GRADIENT ESTIMATES FOR POSITIVE SOLUTIONS OF THE ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 2, February 1999, Pages 619–625 S 0002-9939(99)04578-5

GRADIENT ESTIMATES FOR POSITIVE SOLUTIONS OF THE LAPLACIAN WITH DRIFT ´ BENITO J. GONZALEZ AND EMILIO R. NEGRIN (Communicated by Palle E. T. Jorgensen) Abstract. Let M be a complete Riemannian manifold of dimension n without boundary and with Ricci curvature bounded below by −K, where K ≥ 0. If b is a vector field such that kbk ≤ γ and ∇b ≤ K∗ on M, for some nonnegative constants γ and K∗ , then we show that any positive C ∞ (M ) solution of the equation ∆u(x) + (b(x)|∇u(x)) = 0 satisfies the estimate γ2 n(K + K∗ ) k∇uk2 + , ≤ u2 w w(1 − w) on M , for all w ∈ (0, 1). In particular, for the case when K = K∗ = 0, this estimate is advantageous for small values of kbk and when b ≡ 0 it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201–228).

1. Introduction In this paper we investigate the behaviour of positive C ∞ (M ) solutions of the equation (1.1)

∆u(x) + (b(x)|∇u(x)) = 0

on M , where M is an n-dimensional complete Riemannian manifold without boundary. We require smoothness of the manifold, uniform bound on the norm of the vector field b as well as lower bounds on the tensor fields of the Ricci curvature and ∇b where (1.2)

∇b(X, Y ) = (∇X b|Y ),

∀X, Y ∈ X(M ),

where X(M ) denotes the Lie algebra of vectors fields on M and ∇X b the associated (Levi-Civita) Riemannian covariant derivative of b with respect to X. Our main result is a gradient estimate for positive C ∞ (M ) solutions of equation (1.1), namely, 2

(1.3)

γ2 n(K + K∗ ) k∇uk + , ≤ u2 w w(1 − w)

Received by the editors May 27, 1997; part of the results of this paper have been presented to Equadiff 95, Lisboa, July 24–29, 1995. 1991 Mathematics Subject Classification. Primary 58G11. Key words and phrases. Gradient estimate, Laplacian with drift, Bochner-Lichn` erowiczWeitzenb¨ ock formula, Liouville theorem. c

1999 American Mathematical Society

619

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´ BENITO J. GONZALEZ AND EMILIO R. NEGRIN

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on M , for any w ∈ (0, 1), where the Ricci curvature is bounded below by −K, ∇b ≤ K∗ and kbk ≤ γ for some nonnegative constants K, K∗ and γ. For the particular case when K = K∗ = 0 inequality (1.3) yields k∇uk2 ≤ 4γ 2 . u2 Note that this simple estimation is independent of the dimension of M and for the case when b ≡ 0 it recovers the Liouville theorem of Yau [10]. The proof of (1.3), and thus of (1.4), is essentially along the lines of Li and Yau [7] and Davies [3, Chap. 5]. This method, originated first in Yau [10] and Cheng and Yau [2], has been developed by several authors (cf. [3], [6], [7], [8], and [9], amongst others). More specifically, for the case when b = ∇φ, and φ ∈ C ∞ (M ), a gradient estimate for any positive C ∞ (M ) solution of (1.1) has been obtained by Setti in [9]. In order to start, however, we need an extension of the Bochner-LichnèrowiczWeitzenb¨ ock formula for the operator Lb = ∆ + (b|∇ ). This remarkable fact is proved as an independent lemma. It is known for drift vectors b = ∇φ of gradient form; see e.g. the monograph of Deuschel and Stroock (cf. [4], §6.2). (1.4)

2. Gradient estimates revisited In the derivation of the main results, a central role will be played by the next formula. Lemma 2.1 (Bochner-Lichnèrowicz-Weitzenböck formula for Lb ). Let M be a Riemannian manifold and assume that f ∈ C ∞ (M ). Then, (2.1) Lb (k ∇f k2 ) = 2k Hess(f ) k2H.S. + 2(∇f |∇(Lb f )) + 2( Ric − ∇b)(∇f, ∇f ), where k Hess(f ) kH.S. denotes the Hilbert-Schmidt norm of Hess(f ) (cf. [4, p. 262]), Ric denotes the Ricci curvature and ∇b denotes the tensor field given by (1.2). Proof. Applying the well-known Bochner-Lichnèrowicz-Weitzenböck formula for the Laplacian, one obtains Lb (k ∇f k2 ) = 2k Hess(f ) k2H.S. + 2(∇f |∇(∆f )) + 2 Ric(∇f, ∇f ) + (b|∇(k ∇f k2 )). So, in order to prove (2.1) first we establish that 1 2 ∇b(∇f, ∇f ) = (∇((b|∇f ))|∇f ) − (b|∇(k ∇f k )). 2

(2.2) Now, one has

∇b(∇f, ∇f ) = (∇∇f b|∇f ) = ∇∇f ((b|∇f )) − (b|∇∇f ∇f ) = −(b|∇∇f ∇f ) + ∇f ((b|∇f )) = −(b|∇∇f ∇f ) + (∇((b|∇f ))|∇f ). Thus, all that remains is to show that 1 2 (2.3) ∇∇f ∇f = ∇(k ∇f k ). 2 In order to check (2.3) let any Z ∈ X(M ); then 2

2

(∇(k ∇f k ), Z) = Z(k ∇f k ) = Z((∇f |∇f )) = 2(∇Z ∇f |∇f ) = 2((∇∇f Z + [Z, ∇f ])|∇f ) = 2(∇∇f Z|∇f ) + 2([Z, ∇f ]|∇f )


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= 2{∇f ((Z|∇f )) − (Z|∇∇f )} + 2([Z, ∇f ]|∇f ) = 2{∇f (Zf ) − (Z|∇∇f ∇f )} + 2([Z, ∇f ]|∇f ) = 2{[∇f, Z]f + Z((∇f )f ) − (Z|∇∇f )} + 2([Z, ∇f ]|∇f ) 2

= −2(Z|∇∇f ∇f ) + 2Z(k ∇f k ), where, for any X, Y ∈ X(M ), [X, Y ] ≡ XY − Y X is the commutator of X and Y . Therefore, 2

2

(∇(k ∇f k )|Z) = −2(Z|∇∇f ∇f ) + 2(∇(k ∇f k )|Z), and so

2

(∇(k ∇f k )|Z) = 2(∇∇f ∇f |Z), ∀Z ∈ X(M ), from which (2.3) follows. Remark 2.1. For the case when b = −∇U, U ∈ C ∞ (M ), denoting LU = L−∇U and taking into account that Hess(U )(X, Y ) = (∇X (∇U )|Y ), for all X, Y ∈ X(M ) (cf. [4, p. 261]), formula (2.1) is written as 2

2

LU (k ∇f k ) = 2k Hess(f ) kH.S. + 2(∇f |∇(LU f )) + 2(Ric + Hess(U ))(∇f, ∇f ), which agrees with the Bochner-Lichnèrowicz-Weitzenböck formula derived in [4, p. 262] and [5, p. 32]. Now, formula (2.1) enables us to prove the next local gradient estimate. Theorem 2.1. Let M be a complete Riemannian manifold of dimension n without boundary. Let Bp (2R) be a geodesic ball of radius 2R around p ∈ M and denote by −K(2R), with K(2R) ≥ 0, a lower bound on Bp (2R) of the Ricci curvature. Set b a vector field on M and denote by γ(2R) and K∗ (2R) some nonnegative constants satisfying k b k≤ γ(2R) and ∇b ≤ K∗ (2R) on Bp (2R), where ∇b is the tensor field given by (1.2). If u(x, t) is a positive C ∞ solution of the equation ∂u(x, t) , ∂t on M × [0, ∞), then for any α > 1 and any w ∈ (0, 1), the estimate (2.4)

∆u(x, t) + (b(x)|∇u(x, t)) =

ut (x, t) k ∇u k2 (x, t) −α u2 (x, t) u(x, t) ( √ nα2 22 nα2 (n − 1)(1 + R K) ν K + K∗ + ≤ + + 2+ 2 2 2wt 2w R R R 2(α − 1) (2.5)

2 ) n α γ 2γ + + + R 8(1 − w)(α − 1) n R

holds on Bp (R) × (0, ∞), where > 0 and ν > 0 are some constants. Proof. Observe that the function f (x, t) = log u(x, t) satisfies the equation Lb f + k ∇f k2 = ft . Now, using formula (2.1), it follows that F (x, t) = t{k ∇f k2 (x, t) − αft (x, t)},



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satisfies the estimate Lb F − Ft + 2(∇f | ∇F ) + t−1 F = t Lb (k ∇f k2 ) − αLb ft − 2(∇f | ∇ft ) + αftt

+2(∇f | ∇(k ∇f k2 )) − 2α(∇f | ∇ft ) 2(∆f )2 − 2(K + K∗ ) k ∇f k2 ≥t n 2 2 2 2 {k ∇f k +(b | ∇f ) − ft } − 2(K + K∗ ) k ∇f k =t n on Bp (2R) × (0, ∞), where we have used the inequalities kHess f k2H.S.≥ (∆f )2 /n and (Ric −∇b) ≥ −(K + K∗ ). Let ψ be a C ∞ (R) function such that 1 if r ∈ (−∞, 1], ψ(r) = 0 if r ∈ [2, ∞), and 0 ≤ ψ(r) ≤ 1, ∀r ∈ R. Denote by > 0 and ν > 0 some constants with d 0 ≥ ψ −1/2 (r) ψ(r) ≥ − dr and d2 ψ(r) ≥ −ν. 2 dr d(p, x) , where d(p, x) is the distance between p and x. Now we set φ(x) = ψ R Using an argument of Calabi [1] (see also Cheng and Yau [2] and Setti [9]), we can assume without loss of generality that the function φ, with support in Bp (2R), is of class C 2 . Let (a, s) be the point in Bp (2R) × [0, t] at which φF takes its maximum value, and assume that this value is positive (otherwise the proof is trivial). Then at (a, s) one has ∇(φF ) = 0, ∆(φF ) ≤ 0, Ft ≥ 0. Therefore at (a, s) one has φ∆F + F ∆φ − 2F k ∇φ k2 φ−1 ≤ 0. This inequality together with the estimates k ∇φ k2 ≤

(2.6) and

√ (n − 1)(1 + R K)2 ν ∆φ ≥ − − 2 R2 R

(2.7) (cf. [1]) yields (2.8)

2 φ , R2

φ∆F ≤

! √ (n − 1)(1 + R K) ν 22 + + 2 F, at (a, s). R2 R2 R

Inequalities (2.6) and (2.7) at (a, s) imply that φ∆F − (b|∇φ)F − φFt − 2(∇f |∇φ)F + s−1 φF


GRADIENT ESTIMATES

≥

(2.9)

623

2 2 k ∇f k2 +(b | ∇f ) − ft − 2(K + K∗ ) k ∇f k2 sφ. n

From (2.8) the left-hand side of (2.9) satisfies φ∆F − (b | ∇φ)F − φFt − 2(∇f | ∇φ)F + s−1 φF ! √ (n − 1)(1 + R K) ν 22 + + 2 F − (b | ∇φ)F − 2(∇f | ∇φ)F + s−1 φF. R2 R2 R

≤

Denoting µ =

k ∇f k2 (a, s) , using (2.9) and the last inequality, we obtain F (a, s) ! √ ν 22 (n − 1)(1 + R K) + + 2 F R2 R2 R

2(µφ)1/2 F 3/2 γφ1/2 F + + s−1 φF R R ( 2 ) µs − 1 µs − 1 4(µF )1/2 γ 2 2 µ− µ− F sφ − 2 (K + K∗ ) sµφF. ≥ F − n αs n αs +

Multiplying this inequality by sφ and since φ2 ≤ 1, we obtain 2(1 + (α − 1)µs)2 (φF )2 2sγµ1/2 (1 + (α − 1)µs) sµ1/2 − 2 + (φF )3/2 (2.10) α2 n αn R ! ( √ ν (n − 1)(1 + R K)R2 22 + 2 s+1 − + R2 R γs + 2 (K + K∗ ) µs2 (φF ) ≤ 0. + R On the other hand, for any w ∈ (0, 1) we have 2sγµ1/2 (1 + (α − 1)µs) sµ1/2 + (φF )3/2 −2 αn R ≥− (2.11) −

2(1 − w)(1 + (α − 1)µs)2 (φF )2 α2 n

2 n 2sγµ1/2 (1 + (α − 1)µs) sαµ1/2 + (φF ). 2(1 − w)(1 + (α − 1)µs)2 n R

From (2.11) inequality (2.10) becomes A1 λ2 − 2A2 λ ≤ 0, where λ = φF, and 2A2 =

A1 =

2w (1 + (α − 1)µs)2 , α2 n

! √ γs (n − 1)(1 + R K) ν 22 + 2 (K + K∗ ) µs2 + + 2 s+1+ R2 R2 R R

2 n 2sγµ1/2 (1 + (α − 1)µs) sαµ1/2 + + . 2(1 − w)(1 + (α − 1)µs)2 n R



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As in [3, Lemma 5.3.3], we use the estimate s µs2 , ≤ 2 (1 + (α − 1)µs) 4(α − 1) and so we obtain nα2 s nα2 2A2 + ≤ A1 2w 2w (2.12)

(

! √ (n − 1)(1 + R K) ν 22 + + 2 R2 R2 R

2 ) γ n α K + K∗ 2γ + + + . + 2(α − 1) R 8(1 − w)(α − 1) n R

Now, since λ ≤ 2A2 /A1 , s ∈ [0, t] and using (2.12), estimate (2.5) holds. From Theorem 2.1 one obtains the next global gradient estimate Corollary 2.1. Let M be a complete Riemannian manifold of dimension n without boundary and assume that the Ricci curvature of M is bounded from below by −K with K ≥ 0. Also we suppose that the vector field b satisfies kbk ≤ γ and that the tensor field ∇b, given by (1.2), is bounded from above by K∗ , for some nonnegative constants γ and K∗ . If u(x) is a positive C ∞ (M ) solution of equation (1.1), then for any w ∈ (0, 1), the following estimate holds on M : 2

γ2 n(K + K∗ ) k∇uk + . ≤ 2 u w w(1 − w) Proof. Letting R → ∞ and t → ∞ in (2.5) one has (2.13)

α2 γ 2 nα2 (K + K∗ ) k∇uk2 + , ≤ u2 4w(α − 1) 4w(1 − w)(α − 1)

on M . Setting α = 2 (which minimizes the right-hand side of (2.13)), the result holds. Remark 2.2. If u(x) is a positive C ∞ (M ) solution of ∆u(x) + (b(x) | ∇u(x)) = 0, and assuming that Ric ≥ 0, ∇b ≤ 0 and k b k≤ γ, for some γ ≥ 0, it follows from Corollary 2.1 above that (2.14)

γ2 k∇uk2 , ≤ u2 w(1 − w)

for any w ∈ (0, 1). Setting w = 1/2 (which minimizes the right-hand side of (2.14)) one obtains 2 k∇uk ≤ 4γ 2 . u2 Remark 2.3. Let M = R be the one-dimensional Euclidean space with its standard Riemannian metric. It is a complete Riemannian manifold without boundary and with Ricci curvature identically zero. In this setting consider the equation (2.15)

u00 (x) + bu0 (x) = 0,

where b is a real constant. It is clear that u(x) = e−bx is a positive C ∞ (R) solution 2 k∇uk of (2.15), such that = b2 . This case is contemplated by Corollary 2.1 with u2 2 k∇uk ≤ 4b2 . K = K∗ = 0 and γ = |b|, which establishes that u2


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On the other hand, the equation (2.16)

u00 (x) − (1 + ex )u0 (x) = 0 2

k∇uk is u2 x 0 unbounded. Note that the function b(x) = −(1 + e ) satisfies b ≤ 0 and b is unbounded. Thus, we see that the assumption of the boundedness of kbk is needed for the kind of results obtained here. x

has the function u(x) = ee as a positive C ∞ (R) solution such that

References [1] E. Calabi, An extension of E. Hopf ’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1957), 45–56. MR 19:1056e [2] S.-Y. Cheng and S.-T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333–354. MR 52:6608 [3] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, UK, 1990. MR 92a:35035 [4] J.-D. Deuschel and D.W. Stroock, Large Deviations, Academic Press, Boston, 1989. MR 90h:60026 [5] J.-D. Deuschel and D.W. Stroock, Hypercontractivity and Spectral Gap of Symmetric Diffusions with Applications to the Stochastic Ising Models, J. Funct. Anal. 92 (1990), 30–48. MR 91j:58174 [6] J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal. 100 (1991), 233–256. MR 92k:58257 [7] P. Li and S.-T. Yau, On the parabolic kernel of the Schr¨ odinger operator, Acta Math. 156 (1986), 153–201. MR 87f:58156 [8] E.R. Negrin, Gradient estimates and a Liouville type theorem for the Schr¨ odinger operator, J. Funct. Anal. 127 (1995), 198–203. MR 96a:58175 [9] A.G. Setti, Gaussian estimates for the heat kernel of the weighted Laplacian and fractal measures, Canad. J. Math. 44 (5) (1992), 1061–1078. MR 94f:58124 [10] S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 55:4042 ´ lisis Matema ´ tico, Universidad de La Laguna, 38271 Canary IsDepartamento de Ana lands, Spain E-mail address: [email protected] E-mail address: [email protected]
