Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 93, pp. 1–24.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
A GRADIENT ESTIMATE FOR SOLUTIONS TO PARABOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
JISHAN FAN, KYOUNGSUN KIM, SEI NAGAYASU, GEN NAKAMURA
Abstract. Li-Vogelius and Li-Nirenberg gave a gradient estimate for solu- tions of strongly elliptic equations and systems of divergence forms with piece- wise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be given by manifolds of codimension 1, which we called them manifolds of discontinuities. Their gradient estimate is independent of the distances between manifolds of discontinuities. In this paper, we gave a para- bolic version of their results. That is, we gave a gradient estimate for parabolic equations of divergence forms with piecewise smooth coefficients. The coef- ficients are assumed to be independent of time and their discontinuities are likewise the previous elliptic equations. As an application of this estimate, we also gave a pointwise gradient estimate for the fundamental solution of a para- bolic operator with piecewise smooth coefficients. Both gradient estimates are independent of the distances between manifolds of discontinuities.
1. Introduction
For strongly elliptic, second order scalar equations with real coefficients, it is well known that their solutions have the H¨older continuity even in the case that the coefficients are only bounded measurable functions. However, the solutions do not have the Lipschitz continuity in general. For example, Piccinini-Spagnolo [17, p. 396, Example 1] and Meyers [14, p. 204] gave the following case:
Example 1.1 ([14, 17]). LetB1:={x∈Rn :|x|<1} and eachaij ∈L∞(B1) be defined as
a11=M x21+x22
|x|2 , a22= x21+M x22
|x|2 , a12=a21= (M−1)x1x2
|x|2
with a constantM >1. Then, if we defineuas u(x) =
(|x|1/
√M x1
|x| ifx6= 0,
0 ifx= 0, (1.1)
2000Mathematics Subject Classification. 35K10, 35B65.
Key words and phrases. Parabolic equations; discontinuous coefficients; gradient estimate.
c
2013 Texas State University - San Marcos.
Submitted November 11, 2012. Published April 11, 2013.
1
it is easy to see that the H¨older exponent ofuis at least less than or equal to 1/√ M (indeed, forx= (x1,0) we have|u(x)−u(0)|=|x|1/
√
M. Hence we have
|u(x)−u(0)|
|x|(1/√M)+ε =|x|−ε→+∞ asx→0
for anyε >0.) andusatisfies the strongly elliptic scalar equation with real coeffi- cients
2
X
i,j=1
∂
∂xi
aij
∂u
∂xj
= 0. (1.2)
The same thing can be said also to the parabolic equation
∂u
∂t −
2
X
i,j=1
∂
∂xi
aij
∂u
∂xj
= 0, (1.3)
becauseugiven by (1.1) satisfies this equation.
This example shows that we cannot expect gradient estimates of solutions to equations (1.2) and (1.3) in the caseaij ∈L∞(B1), but we may have the estimates in the case of piecewiseCµ (see (1.5) below) coefficients.
The fact that the gradient estimate of solutions is independent of the distances between manifolds of discontinuities was first observed by Babuˇska-Andersson- Smith-Levin [2] numerically for certain homogeneous isotropic linear systems of elasticity, that is |∇u| is bounded independently of the distances between mani- folds of discontinuities. They considered that this numerical property of solutions is mathematically true. This is the so-called Babuˇska’s conjecture. Recently, proofs for this conjecture appeared in [13] and [12]. In elasticity, a small static deformation of an elastic medium with inclusions can be described by an elliptic system of di- vergence form with piecewise smooth coefficients. The discontinuities of coefficients form the boundaries of inclusions. Similar physical interpretation is also possible for heat conductors. Our main theorem 1.5 given below ensures that this property also holds for parabolic equations of the form (1.3). The details of result given in [13] and [12] for scalar equations will be given below as Theorem 1.2.
To state our main theorem, we begin with introducing several notations which will be used throughout this paper. Let D ⊂ Rn be a bounded domain with a C1,α boundary for some 0< α <1, which means that the domain D contains L disjoint subdomainsD1, . . . , DL withC1,α boundaries, i.e. D= (SL
m=1Dm)\∂D, and we also assume that Dm⊂D for 1≤m≤L−1. Physically, D is a material and Dm(1 ≤ m≤ L−1) are considered as inclusions in D. We define the C1,α norm (resp. C1,α seminorm) ofC1,α domainDm in the same way as in [12], that is, as the largest positive numberasuch that in thea-neighborhood of every point of∂Dm, identified as 0 after a possible translation and rotation of the coordinates so that xn = 0 is the tangent to∂Dm at 0, ∂Dm is given by the graph of aC1,α functionψm, defined in |x0|<2a (x0 = (x1, . . . , xn−1)), the 2a-neighborhood of 0 in the tangent plane, and it satisfies the estimate kψmkC1,α(|x0|<2a) ≤1/a (resp.
[ψm]C1,α(|x0|<2a)≤1/a), where
[ψ]C1,α(|x0|<2a):= sup
|x0|,|ξ0|<2a
|∇0ψ(x0)− ∇0ψ(ξ0)|
|x0−ξ0|α , kψkC1,α(|x0|<2a):=kψkC1(|x0|<2a)+ [ψ]C1,α(|x0|<2a).
Further, let (aij) be a symmetric, positive definite matrix-valued function defined onD satisfying
λ|ξ|2≤
n
X
i,j=1
aij(x)ξiξj≤Λ|ξ|2. (1.4) Here eachaij is piecewiseCµ inD, 0< µ <1; that is,
aij(x) =a(m)ij (x) forx∈Dm, 1≤m≤L (1.5) witha(m)ij ∈Cµ(Dm).
As we have already mentioned above, we will discuss in this paper a gradient estimate for solutions to parabolic equations with piecewise smooth coefficients.
Our result is a parabolic version for the results of Li-Vogelius [13] and the scalar equations version of Li-Nirenberg [12]. They showed that solutionsu∈H1(D) to the elliptic equation
n
X
i,j=1
∂
∂xi
aij ∂u
∂xj
=h+
n
X
i=1
∂gi
∂xi
, (1.6)
where h∈L∞(D) and each gi is defined inD such thatgi|Dm (1≤m≤L) have continuous extensions ∈ Cµ(Dm), 0 < µ < 1 up to ∂Dm have globalW1,∞ and piecewiseC1,α0 estimates (see (1.7) below). These estimates are independent of the distances between inclusions when a material has inclusions.
We first give the result of Li-Nirenberg [12] for scalar equations.
Theorem 1.2 ([12, Theorem 1.1]). For any ε >0, there exists a constant C]>0 such that for anyα0 satisfying
0< α0<min
µ, α
2(α+ 1) , we have
L
X
m=1
kukC1,α0(Dm∩Dε)≤C]
kukL2(D)+khkL∞(D)+
L
X
m=1 n
X
i=1
kgikCα0(Dm)
, (1.7) where we denote
Dε:={x∈D: dist(x, ∂D)> ε}
and a positive constant C] depends only onn, L, µ, α, ε, λ,Λ,kaijkCα0
(Dm) and the C1,α0 norms of Dm.
Remark 1.3. The constant C] >0 is independent of the distances between in- clusions Dm. Therefore, the estimate (1.7) holds even in the case that some of inclusions touch another inclusions as in Figure 1.
Now, we consider the parabolic equation
∂u
∂t −
n
X
i,j=1
∂
∂xi
aij
∂u
∂xj
=f−
n
X
i=1
∂fi
∂xi
inQ, (1.8)
where
f ∈L∞(Q), ∂f
∂t ∈Lκ(Q), fi∈Lp(Q), ∂fi
∂t ∈Lp(Q) and fi=fi(m) onDm×(0, T],
D1
D2
D3
D4 D5
D6
D7
Figure 1. The case that an inclusion touches another inclusion.
(L= 7)
withp > n+ 2,κ=p(n+ 2)/(n+ 2 +p),Q:=D×(0, T],fi(m)∈L∞(0, T;Cµ(Dm)).
Now we define a weak solution to the equation (1.8).
Definition 1.4. We callu∈V21,0(Q) :=L2(0, T;H1(D))∩C([0, T];L2(D)) a weak solution to the equation (1.8) when
Z
D
u(x, t0)ϕ(x, t0)dx− Z t0
0
Z
D
u(x, t)∂ϕ
∂t(x, t)dx dt (1.9)
+ Z t0
0
Z
D n
X
i,j=1
aij(x) ∂u
∂xj(x, t) ∂ϕ
∂xi(x, t)dx dt (1.10)
= Z t0
0
Z
D
f(x, t)ϕ(x, t)dx dt+ Z t0
0
Z
D n
X
i=1
fi(x, t) ∂ϕ
∂xi
(x, t)dx dt (1.11) for any ϕ ∈L2(0, T;H01(D))∩H1(0, T;L2(D)) with ϕ(·,0) = 0 and 0 < t0 ≤T, whereH01(D) is the usualL2-Sobolev space with supports inD.
Our main result is as follows.
Theorem 1.5(Main theorem). Any weak solutionsu∈V21,0(Q) to(1.8)have the following up to the inclusion boundary regularity estimate: For any ε > 0, there exists a constantC]0 >0such that for any α0 satisfying
0< α0<min
µ, α
2(α+ 1) , (1.12)
we have
L
X
m=1
sup
ε2<t≤T
ku(·, t)kC1,α0(Dm∩Dε)≤C]0 kukL2(Q)+F∗+F∗∗
, where
F∗:=kfkLκ(Q)+kfkLmax{2,κ}(Q)+kfkL∞(Q)+k∂f
∂tkLκ(Q), F∗∗:=
n
X
i=1
kfikLp(Q)+k∂fi
∂tkL2(Q)+k∂fi
∂t kLp(Q)+
L
X
m=1
sup
0<t≤T
kfi(·, t)kCα0
(Dm)
and C]0 depends only on n, L, µ, α, ε, λ,Λ, p,kaijkCα0(Dm) and the C1,α0 norms of Dm.
Remark 1.6. (i) Again, the constant C]0 > 0 is independent of the distances between inclusionsDm. Then Theorem 1.5 holds even in the case that an inclusion touches another inclusion as Figure 1.
(ii) It is easy to obtain
F∗≤C∗
kfkL∞(Q)+k∂f
∂tkLκ(Q)
, F∗∗ ≤C∗
n
X
i=1
XL
m=1
sup
0<t≤T
kfi(·, t)kCα0
(Dm)+k∂fi
∂t kLp(Q)
. However, a constantC∗>0 depends onT and D, unfortunately.
For heat conductive materials with inclusions, the solutionuof the initial bound- ary value problem for (1.8) with heat flux given on ∂D×(0, T] and zero initial temperature at t = 0 in D describes the temperature distribution in D. Inject- ing the heat flux and measuring the temperature distribution on ∂D×(0, T] is the measurement of the so called active thermography. This is an non-destructive testing to identify unknown cracks, cavities and inclusions inside a heat conductor from the measurement. As a mathematical study of the active thermography, a method called the dynamical probe method has been given ([7]). It can approx- imately identify for instance inclusions by one measurement. For identifying the inclusions precisely, it needs infinitely many measurements. Further, it uses the gra- dient estimate of the fundamental solution of the heat equation with discontinuous conductivities.
The dynamical probe method has been developed only for the case that the inclusions do not touch another inclusions. So, it is interesting to consider the case when some of them touch. For the first task to handle this case, we need to have the gradient estimate of the fundamental solution. Our main result has given an answer to this. Similar situation can be considered for active thermography and non-destructive testing using acoustic waves. For example, [16] and [18] effectively used a result of Li-Vogelius [13] to give a procedure of enclosing the inclusions by the enclosure method (see [6], for example). What is interested about their arguments is that, by adding further arguments, we can even enclose the inclusions in the case that they can touch another inclusions [15]. More precisely for an increasing sets of inclusions, these inclusions can touch at a point of the boundary of the largest inclusion. Therefore, we believe that our gradient estimates will be useful for inverse problems identifying unknown inclusions.
This kind of gradient estimates stated in Theorem 1.5 for solutions of parabolic equations was initiated by Li [10] in his doctor thesis written in Chinese and was completed recently in Li-Li [11]. In [11], they even discussed the interior gradient estimates of solutions of a second order parabolic system of divergence form with inclusions which can touch another inclusions. They also allowed that the coeffi- cients can depend on the time. However, it should be noted that they could not allow the inclusions to depend on the time. Hence, showing the interior gradient estimate for this case is still opened.
We have to emphasize the following two things. The one is that we independently obtained our results. After we finished our paper and posted our paper in the
preprint server arXiv, Li sent the paper [11] to one of us. But we still did not know the paper [10] until very recently by a chance. The results of [11] was also posted in arXiv after us.
The another is that our proof of Theorem 1.5 is totally different from the proofs given in [10] and [11]. We first reduce the problem of the interior gradi- ent estimates for solutions of parabolic equations to that of elliptic parts of the equations using by the idea in [8], and directly apply the result [12] for elliptic equations. By this method, we estimate not only the L∞-norm of ∇u but also supε2<t≤Tku(·, t)kC1,α0
(Dm∩Dε) more easily. We also remark that the constantC]0 in Theorem 1.5 is independent of T. In [10], this property is not obvious at least even for the case when the right-hand sidef andfi on (1.8) are identically equal zero.
The rest of this paper is organized as follows. In Section 2, we prove our main theorem, i.e. Theorem 1.5 by applying Lemma 2.1. We prove Lemma 2.1 in Sec- tion 3. In Section 4, we consider a pointwise gradient estimate for the fundamental solution of parabolic operators with piecewise smooth coefficients by applying The- orem 1.5.
2. Proof of main result
In this section, we prove our main theorem. We first state some estimates in Lemma 2.1 which we need to prove our main theorem. We prove Lemma 2.1 in Section 3.
Lemma 2.1. Let(aij)be a matrix-valued function defined onD. Assume that(aij) is symmetric, positive definite, and satisfies the condition (1.4). Let Q as before andQbε:=Dε×(ε2, T]. Then for p > n+ 2, a weak solution u∈V21,0(Q)to (1.8) satisfies the following estimates:
sup
ε2<t≤T
ku(·, t)kL2(Dε)≤C kukL2(Q)+F0
, (2.1)
kukL∞(Qbε)≤C kukL2(Q)+F0
, (2.2)
k∂u
∂tkL2(Qbε)≤C kukL2(Q)+F1
, (2.3)
where we set
F0:=kfk
L
p(n+2) n+2+p(Q)
+
n
X
i=1
kfikLp(Q), (2.4)
F1:=kfk
Lmax{2,
p(n+2) n+2+p}
(Q)
+
n
X
i=1
kfikLp(Q)+k∂fi
∂tkL2(Q)
, (2.5)
andC >0 depends only onn, λ,Λ, p andε.
Now we prove our main theorem by applying Lemma 2.1. This proof is inspired by [8].
Proof of Theorem 1.5. Before going into the proof, we remark that a general con- stant C which we used below in our estimates depends only on n, λ,Λ, p and εj
(j= 1,2,3). To begin with the proof, let 0< ε1< ε2< ε3. Then we have sup
ε22<t≤T
ku(·, t)kL2(Dε2)≤C kukL2(Q)+F0
(2.6)
and
k∂u
∂tkL2(Qbε1)≤C kukL2(Q)+F1
(2.7) by (2.1) and (2.3) in Lemma 2.1, whereF0,F1 are defined by (2.4) and (2.5). On the other hand,ut=∂u/∂tsatisfies the equation
∂ut
∂t −
n
X
i,j=1
∂
∂xi
aij(x)∂ut
∂xj
=∂f
∂t −
n
X
i=1
∂
∂xi
∂fi
∂t
by applying∂/∂tto (1.8) (also see Remark 2.2). Hence we have kutkL∞(Qbε2)≤C
kutkL2(Qbε1)+F00
(2.8) by Lemma 2.1 (2.2), where we define
F00 :=k∂f
∂tk
L
p(n+2) n+2+p(Q)
+
n
X
i=1
k∂fi
∂tkLp(Q).
In particular, ut(·, t) ∈ L∞(Dε2) holds for a.e. t ∈ (ε22, T]. Now we regard the equation (1.8) as the elliptic equation
n
X
i,j=1
∂
∂xi
aij(x)∂u
∂xj
=∂u
∂t −f +
n
X
i=1
∂fi
∂xi (2.9)
by fixingt∈(ε22, T]. We remark that∂u/∂t−f ∈L∞(Dε2). Then, for anyα0 with the condition (1.12), we have the estimate
L
X
m=1
ku(·, t)kC1,α0(Dm∩Dε3)≤C]
ku(·, t)kL2(Dε2)+k∂u
∂t(·, t)kL∞(Dε2)
+kf(·, t)kL∞(Dε
2)+
L
X
m=1 n
X
i=1
kfi(·, t)kCα0(Dm)
(2.10) by Theorem 1.2, whereC]>0 depends only onn, L, µ, α, ε, λ,Λ,kaijkCα0(Dm)and theC1,α0 norms ofDm. Taking the supremum of the inequality (2.10) over (ε22, T] with respect tot, and using (2.6), (2.7) and (2.8), we have
L
X
m=1
sup
ε22<t≤T
ku(·, t)kC1,α0
(Dm∩Dε3)
≤C]
sup
ε22<t≤T
ku(·, t)kL2(Dε2)+k∂u
∂tkL∞(Qbε2)+kfkL∞(Qbε2)
+
L
X
m=1 n
X
i=1
sup
ε22<t≤T
kfi(·, t)kCα0(Dm)
≤C]C
kukL2(Q)+F0+F1+F00+kfkL∞(Qbε2)
+
L
X
m=1 n
X
i=1
sup
ε22<t≤T
kfi(·, t)kCα0
(Dm)
,
which is the estimate we want to obtain.
Remark 2.2. Since we assume thatubelongs only inV21,0(Q) with respect to the regularity of a weak solution, one may think that we cannot apply∂/∂t directly.
However, it is enough to consider the Steklov mean function and to make htend to 0, where we define the Steklov mean functionvh ofv by
vh(x, t) = 1 h
Z t+h
t
v(x, τ)dτ.
Hereafter we omit the detail with respect to this remark although we often apply this argument. Also see [9, III§2 p. 141] and (62) in [8, p. 152], for example.
3. Some estimates
In this section, we prove Lemma 2.1. The estimates (2.1) and (2.2) are well known, but we give these proofs in Appendix for readers’ convenience. To show the estimate (2.3), we prepare some necessary lemmas for its proof.
Throughout this section, C >0 denotes a general constant depending only on n, λ,Λ. Also, we assume that the coefficient (aij) is a matrix-value function defined onD, symmetric, positive definite, and satisfies the condition (1.4). Moreover, we setQr:=Br(x0)×(t0−r2, t0], and assume thatQ2ρ⊂D×(0, T] with 0< ρ≤1.
The following two lemmas are essentially shown in [8]. We give their proofs here for the sake of completeness.
Lemma 3.1 ([8, Lemma 3]). Let1< r <∞ and1/r+ 1/r0 = 1. Then a solution uto (1.8)satisfies the estimate
k∇ukL2(Qρ)≤Ch
(ρn/2+ρ(n+2)/r0) oscQ2ρu+kfkLr(Q2ρ)+
n
X
i=1
kfikL2(Q2ρ)
i. (3.1) Proof. Letζbe a smooth cut-off function onQ2ρ satisfyingζ≡1 onQρ,ζ≡0 on Q2ρ\Q3ρ/2, 0≤ζ≤1 onQ2ρ, and|∂ζ/∂t|+|∇ζ|2≤Cρ−2onQ2ρ. Letu0 be the average value ofuinQ2ρ:
u0:= 1
|Q2ρ| Z Z
Q2ρ
u(x, t)dx dt,
where|Q2ρ|denotes the measure ofQ2ρ. Testing (1.8) by (u−u0)ζ2and integrating by parts (i.e. takingϕ= (u−u0)ζ2 for (1.11). Also see Remark 2.2), we have
1 2
Z
B2ρ(x0)
(u−u0)2ζ2
(x, t0)dx− Z Z
Q2ρ
(u−u0)2ζ∂ζ
∂tdx dt +
Z Z
Q2ρ
n
X
i,j=1
aij
∂u
∂xj
∂u
∂xiζ2dx dt+ 2 Z Z
Q2ρ
n
X
i,j=1
aij
∂u
∂xj(u−u0)ζ ∂ζ
∂xidx dt
= Z Z
Q2ρ
f(u−u0)ζ2dx dt+
n
X
i=1
Z Z
Q2ρ
h fi
∂u
∂xiζ2+ 2fi(u−u0)ζ ∂ζ
∂xi i
dx dt.
Hence we have 1 2
Z
B2ρ(x0)
(u−u0)2ζ2
(x, t0)dx+λ Z Z
Q2ρ
|∇u|2ζ2dx dt
≤ 1 2
Z
B2ρ(x0)
(u−u0)2ζ2
(x, t0)dx+ Z Z
Q2ρ n
X
i,j=1
aij ∂u
∂xj
∂u
∂xiζ2dx dt
= Z Z
Q2ρ
(u−u0)2ζ∂ζ
∂tdx dt−2 Z Z
Q2ρ n
X
i,j=1
aij ∂u
∂xj
(u−u0)ζ ∂ζ
∂xi
dx dt
+ Z Z
Q2ρ
f(u−u0)ζ2dx dt +
n
X
i=1
Z Z
Q2ρ
h fi
∂u
∂xiζ2+ 2fi(u−u0)ζ ∂ζ
∂xi i
dx dt
≤ Z Z
Q2ρ
(u−u0)2ζ
∂ζ
∂t
dx dt+ε1
Z Z
Q2ρ
|∇u|2ζ2dx dt +C
ε1
Z Z
Q2ρ
|u−u0|2|∇ζ|2dx dt+1 2
Z Z
Q2ρ
|f ζ|rdx dt2/r +1
2 Z Z
Q2ρ
|(u−u0)ζ|r0dx dt2/r0
+ε1 Z Z
Q2ρ
|∇u|2ζ2dx dt
+ 1 ε1
+ 1 Z Z
Q2ρ n
X
i=1
|fi|2ζ2dx dt+ Z Z
Q2ρ
|u−u0|2|∇ζ|2dx dt.
We now takeε1>0 small enough. Then, we have Z Z
Qρ
|∇u|2dx dt
≤ Z Z
Q2ρ
|∇u|2ζ2dx dt
≤C Z Z
Q2ρ
(u−u0)2h ζ|∂ζ
∂t|+|∇ζ|2i
dx dt+CZ Z
Q2ρ
|(u−u0)ζ|r0dx dt2/r0
+CZ Z
Q2ρ
|f ζ|rdx dt2/r
+C Z Z
Q2ρ
n
X
i=1
|fi|2ζ2dx dt
≤Ch
ρn+ρ2(n+2)/r0
oscQ2ρu2
+kfk2Lr(Q2ρ)+
n
X
i=1
kfik2L2(Q2ρ)
i ,
because |u(x, t)−u0| ≤ oscQ2ρu holds for any (x, t) ∈ Q2ρ. This completes the
proof.
Lemma 3.2 ([8, Lemma 5]). A solutionuto (1.8)satisfies the estimate k∂u
∂tkL2(Qρ)≤Ch
ρ−1k∇ukL2(Q2ρ)+kfkL2(Q2ρ)
+
n
X
i=1
ρ−1kfikL2(Q2ρ)+k∂fi
∂tkL2(Q2ρ)
i (3.2)
Proof. We first take the same smooth cut-off functionζas in the proof of Lemma 3.1.
Testing (1.8) by (∂u/∂t)ζ2and integrating by parts (also see Remark 2.2), we have 1
2 Z
B2ρ(x0) n
X
i,j=1
aij ∂u
∂xi
∂u
∂xj
ζ2
(x, t0)dx
+ Z Z
Q2ρ
h|∂u
∂t|2ζ2−
n
X
i,j=1
aij ∂u
∂xi
∂u
∂xjζ∂ζ
∂t + 2
n
X
i,j=1
aij ∂u
∂xj
∂u
∂tζ∂ζ
∂xi i
dx dt
= Z Z
Q2ρ
f∂u
∂tζ2dx dt+
n
X
i=1
hZ
B2ρ(x0)
fi∂u
∂xi
ζ2
(x, t0)dx +
Z Z
Q2ρ
−∂fi
∂t
∂u
∂xiζ2−2fi
∂u
∂xiζ∂ζ
∂t + 2fi
∂u
∂tζ∂ζ
∂xi i
dx dt due to
n
X
i,j=1
aij
∂2u
∂t∂xi
∂u
∂xj
ζ2=1 2
∂
∂t Xn
i,j=1
aij
∂u
∂xi
∂u
∂xj
ζ2
−
n
X
i.j=1
aij
∂u
∂xi
∂u
∂xj
ζ∂ζ
∂t and
fi ∂2u
∂t∂xi
ζ2= ∂
∂t
fi ∂u
∂xi
ζ2
− ∂u
∂xi
∂
∂t(fiζ2).
Hence we have λ 2 Z
B2ρ(x0)
|∇u|2ζ2
(x, t0)dx+ Z Z
Q2ρ
|∂u
∂t|2ζ2dx dt
≤1 2
Z
B2ρ(x0)
Xn
i,j=1
aij ∂u
∂xi
∂u
∂xj
ζ2
(x, t0)dx+ Z Z
Q2ρ
|∂u
∂t|2ζ2dx dt
= Z Z
Q2ρ
h Xn
i,j=1
aij ∂u
∂xi
∂u
∂xj
ζ∂ζ
∂t −2
n
X
i,j=1
aij ∂u
∂xj
∂u
∂tζ∂ζ
∂xi
i dx dt
+ Z Z
Q2ρ
f∂u
∂tζ2dx dt+
n
X
i=1
hZ
B2ρ(x0)
fi
∂u
∂xiζ2
(x, t0)dx +
Z Z
Q2ρ
−∂fi
∂t
∂u
∂xi
ζ2−2fi
∂u
∂xi
ζ∂ζ
∂t + 2fi
∂u
∂tζ ∂ζ
∂xi
i dx dt
≤C Z Z
Q2ρ
|∇u|2ζ|∂ζ
∂t|dx dt+ε2
Z Z
Q2ρ
|∂u
∂t|2ζ2dx dt +C
ε2
Z Z
Q2ρ
|∇u|2|∇ζ|2dx dt+ε2
Z Z
Q2ρ
|∂u
∂t|2ζ2dx dt +C
ε2
Z Z
Q2ρ
|f|2ζ2dx dt+ε2
Z
B2ρ(x0)
|∇u|2ζ2
(x, t0)dx + C
ε2
Z
B2ρ(x0)
Xn
i=1
|fi|2ζ2
(x, t0)dx
+C Z Z
Q2ρ
|∇u|2ζ2dx dt+C Z Z
Q2ρ n
X
i=1
∂fi
∂t
2ζ2dx dt
+C Z Z
Q2ρ
|∇u|2ζ|∂ζ
∂t|dx dt+C Z Z
Q2ρ n
X
i=1
|fi|2ζ|∂ζ
∂t|dx dt +ε2
Z Z
Q2ρ
|∂u
∂t|2ζ2dx dt+ C ε2
Z Z
Q2ρ
n
X
i=1
|fi|2|∇ζ|2dx dt.
We remark that Z
B2ρ(x0)
(fiζ)2(x, t0)dx= Z
B2ρ(x0)
Z t0
t0−(2ρ)2
∂
∂t (fiζ)2
(x, t)dt dx
≤C Z Z
Q2ρ
h|fi|2
ζ2+ζ|∂ζ
∂t| +|∂fi
∂t |2ζ2i dx dt.
Therefore, by takingε2>0 small enough, we have Z Z
Qρ
|∂u
∂t|2dx dt≤ Z
B2ρ(x0)
|∇u|2ζ2
(x, t0)dx+ Z Z
Q2ρ
|∂u
∂t|2ζ2dx dt
≤C Z Z
Q2ρ
|∇u|2
ζ2+ζ|∂ζ
∂t|+|∇ζ|2
dx dt+C Z Z
Q2ρ
|f|2ζ2dx dt
+C Z Z
Q2ρ n
X
i=1
h|fi|2
ζ2+ζ|∂ζ
∂t|+|∇ζ|2 +|∂fi
∂t|2ζ2i dx dt
≤Cρ−2k∇uk2L2(Q2ρ)+Ckfk2L2(Q2ρ)+Cρ−2
n
X
i=1
kfik2L2(Q2ρ)
+C
n
X
i=1
k∂fi
∂tk2L2(Q2ρ).
We obtain the estimate (2.3) from Lemmas 5.6 (given in Appendix), 3.1 and 3.2.
4. A gradient estimate of the fundamental solution
In this section, we consider a gradient estimate of the fundametal solution of parabolic operators. We first state some facts. It is known that if coefficient (aij) is a symmetric and positive definite matrix-valueL∞(Rn) function satisfying (1.4), then there exists a fundamental solution Γ(x, t;y, s) of the parabolic operator
∂
∂t−
n
X
i,j=1
∂
∂xi
aij
∂
∂xj
(4.1) with the estimate
|Γ(x, t;y, s)| ≤ C∗
(t−s)n/2exp
−c∗|x−y|2 t−s
χ[s,∞)(t) (4.2) for allt, s∈R, and a.e.x, y∈Rn, whereC∗, c∗>0 depend only onn, λ,Λ (see [1]
or [4], for example). In particular, the constantsC∗ andc∗are independent of the distance between inclusions. If the coefficients (aij) is not piecewise smooth but H¨older continuous in the whole spaceRn, then the pointwise gradient estimate
|∇xΓ(x, t;y, s)| ≤ C∗
(t−s)(n+1)/2exp
−c∗|x−y|2 t−s
χ[s,∞)(t) holds fort, s∈R, a.e.x, y∈Rn (see [9, Chapter IV§11–13], for example).
Now, the aim of this section is to show the gradient estimate (4.8) in Theorem 4.3 even if the coefficients are piecewiseCµ in D. We assume that (aij) defined inD satisfies the conditions (1.4) and (1.5), and extend it to the wholeRn by defining (aij)≡ΛIinRn\D, whereIis the identity matrix. We remark that this extension does not destroy the conditions (1.4) and (1.5). Then there exists a fundamental solution Γ(x, t;y, s) of the parabolic operator (4.1) with the estimate (4.2) as we stated above.
To prove our gradient estimate of the fundamental solution, we apply the fol- lowing corollary from Theorem 1.5.
Corollary 4.1. Let 0< ρ≤1. Then a solutionuto the parabolic equation
∂u
∂t −
n
X
i,j=1
∂
∂xi
aij ∂u
∂xj
= 0 in Bρ(x0)×(t0−ρ2, t0] (4.3) has the estimate
k∇ukL∞(Bρ/2(x0)×(t0−(ρ/2)2, t0]))≤ C]0
ρn/2+2kukL2(Bρ(x0)×(t0−ρ2, t0]), (4.4) whereC]0>0depends only onn, L, µ, α, λ,Λ, andkaijkCα0
(Dm)and theC1,α0norms of Dm for someα0 with (1.12).
Proof. It is sufficient to apply the scaling argument. Letρy=x−x0,ρ2(s−1) = t−t0 and
u(y, s) :=e u(x, t) =u ρy+x0, ρ2(s−1) +t0 , eaij(y) :=aij(x) =aij(ρy+x0),
Dem:=1
ρ(x−x0) :x∈Dm .
(4.5)
Then we have
∂eu
∂s−
n
X
i,j=1
∂
∂yi
eaij ∂ue
∂yj
= 0 in B1(0)×(0,1]. (4.6) Therefore, by noting Remark 4.2, we have
k∇uke L∞(B1/2(0)×(3/4,1])≤C]0kuke L2(B1(0)×(0,1))
by Theorem 1.5, where C]0 depends only on n, L, µ, α, λ,Λ,kaijkCα0(Dm), and the C1,α0 seminorms ofDm. By this estimate and the definition (4.5), we obtain the
estimate (4.4).
Remark 4.2. One may think that C]0 depends also on ρ sincekeaijk
Cα0(Dem) and theC1,α0 norms ofDemdepend onρ. However, we can takeC]0 independent ofρby taking the following into consideration.
First we consider keaijk
Cα0(Dem)=keaijk
C0(Dem)+ [eaij]
Cα0(Dem)
:= sup
y∈Dem
|eaij(y)|+ sup
y,η∈Dem
|eaij(y)−eaij(η)|
|y−η|α0 . It is easy to show
keaijk
C0(Dem)=kaijkC0(Dm)
and
[eaij]
Cα0(Dem)=ρα0[aij]Cα0(Dm)≤[aij]Cα0(Dm). Then we have
keaijk
Cα0(Dem)≤ kaijkCα0
(Dm).
Next we consider the C1,α0 norms ofDem. We need to recall the proofs of the results of [12] and [13] more carefully. In the case when we consider theL∞-norm of ∇eufor a solution ue to the equation (4.6), the influence of the C1,α0 norms of
subdomains Dem appears only in the following constant C in (4.7): We estimate O |x0|1+α
in the equation (49) in [13, p. 118], i.e.
fm(x0) =fm(00) +∇fm(00)x0+O |x0|1+α
(49) as
O |x0|1+α
≤C|x|1+α (4.7)
(See also [12, Lemma 4.3]). HereC1,αfunctionsfmare defined in the cube (−1,1)n, and the graphs of fm describe ∂Dm. Now we remark that the constant C in (4.7) depends only on theC1,α seminorms offm. We consider the variable change ρy=x. Then the graphxn =fm(x0) is changed toyn=fem(y0), wherefem(y0) :=
ρ−1fm(ρy0), and we have
[fem]C1,α((−1,1)n)≤[fem]C1,α((−1/ρ,1/ρ)n)
=ρα[fm]C1,α((−1,1)n)≤[fm]C1,α((−1,1)n).
Hence, even when we consider the variable changeρy=x, we can take the constant C in (4.7) independent ofρ.
Considering the circumstances mentioned above, we can takeC]0 >0 independent ofρ.
Now we state the estimate of∇xΓ(x, t;y, s).
Theorem 4.3. We have
|∇xΓ(x, t;y, s)| ≤ C
(t−s)(n+1)/2exp
−c|x−y|2 t−s
(4.8) for a.e.x, y∈Rn andt > s with|x−y|2+t−s≤16, whereC, c >0 depend only on n, L, µ, α, λ,Λ,kaijkCα0(Dm) and the C1,α0 seminorms ofDm for some α0 with (1.12).
We prove Theorem 4.3 in the same way as the proof of [3, Proposition 3.6]. We first show the following lemmas.
Lemma 4.4. Let ρ:= (|x0−ξ|2+t0−τ)1/2/4. Then Z t0
t0−ρ2
Z
Bρ(x0)
|Γ(x, t;ξ, τ)|2dx dt≤ (C∗0)2ρn (t0−τ)n−1exp
−2c0∗|x0−ξ|2 t0−τ
fort0> τ, whereC∗0, c0∗>0 depend only onn, λ,Λ.
Proof. By (4.2), it is sufficient to obtain the estimate I0:=
Z t0
t0−ρ2
Z
Bρ(x0)
1
(t−τ)nexp
−2c∗|x−ξ|2 t−τ
χ[τ,∞)(t)dx dt
≤ (C∗0)2ρn
(t0−τ)n−1exp
−2c0∗|x0−ξ|2 t0−τ
.
(4.9)
We consider the following three cases:
(i) t0−ρ2≤τ < t0, (i) t0−2ρ2≤τ ≤t0−ρ2, (i) τ≤t0−2ρ2.
Now we consider the case (i). Then we have (√
15−1)ρ≤ |x−ξ|for anyx∈Bρ(x0), because|x0−ξ| ≥√
15ρ. Hence we have I0≤
Z t0
τ
Z
Bρ(x0)
1
(t−τ)nexp
− c1ρ2 t−τ
dx dt=|B1(0)|ρn Z t0−τ
0
ϕ1(s)ds, whereϕ1(s) :=s−nexp(−c1ρ2/s) andc1:= 2(√
15−1)2c∗. If 0< t0−τ≤c1ρ2/n, then we have
Z t0−τ
0
ϕ1(s)ds≤ Z t0−τ
0
ϕ1(t0−τ)ds= (t0−τ)−n+1exp
− c1ρ2 t0−τ
, because ϕ1(s) ≤ ϕ1(t0−τ) holds for any s ∈ [0, t0−τ]. On the other hand, if c1ρ2/n≤t0−τ ≤ρ2, then we have
Z t0−τ
0
ϕ1(s)ds≤ Z t0−τ
0
ϕ1
c1ρ2 n
ds= n c1
n
(t0−τ)ρ−2nexp(−n)
≤ n c1
n
(t0−τ)1−nexp
− c1ρ2 t0−τ
, where we used the properties that
ϕ1(s)≤ϕ1c1ρ2 n
for any 0< s≤t0−τ; n≥ c1ρ2
t0−τ and ρ2≥t0−τ.
Summing up, we have I0≤max
1, n c1
n
|B1(0)|ρn(t0−τ)1−nexp
− c1ρ2 t0−τ
. Let us consider the case (ii). Then we have (√
14−1)ρ≤ |x−ξ|for allx∈Bρ(x0), because|x0−ξ| ≥√
14ρ. Hence we have I0≤
Z t0
t0−ρ2
Z
Bρ(x0)
1
(t−τ)nexp
− c2ρ2 t−τ
dx dt=|B1(0)|ρn Z t0−τ
t0−ρ2−τ
ϕ2(s)ds, where ϕ2(s) :=s−nexp(−c2ρ2/s) and c2:= 2(√
14−1)2c∗. In a similarly way as the case (i), ifρ2≤t0−τ≤c2ρ2/n, then we have
Z t0−τ
t0−ρ2−τ
ϕ2(s)ds≤ Z t0−τ
t0−ρ2−τ
ϕ2(t0−τ)ds=ρ2(t0−τ)−nexp
− c2ρ2 t0−τ
≤(t0−τ)−n+1exp
− c2ρ2 t0−τ
,
becauseϕ2(s)≤ϕ(t0−τ) for anys∈[t0−ρ2−τ, t0−τ], and we haveρ2≤t0−τ.
On the other hand, ifc2ρ2/n≤t0−τ ≤2ρ2, then we have Z t0−τ
t0−ρ2−τ
ϕ2(s)ds≤ Z t0−τ
t0−ρ2−τ
ϕ2 c2ρ2
n
ds= n
c2
n
ρ−2n+2exp(−n)
≤2n−1 n
c2
n
(t0−τ)1−nexp
− c2ρ2 t0−τ
, where we used the properties that
ϕ2(s)≤ϕ2c2ρ2 n
for anyt0−ρ2−τ≤s≤t0−τ;
