Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces
-
Ali Taheri
and
Vahideh Vahidifar
Abstract
In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.
1 Introduction
Unravelling the interplay between the analytic, stochastic and geometric properties of Riemannian manifolds on the one hand and the dynamics of the solutions to nonlinear evolution equations on the other lies at the heart of geometric analysis. Some particular questions include: short and long time existence, smoothing properties, convergence to equilibrium, spectral gaps, blow-ups, self-similar solutions and their profiles, analysis of singularities, critical exponents, description and classification of ancient and eternal solutions as well as Liouville or global constancy type results to mention a few.
The ubiquitous gradient estimates play a significant role here and their derivation is often the main step in confronting such challenging tasks. For parabolic and heat-type equations, there are several types of gradient estimates and techniques in literature, each having their own strength and utility. In this paper we extend such techniques and results to the context of nonlinear equations on smooth metric measure spaces where understanding the interaction between the nonlinearity and curvature on the one hand and the different terms in the estimate on the other is the main driving force (see [1]–[36] for more).
Towards this end let (M, g) be a smooth Riemannian manifold of dimension n ≥ 2 and let dμ = e−ϕ dv g denote the weighted measure associated with a potential ϕ and the Riemannian volume measure dv g on M. The triple (M, g, dμ) is referred to as a smooth metric space or a weighted manifold. In this paper we develop gradient estimates of elliptic and parabolic types, more specifically, of Souplet-Zhang, Hamilton and Li-Yau types, along with some of their implications, for positive smooth solutions w = w(x, t) to the nonlinear parabolic equation on (M, g, dμ) given by:
The operator Δ
ϕ
appearing on the right in (1.1) is the Witten Laplacian associated with the smooth metric measure space (M, g, dμ) (also called the weighted or drifting or ϕ-Laplacian) whose action on functions
Here Δ, div and ∇ are the usual Laplace-Beltrami, divergence and gradient operators associated with the metric g. The nonlinearity
Gradient estimates for positive solutions to linear and nonlinear heat type equations have been studied extensively starting from the seminal paper of Li and Yau [15] (see also [14]). In the nonlinear setting perhaps the first equation to be considered is the one with a logarithmic type nonlinearity (see, e.g., [37]) [specifically,
The interest in such problems originates partly from its natural links with gradient Ricci solitons and partly from links with geometric and functional inequalities on manifolds, notably, the logarithmic Sobolev and energy-entropy inequalities [25], [26], [38]–[42]. Recall that a Riemannian manifold (M, g) is said to be a gradient Ricci soliton if there exists a smooth function ϕ on M and a constant
The notion is a generalisation of an Einstein manifold and has a fundamental role in the analysis of singularities of the Ricci flow [9], [36], [44], [45]. Taking trace from both sides of (1.4) and using the contracted Bianchi identity leads one to a simple form of (1.3) with constant coefficients.
Another prominent class of nonlinear equations rooted in conformal geometry and studied extensively in this setting are Yamabe-type equations (see, [4], [11], [46], [47]). In the context of smooth metric measure spaces these equations can be broadly formulated as (see, e.g., [35], [48], [49])
Incidentally, the case
Here
with p, q real exponents,
Another case of recent interest arises from iterated logarithms (cf. [55]) associated with a string of parameters d,
where log k w = log log k−1 w for k ≥ 2 and log1 w = logw. This can be considered but with due care, e.g., subject to the assumption of w being sufficiently large, specifically, with respect to iterated exponentials of k 1, …, k d (as otherwise the repeated logarithm is meaningless due to the possibility of log k−1 w being non-positive hence making log k w undefined). Naturally one can also consider variations of the same theme, for example, by replacing log k with either of its close relatives
However, one needs to observe that the function Γ thus obtained is only
Another related and yet more general form of Yamabe-type equations is the Einstein-scalar field Lichnerowicz equation (see Choquet-Bruhat [57], Chow [9] and Zhang [36]). In the context of smooth metric measure spaces a generalisation of the Einstein-scalar field Lichnerowicz equation with space-time dependent coefficients can be described as:
For gradient estimates, Harnack inequalities, Liouville type theorems and other related results in this direction see [27], [28], [30], [35], [49], [58], [59] and the references therein.
1.1 Plan and layout of paper
Let us conclude this introduction by briefly describing the plan of the paper. In Section 2 we present some background material on smooth metric measure spaces and the associated generalised Ricci curvature tensors as required for the development of the paper. In Section 3 we present the main results of the paper along with some related discussion. The remainder of the paper is then devoted to the detailed proofs of these results. In Section 4 we present the proof of the local Souplet-Zhang type gradient estimate in Theorem 3.1 and in Section 5 we give the proof of the local and global elliptic Harnack inequality in Theorem 3.3. In Section 6 we present the proof of the local Hamilton-type gradient estimate in Theorem 3.4 and in Section 7 we give the proof of the various Liouville-type results formulated in Theorems 3.7 and 3.8. In Section 8 we present the proof of the local Li-Yau differential Harnack estimate in Theorem 3.12 and in Section 9 we establish the local and global parabolic Harnack inequalities as formulated in Theorem 3.14. Section 10 is devoted to the proof of the general Liouville result in Theorem 3.16 and its subsequent corollaries.
For the convenience of the reader and future reference we describe at the end of this section some of the key notation and quantities used in the paper.
1.2 Notation
We denote by d = d(x, y) the geodesic distance between x, y in M. For the sake of the local estimates below we typically fix a reference point x
0 in M and denote by r = r(x) the geodesic radial variable measuring the distance between x, x
0 in M. We write
When t 0 = T > 0 it is more convenient to write (1.13) as
When the choices of x
0 and t
0 are clear from the context, in the interest of brevity we simply write
2 Preliminaries and background
A smooth metric measure space or a weighted manifolds is a triple (M, g, dμ) where (M, g) is a smooth Riemannian manifold of dimension n ≥ 2 and dμ = e−ϕ dv g is the weighted measure associated with the potential ϕ (that is, the positive weight w = e−ϕ ) and the Riemannian volume measure dv g on M. The notion is an important one in geometric analysis and arises in various contexts (see, e.g., [12], [16]–[19], [22], [60], [61]).
Associated with the triple (M, g, dμ) there is a ϕ-Laplacian (also called the weighted, drifting or Witten Laplacian) defined by (1.2). The ϕ-Laplacian is a symmetric diffusion operator with respect to the invariant weighted measure dμ and reduces to the ordinary Laplacian (the Laplace-Beltrami operator) precisely when the potential ϕ is a constant function. By an application of the integration by parts formula it can be seen that for any compactly supported smooth functions
As for the geometry and curvature properties of the triple (M, g, dμ) we introduce the generalised Ricci curvature tensor field on M by writing
(the Bakry-Èmery m-Ricci curvature) where
The weighted Bochner-Weitzenböck formula in this context asserts that for every function
Making note of the basic inequality (Δw)2/n ≤ |∇∇w|2 and by recalling the defining relation Δ ϕ w = Δw − ⟨∇ϕ, ∇w⟩ it is then easily seen that
Subsequently it follows from (2.2), (2.4) and (2.5) that here we have the inequality
Now a curvature lower bound in the form
Higher order iterates are then defined inductively by replacing the product operation with Γ respectively (cf. [39]). As a matter of fact, the second order iterated carre du champ operator Γ2[L], can be seen to be given by,
By a direct calculation, it is now observed that for the diffusion operator L = Δ ϕ as in (1.2) and with Γ[L] and Γ2[L] as in (2.7) and (2.8), we have the first and second order relations
and
respectively. Hence, in light of (2.6), (2.9) and (2.10) it follows from the curvature lower bound
In contrast, a curvature lower bound in the form
3 Statement of the main results
Here we present the main results of the paper leaving the proofs, technicalities and further details involved to the subsequent sections. For the convenience of the reader and clarity of presentation, below we have grouped the results into four subsections.
3.1 Two gradient estimates of elliptic types for
(
P
)
We begin by presenting two estimates of elliptic type for positive solutions to
Theorem 3.1.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Here C > 0 is a constant depending only on the dimension n,
with
The local estimate above has a global counterpart subject to the prescribed bounds in the theorem being global. The proof follows by passing to the limit R → ∞ in (3.1) and making note of the constants being independent of R whilst 1/R,
Theorem 3.2.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Here C > 0 is a constant depending only on the dimension n.
One of the useful consequences of the estimates above is the following elliptic Harnack inequality for bounded positive solutions to
Theorem 3.3.
Under the assumptions and local bounds in Theorem 3.1, for every x
1, x
2 in
where the exponent α = α(R) ∈ (0, 1) is given explicitly by
If the prescribed bounds are global as in Theorem 3.2, then for every x 1, x 2 in M and t 0 − T < t ≤ t 0, the same inequality (3.4) holds now with the exponent α = α(M) given by
The second type of gradient estimates to be considered in this paper are of Hamilton type. Here again x
0 ∈ M,
Theorem 3.4.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
where C = C(α, n) > 0 and
Remark 3.1.
It is useful to point out that under the stronger curvature condition
that as will be seen in the proofs, arises from an application of the Wei-Wiley weighted Laplace comparison theorem under the lower bound on
Again, the local estimate above has a global counterpart, when the asserted bounds in the theorem are global. The proof follows by passing to the limit R → ∞ in (3.7). This is the content of the following theorem.
Theorem 3.5.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
where C = C(α, n) > 0.
3.2 Applications to Liouville-type results (I):
R
i
c
ϕ
(
g
)
≥
0
Let us now move on to discussing some applications of the gradient estimates above. Here we focus mainly on applications to Liouville and global constancy type results for both the parabolic and elliptic (stationary states) equations associated with
Towards this end let us first suppose that w is a smooth bounded positive solution to the elliptic equation
Our main goal is to present two independent elliptic Liouville-type results for positive solutions to (3.10). Note that since a constant solution to (3.10) must be a zero of
Theorem 3.6.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
The second Liouville-type result of interest as mentioned above is formulated below. The proof of both these theorems will be presented in Section 7.
Theorem 3.7.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Let us now discuss implications and some examples of the above theorems to specific nonlinearities of interest arising in various contexts within geometry and mathematical physics. For further discussion on this see the introduction to the paper.
Here we focus on nonlinearities in the “split” form
Theorem 3.8.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Suppose X′(w) ≤ 0 and X(w) − wX′(w) ≥ 0. Moreover, when Y ≢ 0, depending on w satisfying w ≥ 1 or 0 < w ≤ 1 everywhere on M, assume that:
Then w is a constant. In particular X(w) + w r Y(log w) = 0.
Regarding the assumptions in the theorem note that the examples
As another example consider the case where the nonlinearity is a superposition of power-like terms in the form
Here
Theorem 3.9.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Assume
As a final application consider
Theorem 3.10.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Assume that along the solution w we have the inequality Y′ + [(β/2 + 1)/α + r − 1]Y ≤ 0 along with
Note that the assumption Y′ + [(β/2 + 1)/α + r − 1]Y ≤ 0 in Theorem 3.10 is implied by either of
We end this section with an application of the estimates above to parabolic Liouville-type theorems. Here by an ancient solution w = w(x, t) to
Theorem 3.11.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
does not have any positive ancient solutions satisfying
3.3 A parabolic Li-Yau type gradient estimate for
(
P
)
Let us turn to our final gradient estimate for positive solutions to
Here as before subscripts denote partial derivatives. Moreover
For the first estimate we fix x
0 ∈ M, R > 0, T > 0 and formulate an upper bound for the Harnack quantity [see the left-hand side of (3.28)] on the cylinder
Theorem 3.12.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
The constants c 1, c 2 > 0 in (3.28) are those appearing in the bounds (8.15) in Lemma 8.3 and the γ-quantities are as in (3.24)–(3.27) (with 2R replacing R).
The local estimate above has a global in space counterpart subject to the prescribed bounds in the theorem being global in space. The proof follows by passing R → ∞ in (3.28) and taking into account the vanishing of the term
in the limit due to the constants being independent of R. The precise formulation of this global estimate is given in the theorem below.
Theorem 3.13.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Here the γ-quantities on the right in (3.30) are as given by (3.24)–(3.27).
An immediate consequence of the local and global estimates in Theorems 3.12 and 3.13 is the following parabolic Harnack inequality on the solutions.
Theorem 3.14.
Under the assumptions of Theorem 3.12 let w be a positive solution to
Here
where
3.4 Applications to Liouville-type results (II):
R
i
c
ϕ
m
(
g
)
≥
0
We complement the Liouville-type results given earlier by invoking the Li-Yau gradient estimates above which were obtained under a different curvature condition. We begin by the following theorem on the stationary version of
Theorem 3.15.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Theorem 3.16.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
for some α > 1, everywhere on M. Then w must be a constant and
Theorem 3.16 leads to the following conclusion on Yamabe equations. As indicated earlier, since a constant solution to (3.10) must be a zero of
Theorem 3.17.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
If
As another application consider a nonlinearity in the form of a superposition of a logarithmic and a power-like nonlinearity with constant coefficients
The following theorem now directly results from Theorem 3.16.
Theorem 3.18.
Let (M, g, dμ) with dμ = e−ϕ
dv
g
be a complete smooth metric measure space satisfying
Assume that along the solution w we have Y ≥ 0, Y′ ≤ 0 and αY″ + (α − 1)Y′ ≥ 0 for some α > 1. Furthermore assume
4 Proof of the local Souplet-Zhang estimate in Theorem 3.1
We now attend to the proof of the results formulated in Section 3. Throughout this section we confine to the proof of Theorem 3.1 and all of its necessary ingredients. The first two subsections develop the necessary apparatus, including an evolution inequality on a quantity (below called H) built out of the solution w, and the last section completes the proof by putting these together, using a localisation argument and finally maximum principle.
4.1 Evolution inequality for H = |∇[log(1 − h)]|2 with h = log(w/D)
From the positive solution w satisfying the bounds 0 < w ≤ D we first define a non-positive function by putting h = log(w/D) and then H = |∇h|2/(1 − h)2. The evolution of H under the equation
Lemma 4.1.
Let w be a positive and bounded solution to
In particular, if
Proof.
As w is a solution to
As a result, by noting of
Now moving on to the function H = |∇h|2/(1 − h)2 = |∇log(1 − h)|2 it is easily seen that
Likewise we have ∇H = [∇|∇h|2]/(1 − h)2 + [2|∇h|2∇h]/(1 − h)3 and so by recalling the formulation Δ ϕ H = ΔH − ⟨∇ϕ, ∇H⟩ it follows that
Putting (4.5) and (4.6) together and taking into account the necessary cancellations give
Now by making use of the weighted Bochner–Weitzenböck formula we obtain
and therefore a rearrangement of terms and basic considerations leads to
Finally by making note of the relation (1 − h)3⟨∇h, ∇H⟩ = (1 − h)⟨∇h, ∇|∇h|2⟩ + 2|∇h|4 the expression on the second line on the right simplifies to 2h⟨∇h, ∇H⟩/(1 − h) and so a reference to H = |∇h|2/(1 − h)2 leads at once to the desired conclusion.
The inequality (4.2) now follows from the above by using the Ricci curvature lower bound
4.2 Construction of space-time cut-offs and cylindrical localisation
In order to prove the estimate in Theorem 3.1 we first need to establish a local version of the evolution inequality in Lemma 4.1 suitable for the application of maximum principle. This will be achieved through the use of suitable space-time cut-off functions. Towards this end, let us fix x
0 ∈ M,
Lemma 4.2.
Fix
there exists c > 0 such that
(4.10)in [0, ∞) × [t 0 − T, t 0] and
Having the above lemma at our disposal we now move on to introducing a smooth space-time cut-off function η = η(x, t) by setting, for R ≥ 2, T > 0 and t 0 − T < τ ≤ t 0 (the reason for the choice R ≥ 2 will be clear later),
It is plain that η is supported in the compact space-time cylinder Q
R,T
⊂ M × [t
0 − T, t
0]. Furthermore in view of (4.11) we have
Let us now move on to the following useful product identity for the action of the weighted heat operator on the space-time localised function ηw.
Lemma 4.3.
Let η = η(x, t) be as above and let H = H(x, t) be a space-time function of class
Proof.
Firstly [Δ ϕ − ∂ t ](ηH) = η[Δ ϕ − ∂ t ]H + 2⟨∇H, ∇η⟩ + H[Δ ϕ − ∂ t ]η as is easily seen by direct differentiation. Next, ⟨∇η, ∇(ηH)⟩ = ⟨∇η, H∇η + η∇H⟩ = H|∇η|2 + η⟨∇η, ∇H⟩ and therefore using the latter relation to substitute for the middle term 2⟨∇H, ∇η⟩ in the first identity gives the desired result. □
4.3 Finalising the proof of Theorem 3.1
Let us now fix τ ∈ (t
0 − T, t
0] and put
Hence for the localised function ηH we have upon invoking Lemma 4.3 the inequality
Assume that the localised function ηH is maximal at the point (x 1, t 1) in the compact space-time cylinder {(x, t): d(x, x 0) ≤ R, t 0 − T ≤ t ≤ τ}. Suppose also that x 1 is not in the cut locus of M by Calabi’s argument [15] and that (ηH)(x 1, t 1) > 0 as otherwise the result is trivial with w(x, τ) ≤ 0 for all d(x, x 0) ≤ R/2. It then follows that t 1 > t 0 − T at (x 1, t 1) and subsequently by maximum principle
In particular at (x 1, t 1) we have [Δ ϕ − ∂ t ](ηH) ≤ 0 and therefore from (4.14) we obtain the inequality
As a result dividing through by 2(1 − h) ≥ 0 it follows that
The goal is now to utilise (4.17) to establish the required estimate at (x, τ). Towards this end we proceed by considering two alternatives: d(x 1, x 0) ≤ 1 and d(x 1, x 0) ≥ 1.
Alternative 1.
(d(x
1, x
0) ≤ 1). Since here η is a constant function in the space [for all x with d(x, x
0) ≤ R/2 and R ≥ 2 by property (ii)] the terms involving space derivatives of η at (x
1, t
1) vanish (that is, ∇η = 0, Δ
ϕ
η = 0 and
and so
Next by an application of Cauchy-Schwarz and Young’s inequality and after rearranging terms it follows that for suitable C = C(n) > 0 we have
As η ≡ 1, when d(x, x
0) ≤ R/2 and τ ≤ t ≤ t
0 (and so in particular for when t = τ) by (i), we have
This together with the arbitrariness of τ > t 0 − T is now seen to give a special case of (3.1).
Alternative 2.
(d(x
1, x
0) ≥ 1). Upon referring to the right-hand side of (4.17), and noting the properties of
Towards this end dealing with the first term first, we have
(4.22)Here we utilise the Wei-Wylie weighted Laplacian comparison theorem along with the lower curvature bound on
(4.23)where we have used Theorem 3.1 in Ref. [66] to write
(4.24)and in a similar way using (iii) in Lemma 4.2,
(4.25)In much the same way regarding the third term involving
(4.26)Likewise for the subsequent terms, upon noting −1 ≤ h/(1 − h) ≤ 0, h ≤ 0 and 0 ≤ η ≤ 1, we have
(4.27)
Now referring to (4.17) noting the inequality 1 − h ≥ 1 and making use of the bounds obtained in (4.22)–(4.27), it follows the following upper bound holds for ηH 2 at (x 1, t 1),
Recalling the maximality of ηH at (x 1, t 1) along with η ≡ 1 when d(x, x 0) ≤ R/2 and τ ≤ t ≤ t 0, it follows that H 2(x, τ) = (η 2 H 2)(x, τ) ≤ (η 2 H 2)(x 1, t 1) ≤ (ηH 2)(x 1, t 1) when d(x, x 0) ≤ R/2. hence upon noting H = |∇h|2/(1 − h)2, the above gives
Thus in either case we have shown the estimate is true at (x, τ). The desired conclusion now follows from the arbitrariness of τ ∈ (t 0 − T, t 0]. The proof is thus complete.
5 Proof of the elliptic Harnack inequality in Theorem 3.3
We now come to the proof of the Harnack inequality in Theorem 3.3. To this end pick x
1, x
2 in M and t
0 − T < t < t
0. Let ζ = ζ(s) with 0 ≤ s ≤ 1 be a shortest geodesic curve with respect to g, lying completely in the closed ball
Henceforth we shall assume the
where in the language of (3.5) the last expression on the right is −log α(R). Moreover here we have used |∇h|/(1 − h) = |∇log w|/[1 − log(w/D)]. Therefore exponentiating (5.1) results in
and so the claim follows at once by a further exponentiation and rearranging terms.
6 Proof of the local Hamilton estimate in Theorem 3.4
This section is devoted to the proof of Theorem 3.4 and its ingredients. The first subsection develops an evolution inequality on a quantity (below called
6.1 Evolution inequality for
F
β
α
=
w
(
β
+
2
)
/
α
−
2
|
∇
w
|
2
/
α
2
From the positive solution w we first define a positive function through f = w
1/α
(with α > 1 fixed) and then
Lemma 6.1.
Let w be a positive solution to
In particular, if
Proof.
It follows from the defining relation f = w
1/α
that ∂
t
w = αf
α−1∂
t
f and Δ
ϕ
w = αf
α−1Δ
ϕ
f + α(α − 1)f
α−2|∇f|2. Thus, from
Next, an application of the weighted Bochner-Weitzenböck formula and the evolution of f described by (6.3) gives,
In a similar way we have
As by a straightforward calculation
it then follows by putting (6.4)–(6.6) together and rearranging the equation by moving the terms involving the nonlinearity
Next, by virtue of
Therefore by substituting (6.8) back into the (6.7), rearranging terms and recalling the relation F = f β |∇f|2 we can write
which is the required conclusion as stated in (6.1).
To justify the inequality in the second part we make use of (6.1) together with the Ricci curvature lower bound
resulting in turn from “completing the square” ideneity
Therefore substituting in (6.1) and again recalling F = f β |∇f|2 it follows that
By making note of ⟨∇f, ∇F⟩ = ⟨∇f, ∇(f β |∇f|2)⟩ = βf β−1|∇f|4 + f β ⟨∇f, ∇|∇f|2⟩ and substituting back in (6.10) for F we thus conclude
which is the required conclusion as stated in (6.2). This completes the proof. □
6.2 Finalising the proof of Theorem 3.4
Let us now turn on to completing the proof of the local estimate in Theorem 3.4. Here, for reasons that will become clear shortly, the ranges of the parameters α, β will be restricted to β ≥ 0, α > 1 + β (see the last stage of the proof). Towards this end recalling the inequality (6.2) in Lemma 6.1 we have
Next, localising by taking a space-time cut-off function η as in (4.11) and following similar principles to those used in the proof of Theorem 3.1, we can write
For fixed τ ∈ (t 0 − T, t 0] let (x 1, t 1) be a maximum point for the localised function ηF in the compact set {(x, t): d(x, x 0) ≤ R, t 0 − T ≤ t ≤ τ}. Then without loss of generality we can take t 1 > t 0 − T and for the sake of establishing the estimate at (x, τ) in Q R/2,T it suffices to confine to the case d(x 1, x 0) ≥ 1. Now at (x 1, t 1) we have the inequalities
Therefore applying these to (6.13) and rearranging the inequality result in
We now proceed onto bounding from above each of the terms on the right-hand side of (6.15). Again the argument proceeds by considering the two case d(x 1, x 0) ≤ 1 and d(x 1, x 0) ≥ 1. However in view of certain similarities with the proof of Theorem 3.1, we shall remain brief, focusing on case two only and mainly on the differences.
First the sum on the first line on the right-hand side of (6.15) is bounded directly in modulus by using the Cauchy-Schwarz followed by Young’s inequality:
(6.16)where we have used
(6.17)For the terms involving Δ ϕ η and ∂ t η as in the proof of Theorem 3.1 we have the bounds
Hence, by substituting and making use of Young’s inequality we have,
(6.18)Making note of
(6.19)
Now putting together (6.16)–(6.19) it follows from (6.15) after some basic calculations and rearrangement of terms that at the space-time point (x 1, t 1) we have
By the maximality of ηF at (x
1, t
1) we have for any x with d(x, x
0) ≤ R/2 the chain of inequalities
where we have set
Note that the condition 0 < (1 + β) < α guarantees that (1 + β)[1 − (1 + β)/α] > 0 and so in turn s(α, β) > 0. This upon rearranging terms, taking square roots and noting the arbitrariness of τ gives the desired estimate for every
7 Proof of the Liouville theorems and implications (I):
R
i
c
ϕ
(
g
)
≥
0
This section is devoted to the proofs of the Liouville constancy results formulated in Theorem 3.6 through to Theorem 3.11. Note that in Theorems 3.6–3.10 it is the elliptic counterpart of
Proof of Theorem 3.6.
Let h = log(w/D) and note that in view of 0 < w ≤ D we have h ≤ 0 and so 1/(1 − h) ≤ 1. Referring to the estimate (3.1), making note of
From
Proof of Theorem 3.7.
As the methodology and main ideas here are to some extenet similar to those used in the proof of Theorem 3.6, we shall remain brief, focusing mainly on the differences. Referring to the estimate (3.7), making note of
Now
Proof of Theorem 3.8.
Consider the function
By inspection it is seen that (1 − h)wX′ + hX = wX′ − h(wX′ − X) ≤ 0 for w > 0 as a result of the assumptions on X in the theorem. Therefore
Next we have [r + h/(1 − h)]Y(log w) + Y′(log w) ≤ 0 for w ≥ 1, r ≥ 1 when Y satisfies
Similarly from
by writing h = log(w/D) and choosing D ≥ 1. Therefore again here we have
Thus referring to in (7.3) we have
Proof of Theorem 3.9.
The idea is to invoke Theorem 3.7 with the choices X = X(w) as in (3.14) and Y ≡ 0. Indeed referring to the formulation of Theorem 3.7 it is seen by a direct calculation that
and thus
This is then easily seen to be non-negative, as required by Theorem 3.7, upon suitably restricting the ranges of
Remark 7.1.
Note that the upper and lower bounds on the exponents p
j
, q
j
are given by the same quantity 1 − (β/2 + 1)/α which can be adjusted by optimising the parameters α, β within their respective range. In particular if
Proof of Theorem 3.10.
Writing
The conclusion follows from an application of the first part of Theorem 3.7 upon noting that according to assumptions
Proof of Theorem 3.11.
Fix a space-time point (x
0, t
0). Then with the quantities k = 0, t = t
0 and T = R in (3.1) and upon writing
where as in the proof of Theorem 3.8 we have deduced from the assumptions
8 Proof of the local Li-Yau estimate in Theorem 3.12
We now attend to the proof of the local Li-Yau estimate (also called a differential Harnack inequality) as formulated in Theorem 3.12. The first two subsections develop the necessary tools including an evolution inequality on a suitably defined Harnack quantity (below called F) built out of the solution w, and involving a parameter α > 1, and the last section completes the proof by putting these together, using a localisation argument and eventually maximum principle.
8.1 Evolution of the Harnack quantity
F
=
t
|
∇
w
|
2
/
w
2
−
α
∂
t
w
/
w
+
α
G
/
w
From the positive solution w we first define the function f = log w and then define the Harnack quantity
Lemma 8.1.
Let w be a positive solution to
Then F satisfies the evolution equation
In particular, if
Proof.
From
Thus using (8.1) and (8.4) it is seen that
Next, an application of Δ ϕ to F in (8.1) gives
where in concluding the second line we have used the weighted Bocnher-Weitzenböck formula as applied to f. Hence by virtue of the (8.4) and (8.5) this can be simplified and re-written as
Rearranging this now lead to
which is the desired conclusion.
For the inequality in (8.3) we first note that by an application of the Cauchy-Schwarz and Young inequalities we have
Hence the desired conclusion follows by substituting the above in (8.8) and applying the Ricci curvature lower bound
In the course of the proof of Theorem 3.12 we make use of the following lemma. As before we use the notation
Lemma 8.2.
Suppose
Proof.
Firstly, it is seen, by calculating in local coordinates or directly otherwise that with
where we have abbreviated the arguments (t, x, w) for convenience. Now differentiating further with the aim of calculating the Laplacian (and noting
Subsequently calculating the Witten Laplacian, by utilising the above fragments we have,
which is the required result. □
8.2 Construction of spatial cut-offs and localisation in space
The next and final ingredient needed in the proof of Theorem 3.12 is the construction of a suitable cut-off function in space. To this end we pick a reference point x 0 ∈ M and fix R > 0 and with r(x) = d(x, x 0) being the geodesic radial variable with respect to x 0 we set
The function
Lemma 8.3.
There exists a function
- (8.15)
It is evident that here ζ ≡ 1 for when 0 ≤ r(x) ≤ R and ζ ≡ 0 for when r(x) ≥ 2R. Furthermore a straightforward calculation starting from (8.14) gives
and so
8.3 Finalising the proof of Theorem 3.12
We now come to the proof of the main result in the section which is the local Li-Yau estimate in Theorem 3.12. To this end we consider the spatially localised function ζF where F is as in (8.1). We denote by (x 1, t 1) the point where this function attains its maximum over the compact cylinder {d(x, x 0) ≤ 2R, 0 ≤ t ≤ T}. We also assume that [ζF](x 1, t 1) > 0 as otherwise the desired estimate is trivially true as a result of F ≤ 0. It thus follows that t 1 > 0 and d(x 1, x 0) < 2R and so at the maximum point (x 1, t 1) we have the relations
From the inequalities above we deduce that
Now making note of [Δ ϕ − ∂ t ]ζ = Δ ϕ ζ (as ∂ t ζ ≡ 0) it follows from the above that at the maximum point (x 1, t 1) we have
Next by referring to (8.16), (8.17) and
Moreover upon noting
Likewise, referring to (8.15) and (8.16), a straightforward calculation gives,
Now returning to (8.20), invoking (8.3) and making note of (8.22) and (8.23), we obtain, at the maximum point (x 1, t 1), the inequality
We point out that here and below we abbreviate the arguments of
Now upon utilising the bound
Referring next to the expression on the last line in (8.26) by working on the sum inside the brackets we can write
Here we have made note of the relation
Next, since according to (8.5) we have,
upon substituting this back in (8.27) it follows that
Therefore, by taking into account the relevant cancellations, after simplifying terms and using basic inequalities, we can write
As a result making use of the relations (8.11)–(8.13) and the inequality (8.30) above and substituting all back into (8.26) and recalling 0 ≤ ζ ≤ 1 we obtain:
In order to obtain the desired bounds out of this it is more efficient to introduce the quantities y, z and
Now substituting the quantities (8.32) back in (8.31) and recalling 0 ≤ ζ ≤ 1, it follows that
In order to proceed further we now state and prove the following lemma.
Lemma 8.4.
Suppose
Proof.
Starting from the expression on the left-hand side in (8.34) we can write for any δ, ɛ by basic considerations
In particular setting δ = (1/α − 1)2 and ɛ = 2 − 2/α − 2(1/α − 1)2 = 2(α − 1)/α
2 gives 1 − ɛ − δ − 1/α + ɛ/2 = 0 and 1 − α + ɛα
2/2 = 0 and so by making note of the inequality
Next, considering the last three terms only we can write, for any ɛ ∈ (0, 1),
where above we have made use of (1 − ɛ)(α − 1)2
y
2/α
2 − by ≥−(α
2
b
2)/[4(1 − ɛ)(α − 1)2] and
Now returning to the inequality (8.33) and making use of Lemma 8.4 it follows that
We now aim to rewrite (8.38) as an inequality involving a quadratic expression in powers of the localised function ζF. To this end upon setting
and
where
it is seen that (8.38) can be expressed as
Now basic considerations using (8.42) on quadratics result in
Since ζ ≡ 1 for d(x, x 0) ≤ R and (x 1, t 1) is the point where ζF attains its maximum on d(x, x 0) ≤ 2R we have
Therefore recalling (8.1), substituting for
Finally using the arbitrariness of 0 < τ ≤ T it follows after reverting back to w upon noting the relation f = log w and rearranging terms that
Using (3.20) and (3.21) in conjunction with the bounds (3.24), (3.25) and making note of (8.32) it is seen that
which upon substitution in (8.46) gives the desired estimate as formulated in (3.28).
9 Proof of the parabolic Harnack inequality in Theorem 3.14
This will be shown as a consequence of the local estimate in Theorem 3.12. Here the main idea is to integrate (3.28) along suitable space-times curves described below. Towards this end and upon referring to the expression on the right-hand side of (3.28) let us introduce the constant
where in analogy with the other γ-quantities occurring earlier, the first term on the right stands for
It then follows from Theorem 3.12 that in the space-time cylinder H R,T we have the inequality
Note that in (9.1) we can replace
Therefore integrating the above inequality gives
Hence upon exponentiating we have
or after rearranging terms and rescaling the integral:
where
This gives the parabolic Harnack inequality in its local form. Now if the bounds are global as in Theorem 3.13 then by passing to the limit R → ∞ we obtain the global counterpart of the inequality with the constant
10 Proof of the Liouville theorems and implications (II):
R
i
c
ϕ
m
(
g
)
≥
0
In this section we give the proofs of the Liouville-type results formulated in Theorem 3.16 and its implications as given in Theorems 3.17 and 3.18. The basis for all these proofs is the elliptic estimate in Theorem 3.15 below that in term is a consequence of the parabolic Li-Yau estimates in Theorems 3.12 and 3.13. Towards this end we begin by presenting the proof of the estimate (3.33). We emphasise that in all these results we are assuming the curvature condition
Proof of Theorem 3.15.
The proof of (3.33) for the elliptic equation
thus giving
Putting the above fragments together gives the desired estimate. □
Proof of Theorem 3.16.
Since here
Next, since by assumption
thus giving
Since
Proof of Theorem 3.17.
A direct calculation gives
Now if
Proof of Theorem 3.18.
Here
Acknowledgments
The authors wish to thank the anonymous reviewers for a careful reading of the paper and useful comments.
-
Research ethics: Not applicable.
-
Author contributions: Both authors have equal contribution in this research, have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Competing interests: Authors state no conflict of interest.
-
Research funding: The authors gratefully acknowledge financial support from the Engineering and Physical Sciences Research Council (EPSRC) through the research grant EP/V027115/1.
-
Data availability: All data in this research is provided in full in the results section. Additional data is in the public domain at locations cited in the reference section.
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This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- Research Articles
-
A note on the classification of positive solutions to the critical p-Laplace equation in
- Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces
- A degenerate migration-consumption model in domains of arbitrary dimension
- Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials
- Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system
- On the large solutions to a class of k-Hessian problems
- New multiplicity results in prescribing Q-curvature on standard spheres
-
Michael-Simon type inequalities in hyperbolic space
- Two solutions for Dirichlet double phase problems with variable exponents
- Limiting behavior of quasilinear wave equations with fractional-type dissipation
- Global structure of positive and sign-changing periodic solutions for the equations with Minkowski-curvature operator
Articles in the same Issue
- Frontmatter
- Research Articles
-
A note on the classification of positive solutions to the critical p-Laplace equation in
- Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces
- A degenerate migration-consumption model in domains of arbitrary dimension
- Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials
- Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system
- On the large solutions to a class of k-Hessian problems
- New multiplicity results in prescribing Q-curvature on standard spheres
-
Michael-Simon type inequalities in hyperbolic space
- Two solutions for Dirichlet double phase problems with variable exponents
- Limiting behavior of quasilinear wave equations with fractional-type dissipation
- Global structure of positive and sign-changing periodic solutions for the equations with Minkowski-curvature operator
