (p))\ :\ p\in V_k \cap \partial \Omega \right\} \subset W \quad \forall t \in [0,\epsilon _k]. \end{aligned}$$

Moreover, up to enlarge $V_k$, we have

$$\begin{aligned} S_{k,t}=\left( \exp \circ A\right) (Z_{k,t})\cap U_k. \end{aligned}$$

Since $V_k \cap \partial \Omega$ (and hence $Z_{k,t}$) is smooth and $\left( \exp \circ A\right) \Big |_{Z_{k,t}}$ is a diffeomorphism onto its image, it follows that $S_{k,t}=\left( \exp \circ A\right) (Z_{k,t})\cap U_k$ is a smooth (possibly not connected) hypersurface for every $t \in [0,\epsilon _k]$.

Now define

$$\begin{aligned} \Omega _{k,t}:=[\Omega \cap U_k] \setminus \bigcup _{0<s<t} S_{k,s} \end{aligned}$$

and, up to decrease $\epsilon _k$, suppose

$$\begin{aligned} \Omega _{k,t}\ne \emptyset \quad \forall t \in [0,\epsilon _k]. \end{aligned}$$

By construction

$$\begin{aligned} \bigcup _{0<t<\epsilon _k} \Omega _{k,t}=\Omega \cap U_k. \end{aligned}$$

Multiplying (23) by $\sigma ^+ \rho _k^2$ and integrating over $\Omega _{k,t}$, by the divergence theorem we get

$$\begin{aligned} \int _{\partial \Omega _{k,t}} \sigma ^+&\rho _k^2 e^\eta \varphi ^2 g(\nu ,A\cdot \nabla \sigma )\ \text {da} - \int _{\Omega _{k,t}} g\left( \nabla \left( \sigma ^+\rho _k^2\right) ,A\cdot \nabla \sigma \right) e^\eta \varphi ^2 \ \text {dv}\\&\ge \lambda _1 \int _{\Omega _{k,t}} e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \ \text {dv}. \end{aligned}$$

Observe that

$$\begin{aligned} \int _{\partial \Omega _{k,t}} \sigma ^+ \rho _k^2 e^\eta \varphi ^2 g(\nu ,A\cdot \nabla \sigma )\ \text {da}=\int _{S_{k,t}\cap \text {supp}(\rho _k)} \sigma ^+ \rho _k^2 e^\eta \varphi ^2 g(\nu ,A\cdot \nabla \sigma )\ \text {da} \end{aligned}$$

since $\rho _k\equiv 0$ on $\partial \Omega _{k,t}{\setminus } \left( S_{k,t}\cap \text {supp}(\rho _k)\right)$. Moreover, since A is symmetric,

$$\begin{aligned}&g \left( \nabla \left( \rho _k^2 \sigma ^+\right) , A\cdot \nabla \sigma \right) \\&\quad = g\left( \nabla \left( \rho _k \sigma ^+\right) ,A\cdot \nabla \left( \rho _k \sigma ^+\right) \right) - g\left( \nabla \rho _k,A\cdot \nabla \rho _k \right) (\sigma ^+)^2\\&\quad \ge - g\left( \nabla \rho _k,A\cdot \nabla \rho _k \right) (\sigma ^+)^2, \end{aligned}$$

obtaining

$$\begin{aligned}{} & {} \int _{\partial \Omega _{k,t}} \sigma ^+ \rho _k^2 e^\eta \varphi ^2 g(\nu ,A\cdot \nabla \sigma )\ \text {da}+ \int _{\Omega _{k,t}} g\left( \nabla \rho _k,A\cdot \nabla \rho _k\right) (\sigma ^+)^2 e^\eta \varphi ^2 \ \text {dv}\nonumber \\{} & {} \quad \ge \lambda _1 \int _{\Omega _{k,t}} e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2\ \text {dv}. \end{aligned}$$

(26)

The next step is to study the behaviour of previous integrals as $t\rightarrow 0$. Since

$$\begin{aligned} 0\le \lambda _1 e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \chi _{\Omega _{k,t}}\le \lambda _1 e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \end{aligned}$$

and

$$\begin{aligned} \lambda _1 e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \chi _{\Omega _{k,t}}\rightarrow \lambda _1 e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \quad \text {a.e. in } \Omega \text { as } \, t \rightarrow 0, \end{aligned}$$

by dominated convergence theorem we get

$$\begin{aligned}{} & {} \lambda _1 \int _{\Omega _{k,t}} e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \ \text {dv}\nonumber \\{} & {} \quad =\lambda _1 \int _{\Omega } e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \chi _{\Omega _{k,t}} \ \text {dv}\xrightarrow []{t \rightarrow 0}\lambda _1 \int _{\Omega } e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \ \text {dv}. \end{aligned}$$

(27)

Similarly, using the fact that A is positive definite, we obtain

$$\begin{aligned} \int _{\Omega _{k,t}} g\left( \nabla \rho _k,A\cdot \nabla \rho _k\right) (\sigma ^+)^2 e^\eta \varphi ^2 \ \text {dv}\xrightarrow []{t \rightarrow 0} \int _{\Omega } g\left( \nabla \rho _k,A\cdot \nabla \rho _k\right) (\sigma ^+)^2 e^\eta \varphi ^2 \ \text {dv}. \end{aligned}$$

(28)

Lastly, for $F:=\sigma ^+ \rho _k^2 e^\eta \varphi ^2\,g(\nu ,A\cdot \nabla \sigma )$ we have

$$\begin{aligned} \int _{\partial \Omega _{k,t}} F(y)\ \text {da}&=\int _{S_{k,t}} F(y)\ \text {da} \\&=\int _{\partial \Omega } F\left( \text {exp}_x(-t A(x)\cdot \nu (x))\right) \ \text {da}_t, \end{aligned}$$

where the last integration is performed with respect to Riemannian measure $\text {da}_t$ associated to the pull-back metric $\Phi _t^* g$, denoting by $\Phi _t$ the map

$$\begin{aligned} \Phi _t:\partial \Omega&\rightarrow S_{k,t}\\ x&\mapsto \exp _x(-tA(x)\cdot \nu (x)). \end{aligned}$$

Observe that if $\text {da}$ is the Riemannian measure induced by g on $S_{k,t}$, then in local coordinates

$$\begin{aligned} \text {da}_t=|\det (\text {d}\Phi _t)|\ \text {da}. \end{aligned}$$

In particular, by the smoothness of $\Phi _t$ and the boundedness of the support of $\rho _k$, it follows that $|\det (\text {d}\Phi _t)|$ is bounded on $\text {supp}(\rho _k)\cap \partial \Omega$ for every $t\in [0,\epsilon _k]$. Since by (24)

$$\begin{aligned} F\left( \text {exp}_x(-t A(x)\cdot \nu (x))\right) \xrightarrow []{t \rightarrow 0} 0 \end{aligned}$$

for every $x\in \partial \Omega$, using the dominated convergence theorem we get

$$\begin{aligned} \int _{\partial \Omega _{k,t}} \sigma ^+ \rho _k^2 e^\eta \varphi ^2 g(\nu ,A\cdot \nabla \sigma )\ \text {da}=\int _{\partial \Omega _{k,t}} F(y)\ \text {da}\xrightarrow []{t \rightarrow 0} 0. \end{aligned}$$

(29)

Letting $t\rightarrow 0$ in (26) and using (27)–(29), it follows that

$$\begin{aligned} \int _{\Omega } g\left( \nabla \rho _k,A\cdot \nabla \rho _k\right) (\sigma ^+)^2 e^\eta \varphi ^2 \ \text {dv}\ge \lambda _1 \int _{\Omega } e^\eta \varphi ^2 (\sigma ^+)^2 \rho _k^2 \ \text {dv}, \end{aligned}$$

obtaining the claim, since $\sigma ^+\varphi =v^+$. $\square$

We are finally ready to prove Theorem B.

Proof of Theorem B

Let $u\in C^2(\overline{\Omega })$ be a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}u\ge 0 &{} \text {in}\ \Omega \\ u\le 0 &{} \text {on}\ \partial \Omega \end{array} \right. \end{aligned}$$

and satisfying (5) for a certain $m\in \mathbb {N}$. Suppose by contradiction that $u^+\not \equiv 0$. By Lemma 3.23

$$\begin{aligned} \lambda _1\le \frac{\int _\Omega g\left( \nabla \rho _k,A\cdot \nabla \rho _k\right) e^\eta (u^+)^2\ \text {dv}}{\int _\Omega \rho _k^2 e^\eta (u^+)^2\ \text {dv}} \end{aligned}$$

for k large enough so that the denominator does not vanish. Now consider the function $w=e^{\eta /2}u^+$. We get

$$\begin{aligned} \frac{g(\nabla \rho _k,A\cdot \nabla \rho _k) w^2}{\int _{\Omega } \rho _k^2 w^2}&\le C_0 \frac{g(\nabla \rho _k,\nabla \rho _k) w^2}{\int _\Omega \rho _k^2 w^2} \le C_0 \frac{E^2}{k^2} \frac{w^2\chi _{\Omega \cap B_{2k}(x_0)}}{\int _{\Omega } \rho _k^2 w^2 } \end{aligned}$$

that implies

$$\begin{aligned} \lambda _1&\le \frac{\int _\Omega g(\nabla \rho _k,A\cdot \nabla \rho _k) e^\eta (u^+)^2\ \text {dv}}{\int _\Omega \rho _k^2 e^\eta (u^+)^2\ \text {dv}} \end{aligned}$$

(30)

$$\begin{aligned}&\le C_0 \frac{E^2}{k^2} \frac{\int _{\Omega \cap B_{2k}(x_0)} w^2\ \text {dv}}{\int _{\Omega } \rho _k^2 w^2\ \text {dv}} \end{aligned}$$

(31)

$$\begin{aligned}&\le C_0 \frac{E^2}{k^2} \frac{\int _{\Omega \cap B_{2k}(x_0)} w^2\ \text {dv}}{\int _{\Omega \cap B_{k}(x_0)} w^2\ \text {dv}}. \end{aligned}$$

(32)

By assumption, there exist two constants $\alpha >0$ and $m\in \mathbb {N}$ so that

$$\begin{aligned} \int _{\Omega \cap B_R(x_0)} w^2 \ \text {dv}\le \alpha R^m \end{aligned}$$

(33)

and, following the strategy adopted in the proof of Nordmann [13, Lemma 4], this implies that

$$\begin{aligned} \liminf _{k\rightarrow \infty } \frac{\int _{\Omega \cap B_{2k}(x_0)} w^2 \ \text {dv}}{\int _{\Omega \cap B_{k}(x_0)} w^2 \ \text {dv}} <+\infty . \end{aligned}$$

Indeed, if by contradiction there exists a monotone increasing function $\beta :\mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{>0}$ so that

$$\begin{aligned} \lim _{r\rightarrow + \infty } \beta (r)=+\infty \quad \text {and}\quad \frac{\int _{\Omega \cap B_{2k}(x_0)} w^2\ \text {dv}}{\int _{\Omega \cap B_{k}(x_0)} w^2\ \text {dv}} \ge \beta (k), \end{aligned}$$

then, by iterating, for any $i\in \mathbb {N}$ we have

$$\begin{aligned} \int _{\Omega \cap B_{k}(x_0)} w^2 \ \text {dv}&\le \frac{\int _{\Omega \cap B_{2k}(x_0)} w^2 \ \text {dv}}{\beta (k)}\\&\le \frac{\int _{\Omega \cap B_{2^i k}(x_0)} w^2 \ \text {dv}}{(\beta (k))^i}\\&\overset{\text {by}\ (\text {33})}{\le } \frac{\alpha (2^i k)^m}{(\beta (k))^i}\\&= \alpha \left( \frac{2^m}{\beta (k)}\right) ^i k^m. \end{aligned}$$

Fixing k large enough so that $\beta (k)>2^m$, we get

$$\begin{aligned} \int _{\Omega \cap B_{k}(x_0)} w^2 \ \text {dv}\le \alpha \left( \frac{2^m}{\beta (k)}\right) ^i k^m\xrightarrow []{i\rightarrow +\infty } 0, \end{aligned}$$

contradicting the assumption $w\not \equiv 0$ in $\Omega \cap B_k(x_0)$. Hence, up to a subsequence,

$$\begin{aligned} \lim _{k\rightarrow +\infty } \frac{\int _{\Omega \cap B_{2k}(x_0)} w^2 \ \text {dv}}{\int _{\Omega \cap B_{k}(x_0)} w^2 \ \text {dv}} <+\infty . \end{aligned}$$

By (30) it follows that

$$\begin{aligned} \lambda _1 \le C_0 \frac{E^2}{k^2} \frac{\int _{\Omega \cap B_{2k}(x_0)} w^2 \ \text {dv}}{\int _{\Omega \cap B_{k}(x_0)} w^2 \ \text {dv}} \xrightarrow []{k\rightarrow +\infty }0 \end{aligned}$$

obtaining a contradiction. $\square$

4 Some applications of the maximum principle in unbounded domains

Now we are going to apply Theorem B to generalize the symmetry results contained in [6].

4.1 Strongly stable solutions in homogeneous domains

To start with, consider a complete Riemannian manifold (M, g). We recall that an isoparametric domain $\Omega \subseteq M$ is a domain endowed with a singular Riemannian foliation $\overline{\Omega }=\bigcup _t \Sigma _t$ whose regular leaves are connected parallel hypersurfaces with constant mean curvature $H^{\Sigma _t}$. Now let $\Psi :M\rightarrow \mathbb {R}$ be a smooth function and consider the weighted Riemannian manifold $M_\Psi :=(M,g,\text {dv}_\Psi )=(M,g,e^\Psi \text {dv})$. We say that $\Omega \subseteq M_\Psi$ is a $\Psi$-isoparametric domain if $\overline{\Omega }$ is foliated by parallel hypersurfaces $\Sigma _t$ of constant weighted mean curvature, i.e. so that

$$\begin{aligned} H_\Psi ^{\Sigma _t}=H^{\Sigma _t}-g(\nabla \Psi , \vec {\nu })\equiv const. \end{aligned}$$

where $\vec {\nu }$ is the unit vector field normal to $\Sigma _t$. Lastly, we say that $\Omega \subseteq M$ is an homogeneous domain if $\Omega$ is an isoparametric domain whose regular leaves are orbits of the action of a closed subgroup of $\text {Iso}_0(M)$, the identity component of $\text {Iso}(M)$.

Definition 4.1

($\Psi$-homogeneous domain) Given a weighted Riemannian manifold $M_\Psi$, we say that $\Omega \subseteq M_\Psi$ is a $\Psi$-homogeneous domain if it is a $\Psi$-isoparametric domain and a homogeneous domain simultaneously.

For further details about isoparametric and homogeneous domains, see [6]. We only recall that

• given an homogeneous domain $\overline{\Omega }$ of a complete Riemannian manifold M, there always exists a (finitely generated) integral distribution $\{X_1,...,X_k\}$ of Killing vector fields of M spanning pointwise every tangent space to al leaves $\Sigma _t$ of the foliation of $\overline{\Omega }$;

• if $\overline{\Omega }$ is homogeneous and $\Psi :M\rightarrow \mathbb {R}_{>0}$ is a symmetric (at least on $\overline{\Omega }$) smooth weight, then the symmetry of $\Psi$ turns $\overline{\Omega }$ into e $\Psi$-homogeneous domain.

Before proceeding with the first symmetry result of this section, we recall that on the weighted manifold $M_{\Psi }$ we have a natural counterpart to the standard Laplacian. It is the weighted Laplacian, also called $\Psi$-Laplacian, which is defined by the formula

$$\begin{aligned} \Delta _{\Psi } u = e^{\Psi } \text {div}(e^{-\Psi } \nabla u) = \Delta u - g(\nabla \Psi ,\nabla u). \end{aligned}$$

We also recall that

Definition 4.2

Given $f\in C^1(\Omega )$, a function $u\in C^3(\Omega )\cap C^1(\overline{\Omega })$ is said to be a stable (respectively strongly stable) solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _\Psi u=f(u) &{} in\ \Omega \\ u=c_j &{} on\ (\partial \Omega )_j \end{array} \right. \end{aligned}$$

if

$$\begin{aligned} \lambda _1^{-\Delta _\Psi + f'(u)}(\Omega ):= \underset{\varphi \not \equiv 0}{\inf _{\varphi \in C^\infty _c(\Omega ),}} \frac{\int _\Omega \left( |\nabla \varphi |^2+f'(u) \varphi ^2 \right) \text {dv}_\Psi }{\int _\Omega \varphi ^2 \text {dv}_\Psi }\ge 0 \ \ \ \ (resp.\ >0). \end{aligned}$$

Definition 4.3

If $\overline{\Omega }$ is a $\Psi$-isoparametric domain inside the weighted manifold $M_\Psi$ and $\Sigma$ is a fixed leaf of the isoparametric foliation of $\overline{\Omega }$, let $d:M\rightarrow \mathbb {R}$ be the signed distance function from $\Sigma$

$$\begin{aligned} d:x\mapsto \text {dist}(x,\Sigma ). \end{aligned}$$

A function u on $\overline{\Omega }$ is said to be

• symmetric if there exists $\widehat{u}:\mathbb {R}\rightarrow \mathbb {R}$ so that

  $$\begin{aligned} u(x)=\widehat{u}(d(x)); \end{aligned}$$

• locally symmetric if $u\in C^1(\overline{\Omega })$ and $X(u)\equiv 0$ for any smooth vector field $X\in \mathcal {D}$.

The fixed leaf $\Sigma$ is also called the soul of the $\Psi$-isoparametric domain $\overline{\Omega }$.

We stress that the property of a function to be symmetric does not depend on the fixed leaf of the foliation. Moreover, by Bisterzo and Pigola [6, Lemma 3.7], the notions of symmetry and local symmetry coincide in our setting.

The first theorem, stated below, provides an adaptation of the symmetry result [6, Theorem 5.1] to (possibly) noncompact $\Psi$-homogeneous domains. To achieve this goal we make use of Theorem B in order to replace the nodal domain theorem used by the author and Pigola [6]. However, this leads to more restrictive assumptions on the solution, namely that it has to be strongly stable.

Theorem 4.4

Let $\overline{\Omega }$ be a (possibly noncompact) $\Psi$-homogeneous domain, where $\Psi$ is bounded below and symmetric (at least on $\overline{\Omega }$). Moreover, consider $f\in C^1(\mathbb {R})$ and denote with $\mathcal {D}=\{X_1,...,X_k\}$ the integrable distribution of Killing vector fields associated to the foliation of $\overline{\Omega }$.

If $u\in C^3(\overline{\Omega })$ is a strongly stable solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _\Psi u=f(u) &{} in\ \Omega \\ u=c_j &{} on\ (\partial \Omega )_j \end{array} \right. \end{aligned}$$

(34)

so that for every $\alpha \in \{1,...,k\}$ there exist $m\in \mathbb {N}$ and $x_0\in M$ so that

$$\begin{aligned} \int _{\Omega \cap B_R(x_0)} \left( X_\alpha (u)\right) ^2 \ \text {dv}_\Psi = O(R^m)\quad as\ R\rightarrow +\infty , \end{aligned}$$

(35)

then u is symmetric.

Remark 4.5

In [6, Theorem 5.1] the authors proved a symmetry result for (regular enough) stable solutions to

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _\Psi u=f(u) &{} \text {in}\ \Omega \\ u=c_j &{} \text {on}\ (\partial \Omega )_j \end{array} \right. \end{aligned}$$

in case $\overline{\Omega }$ is a compact $\Psi$-homogeneous domain with associated Killing distribution $\{X_\alpha \}_{\alpha \in A}$ and the weight $\Psi$ satisfies the compatibility condition

$$\begin{aligned} g(X_\alpha ,\Psi )\equiv const. \quad \text {on}\ \Omega \quad \forall \alpha \in A. \end{aligned}$$

In fact, the preceding compatibility condition implies that the weight has to be symmetric (at least on $\Omega$). Indeed, if $X_\alpha \in \mathcal {D}$, denoting $C_\alpha :=g(X_\alpha ,\nabla \Psi )$ we have

$$\begin{aligned} \int _\Omega \text {div}(\Psi X_\alpha ) \ \text {dv}&= \int _\Omega \underbrace{g(X_\alpha ,\nabla \Psi )}_{C_\alpha } \ \text {dv}+ \int _\Omega \Psi \underbrace{\text {div}\left( X_\alpha \right) }_{=0} \ \text {dv}\\&= C_\alpha |\Omega |, \end{aligned}$$

while, by the divergence theorem,

$$\begin{aligned} \int _\Omega \text {div}(\Psi X_\alpha ) \ \text {dv}= \int _{\partial \Omega } \Psi \ \underbrace{g(X_\alpha , \nu )}_{=0}\ \text {dv}=0, \end{aligned}$$

where $\nu$ denotes the unit vector field normal to $\partial \Omega$. Putting together previous equalities, we obtain $C_\alpha =0$ for every $\alpha \in A$. By previous remark, this exactly means that $\Psi$ is symmetric.

Proof of Theorem 4.4

Let $X=X_j\in \mathcal {D}$ and define

$$\begin{aligned} v:=X(u). \end{aligned}$$

Since u is locally constant on $\partial \Omega$ and $X|_{\partial \Omega }$ is tangential to $\partial \Omega$, we have

$$\begin{aligned} v=0 \ \ \ \ \text {on}\ \partial \Omega . \end{aligned}$$

By Bisterzo and Pigola [6, Lemma 5.4]

$$\begin{aligned} \Delta _\Psi v=f'(u) v \end{aligned}$$

implying that $v\in C^2(\overline{\Omega })$ is a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \left( \Delta _\Psi -f'(u)\right) v=0 &{} \text {in}\ \Omega \\ v=0 &{} \text {on}\ \partial \Omega \end{array} \right. \end{aligned}$$

satisfying condition (35). Since $\lambda _1^{-\Delta _\Psi +f'(u)}(\Omega )>0$, by applying Theorem B to both v and $-v$ it follows

$$\begin{aligned} v\equiv 0 \ \ \ \ \text {in}\ \Omega . \end{aligned}$$

(36)

We have thus proved that $X_\alpha (u)\equiv 0$ in $\overline{\Omega }$ for every $\alpha \in A$. Thanks to the fact that $\mathcal {D}$ generates every tangent space to all leaves of the foliation of $\overline{\Omega }$, it follows that u is locally symmetric, and hence symmetric, on $\overline{\Omega }$. $\square$

4.2 Strongly stable solutions in non-homogeneous domains in warped product manifolds

Now consider a weighted warped product manifold

$$\begin{aligned} M_\Psi =(I\times _\sigma N)_\Psi \end{aligned}$$

where $I\subseteq \mathbb {R}$ is an interval, $(N,g^N)$ is a (possibly noncompact) Riemannian manifold without boundary and $\Psi$ is a smooth weight function of the form

$$\begin{aligned} \Psi (r,\xi )= \Phi (r)+\Gamma (\xi ) \end{aligned}$$

for $(r,\xi )\in I\times N$. The second result we want to deal with concerns the case when the domain is an annulus in $\overline{A}(r_1,r_2)\subseteq M$ and there are not enough Killing vector fields tangential to N (and thus there are not enough local isometries acting on the leaves of the annulus).

Despite this lack of symmetries on the domain, in [6, Theorem 6.5] the authors showed that, requiring the finiteness of $\text {vol}_\Gamma (N)$, some potential theoretic tools can be used to recover a symmetry result under a stability-like assumption on the solution. More in details, they showed that if $f'(t)\le 0$ and u is a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _\Psi u=f(u) &{} \text {in}\ A(r_1,r_2) \\ u\equiv c_1 &{} \text {on}\ \{r_1\}\times N \\ u\equiv c_2 &{} \text {on}\ \{r_2\}\times N \end{array} \right. \end{aligned}$$

so that $\left| \left| u \right| \right| _{C^2_{rad}}<+\infty$ and $f'(u)\ge -B$ for some nonnegative constant B satisfying

$$\begin{aligned} 0\le B < \left( \int _{r_1}^{r_2} \frac{\int _{r_1}^s e^{-\Phi (z) \sigma ^{m-1}(z) \ \text {d}z}}{e^{-\Phi (s)} \sigma ^{m-1}(s)} \ \text {d}s \right) ^{-1}, \end{aligned}$$

(37)

then $u(r,\xi )=\widehat{u}(r)$ is symmetric.

Remark 4.6

As already observed by the authors, as a consequence of condition (37) we get the existence of a positive smooth supersolution of the stability operator $-\Delta _\Psi +f'(u)$ in $\text {int} M$, that implies the stability of the solution u.

We stress that, as already claimed, the second result we present in this section is based on some potential theoretic tools. The first notion we need is Neumann-counterpart of the Dirichlet parabolicity. We say that a connected weighted Riemannian manifold $M_\Psi$ with (possibly empty) boundary $\partial M$ is Neumann parabolic (or $\mathcal {N}$-parabolic) if for any given $u\in C^0(M)\cap W^{1,2}_{loc}(\text {int} M, \text {dv}_\Psi )$ satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _\Psi u \ge 0 &{} \text {in}\ \text {int}M \\ \partial _{\nu }u\le 0 &{} \text {on}\ \partial M \\ \sup _M u<+\infty \end{array} \right. \end{aligned}$$

it holds

$$\begin{aligned} u\equiv const., \end{aligned}$$

where $\nu$ is the outward pointing unit normal to $\partial M$. In the case $\partial M=\emptyset$, the normal derivative condition is void.

As an application of Theorem B, we can replace (37) in [6, Theorem 6.5] with the (simpler) strong stability condition of u. Moreover, we only need the manifold $N_\Gamma$ to be $\mathcal {N}$-parabolic, avoiding the assumption on the finiteness of its volume (originally required in [6]).

Theorem 4.7

Let $M_\Psi =(I\times _\sigma N)_\Psi$ where $(N,g^N)$ is a complete (possibly noncompact), connected, $(n-1)$-dimensional Riemannian manifold without boundary. Moreover, assume that $N_\Gamma$ is $\mathcal {N}$-parabolic.

Let $u\in C^4\left( \overline{A}(r_1,r_2)\right)$ be a solution of the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _\Psi u=f(u) &{} \text {in}\ A(r_1,r_2) \\ u\equiv c_1 &{} \text {on}\ \{r_1\}\times N \\ u\equiv c_2 &{} \text {on}\ \{r_2\}\times N \end{array} \right. \end{aligned}$$

where $c_j\in \mathbb {R}$ are given constants and the function f(t) is of class $C^2$ and satisfies $f''(t)\le 0$. If u is strongly stable so that

$$\begin{aligned} \left| \left| u \right| \right| _{C^2_{rad}}=\sup _{A(r_1,r_2)} |u|+\sup _{A(r_1,r_2)} |\partial _r u|+\sup _{A(r_1,r_2)} |\partial _r^2 u|<+\infty \end{aligned}$$

and there exist $m\in \mathbb {N}$ and $x_0\in M$ so that

$$\begin{aligned} \int _{A(r_1,r_2)\cap B_R(x_0)} \left( (\Delta _\Gamma ^N u)^+ \right) ^2 \ \text {dv}_\Psi = O(R^m) \quad as\ R\rightarrow +\infty , \end{aligned}$$

then u is symmetric.

Proof of Theorem 4.7

Let us consider the function

$$\begin{aligned} v(r,\xi ):=\Delta ^N_{\Gamma } u(r,\xi ) \end{aligned}$$

which vanishes on $\partial A(r_1,r_2)$. By a direct calculation we have $[\Delta ^M_\Psi , \Delta ^N_\Gamma ]=0$, that implies

$$\begin{aligned} \Delta ^M_\Psi v&=\Delta ^N_\Gamma f(u)\\&=f''(u) |\nabla ^N u|^2_N+f'(u)v\\&\le f'(u) v. \end{aligned}$$

It follows that v is a solution to

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _\Psi (-v)\ge f'(u) (-v) &{} \text {in}\ A(r_1,r_2) \\ -v=0 &{} \text {on}\ \partial A(r_1,r_2). \end{array} \right. \end{aligned}$$

Using the strong stability assumption on u and the fact that it satisfies (5), by Theorem B we get

$$\begin{aligned} v\ge 0 \ \text {in}\ A(r_1,r_2). \end{aligned}$$

(38)

On the other hand, thanks to the parabolicity of $N_\Gamma$, we can apply [9, Proposition 3.1] and [6, Lemma 6.12] obtaining

$$\begin{aligned} \int _{A(r_1,r_2)} v\ \text {dv}_\Psi&= \int _{r_1}^{r_2} \left( \int _{\{t\}\times N} \Delta ^N_\Gamma u (t,\xi )\ \text {dv}_\Gamma (\xi ) \right) e^{-\Phi (t)} \sigma ^{m-1}(t) \ \text {d}t=0 \end{aligned}$$

that, together with (38), implies $v\equiv 0$ in $A(r_1,r_2)$.

It follows that for every fixed $\overline{r}\in [r_1,r_2]$ the function $\xi \mapsto v(\overline{r},\xi )$ is constant on N and thus $\xi \mapsto u(\overline{r},\xi )$ is a bounded harmonic function on the parabolic manifold $N_\Gamma$. By definition of parabolicity, this implies that $u(\overline{r},\cdot )$ is constant in $N_\Gamma$, as claimed. $\square$