行政院國家科學委員會專題研究計畫 成果報告
在 time scales 上的衝擊動態方程的週期邊界值問題
研究成果報告(精簡版)
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 99-2115-M-004-002- 執 行 期 間 ： 99 年 08 月 01 日至 100 年 07 月 31 日 執 行 單 位 ： 國立政治大學應用數學學系 計 畫 主 持 人 ： 符聖珍 計畫參與人員： 碩士班研究生-兼任助理人員：陳家盛 處 理 方 式 ： 本計畫可公開查詢中 華 民 國 100 年 10 月 14 日
1 Introduction
Maximum principles are an important tool in the study of partial differ-ential and difference equations. For example, they can be used to obtain the existence and uniqueness of solutions and to approximate it. Consequently the theory of maximum principles in difference and differential equations has been investigated extensively, see for example [1] and [2] and the references cited therein.
In recent years, the study of dynamic equations on time scales has received a lot of attentions since it not only can unify the calculation of difference and differential equations but also has various applications. In particular, the maximum principles have been established in [4] for the second order ordinary dynamic operator and [5] for the elliptic dynamic operator. Motivated by the above work, in this paper, we study the maximum principles for the elliptic dynamic operator L[u] := n X i=1 (u∇i∆i + B iu∆i + Ciu∇i)
and the parabolic dynamic operator
L[u] := n X i=1 (u∇i∆i + ˜B iu∆i + ˜Ciu∇i) − u∇n+1.
Our results improve the results in [5].
This paper is organized as follows. Section 2 contains some basic defini-tions and the necessary results about time scales. In Section 3, we present the maximum principles for the elliptic dynamic operators. Finally, in section 4, we establish the maximum principles for parabolic operators, and apply it to obtain some useful applications.
2 Preliminary
For completeness, we state some fundamental definitions and results concern-ing partial dynamic equations on time scales that we will use in the sequel. It can be regarded as a generalization of the one-dimensional case. More details can be found in [6], [7], [8], and [9].
A time scale is an arbitrary nonempty closed subset of R. Throughout this paper, we denote I = {1, 2, · · · , n}, where n ∈ N, and we assume that Ti, for
each i ∈ I, is a time scale and the set
Λ = T1× T2× · · · × Tn = {t = (t1, t2, · · · , tn) | ti ∈ Ti for each i ∈ I},
defined by the Cartesian product is an n-dimensional time scale.
Definition 2.1 For each i ∈ I, the mappings σi, ρi : Ti → Ti defined by
σi(u) :=
inf{v ∈ Ti | v > u}, if u 6= max Ti,
max Ti, if u = max Ti, and ρi(u) :=
sup{v ∈ Ti | v < u}, if u 6= min Ti,
min Ti, if u = min Ti,
are called the ith forward and backward jump operators respectively. In this definition, the corresponding graininess functions µi, νi : Ti → [0, ∞) are
defined by
µi(u) := σi(u) − u, νi(u) := u − ρi(u).
For convenience, we define the functions ˆσi, ˆρi : Λ → Λ by
ˆ
and
ˆ
ρi(t) = (t1, t2, · · · , ti−1, ρi(ti), ti+1, · · · , tn),
for any t ∈ Λ and i ∈ I. In addition, if u : Λ → R is a function, then the functions uσˆi, uρˆi : Λ → R are defined by
uˆσi(t) = u(ˆσ
i(t)) and uρˆi(t) = u( ˆρi(t)),
for any t ∈ Λ and i ∈ I.
Definition 2.2 A point t in Λ is said to be i-right dense if ti < max Ti and
σi(ti) = ti, and i-left dense if ti > min Ti and ρi(ti) = ti. Also, if σi(ti) > ti
then t is called i-right scattered, and if ρi(ti) < tithen t is called i-left scattered.
Moreover, we say that t is i-scattered if it is both i-left scattered and i-right scattered, and i-dense if it is both i-left dense and i-right dense.
Definition 2.3 For each i ∈ I, let
(Ti)K =
Ti\ max Ti, if Ti has a lef t scattered maximum,
Ti, if Ti has a lef t dense maximum.
Then we can define
ΛK= (T1)K× (T2)K× · · · × (Tn)K.
Assume u : Λ → R is a function and let t ∈ ΛK. Then we define u∆i(t) to be
the number (provided it exists) with the property that given any ε > 0, there exists a δ > 0 such that
| [u(ˆσi(t)) − u(s)] − u∆i(t)[ˆσi(t) − s] |≤ ε | ˆσi(t) − s |,
for all s ∈ (t − δei, t + δei) ∩ Λ, where {ei | i ∈ I} denotes the natural basis
for Rn. In this case, we call u∆i(t) the partial delta derivative of u at t with
respect to ti.
function from T1 into R, and we denote the delta derivative of u at t ∈ (T1)K
by u∆(t).
Definition 2.4 For each i ∈ I, let
(Ti)K=
Ti\ min Ti, if Ti has a right scattered minimum,
Ti, if Ti has a right dense minimum.
Then we can define
ΛK= (T1)K× (T2)K× · · · × (Tn)K.
Assume u : Λ → R is a function and let t ∈ ΛK. Then we define u∇i(t) to be
the number (provided it exists) with the property that given any ε > 0, there exists a δ > 0 such that
| [u(ˆρi(t)) − u(s)] − u∇i(t)[ ˆρi(t) − s] |≤ ε | ˆρi(t) − s |,
for all s ∈ (t − δei, t + δei) ∩ Λ. In this case, we call u∇i(t) the partial nabla
derivative of u at t with respect to ti.
In particular, if we choose n = 1 in this definition, then u is a single variable function from T1 into R, and we denote the nabla derivative of u at t ∈ (T1)K
by u∇(t).
For convenience, we denote the intersection of ΛK and ΛK by ΛKK, i.e.,
ΛKK= (T1)KK× (T2)KK× · · · × (Tn)KK.
Definition 2.5 Let the functions U , u : Λ → R satisfy U∆i(t) = u(t) for
all t ∈ ΛK, then we define
Z s
r
u(t)∆it = U (s) − U (r) for all r, s ∈ Λ and
i ∈ I. Similarly, we can define
Z s
r
u(t)∇it = U (s) − U (r) for all r, s ∈ Λ if
U∇i(t) = u(t) for all t ∈ Λ
K and i ∈ I.
and satisfy
1 − ν(t)p(t) 6= 0 f or all t ∈ TK.
Then we define the nabla exponential function by
ˆ ep(t, s) = exp( Z t s g(τ )∇τ ) f or s, t ∈ T, where g(τ ) = p(τ ), if ν(τ ) = 0, − 1 ν(τ )Log(1 − ν(τ )p(τ )), if ν(τ ) 6= 0.
Lemma 2.7 Suppose that α is a negative constant and s, t, u ∈ T, then (a) ˆeα(t, s) > 0 and ˆeα(t, t) ≡ 1;
(b) ˆeα(t, u)ˆeα(u, s) = ˆeα(t, s);
(c) ˆe∇α(t, s) = αˆeα(t, s).
Lemma 2.8 Assume that f : T → R is a single variable function and let t ∈ TKK, then we have the following:
(a) If f is delta or nabla differentiable at t, then f is continuous at t.
(b) If f is continuous at a right-scattered point t, then f is delta differentiable at t with
f∆(t) = f (σ(t)) − f (t) µ(t) .
(c) If t is right-dense, then f is delta differentiable at t if and only if the limit
lim
s→t
f (t) − f (s) t − s
exists. In this case,
f∆(t) = lim
s→t
f (t) − f (s) t − s . (d) If f is delta differentiable at t, then
(e) If f is continuous at a left-scattered point t, then f is nabla differentiable at t with
f∇(t) = f (t) − f (ρ(t)) ν(t) .
(f ) If t is left-dense, then f is nabla differentiable at t if and only if the limit
lim
s→t
f (t) − f (s) t − s
exists. In this case,
f∇(t) = lim
s→t
f (t) − f (s) t − s . (g) If f is nabla differentiable at t, then
f (ρ(t)) = f (t) − ν(t)f∇(t).
Hereafter [a, b]T represents an interval on time scale T, that is, [a, b]T = [a, b] ∩ T. Other types of intervals on a time scale can be represented by the similar way.
Lemma 2.9 Assume that f : T → R is a function, then (a) If f∆> 0 on [a, b]
T, then f is strictly increasing on [a, b]T.
(b) If f > 0 is a continuous function on [a, b]T, then
Z b
a
f (t)∆t > 0 and
Z b
a f (t)∇t > 0, where a, b ∈ T.
Lemma 2.10 Assume that f : T → R is nabla differentiable and f∇ is con-tinuous on TK. Then f is delta differentiable at t and
3 Maximum principles for elliptic dynamic equations
In this section we first consider the dynamic Laplace operator
∆Tu := n X i=1 u∇i∆i. Let Λ = [ρ1(a1), σ1(b1)]T1 × · · · × [ρn(an), σn(bn)]Tn.
We shall study the functions in the set
D(Λ) := {u : Λ → R | u∇i∆i is continuous in ΛK
K for each i ∈ I}.
The following lemma provides some basic properties for an interior maximum point of a function in D(Λ).
Lemma 3.1 Suppose that u ∈ D(Λ) attains its maximum at an interior point m of Λ. Then, for each i ∈ I, we have
u∇i(m) ≥ 0, u∆i(m) ≤ 0, and u∇i∆i(m) ≤ 0.
In particular, if m is i-right dense, then
u∇i(m) = u∆i(m) = 0.
Proof. Since u attains its maximum at an interior point m of Λ, it follows from the definition of u∇i and u∆i that
u∇i(m) ≥ 0 and u∆i(m) ≤ 0, (1)
for each i ∈ I. Let us divide our proof into two cases according to the point type of m with respect to the ith component.
(i) m is i-right dense:
In this case, by applying Lemma 2.10, we have that
u∆i(m) = u∇i(ˆσ
and consequently, together with (1), we conclude that
u∇i(m) = u∆i(m) = 0.
Now we want to show that u∇i∆i(m) ≤ 0. For contradiction, we assume
that u∇i∆i(m) > 0. Then the continuity of u∇i∆i and Lemma 2.9 imply
that there exists a δ > 0 such that u∇i is strictly increasing in t
i on
J , where J denotes the set of all points t ∈ Λ lying on the line segment joining m and m+δei. Since m is i-right dense, without loss of generality,
we may assume that mi + δ ∈ Ti. Since u∇i(m) = 0, it follows that
u∇i(t) > 0 for all t ∈ J . Then, by applying Lemma 2.9, we easily get
Z m+δei
m
u∇i(s)∇
is = u(m + δei) − u(m) > 0,
which contradicts the fact that u(m) is the maximum value on Λ. (ii) m is i-right scattered.
Note that u∇i(ˆσ i(m)) = u(ˆσi(m)) − u( ˆρi(ˆσi(m))) σi(mi) − ρi(σi(mi)) = u(ˆσi(m)) − u(m) σi(mi) − mi = u∆i(m).
Together with (1), we obtain
u∇i∆i(m) = u ∇i(ˆσ i(m)) − u∇i(m) σi(mi) − mi = u ∆i(m) − u∇i(m) σi(mi) − mi ≤ 0. 2 Corollary 3.2 If u ∈ D(Λ) satisfies ∆Tu > 0, in ΛKK, (2)
then u cannot attain its maximum at an interior point of Λ.
Proof. For contradiction, we assume that u attains its maximum at an interior point m of Λ. By applying Lemma 3.1, we have that u∇i∆i(m) ≤ 0 for each
i ∈ I. This implies that ∆Tu(m) = n X i=1 u∇i∆i(m) ≤ 0, which contradicts (2). 2
Next we consider the more general operator which contains the first-derivative terms L[u] := n X i=1 (u∇i∆i + B iu∆i + Ciu∇i) = ∆Tu + n X i=1 (Biu∆i+ Ciu∇i).
Following the statement of Lemma 3.1, for each t ∈ Λ, we define the auxiliary index sets It RD := {i ∈ I : ti = σi(ti)}, It RS := {i ∈ I : ti < σi(ti)}. Corollary 3.3 If u ∈ D(Λ) satisfies L[u] > 0, in ΛKK, (3)
and let Bi and Ci satisfy
Bi(t) ≥ 0, Ci(t) ≤ 0, (4)
for each t ∈ ΛKK which is i-right scattered and i ∈ I. Then u cannot attain its maximum at an interior point of Λ.
Proof. For contradiction, we assume that u attains its maximum at an interior point m of Λ. Lemma 3.1 yields that at the point m, we have
u∆i(m) = 0, u∇i(m) = 0, and u∇i∆i(m) ≤ 0 if i ∈ Im
RD,
u∆i(m) ≤ 0, u∇i(m) ≥ 0, and u∇i∆i(m) ≤ 0 if i ∈ Im
RS.
L[u](m) = n X i=1 (u∇i∆i(m) + B i(m)u∆i(m) + Ci(m)u∇i(m)) = X i∈Im RD u∇i∆i(m) + X i∈Im RS (u∇i∆i(m) + B i(m)u∆i(m) + Ci(m)u∇i(m)) ≤ 0, which contradicts (3). 2
Theorem 3.4 Let u ∈ D(Λ) satisfy the inequality (3) and let Bi and Ci
satisfy 1 + Bi(t)µi(ti) ≥ 0, −1 + Ci(t)µi(ti) ≤ 0, (5)
for each t ∈ ΛKK which is i-right scattered and i ∈ I. Then u cannot attain its maximum at an interior point of Λ.
Proof. For contradiction, we assume that u attains its maximum at an interior point m of Λ. Then, by applying Lemma 3.1, we can rewrite L[u](m) in the following way: L[u](m) = n X i=1 (u∇i∆i(m) + B i(m)u∆i(m) + Ci(m)u∇i(m)) = X i∈Im RD u∇i∆i(m) + X i∈Im RS (u∇i∆i(m) + B i(m)u∆i(m) + Ci(m)u∇i(m)) = X i∈Im RD u∇i∆i(m) + X i∈Im RS (u ∆i(m) − u∇i(m) µi(mi) + Bi(m)u∆i(m) + Ci(m)u∇i(m)). (6) If I = Im
RD, then (6) implies that
L[u](m) = X
i∈Im RD
which contradicts (3). Otherwise, let us define the auxiliary functions ˆ µ(t) := Y j∈It RS µj(tj), µˆ−i(t) := Y j∈It RS j6=i µj(tj). Obviously, if i ∈ It RS we have ˆ µ(t) = ˆµ−i(t)µi(ti). (7)
We multiply both sides of the equality (6) by ˆµ(m) > 0 and use (7) to obtain
ˆ µ(m)L[u](m) = ˆµ(m) X i∈Im RD u∇i∆i(m) +ˆµ−i(m)µi(mi) X i∈Im RS (u ∆i(m) − u∇i(m) µi(mi) + Bi(m)u∆i(m) + Ci(m)u∇i(m)) = ˆµ(m) X i∈Im RD u∇i∆i(m) +ˆµ−i(m) X i∈Im RS [(1 + Bi(m)µi(mi))u∆i(m) + (−1 + Ci(m)µi(mi))u∇i(m)].
Lemma 3.1 together with the assumptions (5), and positivity of ˆµ(m) and ˆ
µ−i(m) imply that
ˆ
µ(m)L[u](m) ≤ 0,
which contradicts (3). Therefore we conclude that u cannot achieve its maxi-mum at an interior point of Λ. 2
4 Maximum principles for parabolic dynamic equations
In this section, we extend our results in the last section to the parabolic dynamic operators. Let Λ be an n-dimensional time scale defined in section 3. Then we define the (n + 1)-dimensional time scale Ω by
Ω = Λ × [0, T ]Tn+1,
where Tn+1 is an arbitrary time scale and 0, T ∈ Tn+1. In addition, we set
B = Λ × {0} and S = ∂Λ × (0, T ]Tn+1,
then we can define the parabolic boundary P Ω by
P Ω = S ∪ B.
Throughout this section, we study the functions in the set
D(Ω) := {u : Ω → R | u∇i∆i is continuous in ΛK K× [0, T ]Tn+1 for each i ∈ I and u∇n+1 is continuous in Λ × ([0, T ] Tn+1)K}. Corollary 4.1 If u ∈ D(Ω) satisfies ∆Tu − u∇n+1 = n X i=1 u∇i∆i − u∇n+1 > 0, in ΛK K× ([0, T ]Tn+1)K, (8)
Then u cannot attain its maximum anywhere other than on the parabolic boundary.
Proof. For contradiction, we assume that u attains its maximum at a point m ∈ Ω \ P Ω. This implies that m ∈ ΛKK× ([0, T ]Tn+1)K. Therefore, by applying
Lemma 3.1, we have
Since u attains its maximum at m, by the definition of partial nabla derivative of u, we obtain u∇n+1(m) ≥ 0. (9) It follows that (∆Tu − u∇n+1)(m) = n X i=1 u∇i∆i(m) − u∇n+1(m) ≤ 0, which contradicts (8). 2
Similarly, we consider the more general operator
L[u] := n X i=1 (u∇i∆i + ˜B iu∆i + ˜Ciu∇i) − u∇n+1. Corollary 4.2 If u ∈ D(Ω) satisfies L[u] > 0, in ΛKK× ([0, T ]Tn+1)K, (10)
and let ˜Bi and ˜Ci satisfy
˜ Bi(t) ≥ 0, ˜ Ci(t) ≤ 0, (11)
for each t ∈ ΛKK× ([0, T ]Tn+1)K which is i-right scattered and i ∈ I. Then u
cannot attain its maximum anywhere other than on the parabolic boundary.
Proof. For contradiction, we assume that u attains its maximum at a point m ∈ Ω \ P Ω. Lemma 3.1 together with the assumptions (11) and (9) imply that L[u](m) = n X i=1 (u∇i∆i(m) + ˜B i(m)u∆i(m) + ˜Ci(m)u∇i(m)) − u∇n+1(m) = X i∈Im RD u∇i∆i(m) + X i∈Im RS (u∇i∆i(m) + ˜B i(m)u∆i(m) + ˜Ci(m)u∇i(m)) − u∇n+1(m) ≤ 0,
which contradicts (10). 2
Theorem 4.3 Let u ∈ D(Ω) satisfy the inequality (10) and let ˜Bi and ˜Ci
satisfy 1 + ˜Bi(t)µi(ti) ≥ 0, −1 + ˜Ci(t)µi(ti) ≤ 0, (12)
for each t ∈ ΛKK× ([0, T ]Tn+1)K which is i-right scattered and i ∈ I. Then u
cannot attain its maximum anywhere other than on the parabolic boundary.
Proof. For contradiction, we assume that u attains its maximum at a point m ∈ Ω \ P Ω. As similar as the proof of Theorem 3.4, we rewrite L[u](m) in the following way:
L[u](m) = X i∈Im RD u∇i∆i(m) + X i∈Im RS (u ∆i(m) − u∇i(m) µi(mi) + ˜Bi(m)u∆i(m) + ˜Ci(m)u∇i(m)) − u∇n+1(m). (13) If I = Im
RD, then (13) and (9) imply that
L[u](m) = X
i∈Im RD
u∇i∆i(m) − u∇n+1(m) ≤ 0,
which contradicts (10). Otherwise, we multiply both sides of the equality (13) by ˆµ(m) > 0 and use (7) and (9) to obtain that
ˆ µ(m)L[u](m) = ˆµ(m) X i∈Im RD u∇i∆i(m) +ˆµ−i(m) X i∈Im RS [(1 + ˜Bi(m)µi(mi))u∆i(m) + (−1 + ˜Ci(m)µi(mi))u∇i(m)] −ˆµ(m)u∇n+1(m) ≤ 0,
which contradicts (10) and the proof is done. 2
Next we consider the operator which contains the non-derivative term
(L + h)[u] := n X i=1 (u∇i∆i + ˜B iu∆i + ˜Ciu∇i) − u∇n+1 + hu.
Theorem 4.4 Let u ∈ D(Ω) satisfy
(L + h)[u] > 0, in ΛKK× ([0, T ]Tn+1)K, (14)
and let ˜Bi and ˜Ci satisfy the inequality (12). Moreover, we suppose that
h(t) ≤ 0, (15)
for each t ∈ ΛKK× ([0, T ]Tn+1)K. Then u cannot attain a nonnegative maximum
anywhere other than on the parabolic boundary.
Proof. For contradiction, we assume that u attains a nonnegative maximum at a point m ∈ Ω \ P Ω. By the proof of Theorem 4.3, we know that
L[u](m) ≤ 0,
if u attains its maximum at the point m. Then, together with the condition h(m)u(m) ≤ 0, we easily see that
which contradicts (14). 2 Corollary 4.5 If u ∈ D(Ω) satisfies n X i=1 (u∇i∆i+ ˜B iu∆i + ˜Ciu∇i + βiuˆσi+ γiuρˆi) − u∇n+1 + hu > 0, (16)
in ΛKK× ([0, T ]Tn+1)K. Further, we assume that
1 + ( ˜Bi(t) + µi(ti)βi(t))µi(ti) ≥ 0, −1 + ( ˜Ci(t) − νi(ti)γi(t))µi(ti) ≤ 0, (17)
for each t ∈ ΛKK× ([0, T ]Tn+1)K which is i-right scattered and i ∈ I, and
h +
n
X
i=1
(βi+ γi) ≤ 0, in ΛKK× ([0, T ]Tn+1)K. (18)
Then u cannot attain a nonnegative maximum anywhere other than on the parabolic boundary.
Proof. Using the formula (d) and (g) in the Lemma 2.8, we can obtain the two analogues equalities:
u(ˆσi(t)) = u(t) + µi(ti)u∆i(t),
u( ˆρi(t)) = u(t) − νi(ti)u∇i(t),
for each t ∈ ΛKK× ([0, T ]Tn+1)K and i ∈ I. Substituting these into (16), we
obtain n X i=1 (u∇i∆i+( ˜B i+µi(ti)βi)u∆i+( ˜Ci−νi(ti)γi)u∇i)−u∇n+1+(h+ n X i=1 (βi+γi))u > 0.
Obviously, this operator has the form of (14), and the assumptions (17) and (18) ensure that the inequalities (12) and (15) hold. Consequently, we can use Theorem 4.4 to verify the statement. 2
Finally, we establish the weak maximum principles for the parabolic operator and apply it to obtain the uniqueness of solutions for the initial boundary
value problem.
Theorem 4.6 Let u ∈ D(Ω) satisfy
L[u] ≥ 0, in ΛKK× ([0, T ]Tn+1)K, (19)
and we assume that ˜Bi be bounded above and ˜Ci ≤ 0 satisfy the inequalities
(12). Then u attains its maximum on the parabolic boundary, i.e.,
sup
Ω
u = sup
P Ω
u. (20)
Proof. Since ˜B1 is bounded above, there exists a negative constant α such that
α + ˜B1 < 0, in ΛKK× ([0, T ]Tn+1)K. (21)
Select any point ˆt ∈ T1. Then, applying Lemma 2.7 and 2.10, we obtain
L[ˆeα(t1, ˆt)] = (ˆeα(t1, ˆt))∇1∆1 + ˜B1(ˆeα(t1, ˆt))∆1 + ˜C1(ˆeα(t1, ˆt))∇1 = (α + ˜B1)ˆe∆α1(t1, ˆt) + α ˜C1eˆα(t1, ˆt) = (α + ˜B1)ˆe∇α1(σ1(t1), ˆt) + α ˜C1eˆα(t1, ˆt) = (α + ˜B1)αˆeα(σ1(t1), ˆt) + α ˜C1ˆeα(t1, σ1(t1))ˆeα(σ1(t1), ˆt) = αˆeα(σ1(t1), ˆt)[α + ˜B1+ ˜C1ˆeα(t1, σ1(t1))]. (22)
The assumption ˜C1 ≤ 0 together with (21), we see that
L[ˆeα(t1, ˆt)] > 0, in ΛKK× ([0, T ]Tn+1)K.
Then for each ε > 0, we have
L[u + εˆeα(t1, ˆt)] = L[u] + εL[ˆeα(t1, ˆt)] > 0, (23) in ΛKK× ([0, T ]Tn+1)K, so that sup Ω (u + εˆeα(t1, ˆt)) = sup P Ω (u + εˆeα(t1, ˆt)), (24)
by applying the Theorem 4.3. Now we want to show that sup
Ω
u = sup
P Ω
u. For contradiction, we assume that sup
Ω
u > sup
P Ω u. Since the time scale T
1 is bounded, this implies that 0 <
ˆ
eα(t1, ˆt) < M for some M > 0. We set K = sup Ω
u − sup
P Ω
u > 0 and take ε = 2MK , then by applying (24) we can deduce that
sup P Ω (u + εˆeα(t1, ˆt)) ≤ sup P Ω (u + εM ) = sup P Ω u + εM = (sup Ω u − K) + K 2 < supΩ u ≤ sup Ω (u + εˆeα(t1, ˆt)) = sup P Ω (u + εˆeα(t1, ˆt)),
which is a contradiction and the proof is done. 2
The above proven maximum principles yields the uniqueness of solutions for the following problem:
n X i=1 (u∇i∆i + ˜B iu∆i + ˜Ciu∇i) − u∇n+1 = f (t) on ΛKK× ([0, T ]Tn+1)K, u(t) = g(t) on B, u(t) = h(t) on S. (25)
Theorem 4.7 Suppose that the assumptions of Theorem 4.6 holds. If u1 and
u2 are solutions of the initial boundary value problem (25), then u1 ≡ u2.
Proof. First of all, we define the auxiliary function v = u1− u2. Since both u1
and u2 are solutions of (25), this implies that
n X i=1 (v∇i∆i+ ˜B iv∆i + ˜Civ∇i) − v∇n+1 = 0 on ΛKK× ([0, T ]Tn+1)K, v(t) = 0 on P Ω (26)
Obviously, we know that −v is also a solution of (26). Then by applying Theorem 4.6, we have that
sup Ω v = sup P Ω v = 0 and sup Ω (−v) = sup P Ω (−v) = 0. It follows that v(t) ≤ 0 and − v(t) ≤ 0,
References
[1] M. Protter, H. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New Jersey, (1967).
[2] H. Kuo, N. Trudinger, On the discrete maximum principle for parabolic difference operators, Math. Model. Numer. Anal. 27 (1993) 719-737.
[3] David Gilbarg, Neil S. Trudinger, Elliptic partial differential equations of second order, Berlin, New York: Springer-Verlag, (1977).
[4] P. Stehlik, B. Thompson, Maximum principles for second order dynamic equations on time scales, J. Math. Anal. Appl. 331 (2007) 913-926.
[5] P. Stehlik, Maximum principles for elliptic dynamic equations, Mathematical and Computer Modelling 51 (2010) 1193-1201.
[6] R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35(1999) 3-22.
[7] M. Bohner and A. Peterson, Dynamic Equation on Time Scales, An Introduction with Application, Birkhauser, Boston (2001).
[8] M. Bohner and A. Peterson, Advances in Dynamic Equation on Time Scales, Birkhauser, Boston (2003).
[9] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math. 186 (2006) 391-415.
國科會補助計畫衍生研發成果推廣資料表
日期:2011/10/14國科會補助計畫
計畫名稱: 在time scales上的衝擊動態方程的週期邊界值問題 計畫主持人: 符聖珍 計畫編號: 99-2115-M-004-002- 學門領域: 微分方程無研發成果推廣資料
99 年度專題研究計畫研究成果彙整表
計畫主持人：符聖珍 計畫編號：99-2115-M-004-002- 計畫名稱：在 time scales 上的衝擊動態方程的週期邊界值問題 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 1 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 1 1 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次其他成果
(
無法以量化表達之成 果如辦理學術活動、獲 得獎項、重要國際合 作、研究成果國際影響 力及其他協助產業技 術發展之具體效益事 項等，請以文字敘述填 列。) 無 成果項目 量化 名稱或內容性質簡述 測驗工具(含質性與量性) 0 課程/模組 0 電腦及網路系統或工具 0 教材 0 舉辦之活動/競賽 0 研討會/工作坊 0 電子報、網站 0 科 教 處 計 畫 加 填 項 目 計畫成果推廣之參與（閱聽）人數 0國科會補助專題研究計畫成果報告自評表
請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價
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、是否適
合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。
1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估
■達成目標
□未達成目標（請說明，以 100 字為限）
□實驗失敗
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□其他原因
說明：
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論文：□已發表 ■未發表之文稿 □撰寫中 □無
專利：□已獲得 □申請中 ■無
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3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價
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500 字為限）
In recent years, the study of dynamic equations on time scales has received a lot of attentions since it not only can unify the calculation of difference and differential equations but also has various applications. In this project, we study the maximum principles for the elliptic and parabolic dynamic equations on multi-dimensional time scales.
