On Diagonal Elliptic and Parabolic Systems with Super-Quadratic Hamiltonians Alain Bensoussan ...

19 Pages·2008·0.17 MB·German

On Diagonal Elliptic and Parabolic Systems
with Super-Quadratic Hamiltonians

Alain Bensoussan, Jens Frehse

no. 375

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-
gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität
Bonn entstanden und als Manuskript vervielfältigt worden.
Bonn, Februar 2008
On Diagonal Elliptic and Parabolic Systems
with Super-Quadratic Hamiltonians ∗
Alain Bensoussan Jens Frehse
University of Texas at Dallas University of Bonn
School of Management Institut fu¨r Angewandte Mathematik
International Center for [email protected]
Decision and Risk Analysis
[email protected]
Abstract
We consider in this article a class of systems of second order partial dif-
ferential equations with non-linearity in the first order derivative and zero
order term which can be super-quadratic. These problems are motivated by
differentialgeometryandstochasticdifferentialgames. Uptonow, inthecase
of systems, only quadratic growth had been considered.
Key Words. Differential Geometry, Stochastic Differential Games, Hamiltonians.
∗This article is dedicated to Professor Philippe G. Ciarlet for his 70th anniversary
1
1 Introduction
In differential geometry and in the theory of stochastic differential games one finds
many examples of systems of the type
n
(cid:88) (cid:0) (cid:1)
(uν)− D a (x)D uν +λ (x,u,∇u) = Hν(x,u,∇u),ν = 1,...,N (1)
t i ik k ν
i,k=1
or equations which can be transformed to this type of system. The right hand side
Hν frequently is called Hamiltonian due to the application in stochastic control
theory. The coefficients a ∈ L∞(Ω) are assumed to satisfy a condition of uniform
ik
ellipticity. For applications to differential geometry see Hildebrandts survey [Hil82],
books on geometric analysis, say [Jos02], for applications to stochastic differential
games see Bensoussan-Frehse [BF02a] and [BF84]. In differential geometry, due to
scaling invariance, the functions Hν(x,u,∇u) are usually quadratic in ∇u. This
is not the case in the theory of stochastic differential games where sub-quadratic,
quadratic and even super-quadratic growth of H (ν,u,∇u) with respect to ∇u may
ν
occur in natural settings.
From the point of view of regularity theory the case of sub-quadratic growth is
simple. The quadratic growth case is already difficult and creates a lot of interesting
problems ([Fre01],[BF02b], [BF84], [BF95], [BF02a]).
Up to now, there is no result at all for the case of systems where the Hamiltonian
grows super-quadratically in ∇u. In the scalar case (N = 1), L∞-bounds for ∇u
can be archived via barrier methods, thus cases of super quadratic Hamiltonians
can be treated. See e.g. [Lio82].
In this note we present a special system of type (1) where the Hamiltonian
Hν = Hν(∇u) may have any polynomial growth and, nevertheless, there exist
regular solutions. We have to confine to a periodic setting. We consider the method
of proof as very simple and believe that there should be a lot of possibilities to
generalize our conditions.
2 Formulation of the Theorem
Let Q = [0,L]n ⊂ Rn be a cube and Hk,q = Hk,q(Q;RN) be the usual Sobolev space
(cid:93) (cid:93)
of L-periodic functions u = (u1,...,uN) : [0,L]n → RN with generalized derivatives
up to order k ∈ N in the Lebesgue space Lq, q ∈ [1,∞]. We look for solutions
1
u ∈ H2,q of the system
(cid:93)
−∆uν +λ uν = Gν(|∇u|2)+Go(|∇u|2)·∇uν +f (x), ν = 1,...,N (2)
ν ν
where
λ = const > 0, (3)
ν
f ∈ H1,∞(Q), (4)
ν (cid:93)
Gν ∈ C1(R),ν = 1,...,N , (5)
Go ∈ C1(R;Rn), (6)
and we assume the growth conditions
(cid:12) ∂ (cid:12) (cid:12) ∂ (cid:12)
(cid:12) Gν(η)(cid:12)+(cid:12) Go(η)(cid:12) ≤ K|η|q−1 +K, η ∈ R, (7)
(cid:12)∂η (cid:12) (cid:12)∂η (cid:12)
with some exponent q ≥ 1 and a constant K. Note that q >> 1 is admitted. From
(7) obviously we have
|Gν(η)|+|Go(η)| ≤ K|η|q +K (8)
with some constants K.
Theorem 2.1 Under the above regularity and growth assumption (3)–(8) for the
data there exists a periodic solution u : Q → RN of the system (2) which is contained
in H2,2 ∩C2+α for α ∈ [0,1).
(cid:93)
There is also a parabolic analogue of Theorem 2.1. We treat initial value problems
uν −∆uν = Gν(|∇u|2)+G0(|∇u|2)·∇uν +f (t,x), ν = 1,...,N (9)
t ν
with Gν,Go as before and
f ∈ L∞(cid:0)0,T;H1,∞(cid:1) (10)
ν (cid:93)
in the space-time cylinder [0,T] × Q and look for smooth solutions
u ∈ C(0,T;L2) ∩ L2(0,T,H1) which are periodic in the space variables and sat-
(cid:93) (cid:93)
isfy the initial condition
u| = u ∈ H2,∞(Q),. (11)
t=0 0 (cid:93)
2
Theorem 2.2 Let Gν; Go, f , u satisfy the regularity (5), (6), (10), (11) and the
ν 0
growth condition (7), (8). Then there exists a solution
u ∈ Lr(cid:0)0,T,H2,r(Q)(cid:1)
(cid:93)
of (9), (11) with
(cid:0) (cid:1)
u ∈ Lr 0,T;Lr(Q)
t (cid:93)
for all r < ∞ and T > 0.
Remark:Even in the case where Gν(|∇u|2) and Go(|∇u|2)∇u grow only quadrati-
cally in |∇u| Theorem 2.1 and Theorem 2.2 seem to be “new” and are not contained
in [BF02b] and [Fre01]
Let us sketch how to get estimates in the case of Neumann boundary conditions.
We pose additional assumptions
G1(η) ≥ c |η|q(cid:48) −K, G2(η) ≥ c |η|q(cid:48) −K.
0 0
For simplicity, q(cid:48) > n/2.
G1(η) ≥ (1−δ )|G0(η)||η|1/2 −K, δ > 0
0 0
G2(η) ≥ (1−δ )|G0(η)||η|1/2 −K, δ > 0
0 0
By maximum principle arguments one can achieve bounds from below
u1 ≥ −K, u2 ≥ −K.
Then one can use the function 1 as a test function, and if the Gν have a
uν+K+1
stronger growth and coerciveness behaviour than G0 we obtain an estimate
(cid:90) |∇uν|2 (cid:90) |∇uν|2q(cid:48)
dx+ dx ≤ K. (12)
(uν +K +1)2 uν +K +1
Ω Ω
Unfortunately, no bound for u itself is available. (We may treat a variational in-
equality with convex set −c ≤ u ≤ c ; in this case the main estimates of our proof
ν
would work.) Approximating the system such that structure is maintained is also a
delicate task. One can approximate the problem adding (cid:15) times the pˆ-Laplacian of
u, pˆ> 2p. Then one has an approximate solution. Unfortunately, we were not able
3
to localize the estimates in the proof of theorem 2.1 and 2.2, yet.
3 Proof of the Theorems
We approximate (8) by replacing the functions Gν(η) and Go(η) with
Gν(η) := (1+δη)−qGν(η)
δ
Go(η) := (1+δη)−q−1Go(η),
δ
where η ≥ 0, δ > 0, δ → 0.
Since Gν and Go(|∇u|2)∇uν are bounded for any, say, H1-function u, if δ is fixed,
δ δ
there is a solution u ∈ H2,p of the system
δ (cid:93)
−∆uν +λ uν = Gν(|∇u |2)+Go(|∇u |2)·∇u +f (x), (13)
δ ν δ δ δ δ δ δ ν
where p ∈ [1,∞).
We will establish a uniform bound for |∇u |2exp(|∇u |2r) in L1, r large enough, as
δ δ
δ → 0, and we will estimate related quantities.
A simple estimate shows, that
(cid:12) ∂ (cid:12) (cid:12) ∂ (cid:12)
(cid:12) Gν(cid:12) ≤ K , (cid:12) Go(cid:12) ≤ K
(cid:12)∂η δ(cid:12) δ (cid:12)∂η δ(cid:12) δ
and, uniformly as δ → 0,
(cid:12) ∂ (cid:12) (cid:12) ∂ (cid:12)
(cid:12) Gν(η)(cid:12)+(cid:12) Go(η)(cid:12) ≤ K|η|q−1 +K.
(cid:12)∂η δ (cid:12) (cid:12)∂η δ (cid:12)
We now differentiate equation (13), i. e. apply D, and use the function
Duν exp(|∇u |2r)
δ δ
asatestfunctionwherer ≥ 1willbechosenlater. HereD standsforthefirstpartial
derivatives D , i = 1,...,n. Note that we have sufficient regularity for justifying
i
these operations since ∇u ∈ L∞∩H1,p due to regularity and imbedding theorems.
δ
4
We obtain, dropping the index δ of the function during the estimates
(cid:90) (cid:90)
1
|∇Duν|2exp(|∇u|2r)dx+ ∇|Duν|2∇exp(|∇u|2r)dx+
2
Q Q
(cid:90)
+λ |Duν|2exp(|∇u|2r)dx =
ν
Q
(14)
(cid:90)
(cid:0) (cid:1)(cid:8) (cid:9)
= D |∇u|2 Gν(cid:48)(|∇u|2)+Go(cid:48)(|∇u|2)·∇uν Duν exp(|∇u|2)dx+
δ δ
Q
(cid:90) (cid:90)
1
+ Go(|∇u|2)∇(Duν)2exp(|∇u|2)dx+ Df Duν exp(|∇u|2r)dx.
2 δ ν
Q Q
We sum with respect to ν = 1,...,M and i = 1,...,n, D = D , and use vector
i
notation and perform simple estimates. This yields, with λ = minλ ,
0 ν
(cid:90) (cid:90)
|∇2u|2exp(|∇u|2r)dx+r (cid:12)(cid:12)∇(cid:0)|∇u|2(cid:1)(cid:12)(cid:12)2|∇u|2r−2exp(|∇u|2r)dx+
Q Q
(cid:90)
+λ |∇u|2exp(|∇u|2r)dx ≤
0
Q
(cid:90)
(cid:12) (cid:0) (cid:1)(cid:12)(cid:8) (cid:9)
≤ (cid:12)∇ |∇u|2 (cid:12) K(|∇u|2q−2)+K (|∇u|+|∇u|2)exp(|∇u|2r)dx+
Q
(cid:90) (cid:90)
(cid:8) (cid:9)(cid:12) (cid:0) (cid:1)(cid:12)
+ K|∇u|2q +K (cid:12)∇ |∇u|2 (cid:12)exp(|∇u|2r)dx+K |∇u|exp(|∇u|2r)dx ≤
Q Q
≤ A+B +C,
(15)
where
(cid:90)
A = K (cid:12)(cid:12)∇(cid:0)|∇u|2(cid:1)(cid:12)(cid:12)2|∇u|2qexp(|∇u|2r)dx,
Q
(cid:90)
(cid:12) (cid:0) (cid:1)(cid:12)
B = K (cid:12)∇ |∇u|2 (cid:12)exp(|∇u|2r)dx,
Q
(cid:90)
C = K |∇u|exp(|∇u|2r)dx,
Q
uniformly as δ → 0, and K does not depend on r.
5
Note that we used
(cid:12)(cid:88)n (cid:88)N (cid:12) (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) Go∇(D uν)2exp(|∇u|2)(cid:12) = (cid:12)Go∇ |∇u|2 (cid:12)exp(|∇u|2) ≤
(cid:12) δ i (cid:12) δ
i=1 ν=1
(cid:12) (cid:0) (cid:1)(cid:12)
≤ (K|∇u|2q +K)(cid:12)∇ |∇u|2 (cid:12)exp(|∇u|2).
(cid:0) (cid:1)
We estimate exp = exp(|∇u|2r) :
(cid:90)
A ≤ r (cid:12)(cid:12)∇(cid:0)|∇u|2(cid:1)(cid:12)(cid:12)2|∇u|2r−2exp dx+
2
(|∇u|≥1)
(cid:90)
K
+ |∇u|4q+4−2r+2exp dx+
2r
(|∇u|≥1)
(cid:90) (cid:90)
+ε |∇2u|2exp dx+K exp dx =
0 ε0
|∇u|≤1 |∇u|≤1
= A +A +A +A .
1 2 3 4
If we choose r ≥ 2q +3 we have that
(cid:90)
K
A ≤ exp(|∇u|2r)dx
2
2r
|∇u|≥1
The term A is absorbed by a corresponding one on the left hand side of (15). The
1
term A (the one with ε ) is absorbed by the left hand side, too. The term A is
3 0 4
obviously bounded. Thus we arrive at
(cid:90) (cid:90)
(1−ε ) |∇2u|2exp dx+ r (cid:12)(cid:12)∇(cid:0)|∇u|2(cid:1)(cid:12)(cid:12)2|∇u|2r−2exp dx+
0
2
Q Q
(16)
(cid:90) (cid:90)
K
+λ |∇u|2exp dx ≤ exp dx+C +K.
0
2r
Q |∇u|≥1
The first summand on the left hand side of (16) can be absorbed by the term with
factor λ if r is chosen larger than 2λ /K since K does not depend on r here. The
0 0
6
term C is estimated by
(cid:90) (cid:90)
(cid:16) (cid:17)
C = K + |∇u|exp dx ≤
|∇u|≤Λ |∇u|≥Λ
(cid:90) (cid:90)
1
≤ K +K |∇u|exp dx ≤ K + λ |∇u|2exp dx
Λ Λ 0
2
|∇u|≥Λ Q
if we choose Λ = 2K/λ .
0
(cid:82)
Again, the term 1λ |∇u|2exp dx is absorbed by a corresponding term at the
2 0
left hand side and we arrive at the inequality
(cid:90)
r (cid:12)(cid:12)∇(cid:0)|∇u|2(cid:1)(cid:12)(cid:12)2|∇u|2r−2exp(|∇u|2r)dx+
2
Q
(cid:90) (cid:90)
1 1
+ |∇2u|2exp(|∇u|2r)dx+ λ |∇u|2exp(|∇u|2r)dx ≤
0
2 4
Q
≤ K.
This holds uniformly for u = u , δ → 0. We need only the summand with factor λ
δ 0
and fix
Proposition 3.1 The approximate solutions
u ∈ H2,p(Q,RN)
δ (cid:93)
of equation (13) obey the uniform bound
(cid:90)
|∇u |2exp(|∇u |2r)dx ≤ K
δ δ
Q
uniformly for δ → 0.
Firstly, integrating the equation over the periodicity cube, one sees that propo-
sition 3.1 yields that the mean values of the uν are bounded. Then one obtains
bounds for (cid:107)u (cid:107) .
δ p
From linear elliptic regularity theory we then obtain
(cid:107)u (cid:107) +(cid:107)∇u (cid:107) +(cid:107)∇2u (cid:107) ≤ K
δ p δ p δ p p
7
uniformly for δ → 0, for any fixed p < ∞. This allows us to subtract a subsequence
(u ) where Γ is a sequence of positive numbers tending to zero such that
δ δ∈Γ
∇2u → ∇2u, weakly
δ
u → u,∇u → ∇u strongly
δ δ
in all Lp, p < ∞.
Thus we may pass to the limit δ → 0 in (13) and obtain that the weak limit
u ∈ H2,p(Q;RN)
(cid:93)
satisfies the primal equation (2). Further regularity −u ∈ C2+α follows from (4)–(6)
and elliptic regularity theory. This proves Theorem 2.1. (cid:122)
Let us indicate the proof of Theorem 2.2 concerning the parabolic system.. With
the new unknown variable
v = e−tu
the function v satisfies the system
(cid:0) (cid:1) (cid:0) (cid:1)
vν −∆vν +vν = e−tGν e2t|∇v|2 +Go e2t|∇v|2 ·∇vν +e−tf (t,x).
t ν
We replace Gν, Go by Gν, Go as in the proof of Theorem 2.1 and obtain a regular
δ δ
solution v = v . Then one applies the operation D = D , j = 1,...,n, i. e. the
δ j
derivatives in space direction, and uses the test function
(cid:0) (cid:1)
Dvν exp |∇v|p .
Then the calculations run as in the proof of Theorem 2.1; we only have to handle
with the factors e−t, e2t, which is simple since only space derivatives are involved in
the calculation.
This finally gives an estimate
T
(cid:90) (cid:90)
(cid:0) (cid:1)
|∇v|2exp |∇v|2p dxdt ≤ K
T
0 Q
uniformly as δ → 0 and the Lr-estimate for ∇u follows. Integrating the equation
with respect to x, we derive a differential equation for the mean value of u and hence
a bound. Thereafter, from the theory of linear parabolic equations we obtain the
8