INTEGRAL MAXIMUM PRINCIPLE AND ITS APPLICATIONS

Alexander Grigor’yan

Abstract.The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

1. Introduction and main results

In the present paper we develop a general approach to some estimates of solutions to heat equation bases on so-called integral maximum principle . Suppose that M is a smooth connected complete non-compact Riemannian manifold and consider some precompact subregion Ω ⊂ M. Suppose also that u(x, t) is a (weak) solution to Dirichlet mixed boundary value problem in a cylinder Ω × (0, T):

u_{t} − Δu = 0, u|_{∂Ω×(0,T)} = 0  (1.1)

As it follows from the maximum principle, the function

sup_{x∈Ω} |u(x, t)|

is decreasing in t. Moreover, it is also well-known, the following integral

∫_{Ω} u^{2}(x, t)dx

is a decreasing function of t too. This fact can be regarded as an integral version of the usual maximum principle.

There is a further development of this idea which has been applied in a series of works to obtain heat kernel estimates (see , for example, [1] , [3] , [7] , [5] ) and consists of the fact that some weighted integral of u^{2} decreases in t. Namely, this is applicable to the integral

I(t) = ∫_{Ω} u^{2}(x, t)e^{ξ}dx  (1.2)

provided the function ξ(x, t) is locally Lipschitz and satisfies the relation

ξ_{t} + \frac{1}{2} |∇ξ|^{2} ≤ 0.  (1.3)

The simplest non-trivial examples of such functions ξ are as follows:

ξ = \frac{d(x)^{2}}{2t}

d(x) being a locally Lipschitz function such that |∇d(x)| ≤ 1 (for instance, a distance function from a set) and

ξ = αd(x) − \frac{α^{2}}{2} t

α being an arbitrary constant.

The following improvement of the maximum principle is proved in Section 2 below.

Grigor’yan A., Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh , 124A (1994) 353-362.

Theorem 1 (Integral maximum principle) Suppose that u(x, t) is a (weak) solution to the mixed problem (1.1) and a locally Lipschitz function ξ satisfies the relation (1.3) in Ω × (0, T), then the function

I(t) exp(2λ_{1}(Ω)t)

is decreasing in t ∈ (0, T) where I(t) is defined by (1.2) and λ_{1}(Ω) is the first Dirichlet eigenvalue of Ω.

B.Davies [4] proved the following universal integral bound for the heat kernel p(x, y, t) being the smallest positive fundamental solution to the heat equation (for details of the definition of the heat kernel see [2] ). Let A and B be two Borel sets in M with finite volumes and let the distance R = dist(A, B) be positive, then

∫_{A} ∫_{B} p(x, y, t)dxdy ≤ \sqrt{μAμB} exp\{−\frac{R^{2}}{4t} \} .  (1.4)

This estimate is of much importance due to its generality: no a priori geometric assumption are needed for (1.4) to be valid. It turns out that Davies’s estimate can be deduced with ease from the integral maximum principle. Moreover, Theorem 1 implies the improved version of (1.4) :

Theorem 2 Let A, B be two Borel subsets in M of a finite volume and R = dist(A, B), then

∫_{A} ∫_{B} p(x, y, t)dxdy ≤ \sqrt{μAμB} exp\{−\frac{R^{2}}{4t} − λ_{1}(M)t\} .  (1.5)

Here λ_{1}(M) is by definition the bottom of the spectrum of the Laplacian in L^{2}(M) that is called the spectral radius and coincides with inf λ_{1}(Ω) over all precompact subregions Ω.

If the spectral radius of a manifold is positive then the estimate of theorem 2 gives the sharp speed of decay of heat kernel as t → ∞ because as it is known

lim_{t→∞} \frac{log p(x, y, t)}{t} = −λ_{1}(M) .

Takeda [8] proved by a probabilistic method another kind of double integral estimate of heat kernel. Let A be an arbitrary compact of a positive volume on M and let us denote by A^{R} the open R-neighbourhood of A where R > 0. Let X_{t} be Brownian motion on manifold M governed by heat equation (1.1) . We shall consider the un-normalised law P_{A} of X_{t} under the condition that the initial point X_{0} is uniformly distributed in A, where ”un-normalised” means that the maximum value of P_{A} is equal to μA rather than to 1. Takeda’s inequality for this setting estimates the probability P (R, T) for X_{t} to exit A^{R} by a time T starting at a point of A that is the function

P (R, T) ≡ P_{A} (∃t ≤ T : X_{t} ∉ A^{R}| X_{0} ∈ A) .

The following sharpened version of Takeda’s inequality is due to T.Lyons [6]

P (R, T) ≤ 16μA^{R} ∫_{R}^{∞} \frac{1}{(4πT)^{\frac{1}{2}}} exp(−\frac{η^{2}}{4T} )dη  (1.6)

In Section 3 we obtain by means of the integral maximum principle an analytic proof of a similar inequality which however doesn’t cover (1.6) but sometimes is sharper.

Theorem 3 Let u(x, t) be a smooth subsolution to the heat equation in the cylinder A^{R} × [0, T] (where A ⊂ M is a compact and R, T are arbitrary positive numbers) i.e.

u_{t} − Δu ≤ 0

and suppose that

0 ≤ u(x, t) ≤ 1 and u(x, 0) = 0 ∀x ∈ A^{R}, t ∈ [0, T],  (1.7)

then

∫_{A} u^{2}(x, T)dx ≤ μ(A^{R} \ A) max(\frac{R^{2}}{2T} , \frac{2T}{R^{2}} ) exp(−\frac{R^{2}}{2T} + 1)  (1.8)

To explain connection of this theorem with inequality (1.6) we first mention that the following function

u(x, t) ≡ P(∃τ ≤ t : X_{τ} ∉ A^{R}|X_{0} = x)

(where P denotes a probability measure) satisfies the heat equation in the cylinder in question and the conditions (1.7) . Thus, Theorem 3 is applicable to this function. Noting that the function P (R, T) is equal to ∫_{A} u(x, T)dx and applying Cauchy-Schwarz inequality we get from (1.8)

P (R, T) ≤ \sqrt{μ(A)μ(A^{R} \ A)} max\{\frac{R}{\sqrt{2T}} , \frac{\sqrt{2T}}{R} \} exp(−\frac{R^{2}}{4T} + \frac{1}{2}) .  (1.9)

Compare this inequality to that of (1.6) . It is easy to check that for all R, T the following estimate is valid

∫_{R}^{∞} exp(−\frac{η^{2}}{4T} )dη ≤ \frac{2T}{R} exp(−\frac{R^{2}}{4T})

and, moreover, the ratio of the left and the right sides here tends to 1 as \frac{R^{2}}{T} → ∞. Therefore, (1.6) implies that

P (R, T) ≤ \frac{16}{\sqrt{π}} μ(A^{R})\frac{\sqrt{T}}{R} exp(−\frac{R^{2}}{4T})  (1.10)

and for large \frac{R^{2}}{T} this inequality is only a bit weaker than (1.6) . On the other hand for \frac{R^{2}}{2T} ≥ 1 (1.9) implies

P (R, T) ≤ \sqrt{\frac{e}{2}}\sqrt{μ(A)μ(A^{R} \ A)}\frac{R}{\sqrt{T}} exp(−\frac{R^{2}}{4T})  (1.11)

or, applying \sqrt{ab} ≤ (a + b)/2 ,

P (R, T) ≤ \sqrt{\frac{e}{8}}μ(A^{R})\frac{R}{\sqrt{T}} exp(−\frac{R^{2}}{4T}) .  (1.12)

Obviously, (1.10) is better for large \frac{R^{2}}{T}, but for intermediate values of \frac{R^{2}}{T} (1.11) and (1.12) may give a sharper estimate, than (1.10) and (1.6) especially when the volume μ(A^{R} \ A) is small.

The inequalities (1.6) and (1.9) imply some estimates of heat kernel. It is obvious from a probabilistic point of view that P (R, t) is an upper bound of the following integral of heat kernel

∫_{A} ∫_{M\A^{R}} p(x, y, t)dydx.

This can be explained from analytic point of view too. Indeed, applying theorem 3 to the function

v(x, t) = ∫_{M\A^{R}} p(x, y, t)dy

we obtain as above

∫_{A} ∫_{M\A^{R}} p(x, y, t)dydx = ∫_{A} v(x, t)dx

≤ \sqrt{μ(A)μ(A^{R} \ A)} max\{\frac{R}{\sqrt{2T}} , \frac{\sqrt{2T}}{R} \} exp(−\frac{R^{2}}{4T} + \frac{1}{2}).

2. Proof of theorems 1 and 2

To prove Theorem 1 consider a time derivative I′(t) of the function

I(t) = ∫_{Ω} u^{2}(x, t)e^{ξ(x,t)}dx  (2.1)

Applying the equation (1.1) and the boundary condition (in a weak sense if the boundary of Ω is not smooth) we have

I′(t) = ∫_{Ω} ξ_{t}e^{ξ}u^{2} + ∫_{Ω} 2ue^{ξ}u_{t} = ∫_{Ω} ξ_{t}e^{ξ}u^{2} − ∫_{Ω} (∇ (2ue^{ξ}) , ∇u)

(here we are applying inequality (1.3) )

≤ −\frac{1}{2} ∫_{Ω} |∇ξ|^{2} e^{ξ}u^{2} − 2 ∫_{Ω}(∇u, ∇ξ)ue^{ξ} − 2 ∫_{Ω} |∇u|^{2} e^{ξ}

= −\frac{1}{2} ∫_{Ω} e^{ξ}\{|∇ξ|^{2} u^{2} + 4(∇u, ∇ξ)u + 4 |∇u|^{2}\} = −\frac{1}{2} ∫_{Ω} e^{ξ} (u∇ξ + 2∇u)^{2}

On the other hand

∇(e^{\frac{ξ}{2}} u) = \frac{1}{2}e^{\frac{ξ}{2}} (u∇ξ + 2∇u),

|∇(e^{\frac{ξ}{2}} u)|^{2} = \frac{1}{4}e^{ξ}(u∇ξ + 2∇u)^{2}

which implies the inequality

I′(t) ≤ −2 ∫_{Ω} |∇(e^{\frac{ξ}{2}} u)|^{2}  (2.2)

We are left to notice that the function v = e^{ξ/2}u as any other function vanishing on the boundary ∂Ω satisfies the relation

∫_{Ω} |∇v|^{2} ≥ λ_{1}(Ω) ∫_{Ω} v^{2}

Substituting into (2.2) we obtain a differential inequality

I′(t) ≤ −2λ_{1}(Ω)I(t)

whence the decreasing of I(t)e^{2λ_{1}(Ω)t} follows. ⊓ ⊔

Remark. One may replace u^{2} in the statement of Theorem 1 by another power or function of the solution. Let f(z) be a smooth function on (0, +∞) such that

f(z) > 0, f(z)′ > 0, f(z)′′ > 0

and

κ = inf_{z>0} \frac{f′′f}{f′^{2}} > 0

Suppose also that the function ξ satisfies the relation

ξ_{t} + \frac{|∇ξ|^{2}}{4κ} ≤ 0

Then the interal

I_{f}(t) = ∫_{Ω} f(u(x, t))e^{ξ}dx

is a decreasing function of t.

For example, if f(z) = z^{p}, p > 1 then κ = \frac{p−1}{p}. Of course it would be interesting to specify a speed of decay of I_{f}(t) as was done for I(t) but the spectral radius seems not to suit this purpose. Proof of theorem 2. It suffices to prove the theorem for the case when A and B are bounded sets. Indeed, if we have proved that, a general case will be reduced to that as follows. Consider a bounded region Ω then, by the hypothesis, we have the inequality

∫_{A∩Ω} ∫_{B∩Ω} p(x, y, t)dxdy ≤ \sqrt{μAμB} exp\{−\frac{R^{2}}{4t} − λ_{1}(M)t\}

Letting Ω → M we obtain (1.5) .

To prove theorem 2 for bounded sets A, B let us consider the distance function d(x) from set A and put ξ(x, t) = αd(x) − \frac{α^{2}}{2} t where constant α > 0 is to be chosen below. Since |∇d| ≤ 1 it follows that ξ satisfies the relation (1.3) . Let Ω be a large region containing both sets A, B and p_{Ω}(x, y, t) be a heat kernel in region Ω (with a vanishing Dirichlet boundary condition). Let us apply theorem 1 in region Ω to the function

u(x, t) = ∫_{A} p_{Ω}(x, y, t)dy

being a solution to Dirichlet mixed value problem in Ω with an initial value u(x, 0) = 1_{A}. We have by theorem 1 that for any t > 0 I(t) ≤ exp(−2λ_{1}(Ω)t)I(0). Note that

I(0) = ∫_{Ω} 1_{A}^{2} exp(αd(x))dx = ∫_{A} exp(αd(x))dx = μA

for d(x)|_{A} = 0. Therefore, we obtain

∫_{Ω} u^{2}(x, t)exp(αd(x) − \frac{α^{2}}{2} t)dx ≤ exp(−2λ_{1}(Ω)t)μA

Reducing the domain of integration to B and taking into account that d(x)|_{B} ≥ dist(A, B) = R we have

∫_{B} u^{2}(x, t)dx ≤ exp\{−αR + \frac{α^{2}}{2} t − 2λ_{1}(Ω)t\} μA.

Finally, applying Cauchy-Schwarz inequality

∫_{B} ∫_{A} p_{Ω}(x, y, t)dydx = ∫_{B} u(x, t)dx ≤\{∫_{B} u^{2}(x, t)dx\}^{\frac{1}{2}} \sqrt{μB}

≤ exp\{−\frac{α}{2}R + \frac{α^{2}}{4} t − λ_{1}(Ω)t\} \sqrt{μAμB} .

Taking here the optimal value α = \frac{R}{t} we obtain

∫_{B} ∫_{A} p_{Ω}(x, y, t)dydx ≤ exp\{−\frac{R^{2}}{4t} − λ_{1}(Ω)t\} \sqrt{μAμB}

We are left to let here Ω → M and to mention that λ_{1}(Ω) ≥ λ_{1}(M) and p_{Ω} → p locally uniformly. ⊓ ⊔

3. Proof of theorem 3

The proof of theorem 3 doesn’t use theorem 1 directly. We shall apply the idea behind the proof of integral maximum principle in another situation. The proof will be split into three steps.

STEP 1. Let us denote by d(x) the distance from x to the set A and consider a cut-off function φ(x) such that

φ|_{A} = 1, suppφ ⊂ A^{R},

then the function η ≡ (1 − δ)ξ satisfies the following inequality:

∫_{A^{R}} u^{2}e^{η(x,T)}φ^{2}(x)dx ≤ \frac{2}{δ} ∫_{0}^{T} ∫_{A^{R}\A} |∇φ|^{2} e^{η(x,t)}dx dt  (3.1)

where δ ∈ (0, 1) is arbitrary.

Proof. We have

\frac{d}{dt} ∫_{A^{R}} u^{2}e^{η(x,t)}φ^{2} = 2 ∫_{A^{R}} uu_{t}e^{η}φ^{2} + ∫_{A^{R}} u^{2}η_{t}e^{η}φ^{2}

≤ 2 ∫_{A^{R}} uΔue^{η}φ^{2} + ∫_{A^{R}} u^{2}η_{t}e^{η}φ^{2}

= −2 ∫_{A^{R}} |∇u|^{2} e^{η}φ^{2} − 2 ∫_{A^{R}} u(∇u, ∇η)e^{η}φ^{2} − 4 ∫_{A^{R}} ue^{η}(∇u, ∇φ)φ + ∫_{A^{R}} u^{2}η_{t}e^{η}φ^{2}.

Applying the inequality

−2u(∇u, ∇φ)φ ≤ δ^{−1}u^{2} |∇φ|^{2} + δφ^{2} |∇u|^{2}

we get

\frac{d}{dt} ∫_{A^{R}} u^{2}e^{η}φ^{2} ≤ \frac{2}{δ} ∫_{A^{R}} u^{2} |∇φ|^{2} e^{η}

−2 ∫_{A^{R}} e^{η}φ^{2}\{(1 − δ) |∇u|^{2} − u |∇u| |∇η| − \frac{1}{2}η_{t}u^{2}\} .

The expression in brackets on the right hand side of this inequality is equal to

(1 − δ)X^{2} − |∇η| XY − \frac{1}{2}η_{t}Y^{2}

where X = |∇u| , Y = u. This quadratic polynomial of X, Y is non-negative if its discriminant is non-positive, i.e.

|∇η|^{2} + 2(1 − δ)η_{t} ≤ 0

which is true due to (1.3) . Recalling that 0 ≤ u ≤ 1 we have

\frac{d}{dt} ∫_{A^{R}} u^{2}e^{η}φ^{2} ≤ \frac{2}{δ} ∫_{A^{R}} u^{2} |∇φ|^{2} e^{η} ≤ \frac{2}{δ} ∫_{A^{R}} |∇φ|^{2} e^{η}.

Integrating this inequality with respect to t and taking into account that |∇φ| = 0 outside A^{R} \ A we obtain (3.1) .

STEP 2. Let us prove the following estimate

∫_{A} u^{2}(x, T)dx ≤\frac{2δ^{−1}μ(A^{R} \ A)}{\{∫_{0}^{R}\{∫_{0}^{T} exp\{(1 − δ)(ζ(ρ, τ) − ζ(0, T))\} dτ\}^{− \frac{1}{2}} dρ\}^{2}}  (3.2)

ζ(ρ, τ) being a Lipschitz function in [0, R] × [0, T] satisfying in the following relation

ζ_{τ} + \frac{1}{2}ζ_{ρ}^{2} ≤ 0.  (3.3)

Let us consider the function

ξ(x, t) = ζ(d(x), t),

and apply (3.1) (note, that this function satisfies the condition (1.3) ). Since for all x ∈ A ξ(x, T) = ζ(0, T) ≡ const we can get from (3.1)

∫_{A} u^{2}(x, T)dx ≤ \frac{2}{δ} ∫_{0}^{T} ∫_{A^{R}\A} |∇φ|^{2} e^{(1−δ)(ξ(x,t)−ζ(0,T))}dx dt.  (3.4)

Let us introduce a function

ω(r) ≡ ∫_{0}^{T} e^{(1−δ)(ζ(r,t)−ζ(0,T))}dt

and suppose that φ depends on d(x) only (i.e. we denote further by φ a function on (0, R)), then we can rewrite (3.4) as follows

∫_{A} u^{2}(x, T)dx ≤ \frac{2}{δ} ∫_{0}^{R} φ′(r)^{2}ω(r)dV (r)

where

V (r) ≡ μ(A^{r} \ A).

Optimizing the last integral over all Lipschitz functions φ on (0, R) under conditions φ(0) = 1, φ(R) = 0 we obtain

∫_{A} u^{2}(x, T)dx ≤ \frac{2}{δ}\{∫_{0}^{R} \frac{dr}{V′(r)ω(r)}\}^{−1} .  (3.5)

Since

∫_{0}^{R} V′(r)dr ∫_{0}^{R} \frac{dr}{V′(r)ω(r)} ≥\{∫_{0}^{R} \frac{dr}{\sqrt{ω(r)}}\}^{2}

we get from (3.5) an inequality

∫_{A} u^{2}(x, T)dx ≤ \frac{2}{δ}V (R)\{∫_{0}^{R} \frac{dr}{\sqrt{ω(r)}}\}^{−2} ,

which implies (3.2) .

STEP 3. Here we shall complete the proof choosing a suitable function ζ. If we take in (3.2) ζ ≡ 0 and δ = 1 we obtain a more or less trivial estimate

∫_{A} u^{2}(x, T)dx ≤ μ(A^{R} \ A)\frac{2T}{R^{2}} .  (3.6)

Applying (3.2) to a function ζ(ρ, τ) = −αρ − \frac{1}{2}α^{2}τ with α = \frac{R}{T} we get after integrating

∫_{A} u^{2}(x, T)dx ≤ \frac{1 − δ}{δ} \frac{μ(A^{R} \ A)}{e^{(1−δ) \frac{R^{2}}{2T}} − 1}.  (3.7)

If \frac{R^{2}}{2T} ≥ 1 then one can take δ = \frac{2T}{R^{2}} and (3.7) gives the following

∫_{A} u^{2}(x, T)dx ≤\{\frac{R^{2}}{2T} − 1\} \frac{μ(A^{R} \ A)}{exp (\frac{R^{2}}{2T} − 1) − 1}

or, applying the inequality

\frac{X − 1}{e^{X−1} − 1} ≤ \frac{X}{e^{X−1}}

which is valid for all X ≥ 1, we finally get that for \frac{R^{2}}{2T} ≥ 1

∫_{A} u^{2}(x, T)dx ≤ μ(A^{R} \ A)\frac{R^{2}}{2T} exp(−\frac{R^{2}}{2T} + 1).  (3.8)

The desired estimate (1.8) follows from (3.8) and (3.6) immediately. ⊓ ⊔

Remark. One could expect that the spectral radius may be put into the estimate of theorem 3 like in theorems 1, 2 but this is not so. Indeed, the integral

∫_{A} u^{2}(x, T)dx

may tend to μA as T → ∞ if u has a boundary value equal to 1 (on the contrary to the case of theorem 1 where a solution under consideration vanishes on a boundary). Hence, any upper bound of this integral cannot contain a term exp(−λ_{1}(M)T) vanishing as T → ∞.

Acknowledgement. I would like to thank T.Lyons for fruitful discussions of the questions touched upon in this paper. I would thank him again for inviting me to Edinburgh University with support from an EEC project where this work was done.

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