L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY
PROBLEMS IS INDEPENDENT OF p
RALPH CHILL AND SACHI SRIVASTAVA
A. If the second order problem ü + Bu̇ + Au = f has L p maximal
regularity for some p ∈ (1, ∞), then it has L p -maximal regularity for
every p ∈ (1, ∞).
1. I
The notion of L p maximal regularity for the abstract linear second order
problem
(1)
ü + Bu̇ + Au = f on [0, T ],
u(0) = u̇(0) = 0,
was first introduced and studied in [4]. Here A and B are two closed linear
operators on a Banach space X, with dense domains DA and DB , respectively. We say that the problem (1) has L p -maximal regularity, if for every
f ∈ L p (0, T ; X) there exists a unique (strong) solution
n
o
u ∈ MR p,T := v ∈ W 2,p (0, T ; X) ∩ L p (0, T ; DA ) : v̇ ∈ L p (0, T ; DB )
of the inhomogeneous problem (1). Strong solution means that u(0) =
u̇(0) = 0 and the differential equation (1) is satisfied almost everywhere.
The space MR p,T is called maximal regularity space.
The definition of L p maximal regularity is similar to that of L p maximal
regularity for the first order problem u̇ + Au = f , and is closely related to
the abstract notion of maximal regularity studied by Da Prato and Grisvard
[5].
In this note, which may be considered as a sequel to [4], we show that
p
L maximal regularity for the second order problem is independent of the
choice of p, 1 < p < ∞. For the examples studied in [4], L p maximal regularity was known to hold for all p ∈ (1, ∞), but at the end of this note
we describe another example for which only L2 maximal regularity was
known. The fact that L p maximal regularity is independent of p is well
known for the first order problem; see for example, Sobolevskii [8], Cannarsa and Vespri [3], Hieber [7].
The second named author thanks the Paul Verlaine University of Metz for local hospitality during her visit to Metz when this work was started.
1
2
RALPH CHILL AND SACHI SRIVASTAVA
2. I  
We first prove that L p -maximal regularity of (1) implies existence and
uniqueness of strong solutions of the initial value Cauchy problem
(2)
ü + Bu̇ + Au = 0 on [0, T ],
u(0) = u0 , u̇(0) = u1 ,
if the pair of initial values belongs to the trace space T r p associated with X,
A and B, defined as
n
o
T r p := (u(0), u̇(0)) : u ∈ MR p,1 .
Subsequently, two consequences of this result are recalled.
The existence and uniqueness theorem was proven in [4] under the additional assumption DA ,→ DB . We present a proof which does not require
this assumption.
Theorem 2.1. (Initial value problem) Let p ∈ (1, ∞). Suppose that (1) has
L p maximal regularity for some T > 0. Then, for every (u0 , u1 ) ∈ T r p and
every T > 0, there exists a unique solution u ∈ MR p,T of the initial value
problem (2).
Proof. Suppose that (1) has L p maximal regularity for T > 0. Let (u0 , u1 ) ∈
T r p be given. By [4, Lemma 6.3 (i), (ii)], for every t ∈ [0, T ],
o
n
(u(t), u̇(t)) : u ∈ MR p,T = T r p .
Hence, there exists v ∈ MR p,T such that u0 = v(0) while u1 = v̇(0). Let
f := v̈ + Bv̇ + Av ∈ L p (0, T ; X). By L p maximal regularity, there exists a
solution w ∈ MR p,T of (1) with f as chosen above. Then w(0) = 0 = ẇ(0).
Let u := v − w ∈ MR p,T . Clearly u is then a solution of (2) with initial values
u(0) = u0 and u̇(0) = u1 . Uniqueness of this solution on [0, T ] follows from
linearity and unique solvability of the problem (1).
This solution of (2) can be extended to a solution on [0, 2T ] . Indeed, by
the same argument as above, for (u(T ), u̇(T )) ∈ T r p , there exists a unique
solution z ∈ MR p,T of the initial value problem (2) satisfying z(0) = u(T ),
ż(0) = u̇(T ). Then setting ũ(t) = u(t) if 0 ≤ t ≤ T and equal to z(t − T ) if
T ≤ t ≤ 2T, we obtain a solution of (2) in MR p,2T . Iterating this argument
we see that the solution u can be extended or restricted to a solution in
MR p,T 0 for any T 0 > 0.
In order to show uniqueness on [0, T 0 ] for any T 0 > 0, let u and v be two
solutions of (2) on [0, T 0 ]. Extending both solutions, if necessary, we can
assume that T 0 = kT for some k ∈ N. Then it suffices to note that u = v on
[0, T ] by uniqueness on the intervall [0, T ], and iterate this argument.
The same arguments as in the proof of [4, Corollary 2.4] can now be used
to show that if the problem (1) has L p maximal regularity for some T > 0
then it has L p maximal regularity for all T > 0. This strengthens the earlier
result as the assumption that DA ,→ DB is no longer required. We record
this statement formally as
L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS
3
Corollary 2.2 (Independence of T > 0). Let p ∈ (1, ∞). Suppose that the
problem (1) has L p maximal regularity on [0, T ] for some 0 < T < ∞. Then
the problem (1) has L p maximal regularity on [0, T ] for every T , 0 < T < ∞.
Recall from [4, Lemma 6.1] that the trace space T r p is the product V0 ×V1
of two Banach spaces which continuously embed into X. Hence, Theorem
2.1 implies that for every x ∈ V1 and every T > 0 the initial value problem
(3)
ü + Bu̇ + Au = 0 on [0, T ],
u(0) = 0, u̇(0) = x,
admits a unique solution u ∈ MR p,T . Setting S (t)x := u(t), where u is
this solution of the preceeding problem, we thus obtain a solution family
(S (t))t≥0 of operators in L(V1 , V0 ) which plays the role of a sine family. In
a similar way one could define the cosine family associated with (2), but
unlike in [1] (where B = 0), the sine family is in general not the primitive
of the cosine family.
As in [4, Proposition 2.2], one can show that the S (t) extend to operators
in L(X), and that for every f ∈ L p (0, T ; X) the convolution S ∗ f is the
unique solution of the inhomogeneous problem (1). Infact, the following is
true.
Corollary 2.3 (Sine family and inhomogeneous problem). Suppose that (1)
has L p maximal regularity and define the sine family (S (t)) as above. Then
S ∈ C (R+ ; L(X)) ∩ C ∞ ((0, ∞); L (X; DA ∩ DB ))
and for every f ∈ L p (0, T ; X) the solution u of the inhomogeneous problem
(1) is given by
Z
u(t) = (S ∗ f )(t) :=
S (t − s) f (s)ds.
R+
3. R  
The following result exhibits the regularity of the solutions of the initial
value problem (2) by invoking an idea used in the proof of [4, Proposition
2.2].
Theorem 3.1. Let p ∈ (1, ∞) and assume that (1) has L p maximal regularity. Let (u0 , u1 ) ∈ T r p , T > 0, and let u ∈ Mr p,T be the unique solution of the
initial value problem (2). Then u ∈ C ∞ ((0, T ); DA ) and u̇ ∈ C ∞ ((0, T ); DB ).
Moreover, if for k ∈ N one defines
uk (t) := tk u(k) (t),
t ∈ [0, T ],
then uk ∈ MR p,T .
Remark 3.2. The proof below will actually show that the solution extends
to an analytic function in a sector around the positive real axis.
4
RALPH CHILL AND SACHI SRIVASTAVA
Proof. By Theorem 2.1 we know that the solution extends to a solution on
R+ . We consider the operator
G : (−1, 1) × MR p,T → L p (0, T ; X) × T r p ,
(λ, v) 7→ (v̈ + (1 + λ)Bv̇ + (1 + λ)2 Av, (v(0) − u0 , v̇(0) − u1 )).
The operator G is clearly analytic (see [9] for the definition of an analytic
function between two Banach spaces). For λ ∈ (−1, 1) we put uλ (t) :=
u(t + λt). Then uλ ∈ MR p,T , u0 = u, and G(λ, uλ ) = 0. Moreover, the partial
derivative
∂G
(0, u) : MR p,T → L p (0, T ; X) × T r p ,
∂v
v 7→ (v̈ + Bv̇ + Av, v(0), v̇(0))
is boundedly invertible by L p -maximal regularity and Theorem 2.1. Hence,
by the implicit function theorem, [9, Theorem 4.B, p. 150], there exists
ε > 0, a neighbourhood U of u in MR p,T , and an analytic function g :
(−ε, ε) → U such that
{(λ, v) ∈ (−ε, ε) × U : G(λ, v) = 0} = {(λ, g(λ)) : λ ∈ (−ε, ε)}.
From this we obtain g(λ) = uλ , that is, the function λ → uλ is analytic in
dk λ
(−ε, ε). In particular, the derivatives dλ
k u |λ=0 exist in MR p,T for every k ∈
dk λ
N. One easily checks that dλk u |λ=0 coincides with the function uk defined
in the statement, so that one part of the claim is proved. The regularity of u
and u̇ is an easy consequence of this part.
Our next Lemma is of a technical nature. We define, for 0 < T < ∞,
k ∈ N0 , and 1 < p < ∞, the spaces Dk,p (0, T ; X) as
n
k,p
Dk,p (0, T ; X) := f ∈ Wloc
((0, T ); X) : f j ∈ L p (0, T ; X) for every 0 ≤ j ≤ k,
o
where f j (t) = t j f ( j) (t) .
Equipped with the norm k · kDk,p given by
k f kDk,p :=
k
X
k f j kL p
j=0
these spaces are Banach spaces.
Lemma 3.3. Suppose the problem (1) has L p maximal regularity for some
p ∈ (1, ∞). Then for every k ∈ N0 the map ψk,T : Dk,p (0, 2T ; X) → MR p,T
given by (ψk,T f )(t) = tk u(k) (t), where u is the unique solution of (1) corresponding to f , is bounded. Moreover, there is a constant Ck > 0 such that
kψk,T k ≤ Ck for all 0 < T ≤ 1.
Proof. In the following, we will write, for convenience ψk,T = ψ, and
only specify the indices when there is a chance of confusion. Let f ∈
L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS
5
Dk,p (0, 2T ; X) and let u denote the unique solution of (1) for this f . Let
MR0p,T be the subspace of MR p,T given by
n
o
MR0p,T = v ∈ MR p,T : v(0) = v̇(0) = 0 .
By considering the C k map
G : (−1, 1) × MR0p,T → L p (0, T ; X)
G(λ, v)(t) = v̈(t) + (1 + λ)Bv̇(t) + (1 + λ)2 Av(t) − (1 + λ)2 f ((1 + λ)t)
and following the same strategy as in the proof of Theorem 3.1 we get an
ε > 0, such that the function λ → uλ is C k from (−ε, ε) into MR p,T ; uλ (t) =
dj λ
u(t + λt). In particular, the derivatives dλ
j u |λ=0 exist in MR p,T for every
dj λ
j ( j)
j ∈ {0, 1, . . . , k}. But dλ j u |λ=0 (t) = t u (t). Therefore ψ maps Dk,p (0, 2T ; X)
into MR p,T . Recall here that the operator that maps f ∈ L p (0, T ; X) to the
unique solution u ∈ MR p,T of problem (1) is bounded. It is straightforward
to check then that ψ is closed. By the closed graph theorem it follows that
ψ is bounded.
Let E be a bounded operator that maps any f ∈ Dk,p (0, 2; X) to an extension E f ∈ Dk,p (0, 4; X) in such a way that (E f )(t) = 0 for t ∈ (3, 4).
For α > 0 and τ > 0, let Dα : Dk,p (0, τ; X) → Dk,p (0, ατ ; X) be the dilation
given by (Dα f )(t) = f (αt). Then, for every τ > 0,
(4)
kDα f kDk,p (0, ατ ;X) =
1
k f kDk,p (0,τ;X) .
α
Let 0 < T ≤ 1. Then for any f ∈ Dk,p (0, 2T ; X) and t ∈ [0, 2] set
(
(D T1 ◦ E ◦ DT f )(t) for t ∈ [0, 4T ] ∩ [0, 2],
(ET f )(t) :=
0
otherwise.
Then ET is bounded extension operator from Dk,p (0, 2T ; X) into Dk,p (0, 2; X),
and we have, on using (4),
kET f kDk,p (0,2;X) ≤ kD T1 ◦ E ◦ DT f kDk,p (0,4T ;X)
= T kE ◦ DT f kDk,p (0,4;X)
≤ T kEk kDT f kDk,p (0,2;X)
= kEk k f kDk,p (0,2T ;X) .
Hence, for all T ∈ (0, 1],
ψ f k,T
MR p,T
≤ ψk,1 (ET f ) MR
p,1
≤ kψk,1 k kET f kDk,p (0,2;X)
≤ kψk,1 k kEk k f kDk,p (0,2T ;X) .
Therefore ψk,T ≤ Ck := kψk,1 k kEk for all T , 0 < T ≤ 1.
6
RALPH CHILL AND SACHI SRIVASTAVA
4. p 
In this section we will establish that if the problem (1) has L p maximal
regularity for some p ∈ (1, ∞), then it has L p maximal regularity for all
p ∈ (1, ∞). The scheme followed will be similar to that in [7], where the
corresponding result for the first order problem is proven. We will make use
of the following theorem, which is a vector valued version of a result due to
Benedek, Calderón and Panzone [2].
Theorem 4.1 ([7], Theorem 4.3). Suppose that T is a bounded operator on
L p (R; X) for some p ∈ (1, ∞) and is represented by
Z
(5)
T f (t) :=
K(t − s) f (s)ds
R
∞
for f ∈ L (R; X) with compact support and t < supp f, and the kernel
1
K ∈ Lloc
(R \ {0}, X) satisfies
Z
kK(t − s) − K(t)kdt ≤ C for every s , 0 and some c > 1.
(6)
|t|>c|s|
Then T admits a bounded extension to Lr (R, X) for every r ∈ (1, ∞), that is,
there exists a constant C̃r such that
k T f kLr (R,X) ≤ C̃r k f kLr (R,X) .
We now state our main result concerning L p maximal regularity for problem (1) and the choice of p ∈ (1, ∞).
Theorem 4.2. Suppose that for some p ∈ (1, ∞) the problem (1) has L p
maximal regularity. Then (1) has L p maximal regularity for every p ∈
(1, ∞).
Proof. : Fix p ∈ (1, ∞). Suppose that problem (1) has L p maximal regularity
and let (S (t)) be the sine family from Corollary 2.3. By L p maximal regularity and Corollary 2.3, the convolution operator f 7→ S ∗ f is a bounded
linear operator from L p (0, T ; X) into MR p,T .
It is enough for our purposes to show that the convolution operator f 7→
S ∗ f extends to a bounded linear operator from Lr to MRr for every r ∈
(1, ∞).
Fix x ∈ X and 1 ≥ T > 0. Set f (s) = x, s ≥ 0. Then f ∈ Dk,p (0, 2T, X)
for all k ∈ N0 . Therefore, applying Lemma 3.3 to f and the corresponding
unique solution u := S ∗ f of the problem (1), with k = 2 we have that
t 7→ t2 u(2) (t) ∈ MR p,T and
kψ2,T f k MR p,T = kt2 u(2) (t)k MR p,T
≤ C2 k f kD2,p (0,2T,X)
1
= C2 (2T ) p kxk,
where C2 is Rthe constant independent
R t of T obtained in Lemma 3.3. Noting
t
that u(t) = 0 S (t − s) f (s)ds = 0 S (s)xds, we therefore obtain from the
L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS
7
above inequality
(7)
1
kt2 AṠ (t)xkL p (0,T ;X) ≤ C2 (2T ) p kxk.
Define the operator-valued kernel K as



AS (t) if t ∈ (0, T ),
K(t) = 

0
otherwise.
Due to the LRp maximal regularity of (1), the convolution operator T A given
by T A f (t) = R K(t−s) f (s) extends to a bounded linear operator on L p (0, T ; X).
+
Thus there is a constant C̃ p such that for all f ∈ L p (R, X),
k T f kL p (R,X) ≤ C̃ p k f kL p (R,X) .
We claim that the kernel K further satisfies condition (6) for some c > 1.
Indeed, for T ≥ t > s > 0, and x ∈ X, we have, on using (7) and Hölder’s
inequality
kK(t − s)x − K(t)xk = kAS (t − s)x − AS (t)xk
Z t
=
AṠ (r)x dr
t−s
Z t
−2 2
r r AṠ (r)x dr
≤
t−s
1
! 1p
!q Z t
Z t
kr2 AṠ (r)xk p dr
≤
r−2q dr
0
t−s
Z
! 1q
t
r−2q dr
≤
1
C2 (2t) p kxk.
t−s
Therefore, for c > 1, and s > 0 we have
Z
Z
1
1 kK(t − s) − K(t)kdt ≤ C
(2t) p t(1−2q) − (t − s)(1−2q) q dt
t>cs
Zt>cs
1
1
1
≤C
(2t) p t( q −2) 1 − (1 − st−1 )(1−2q) q dt
Zt>cs
1 1
≤ C̃
t p s q t−2 dt
t>cs
Z
1
0 1q
≤C s
t( p −2) dt
T >t>cs
≤ M,
where C̃, C 0 , and M are constants depending only on C̃ p , q, c and T.
Therefore, from Theorem 4.1 it follows that T A is a bounded operator
from Lr (0, T ; X) to Lr (0, T ; X), for every r ∈ (1, ∞). Thus, for each r ∈
(1, ∞) there exists a constant C̃r such that for all f ∈ Lr (0, T ; X),
(8)
kAS ∗ f kLr (0,T ;X) = kT A f kLr (0,T ;X) ≤ C̃r k f kLr (0,T ;X) .
8
RALPH CHILL AND SACHI SRIVASTAVA
Now define another operator valued kernel K1 as follows.



 BṠ (t) if t ∈ (0, T ),
K1 (t) = 
0

otherwise.
Let T B be the operator given by T B f (t) := K1 ∗ f. Due to L p maximal
regularity, T B is a bounded operator on L p (0, T ; X). Applying Lemma 3.3
to the constant function f (t) = x, t ∈ (0, T ), successively for k = 0, 1 and 2
we obtain for u = S ∗ f,
kt2 BS̈ (t)xkL p (0,T ;X) ≤ kt2 Ṡ (t)xk MR p,T + 2ktS (t)xk MR p,T + 2k(S ∗ f )(t)xk MR p,T
≤ C2 k f kD2,p + 2C1 k f kD1,p + 2C0 k f kD0,p
1
= C(2T ) p kxk,
where the constant C is independent of T for 0 < T ≤ 1. Using this inequality and the same arguments as before, we can show that the kernel K1 also
satisfies (6) above. Thus, from Theorem 4.1 it follows that f 7→ K1 ∗ f is
a bounded operator on Lr (0, T ; X) for every r ∈ (1, ∞). Therefore, we have
for every f ∈ Lr (0, T ; X),
(9)
kBṠ ∗ f kLr (0,T ;X) = kT B f kLr (0,T ;X) ≤ Cr1 k f kLr (0,T ;X) .
Since u := S ∗ f is a solution of the equation ü(t) + Bu̇(t) + Au(t) = f (t)
a.e , it follows that ü ∈ Lr (0, T ; X) and there exists a constant Cr0 such that
kukW 2,r ≤ Cr0 k f k. This statement, together with (8) and (9) implies that the
problem (1) has Lr maximal regularity for every r ∈ (1, ∞).
There are, as of now, only a few results that ensure that the problem (1)
has L p maximal regularity for some p ∈ (1, ∞). The main ingredient for
showing L p maximal regularity in [4] is a characterization of L p maximal
regularity in terms of Fourier multipliers. In a particular model problem
L p maximal regularity was shown by using the Mikhlin-Weis Fourier multiplier theorem [4, Theorem 4.1]. This then implied L p maximal regularity
for every p ∈ (1, ∞), so that in the examples considered in [4], L p maximal
regularity is independent of p ∈ (1, ∞).
However, consider the following variational setting, not covered by the
results in [4]. Let V and H be two separable Hilbert spaces such that V
embeds densely and continuously into H. We identify H with its dual H 0 so
that H is also densely and continuously embedded into V 0 .
Corollary 4.3. Let A and B be two linear, maximal monotone, symmetric,
not necessarily commuting operators from V to V 0 . Then for every p ∈
(1, ∞) and every f ∈ L p (0, T ; V 0 ), every u0 ∈ V and every u1 ∈ (V 0 , V) 10 ,p
p
there exists a unique solution
u ∈ W 2,p (0, T ; V 0 ) ∩ W 1,p (0, T ; V)
of the problem
(10)
ü + Bu̇ + Au = f on [0, T ],
u(0) = u0 , u̇(0) = u1 .
L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS
9
In other words, the above problem has L p maximal regularity for every
p ∈ (1, ∞).
Proof. By a result of J.-L. Lions [6, Théorème 1, p.670] the problem (10)
has L2 maximal regularity in V 0 . By Theorem 4.2, the problem (10) has
L p maximal regularity for every p ∈ (1, ∞). Solvability of the initial value
problem follows from Theorem 2.1. The fact that the associated trace space
equals V × (V 0 , V) 10 ,p follows from [4, Lemma 6.2].
p
The fact that in the variational setting above the problem (1) has L2 maximal regularity was proved in [6] by the Faedo-Galerkin method and a priori
estimates. The proof thus heavily depends on the Hilbert space setting and
it does not imply L p maximal regularity for p different from 2. We point out
that the conditions on A and B can be considerably relaxed; for the precise
assumptions, see [6].
R
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Math. Z. 251 (2005), 751–781.
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1999, pp. 363–380.
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Akad. Nauk SSSR 157 (1964), 52–55.
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New York, Berlin, Heidelberg, 1990.
U́ P V - M, L  M́  A 
M - CNRS, UMR 7122, B̂. A, I  S, 57045 M C 1, F
E-mail address: chill@univ-metz.fr
D  M, L S R C, L N, N D - 24,
I
Current address: Department of Mathematics, Indian Institute of Science, Bangalore,
India
E-mail address: sachi srivastava@math.iisc.ernet.in