Abstract
We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green’s function vanishing at infinity. On the source function, we assume a sharp pointwise decay depending on the weight appearing in the Poincaré inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold. In comparison with previous works, we can deal with a more general setting on the curvature bounds and without any spectral assumption.
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1 Introduction
The existence of solutions to the Poisson equation
on a complete Riemannian manifold (M, g), for a given function f on M, is a classical problem which has been the object of deep interest in the literature. Malgrange [11] obtained solvability of the Poisson equation for any smooth function f with compact support, as a consequence of the existence of a Green’s function for \(-\Delta\) on every complete Riemannian manifold. Under integrability assumptions on f, existence of solutions has been established by Strichartz [17] and Ni–Shi–Tam [16, Theorem 3.2] (see also [15, Lemma 2.3]). Moreover, in the same paper, the authors proved an existence result for the Poisson problem on manifolds with nonnegative Ricci curvature under a sharp integral assumption involving suitable averages of f. This condition in particular is satisfied if
for some \(C>0\) and \(\alpha >2\), where \(r(x):={\text {dist}}(x,p)\) is the distance function of any \(x\in M\) from a fixed reference point \(p\in M\). In fact, they proved a more general result where the decay rate of f is just assumed to be of order \(1+\varepsilon\). Note that this result is sharp on the flat space \({\mathbb {R}}^{n}\).
From now on let us consider solutions u of the Poisson equation \(-\Delta u=f\) which can be represented as
where G(x, y) is a Green’s function of \(-\Delta\) on M (see Sect. 2 for further details). Muntenau–Sesum [12] addressed the case of manifolds with positive spectrum, i.e., \(\lambda _1(M)>0\), and Ricci curvature bounded from below, obtaining existence of solutions under the pointwise decay assumption
for some \(C>0\) and \(\alpha >1\). Note that this result is sharp on \({\mathbb {H}}^{n}\). Their proof relies on very precise integral estimates on the minimal positive Green’s function, which are inspired by the work of Li–Wang [10]. Note that in [12, 13] the authors also study the behavior of the solution at infinity.
In [4] the authors generalized the existence result in [12], obtaining existence of solutions on manifolds with positive essential spectrum, i.e., \(\lambda _1^{\text {ess}}(M)>0\), for source functions f satisfying
for any \(R>0\), where \(\theta _{R}(m)\) is a function related to a lower bound on the Ricci curvature, locally on geodesic balls with center p and radius \(2R+m\). In particular, the authors showed in [4, Corollary 1.3] existence of solutions on Cartan–Hadamard manifolds with strictly negative Ricci curvature, whenever
for some \(C>0\) and \(\gamma _{1},\gamma _{2}\ge 0\) with \(\alpha >1+\frac{\gamma _{1}}{2}-\gamma _{2}\).
Observe that the results in [4, 12] cannot be used whenever the Ricci curvature tends to zero at infinity fast enough (see [19]) since, in this case, one has \(\lambda _1^{\text {ess}}(M)=0\) (and so \(\lambda _1(M)=0\)). In particular, the case of \({\mathbb {R}}^n\) is not covered. On the other hand, the result in [16] does not apply on manifolds with negative curvature. The purpose of our paper is to obtain a general result which includes, as special cases, both manifolds with strictly negative curvature and manifolds with Ricci curvature vanishing at infinity. Moreover, our result is sharp on spherically symmetric manifolds, and in particular on \({\mathbb {R}}^n\) and \({\mathbb {H}}^n\).
Note that the condition \(\lambda _1(M)>0\) is equivalent to the validity of the Poincaré inequality
for any \(u\in C^\infty _c(M)\). On the other hand, one has positive essential spectrum if and only if, for some compact subset \(K\subset M\), one has \(\lambda _1(M \setminus K)>0\) and
for any \(u\in C^\infty _c(M\setminus K)\). Generalizing the previous inequalities, one says that (M, g) satisfies a weighted Poincaré inequality with a nonnegative weight function \(\rho\) if
for every \(v\in C^\infty _c(M)\). If for any \(R\ge R_0>0\), there exists a nonnegative function \(\rho _R\) defined on M such that (1) holds for every \(v\in C^\infty _c(M\setminus B_R(p))\) and for \(\rho \equiv \rho _R\), we say that (M, g) satisfies a weighted Poincaré inequality at infinity. In addition, inspired by [10], we say that (M, g) satisfies the property \(\left( \mathcal {P}^{\infty }_{w}\right)\), if a weighted Poincaré inequality at infinity holds for the family of weights \(\rho _R\) and the conformal \(\rho _R\)-metric defined by
is complete for every \(R\ge R_0\). The validity of a weighted Poincaré inequality on some classes of manifolds has been investigated in the literature. It is well known that on \({\mathbb {R}}^n\) inequality (1) holds with \(\rho (x)=\frac{(n-2)^2}{4}\frac{1}{r^2(x)}\). It is also called Hardy inequality. More in general, it holds on every Cartan–Hadamard manifold with \(\rho (x)=\frac{C}{r^2(x)}\), for some \(C>0\) (see [1, 3] for some refinement of this result).
In order to state our main results, we need to introduce a (increasing) function \(\omega (s)\) related to the value of the Ricci curvature on the annulus \(B_{\frac{5}{4}s}(p)\setminus B_{\frac{3}{4}s}(p)\) (see (4) for the precise definition). In this paper, we prove the following result.
Theorem 1.1
Let (M, g) be a complete non-compact Riemannian manifold satisfying the property \(\left( \mathcal {P}^\infty _{w}\right)\) w.r.t. the family of weights \(\rho _R\), \(R\ge R_0\), and let f be a locally Hölder continuous function on M. If
then the Poisson equation
admits a classical solution u.
Assume that \(\lambda _1^{\text {ess}}(M)>0\) and
for some \(\gamma \ge 0\). Then, it is direct to see that
for every \(R>0\) and the property \(\left( \mathcal {P}^\infty _{w}\right)\) w.r.t. the family of weights \(\rho _R\), \(R\ge R_0\), holds for every R with \(\rho _R(x)=\lambda _1(M\setminus B_R(p))\). Thus,
therefore, our existence result is in accordance with those in [4, 12].
We recall that by [10, Corollary 1.4, Lemma 1.5] the validity of a weighted Poincaré inequality (1) on M implies the non-parabolicity of the manifold; on the contrary, if (M, g) is non-parabolic, then a weighted Poincaré inequality holds on M, with weight
where G is the minimal positive Green’s function on (M, g). Exploiting this result, using similar techniques as in Theorem 1.1, we obtain the following refined result on complete non-compact non-parabolic manifolds.
Theorem 1.2
Let (M, g) be a complete non-compact non-parabolic Riemannian manifold with minimal positive Green’s function G. Let \(\rho (x)=\frac{|\nabla G(p,x)|^2}{4 G^2(p,x)}\) and let f be a locally Hölder continuous function on M. If
then the Poisson equation
admits a classical solution u.
Remark 1.3
We explicitly observe that in Theorem 1.2 the completeness of the conformal metric \(g_\rho =\rho g\) is not required. As it was observed in [10], the completeness of \(g_\rho\) would hold if \(G(p,x)\rightarrow 0\) as \(r(x)\rightarrow \infty\), a condition that we do not need to assume here.
It is well known that \({\mathbb {R}}^n\) is a non-parabolic manifold if \(n\ge 3\), with minimal positive Green’s function \(G(x,y)=\frac{c_n}{|x-y|^{n-2}}\) for some positive constant \(c_n\). Moreover, the weighted Poincaré – Hardy’s inequality holds on \({\mathbb {R}}^n\) with
In this case, using the definition (4) of the function \(\omega (s)\), it is easy to see that
Hence, we can apply Theorem 1.2, with
and the convergence of the series follows, whenever \(|f(x)|\le C/(1+r(x))^\alpha\) for some \(\alpha >2\). This condition is optimal, as it can be easily verified by explicit computations.
In general, concerning Cartan–Hadamard manifolds, by using Theorem 1.1 we improve [4, Corollary 1.3] allowing the Ricci curvature to approach zero at infinity.
Corollary 1.4
Let (M, g) be a Cartan–Hadamard manifold, and let f be a locally Hölder continuous, bounded function on M. If
for some \(C\ge 1\), \(\gamma _1,\gamma _2\in {\mathbb {R}}\), \(\gamma _{1}\ge \gamma _{2}\), \(\gamma _1\ge 0\) and \(\alpha\) satisfying
then the Poisson equation
admits a classical solution u.
Remark 1.5
In the special case \(\gamma _{1}=\gamma _{2}=\gamma \ge 0\) the condition on \(\alpha\) in the previous corollary becomes
In particular, in (M, g) is the standard hyperbolic space \({\mathbb {H}}^n\), and then \(\gamma =0\). Thus, we need that \(\alpha >1\) and this condition is sharp as observed above. We will consider also the case \(\gamma <0\) in Sect. 6.2 on model manifolds.
The paper is organized as follows: In Sect. 2 we collect some preliminary results and we define precisely the function \(\omega\); in Sect. 3 we prove a refined local gradient estimates for positive harmonic functions; in Sect. 4 we prove key estimates on the positive minimal Green’s function G(x, y) of a non-parabolic manifold, by means of the property \(\left( \mathcal {P}^\infty _{w}\right)\) w.r.t. the family of weights \(\rho _R\), \(R\ge R_0\); in Sect. 5 we prove Theorem 1.1; finally, in Sect. 6 we prove Corollary 1.4 and show the optimality of the assumption in Theorem 1.2 for rotationally symmetric manifolds.
Finally, we note that some results concerning the Poisson equation on some manifolds satisfying a weighted Poincaré inequality have been very recently obtained in [14]. However, their assumptions and results apparently are completely different to ours.
2 Preliminaries
Let (M, g) be a complete non-compact n-dimensional Riemannian manifold. For any \(x\in M\) and \(R>0\), we denote by \(B_{R}(x)\) the geodesic ball of radius R with center x and let \({\mathrm {Vol}}(B_{R}(x))\) be its volume. We denote by \({\mathrm {Ric}}\) the Ricci curvature of g. For any \(x \in M\), let \(\mu (x)\) be the smallest eigenvalue of \({\mathrm {Ric}}\) at x. Thus, for any \(V\in T_{x}M\) with \(|V|=1\), \({\mathrm {Ric}}(V,V)(x) \ge \mu (x)\) and we have \(\mu (x)\ge -\omega (r(x))\) for some \(\omega \in C([0,\infty ))\), \(\omega \ge 0\). Hence, for any \(x\in M\), we have
for some \(\varphi \in C^{\infty }((0,\infty ))\cap C^{1}([0,\infty ))\) with \(\varphi (0)=0\) and \(\varphi '(0)=1\). Note that \(\varphi ,\varphi ',\varphi ''\) are positive in \((0,\infty )\). We set
for \(r(x)>R>1\);
Note that \(Q_{R}(x)\equiv Q_{R}(r(x))\). For any \(z\in M\), let \(\gamma\) be the minimal geodesic connecting p to z. We define the function
for a given \(a>0\). Note that \(t\mapsto \omega (t)\) is increasing and so invertible.
Under (2), we know that
Moreover, let \({\text {Cut}}(p)\) be the cut locus of \(p\in M\).
It is known that every complete Riemannian manifold admits a Green’s function (see [11]), i.e., a smooth function defined in \((M\times M)\setminus \{(x,y)\in M\times M:\,x=y\}\) such that \(G(x,y)=G(y,x)\) and \(\Delta _{y} G(x,y)=-\delta _{x}(y)\). We say that (M, g) is non-parabolic if there exists a minimal positive Green’s function G(x, y) on (M, g), and parabolic otherwise.
We say that (M, g) satisfies a weighted Poincaré inequality with a nonnegative weight function \(\rho\) if
for every \(v\in C^\infty _c(M)\). If for any \(R\ge R_0>0\), there exists a nonnegative function \(\rho _R\) such that (1) holds for every \(v\in C^\infty _c(M\setminus B_R(p))\) and for \(\rho \equiv \rho _R\), we say that (M, g) satisfies a weighted Poincaré inequality at infinity. In addition, inspired by [10], we say that (M, g) satisfies the property \(\left( \mathcal {P}^{\infty }_{\rho _R}\right)\) if a weighted Poincaré inequality at infinity holds for the family of weights \(\rho _R\) and the conformal \(\rho _R\)-metric defined by
is complete. With this metric we consider the \(\rho\)-distance function
where the infimum of the lengths is taken over all curves joining x and y, with respect to the metric \(g_\rho\). For the fixed reference point \(p\in M\), we denote by
Note that \(|\nabla r_\rho (x)|^2 = \rho (x)\). Finally, we denote by
Let \(\lambda _{1}(M)\) be the bottom of the \(L^{2}\)-spectrum of \(-\Delta\). It is known that \(\lambda _{1}(M)\in [0,+\infty )\) and it is given by the variational formula
If \(\lambda _{1}(M)>0\), then (M, g) is non-parabolic (see [6, Proposition 10.1]). Whenever (M, g) is non-parabolic, let \(G_{R}(x,y)\) be the Green’s function of \(-\Delta\) in \(B_{R}(z)\) satisfying zero Dirichlet boundary conditions on \(\partial B_{R}(z)\), for some \(z\in M\). We have that \(R\mapsto G_{R}(x,y)\) is increasing and, for any \(x,y\in M\),
locally uniformly in \((M\times M)\setminus \{(x,y)\in M\times M:\,x=y\}\). We define \(\lambda _{1}(\Omega )\), with \(\Omega\) an open subset of M, to be the first eigenvalue of \(-\Delta\) in \(\Omega\) with zero Dirichlet boundary conditions. It is well known that \(\lambda _{1}(\Omega )\) is decreasing with respect to the inclusion of subsets. In particular, \(R\mapsto \lambda _{1}(B_{R}(x))\) is decreasing and \(\lambda _{1}(B_{R}(x))\rightarrow \lambda _{1}(M)\) as \(R\rightarrow \infty\).
For any \(x\in M\), for any \(s>0\) and for any \(0\le a < b\le +\infty\), we define
3 Local gradient estimate for harmonic functions
In this section, we improve [4, Lemma 3.1]. We set
for \(z\in M\) and \(R>0\);
Lemma 3.1
Let \(R>0\) and \(z\in M\). Let \(u\in C^{2}(B_{R}(z))\) be a positive harmonic function in \(B_{R}(z)\). Then,
for some positive constant \(C>0\).
Proof
Following the classical argument of Yau, let \(v:=\log u\). Then,
Let \(\eta (\xi )=\eta (d(\xi ))\), with \(d(\xi ):={\text {dist}}(\xi ,z)\), a smooth cutoff function such that \(\eta (\xi )\equiv 1\) on \(B_{R/2}(z)\), with support in \(B_{R}(z)\), \(0\le \eta \le 1\) and
Let \(w=\eta ^{2}|\nabla v|^{2}\). Then,
Then, from classical Bochner–Weitzenböch formula and Newton inequality, one has
Moreover, by Laplacian comparison, since \({\mathrm {Ric}}\ge -(n-1)k_R(z)\) in \(B_R(z)\), we have
pointwise in \(B_{R}(z)\setminus (\{z\}\cup {\text {Cut}}(z))\) and weakly on \(B_{R}(z)\). Thus,
Let q be a maximum point of w in \({\overline{B}}_{R}(z)\). Since \(w\equiv 0\) on \(\partial B_{R}(z)\), we have \(q\in B_{R}(z)\). First assume \(q\notin {\text {Cut}}(z)\). At q, we obtain
So
Thus, for any \(\xi \in B_{R/2}(z)\),
We get
for some positive constant \(C>0\). By standard Calabi trick (see [2, 5]), the same estimate can be obtained when \(q\in {\text {Cut}}(z)\). This concludes the proof of the lemma.
\(\square\)
As a corollary, we have the following
Corollary 3.2
Let (M, g) be non-parabolic. If \(r(z)>R>0\), then
for some positive constant \(C>0\).
4 Green’s function estimates
4.1 Pointwise estimate
Lemma 4.1
Let (M, g) be non-parabolic, and let \(a>0\) and \(y\in M\setminus B_{a}(p)\). Then,
where \(A:=\max \left\{ \max _{\partial B_a(p)}G(p,\cdot ), \left( \min _{\partial B_a(p)}G(p,\cdot )\right) ^{-1}\right\}\) and \(B>0\) is a positive constant depending on C from Corollary 3.2.
Proof
Let \(y\in M\setminus \overline{B_{a}(p)}\) with \(a> 0\) and consider the minimal geodesic \(\gamma\) joining p to y and let \(y_{0}\in \partial B_{a}(p)\) be a point of intersection of \(\gamma\) with \(\partial B_{a}(p)\). Since \(G(p,\cdot )\) is harmonic in \(B_{r(z)/4}(z)\), for every \(z\in \gamma\) with \(r(z)\ge a\), by Corollary 3.2 we get
We have
By Gronwall's inequality,
with \(A:=\max \left\{ \max _{\partial B_a(p)}G(p,\cdot ), \left( \min _{\partial B_a(p)}G(p,\cdot )\right) ^{-1}\right\}\) and \(B=C\). Similarly,
\(\square\)
Remark 4.2
One has
for any \(y\in \overline{B_{a}(p)}\). This follows from Lemma 4.1 with \(y\in \partial B_{a}(p)\) and the maximum principle, since \(y\mapsto G(p,y)\) is (weakly) superharmonic in \(B_{a}(p)\). In particular,
Remark 4.3
We also note that
In fact, let \(y\in M\setminus B_a(p)\) and take \(j>r(y)\). Since \(G_{j}(p,y)\le G(p,y)\) and \(G_{j}(p,\cdot )\equiv 0\) on \(\partial B_{j}(p)\), by Lemma 4.1, we have
note that the right-hand side is independent of y. Since \(y\mapsto G_{j}(x,y)\) is harmonic in \(B_{j}(p)\setminus B_{a}(p)\), by maximum principle,
Sending \(j\rightarrow \infty\), by (7), we obtain
and the claim follows.
4.2 Auxiliary estimates
Lemma 4.4
Let (M, g) be non-parabolic. For any \(s>0\), there holds
where dA(y) is the \((n-1)\)-dimensional Hausdorff measure on \(\mathcal {L}_{x}(s)\). As a consequence, by the co-area formula, for any \(0<a<b\), there holds
For the proof see [12]. Moreover, we get the following weighted integrability property for the Green’s function.
Lemma 4.5
Assume that (M, g) satisfies the property \(\left( \mathcal {P}^\infty _{w}\right)\) w.r.t. the family of weights \(\rho _R\), \(R\ge R_0\). Fix \(m\ge R_0\). Then, for any \(R_1>0\) such that \(B_m(p)\subset B^{\rho _m}_{R_1}(p)\), one has
Remark 4.6
Note that \(B_m(p)\subset B^{\rho _m}_{R_1}(p)\) for every \(R_1\) large enough.
Proof
In order to simplify the notation, let \(\rho \equiv \rho _m\). Fix \(R_1>0\) such that \(B_m(p)\subset B^\rho _{R_1}(p)\) and let \(\phi\) be defined as
Let \(R>2R_1\) and \(G^{\rho }_{R}(p,y)\) be the Green’s function of \(-\Delta\) in \(B^{\rho }_{R}(p)\) satisfying zero Dirichlet boundary conditions on \(\partial B^{\rho }_{R}(p)\). Following the proof in [10], since \(G^{\rho }_R\) is harmonic in \(B^{\rho }_{R}(p)\), one has
where the last equality follows by integration by parts and the fact that \(G^{\rho }_{R}(p,y)\) vanishes on \(\partial B^{\rho }_{R}(p)\). Hence, the weighted Poincaré inequality yields
Letting \(R\rightarrow \infty\), by Fatou’s lemma and uniform convergence of \(G_R^\rho \rightarrow G\) on compact subsets, we get
and the thesis follows. \(\square\)
We expect a decay estimate similar to the one in [10, Theorem 2.1]. However, we leave out this refinement since it is not necessary in our arguments.
4.3 Integral estimates on level sets
We begin by noting that using Remark 4.3 and the fact that \(G(p,\cdot )\in L^1_{\text {loc}}(M)\) one has the following integral estimate on large level sets.
Proposition 4.7
Let (M, g) be non-parabolic. Choose A, B as in Lemma 4.1. Then,
For intermediate levels sets, we get the following key inequality.
Proposition 4.8
Assume that (M, g) satisfies the property \(\left( \mathcal {P}^\infty _{w}\right)\) w.r.t. the family of weights \(\rho _R\), \(R\ge R_0\). Then, there exists a positive constant C such that for any function f and any \(0<\delta <1\), \(\varepsilon >0\) satisfying \(\mathcal {L}_p \left( \frac{\delta \varepsilon }{2},2\varepsilon \right) \subset M \setminus B_m(p)\) for some \(m>R_0\), one has
Proof
We follow the general argument in [10, 12]; however, some relevant differences are in order, due to the use of the property \(\left( \mathcal {P}^\infty _{w}\right)\). Let \(\phi :=\chi \psi\) with
and for any fixed \(R>0\)
By the weighted Poincaré inequality at infinity, we get
We estimate
where we used Lemma 4.4 in the last equality. On the other hand,
Now we let \(R\rightarrow \infty\) and use Lemma 4.5. The thesis now follows. \(\square\)
In the special case when M is non-parabolic with positive minimal Green’s function G and with weight \(\rho (x)=\frac{|\nabla G(p,x)|^2}{4 G^2(p,x)}\), we have the following refinement of Proposition 4.8.
Proposition 4.9
Assume that (M, g) is non-parabolic with positive minimal Green’s function G and with weight \(\rho (x)=\frac{|\nabla G(p,x)|^2}{4 G^2(p,x)}\). Then, there exists a positive constant C such that for any function f and any \(0<\delta <1\), \(\varepsilon >0\) one has
Proof
We have
where we have used Lemma 4.4 in the last equality. \(\square\)
5 Proof of Theorem 1.1
In order to prove Theorem 1.1, we will show that
with \(v\in C^{0}(M)\). We divide the proof in two parts, we first consider the case when (M, g) is non-parabolic, and then, the case when it is parabolic.
Proof of Theorem 1.1
Case 1: (M, g) non-parabolic.
By assumption, (M, g) satisfies \(\left( \mathcal {P}^\infty _{w}\right)\) w.r.t. the family of weights \(\rho _R\), \(R\ge R_0\). Let \(x\in M\) and choose \(R=R(x)>R_0\) large enough so that \(x\in B_R (p)\). One has
since \(G(x,\cdot )\in L^1_{\text {loc}}(M)\). Hence, by Harnack’s inequality, we have
where \(C_2(x)\) can be chosen as the constant in the Harnack’s inequality for the ball \(B_{r(x)+1}(p)\). For any \(a>0\), we estimate
By Proposition 4.7, Remark 4.3 we get
for some positive constant \(C_3(a)\). To estimate the first integral, we observe that for any \(m_{0}=m_{0}(x)\ge a\) one has
We need the following lemma.
Lemma 5.1
Choose A, B as in Lemma 4.1. For any \(m\ge m_0\ge a\) one has
Proof
Since \(m_{0}\ge a\), Remark 4.2 implies
If
then by Lemma 4.1
Thus,
and, by monotonicity of \(\omega\), we obtain \(r(z)\ge m\). \(\square\)
In particular, we get
Thus,
Then, since \(G(x,\cdot )\in L^1_{\text {loc}}(M)\), we get
Now, for any \(m\ge m_{0}\), let
By Lemma 5.1,
Hence, we can apply Proposition 4.8 obtaining
where in the last inequality we used Lemma 5.1. The proof of Theorem 1.1 is complete in this case.
Case 2: (M, g) parabolic.
Let G(x, y) be a Green’s function on M (which is positive inside a certain ball, and negative outside). Fix any \(R>0\) and let \(\rho \equiv \rho _{R_0}\). Note that arguing as in the proof of (8), it is sufficient to estimate
since \(G(p,\cdot )\in L^{1}_{\mathrm{{loc}}}(M)\) and f is locally bounded. We have that
where each \(E_{i}\) is an end with respect to \(B^\rho _{R}(p)\). Note that every end \(E_{i}\) is parabolic. In fact, if at least one end \(E_{i}\) is non-parabolic, then (M, g) is non-parabolic (see [8] for a nice overview), but we are in the case that (M, g) is parabolic. Since every \(E_{i}\) is parabolic, every \(E_{i}\) has finite weighted volume (see [9]), i.e.,
Now choose R large enough so that we can apply Lemma 4.5 obtaining
This concludes the proof of Theorem 1.1.
\(\square\)
Proof of Theorem 1.2
We start as in the proof of Theorem 1.1 using (8), (9), (10) and (13). Then, similar to (15), using Proposition 4.9, we obtain
Then
and the proof of Theorem 1.2 is complete. \(\square\)
6 Cartan–Hadamard and model manifolds
We consider Cartan–Hadamard manifolds, i.e., complete, non-compact, simply connected Riemannian manifolds with non-positive sectional curvatures everywhere. Observe that on Cartan–Hadamard manifolds the cut locus of any point p is empty. Hence, for any \(x\in M\setminus \{p\}\) one can define its polar coordinates with pole at p, namely \(r(x) = {\text {dist}}(x, p)\) and \(\theta \in \mathbb S^{n-1}\). We have
for a specific positive function A which is related to the metric tensor [6, Sect. 3]. Moreover, it is direct to see that the Laplace–Beltrami operator in polar coordinates has the form
where \(m(r, \theta ):=\frac{\partial }{\partial r}(\log A)\) and \(\Delta _{\theta }\) is the Laplace–Beltrami operator on \(\partial B_{r}(p)\). We have
Let
We say that (M, g) is a rotationally symmetric manifold or a model manifold if the Riemannian metric is given by
where \({\mathrm{d}}\theta ^2\) is the standard metric on \(\mathbb S^{n-1}\) and \(\varphi \in {\mathcal {A}}\). In this case,
Note that \(\varphi (r)=r\) corresponds to \(M=\mathbb R^n\), while \(\varphi (r)=\sinh r\) corresponds to \(M=\mathbb H^n\), namely the n-dimensional hyperbolic space. The Ricci curvature in the radial direction is given by
6.1 Cartan–Hadamard manifolds
Concerning the validity of the property \(\left( \mathcal {P}^\infty _{w}\right)\) w.r.t. the family of weights \(\rho _R\), \(R\ge R_0\) on a Cartan–Hadamard manifold we have the following result.
Lemma 6.1
Let (M, g) be a Cartan–Hadamard manifold with
for some \(\gamma \in {\mathbb {R}}\), \(C>0\) and any \(x\in M\setminus \{p\}\). Then (M, g) satisfies the property \(\left( \mathcal {P}^\infty _{w}\right)\) with
for all \(R>0\) large enough and some \(C'>0\).
Remark 6.2
As it will be clear from the proof, we have a weighted Poincaré inequality on M if \(\gamma \le 0\) and a the weighted Poincaré inequality for functions with compact support in \(M\setminus B_1(p)\) if \(\gamma >0\).
Proof
We can find \(\varphi \in \mathcal {A}\) given by
for r large enough, \(B>0\) small, \(\delta =\delta (C)>1\) such that \({\mathrm {Ric}}( \nabla r, \nabla r) (x) \le -\frac{\varphi ''(r(x))}{\varphi (r(x))}\). By the Laplacian comparison in a strong form, which is valid only on Cartan–Hadamard manifolds (see [18, Theorem 2.15]), one has
Suppose \(\gamma \le 0\) and let \(\alpha :=\max \{\gamma ,-2\}\le 0\). For any \(u\in C^\infty _c (M)\), since \(|\nabla r|^2=1\), we have
Thus,
and the weighted Poincaré inequality on M follows in this case.
Suppose now \(\gamma >0\). By a Barta-type argument (see, e.g., [7, Theorem 11.17]),
Thus, the Poincaré inequality reads
for any u with compact support in \(M\setminus B_R(p)\). Now let \(R>1\) and, for every \(k\in \mathbb {N}\), define the cutoff functions
Note that \(|\nabla \varphi _k|\le 1\) and for all \(x\in M\setminus B_1(p)\), \(\sum _k \varphi _k =1\) and \(x\in {\text {supp}}\varphi _k\) at most for two integers k. If \({\text {supp}} u \subset M \setminus B_1(p)\), we have
where in the last passage we used (17) with \(R=k-1\). Thus,
where in the last passage we used (17) with \(R=1\). Hence, the weighted Poincaré inequality holds for functions with support in \(M\setminus B_1(p)\).
Finally, the completeness of the metric \(g_{\rho _R}:= {\rho _R}\, g\) follows. In fact, for any curve \(\eta (s)\) parametrized by arclength with \(0\le s \le T\), the length of \(\eta\) with respect tp \(g_{\rho _R}\) is given by
\(\square\)
Let us write some estimates which will be useful both in the proof of Corollary 1.4 and in Sect. 6.2. Choose \(\varphi \in \mathcal {A}\) as in (16) with \(\gamma =\gamma _1\) obtaining
and
for \(r(x)>R>1\). A simple computation shows that for \(R=r(x)/4\), one has
and
Thus,
and, as \(m\rightarrow \infty\),
On the other hand, using Lemma 6.1 with \(\gamma =\gamma _2\), we get the estimate
Proof of Corollary 1.4
For \(\gamma _1\ge \gamma _2\) and \(\gamma _1\ge 0\), we get
and the thesis immediately follows. \(\square\)
6.2 Optimality on rotationally symmetric manifolds
We show that the assumptions in Theorem 1.2 are sharp on model manifolds. Let (M, g) be a rotationally symmetric manifold with \(\varphi \in \mathcal {A}\) defined as in (16) for any \(r>1\). One has
Hence, a solution of \(-\Delta u = f\) in M exists if and only if
Case 1: \(\gamma >-2\). With our choice of \(\varphi\), by the change of variable \(s=t^{1+\frac{\gamma }{2}}\), it is easily seen that for any \(r>0\) sufficiently large
Hence,
Therefore,
This yields that
On the other hand, a direct computation, using (19), shows that
Furthermore, from (18), the assumption of Theorem 1.2 is satisfied if and only if
and the optimality follows in this case.
Case 2: \(\gamma =-2\). We have,
Thus,
Therefore,
and
On the other hand, a direct computation, using (20), shows that
Furthermore, from (18), the assumption of Theorem 1.2 is satisfied if and only if
and the optimality follows in this case.
Case 3: \(\gamma <-2\). We have,
Thus,
Therefore,
and
On the other hand, a direct computation, using (21), shows that
Furthermore, from (18), the assumption of Theorem 1.2 is satisfied if and only if
and the optimality follows in this last case.
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Acknowledgements
Open access funding provided by Politecnico di Milano within the CRUI-CARE Agreement. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first two authors are supported by the PRIN-2015KB9WPT project “Variational methods, with applications to problems in mathematical physics and geometry.” The third author is supported by the PRIN-201758MTR2 project “Direct and inverse problems for partial differential equations: theoretical aspects and applications.”
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Catino, G., Monticelli, D.D. & Punzo, F. The Poisson equation on Riemannian manifolds with weighted Poincaré inequality at infinity. Annali di Matematica 200, 791–814 (2021). https://doi.org/10.1007/s10231-020-01014-0
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DOI: https://doi.org/10.1007/s10231-020-01014-0