GAUSSIAN ESTIMATES FOR FUNDAMENTAL SOLUTIONS OF
SECOND ORDER PARABOLIC SYSTEMS WITH
TIME-INDEPENDENT COEFFICIENTS
SEICK KIM
Abstract. Auscher, McIntosh and Tchamitchian studied the heat kernels
of second order elliptic operators in divergence form with complex bounded
measurable coefficients on Rn. In particular, in the case when n = 2 they
obtained Gaussian upper bound estimates for the heat kernel without imposing
further assumption on the coefficients. We study the fundamental solutions of
the systems of second order parabolic equations in the divergence form with
bounded, measurable, time-independent coefficients, and extend their results
to the systems of parabolic equations.
1. Introduction
In 1967, Aronson [1] proved Gaussian upper and lower bounds for the fundamen-
tal solutions of parabolic equations in divergence form with bounded measurable
coefficients. To establish the Gaussian lower bound Aronson made use of the Har-
nack inequality for nonnegative solutions which was proved by Moser in 1964 (see
[17]). Related to Moser’s parabolic Harnack inequality, we should mention Nash’s
earlier paper [18] where the Hölder continuity of weak solutions to parabolic equa-
tions in divergence form was established. In 1985, Fabes and Stroock [10] showed
that the idea of Nash could be used to establish a Gaussian upper and lower bound
on the fundamental solution. They showed that actually such Gaussian estimates
could be used to prove Moser’s Harnack inequality. We note that Aronson also ob-
tained Gaussian upper bound estimates of the fundamental solution without using
Moser’s Harnack inequality.
In [2], Auscher proposed a new proof of Aronson’s Gaussian upper bound esti-
mates for the fundamental solution of second order parabolic equations with time-
independent coefficients. His method relies crucially on the assumption that the
coefficients are time-independent and thus it does not exactly reproduce Aronson’s
result, which is valid even for the time-dependent coefficients case. However, his
method is interesting in the sense that it carries over to equations with complex
coefficients provided that the complex coefficients are a small perturbation of real
coefficients. Along with this direction, Auscher, McIntosh and Tchamitchian also
showed that the heat kernel of second order elliptic operators in divergence form
with complex bounded measurable coefficients in the two dimensional space has a
Gaussian upper bound (see [3] and also [5]).
We would like to point out that a parabolic equation with complex coefficients
is, in fact, a special case of a system of parabolic equations. From this point of
2000 Mathematics Subject Classification. Primary 35A08, 35B45; Secondary 35K40.
Key words and phrases. Gaussian estimates, a priori estimates, parabolic system.
http://arxiv.org/abs/0704.1372v1
2 SEICK KIM
view, Hofmann and the author showed that the fundamental solution of a parabolic
system has an upper Gaussian bound if the system is a small perturbation of a di-
agonal system, which, in particular, generalized the result of Auscher mentioned
above to the time-dependent coefficients case (see [12]). However, the above men-
tioned result of Auscher, McIntosh and Tchamitchian regarding the heat kernel of
two dimensional elliptic operators with complex coefficients does not follow directly
from our result.
One of the main goals of this article is to provide a proof that weak solutions of
the parabolic system of divergence type with time-independent coefficients associ-
ated to an elliptic system in two dimensions enjoy the parabolic local boundedness
property and to show that its fundamental solution has a Gaussian upper bound.
More generally, we show that if weak solutions of an elliptic system satisfy Hölder
estimates at every scale, then weak solutions of the corresponding parabolic system
with time-independent coefficients also satisfies similar parabolic Hölder estimates
from which, in particular, the parabolic local boundedness property follows easily.
Also, such an argument allows one to derive Hölder continuity estimates for weak
solutions of parabolic equations with time-independent coefficients directly from
De Giorgi’s theorem [7] on elliptic equations, bypassing Moser’s parabolic Harnack
inequality. In fact, this is what Auscher really proved in the setting of complex
coefficients equations by using a functional calculus method (see [2] and also [4],
[5]). Even in those complex coefficients settings, we believe that our approach is
much more straightforward and thus appeals to wider readership.
Finally, we would like to point out that in this article, we are mainly interested
in global estimates and that we do not attempt to treat, for example, the systems
with lower order terms, etc. However, let us also mention that, with some extra
technical details, our methods carry over to those cases as well as to the systems
of higher order; see e.g. [4], [5] for the details, and also Remark 3.5.
The remaining sections are organized in the following way. In Section 2 we give
notations, definitions, and some known facts. We state the main results in Section 3
and give the proofs in Section 4.
2. Notation and definitions
2.1. Geometric notation.
(1) Rn = n-dimensional real Euclidean space.
(2) x = (x1, · · · , xn) is an arbitrary point of R
(3) X = (x, t) denotes an arbitrary point in Rn+1, where x ∈ Rn and t ∈ R.
(4) Br(x) = {y ∈ R
n : |y − x| < r} is an open ball in Rn with center x and
radius r > 0. We sometimes drop the reference point x and write Br for
Br(x) if there is no danger of confusion.
(5) Qr(X) =
(y, s) ∈ Rn+1 : |y − x| < r and t− r2 < s < t
. We sometimes
drop the reference point X and write Qr for Qr(X).
(6) Q∗r(X) =
(y, s) ∈ Rn+1 : |y − x| < r and t < s < t+ r2
(7) Qr,s(X) = {(y, s) ∈ Qr(X)}; i.e., Qr,s(X) = Br(x) × {s} if s ∈ (t − r
2, t)
and Qr,s(X) = ∅ otherwise. We sometimes drop the reference point X and
write Qr,s for Qr,s(X).
(8) For a cylinder Q = Ω× (a, b) ⊂ Rn+1, ∂PQ denotes its parabolic boundary,
namely, ∂PQ = ∂Ω × (a, b) ∪ Ω × {a}, where ∂Ω is the usual topological
boundary of Ω ⊂ Rn and Ω is its closure.
GAUSSIAN ESTIMATES 3
2.2. Notation for functions and their derivatives.
(1) For a mapping from Ω ⊂ Rn to RN , we write f(x) = (f1(x), . . . , fN (x))T
as a column vector.
(2) fQ =
f , where |Q| denotes the volume of Q.
(3) ut = ∂u/∂t.
(4) Dxiu = Diu = uxi = ∂u/∂xi.
(5) Du = (ux1 , . . . , uxn)
T is the spatial gradient of u = u(x, t).
(6) For f = (f1, . . . , fN )T , Df = (Df1, . . . , DfN); that is Df is the n × N
matrix whose i-th column is Df i.
2.3. Function spaces.
(1) For Ω ⊂ Rn and p ≥ 1, Lp(Ω) denotes the space of functions with the
following norms:
‖u‖Lp(Ω) =
|u(x)|
and ‖u‖L∞(Ω) = ess supΩ |u| .
(2) Cµ(Ω) denotes the space of functions that are Hölder continuous with the
exponent µ ∈ (0, 1], and
[u]Cµ(Ω) = sup
x 6=x′∈Ω
|u(x)− u(x′)|
|x− x′|
µ < ∞.
(3) The Morrey space M2,µ(Ω) is the set of all functions u ∈ L2(Ω) such that
‖u‖M2,µ(Ω) = sup
Bρ(x)⊂Ω
Bρ(x)
(4) C
P (Q) denotes the space of functions defined on Q ⊂ R
n+1 such that
[u]Cµ
(Q) = sup
X 6=X′∈Q
|u(X)− u(X ′)|
dP (X,X ′)µ
where dP (X,X
′) = max
|x− x′| ,
|t− t′|
2.4. Elliptic and parabolic systems and their adjoints.
Definition 2.1. We say that the coefficients A
ij (x) satisfy the uniform ellipticity
condition if there exist numbers ν0,M0 > 0 such that for all x ∈ R
n we have
(2.1)
αβ(x)ξβ , ξα
≥ ν0 |ξ|
αβ(x)ξβ ,ηα
≤ M0 |ξ| |η| ,
where we used the following notation.
(1) For α, β = 1, . . . , n, Aαβ(x) are N ×N matrices with (i, j)-entries A
ij (x).
(2) ξα = (ξ
α, · · · , ξ
T and |ξ|
αβ(x)ξβ ,ηα
α,β=1
i,j=1
ij (x)ξ
We emphasize that we do not assume that the coefficients are symmetric.
4 SEICK KIM
Definition 2.2. We say that a system of N equations on Rn
α,β=1
Dxα(A
ij (x)Dxβu
j) = 0 (i = 1, . . . , N)
is elliptic if the coefficients satisfy the uniform ellipticity condition. We often write
the above system in a vector form
(2.2) Lu :=
α,β=1
αβ(x)Dβu) = 0, u = (u
1 . . . , uN )T .
The adjoint system of (2.2) is given by
(2.3) L∗u :=
α,β=1
(Aαβ)∗(x)Dβu
where (Aαβ)∗ = (Aβα)T , the transpose of Aβα.
Definition 2.3. We say that a system of N equations on Rn+1
uit −
α,β=1
Dxα(A
ij (x)Dxβu
j) = 0 (i = 1, . . . , N)
is parabolic if the (time-independent) coefficients satisfy the uniform ellipticity
condition. We often write the above system in a vector form
(2.4) ut − Lu := ut −
α,β=1
αβ(x)Dβu) = 0.
The adjoint system of (2.4) is given by
(2.5) ut + L
∗u := ut +
α,β=1
(Aαβ)∗(x)Dβu
where (Aαβ)∗ = (Aβα)T , the transpose of Aβα.
2.5. Weak solutions. In this article, the term “weak solution” is used in a rather
abusive way. To avoid unnecessary technicalities, we may assume that all the coef-
ficients involved are smooth so that all weak solutions are indeed classical solutions.
However, this extra smoothness assumption will not be used quantitatively in our
estimates. This is why we shall make clear the dependence of constants.
(1) We say that u is a weak solution of (2.2) in Ω ⊂ Rn if u is a (classical)
solution of (2.2) in Ω and u, Du ∈ L2(Ω).
(2) We say that u is a weak solution of (2.4) in a cylinder Q = Ω×(a, b) ⊂ Rn+1
if u is a (classical) solution of (2.2) in Q and u, Du ∈ L2(Q), u(·, t) ∈ L2(Ω)
for all a ≤ t ≤ b, and supa≤t≤b ‖u(·, t)‖L2(Ω) < ∞.
2.6. Fundamental solution. By a fundamental solution (or fundamental matrix)
Γ(x, t; y) of the parabolic system (2.4) we mean anN×N matrix of functions defined
for t > 0 which, as a function of (x, t), is a solution of (2.4) (i.e., each column is a
solution of (2.4)), and is such that
Γ(x, t; y)f(y) dy = f(x)(2.6)
GAUSSIAN ESTIMATES 5
for any bounded continuous function f = (f1, . . . , fN)T , where Γ(x, t; y)f (y) de-
notes the usual matrix multiplication.
2.7. Notation for estimates. We employ the letter C to denote a universal con-
stant usually depending on the dimension and ellipticity constants. It should be
understood that C may vary from line to line. We sometimes write C = C(α, β, . . .)
to emphasize the dependence on the prescribed quantities α, β, . . ..
2.8. Some preliminary results and known facts.
Lemma 2.4 (Energy estimates). Let u be a weak solution of (2.4) in QR = QR(X).
Then for 0 < r < R, we have
t−r2≤s≤t
|u(·, s)|
(R− r)2
Proof. See e.g., [14, Lemma 2.1, p. 139]. �
Lemma 2.5 (Parabolic Poincaré inequality). Let u be a weak solution of (2.4) in
QR = QR(X). Then there is some constant C = C(n,M0) such that
|u− uQR |
≤ CR2
Proof. See e.g., [19, Lemma 3]. �
Lemma 2.6. Let Q2R = Q2R(X0) be a cylinder in R
n+1. Suppose u ∈ L2(Q2R)
and there are positive constants µ ≤ 1 and M such that for any X ∈ QR and any
r ∈ (0, R) we have
Qr(X)
∣u− uQr(X)
≤ M2rn+2+2µ.
Then u is Hölder continuous in QR with the exponent µ and [u]Cµ
(QR) ≤ C(n, µ)M .
Proof. See e.g., [15, Lemma 4.3, p. 50]. �
Definition 2.7 (Local boundedness property). We say that the system (2.4) sat-
isfies the local boundedness property for weak solutions if there is a constant M
such that all weak solutions u of (2.4) in Q2r(X) satisfy the estimates
Qr(X)
|u| ≤ M
|Q2r|
Q2r(X)
Similarly, we say that the adjoint system (2.5) satisfies the local boundedness prop-
erty if the corresponding estimates hold for weak solutions u of (2.5) in Q∗2r(X).
Theorem 2.8 (Theorem 1.1, [12]). Assume that the system (2.4) and its adjoint
system (2.5) satisfy the local boundedness property for weak solutions. Then the
fundamental solution of the system (2.4) has an upper bound
(2.7) |Γ(x, t; y)|op ≤ C0t
−n/2 exp
k0 |x− y|
where |Γ(x, t; y)|op denotes the operator norm of the fundamental matrix Γ(x, t; y).
Here, C0 = C0(n, ν0,M0,M) and k0 = k0(ν0,M0).
6 SEICK KIM
3. Main results
Definition 3.1. We say that an elliptic system (2.2) satisfies the Hölder estimates
for weak solutions at every scale if there exist constants µ0 > 0 and H0 such that
all weak solutions u of the system in B2r = B2r(x0) satisfy the following estimates
(3.1) [u]Cµ0 (Br) ≤ H0r
−(n/2+µ0) ‖u‖L2(B2r) .
Similarly, we say that a parabolic system (2.4) satisfies Hölder estimates for weak
solutions at every scale if there exist constants µ1 > 0 and H1 such that all weak
solutions u of the system in Q2r = Q2r(X0) satisfy the following estimates
(3.2) [u]Cµ1
≤ H1r
−(n/2+1+µ1) ‖u‖L2(Q2r) .
Remark 3.2. Elliptic systems with constant coefficients satisfy the above property,
and in that case, the ellipticity condition (2.1) can be weakened and replaced by the
Legendre-Hadamard condition. De Giorgi’s theorem [7] states that the property is
satisfied if N = 1. The property is also satisfied if n = 2 and it is due to Morrey
(see Corollary 3.6). Some other examples include, for instance, a certain three
dimensional elliptic system which was studied by Kang and the author in [13].
We shall prove the following main results in this paper:
Theorem 3.3. If an elliptic system (2.2) satisfies the Hölder estimates for weak
solutions at every scale, then the corresponding parabolic system (2.4) with time-
independent coefficients also satisfies the Hölder estimates for weak solutions at
every scale.
Theorem 3.4. Suppose that the elliptic system (2.2) and its adjoint system (2.3)
defined on Rn both satisfy the Hölder estimates for weak solutions at every scale
with constants µ0, H0. Let Γ(x, t; y) be the fundamental solution of the parabolic
system (2.4) with the time-independent coefficients associated to the elliptic system
(2.2). Then Γ(x, t; y) has an upper bound
(3.3) |Γ(x, t; y)|op ≤ C0t
−n/2 exp
k0 |x− y|
where C0 = C0(n, ν0,M0, µ0, H0) and k0 = k0(ν0,M0). Here, |Γ(x, t; y)|op denotes
the operator norm of fundamental matrix Γ(x, t; y).
Remark 3.5. We would like to point out that (3.3) is a global estimate. Especially,
the bound (3.3) holds for all time t > 0. Suppose that the elliptic system (2.2) and
its adjoint system (2.3) enjoy the Hölder estimates for weak solutions up to a fixed
scale R0; that is, there is a number R0 > 0 such that if u is a weak solution of
either (2.2) or (2.3) in Br = Br(x) with 0 < r ≤ R0, then u is Hölder continuous
and satisfies
[u]Cµ0 (Br) ≤ H0r
−(n/2+µ0) ‖u‖L2(B2r) .
Then, the statement regarding the bound (3.3) for the fundamental solution should
be localized as follows: For any given T > 0, there are constants k0 = k0(ν0,M0)
and C0 = C0(n, ν0,M0, µ0, H0, R0, T ) such that (3.3) holds for 0 < t ≤ T .
Corollary 3.6. Let Γ(x, t; y) be the fundamental solution of the parabolic system
(2.4) with time-independent coefficients associated to an elliptic system (2.2) defined
on R2. Then Γ(x, t; y) has an upper bound (3.3) with the constants C0, k0 depending
only on the ellipticity constants ν0,M .
GAUSSIAN ESTIMATES 7
Proof. First, let us recall the well known theorem of Morrey which states that any
two dimensional elliptic system (2.2) with bounded measurable coefficients satisfies
the Hölder estimates for weak solutions at every scale, with the constants µ0, H0
depending only on the ellipticity constants (see, [16, pp. 143–148]). Next, note that
the ellipticity constants ν0,M0 in (2.1) remain unchanged for Ã
ij (x) = A
ji (x).
Therefore, the corollary is an immediate consequence of Theorem 3.4. �
Remark 3.7. In fact, the converse of Theorem 2.8 is also true (see [12, Theorem 1.2]).
Therefore, in order to extend the above corollary to the parabolic system with
time-dependent coefficients, one needs to show that the system satisfies the local
boundedness property for weak solutions. Unfortunately, we do not know whether
it is true or not if the coefficients are allowed to depend on the time variable. If
n ≥ 3, it is not true in general, even for the time-independent coefficients case since
there is a famous counter-example due to De Giorgi (see [8]).
4. Proof of Main Results
4.1. Some technical lemmas and proofs.
Lemma 4.1. If u is a weak solution of the parabolic system with time-independent
coefficients (2.4) in QR = QR(X0), then ut ∈ L
2(Qr) for r < R and satisfies the
estimates
(4.1) ‖ut‖L2(Qr) ≤ C(R − r)
−1 ‖Du‖L2(QR) .
In particular, if u is a weak solution of (2.4) in Q2r, then the above estimates
together with the energy estimates yield
(4.2) ‖ut‖L2(Qr) ≤ Cr
−2 ‖u‖L2(Q2r) .
Proof. We first note that if the coefficients are symmetric, (i.e., A
ij = A
ji ) this
is a well known result; a proof for such a case is found, for example, in [14, pp.
172–181] or in [9, pp. 360–364]. However, the standard proof does not carry over to
the non-symmetric coefficients case and for that reason, we provide a self-contained
proof here.
Fix positive numbers σ, τ such that σ < τ ≤ R. Let ζ be a smooth cut-off
function such that ζ ≡ 1 in Qσ, vanishes near ∂PQτ , and satisfies
0 ≤ ζ ≤ 1 and |ζt|+ |Dζ|
≤ C(τ − σ)−2.
Note that on each slice Qτ,s, we have
ut −Dα(A
αβDβu)
· ζ2ut
ζ2 |ut|
αβDβu, Dαut
αβDβu, Dαζut
8 SEICK KIM
Therefore, we find by using the Cauchy-Schwarz inequality that
ζ2 |ut|
ζ2 |Du| |Dut|+ C
ζ |Du| |Dζ| |ut|
ζ2 |Dut|
ζ2 |Du|
ζ2 |ut|
Thus we have
(4.3)
ζ2 |ut|
ζ2 |Dut|
ζ2 |Du|
Since ut also satisfies (2.4), the energy estimates yield
(4.4)
ζ2 |Dut|
(τ − σ)2
This is the part where we exploit the assumption that the coefficients are time-
independent. Combining (4.3) and (4.4), we have
(τ − σ)2
(τ − σ)2
If we set ǫ = (τ − σ)2/2C0, we finally obtain
(τ − σ)2
Here, we emphasize that C is a constant independent of σ, τ . Then by a standard
iteration argument (see e.g. [11, Lemma 3.1, pp. 161]), we have
(4.5)
(R− r)2
for 0 < r < R.
The proof is complete. �
Lemma 4.2. If u is a weak solution of the parabolic system with time-independent
coefficients (2.4) in Q2r = Q2r(X0), then Du(·, s),ut(·, s) ∈ L
2(Qr,s) for all s ∈
[t0 − r
2, t0], and satisfy the following estimates uniformly in s ∈ [t0 − r
2, t0].
‖Du(·, s)‖L2(Qr,s) ≤ Cr
−2 ‖u‖L2(Q2r) ,(4.6)
‖ut(·, s)‖L2(Qr,s) ≤ Cr
−3 ‖u‖L2(Q2r) .(4.7)
Proof. By the energy estimates applied to ut we obtain
(4.8) sup
t0−r2≤s≤t0
|ut(·, s)|
Q3r/2
On the other hand, the estimates (4.5) and the energy estimates (this time, applied
to u itself) yield
Q3r/2
Q7r/4
.(4.9)
Combining (4.8) and (4.9) together, we have the estimates (4.7).
GAUSSIAN ESTIMATES 9
Next, assume that u is a weak solution of (2.4) in Q4r = Q4r(X0). Let ζ be a
smooth cut-off function such that ζ ≡ 1 in Qr, vanishes near ∂PQ2r, and satisfies
(4.10) 0 ≤ ζ ≤ 1 and |ζt|+ |Dζ|
≤ Cr−2.
Note that on each slice Q2r,s, we have
Q2r,s
ut −Dα(A
αβDβu)
· ζ2u
Q2r,s
ζ2ut · u+
Q2r,s
αβDβu, Dαu
αβDβu, Dαζu
Using the ellipticity condition and the Cauchy-Schwarz inequality, we find
Q2r,s
ζ2 |Du|
Q2r,s
ζ2 |ut| |u|+ C
Q2r,s
ζ |Du| |Dζ| |u|
Q2r,s
ζ2 |ut|
Q2r,s
ζ2 |u|
Q2r,s
Q2r,s
ζ2 |Du|
Then by (4.10), (4.7), and the energy estimates, for all s ∈ [t0 − r
2, t0], we have
Q2r,s
Q2r,s
Q2r,s
(4.11)
If we set ǫ = r2, then the above estimates (4.11) now become
from which the estimates (4.6) follows by a well known covering argument. �
Lemma 4.3. Assume that the elliptic system (2.2) satisfies the Hölder estimates
for weak solutions at every scale with constants µ0, H0. Let u be a weak solution of
the inhomogeneous elliptic system
(4.12) Dα(A
αβ(x)Dβu) = f in B2 = B2(x0),
where f belongs to the Morrey space M2,λ(B2) with λ ≥ 0. Then, for any γ ≥ 0
with γ < γ0 = min(λ + 4, n + 2µ0) (we may take γ = γ0 if γ0 < n) there exists
C = C(n, ν0,M0, µ0, H0, λ, γ) such that u satisfies the following local estimates
(4.13)
Br(x)
+ rγ−2 ‖f‖
M2,λ(B2)
uniformly for all x ∈ B1 = B1(x0) and 0 < r ≤ 1. Moreover, if γ < n, then u
belongs to the Morrey space M2,γ(B1) and
(4.14) ‖u‖M2,γ (B1) ≤ C
‖u‖L2(B2) + ‖Du‖L2(B2) + ‖f‖M2,λ(B2)
10 SEICK KIM
Proof. First, we note that the property (3.1) implies that for all 0 < ρ < r and
x ∈ Rn, we have
Bρ(x)
≤ C ·H0
)n−2+2µ0
Br(x)
In the light of the above observation, the estimates (4.13) is quite standard and is
found, for example, in [11, Chapter 3]. Then, by Poincaré inequality we have
(4.15)
Br(x)
∣u− uBr(x)
≤ Crγ
L2(B2)
+ ‖f‖
M2,λ(B2)
uniformly for all x ∈ B1 = B1(0) and 0 < r ≤ 1. It is well known that if γ < n,
then the estimates (4.15) yield (4.14) (see e.g. [11, Chapter 3]). �
4.2. Proof of Theorem 3.3. Let u be a weak solution of (2.4) in a cylinder
Q4 = Q4(0). We rewrite (2.4) as Lu = ut. By Lemma 4.2, we find that ut(·, s) is
in L2(Q2,s) and satisfies
‖ut(·, s)‖L2(Q2,s) ≤ C ‖u‖L2(Q4) for all − 4 ≤ s ≤ 0.
Therefore, we may apply Lemma 4.3 with f = ut and λ = 0, and then apply
Lemma 4.2 to find that for all x ∈ B1(0) and 0 < r ≤ 1, we have
Br(x)
|Du(·, s)|
≤ Crγ−2
‖Du(·, s)‖
L2(Q2,s)
+ ‖ut(·, s)‖
L2(Q2,s)
≤ Crγ−2 ‖u‖
L2(Q4)
uniformly in s ∈ [−4, 0]
(4.16)
for all γ < min(4, n+ 2µ0).
By Lemma 2.5 and then by (4.16) we find that for all X = (x, t) ∈ Q1 and r ≤ 1
Qr(X)
∣u− uQr(X)
≤ Cr2
Br(x)
|Du(y, s)|
dy ds
≤ Cr2+γ ‖u‖
L2(Q4)
(4.17)
Note that if n ≤ 3, then we may write γ = n + 2µ for some µ > 0. In that case,
(4.17) now reads
(4.18)
Qr(X)
∣u− uQr(X)
≤ Crn+2+2µ ‖u‖
L2(Q4)
for all X ∈ Q1 and r ≤ 1. Therefore, if n ≤ 3, then Lemma 2.6 yields the estimates
(4.19) [u]Cµ
(Q1/2)
≤ C ‖u‖L2(Q4) .
We have thus shown that in the case when n ≤ 3, any weak solution u of (2.4) in a
cylinder Q4 = Q4(0) satisfies the above a priori estimates (4.19) provided that the
associated elliptic system satisfies the Hölder estimates for weak solutions at every
scale. The general case is recovered as follows. For given X0 = (x0, t0) and r > 0,
let us consider the new system
(4.20) ut − L̃u := ut −
α,β=1
Dα(Ã
αβ(x)Dβu) = 0,
where Ãαβ(x) = Aαβ(x0 + rx). Note that the associated elliptic system L̃u = 0
also satisfies the Hölder estimates for weak solutions at every scale. Moreover, the
ellipticity constants ν0,M0 remain the same for the new coefficients Ã
αβ . Let u be
GAUSSIAN ESTIMATES 11
a weak solution of (2.4) in Q4r(X0). Then ũ(X) = ũ(x, t) := u(x0 + rx, t0 + r
is a weak solution of (4.20) in Q4(0) and thus ũ satisfies the estimates (4.19). By
rescaling back to Q4r(X0), the estimates (4.19) become
(4.21) [u]Cµ
(Qr/2)
≤ Cr−(n/2+1+µ) ‖u‖L2(Q4r) .
Thus, when n ≤ 3, the theorem now follows from a well known covering argument.
In the case when n ≥ 4, we invoke a bootstrap argument. For the sake of
simplicity, let us momentarily assume that 4 ≤ n ≤ 7. Let u be a weak solution
of (2.4) in Q8 = Q8(0). Let us fix X0 = (x0, t0) ∈ Q2(0) and observe that ut also
satisfies the system (2.4) in Q4(X0). Thus, by a similar argument that led to (4.16),
we find that for all x ∈ B1(x0) and 0 < r ≤ 1 we have
(4.22)
Br(x)
|Dut(·, s)|
≤ Crγ−2 ‖ut‖
L2(Q4(X0))
uniformly in s ∈ [t0 − 4, t0],
for all γ < 4 (we may take γ = 4 if n > 4). Then, by (4.14) in Lemma 4.3,
Lemma 4.1, and Lemma 4.2 we conclude that
(4.23) ‖ut(·, s)‖M2,γ (B1(x0)) ≤ C ‖u‖L2(Q8(0)) for all s ∈ [t0 − 4, t0].
Since the above estimates (4.23) hold for all X0 = (x0, t0) ∈ Q2(0), we find that, in
particular, ut(·, s) belongs to M
2,γ(B2(0)) for all −4 ≤ s ≤ 0, and satisfies
(4.24) ‖ut(·, s)‖M2,γ (B2(0)) ≤ C ‖u‖L2(Q8(0)) for all s ∈ [−4, 0],
where we also used (4.7) of Lemma 4.2.
The above estimates (4.24) for ut now allows us to invoke Lemma 4.3 with f = ut
and λ = γ. Then, by Lemma 4.3 and Lemma 4.2, we find that for all x ∈ B1(0)
and 0 < r ≤ 1, we have
Br(x)
|Du(·, s)|
≤ Crγ−2 ‖u‖
L2(Q8(0))
uniformly in s ∈ [−4, 0]
for all γ < min(γ+4, n+2µ0). Since we assume that n ≤ 7, we may write γ = n+2µ
for some µ > 0. By the exactly same argument we used in the case when n ≤ 3,
we derive the estimates
[u]Cµ
(Q1/2)
≤ C ‖u‖L2(Q8) ,
and the theorem follows as before.
Finally, if n ≥ 8, we repeat the above process; if u is a weak solution of (2.4) in
Q16(0), then ut(·, s) is in M
2,γ(B1(0)) for all γ < 8 and so on. The process cannot
go on indefinitely and it stops in k = [n/4] + 1 steps. The proof is complete. �
4.3. Proof of Theorem 3.4. The proof is based on Theorem 2.8, the proof of
which, in turn, is found in [12]. By Theorem 2.8, we only need to establish the
local boundedness property for weak solutions of the parabolic system (2.4) and for
those of its adjoint system (2.5).
From the hypothesis that the elliptic system (2.2) satisfies the Hölder estimates
for weak solutions at every scale, we find, by Theorem 3.3, that the parabolic
system (2.4) with the associated time-independent coefficients also satisfies the
Hölder estimates for weak solutions at every scale; that is, there exist some constants
µ > 0 and C, depending on the prescribed quantities, such that if u is a weak
solution of (2.4) in Q4r(X), then it satisfies the estimates
[u]Cµ
(Q2r) ≤ Cr
−(n/2+1+µ) ‖u‖L2(Q4r) .
12 SEICK KIM
Let us fix Y ∈ Qr = Qr(X). Then, for all Z ∈ Qr(Y ) ⊂ Q2r(X), we have
(4.25) |u(Y )| ≤ |u(Z)|+dP (Y, Z)
µ · [u]Cµ
(Q2r) ≤ |u(Z)|+Cr
−(n/2+1) ‖u‖L2(Q4r) .
By averaging (4.25) over Qr(Y ) with respect to Z, we derive (note |Qr| = Cr
|u(Y )| ≤ Cr−(n+2) ‖u‖L1(Qr(Y )) + Cr
−(n/2+1) ‖u‖L2(Q4r) .
Since Y ∈ Qr is arbitrary, we find, by Hölder’s inequality, that u satisfies
‖u‖L∞(Qr) ≤ Cr
−(n/2+1) ‖u‖L2(Q4r)
for some constant C = C(n, ν0,M0, µ0, H0).
To finish the proof, we also need to show that if u is a weak solution of the adjoint
system (2.5) in Q∗4r = Q
4r(X), then it satisfies the local boundedness property
(4.26) ‖u‖L∞(Q∗r)
≤ Cr−(n/2+1) ‖u‖L2(Q∗
The verification of (4.26) requires only a slight modification of the previous argu-
ments (mostly, one needs to replace Qr by Q
r and so on), but it is rather routine
and we skip the details. �
References
1. Aronson, D. G. Bounds for the fundamental solution of a parabolic equation. Bull. Amer.
Math. Soc. 73 (1967) 890–896.
2. Auscher, P. Regularity theorems and heat kernel for elliptic operators. J. London Math. Soc.
(2) 54 (1996), no. 2, 284–296.
3. Auscher, P.; McIntosh, A.; Tchamitchian, Ph. Heat kernels of second order complex elliptic
operators and applications. J. Funct. Anal. 152 (1998), no. 1, 22–73.
4. Auscher, P.; Qafsaoui, M. Equivalence between regularity theorems and heat kernel estimates
for higher order elliptic operators and systems under divergence form. J. Funct. Anal. 177
(2000), no. 2, 310–364.
5. Auscher, P.; Tchamitchian, Ph. Square root problem for divergence operators and related
topics. Astérisque No. 249 (1998)
6. Davies, E. B. Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math.
109 (1987), no. 2, 319–333.
7. De Giorgi, E. Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli
regolari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43.
8. De Giorgi, E. Un esempio di estremali discontinue per un problema variazionale di tipo
ellittico. (Italian) Boll. Un. Mat. Ital. (4) 1 (1968), 135–137.
9. Evans, L. C. Partial differential equations. American Mathematical Society, Providence, RI,
1998.
10. Fabes, E. B.; Stroock, D. W. A new proof of Moser’s parabolic Harnack inequality using the
old ideas of Nash. Arch. Rational Mech. Anal. 96 (1986), no. 4, 327–338.
11. Giaquinta, M. Multiple integrals in the calculus of variations and nonlinear elliptic systems.
Princeton University Press:Princeton, NJ, 1983.
12. Hofmann, S.; Kim, S. Gaussian estimates for fundamental solutions to certain parabolic
systems. Publ. Mat. 48 (2004), 481–496.
13. Kang, K.; Kim, S. On the Hölder continuity of solutions of a certain system related to
Maxwell’s equations. SIAM J. Math. Anal. 34 (2002), no. 1, 87–100 (electronic).
14. Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N. Linear and quasilinear equations of
parabolic type. American Mathematical Society: Providence, RI, 1967.
15. Lieberman, G. M. Second order parabolic differential equations. World Scientific Publishing
Co., Inc., River Edge, NJ, 1996
16. Morrey, C. B., Jr. Multiple integrals in the calculus of variations. Springer-Verlag New York,
Inc., New York, 1966
17. Moser, J. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math.
17 (1964) 101–134; Correction: Comm. Pure Appl. Math. 20 (1967) 231–236.
GAUSSIAN ESTIMATES 13
18. Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958)
931–954.
19. Struwe, M. On the Hölder continuity of bounded weak solutions of quasilinear parabolic sys-
tems. Manuscripta Math. 35 (1981), no. 1-2, 125–145.
Mathematics Department, University of Missouri, Columbia, Missouri 65211
E-mail address: seick@math.missouri.edu
	1. Introduction
	2. Notation and definitions
	2.1. Geometric notation
	2.2. Notation for functions and their derivatives
	2.3. Function spaces
	2.4. Elliptic and parabolic systems and their adjoints
	2.5. Weak solutions
	2.6. Fundamental solution
	2.7. Notation for estimates
	2.8. Some preliminary results and known facts
	3. Main results
	4. Proof of Main Results
	4.1. Some technical lemmas and proofs
	4.2. Proof of Theorem ??
	4.3. Proof of Theorem ??
	References

  Auscher, McIntosh and Tchamitchian studied the heat kernels of second order
elliptic operators in divergence form with complex bounded measurable
coefficients on $\mathbb{R}^n$. In particular, in the case when $n=2$ they
obtained Gaussian upper bound estimates for the heat kernel without imposing
further assumption on the coefficients. We study the fundamental solutions of
the systems of second order parabolic equations in the divergence form with
bounded, measurable, time-independent coefficients, and extend their results to
the systems of parabolic equations.

Introduction
In 1967, Aronson [1] proved Gaussian upper and lower bounds for the fundamen-
tal solutions of parabolic equations in divergence form with bounded measurable
coefficients. To establish the Gaussian lower bound Aronson made use of the Har-
nack inequality for nonnegative solutions which was proved by Moser in 1964 (see
[17]). Related to Moser’s parabolic Harnack inequality, we should mention Nash’s
earlier paper [18] where the Hölder continuity of weak solutions to parabolic equa-
tions in divergence form was established. In 1985, Fabes and Stroock [10] showed
that the idea of Nash could be used to establish a Gaussian upper and lower bound
on the fundamental solution. They showed that actually such Gaussian estimates
could be used to prove Moser’s Harnack inequality. We note that Aronson also ob-
tained Gaussian upper bound estimates of the fundamental solution without using
Moser’s Harnack inequality.
In [2], Auscher proposed a new proof of Aronson’s Gaussian upper bound esti-
mates for the fundamental solution of second order parabolic equations with time-
independent coefficients. His method relies crucially on the assumption that the
coefficients are time-independent and thus it does not exactly reproduce Aronson’s
result, which is valid even for the time-dependent coefficients case. However, his
method is interesting in the sense that it carries over to equations with complex
coefficients provided that the complex coefficients are a small perturbation of real
coefficients. Along with this direction, Auscher, McIntosh and Tchamitchian also
showed that the heat kernel of second order elliptic operators in divergence form
with complex bounded measurable coefficients in the two dimensional space has a
Gaussian upper bound (see [3] and also [5]).
We would like to point out that a parabolic equation with complex coefficients
is, in fact, a special case of a system of parabolic equations. From this point of
2000 Mathematics Subject Classification. Primary 35A08, 35B45; Secondary 35K40.
Key words and phrases. Gaussian estimates, a priori estimates, parabolic system.
http://arxiv.org/abs/0704.1372v1
2 SEICK KIM
view, Hofmann and the author showed that the fundamental solution of a parabolic
system has an upper Gaussian bound if the system is a small perturbation of a di-
agonal system, which, in particular, generalized the result of Auscher mentioned
above to the time-dependent coefficients case (see [12]). However, the above men-
tioned result of Auscher, McIntosh and Tchamitchian regarding the heat kernel of
two dimensional elliptic operators with complex coefficients does not follow directly
from our result.
One of the main goals of this article is to provide a proof that weak solutions of
the parabolic system of divergence type with time-independent coefficients associ-
ated to an elliptic system in two dimensions enjoy the parabolic local boundedness
property and to show that its fundamental solution has a Gaussian upper bound.
More generally, we show that if weak solutions of an elliptic system satisfy Hölder
estimates at every scale, then weak solutions of the corresponding parabolic system
with time-independent coefficients also satisfies similar parabolic Hölder estimates
from which, in particular, the parabolic local boundedness property follows easily.
Also, such an argument allows one to derive Hölder continuity estimates for weak
solutions of parabolic equations with time-independent coefficients directly from
De Giorgi’s theorem [7] on elliptic equations, bypassing Moser’s parabolic Harnack
inequality. In fact, this is what Auscher really proved in the setting of complex
coefficients equations by using a functional calculus method (see [2] and also [4],
[5]). Even in those complex coefficients settings, we believe that our approach is
much more straightforward and thus appeals to wider readership.
Finally, we would like to point out that in this article, we are mainly interested
in global estimates and that we do not attempt to treat, for example, the systems
with lower order terms, etc. However, let us also mention that, with some extra
technical details, our methods carry over to those cases as well as to the systems
of higher order; see e.g. [4], [5] for the details, and also Remark 3.5.
The remaining sections are organized in the following way. In Section 2 we give
notations, definitions, and some known facts. We state the main results in Section 3
and give the proofs in Section 4.
2. Notation and definitions
2.1. Geometric notation.
(1) Rn = n-dimensional real Euclidean space.
(2) x = (x1, · · · , xn) is an arbitrary point of R
(3) X = (x, t) denotes an arbitrary point in Rn+1, where x ∈ Rn and t ∈ R.
(4) Br(x) = {y ∈ R
n : |y − x| < r} is an open ball in Rn with center x and
radius r > 0. We sometimes drop the reference point x and write Br for
Br(x) if there is no danger of confusion.
(5) Qr(X) =
(y, s) ∈ Rn+1 : |y − x| < r and t− r2 < s < t
. We sometimes
drop the reference point X and write Qr for Qr(X).
(6) Q∗r(X) =
(y, s) ∈ Rn+1 : |y − x| < r and t < s < t+ r2
(7) Qr,s(X) = {(y, s) ∈ Qr(X)}; i.e., Qr,s(X) = Br(x) × {s} if s ∈ (t − r
2, t)
and Qr,s(X) = ∅ otherwise. We sometimes drop the reference point X and
write Qr,s for Qr,s(X).
(8) For a cylinder Q = Ω× (a, b) ⊂ Rn+1, ∂PQ denotes its parabolic boundary,
namely, ∂PQ = ∂Ω × (a, b) ∪ Ω × {a}, where ∂Ω is the usual topological
boundary of Ω ⊂ Rn and Ω is its closure.
GAUSSIAN ESTIMATES 3
2.2. Notation for functions and their derivatives.
(1) For a mapping from Ω ⊂ Rn to RN , we write f(x) = (f1(x), . . . , fN (x))T
as a column vector.
(2) fQ =
f , where |Q| denotes the volume of Q.
(3) ut = ∂u/∂t.
(4) Dxiu = Diu = uxi = ∂u/∂xi.
(5) Du = (ux1 , . . . , uxn)
T is the spatial gradient of u = u(x, t).
(6) For f = (f1, . . . , fN )T , Df = (Df1, . . . , DfN); that is Df is the n × N
matrix whose i-th column is Df i.
2.3. Function spaces.
(1) For Ω ⊂ Rn and p ≥ 1, Lp(Ω) denotes the space of functions with the
following norms:
‖u‖Lp(Ω) =
|u(x)|
and ‖u‖L∞(Ω) = ess supΩ |u| .
(2) Cµ(Ω) denotes the space of functions that are Hölder continuous with the
exponent µ ∈ (0, 1], and
[u]Cµ(Ω) = sup
x 6=x′∈Ω
|u(x)− u(x′)|
|x− x′|
µ < ∞.
(3) The Morrey space M2,µ(Ω) is the set of all functions u ∈ L2(Ω) such that
‖u‖M2,µ(Ω) = sup
Bρ(x)⊂Ω
Bρ(x)
(4) C
P (Q) denotes the space of functions defined on Q ⊂ R
n+1 such that
[u]Cµ
(Q) = sup
X 6=X′∈Q
|u(X)− u(X ′)|
dP (X,X ′)µ
where dP (X,X
′) = max
|x− x′| ,
|t− t′|
2.4. Elliptic and parabolic systems and their adjoints.
Definition 2.1. We say that the coefficients A
ij (x) satisfy the uniform ellipticity
condition if there exist numbers ν0,M0 > 0 such that for all x ∈ R
n we have
(2.1)
αβ(x)ξβ , ξα
≥ ν0 |ξ|
αβ(x)ξβ ,ηα
≤ M0 |ξ| |η| ,
where we used the following notation.
(1) For α, β = 1, . . . , n, Aαβ(x) are N ×N matrices with (i, j)-entries A
ij (x).
(2) ξα = (ξ
α, · · · , ξ
T and |ξ|
αβ(x)ξβ ,ηα
α,β=1
i,j=1
ij (x)ξ
We emphasize that we do not assume that the coefficients are symmetric.
4 SEICK KIM
Definition 2.2. We say that a system of N equations on Rn
α,β=1
Dxα(A
ij (x)Dxβu
j) = 0 (i = 1, . . . , N)
is elliptic if the coefficients satisfy the uniform ellipticity condition. We often write
the above system in a vector form
(2.2) Lu :=
α,β=1
αβ(x)Dβu) = 0, u = (u
1 . . . , uN )T .
The adjoint system of (2.2) is given by
(2.3) L∗u :=
α,β=1
(Aαβ)∗(x)Dβu
where (Aαβ)∗ = (Aβα)T , the transpose of Aβα.
Definition 2.3. We say that a system of N equations on Rn+1
uit −
α,β=1
Dxα(A
ij (x)Dxβu
j) = 0 (i = 1, . . . , N)
is parabolic if the (time-independent) coefficients satisfy the uniform ellipticity
condition. We often write the above system in a vector form
(2.4) ut − Lu := ut −
α,β=1
αβ(x)Dβu) = 0.
The adjoint system of (2.4) is given by
(2.5) ut + L
∗u := ut +
α,β=1
(Aαβ)∗(x)Dβu
where (Aαβ)∗ = (Aβα)T , the transpose of Aβα.
2.5. Weak solutions. In this article, the term “weak solution” is used in a rather
abusive way. To avoid unnecessary technicalities, we may assume that all the coef-
ficients involved are smooth so that all weak solutions are indeed classical solutions.
However, this extra smoothness assumption will not be used quantitatively in our
estimates. This is why we shall make clear the dependence of constants.
(1) We say that u is a weak solution of (2.2) in Ω ⊂ Rn if u is a (classical)
solution of (2.2) in Ω and u, Du ∈ L2(Ω).
(2) We say that u is a weak solution of (2.4) in a cylinder Q = Ω×(a, b) ⊂ Rn+1
if u is a (classical) solution of (2.2) in Q and u, Du ∈ L2(Q), u(·, t) ∈ L2(Ω)
for all a ≤ t ≤ b, and supa≤t≤b ‖u(·, t)‖L2(Ω) < ∞.
2.6. Fundamental solution. By a fundamental solution (or fundamental matrix)
Γ(x, t; y) of the parabolic system (2.4) we mean anN×N matrix of functions defined
for t > 0 which, as a function of (x, t), is a solution of (2.4) (i.e., each column is a
solution of (2.4)), and is such that
Γ(x, t; y)f(y) dy = f(x)(2.6)
GAUSSIAN ESTIMATES 5
for any bounded continuous function f = (f1, . . . , fN)T , where Γ(x, t; y)f (y) de-
notes the usual matrix multiplication.
2.7. Notation for estimates. We employ the letter C to denote a universal con-
stant usually depending on the dimension and ellipticity constants. It should be
understood that C may vary from line to line. We sometimes write C = C(α, β, . . .)
to emphasize the dependence on the prescribed quantities α, β, . . ..
2.8. Some preliminary results and known facts.
Lemma 2.4 (Energy estimates). Let u be a weak solution of (2.4) in QR = QR(X).
Then for 0 < r < R, we have
t−r2≤s≤t
|u(·, s)|
(R− r)2
Proof. See e.g., [14, Lemma 2.1, p. 139]. �
Lemma 2.5 (Parabolic Poincaré inequality). Let u be a weak solution of (2.4) in
QR = QR(X). Then there is some constant C = C(n,M0) such that
|u− uQR |
≤ CR2
Proof. See e.g., [19, Lemma 3]. �
Lemma 2.6. Let Q2R = Q2R(X0) be a cylinder in R
n+1. Suppose u ∈ L2(Q2R)
and there are positive constants µ ≤ 1 and M such that for any X ∈ QR and any
r ∈ (0, R) we have
Qr(X)
∣u− uQr(X)
≤ M2rn+2+2µ.
Then u is Hölder continuous in QR with the exponent µ and [u]Cµ
(QR) ≤ C(n, µ)M .
Proof. See e.g., [15, Lemma 4.3, p. 50]. �
Definition 2.7 (Local boundedness property). We say that the system (2.4) sat-
isfies the local boundedness property for weak solutions if there is a constant M
such that all weak solutions u of (2.4) in Q2r(X) satisfy the estimates
Qr(X)
|u| ≤ M
|Q2r|
Q2r(X)
Similarly, we say that the adjoint system (2.5) satisfies the local boundedness prop-
erty if the corresponding estimates hold for weak solutions u of (2.5) in Q∗2r(X).
Theorem 2.8 (Theorem 1.1, [12]). Assume that the system (2.4) and its adjoint
system (2.5) satisfy the local boundedness property for weak solutions. Then the
fundamental solution of the system (2.4) has an upper bound
(2.7) |Γ(x, t; y)|op ≤ C0t
−n/2 exp
k0 |x− y|
where |Γ(x, t; y)|op denotes the operator norm of the fundamental matrix Γ(x, t; y).
Here, C0 = C0(n, ν0,M0,M) and k0 = k0(ν0,M0).
6 SEICK KIM
3. Main results
Definition 3.1. We say that an elliptic system (2.2) satisfies the Hölder estimates
for weak solutions at every scale if there exist constants µ0 > 0 and H0 such that
all weak solutions u of the system in B2r = B2r(x0) satisfy the following estimates
(3.1) [u]Cµ0 (Br) ≤ H0r
−(n/2+µ0) ‖u‖L2(B2r) .
Similarly, we say that a parabolic system (2.4) satisfies Hölder estimates for weak
solutions at every scale if there exist constants µ1 > 0 and H1 such that all weak
solutions u of the system in Q2r = Q2r(X0) satisfy the following estimates
(3.2) [u]Cµ1
≤ H1r
−(n/2+1+µ1) ‖u‖L2(Q2r) .
Remark 3.2. Elliptic systems with constant coefficients satisfy the above property,
and in that case, the ellipticity condition (2.1) can be weakened and replaced by the
Legendre-Hadamard condition. De Giorgi’s theorem [7] states that the property is
satisfied if N = 1. The property is also satisfied if n = 2 and it is due to Morrey
(see Corollary 3.6). Some other examples include, for instance, a certain three
dimensional elliptic system which was studied by Kang and the author in [13].
We shall prove the following main results in this paper:
Theorem 3.3. If an elliptic system (2.2) satisfies the Hölder estimates for weak
solutions at every scale, then the corresponding parabolic system (2.4) with time-
independent coefficients also satisfies the Hölder estimates for weak solutions at
every scale.
Theorem 3.4. Suppose that the elliptic system (2.2) and its adjoint system (2.3)
defined on Rn both satisfy the Hölder estimates for weak solutions at every scale
with constants µ0, H0. Let Γ(x, t; y) be the fundamental solution of the parabolic
system (2.4) with the time-independent coefficients associated to the elliptic system
(2.2). Then Γ(x, t; y) has an upper bound
(3.3) |Γ(x, t; y)|op ≤ C0t
−n/2 exp
k0 |x− y|
where C0 = C0(n, ν0,M0, µ0, H0) and k0 = k0(ν0,M0). Here, |Γ(x, t; y)|op denotes
the operator norm of fundamental matrix Γ(x, t; y).
Remark 3.5. We would like to point out that (3.3) is a global estimate. Especially,
the bound (3.3) holds for all time t > 0. Suppose that the elliptic system (2.2) and
its adjoint system (2.3) enjoy the Hölder estimates for weak solutions up to a fixed
scale R0; that is, there is a number R0 > 0 such that if u is a weak solution of
either (2.2) or (2.3) in Br = Br(x) with 0 < r ≤ R0, then u is Hölder continuous
and satisfies
[u]Cµ0 (Br) ≤ H0r
−(n/2+µ0) ‖u‖L2(B2r) .
Then, the statement regarding the bound (3.3) for the fundamental solution should
be localized as follows: For any given T > 0, there are constants k0 = k0(ν0,M0)
and C0 = C0(n, ν0,M0, µ0, H0, R0, T ) such that (3.3) holds for 0 < t ≤ T .
Corollary 3.6. Let Γ(x, t; y) be the fundamental solution of the parabolic system
(2.4) with time-independent coefficients associated to an elliptic system (2.2) defined
on R2. Then Γ(x, t; y) has an upper bound (3.3) with the constants C0, k0 depending
only on the ellipticity constants ν0,M .
GAUSSIAN ESTIMATES 7
Proof. First, let us recall the well known theorem of Morrey which states that any
two dimensional elliptic system (2.2) with bounded measurable coefficients satisfies
the Hölder estimates for weak solutions at every scale, with the constants µ0, H0
depending only on the ellipticity constants (see, [16, pp. 143–148]). Next, note that
the ellipticity constants ν0,M0 in (2.1) remain unchanged for Ã
ij (x) = A
ji (x).
Therefore, the corollary is an immediate consequence of Theorem 3.4. �
Remark 3.7. In fact, the converse of Theorem 2.8 is also true (see [12, Theorem 1.2]).
Therefore, in order to extend the above corollary to the parabolic system with
time-dependent coefficients, one needs to show that the system satisfies the local
boundedness property for weak solutions. Unfortunately, we do not know whether
it is true or not if the coefficients are allowed to depend on the time variable. If
n ≥ 3, it is not true in general, even for the time-independent coefficients case since
there is a famous counter-example due to De Giorgi (see [8]).
4. Proof of Main Results
4.1. Some technical lemmas and proofs.
Lemma 4.1. If u is a weak solution of the parabolic system with time-independent
coefficients (2.4) in QR = QR(X0), then ut ∈ L
2(Qr) for r < R and satisfies the
estimates
(4.1) ‖ut‖L2(Qr) ≤ C(R − r)
−1 ‖Du‖L2(QR) .
In particular, if u is a weak solution of (2.4) in Q2r, then the above estimates
together with the energy estimates yield
(4.2) ‖ut‖L2(Qr) ≤ Cr
−2 ‖u‖L2(Q2r) .
Proof. We first note that if the coefficients are symmetric, (i.e., A
ij = A
ji ) this
is a well known result; a proof for such a case is found, for example, in [14, pp.
172–181] or in [9, pp. 360–364]. However, the standard proof does not carry over to
the non-symmetric coefficients case and for that reason, we provide a self-contained
proof here.
Fix positive numbers σ, τ such that σ < τ ≤ R. Let ζ be a smooth cut-off
function such that ζ ≡ 1 in Qσ, vanishes near ∂PQτ , and satisfies
0 ≤ ζ ≤ 1 and |ζt|+ |Dζ|
≤ C(τ − σ)−2.
Note that on each slice Qτ,s, we have
ut −Dα(A
αβDβu)
· ζ2ut
ζ2 |ut|
αβDβu, Dαut
αβDβu, Dαζut
8 SEICK KIM
Therefore, we find by using the Cauchy-Schwarz inequality that
ζ2 |ut|
ζ2 |Du| |Dut|+ C
ζ |Du| |Dζ| |ut|
ζ2 |Dut|
ζ2 |Du|
ζ2 |ut|
Thus we have
(4.3)
ζ2 |ut|
ζ2 |Dut|
ζ2 |Du|
Since ut also satisfies (2.4), the energy estimates yield
(4.4)
ζ2 |Dut|
(τ − σ)2
This is the part where we exploit the assumption that the coefficients are time-
independent. Combining (4.3) and (4.4), we have
(τ − σ)2
(τ − σ)2
If we set ǫ = (τ − σ)2/2C0, we finally obtain
(τ − σ)2
Here, we emphasize that C is a constant independent of σ, τ . Then by a standard
iteration argument (see e.g. [11, Lemma 3.1, pp. 161]), we have
(4.5)
(R− r)2
for 0 < r < R.
The proof is complete. �
Lemma 4.2. If u is a weak solution of the parabolic system with time-independent
coefficients (2.4) in Q2r = Q2r(X0), then Du(·, s),ut(·, s) ∈ L
2(Qr,s) for all s ∈
[t0 − r
2, t0], and satisfy the following estimates uniformly in s ∈ [t0 − r
2, t0].
‖Du(·, s)‖L2(Qr,s) ≤ Cr
−2 ‖u‖L2(Q2r) ,(4.6)
‖ut(·, s)‖L2(Qr,s) ≤ Cr
−3 ‖u‖L2(Q2r) .(4.7)
Proof. By the energy estimates applied to ut we obtain
(4.8) sup
t0−r2≤s≤t0
|ut(·, s)|
Q3r/2
On the other hand, the estimates (4.5) and the energy estimates (this time, applied
to u itself) yield
Q3r/2
Q7r/4
.(4.9)
Combining (4.8) and (4.9) together, we have the estimates (4.7).
GAUSSIAN ESTIMATES 9
Next, assume that u is a weak solution of (2.4) in Q4r = Q4r(X0). Let ζ be a
smooth cut-off function such that ζ ≡ 1 in Qr, vanishes near ∂PQ2r, and satisfies
(4.10) 0 ≤ ζ ≤ 1 and |ζt|+ |Dζ|
≤ Cr−2.
Note that on each slice Q2r,s, we have
Q2r,s
ut −Dα(A
αβDβu)
· ζ2u
Q2r,s
ζ2ut · u+
Q2r,s
αβDβu, Dαu
αβDβu, Dαζu
Using the ellipticity condition and the Cauchy-Schwarz inequality, we find
Q2r,s
ζ2 |Du|
Q2r,s
ζ2 |ut| |u|+ C
Q2r,s
ζ |Du| |Dζ| |u|
Q2r,s
ζ2 |ut|
Q2r,s
ζ2 |u|
Q2r,s
Q2r,s
ζ2 |Du|
Then by (4.10), (4.7), and the energy estimates, for all s ∈ [t0 − r
2, t0], we have
Q2r,s
Q2r,s
Q2r,s
(4.11)
If we set ǫ = r2, then the above estimates (4.11) now become
from which the estimates (4.6) follows by a well known covering argument. �
Lemma 4.3. Assume that the elliptic system (2.2) satisfies the Hölder estimates
for weak solutions at every scale with constants µ0, H0. Let u be a weak solution of
the inhomogeneous elliptic system
(4.12) Dα(A
αβ(x)Dβu) = f in B2 = B2(x0),
where f belongs to the Morrey space M2,λ(B2) with λ ≥ 0. Then, for any γ ≥ 0
with γ < γ0 = min(λ + 4, n + 2µ0) (we may take γ = γ0 if γ0 < n) there exists
C = C(n, ν0,M0, µ0, H0, λ, γ) such that u satisfies the following local estimates
(4.13)
Br(x)
+ rγ−2 ‖f‖
M2,λ(B2)
uniformly for all x ∈ B1 = B1(x0) and 0 < r ≤ 1. Moreover, if γ < n, then u
belongs to the Morrey space M2,γ(B1) and
(4.14) ‖u‖M2,γ (B1) ≤ C
‖u‖L2(B2) + ‖Du‖L2(B2) + ‖f‖M2,λ(B2)
10 SEICK KIM
Proof. First, we note that the property (3.1) implies that for all 0 < ρ < r and
x ∈ Rn, we have
Bρ(x)
≤ C ·H0
)n−2+2µ0
Br(x)
In the light of the above observation, the estimates (4.13) is quite standard and is
found, for example, in [11, Chapter 3]. Then, by Poincaré inequality we have
(4.15)
Br(x)
∣u− uBr(x)
≤ Crγ
L2(B2)
+ ‖f‖
M2,λ(B2)
uniformly for all x ∈ B1 = B1(0) and 0 < r ≤ 1. It is well known that if γ < n,
then the estimates (4.15) yield (4.14) (see e.g. [11, Chapter 3]). �
4.2. Proof of Theorem 3.3. Let u be a weak solution of (2.4) in a cylinder
Q4 = Q4(0). We rewrite (2.4) as Lu = ut. By Lemma 4.2, we find that ut(·, s) is
in L2(Q2,s) and satisfies
‖ut(·, s)‖L2(Q2,s) ≤ C ‖u‖L2(Q4) for all − 4 ≤ s ≤ 0.
Therefore, we may apply Lemma 4.3 with f = ut and λ = 0, and then apply
Lemma 4.2 to find that for all x ∈ B1(0) and 0 < r ≤ 1, we have
Br(x)
|Du(·, s)|
≤ Crγ−2
‖Du(·, s)‖
L2(Q2,s)
+ ‖ut(·, s)‖
L2(Q2,s)
≤ Crγ−2 ‖u‖
L2(Q4)
uniformly in s ∈ [−4, 0]
(4.16)
for all γ < min(4, n+ 2µ0).
By Lemma 2.5 and then by (4.16) we find that for all X = (x, t) ∈ Q1 and r ≤ 1
Qr(X)
∣u− uQr(X)
≤ Cr2
Br(x)
|Du(y, s)|
dy ds
≤ Cr2+γ ‖u‖
L2(Q4)
(4.17)
Note that if n ≤ 3, then we may write γ = n + 2µ for some µ > 0. In that case,
(4.17) now reads
(4.18)
Qr(X)
∣u− uQr(X)
≤ Crn+2+2µ ‖u‖
L2(Q4)
for all X ∈ Q1 and r ≤ 1. Therefore, if n ≤ 3, then Lemma 2.6 yields the estimates
(4.19) [u]Cµ
(Q1/2)
≤ C ‖u‖L2(Q4) .
We have thus shown that in the case when n ≤ 3, any weak solution u of (2.4) in a
cylinder Q4 = Q4(0) satisfies the above a priori estimates (4.19) provided that the
associated elliptic system satisfies the Hölder estimates for weak solutions at every
scale. The general case is recovered as follows. For given X0 = (x0, t0) and r > 0,
let us consider the new system
(4.20) ut − L̃u := ut −
α,β=1
Dα(Ã
αβ(x)Dβu) = 0,
where Ãαβ(x) = Aαβ(x0 + rx). Note that the associated elliptic system L̃u = 0
also satisfies the Hölder estimates for weak solutions at every scale. Moreover, the
ellipticity constants ν0,M0 remain the same for the new coefficients Ã
αβ . Let u be
GAUSSIAN ESTIMATES 11
a weak solution of (2.4) in Q4r(X0). Then ũ(X) = ũ(x, t) := u(x0 + rx, t0 + r
is a weak solution of (4.20) in Q4(0) and thus ũ satisfies the estimates (4.19). By
rescaling back to Q4r(X0), the estimates (4.19) become
(4.21) [u]Cµ
(Qr/2)
≤ Cr−(n/2+1+µ) ‖u‖L2(Q4r) .
Thus, when n ≤ 3, the theorem now follows from a well known covering argument.
In the case when n ≥ 4, we invoke a bootstrap argument. For the sake of
simplicity, let us momentarily assume that 4 ≤ n ≤ 7. Let u be a weak solution
of (2.4) in Q8 = Q8(0). Let us fix X0 = (x0, t0) ∈ Q2(0) and observe that ut also
satisfies the system (2.4) in Q4(X0). Thus, by a similar argument that led to (4.16),
we find that for all x ∈ B1(x0) and 0 < r ≤ 1 we have
(4.22)
Br(x)
|Dut(·, s)|
≤ Crγ−2 ‖ut‖
L2(Q4(X0))
uniformly in s ∈ [t0 − 4, t0],
for all γ < 4 (we may take γ = 4 if n > 4). Then, by (4.14) in Lemma 4.3,
Lemma 4.1, and Lemma 4.2 we conclude that
(4.23) ‖ut(·, s)‖M2,γ (B1(x0)) ≤ C ‖u‖L2(Q8(0)) for all s ∈ [t0 − 4, t0].
Since the above estimates (4.23) hold for all X0 = (x0, t0) ∈ Q2(0), we find that, in
particular, ut(·, s) belongs to M
2,γ(B2(0)) for all −4 ≤ s ≤ 0, and satisfies
(4.24) ‖ut(·, s)‖M2,γ (B2(0)) ≤ C ‖u‖L2(Q8(0)) for all s ∈ [−4, 0],
where we also used (4.7) of Lemma 4.2.
The above estimates (4.24) for ut now allows us to invoke Lemma 4.3 with f = ut
and λ = γ. Then, by Lemma 4.3 and Lemma 4.2, we find that for all x ∈ B1(0)
and 0 < r ≤ 1, we have
Br(x)
|Du(·, s)|
≤ Crγ−2 ‖u‖
L2(Q8(0))
uniformly in s ∈ [−4, 0]
for all γ < min(γ+4, n+2µ0). Since we assume that n ≤ 7, we may write γ = n+2µ
for some µ > 0. By the exactly same argument we used in the case when n ≤ 3,
we derive the estimates
[u]Cµ
(Q1/2)
≤ C ‖u‖L2(Q8) ,
and the theorem follows as before.
Finally, if n ≥ 8, we repeat the above process; if u is a weak solution of (2.4) in
Q16(0), then ut(·, s) is in M
2,γ(B1(0)) for all γ < 8 and so on. The process cannot
go on indefinitely and it stops in k = [n/4] + 1 steps. The proof is complete. �
4.3. Proof of Theorem 3.4. The proof is based on Theorem 2.8, the proof of
which, in turn, is found in [12]. By Theorem 2.8, we only need to establish the
local boundedness property for weak solutions of the parabolic system (2.4) and for
those of its adjoint system (2.5).
From the hypothesis that the elliptic system (2.2) satisfies the Hölder estimates
for weak solutions at every scale, we find, by Theorem 3.3, that the parabolic
system (2.4) with the associated time-independent coefficients also satisfies the
Hölder estimates for weak solutions at every scale; that is, there exist some constants
µ > 0 and C, depending on the prescribed quantities, such that if u is a weak
solution of (2.4) in Q4r(X), then it satisfies the estimates
[u]Cµ
(Q2r) ≤ Cr
−(n/2+1+µ) ‖u‖L2(Q4r) .
12 SEICK KIM
Let us fix Y ∈ Qr = Qr(X). Then, for all Z ∈ Qr(Y ) ⊂ Q2r(X), we have
(4.25) |u(Y )| ≤ |u(Z)|+dP (Y, Z)
µ · [u]Cµ
(Q2r) ≤ |u(Z)|+Cr
−(n/2+1) ‖u‖L2(Q4r) .
By averaging (4.25) over Qr(Y ) with respect to Z, we derive (note |Qr| = Cr
|u(Y )| ≤ Cr−(n+2) ‖u‖L1(Qr(Y )) + Cr
−(n/2+1) ‖u‖L2(Q4r) .
Since Y ∈ Qr is arbitrary, we find, by Hölder’s inequality, that u satisfies
‖u‖L∞(Qr) ≤ Cr
−(n/2+1) ‖u‖L2(Q4r)
for some constant C = C(n, ν0,M0, µ0, H0).
To finish the proof, we also need to show that if u is a weak solution of the adjoint
system (2.5) in Q∗4r = Q
4r(X), then it satisfies the local boundedness property
(4.26) ‖u‖L∞(Q∗r)
≤ Cr−(n/2+1) ‖u‖L2(Q∗
The verification of (4.26) requires only a slight modification of the previous argu-
ments (mostly, one needs to replace Qr by Q
r and so on), but it is rather routine
and we skip the details. �
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GAUSSIAN ESTIMATES 13
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Mathematics Department, University of Missouri, Columbia, Missouri 65211
E-mail address: seick@math.missouri.edu
	1. Introduction
	2. Notation and definitions
	2.1. Geometric notation
	2.2. Notation for functions and their derivatives
	2.3. Function spaces
	2.4. Elliptic and parabolic systems and their adjoints
	2.5. Weak solutions
	2.6. Fundamental solution
	2.7. Notation for estimates
	2.8. Some preliminary results and known facts
	3. Main results
	4. Proof of Main Results
	4.1. Some technical lemmas and proofs
	4.2. Proof of Theorem ??
	4.3. Proof of Theorem ??
	References
