on the occasion of his 70th birthday
ON SECOND ORDER OVERDETERMINED DIFFERENTIAL SYSTEMS
MONA-MIHAELA DOROFTEI
Overdetermined second order systems are analyzed using a reducing method to a representation of gradient systems in locally of finite type Lie algebras with supplementary hypotheses.
AMS 2000 Subject Classification: 35A30, 35N10, 58J45.
Key words: Lie algebras (locally) of finite type, gradient systems, integral mani- fold.
1. INTRODUCTION
This paper relay on the studies made by Varsan [5] in the case of fi- nite generated over R Lie algebra and its applications. Using techniques of gradient systems and algebraic representation some results on the existence of nontrivial solutions for second order overdetermined systems are obtained.
Similar computational methods are applied in papers [1], [2], [3]. The first section of this paper analyzes second order linear and semilinear equations determined by a finite number of derivations, under degeneracy conditions of the Lie algebras generated by the vector fields determining the systems. In the second section of this paper the operator ←−
P =∆ ∂t+ ∆ and equations of parabolic-hyperbolic type are considered. For the solution of the Kolmogorov equation with inverted time we used computations from [4].
2. OVERDETERMINED SECOND ORDER SYSTEMS
We consider{g1, g2, . . . , gm} ⊆C∞(Rn;Rn)=∆Fnand let~gj ∈Der (Rn), j ∈ {1,2, . . . , m}be the associated derivatives. Letg0∈C2(Rn,Rn) and con- sider the second order homogeneous linear system
(2.1) ~g1
∂S
∂x(x), g0(x)
= 0, . . . , ~gm
∂S
∂x(x), g0(x)
= 0.
MATH. REPORTS12(62),3 (2010), 235–243
Definition 2.1. A C2 function S(x) :B(x0, ρ) ⊆Rn→ Rwill be called a nontrivial solution for (2.1) if ~g0S(x), x ∈ B(x0, ρ) is not constant and it verifies the system (2.1) for any x in an integral manifoldMx0 ⊆B(x0, ρ) of dimension kassociated with the Lie algebraL(g1, . . . gm)⊆Fn.
The existence of the nontrivial solution for system (2.1) is assured if the first order homogeneous linear system
(2.2) ~gi(V)(x) = ∂V
∂x(x), gi(x)
= 0, i∈ {1, . . . , m}
admits a nontrivial solution V(x) :B(x0, ρ)→R.
In order to obtain this, we assume the hypothesis
(h1) The Lie algebra L(g1, . . . , gm) ⊆ Fn is (locally) of finite type, g0(x0)6= 0 and dimL(g1, . . . , gm) (x0) =k < n, for fixedx0∈Rn.
If hypothesis (h1) is fulfilled then there exists a smooth and nonconstant function Ve(x) : B(x0, ρ) ⊆ Rn → R verifying the system (2.2) for any x in an integral manifold Mx0 ⊆B(x0, ρ) of dimension k associated with the Lie algebra L(g1, . . . , gm). A nontrivial solution for the second order system (2.1) is defined solving the first order linear equation
(2.3)
∂S
∂x(x), g0(x)
=Ve(x), x∈B(x0, ρ1),
where 0 < ρ1 ≤ ρ, and Ve(x), x ∈ B(x0, ρ) is the nontrivial solution of system (2.2).
Theorem 2.1. Let {g1, . . . , gm} ⊆ Fn and g0 ∈ C2(Rn;Rn) be given.
Consider a fixed x0 ∈ Rnsuch that hypothesis (h1) is fulfilled. Then the ho- mogeneous second order linear system (2.1)admits a nontrivial solution which is determined from equation (2.3), where Ve(x), x∈B(x0, ρ1), is a nontrivial solution for the homogeneous hyperbolic linear system (2.2).
Proof. The equation (2.3) is solved in the standard way if we consider the associated characteristics system
(2.4)
dG0
dt (t, λ) =g0(G0(t, λ)), deh
dt (t, λ) =Ve(G0(t, λ)), t∈[0, T], G0(0, λ) =λ∈B(x0,ρ)e ⊆B(x0, ρ), eh(0, λ) = 0.
We consider bλ = (λ2, . . . , λn), F(t,bλ) = G0(t, x01, λ2, . . . , λn) and the algebraic system
(2.5) F(t,bλ) =x∈B(x0, ρ1)⊆B(x0, ρ), where, without loss of generality, we suppose that the matrix
[g0(x0), e2, . . . , en]=∆
∂F
∂t (0,λ)b ∂F
∂λb(0,bλ)
is nonsingular. Solving system (2.5) we obtain a unique C1 solution
(2.6) t=τ(x) :B(x0, ρ1)→Randbλ= Ψ (x) :B(x0, ρ)→V(bx0)⊆Rn−1, where bx0 = (x∆ 01, . . . , x0n), such that
(2.7)
F(τ(x),Ψ (x)) =x∈B(x0, ρ1), τ F(t,bλ)
=t, Ψj F(t,bλ)
=λj, t∈[0, T], j ∈ {2, . . . , n}, Ψ= (Ψ∆ 2, . . . ,Ψn).
We define
(2.8) S(x) =e eh(τ(x), x01,Ψ2(x), . . . ,Ψn(x)), x∈B(x0, ρ1),
where bh(t, λ) is the solution in (2.4), and t =τ(x), Ψj(x), x ∈ B(x0, ρ1) of C1,j ∈ {2, . . . , n}, verify equations (2.7).
By definitionS(x) :e B(x0, ρ1)→R isC2 and using (2.7) we obtain (2.9) S Fe (t,bλ)
=eh(t, x01, λ2, . . . , λn), t∈[0, T], λb∈V (xb0)⊆Rn−1. Taking derivatives with respect to tin (2.9) we obtain
(2.10)
*∂Se
∂x F(t,bλ)
, g0 F(t,λ)b +
=V Fe (t,bλ) , t∈[0, T], bλ∈V(bx0)⊆Rn−1.
Fort=τ(x),bλ= Ψ(x),x∈B(x0, ρ1), from (2.10) we find the equation (2.11)
*∂Se
∂x(x), g0(x) +
=Ve(x), ∀x∈B(x0, ρ1)
and S(x),e x ∈ B(x0, ρ1), defined in (2.8), is a nontrivial solution for the hyperbolic second order system (2.1).
Remark 2.1. If we admit that the field g0 is a smooth field (g0 ∈ Fn), then system (2.1) can be formally written as
(2.10) ~g1(~g0(S))(x) = 0, . . . , ~gm(~g0(S))(x) = 0,
and a nontrivial solution for (2.10) will be determined in the conditions of hypothesis (h1) as a C∞ functionS(x) :e B(x0, ρ)⊆Rn→R.
If we change the order of derivatives
(2.100) ~g0(~g1(S))(x) = 0, . . . , ~g0(~gm(S))(x) = 0,
then the existence of the non trivial solution for (2.100) requires supplementary compatibility conditions. One should mention that the two systems (2.10) and (2.100) are different, because we do not assume a commuting condition for the fields g0 and gi, i ∈ {1, . . . , m}. Returning to system (2.100) and denoting
~
gi(S)(x) =Vi(x), we will obtain m first integrals for the field g0, so we need to study existence conditions for m ≤ n−1 functionally independent first integrals together with some compatibility conditions, that can assure the existence of a nontrivial solution S for the overdetermined nonhomogeneous linear system ~gi(S)(x) =Vi(x),i∈ {1, . . . , m}.
For the second part of this section we will consider a semilinear second order system
(2.12)
~g1
∂S
∂x(x), g0(x)
=l1
x,
∂S
∂x(x), g0(x)
,
· · ·
~ gm
∂S
∂x(x), g0(x)
=lm
x, ∂S
∂x(x), g0(x)
,
associated with the derivatives ~g1, . . . , ~gm ⊆ Der (Rn) and the field g0 ∈ C2(Rn,Rn), where lj(x, v) : Rn×R → R, j ∈ {1,2, . . . , m} are given C∞ functions.
Definition 2.2. A nontrivial solution for (2.12) is a C2 function S(x) : B(x0, ρ) ⊆ Rn → R such that V(x) =∆ ∂S
∂x(x), g0(x)
: B(x0, ρ) → R is a nontrivial ∂V∂x(x0)6= 0
solution for the first order semilinear system
(2.13)
~g1(V)(x)=∆ ∂V
∂x(x), g1(x)
=l1(x, V(x)),
· · ·
~
gm(V)(x)=∆ ∂V
∂x(x), gm(x)
=lm(x, V(x)).
To obtain a nontrivial solution for (2.13), we must assume a compatibility condition. Denote
f1(y) =
l1(x, v) g1(x)
∈Rn+1, . . . , fm(y) =
lm(x, v) gm(x)
∈Rn+1, y= (x, v)∆
and consider the real Lie algebras L(g1, . . . , gm) ⊆ Fn and L(f1, . . . , fm) ⊆ Fn+1.
Consider the hypothesis
(h2) The Lie algebras L(g1, . . . , gm), L(f1, . . . , fm) are locally of finite type and k = dimL(g1, . . . , gm) (x0) = dimL(f1, . . . , fm) (y0), where y0 = (x0,0) andx0∈Rn is fixed.
Furthermore, if we consider that l(y) = (l∆ 1(x, v), . . . , lm(x, v)) verifies l(y0) 6= 0, then system (2.13) admits a smooth nontrivial solution Ve(x) : B(x0, ρ) → R, ∂∂xVe (x0) 6= 0, such that (2.13) is fulfilled for any x in an integral manifold Mx0 ⊆B(x0, ρ) of dimension k≤nassociated with the Lie algebra L(g1, . . . , gm). With this, to find a nontrivial solution for (2.12) we must construct a solution for the first order linear equation
(2.14)
∂S
∂x(x), g0(x)
=Ve(x), x∈B(x0, ρ1)⊆B(x0, ρ),
by repeating the characteristics method presented in the analysis of the linear equation (2.3).
As a solution of (2.14) one can obtain theC2functionS(x) :B(x0, ρ1)→ R which verifies (2.14) for any x ∈ B(x0, ρ1), 0 < ρ1 ≤ ρ, where Ve(x), x∈B(x0, ρ) is a nontrivial solution
∂Ve
∂x(x0)6= 0
for system (2.13).
Consequently, if we assume hypothesis (h2) and l(y0) = (l1(x0,0), . . . , lm(x0,0)) 6= 0 then the semilinear second order system (2.12) admits a non- trivial solution which can be obtained from (2.14), where Ve(x),x ∈B(x0, ρ) is a nontrivial solution for (2.13). We obtained
Theorem2.2. Let {g1, . . . , gm} ⊆Fn, g0 ∈C2(Rn;Rn) andl= (l1, . . . , lm)∈C∞(Rn×R;Rm) be given. Letx0∈Rn be fixed such that l(x0,0)6= 0 and assume that (h2) is fulfilled.
Then the semilinear second order system (2.12) admits a nontrivial so- lution which can be determined from (2.14), where Ve(x), x ∈ B(x0, ρ), is a nontrivial solution of the hyperbolic system (2.13).
3. SECOND ORDER SYSTEMS OF PARABOLIC-HYPERBOLIC TYPE
We consider the parabolic operator←−
P =∆∂t+∆,∂tϕ= ∂ϕ∂t, ∆ϕ=
n
P
i=1
∂i2ϕ,
∂iϕ= ∂x∂ϕ
i,i∈ {1, . . . , n}where the functionϕ(t, x) :I×Ω→Ris continuous together with the first order derivative with respect to t ∈ I ⊆ R and with the derivatives of any order with respect to x in the domain Ω ⊆ Rn. The functions ϕ which are C1 in variable t∈I and C∞ in x∈Ω will be denoted by Φ (I×Ω).
Let {g1, . . . , gm} ⊆ Fn and consider the linear homogeneous system of parabolic-hyperbolic type
(3.1) ~gi(∂tS+ ∆S) (t, x) = 0, i∈ {1,2, . . . , m}.
Definition 3.1. A nontrivial solution for system (3.1) is a function S ∈ (Φ [0, T]×B(x0, ρ)) for which
∂tS+ ∆S=∆V(x), x∈B(x0, ρ0), t∈[0, T),
is non constant and verifies the system (3.1) for any x ∈Mx0 ⊆B(x0, ρ0)⊆ Rn, whereMx0 is an integral manifold associated with the Lie algebraL(g1, . . . , gm)⊆Fn and
S(T, x) =u(x), where u∈C∞(B(x0, ρ),R),ρ > ρ0, is fixed.
The sufficient existence conditions for a non trivial solution in (3.1) will be formulated such that the first order system
(3.2) ~gi(V) = ∂V
∂x(x), gi(x)
= 0, i∈ {1,2, . . . , m}, admits a solution Ve(x) : B(x0, ρ) ⊆ Rn → R nontrivial
∂Ve
∂x (x0)6= 0 . In order to obtain this result we need the hypothesis
(h1) The real Lie algebraL(g1, . . . , gm) is (locally) of finite type and for fixed x0 ∈Rnwe have dimL(g1, . . . , gm) (x0) =k < n.
Under hypothesis (h1) one can apply theorems on gradient system to the linear system of first order (3.2) to obtain the existence of a C∞ function Ve(x) : B(x0, ρ) → R, with ∂∂xVe (x0) 6= 0 (eV nonconstant), such that the system (3.2) is verified in a domain Mx0 ⊆ B(x0, ρ) described as an integral manifold of dimension k < n passing through x0 ∈ Rn, associated with the Lie algebra L(g1, . . . , gm). The linear system
(3.3)
*∂Ve
∂x(x), gi(x) +
= 0, ∀x∈Mx0 ⊆B(x0, ρ), i∈ {1,2, . . . , m}, is verified, where Ve(x) :B(x0, ρ)→R isC∞ and nonconstant.
A nontrivial solution Se(t, x) : [0, T]×B(x0, ρ) → R, 0 < ρ0 < ρ, Se∈ Φ ([0, T]×B(x0, ρ0)) for system (3.1) is obtained from Ve(x),x∈B(x0, ρ), if we consider the parabolic equation with inverted time (Kolmogorov)
(3.4)
( ∂tS(t, x) + ∆S(t, x) =Ve(x), S(T, x) =u(x),e
t ∈ [0, T), x ∈ B(x0, ρ0), 0 < ρ0 < ρ, where eu ∈ C∞(B(x0, ρ) ;R) is fixed.
For the following computations see [4].
For this, consider eu0(x) and Ve0(x), x∈Rn, the extensions to the whole space Rn of the C∞ functionsu(x),e Ve(x), x∈B(x0, ρ) such that it verifies (3.5) u(x) =e ue0(x), Ve(x) =Ve0(x),
if x ∈ B(x0, ρ0), 0 < ρ0 < ρ, ue0(x) = 0 and Ve0(x) = 0, if x ∈ Rn\B(x0, ρ).
Here the spheres B(x0, ρ), B(x0, ρ0) are defined as open sets.
With these considerations, the solution of equation (3.4) can be repre- sented as
(3.6) Se(t, x) =ue0(eyt,x(T)) + Z T
t
Ve0(yet,x(s)) ds, t∈[0, T],
whereyet,x(s) =x+w(s)−w(t),s∈[t, T], and{w(t),t≥0}isn-dimensional Wiener standard process. The function
Se(t, x), t∈[0, T], x∈Rn
defined by (3.6) is in Φ ([0, T)×Rn) and verifies the Kolmogorov equation with inverted time
(3.7)
( ∂tSe(t, x) + ∆Se(t, x) =Ve0(x), ∀t∈[0, T), x∈Rn, Se(T, x) =ue0(x), x∈Rn.
In particular, using (3.5) for x ∈ B(x0, ρ0), from (3.7) one can obtain the equations
(3.8)
( ∂Se(t, x) + ∆Se(t, x) =Ve(x), t∈[0, T), x∈B(x0, ρ0) Se(T, x) =u(x), xe ∈B(x0, ρ0), where 0< ρ0 < ρ.
LetSe(t, x),t∈[0, T]×B(x0, ρ0) verifying (3.8) and defined in (3.6).
Then Se ∈ Φ ([0, T)×B(x0, ρ0)) and Se is a nontrivial solution for the parabolic-hyperbolic system of second order (3.1). With this we obtained the proof for the existence below.
Theorem3.1. Let{g1, . . . , gm} ⊆Fnandu∈C∞(B(x0, ρ),R)be given.
Let x0∈Rn a fixed point such that hypothesis(h1) is fulfilled.
Then the linear homogeneous parabolic-hyperbolic system (3.1)admits a nontrivial solution which can be determined from the parabolic equation with inverted time(3.4)forue=u,whereVe(x),x∈B(x0, ρ),is a nontrivial solution for the hyperbolic homogeneous linear system (3.2).
The presented algorithm for solving the system (3.1) is useful for semi- linear systems of the form
(3.9) −→gi(∂tS+ ∆S) (t, x) =li(x,(∂tS+ ∆S) (t, x)), i∈ {1,2, . . . , m}, where li(x, v) : Rn×R→ R are givenC∞ functions and {g1, . . . , gm} ⊆Fn are fixed.
Definition3.2. A nontrivial solution for (3.9) is a functionS∈Φ ([0, T)× B(x0, ρ0) such that
•S(T, x) =u(x), where u∈C∞(B(x0, ρ),R) is fixed and
•∂tS+ ∆S =V(x),x∈B(x0, ρ), 0< ρ0 < ρ, t∈[0, T), and V(x) is a nontrivial solution for the semilinear system
(3.10) ~gi(V)(x) = ∂V
∂x(x), gi(x)
=li(x, V(x)), i∈ {1,2, . . . , m}. To determine a solution Ve(x) :B(x0, ρ) →R nontrivial
∂Ve
∂x(x0)6= 0 for system (3.10) we must consider a compatibility hypothesis.
We definey= (x, v)∈Rn+1,fi(y) =
li(y) gi(x)
∈Rn+1,i∈ {1, . . . , m}, l(y)∆=(l1(y), . . . , lm(y)) and lety0 ∆
= (x0,0) wherex0∈Rnis fixed. We assume that the semilinear system (3.10) fulfills the hypothesis
(h2) The real Lie algebrasL(g1, . . . , gm)⊆FnandL(f1, . . . , fm)⊆Fn+1
are (locally) of finite type and
k= dimL(g1, . . . , gm) (x0) = dimL(f1, . . . , fm) (y0) and also l(y0)6= 0.
Under the hypothesis (h2) one can apply gradient theorems to the system (3.10) to obtain the existence of the solution Ve(x) :B(x0, ρ)→R nontrivial ∂Ve
∂x(x0)6= 0
verifying (3.10) on a domainx∈Mx0 ⊆B(x0, ρ), that is, (3.11)
*∂Ve
∂x(x), gi(x) +
=li x,Ve(x)
, x∈Mx0, i∈ {1, . . . , m}, where Mx0 ⊆ B(x0, ρ) is an integral manifold associated with L(g1, . . . , gm) of dimensionk= dimL(g1, . . . , gm) (x0).
As in the previous theorem we reconsider the algorithm, using the so- lution of system (3.11) to solve the equation with inverted time (3.4). Thus we obtain
Theorem3.2. Let {g1, . . . , gm} ⊆ Fn and l(y) = (l∆ 1(y), . . . , lm(y)), y∈ Rn+1be given such that hypothesis(h2)is fulfilled, with fixed y0 = (x0,0),x0∈ Rn. Let u∈C∞(Rn;R)be given. Then the semilinear second order parabolic- hyperbolic system (3.9)admits a nontrivial solution Se∈Φ([0, T)×B(x0, ρ0)).
Remark 3.1. The computations used in the proof of Theorems 1 and 2 remain the same if we replace the parabolic operator←−
P =∂t+∆ by−→
P =−∂t+
∆, with the condition that in the definition of the nontrivial solution the initial condition must be Se(0, x) = u(x), x ∈ B(x0, ρ) instead of S(T, x) = u(x), x ∈ B(x0, ρ). The nontrivial solution S determined in the above theorems
becomes the nontrivial solution for the modified problem by the transformation Se(t, x) =S(T−t, x),t∈(0, T),x∈B(x0, ρ).
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Received 28 April 2009 University “Politehnica” of Bucharest Department of Mathematics I
Splaiul Independent¸ei 313 060032 Bucharest, Romania
mdoroftei@yahoo.com
