Gaussian lower bound for the Neumann Green function of a general parabolic operator

Abstract

Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish a Gaussian lower bound for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for Gaussian two-sided bounds for the fundamental solution.

1 Introduction

Let $\Omega$ be a bounded domain of $\mathbb {R}^n$ with $C^{1,1}$-smooth boundary. Let $t_0<t_1$, we set $Q=\Omega \times (t_0,t_1)$ and we consider the second order differential operator

$$\begin{aligned} L=a_{ij}(x,t)\partial _{ij}^2+b_k(x,t)\partial _k+c(x,t)-\partial _t. \end{aligned}$$

Here and henceforth we use the usual Einstein summation convention for repeated indices.

We make the following assumptions on the coefficients of $L$:

• (i) the matrix $(a_{ij}(x,t))$ is symmetric for any $(x,t)\in \overline{Q}$,

• (ii) $a_{ij}\in W^{1,\infty }(Q)$, $b_k$, $c \in C([t_0,t_1], C^1 (\overline{\Omega }))$,

• (iii) $a_{ij}(x)\xi _i\xi _j\ge \lambda |\xi |^2, (x,t)\in \overline{Q} , \xi \in \mathbb {R}^n$,

• (iv) $\Vert a_{ij}\Vert _{W^{1,\infty }(Q)}+\Vert b_k\Vert _{L^\infty (Q)}+\Vert c\Vert _{L^\infty (Q)}\le A,$

where $\lambda >0$ and $A>0$ are two given constants.

These assumptions are surely not the best possible if one wants to construct a fundamental solution or a Green function. But they are sufficient to carry out our analysis.

Since we will use the fundamental solution in the whole space, we begin by extending the coefficients of $L$ in a neighborhood $\widetilde{\Omega }$ of $\overline{\Omega }$ to coefficients having the same regularity. We observe that this is possible in view of the regularity of $\Omega$. For sake of simplicity, we keep the same symbols for the extended coefficients. We may also assume that the ellipticity condition holds for the extended coefficients with the same constant $\lambda$. Pick $\psi \in C_0^\infty (\widetilde{\Omega })$ satisfying $0\le \psi \le 1$ and $\psi =1$ in a neighborhood of $\overline{\Omega }$. We set

$$\begin{aligned} \widetilde{a}_{ij}=a_{ij}\psi +\lambda \delta _{ij}(1-\psi ),\quad \widetilde{b}_k=b_k\psi ,\quad \widetilde{c}=c\psi \end{aligned}$$

and

$$\begin{aligned} \widetilde{L}=\widetilde{a}_{ij}(x,t)\partial _{ij}^2+\widetilde{b}_k(x,t)\partial _k+\widetilde{c}(x,t)-\partial _t. \end{aligned}$$

Clearly, the coefficients of $\widetilde{L}$ satisfy the same assumptions as those of $L$. So in the sequel we will use the same symbol $L$ for $L$ or its extension $\widetilde{L}$.

We recall that the function

$$\begin{aligned} \fancyscript{G}(x,t)=(4\pi t)^{-n/2}e^{-\frac{|x|^2}{4t}},\quad x\in \mathbb {R}^n,\; t>0, \end{aligned}$$

is usually called the Gaussian kernel. We set

$$\begin{aligned} \fancyscript{G}_c(x,t)=c^{-1}\fancyscript{G}(\sqrt{c}x,t),\quad c>0. \end{aligned}$$

It is important to observe that the map $c\rightarrow \fancyscript{G}_c$ is non increasing.

We are interested in establishing a Gaussian lower bound for the Neumann Green function associated to the operator $L$. More specifically, denoting by $G$ the Neumann Green function for $L$, we want to prove an estimate of the form

$$\begin{aligned} \fancyscript{G}_C(x-\xi ,t-\tau )\le G(x,t;\xi ,\tau ),\quad (x,t;\xi ,\tau )\in Q^2,\quad t>\tau , \end{aligned}$$

where the constant $C$ depends only on $\Omega$, $\lambda$, $T=t_1-t_0$ and $A$.

We succeed in proving that the above Gaussian lower bound holds true provided that $\Omega$ satisfies the chain condition. That is, there exists a constant $c>0$ such that for any two points $x$, $y\in \Omega$ and for any positive integer $m$ there exists a sequence $(x_i)_{0\le i\le m}$ of points in $\Omega$ such that $x_0=x$, $x_m=y$ and

$$\begin{aligned} |x_{i+1}-x_i|\le \frac{c}{m}|x-y|,\quad i=0,\ldots ,m-1. \end{aligned}$$

The sequence $(x_i)_{0\le i\le m}$ is referred to as a chain connecting $x$ and $y$.

We see that any convex subset of $\mathbb {R}^n$ satisfies the chain condition with $c=1$. In two dimensional case, the spherical shell $\mathcal {C}=B(0,2){\setminus } \overline{B(0,1)}$ has the chain property with $c=\sqrt{2}$. This follows from the fact that any two points of $\mathcal {C}$ can be connected by a broken line consisting of two segments parallel to axes of coordinates.

We point out that a $C^{1,1}$-smooth domain does not possess necessarily the chain condition.

To our knowledge a Gaussian lower bound has never been established before for the Neumann Green function of a general parabolic operator. Moreover, even in the case of parabolic operators with time-independent coefficients, we can quote only three references: Chavel [5] when the domain is convex, Choulli et al. [10] for smooth domains and Li and Yau [22] for a compact Riemannian manifold with boundary whose Ricci curvature is bounded from above and its boundary is convex.

A Gaussian upper bound for a general parabolic operator in divergence form was proved by Daners [11]. Choi and Kim [8] obtained a Gaussian upper bound for a system of operators in divergence form under the assumption that the corresponding Neumann boundary value problem possesses a De Giorgi–Nash–Moser type estimate at the boundary. In [4], the authors established a gaussian upper bound for a Neumann Green function corresponding to a time-dependent domain.

The problem is quite different for a Dirichlet Green function since the latter vanishes on the boundary. One can prove in an obvious manner, with the help of the parabolic maximum principle, that a Dirichlet Green function is non negative and dominated pointwise by a fundamental solution and so it has a Gaussian upper bound. Aronson [2, Theorem 8, page 670] established an interior Gaussian lower bound for a Dirichlet Green function. It is worthwhile to mention that [2, Theorem 8, page 670] can be used to extend the results of [15, Section 3] to a general parabolic operator. In other words, one can obtain a proof of a continuity theorem by Nash [25] and Moser–Harnack inequality [24] for a general divergence form parabolic operator, since they rely on two-sided Gaussian bounds for the fundamental solution. Later, Cho [6], Cho et al. [7] extended this result to a global weighted Gaussian lower bound involving the distance to the boundary. A Gaussian lower bound for a Dirichlet Green function when the Euclidean distance is changed by a geodesic distance was proved by van den Berg [30, 31].

For parabolic operators with time-independent coefficients, a fundamental solution or a Green function is reduced to a heat kernel. We mention that there is a tremendous literature dealing with Gaussian bounds for heat kernels. We quote the classical books by Davies [12], Grigor’yan [17], Ouhabaz [27] Saloff-Coste [28] and Stroock [29], but of course there are many other references on the subject.

As we said in the summary, the main ingredient in our analysis relies on the classical construction of the fundamental solution by means of the so-called parametrix method. We revisit this construction in the next section and we derive from it Gaussian two-sided bounds for the fundamental solution. In Sect. 3, we prove a Gaussian lower bound for the Neumann Green function. To do so, we construct the Neumann Green function as a perturbation of the fundamental solution by a single-layer potential. The Gaussian lower bound is then derived from the smoothing effect of the single-layer potential.

2 The parametrix method revisited

We are concerned in this section with Gaussian two-sided bounds for the fundamental solution of $Lu=0$. For a systematic study of fundamental solutions, we refer to the classical monographs by Friedman [16] and Ladyzhenskaja et al. [20].

In the sequel $P=\mathbb {R}^n\times (t_0,t_1)$.

We recall that a fundamental solution of $Lu=0$ in $P$ is a function $E (x,t;\xi ,\tau )$ which is $C^{2,1}$ in $P^2\cap \{t>\tau \}$, which satisfies

$$\begin{aligned} LE (\cdot \, ,\cdot ;\xi ,\tau )=0\quad \text{ in }\; \mathbb {R}^n \times \{\tau <t\le t_1\},\quad \text{ for } \text{ any }\; (\xi ,\tau )\in \mathbb {R}^n \times [t_0,t_1[ \end{aligned}$$

and, for any $f\in C_0(\mathbb {R}^n)$,

$$\begin{aligned} \lim _{t\searrow \tau }\int _{\mathbb {R}^n} E (x,t;\xi ,\tau )f(\xi )d\xi =f(x),\quad x\in \mathbb {R}^n. \end{aligned}$$

In this definition, we can also take a larger class of functions $f$. Namely, a class of continuous functions satisfying a certain growth condition at infinity (see for instance [16], formulas $(6.1)$ and $(6.2)$, page 22]).

The construction of a fundamental solution by means of the so-called parametrix method was initiated by Levi [21]. Let $a=(a^{ij})$ be the inverse matrix of $(a_{ij})$, $|a|$ the determinant of $a$ and

$$\begin{aligned} Z(x,t;\xi ,\tau )&=[4\pi (t-\tau )]^{-n/2}\sqrt{|a(\xi ,\tau )|}e^{-\frac{a(\xi ,\tau )(x-\xi )\cdot (x-\xi )}{4(t-\tau )}},\\&\quad (x,t;\xi ,\tau )\in P^2\cap \{t>\tau \}. \end{aligned}$$

This function is called the parametrix. It satisfies

$$\begin{aligned} L_0Z(\cdot \, ,\cdot \, ,\xi ,\tau )=0\quad \text{ in }\; \mathbb {R}^n\times \{\tau <t\le t_1\}\quad \text{ for } \text{ any }\; (\xi ,\tau )\in \mathbb {R}^n \times [t_0,t_1[, \end{aligned}$$

(2.1)

where

$$\begin{aligned} L_0=a_{ij}(\xi ,\tau )\partial ^2_{ij} -\partial _t. \end{aligned}$$

When $(\xi ,\tau )$ are fixed, $L_0$ is considered as a constant coefficients operator with respect to $(x,t)$.

In the parametrix method we seek $E$, a fundamental solution of $Lu=0$ in $P$, of the form

$$\begin{aligned} E (x,t;\xi ,\tau )=Z (x,t;\xi ,\tau )+\int _\tau ^t\int _{\mathbb {R}^n} Z (x,t;\eta ,\sigma )\Phi (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma , \end{aligned}$$

(2.2)

where $\Phi$ is to be determined in order to satisfy $LE (\cdot \, ,\cdot \, ;\xi ,\tau )=0\; \text{ for } \text{ any }\; (\xi ,\tau )\in \mathbb {R}^n \times [t_0,t_1[$.

Following [16], Formulas $(4.4)$ and $(4.5)$, page 14], $\Phi$ is given by the series

$$\begin{aligned} \Phi =\sum _{\ell =1}^\infty \Phi _\ell , \end{aligned}$$

where $\Phi _1(x,t;\xi ,\tau )=LZ(x,t;\xi ,\tau )$ and

$$\begin{aligned} \Phi _{\ell +1}(x,t;\xi ,\tau )=\int _\tau ^t\int _{\mathbb {R}^n} \Phi _1(x,t ;\eta ,\sigma )\Phi _\ell (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma ,\quad \ell \ge 1. \end{aligned}$$

Here, for simplicity, we write $LZ(x,t;\xi ,\tau )$ instead of $[LZ(\cdot \, , \cdot \, ; \xi ,\tau )](x,t)$.

Let $d_i$, $1\le i\le n$, given by

$$\begin{aligned} d_i=d_i(x,t;\xi ,\tau )=-\frac{a^{ij}(\xi ,\tau )(x_j-\xi _j)}{2(t-\tau )},\quad (x,t;\xi ,\tau )\in P^2\cap \{t>\tau \}. \end{aligned}$$

Then

$$\begin{aligned} \partial _i Z=d_iZ,\quad \partial _{ij}^2Z=\left[ -\frac{a^{ij}(\xi ,\tau )}{2(t-\tau )}+d_jd_i\right] Z. \end{aligned}$$

Therefore, taking into account (2.1), we get

$$\begin{aligned} LZ=LZ-L_0Z=\left\{ \left( a_{ij}(x,t)- a_{ij}(\xi ,\tau )\right) \left[ -\frac{a^{ij}(\xi ,\tau )}{2(t-\tau )}+d_jd_i\right] +b_kd_k +c\right\} Z. \end{aligned}$$

We write $LZ=\Psi Z$, where

$$\begin{aligned} \Psi =\left( a_{ij}(x,t)- a_{ij}(\xi ,\tau ) \right) \left[ -\frac{a^{ij}(\xi ,\tau )}{2(t-\tau )}+d_jd_i \right] +b_kd_k +c. \end{aligned}$$

Let

$$\begin{aligned} M=\max _{i,j}\Vert a_{ij}\Vert _{W^{1,\infty }(Q)},\quad N=\max \left( \max _k\Vert b_k\Vert _{L^\infty (Q)},\Vert c\Vert _{L^\infty (Q)},1\right) .\end{aligned}$$

Since

$$\begin{aligned} |d_i|&\le \frac{|x-\xi |}{2\lambda (t-\tau )},\\ |a_{ij}(x,t)- a_{ij}(\xi ,\tau )|&\le M(|x-\xi |+t-\tau ), \end{aligned}$$

we have

$$\begin{aligned} |\Psi (x,t;\xi ,\tau )|\le N\frac{1}{\sqrt{t-\tau }}\fancyscript{P}\left( \frac{|x-\xi |}{\sqrt{t-\tau }}\right) . \end{aligned}$$

(2.3)

Here $\fancyscript{P}$ is a polynomial function of degree less than three whose coefficients depend only on $M$.

Unless otherwise stated, all the constants we use now do not depend on $N$.

In light of (2.3) we obtain

$$\begin{aligned} |LZ|&\le CN(t-\tau )^{-(n+1)/2}P(\rho )e^{-(\lambda /4)\rho ^2}\\&= CN(t-\tau )^{-(n+1)/2}\left[ P(\eta )e^{-(\lambda /8)\rho ^2}\right] e^{-(\lambda /8)\rho ^2}, \end{aligned}$$

with

$$\begin{aligned} \rho =\frac{|x-\xi |}{\sqrt{t-\tau }}. \end{aligned}$$

But the function $\rho \in (0,+\infty ) \longrightarrow P(\rho )e^{-(\lambda /8)\rho ^2}$ is bounded. Consequently,

$$\begin{aligned} |\Phi _1(x,t;\xi ,\tau )|=|LZ(x,t;\xi ,\tau )|\le N\widetilde{C}(t-\tau )^{-(n+1)/2}e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }}, \end{aligned}$$

(2.4)

where $\lambda ^*=\lambda /8$.

The following lemma will be useful in the sequel. Its proof is given in [16, page 15].

Lemma 2.1

Let $c>0$ and $-\infty <\gamma ,\beta <n/2 +1$. Then

$$\begin{aligned}&\int _\tau ^t\int _{\mathbb {R}^n}(t-\sigma )^{-\gamma }e^{-\frac{c|x-\eta |^2}{t-\sigma }}(\sigma -\tau )^{-\beta }e^{-\frac{c|\eta -\xi |^2}{\sigma -\tau }}d\eta d\sigma \\&\quad =\left( \frac{4\pi }{c}\right) ^{n/2}B(n/2-\gamma +1,n/2-\beta +1)(t-\tau )^{n/2+1-\gamma -\beta }e^{-\frac{c|x-\xi |^2}{t-\tau }}, \end{aligned}$$

where $B$ is the usual beta function.

We want to show

$$\begin{aligned} |\Phi _\ell (x,t;\xi ,\tau )|\le (N\widetilde{C})^\ell \widehat{C}^{\ell -1}(t-\tau )^{-(n+2-\ell )/2}\prod _{j=1}^{\ell -1}B(1/2,j/2)e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }},\quad \ell \ge 2. \end{aligned}$$

(2.5)

Here $\widetilde{C}$ is the same constant as in (2.4) and $\widehat{C}=\left( \frac{4\pi }{\lambda ^*}\right) ^{n/2}$.

As

$$\begin{aligned} \Phi _2(x,t;\xi ,\tau )=\int _\tau ^t\int _{\mathbb {R}^n}\Phi _1(x,t;\eta ,\sigma )\Phi _1(\eta ,\sigma ;\xi ,\tau )d\eta d\sigma , \end{aligned}$$

estimate (2.4) and Lemma 2.1 with $\gamma =\beta =n/2+1$ show that (2.5) holds true with $\ell =2$. The general case follows by an induction argument in $\ell$. Indeed, using

$$\begin{aligned} \Phi _{\ell +1}(x,t;\xi ,\tau )=\int _\tau ^t\int _{\mathbb {R}^n}\Phi _1(x,t;\eta ,\sigma )\Phi _\ell (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma , \end{aligned}$$

(2.4), (2.5) for $\ell$ and Lemma 2.1 with $\gamma =n/2+1$ and $\beta =(n+2-\ell )/2$, we obtain easily that (2.5) holds true with $\ell +1$ in place of $\ell$.

If $\Gamma$ is the usual gamma function, we recall that

$$\begin{aligned} B(1/2,j/2)=\frac{\Gamma (1/2)\Gamma (j/2)}{\Gamma ((j+1)/2)}. \end{aligned}$$

Therefore

$$\begin{aligned} \prod _{j=1}^{\ell -1}B(1/2,j/2)=\frac{\Gamma (1/2)^\ell }{\Gamma (\ell /2)}=\frac{\sqrt{\pi }^\ell }{\Gamma (\ell /2)}. \end{aligned}$$

(2.6)

Hence, (2.4)–(2.6) entail

$$\begin{aligned} |\Phi (x,t;\xi ,\tau )|\le \sum _{\ell \ge 1}|\Phi _\ell (x,t;\xi ,\tau )|\le N\widetilde{C}(1+S)(t-\tau )^{-(n+1)/2}e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }},\quad \end{aligned}$$

(2.7)

with

$$\begin{aligned} S=\sum _{\ell \ge 1}\left[ CN (t-\tau )^{1/2}\right] ^\ell /\Gamma ((\ell +1)/2). \end{aligned}$$

We have $\Gamma ((\ell +1)/2)=\Gamma (m+1/2)\ge \Gamma (m)=(m-1)!$ if $\ell =2m$ and $\Gamma ((\ell +1)/2)=\Gamma (m+1)=m!$ if $\ell =2m+1$. Then

$$\begin{aligned} S&=\frac{1}{\Gamma (3/2)}[CN(t-\tau )^{1/2}]^2+\sum _{m \ge 2}\frac{1}{\Gamma (m+1/2)}\left[ CN (t-\tau )^{1/2}\right] ^{2m}\\&\quad +\sum _{m \ge 0}\frac{1}{\Gamma (m+1)}\left[ CN (t-\tau )^{1/2}\right] ^{2m+1}\\&\le \frac{1}{\Gamma (3/2)}[CN(t-\tau )^{1/2}]^2+\sum _{m \ge 2}\frac{1}{(m-1)!}\left[ CN (t-\tau )^{1/2}\right] ^{2m} \\&\quad +\sum _{m \ge 0}\frac{1}{m!}\left[ CN (t-\tau )^{1/2}\right] ^{2m+1} . \end{aligned}$$

Whence

$$\begin{aligned} 1+S\le \widetilde{C}e^{\widetilde{C}N^2(t-\tau )}. \end{aligned}$$

Plugging this estimate into (2.7), we obtain

$$\begin{aligned} |\Phi (x,t;\xi ,\tau )|\le \widetilde{C}N(t-\tau )^{-(n+1)/2}e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }+\widetilde{C}N^2(t-\tau )}. \end{aligned}$$

(2.8)

With the help of Lemma 2.1, estimate (2.8) yields

$$\begin{aligned} \left| \int _\tau ^t\int _{\mathbb {R}^n} Z (x,t;\eta ,\sigma )\Phi (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma \right| \le \widetilde{C}N(t-\tau )^{-(n-1)/2}e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }+\widetilde{C}N^2(t-\tau )}. \end{aligned}$$

(2.9)

Noting that this inequality can be rewritten as

$$\begin{aligned}&\left| \int _\tau ^t\int _{\mathbb {R}^n} Z (x,t;\eta ,\sigma )\Phi (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma \right| \\&\quad \le \widetilde{C}\left[ N(t-\tau )^{1/2}e^{-\widetilde{C}N^2(t-\tau )}\right] (t-\tau )^{-n/2}e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }+2\widetilde{C}N^2(t-\tau )} \end{aligned}$$

and, using that $\rho \rightarrow \rho e^{-\widetilde{C}\rho ^2}$ is a bounded function on $[0,+\infty )$, we obtain

$$\begin{aligned} \left| \int _\tau ^t\int _{\mathbb {R}^n} Z (x,t;\eta ,\sigma )\Phi (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma \right| \le \widehat{C}(t-\tau )^{-n/2}e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }+2\widetilde{C}N^2(t-\tau )}. \end{aligned}$$

(2.10)

An immediate consequence of (2.10) is

$$\begin{aligned} |E(x,t;\xi ,\tau )|\le \widetilde{C}(t-\tau )^{-n/2}e^{-\frac{\lambda ^*|x-\xi |^2}{t-\tau }+\widetilde{C}N^2(t-\tau )}. \end{aligned}$$

(2.11)

In the rest of this section, we forsake the explicit dependence on $N$. So the constants below may depend on $\Omega$, $\lambda$, $A$, and $T$.

From (2.9) we deduce in a straightforward manner that there exists $\delta >0$ such that

$$\begin{aligned} E (x,t;\xi ,\tau )\ge \widehat{C}(t-\tau )^{-n/2},\quad (x,t;\xi ,\tau )\in P^2,\; t>\tau ,\quad \widetilde{C}|x-\xi |^2<t-\tau \le \delta . \end{aligned}$$

(2.12)

By [16, Theorem 11, page 44], $E$ is positive. Moreover, $E$ satisfies the following identity, usually called the reproducing property,

$$\begin{aligned} E (x,t; \xi ,\tau )\!=\!\int _{\mathbb {R}^n} E (x,t ;\eta ,\sigma )E (\eta ,\sigma ;\xi ,\tau )d\eta , \; x,\xi \in \mathbb {R}^n,\quad t_0\le \tau <\sigma <t\le t_1. \end{aligned}$$

(2.13)

We can then paraphrase the proof of [15, Theorem 2.7, page 334] to get a Gaussian lower bound for $E$ when $0<t-\tau \le \delta$. To pass from $t-\tau \le \delta$ to $t-\tau \le T$, we use again an argument based on the reproducing property. We detail the same argument in the proof of Theorem 3.1.

We sum up our analysis in the following theorem.

Theorem 2.1

The fundamental solution $E$ satisfies the Gaussian two-sided bounds:

$$\begin{aligned} \fancyscript{G}_C(x-\xi , t-\tau )\le E (x,t;\xi ,\tau )\le \fancyscript{G}_{\widetilde{C}}(x-\xi , t-\tau ),\quad (x,t;\xi ,\tau )\in P^2\cap \{t>\tau \}. \end{aligned}$$

(2.14)

Remark 2.1

Let us assume that conditions (i)–(iv) above hold in all of the whole space $\mathbb {R}^n\times \mathbb {R}$ instead of $Q$ only. Taking into account the exponential term in $N^2$ in (2.11), we prove, once again with the help of the reproducing property, the following global estimate in time:

$$\begin{aligned} e^{-\kappa N^2(t-\tau )}\fancyscript{G}_C(x-\xi , t-\tau )\le E (x,t;\xi ,\tau )\le e^{\kappa N^2(t-\tau )}\fancyscript{G}_{\widetilde{C}}(x-\xi , t-\tau ), \end{aligned}$$

(2.15)

for some constant $\kappa >0$, where $(x,t;\xi ,\tau )\in (\mathbb {R}^n\times \mathbb {R})^2$.

We point out that (2.15) does not give the two-sided Gaussian bounds by Fabes and Stroock [15] for the divergence form operator $\partial _i(a_{ij}(x,t)\partial _j\, \cdot \, )-\partial _t$ with ($C^\infty$-) smooth coefficients. This is not surprising since the arguments we used for proving (2.15) are not well adapted to divergence form operator. We note however that the approach developed in [15] for establishing Gaussian two-sided bounds is more involved.

Gaussian two-sided bounds were obtained by Eidel’man and Porper [13] when the coefficients of $L$ satisfy the uniform Dini condition with respect to $x$. The main tool in [13] is a parabolic Harnack inequality. We refer also to [1, 14, 19, 26], where the reader can find various results on bounds for the fundamental solution.

We mentioned in the Introduction that the Moser–Harnack inequality in [15] can be extended to a general divergence form parabolic operator. Let us show briefly how this Moser–Harnack inequality still holds for a general parabolic operator. First, we recall that a Dirichlet Green function was constructed in [20], formula (16.7), page 408] as a perturbation of the fundamental solution by a double-layer potential. Therefore, in light of [20], formula (16.10), page 409] and [20], estimate (16.14), page 411], we can assert that [15, Lemma 5.1] remains true for our $L$. Next, paraphrasing the proofs of [15, Lemma 5.2 and Theorem 5.4] (more detailed proofs are given in [29]), we can state the following Moser–Harnack inequality.

Theorem 2.2

Let $\eta ,\mu ,\varrho \in (0,1)$. Then there is $M>0$, depending on $n$, $\lambda$, $A$, $\eta$, $\mu$ and $\varrho$ such that for all $(x,s)\in \mathbb {R}^n\times \mathbb {R}$, all $R>0$ and all non negative $u\in C^{2,1}(\overline{B}(x, R)\times [s- R^2,s])$ satisfying $Lu=0$ one has

$$\begin{aligned} u(y,t)\le Mu(x,s)\quad \text{ for } \text{ all }\; (y,t)\in \overline{B}(x,\varrho R)\times [s-\eta R^2,s-\mu R^2]. \end{aligned}$$

3 Gaussian lower bound for the Neumann Green function

We recall that the derivative of $U=U(x,t)$ at $(x,t)\in \partial \Omega \times [t_0,t_1]$ in the conormal direction is given by

$$\begin{aligned} \partial _\nu U(x,t)= a_{ij}(x,t)\mathbf {n}_j(x)\partial _iU(x,t), \end{aligned}$$

where $\mathbf {n}(x)=(\mathbf {n}_1(x), \ldots ,\mathbf {n}_n(x))$ is the unit outward normal vector at $x$.

For $\tau \in [t_0,t_1[$, we set $Q_\tau =\Omega \times (\tau , t_1)$, $\Sigma _\tau =\partial \Omega \times (\tau ,t_1)$ and we consider the Neumann initial-boundary value problem (abbreviated to IBVP in the sequel) for the operator $L$:

$$\begin{aligned} \left\{ \begin{array}{ll} Lu =0 &{}\quad \text{ in }\; Q_\tau ,\\ u(\cdot \, ,\tau )=\psi &{}\quad \text{ in }\; \Omega ,\\ \partial _\nu u=0 &{}\quad \text{ on }\; \Sigma _\tau . \end{array} \right. \end{aligned}$$

(3.1)

From [16, Theorem 2, page 144] and its proof, for any $\psi \in C_0^\infty (\Omega )$, the IBVP (3.1) has a unique solution $u\in C^{1,0}(\overline{Q}_\tau )\cap C^{2,1}(Q_\tau )$ given by

$$\begin{aligned} u(x,t)=\int _\tau ^t\int _{\partial \Omega }E (x,t;\xi ,\sigma )\varphi (\xi ,\sigma )d \xi d\sigma +\int _\Omega E (x,t;\xi ,\tau )\psi (\xi )d\xi . \end{aligned}$$

(3.2)

Here

$$\begin{aligned} \varphi (x,t)=F_\tau (x,t)-2\sum _{\ell \ge 1}\int _\tau ^t\int _{\partial \Omega } M_\ell (x,t;\xi ,\sigma )F_\tau (\xi ,\sigma )d\xi d\sigma , \end{aligned}$$

(3.3)

with

$$\begin{aligned} F_\tau (x,t)&=-2\int _\Omega \partial _\nu E (x,t;\xi ,\tau ) \psi (\xi )d\xi ,\\ M_1&=-2\partial _\nu E ,\\ M_{\ell +1}(x,t;\xi ,\tau )&=\int _\tau ^t\int _{\partial \Omega }M_1(x,t;\eta ,\sigma )M_\ell (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma .\end{aligned}$$

For $(x,t)\in \Sigma _\tau$ and $\xi \in \Omega$, let

$$\begin{aligned} \mathcal {N}(x,t; \xi ,\tau )&= -2\partial _\nu E (x,t;\xi ,\tau )\\&\quad -2\sum _{\ell \ge 1}\int _\tau ^t\int _{\partial \Omega }M_\ell (x,t;\eta ,\sigma )\partial _\nu E (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma . \end{aligned}$$

Assume for the moment (see the proof below) that

$$\begin{aligned} \varphi (x,t)=\int _\Omega \mathcal {N}(x,t; \xi ,\tau )\psi (\xi )d\xi . \end{aligned}$$

(3.4)

We set

$$\begin{aligned} G(x,t,\xi ,\tau )= \int _\tau ^t\int _{\partial \Omega }E (x,t;\eta ,\sigma )\mathcal {N}(\eta ,\sigma ;\xi ,\tau )d \eta d\sigma +E (x,t;\xi ,\tau ). \end{aligned}$$

(3.5)

It follows from Fubini’s theorem that

$$\begin{aligned} u (x,t)=\int _\Omega G(x,t;\xi ,\tau )\psi (\xi )d\xi . \end{aligned}$$

(3.6)

The function $G$ is called the Neumann Green function associated to the equation $Lu=0$ in $Q$.

We have, for any $0\le \psi \in C_0^\infty (\Omega )$, $u \ge 0$, according to the maximum principle (see for instance [23, Theorem 2.9, page 15] and remarks following it). Whence, $G\ge 0$.

From the uniqueness of the solution of the IBVP (3.1), we have also

$$\begin{aligned} \int _\Omega G(x,t;\xi ,\tau )\psi (\xi )d\xi&= \int _\Omega G(x,t;\eta ,\sigma )d\eta \int _\Omega G(\eta ,\sigma ;\xi ,\tau )\psi (\xi )d\xi \\&\quad \text{ for } \text{ any }\; \psi \in C_0^\infty (\Omega ),\; \tau <\sigma <t. \end{aligned}$$

Therefore,

$$\begin{aligned} G(x,t;\xi ,\tau )=\int _\Omega G(x,t;\eta ,\sigma )G(\eta ,\sigma ;\xi ,\tau )d\eta ,\quad \tau <\sigma <t. \end{aligned}$$

(3.7)

That is, $G$ has the reproducing property.

We note that when $c=0$, $G$ satisfies in addition

$$\begin{aligned} \int _\Omega G(x,t;\xi ,\tau )d\xi =1. \end{aligned}$$

The key point in the proof of our Gaussian lower bound for $G$ is the following lemma.

Lemma 3.1

For $1/2<\mu < \frac{n}{2}$, we have

$$\begin{aligned} |\mathcal {N}(x,t;\xi ,\tau )|\le C(t-\tau )^{-\mu }|x-\xi |^{-n+2\mu },\quad (x,t)\in \Sigma _\tau ,\; \xi \in \Omega ,\; x\ne \xi . \end{aligned}$$

(3.8)

The lemma below appears in [16, page 137] as Lemma 1. It is needed for proving Lemma 3.1.

Lemma 3.2

Let $0< a,b <n-1$ with $a+b\ne n-1$. Then

$$\begin{aligned} \int _{\partial \Omega }|x-\eta |^{-a}|\eta -\xi |^{-b}d\eta \le \left\{ \begin{array}{ll} \widehat{C}|x-\xi |^{n-1-(a+b)} &{}\quad {if}\; a+b>n-1\\ \widehat{C} &{}\quad {if}\; a+b<n-1. \end{array} \right. \end{aligned}$$

(3.9)

Proof of Lemma 3.1

Since $\Omega$ is of class $C^{1,1}$, we obtain by paraphrasing the proof of [16], formula $(2.12)$, page 137]:

$$\begin{aligned} |\partial _\nu E(x,t;\xi ,\tau )|\le C(t-\tau )^{-\mu }|x-\xi |^{-n+2\mu }, \end{aligned}$$

for any $\mu >0$, and then

$$\begin{aligned} | M_1(x,t;\xi ,\tau )|\le C(t-\tau )^{-\mu }|x-\xi |^{-n+2\mu }. \end{aligned}$$

(3.10)

We assume first that $1/2<\mu <1$. Since

$$\begin{aligned} |M_2(x,t;\xi ,\tau )|\le \int _\tau ^t\int _{\partial \Omega }|M_1(x,t;\eta ,\sigma )||M_1(\eta ,\sigma ;\xi ,\tau )|d\eta d\sigma , \end{aligned}$$

(3.10) leads

$$\begin{aligned} |M_2(x,t;\xi ,\tau )|\le C^2\int _\tau ^t(t-\sigma )^{-\mu }(\sigma -\tau )^{-\mu }d\sigma \int _{\partial \Omega }|x-\eta |^{-n+2\mu }|\xi -\eta |^{-n+2\mu }d\eta . \end{aligned}$$

(3.11)

By Lemma 3.2,

$$\begin{aligned}&\int _{\partial \Omega }|x-\eta |^{-n+2\mu }|\xi -\eta |^{-n+2\mu }d\eta \nonumber \\&\quad \le \left\{ \begin{array}{ll} \widehat{C}|x-\xi |^{-n+4\mu -1} &{}\quad \text {if}\; n\ge 3\quad \text {or}\quad n=2 \quad \text {and}\quad \frac{1}{2}<\mu <\frac{3}{4},\\ \widehat{C} &{}\quad \text {if}\; n =2\quad \text {and}\quad \frac{3}{4}<\mu <1. \end{array}\right. \end{aligned}$$

(3.12)

On the other hand

$$\begin{aligned} \int _\tau ^t(t-\sigma )^{-\mu }(\sigma -\tau )^{-\mu }d\sigma&= (t-\tau )^{-\mu +(1-\mu )}\int _0^1s^{-\mu }(1-s)^{1-\mu }ds\nonumber \\&= (t-\tau )^{-\mu +(1-\mu )}B(1-\mu ,1-\mu ). \end{aligned}$$

(3.13)

We plug (3.12) and (3.13) into (3.11), and we obtain

$$\begin{aligned} |M_2(x,t;\xi ,\tau )|&\le C^2\widehat{C}(t-\tau )^{-\mu +(1-\mu )}B(1-\mu ,1-\mu )|x-\xi |^{-n+2\mu +(2\mu -1)},\\&\text {if } n\ge 3 \text { or } n=2 \text { and } \frac{1}{2}<\mu <\frac{3}{4} \end{aligned}$$

and

$$\begin{aligned} |M_2(x,t;\xi ,\tau )|\le C^2\widehat{C}(t-\tau )^{-\mu +(1-\mu )}B(1-\mu ,1-\mu ),\quad \text {if}\; n =2\; \text {and}\; \frac{3}{4}<\mu <1. \end{aligned}$$

Let $\ell (n)$ be the smallest integer $\ell$ so that $n+1<2\ell$ and fix $\frac{n+1}{2\ell (n)}<\mu <1$. Then an induction argument yields

$$\begin{aligned} |M_\ell (x,t;\xi ,\tau )|\le C^\ell \widehat{C}^{\ell -1}(t-\tau )^{-\mu +(\ell -1)(1-\mu )}\frac{\Gamma (1-\mu )^\ell }{\Gamma (\ell (1-\mu ))},\quad \ell \ge \ell (n). \end{aligned}$$

By Stirling’s formula

$$\begin{aligned} \Gamma (\ell (1-\mu ))\sim (e^{-1}(\ell (1-\mu )-1))^{\ell (1-\mu )-1}\sqrt{2\pi (\ell (1-\mu )-1))}\qquad \text{ as }\; \ell \rightarrow +\infty , \end{aligned}$$

implying that the series

$$\begin{aligned} S= \sum _{\ell \ge \ell (n)}C^\ell \widehat{C}^{\ell -1}T^{-\mu +(\ell -1)(1-\mu )}\frac{\Gamma (1-\mu )^\ell }{\Gamma (\ell (1-\mu ))} \end{aligned}$$

converges.

Clearly,

$$\begin{aligned} |\mathcal {N}(x,t;\xi ,\tau )| \le \ \sum _{\ell =1}^{\ell (n)-1}|M_\ell (x,t;\xi ,\tau )|+S. \end{aligned}$$

Therefore, it is enough to prove that

$$\begin{aligned} \widetilde{\mathcal {N}}(x,t;\xi ,\tau )= \sum _{\ell =1}^{\ell (n)-1}|M_\ell (x,t;\xi ,\tau )| \end{aligned}$$

satisfies (3.8). To this end, we observe that

$$\begin{aligned} |M_2(x,t;\xi ,\tau )|&\le \int _\tau ^{(\tau +t)/2}\int _{\partial \Omega }|M_1(x,t;\eta ,\sigma )||M_1(\eta ,\sigma ;\xi ,\tau )|d\eta d\sigma \\&\quad + \int _{(\tau +t)/2} ^t\int _{\partial \Omega }|M_1(x,t;\eta ,\sigma )||M_1(\eta ,\sigma ;\xi ,\tau )|d\eta d\sigma . \end{aligned}$$

Assume that $1/2<\mu < n/2$ and pick $1/2<\alpha <\min (1,(n/2-\mu )+1/2)$. From Lemma 3.2, we have

$$\begin{aligned}&\int _\tau ^{(\tau +t)/2}\int _{\partial \Omega }|M_1(x,t;\eta ,\sigma )||M_1(\eta ,\sigma ;\xi ,\tau )|d\eta d\sigma \\&\quad \le C^2\int _\tau ^{(\tau +t)/2}(\sigma -\tau )^{-\alpha }(t-\sigma )^{-\mu }d\sigma \\&\quad \quad \int _{\partial \Omega }|x-\eta |^{-n+2\mu }|\xi -\eta |^{-n+2\alpha }d\eta \\&\quad \le C^2\left( \frac{t-\tau }{2}\right) ^{-\mu }\int _\tau ^{(\tau +t)/2}(\sigma -\tau )^{-\alpha }d\sigma |x-\xi |^{-n+2\mu +2\alpha -1}\\&\quad \le C^2\left( \frac{t-\tau }{2}\right) ^{-\mu -\alpha +1} |x-\xi |^{-n+2\mu +2\alpha -1}\\&\quad \le C'(t-\tau )^{-\mu } |x-\xi |^{-n+2\mu }. \end{aligned}$$

Similarly,

$$\begin{aligned} \int _{(\tau +t)/2} ^t\int _{\partial \Omega }|M_1(x,t;\eta ,\sigma )||M_1(\eta ,\sigma ;\xi ,\tau )|d\eta d\sigma \le C(t-\tau )^{-\mu } |x-\xi |^{-n+2\mu }. \end{aligned}$$

Thus, $M_2$ satisfies (3.8). We repeat the previous argument to deduce that also $\widetilde{\mathcal {N}}$ obeys (3.8). $\square$

Proof of (3.4)

Let

$$\begin{aligned} \mathcal {N}_k(x,t; \xi ,\tau )=&-2\partial _\nu E (x,t;\xi ,\tau )\\&-2\sum _{\ell \ge 1}^k\int _\tau ^t\int _{\partial \Omega }M_\ell (x,t;\eta ,\sigma )\partial _\nu E (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma ,\\ \varphi _k (x,t)=&-2F_\tau (x,t)-2\sum _{\ell \ge 1}^k\int _\tau ^t\int _{\partial \Omega } M_\ell (x,t;\xi ,\sigma )F_\tau (\xi ,\sigma )d\xi d\sigma . \end{aligned}$$

In light of Lemma 3.2 and with the help of Lebesgue’s dominated convergence theorem, we can assert that

$$\begin{aligned} \int _\Omega \mathcal {N}_k(x,t; \xi ,\tau )\psi (\xi )d\xi \longrightarrow \int _\Omega \mathcal {N}(x,t; \xi ,\tau )\psi (\xi )d\xi \quad \text{ as }\; k\longrightarrow +\infty . \end{aligned}$$

According to Funini’s theorem

$$\begin{aligned} \varphi _k (x,t)= \int _\Omega \mathcal {N}_k(x,t; \xi ,\tau )\psi (\xi )d\xi . \end{aligned}$$

But $\varphi _k (x,t)\rightarrow \varphi (x,t)$ when $k$ tends to infinity. Then the uniqueness of the limit yields

$$\begin{aligned} \varphi (x,t)=\int _\Omega \mathcal {N}(x,t; \xi ,\tau )\psi (\xi )d\xi . \end{aligned}$$

$\square$

We are now ready to prove

Theorem 3.1

Under the assumption that $\Omega$ obeys the chain condition, the Neumann Green function $G$ satisfies the Gaussian lower bound:

$$\begin{aligned} \fancyscript{G}_C(x-\xi ,t-\tau )\le G(x,t;\xi ,\tau ),\quad (x,t;\xi ,\tau )\in Q^2\cap \{t>\tau \}. \end{aligned}$$

(3.14)

Proof

Let

$$\begin{aligned} G_0(x,t;\xi ,\tau )=\int _\tau ^t\int _{\partial \Omega }E (x,t;\eta ,\sigma )\mathcal {N}(\eta ,\sigma ;\xi ,\tau )d \eta d\sigma . \end{aligned}$$

From the Gaussian upper bound for $E$ we obtain in a straightforward way that, for any $\beta >0$,

$$\begin{aligned} |E(x,t;\xi ,\tau )|\le C(t-\tau )^{-\beta }|x-\xi |^{-n+2\beta }. \end{aligned}$$

On the other hand, by Lemma 3.1,

$$\begin{aligned} | \mathcal {N}(\eta ,\sigma ;\xi ,\tau )|\le C(t-\tau )^{-\mu }|x-\xi |^{-n+2\mu }. \end{aligned}$$

where $\frac{1}{2}<\mu < \frac{n}{2}$.

We fix $0<\epsilon <\frac{1}{2}$ and $0<\alpha <\frac{1}{2}$. In the preceding inequalities, we take $\mu =\frac{n}{2}-\epsilon$ and $\beta =1+\epsilon -\alpha$. In light of the fact that $-n+2\mu +2\beta -1=1-2\alpha >0$, we get from Lemma 3.2

$$\begin{aligned} |G_0(x,t;\xi ,\tau )|\le C(t-\tau )^{-n/2 +\alpha } . \end{aligned}$$

But, we know from (2.12) that

$$\begin{aligned} E (x,t;\xi ,\tau )\ge C(t-\tau )^{-n/2},\quad (x,t;\xi ,\tau )\in P^2,\; t>\tau ,\quad \widehat{C}|x-\xi |^2<t-\tau . \end{aligned}$$

Hence,

$$\begin{aligned} G(x,t;\xi ,\tau )&\ge E (x,t;\xi ,\tau )- |G_0(x,t;\xi ,\tau )|\\&\ge C(t-\tau )^{-n/2}(1-\widetilde{C}(t-\tau )^{\alpha }),\; t>\tau ,\quad \widehat{C}|x-\xi |^2<t-\tau . \end{aligned}$$

Consequently, we find $\delta >0$ so that

$$\begin{aligned} G (x,t;\xi ,\tau )&\ge C(t-\tau )^{-n/2},\quad (x,t;\xi ,\tau )\in Q^2,\; 0<t- \tau \le \delta ,\\&\widetilde{C}|x-\xi |^2<t-\tau . \end{aligned}$$

Or equivalently

$$\begin{aligned} G (x,t;\xi ,\tau )&\ge C(t-\tau )^{-n/2},\quad (x,t;\xi ,\tau )\in Q^2,\; 0<t- \tau \le \delta ,\nonumber \\&|x-\xi |<\widehat{C}(t-\tau )^{1/2}. \end{aligned}$$

(3.15)

As $\Omega$ has the chain condition, there exists a constant $c>0$, independent on $x$ and $\xi$, such that for any positive integer $k$ there exists a sequence $(x_i)_{0\le i\le k}$ of points in $\Omega$ so that $x_0=x$, $x_k=\xi$ and

$$\begin{aligned} |x_{i+1}-x_i|\le \frac{c}{k}|x-\xi |,\quad 0\le i\le k-1. \end{aligned}$$

(3.16)

When $2c|x-\xi |\le \widehat{C}(t-\tau )^{1/2}$ (implying $|x-\xi |\le \widehat{C}(t-\tau )^{1/2}$), Eq. (3.14) follows immediately from (3.15). Therefore we may assume that $2c|x-\xi | >\widehat{C}(t-\tau )^{1/2}$. We choose $m\ge 2$ to be the smallest integer satisfying

$$\begin{aligned} 2c\frac{|x-y|}{m^{1/2}}\le \widehat{C}(t-\tau )^{1/2}. \end{aligned}$$

Let $(x_i)_{0\le i\le m}$ be the sequence given by (3.16) when $k=m$ and

$$\begin{aligned} r=\frac{1}{4}\widehat{C}\left( \frac{t-\tau }{m}\right) ^{1/2}. \end{aligned}$$

In light of the reproducing property and the positivity of $G$, we obtain

$$\begin{aligned} G(x,t;\xi ,\tau )&=\int _\Omega \ldots \int _\Omega G\left( x,t;\xi _1, \frac{(m-1)t+\tau }{m}\right) \nonumber \\&\quad \ldots G\left( \xi _{m-1}, \frac{t+(m-1)\tau }{m};\xi ,\tau \right) d\xi _1\ldots d\xi _{m-1}\nonumber \\&\ge \int _{B(x_1,r)\cap \Omega } \ldots \int _{B(x_{m-1},r)\cap \Omega } G\left( x,t;\xi _1, \frac{(m-1)t+\tau }{m}\right) \nonumber \\&\quad \ldots G\left( \xi _{m-1}, \frac{t+(m-1)\tau }{m} ;\xi ,\tau \right) d\xi _1\ldots d\xi _{m-1}. \end{aligned}$$

(3.17)

Using that $\Omega$ is $C^{1,1}$-smooth, we obtain from the result in Appendix A: there exist two positive constants $d$ and $r_0$ such that, for any $z\in \overline{\Omega }$ and $0<\rho \le r_0$,

$$\begin{aligned} d\rho ^n\le |B(z,\rho )\cap \Omega |. \end{aligned}$$

(3.18)

We mention that Choi and Kim [8] observed that this condition is necessary for domains having a De Giorgi–Nash–Moser type estimate at the boundary.

In the sequel, replacing $\widehat{C}$ by a smaller constant, we may assume that $r\le r_0$.

Let $\xi _0=x$, $\xi _i \in B(x_i,r)$ and $\xi _m=\xi$. Then we have

$$\begin{aligned} |\xi _{i+1}-\xi _i|&\le |x_{i+1}-x_i| +2r\le c\frac{|x-\xi |}{m}+2r\\&\le c\frac{|x-\xi |}{m^{1/2}}+2r\le 4r,\quad 0\le i\le m-1 . \end{aligned}$$

Whence,

$$\begin{aligned} |\xi _{i+1}-\xi _i|\le \widehat{C}\left( \frac{t-\tau }{m}\right) ^{1/2},\quad 0\le i\le m-1 . \end{aligned}$$

It follows from (3.15) and (3.18) that

$$\begin{aligned} G(x,t;\xi ,\tau )&\ge \int _{B(x_1,r)\cap \Omega } \ldots \int _{B(x_{m-1},r)\cap \Omega }C^m\left( \frac{t-\tau }{m}\right) ^{-nm/2}d\xi _1\ldots d\xi _{m-1}\\&\ge (dr^n)^{m-1}C^m\left( \frac{t-\tau }{m}\right) ^{-nm/2}\\&\ge d^{m-1}\left[ \frac{\widehat{C}^2}{16}\left( \frac{t-\tau }{m}\right) \right] ^{n(m-1)/2}C^m\left( \frac{t-\tau }{m}\right) ^{-nm/2}\\&\ge \widetilde{C}C^m(t-\tau )^{-n/2}. \end{aligned}$$

Hence

$$\begin{aligned} G(x,t;\xi ,\tau )\ge \widetilde{C}e^{-Cm}(t-\tau )^{-n/2}. \end{aligned}$$

(3.19)

From the definition of $m$, we have

$$\begin{aligned} m-1\le \left( \frac{2c}{\widehat{C}}\right) ^2\frac{|x-y|^2}{t-\tau }. \end{aligned}$$

(3.20)

Finally, a combination of (3.19) and (3.20) leads to (3.14) when $t-\tau \le \delta$.

We complete the proof by showing that we can remove the assumption $t- \tau \le \delta$ in (3.15). Let then $0<t-\tau \le T$, such that $t-\tau >\delta$, and let $m\ge 2$ be the smallest integer such that $\delta ^{-1}(t-\tau )\le m$. We set

$$\begin{aligned} r=r_0T^{-1/2}m^{-1/2}(t-\tau )^{1/2} \end{aligned}$$

(3.21)

and we denote by $p$ the smallest integer satisfying

$$\begin{aligned} \frac{2cD}{rm}\le p,\quad \text{ with }\; D=\text{ diam }(\Omega ). \end{aligned}$$

If we choose $k=pm$ in (3.16), we obtain

$$\begin{aligned} |x_{i+1}-x_i|\le \frac{c|x-\xi |}{pm}\le \frac{cD}{pm}\le {r/2}. \end{aligned}$$

Let us denote by (3.17$^*)$ the inequality (3.17) in which we take $r/2$ in place of $r$, with $r$ given as in (3.21), and $m$ changed by $pm$.

Taking into account that

$$\begin{aligned} p<1+2cDr_0^{-1}T^{1/2}\delta ^{-1/2}=p^*, \end{aligned}$$

we get, for $\xi _i\in B(x_i,r)$, $1\le i\le m-1$.

$$\begin{aligned} \frac{pm|\xi _{i+1}-\xi _i|^2}{t-\tau }\le \frac{pmr^2}{t-\tau }<p^*r_0^2T^{-1}. \end{aligned}$$

(3.22)

As $(pm)^{-1}(t-\tau )\le \delta$, (3.14) holds true. Therefore, in light of (3.22), we obtain from $(3.17 ^*)$

$$\begin{aligned} G(x,t;\xi ,\tau )&\ge \!\left( d\!\left[ 2^{-1}r_0T^{-1/2}m^{-1/2}(t-\tau )^{1/2}\!\right] ^n\right) ^{pm-1}C^{pm}\!\left[ \!(pm)^{-1}(t-\tau )\!\right] ^{-pnm/2}\\&\ge \left( d\left[ 2^{-1}r_0T^{-1/2}\right] ^n\right) ^{pm-1}C^{pm}m^{n/2}p^{pnm/2}(t-\tau )^{-n/2}\\&\ge \left( d\left[ 2^{-1}r_0T^{-1/2}\right] ^n\right) ^{pm-1}(t-\tau )^{-n/2}\\&\ge \widetilde{C}\widehat{C}^{pm}(t-\tau )^{-n/2}\\&\ge \widetilde{C}e^{-{pm}|\ln \widehat{C}|}(t-\tau )^{-n/2}\\&\ge \widetilde{C}e^{-p^*m^*|\ln \widehat{C}|}(t-\tau )^{-n/2},\quad \text{ with }\; m^*=\delta ^{-1}T+1. \end{aligned}$$

This estimate completes the proof. $\square$

Theorem 3.1 can be easily extended to a Robin Green function. Indeed, if we replace the Neumann boundary condition by the following Robin boundary condition:

$$\begin{aligned} \partial _\nu u+q(x,t)u=0\quad \text{ in }\; \Sigma _\tau , \end{aligned}$$

where $q\in C(\Sigma _\tau )$, then $\mathcal {N}$ has to be changed by

$$\begin{aligned} \mathcal {N}_q(x,t; \xi ,\tau )=&-2[\partial _\nu E (x,t;\xi ,\tau )+ q(x,t)E (x,t;\xi ,\tau )]\\&-2\sum _{\ell \ge 1}\int _\tau ^t\int _{\partial \Omega }M_\ell (x,t;\eta ,\sigma )[\partial _\nu E (\eta ,\sigma ;\xi ,\tau )\\&+q(\eta ,\sigma )E (\eta ,\sigma ;\xi ,\tau )]d\eta d\sigma . \end{aligned}$$

Here

$$\begin{aligned} M_1(x,t;\xi ,\tau )&=-2[\partial _\nu E (x,t;\xi ,\tau )+ q(x,t)E (x,t;\xi ,\tau )]\\ M_{\ell +1}(x,t;\xi ,\tau )&=\int _\tau ^t\int _{\partial \Omega }M_1(x,t;\eta ,\sigma )M_\ell (\eta ,\sigma ;\xi ,\tau )d\eta d\sigma ,\quad \ell \ge 1. \end{aligned}$$

Apart the positivity of the Green function, which can be obtained by an adaptation of [9, Proposition 3.2], one can see without any difficulty, that the rest of our analysis holds true when $\mathcal {N}$ is replaced by $\mathcal {N}_q$.

We already mentioned that, for parabolic operators with time-independent coefficients, a Neumann Green function is nothing else but a Neumann heat kernel. Let us then consider a parabolic operator of the form

$$\begin{aligned} \mathcal {L}=\partial _j (a_{ij}(x)\partial _i\, \cdot \, )+b_k(x)\partial _k+c(x)-\partial _t, \end{aligned}$$

(3.23)

so that the following assumptions are satisfied:

(i$'$):

the matrix $(a_{ij}(x))$ is symmetric for any $x\in \overline{\Omega }$,

(ii$'$):

$a_{ij}\in W^{1,\infty }(\Omega ),\; \partial _ka_{ik},\; b_k,\; c \in C^1 (\overline{\Omega })$,

(iii$'$):

$a_{ij}(x)\xi _i\xi _j\ge \lambda |\xi |^2,\; (x,t)\in \overline{\Omega } ,\; \xi \in \mathbb {R}^n$,

(iv$'$):

$\Vert a_{ij}\Vert _{W^{1,\infty }(\Omega )}+\Vert \partial _ka_{ik}+b_k\Vert _{L^\infty (\Omega )}+\Vert c\Vert _{L^\infty (\Omega )}\le A$,

where $\lambda >0$ and $A>0$ are two given constants.

Here again, the assumptions on the coefficients of $\mathcal {L}$ are not necessarily the best possible.

Let $\mathfrak {a}$ be the unbounded bilinear form defined by $D(\mathfrak {a})=H^1(\Omega )$ and

$$\begin{aligned} \mathfrak {a}(u,v)=\int _\Omega a_{ij}\partial _iu\partial _jvdx+\int _\Omega b_k\partial _kuvdx+\int _\Omega cuvdx,\quad u,v\in D(\mathfrak {a}). \end{aligned}$$

We associate to $\mathfrak {a}$ the unbounded operator $\mathcal {A}$ given by

$$\begin{aligned} D(\mathcal {A})=\{ u\in L^2(\Omega );\; \exists v\in L^2(\Omega ):\mathfrak {a}(u,\varphi )=(v,\varphi )_2,\; \varphi \in H^1(\Omega )\},\quad \mathcal {A}u:=v. \end{aligned}$$

Here $(\cdot ,\cdot )_2$ denotes the usual scalar product of $L^2(\Omega )$.

We have

$$\begin{aligned} \int _\Omega b_k\partial _ku u \le \frac{2}{\lambda }\int _\Omega |\nabla u |^2+\left( \frac{8\sup _k\Vert b_k\Vert _{L^\infty (\Omega )}}{\lambda }\right) \int _\Omega u^2,\quad u\in H^1(\Omega ). \end{aligned}$$

This and (iii$'$) entail that $\mathfrak {a}+\kappa$ is accretive for a sufficiently large $\kappa >0$. Since $\mathfrak {a}$ is clearly densely defined and continuous on $H^1(\Omega )$, we derive from [27, Theorem 1.52, page 29] that $-\mathcal {A}$ is the generator of an holomorphic semigroup $e^{-t\mathcal {A}}$.

Let $Q=\Omega \times (0,T)$, $\Sigma =\partial \Omega \times (0,T)$, $\psi \in C_0^\infty (\Omega )$ and $u\in C^{1,0}(\overline{Q})\cap C^{2,1}(Q)$ ([16, Theorem2, page 144]) be the unique solution of the IBVP

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}u =0 &{}\quad \text{ in }\; Q,\\ u(\cdot \, ,0 )=\psi &{}\quad \text{ in }\; \Omega ,\\ \partial _\nu u=0 &{}\quad \text{ on }\; \Sigma . \end{array} \right. \end{aligned}$$

(3.24)

By [3, Theorem 10.9, page 341], $u\in L^2(0,T;H^1(\Omega ))\cap C([0,T];L^2(\Omega ))$, $u'\in L^2(0,T; [H^1(\Omega )]')$ and it is the unique solution of

$$\begin{aligned} \left\{ \begin{array}{l} \langle u'(t),v\rangle +\mathfrak {a}(u(t),v)=0\quad \text {a.e.}\; t\in [0,T],\; v\in H^1(\Omega ),\\ u(0)=\varphi \end{array} \right. \end{aligned}$$

(3.25)

where $\langle \cdot ,\cdot \rangle$ is the duality pairing between $H^1(\Omega )$ and its dual space $[H^1(\Omega )]'$.

We set $\widetilde{u}(t)=e^{-t\mathcal {A}}\psi$, $t\ge 0$. By using $\widetilde{u}'(t)=\mathcal {A}u(t)$, $t\in [0,T]$, we obtain in a straightforward manner

$$\begin{aligned} \langle \widetilde{u}'(t),v\rangle +\mathfrak {a}(\widetilde{u}(t),v)=0,\quad t\in [0,T],\; v\in H^1(\Omega ), \end{aligned}$$

Using that $u(0)=\widetilde{u}(0)$, we get from the uniqueness of the solution of problem (3.25) that $u=\widetilde{u}$. Hence,

$$\begin{aligned} e^{-t\mathcal {A}}\psi (x)=\int _\Omega G(x,t;\xi ,0)\psi (\xi )d\xi ,\quad 0<t \le T. \end{aligned}$$

We rewrite this equality as follows:

$$\begin{aligned} e^{-t\mathcal {A}}\psi (x)=\int _\Omega K(x,\xi ,t)\psi (\xi )d\xi ,\quad 0<t \le T. \end{aligned}$$

The function

$$\begin{aligned} K(x,\xi ,t)= G(x,t;\xi ,0) \end{aligned}$$

is usually called the heat kernel of the semigroup $e^{-t\mathcal {A}}$.

We have as an immediate consequence of Theorem 3.1:

Corollary 3.1

When $\Omega$ possesses the chain condition, the Neumann heat kernel $K$ satisfies the Gaussian lower bound:

$$\begin{aligned} \fancyscript{G}_C(x-\xi ,t)\le K(x,\xi ,t),\quad (x,\xi )\in \Omega ^2 ,\quad 0<t\le T. \end{aligned}$$

(3.26)

A Gaussian lower bound for the Neumann heat kernel was proved in [10] when $L$ is the Laplace operator. The key point is the Hölder continuity of $x\longrightarrow K(x,\xi ,t)$ which relies on the fact that $\mu -\mathcal {A}$ is an isomorphism from $H^s(\Omega )$ into $H^{s-2}(\Omega )$, for large $\mu$ and $s>n/2+1$. We note that a quick examination of the proof in [10] shows that this result can be extended to a divergence form operator with $C^\infty$-smooth coefficients.

We end this section by showing that we can obtain a strong maximum from Theorem 3.1. Let $\psi \in C(\overline{\Omega })$, $f\in C(\overline{Q}_\tau )$, $g\in C(\overline{\Sigma } _\tau )$ and consider the IBVP

$$\begin{aligned} \left\{ \begin{array}{ll} Lu =f &{}\quad \text{ in }\; Q_\tau ,\\ u(\cdot \, ,\tau )=\psi &{}\quad \text{ in }\; \Omega ,\\ \partial _\nu u=g &{}\quad \text{ on }\; \Sigma _\tau . \end{array} \right. \end{aligned}$$

(3.27)

Corollary 3.2

We assume that $\psi \ge 0$, $f\ge 0$, $g\ge 0$ and at least one of the functions $\psi$, $f$ and $g$ is non identically equal to zero. If $u\in C^{0,1}(\overline{Q}_\tau )\cap C^{2,1}(Q_\tau )$ is the solution of the IBVP (3.27), then

$$\begin{aligned} u>0\quad \text{ in }\; \Omega \times ]\tau , t_1]. \end{aligned}$$

Proof

Follows from Theorem 3.1 since (see for instance [16], formula (3.5), page 144])

$$\begin{aligned} u(x,t)&= \int _\Omega G(x,t;\xi ,\tau )\psi (\xi )d\xi +\int _\tau ^t\int _\Omega G(x,t;\xi ,s)f(\xi ,s)d\xi ds\\&+\int _\tau ^t\int _{\partial \Omega } G(x,t;\xi ,s)g(\xi ,s)dS_\xi ds. \end{aligned}$$

$\square$

Appendix A

In this appendix we prove (3.18). Henceforth, $\Omega$ is a bounded domain of $\mathbb {R}^n$ with boundary $\Gamma$.

Following [18, Definition 2.4.1, page 50], we introduce the notation

$$\begin{aligned} \fancyscript{C}(y,\xi ,\epsilon )=\{ z\in \mathbb {R}^n;\; (z-y)\cdot \xi \ge (\cos \epsilon )|z-y|,\ 0<|y-z|<\epsilon \}, \end{aligned}$$

where $y\in \mathbb {R}^n$, $\xi \in \mathbb {S}^{n-1}$ and $0<\epsilon$. That is, $C(y,\xi ,\epsilon )$ is the cone, of dimension $\epsilon$, with vertex $y$, aperture $\epsilon$ and directed by $\xi$.

We say that $\Omega$ has the $\epsilon$-cone property if

$$\begin{aligned} \text {for any}\; x\in \Gamma ,\; \text {there exists}\; \xi _x\in \mathbb {S}^{n-1}\; \text {so that, for all}\; y\in \overline{\Omega }\cap B(x,\epsilon ),\; \fancyscript{C}(y,\xi _x,\epsilon )\subset \Omega . \end{aligned}$$

(4.1)

Assume that $\Omega$ has the $\epsilon$-cone property, for some $0<\epsilon$. By using the compactness of $\Gamma$, we find a finite number of points of $\Gamma$, $x_1,\ldots x_p$, so that $\Gamma =\bigcup _k\left[ \Gamma \cap B(x_k,\epsilon /2)\right]$ and (4.1) is satisfied for each $x_i$, $i=1,\ldots ,p$. Let $K=\overline{\Omega }{\setminus } \bigcup _k B(x_k,\epsilon /2)$. Then, $0<\varrho =\text {dist}(K,\Gamma ) (<\epsilon )$ and therefore, for each $x\in K$, we have $B(x,\varrho )\subset \Omega$. We deduce from this observation that, for each $x\in \overline{\Omega }$ and $0<r<\varrho$, $\Omega \cap B(x,r)$ contains a cone of dimension $r$ and aperture $\epsilon$. It is then straightforward to get the following inequality:

$$\begin{aligned} |\Omega \cap B(x,r)|\ge cr^n,\quad 0<r<\varrho , \end{aligned}$$

for some constant $c=c(n,\varrho )$.

We complete the proof of (3.18) by using the following theorem.

Theorem 4.1

$\Omega$ has the $\epsilon$-cone property, for some $0<\epsilon$, if and only if its boundary $\Gamma$ is Lipschitz.

We refer to [18, Theorem 2.4.7, page 53] for a detailed proof of this theorem.