Rend. Mat. Acc. Lincei
s. 9, v. 16:125-132 (2005)
Equazioni a derivate parziali. Ð Characterization of the interior regularity for parabolic
systems with discontinuous coefficients. Nota di DIAN K. PALAGACHEV e LUBOMIRA G.
SOFTOVA, presentata (*) dal Socio G. Da Prato.
ABSTRACT. Ð We deal in this Note with linear parabolic (in sense of Petrovskij) systems of order 2b with
discontinuous principal coefficients belonging to VMO \ L1 : By means of a priori estimates in Sobolev-Morrey
spaces we give a precise characterization of the Morrey, BMO and HoÈlder regularity of the solutions and their
derivatives up to order 2b 1:
KEY WORDS: Parabolic systems; A priori estimates; Morrey spaces; HoÈlder regularity; VMO.
RIASSUNTO. Ð Caratterizzazione della regolaritaÁ all'interno per sistemi parabolici con coefficienti discontinui.
In questa Nota si studiano dei sistemi lineari parabolici (in senso di Petrovskij) di ordine 2b con coefficienti
principali appartenenti a VMO \ L1 : Per mezzo di stime a priori negli spazi di Sobolev-Morrey si propone una
caratterizzazione precisa della regolaritaÁ (Morrey, BMO e HoÈlder) all'interno delle soluzioni e le loro derivate
fino all'ordine 2b 1:
1. INTRODUCTION
A celebrated 1958's paper by J. Nash asserts HoÈlder continuity of weak solutions to
second-order, divergence form linear parabolic equations with measurable and essentially
bounded coefficients, providing this way for a parabolic counterpart to the famous elliptic
result of E. De Giorgi. By employing completely different technique, N.V. Krylov and
M.V. Safonov proved in the beginning of 1980's HoÈlder continuity of the strong solutions
to second-order, non-divergence form equations with L1 coefficients. Their results opened
the way for the current blossom of the theory of fully nonlinear equations with all its
applications in differential geometry, stochastic control, nonlinear optimization, etc. On
the other hand, it is well known that this kind of results cannot hold, in general, neither for
systems nor for single equations of order greater than 2 if the principal coefficients are
merely bounded. A notable exception is given by operators for which the eigenvalues of
the principal part do not scatter too much ± a condition originally introduced by
H.O. Cordes (see [12, and the references therein]).
It turns out that appropriate regularity of the principal coefficients not only
guarantees HoÈlder's continuity of strong solutions but even permits to develop a
relevant theory of elliptic/parabolic systems in Sobolev classes built on Lp for p > 1:
The background for these results is ensured by the possibility to estimate a priori the
Lp -norms of solution's highest order derivatives (say D2b u) by means of the data. The
method, associated to the names of A. CalderoÂn and A. Zygmund, uses explicit
representation formulas for D2b u in terms of singular integral acting on the known
right-hand side plus another one acting on the very same derivatives D2b u: Fortunately,
(*) Nella seduta del 14 gennaio 2005.
126
D.K. PALAGACHEV - L.G. SOFTOVA
these derivatives appear in a singular commutator the norm of which can be made
small if the coefficients have small oscillation over small balls. This way, continuity of
the principal coefficients is a sufficient condition ensuring boundedness of the singular
integral operators and therefore validity of the Lp a priori estimate. The desired
regularity of u then follows by known embeddings between Sobolev and HoÈlder spaces
for suitable values of p: We refer the reader to the seminal works [1, 2, 6] for what
concerns elliptic systems, and to [7, 9, 15] in the case of parabolic operators. A relevant
Lp -theory of uniformly elliptic systems with discontinuous principal coefficients was
developed in [4]. The discontinuity is expressed in terms of appartenance to the class
of functions with vanishing mean oscillation VMO which contains as a proper subset
the space of uniformly continuous functions.
This Note deals with linear systems of order 2b which are parabolic in sense of Petrovskij.
Our aim is to extend the aforementioned results to such systems with discontinuous
principal coefficients akj
a (x; t): We deal with strong solutions belonging to Sobolev2b;1
(Q) where Q is a cylindrical domain in Rn R‡ : It is proved that
Morrey's class Wp;l;loc
1
is a sufficient condition ensuring local HoÈlder regularity of such
akj
a (x; t) 2 VMO \ L
solutions and all their spatial derivatives up to order 2b 1 for appropriate values of p and l:
Our approach makes use of the CalderoÂn-Zygmund method of expressing highest order
solution's derivatives in terms of Gaussian-type potentials. These turn out to be singular
integrals with kernels of mixed homogeneity of degree n 2b (strongly defined by the
system itself) and their commutators with the multiplication by the VMO functions akj
a
which have small integral oscillation over small cylinders. Employing results on
boundedness of these singular integral operators in Lp;l (recently proven by the authors
in [13]), we derive a priori bounds of Caccioppoli type which yield estimates for the strong
p;l
2b;1
(Q) by means of the Lloc (Q) norm of the right-hand side plus a weaker
solutions in Wp;l;loc
norm of the solution itself. By virtue of embedding properties of the Sobolev-Morrey spaces
into HoÈlder ones, these a priori bounds lead to a complete characterization of the Morrey,
BMO and HoÈlder regularity of the solution and its spatial derivatives up to order 2b 1:
2. MAIN RESULTS
Let V Rn ; n 2; be a domain and define Q ˆ V (0; T) with T > 0: We consider
the linear system
X
1†
Aa (x; t)Da u(x; t) ˆ f (x; t)
Lu :ˆ Dt u(x; t)
jajˆ2b
for the vector-valued function u: Q ! Rm given by the transpose u(x; t) ˆ
T
ˆ u1 (x;t); . . . ; um (x; t) ; f ˆ (f1 ; . . . ; fm )T ; where Aa (x; t) is the m m matrix
m
jk
aa (x; t) j;kˆ1 of the measurable coefficients ajk
a : Q ! R: Hereafter b 1 is a fixed
a
and D Dax :ˆ Da11 . . . Dann with Di :ˆ @=@xi :
integer, a ˆ (a1 ; . . . ; an ); Dt :ˆ @=@t
T
and Ds u substitutes any derivative Da u with
Later on, Da u :ˆ Da u1 ; . . . ; Da um
jaj ˆ s 2 N:
127
CHARACTERIZATION OF THE INTERIOR REGULARITY FOR PARABOLIC ...
Suppose the system (1) is uniformly parabolic in sense of Petrovskij (see [7, 9, 15]). That is,
for a.a. (x; t) 2 Q and any j 2 Rn set ps (x; t; j); s ˆ 1; . . . ; m; for the eigenvalues of the
P
m m matrix ( 1)b jajˆ2b Aa (x; t)j a where j a :ˆ j a11 j a22 j ann : Then parabolicity of (1)
means
2†
9 d > 0:
Re ps (x; t; j) djjj2b
a:a: (x; t) 2 Q; 8j 2 Rn ; 8s ˆ 1; . . . ; m:
Endow Rn‡1 ˆ Rnx Rt with the parabolic metric r(x; t) ˆ maxfjxj; jtj1=2b g and
consider the collection of parabolic cylinders
3†
Cr (x0 ; t0 ) :ˆ Br (x0 ) (t0
Br (x0 ) :ˆ fx 2 Rn : jx
r 2b ; t0 );
x0 j < r g
each having Lebesgue measure jCr j comparable to r n‡2b :
DEFINITION 2.1. Set Qr :ˆ Cr (x0 ; t0 ) \ Q and suppose there is a positive constant A
such that jQr j Ar n‡2b whenever (x0 ; t0 ) 2 Q (for example, this is clearly verified if V
has the interior cone property). Let p 2 (1; 1) and l 2 (0; n ‡ 2b): The function
u 2 Lp (Q) belongs to the parabolic Morrey space Lp;l (Q) if
0
11=p
Z
1
B
C
ju(x; t)jp dxdtA < 1:
kukp;l;Q ˆ @sup sup l
r
r>0 (x0 ;t0 )2Q
Qr
2b;1
The Sobolev-Morrey space Wp;l
(Q) consists of all functions u: Q ! R belonging to the
2b;1
Sobolev space Wp (Q) with derivatives Dt u; Dax u; jaj 2b; lying in Lp;l (Q): The norm in
P P
2b;1
a
(Q) is given by kukW 2b;1 (Q) :ˆ kDt ukp;l;Q ‡ 2b
Wp;l
sˆ0
jajˆs kDx ukp;l;Q :
p;l
For the sake of brevity, we denote the cross-product of m copies of Lp;l (Q) by the
same symbol. Thus, u ˆ (u1 ; . . . ; um )T 2 Lp (Q) means uk 2 Lp (Q) for all k ˆ 1; . . . ; m;
P
2b;1
2b;1
0
and kukp;l;Q :ˆ m
kˆ1 kuk kp;l;Q : Further on, u 2 Wp;l;loc (Q) if u 2 Wp;l (V (0; T)) for
0
any V ! V:
For an integrable function f : Q ! R define
Z
1
hf (R) :ˆ sup sup
j f (y; t)
rR (x0 ;t0 )2Q jQr j
fQr jdydt
8 R > 0;
Qr
R
where fQr is the average jQr j 1 Qr f (y; t)dydt: Then:
f 2 BMO(Q) (cf. [10]) if k f k;Q :ˆ supR hf (R) < 1;
f 2 VMO(Q)(see [14]) if f 2 BMO(Q) and limR#0 hf (R) ˆ 0: The quantity hf (R) is
referred to as VMO-modulus of f :
Our main result provides an a priori estimate in the Sobolev-Morrey scale for each
strong solution of the system (1) with zero initial trace on ft ˆ 0g:
1
THEOREM 2.2. Suppose …2†; ajk
a 2 VMO(Q) \ L (Q); p 2 (1; 1); l 2 (0; n ‡ 2b) and
2b;1
let u 2 Wp;l;loc (Q) be a strong solution to (1) such that u(x; 0) ˆ 0:
Then, for any Q0 ˆ V 0 (0; T) and Q00 ˆ V 00 (0; T); V 0 ! V 00 ! V; there is a constant
128
D.K. PALAGACHEV - L.G. SOFTOVA
0
00
C ˆ C(n; m; b; d; p; l; kajk
a k1;Q ; hajk ; dist (V ; @V )) such that
a
4†
kukW 2b;1 (Q0 ) C(kLukp;l;Q00 ‡ kukp;l;Q00 ):
p;l
As consequence of (4) we obtain a precise characterization of the Morrey, BMO and
HoÈlder regularity of the derivatives Ds u with s 2 f0; 1; . . . ; 2b 1g: Namely,
COROLLARY 2.3. Under the hypotheses of Theorem 2:2 set Kp;l;Q00 (u) :ˆ kukp;l;Q00 ‡
‡kLukp;l;Q00 and fix an s 2 f0; 1; . . . ; 2b 1g: There is a constant C such that:
n ‡ 2b l
a) if p 2 1;
then Ds u 2 Lp;(2b s)p‡l (Q0 ) and kDs ukp;(2b s)p‡l;Q0 2b s
CKp;l;Q00 (u);
n ‡ 2b l
then Ds u 2 BMO(Q0 ) and kDs uk;Q0 CKp;l;Q00 (u);
2b s
n ‡ 2b l n ‡ 2b l 1
;
c) if p 2
( ) then Ds u 2 C ss ;ss =2b (Q0 ) with ss ˆ 2b
2b s
2b s 1
n ‡ 2b l
jDs u(x; t) Ds u(x0 ; t0 )j
and
sup
CKp;l;Q00 (u):
s
p
x0 j ‡ jt t0 j1=2b )ss
(x;t); (x0 ;t0 )2Q0 ;(x;t)6ˆ(x0 ;t0 ) (jx
b) if p ˆ
(1) This inclusion rewrites as p 2 (n ‡ 2b
l; 1) when s ˆ 2b
1:
CHARACTERIZATION OF THE INTERIOR REGULARITY FOR PARABOLIC ...
129
The picture illustrates geometrically the results of Corollary 2:3 with the couple (p; l)
lying in the semistrip f(p; l): p > 1; 0 < l <n ‡ 2bg and s 2 f0; 1; . . . ; 2b 1g: The
n ‡ 2b
; 0 ; B ˆ (1; 0); and As ˆ (1; n ‡ s) is the
points Bs on the p-axis are Bs ˆ
2b s
intersection of the line passing through (0; n ‡ 2b) and Bs with the vertical line fp ˆ 1g:
We have Ds u 2 Lp;(2b s)p‡l (Q0 ) if (p; l) belongs to the open right triangle BBs As
(case a)). In particular, (p; l) 2 4BB0 A0 yields u 2 Lp;2bp‡l (Q0 ):
Let s 2 f0; . . . ; 2b 2g and take (p; l) in the interior of the quadrilateral
Rs :ˆ Bs Bs‡1 As‡1 As : Then the spatial derivatives Ds u are HoÈlder continuous with
exponent ss (case c)), while Ds‡1 u 2 Lp;(2b s 1)p‡l (Q0 ): Moreover, ss is the length jCAs j
of the segment CAs where C ˆ C(p; l) is the intersection of the vertical line fp ˆ 1g with
the line connecting the points (p; l) and (0; n ‡ 2b): If (p; l) 2 As Bs (the open line segment)
we have Ds u 2 BMO(Q0 ) (case b)), whereas (p; l) 2 As‡1 Bs‡1 implies Ds‡1 u 2 BMO(Q0 ):
Similarly, let R2b 1 be the shadowed unbounded region on the picture. Then
(p0 ; l0 ) 2 R2b 1 gives that D2b 1 u are HoÈlder continuous with exponent s2b 1 ˆ
ˆ jC 0 A2b 1 j; C 0 ˆ C 0 (p0 ; l0 ); while (p0 ; l0 ) 2 A2b 1 B2b 1 yields D2b 1 u 2 BMO(Q0 ):
3. SKETCH OF THE PROOFS
PROOF OF THEOREM 2.2. Set G for the fundamental matrix of the operator L given by
(see [7, 15])
Z
P
8
1
i(yt 1=2b j)‡( 1)b
A (x;t)j a
>
jajˆ2b a
e
dj for t > 0,
<
n n=2b
G(x; t; y; t) ˆ (2p) t
n
R
>
:
0
for t 0.
Let v 2 C 1 (Rn‡1 ) be a vector-valued function which is compactly supported in x and
v(x; 0) ˆ 0: Employing standard arguments (freezing aija 's at a point (x0 ; t0 ) 2 supp v;
expressing v as a Gaussian potential, taking the D2b -derivatives of v and then unfreezing
the coefficients, cf. [1, 2, 4, 6] in case of elliptic systems and [7, 9, 12, 15] in the parabolic
case), it is not hard to get the representation
Z
5†
Da G(x; t; x y; t t)Lv(y; t)dydt
Da v(x; t) ˆ p:v:
Rn‡1
|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}
ˆ: Ka (Lv)
Z
X
0
‡
p:v:
Da G(x; t; x y; t t) Aa0 (y; t) Aa0 (x; t) Day v(y; t)dydt
ja0 jˆ2b
Rn‡1
|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}
0
ˆ: Ca [Aa0 ; Da v]
Z
s
‡
Db G(x; t; y; t)ys ds(y;t) Lv(x; t)
8a: jaj ˆ 2b
Sn
130
D.K. PALAGACHEV - L.G. SOFTOVA
where the derivatives Da G(; ; ; ) are taken with respect to the third variable,
bs :ˆ (a1 ; . . . ; as 1 ; as 1; as‡1 ; . . . ; an ) and ys stands for the s-th entry of the outward
normal to the unit Euclidean sphere Sn :ˆ f(x; t) 2 Rn‡1 : jxjR2 ‡ t2 ˆ 1g: Further on,
Da G(x; t; y; t)ds(y;t) ˆ 0
Da G(x; t; my; m2b t) ˆ m (n‡2b) Da G(x; t; y; t) 8m > 0 and
Sn
whence each entry of the m m matrix Da G(x; t; y; t) is a variable kernel of CalderoÂnZygmund type with mixed homogeneity (see [8, 13]). The singular integral operators Ka
and Ca [Aa0 ; ] are bounded on Lp (Q) because of Aa 2 L1 (Q) (cf. [8]) and therefore the
representation formula (5) still holds true (a.e. in Q) for compactly supported in x functions
v 2 Wp2b;1 such that v(x; 0) ˆ 0: Moreover, Ka and Ca [Aa0 ; ] act continuously from
Lp;l (Q) into itself and the norm of the commutator Ca [Aa0 ; ] is comparable to kAa0 k;Q
(see [13, Theorem 2.1, Corollary 2.7]).
Fix an arbitrary point (x0 ; t0 ) 2 Q and consider the parabolic cylinder Cr :ˆ Cr (x0 ; t0 )
2b;1
(Cr ); supp v Cr and v(x; t0 r 2b ) ˆ 0: Since Aa 2 VMO; the
(cf. (3)). Let v 2 Wp;l
0
a
norm kCa [Aa0 ; D v]kp;l;Cr can be made arbitrary small if Cr shrinks to (x0 ; t0 ): Therefore,
for each e > 0 there is an r0 depending on e and on the VMO-moduli hAa of the
coefficients, such that kD2b vkp;l;Cr C(kLvkp;l;Cr ‡ ekD2b vkp;l;Cr ) whenever r < r0 (cf.
[13, Corollary 2.8]). Hence, choosing e small enough we get
6†
kD2b vkp;l;Cr CkLvkp;l;Cr :
Let r 2 (0; r0 ); u 2 (0; 1); u0 ˆ u(3 u)=2 > u and define the cut-off function
W(x; t) :ˆ W1 (x)W2 (t); 0 W 1; with W1 2 C01 (Br (x0 )) and W2 2 C 1 (R) such that
(
(
1 x 2 Bur (x0 )
1 t 2 (t0 (ur)2b ; t0 ]
W1 (x) ˆ
W2 (t) ˆ
0 x2
= Bu0 r (x0 )
0
t < t0 (u0 r)2b :
Since u0 u ˆ u(1 u)=2; we have jDsx Wj C(s)[u(1 u)r] s for any 1 s 2b and
jDt Wj C[u(1 u)r] 2b : Thus, applying (6) to v :ˆ Wu; we obtain
!
2X
b 1
U2b C r 2b jLujp;l;Cr ‡
Us ‡ U0 ;
sˆ1
where Us stands for the Morrey seminorm Us :ˆ sup [u(1
0<u<1
u)r]s kDs ukp;l;Cur ;
s 2 f0; . . . ; 2bg: Now interpolate the intermediate seminorms above (cf. [15, eq. (5.6)]
and [13, Proposition 3.2]) and fix u ˆ 1=2; in order to get the next Caccioppoli-type
estimate
7†
kD2b ukp;l;Cr=2 C(kLukp;l;Cr ‡ Cr
2b
kukp;l;Cr );
which holds for kDt ukp;l;Cr=2 as well by virtue of the parabolic structure of (1).
Therefore (7) implies (4) by means of a finite covering of Q by cylinders Cr=2 ;
r < dist (V 0 ; @V 00 ):
REMARK 3.1. Employing the representation formula (5) and suitable homotopy
arguments (cf. [13, Section 3]) it is possible to prove also that the operator L improves
CHARACTERIZATION OF THE INTERIOR REGULARITY FOR PARABOLIC ...
131
integrability. Namely, being in the framework of Theorem 2.2 let q 2 [p; 1) and suppose
2b;1
(Q) be a strong solution of (1) such that u(x; 0) ˆ 0: Then Lu 2 Lp;l
u 2 Wq;loc
loc (Q)
2b;1
implies u 2 Wp;l;loc
(Q):
PROOF OF COROLLARY 2.3. It relies on the next result which is a parabolic version of
the classical Poincare inequality.
LEMMA 3.2. Let u 2 Wp2b;1 (Cr ) where Cr is any parabolic cylinder (3). Then for each
s 2 f0; 1; . . . ; 2b 1g there is a constant C ˆ C(p; m; n; s) such that
Z
jDs u(x; t) (Ds u)Cr jp dxdt C(r (2b s)p (kD2b ukpLp (Cr ) ‡ kDt ukpLp (Cr ) ) ‡ r p kDs‡1 ukpLp (Cr ) ):
Cr
To begin with, let s ˆ 2b 1: Direct calculations based on Lemma 3:2 lead to
Z
1
p
p
jD2b 1 u(x; t) (D2b 1 u)Qr jp dxdt C(kD2b ukp;l;Q~ 0 ‡ kDt ukp;l;Q~ 0 );
r p‡l
Qr
~0 ˆ V
~ 0 (0; T) and V 0 ! V
~ 0 ! V 00 : Taking
where Qr ˆ Cr \ Q0 with 2r < dist (V 0 ; @V 00 ); Q
the supremum with respect to r above we get the Campanato seminorm of D2b 1 u on the
left-hand side which, in view of (4), turns out to be bounded by the Morrey norms of Lu
and u in Q00 : Now, employing the embedding properties of Campanato classes on spaces
of homogeneous type recently proved in [11] (cf. also [3, 5] in the case b ˆ 1) we obtain
as follows. If p ‡ l < n ‡ 2b then D2b 1 u 2 Lp;p‡l (Q0 ) and kD2b 1 ukp;p‡l;Q0 is controlled
in terms of kLukp;l;Q00 and kukp;l;Q00 : If p ‡ l > n ‡ 2b then D2b 1 u 2 C s2b 1 ;s2b 1 =2b (Q0 )
n ‡ 2b l
(cf. [11, Corollary 1]). Finally, if p ‡ l ˆ n ‡ 2b we first
with s2b 1 ˆ 1
p
apply the HoÈlder inequality to kD2b 1 u (D2b 1 u)Cr kL1 (Qr ) and then Lemma 3:2 in order
to get D2b 1 u 2 BMO(Q0 ):
The proof of Corollary 2:3 completes in the same manner, running induction for
decreasing values of s:
REMARK 3.3. The above results are easily extendable, modulo unessential
technicalities, to parabolic systems with lower order terms
X
X
Aa (x; t)Da u ‡
Bb (x; t)Db u ˆ f (x; t)
Dt u
jajˆ2b
with Bb (x; t) ˆ
m
fbkj
b (x; t)gk; jˆ1
jbj2b 1
and
bkj
b
belonging to suitable Morrey classes.
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______________
Pervenuta il 22 novembre 2004,
in forma definitiva il 30 dicembre 2004.
D.K. Palagachev:
Dipartimento di Matematica
Politecnico di Bari
Via E. Orabona, 4 - 70125 BARI
dian@dm.uniba.it
L.G. Softova:
Bulgarian Academy of Sciences
Institute of Mathematics and Informatics
SOFIA (Bulgaria)
luba@dm.uniba.it