Di Fazio, Giuseppe

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 163, 157-167 (1992) 
Poisson Equations and Morrey Spaces 
GIUSEPPE DI FAZIO 
Dipartimento di Matematica Viale A. Doria, 6 95100 Catania, Ital), 
Submitted by Avner Friedman 
Received September 25, 1989 
INTRODUCTION 
In this paper we study the Dirichlet problem for equation 
Lu= -(aiju.,),,=,L (*I 
where L is a second-order uniformly elliptic operator in a bounded domain 
a and f is taken in the Morrey space L’,“, 0 < i. < n. (For the precise 
statement see Section 1). 
The solution we consider is a very weak one introduced in [LSW] 
because in general the Dirichlet problem for Eq. (*) does not have a weak 
(variational) solution under our assumption onf. 
The purpose of our work is to study the regularity properties of the solu- 
tion as the parameter A increases from 0 to n. In fact we prove Lp regularity 
for 0 < E. < n - 2, and that the solution belongs (locally) to the space BMO 
if i=n -2. For A >n - 2 we prove the local Holder continuity of the 
solution. 
Furthermore, we consider the cases in which f belongs to some spaces 
related to the so-called Stummel-Kato classes which fall in between L’-” ’ 
and L’,“(l> n - 2) obtaining boundedness and continuity of the solution. 
The study of the Stumel and L’,‘(A> n - 2) cases is much along the lines 
of [CFG] and our previous work [D]. 
We also show that if f is taken in the Stummel class S (in fact in a 
slightly larger one) the solution has its gradient in L*(Q) and then it is a 
variational solution. The same is obviously true in the case L’*” (A > n - 2) 
because of the inclusion L’,” E S (A > n - 2). 
Finally we give some regularity results for the gradient of the solution for 
A> n - 2 under some regularity hypotheses on Sz and (Us). 
409,1163!1-II 
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CopyrIght ,_ 1992 by Academx Press. Inc 
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158 GIUSEPPE DI FAZIO 
1. PRELIMINARY RESULTS 
Let 52 be an open bounded subset of R” (n > 3). In what follows some 
regularity on dQ is needed. Precise assumptions are made in the statement 
of the theorems. However, we require at least the following condition to be 
satisfied in 
3A E IO, 1 C : ID,(x)l2 A IB,(x)l O<r<diamQ, (1.1) 
where B,(x) is the closed ball centered at x with radius Y and 
G,(x) := B,(x) n Sz. 
Here and in the following we set (E( for the Lebesgue measure of a 
measurable set E E R”. 
For O<J<n and MEL’ we let JJf))I,i.:=supO~r<diamR,xcnr-i 
Jn,Cxj If(y)1 dy. L’,“(G) := {f~ L’(Q) : [l.fll I,i. < + cc 1 is the classical 
Morrey space. We also use some other function spaces. 
For f~ ,$,,(s2) we let IlfIIBMo := sup fB If-fJ dx, where the sup is 
taken on the closed balls BE Q. Also the symbol f stands for the average 
and fB = fsf(x) dx. BMO := (f~ L:,,(Q) : llfllBMo < + co }. We also set 
S(n):=(f~L’(f2): sup [ If(v)1 Ix-ylzp”dy< +co 
and 
Ix-y12-“dydrl(r) 
I 
for some increasing function q: 10, diam !2[ --t lR+ that lim,,, v(r) = 0. 
Obviously S _C 3. S and 3 are variants of the classical Stummel-Kato 
classes. 
It is easy to show that (see [D]) 
L’~+2) c S(l2) c S(52) E L’qf-2) for O<p<n-2<A<n. 
We also use the Sobolev space HkP(0)l <p< + co and their duals 
H-‘,4(Q) (I/p + l/q = 1). Also H;‘(Q) E Hi(Q) and H-‘,*(Q) z H-‘(Q). 
We consider in Sz the Dirichlet problem for the divergence form equation 
La-= -hj(X)U,,L,=f, (1.2) 
where we assume 
a&) E L”(Q), a(j(x) = uji(x) for i, j= 1, . . . . n 
3vER+ :v-1 1412 < U,i(X)ti5j< v ItI* vt E R”, 
a.e. x in 52. 
(1.3) 
POISSON EQUATIONS AND MORREY SPACES 159 
We make various assumptions on the known term ,f which, for the time 
being, we suppose to be an L’(O) function. 
Under this assumption there is, in general, no hope for the existence of 
a variational solution to the Dirichlet problem for (1.2). 
Then, following [LSW], we give the 
DEFINITION 1. Let p be a bounded variation measure in Q. We say that 
u E L’(Q) is a very weak soution to the Dirichlet problem in D for equation 
Lu=p iff 
.r uLq dx = s cp dp Vq E HA(Q) n P(i2) a 
such thatQLq E C’(Q). (1.4) 
It is well known (see [ST]) that under our assumptions ( 1 .l ), ( 1.3) 
there is a unique very weak solution. In the case p = 6,., y E Sz, the corre- 
sponding very weak solution g(x, y) is called the Green function for L and 
we have (see [ST]) the representation formula 
u(x)= i,, dx, Y) 4 a.e. in R. 
It is also clear, from the definitions, that a weak solution to the Dirichlet 
problem, i.e., 
whenever it exists, is the very weak solution in the sense of the above 
definition. 
It is also known [LSW, p. 671 that g(x, v) satisfies the estimate 
c, Ix-y12- “Gg(x, y)QC, lx-.Y12 ‘), (1.5) 
where C, and C2 are positive constants depending only on n and v and the 
right-hand side estimate is true under the assumption Ix -Y\ < l/4 d(y, iX2). 
In their paper [GW] Griiter and Widman give a different definition of 
Green function establishing many useful estimates which we need in the 
following. 
Precisely Griiter and Widman define, in their theorem ( 1.1 ), as a Green 
function 2(x, y) for the operator L the unique nonnegative function such 
that 
VY E Q ETC., y) E H’(Q\B,(y)) n Hk’(Q) Vr E 10, dist( y, &2)[ 
160 GIUSEPPE DI FAZIO 
and satisfying 
J 4jb) &+(x3 YbL,W dx = P(Y) vcp Ec;(Q). R (1.6) 
LEMMA 1.1. Assume (l.l), (1.3), and suppose Q(X) are so smooth that 
for Vq E H,$“(SZ) the weak solution cp to the Dirichlet problem 
JQfQ=v 
cliff:, 
is in Hip for some p > n. 1 (1.7) 
Then 
‘dx, Y) =a-? Y). 
Proof: Let cp be the solution of (1.7). Then Sna,i(x)40,(~)~~~,(x)dx= 
jnq(x)$i.(x) dx for any IC/E HA(Q) and also, by density, for all 
$ E H’*P’(S2) (l/p + l/p’= 1, p’< n/n - 1). On the other hand by [GW, 
Theorem 1.1, (1.7)] we know that g( ., y) is in H$“(52) for any 
s~[l,n/n-11. Hence 
j a&h,(x) iL,(x, v) dx= 1 v(x) 8(x, y) dx. R R 
By definition (1.6) the solution 4p~ Hip(Q) (p > n) of (1.7) satisfies 
s R @+P,,(x) L,k v) d  =rp(wv) 
and then 
s v(x) 0, Y) dx = CP(Y). R 
By the [LSW] definition of g(x, v) using Lq = q we also have 
5 v(x) dx, Y) d-x = NV). R 
Finally, by (1.8) and (1.9), 
1 r(x) Ax, v) dx = j- v(x) i!(x, Y) dx V~EH;“(SZ) 
R D 
(1.8) 
(1.9) 
and this in turn implies g(x, v) = g(x, -v). 
’ au(x) E cO(r2) will dice. 
POISSON EQUATIONSAND MORREY SPACES 161 
We now quote an estimate from [GW] which is useful in the next 
section. 
LEMMA 1.2 [GW, Theorem 3.3). Suppose s1 satisfies a uniform exterior 
sphere condition. Assume (1.3) are satisfied and ,furthermore suppose au are 
Dini-continuous. Then 
Ig,(x, y)l ~Klx-.Y’-” vx, y E Q, (1.10) 
where K is a positive constant depending only on n, V, 52 and the modulus at‘ 
continuity of the coefficients. 
Observe that in the above lemma we denoted by g the Green function 
using Lemma 1. Lemma 1.2 enables us to prove 
LEMMA 1.3. Consider the very weak solution u to the Dirichlet problem 
for Lu = f, where we suppose the assumption of Lemma 1.2 to be satisfied by 
Q and ai, and,f is any L’(Q) function. Then 
u,(x) = j g\-,(x, y)f(.~) &. 
R 
Proof Let cp E C:(Q) and consider 
! j $, * g(x, Y)f(Y) dY, a) ) 
=- 
g(x, y ) 2 dx dy 
’ I 
= (J R g.r,(x, Y)~(Y) 4s v(x) vcp E C,“(Q). 
In the above calculation we exchanged twice the order of integrations. 
162 GIUSEPPE DI FAZIO 
This is possible because both g(x, y)f(y)q.,.,(x) and g-,(x, y)f(y)cp(x) 
are in L’(Q x Sz). Indeed, using (1.5) 
II Id4 Y)f(Y)cp,(X)l 4 dx RxR 
6Cmax I9.&)l jj If( Ix-~l’-“d~dx< +m. R RxR 
Similarly, using ( 1.10) 
J.i l&L,(x, Y)f(Y)9(Y)ldY dx RxR 
<Cm= 19(x)1 jj If(y)1 lx-yll~“dydx~ +a. 
a nxn 
We conclude this section with a simple result: 
LEMMA 1.4. (2) S(l2)cH-‘(Q). In particular S(Q) and L’*’ (l>n-2) 
are also contained in H-‘(Q). 
Proof Let cp E C,“(Q), f~ S(a). 
=Cj 
R 
IVcp(y)l (j 
R 
If(x)1 Ir-yll”dx)dy 
d c II IV91 lILZ(c2) Il4(fk(L2)~ 
where we set I,(f)(x) = ln If(x)1 Ix - yl ’ --n dx. 
But 
II~l(S)II L2(R) = If( lx-.A’-“dx > 
lw4-“d+9]“2 
>I1 
112 
X Ix-yyl’-“lz-yl’-“dy dx dz . 
L We thank Prof. C. Simader for pointing out the proof of the lemma 
POISSON EQUATIONS AND MORREY SPACES 163 
Using well-known properties of the Riesz kernels (see, e.g., [LA, 
p. 45 (1.1.3)]) we estimate the inner integral by a multiple of 1.x - ~1’ ‘I 
obtaining 
By the assumption f~ 3 the inner integral is uniformly bounded in Q and 
the conclusion follows. The quantity on the right-hand side of (1 .l 1) is the 
energy of the measure If(x)\ (see, e.g., [LA, p. 77 ( 1.4.1)]). 
2. REGULARITY RESULTS 
In this section we consider the very weak solution (see (1.4)) to the 
Dirichlet problem 
Lu =,f 
u;, = 0. 
(2.1 ) 
Here and in the following L is defined by (1.2) and (1.1 ), (1.3) are satisfied. 
f belongs to some L’,“(Q), 0 < il <n. 
THEOREM 2.1. Let AE 10, n-2[,f~ L’~‘(Q). Then the soh&on u to (2.1) 
is in the weak L”*(Q) space, where l/p,= 1 - 2/(n - A). In particular 
24 E Lp(Q) for all p <pi.. 
Proqf: We have 
and the conclusion follows by known properties of the Newtonian potential 
(see, e.g., [A] or [CF]). 
THEOREM 2.2. Let E, = n - 2. Then the solution u to (2.1) is locally in 
BMO(Q), i.e., VQ’ c R, d :=dist(Q’, XJ), there is a positive constant 
C = C(n, v, d) such that for any B,(x), with x E Sz’ and 0 < r < d/2 
f 14-x) - us,(,)1 dx G C. B,( x ) 
164 GIUSEPPE DI FAZIO 
Proof: Let B s B,(x,) one of the balls in the conclusion of the theorem. 
Set 
fl :=fxe.> f2 :=f(l -xB*)P where B* s II,,( 
Then we have u=ur +u,, where 44 = ji dx, ~)fh9 4 and u2W = 
jn g(x, y)f*(y) dy are the very weak solutions of 
Lu =fi 
=o Ulan 
and Lu2 =f2 
%n =o 
respectively. 
We now estimate the BMO norm of u1 and u2. 
As for u, we have 
f (ul(x)-qB( dxG2 (Ul(B<2C B 
jB* If( s, lx-Y12-ndx4 
d Cr2-” 5 'f(v)1 4 G wllL',n-*(n), B* 
where we used estimate (1.5). 
The estimate for u2 is slightly more delicate. 
f luz(x) - u2gI dx B 
= dz, Y)~Z f(y)& dx 1 I 
d Ax, v,--j AZ, v)dz If(wv)l&dx 
= dz, ,+I dxdy= jDd.- + jQd.-y 
where we set Jz,= {v: (x0-y[ <d} n (Q\B*) and Od= {,v : (x0-y()d) n 
(Q\B*). To estimate the first integral we use the fact that, when restricted 
to B, g( ., JJ) is a weak solution to the equation Lu = 0, and then we can 
apply De Giorgi-Nash’s ‘theorem to obtain 
dx, v) -s, g(z, Y)dz dx 
POISSON EQUATIONS AND MORREY SPACES 165 
Now, using Harnack’s inequality together with estimate (1x5), we get 
fl g(x, y)-{ g(z, y)dz dx<C(n,v)r” (x0-yl’ ’ ‘. B B 
Then, for the first integral, we have 
f~dIp(y)IfB 1 B / g(x, Y)-f g(z, Y)~Z dxdy 
To bound the second term we observe that for any x E B /x - yI >, 
lx,--yl-r>d/2. 
< Cr” s If(y)1 Ixo-y12-‘i-1dy 
Qd 
and, settingSZ,,:=(yEJZd:2krd(~g--yl~2k+’r, kEiW), 
r If(~)1 lxo-y12-n-a dv 
Hence, once more using (1.5), 
c,d If(‘)’ (fB j g(x, y)-j, &, Y) dz 
< Cd’-” 
I nd If( dy < C IISII Ll.nmqfi) 
and this completes the proof. 
We now turn to the study of the regularity properties of the solution u 
iffis in 3 and in its subspaces S and L’,‘(sZ), I7 n - 2. Let us remark that, 
because of Lemma 1.4, IVul E L’(Q) and the very weak solution is in fact 
the weak (variational) solution to the Dirichlet problem (2.1) 
THEOREM 2.3. Let f~ 3. Then the solution u to (2.1) is bounded in Q. 
Proof We have 
6 c sup s lf(~)I L--YI*~~ dy. J E R QrC.r) r>o 
166 GIUSEPPE DI FAZIO 
THEOREM 2.4. Let f E S. Then the solution u to (2.1) is continuous in Q. 
THEOREM 2.5. Let f E L’,’ (n - 2 < 2 -C n). Then the solution u to (2.1) is 
locally Holder-continuous in Q. 
Proof of Theorems 2.4,2.5. The proof very closely follows the proof of 
Theorem 3.1 in [CFG] and Theorem 3.1 in [D]. 
In both the cases, obviously, we must substitute f in place of Vu in the 
equations considered. 
We conclude this section with some results on the regularity of the 
gradient of the solution u to (2.1). 
THEOREM 2.6. Suppose Q satisfies a uniformly exterior sphere condition. 
Assume (1.3) are satisfied and a, are Dini-continuous in 0. Let f E L’,” 
1E]n-2,n-l[. 
Then the gradient of the solution u to (2.1) is in the weak Lpi space where 
pn = (n - A)/(n - 1- 1). 
Proof. We have, by Lemmas 1.2 and 1.3, 
luxj G/Q Igx,(x, Y)I If( d,GK/Q If( lx-yl’-“dy 
and the right-hand side is in the weak Lpi space of the conclusion by a 
theorem of Adams (see [A] or [CF)]. 
THEOREM 2.7. Suppose 52 satisfies a untform exterior sphere condition. 
Assume (1.3) are satisfied with aq Holder-continuous. Let f E L’*“- ‘. Then 
the gradient of the solution u to (2.1) is locally in BMO (see Theorem 2.2. 
above). 
Indeed, take fi and f2 as in Theorem 2.2. By the linearity and Lemma 1.3 
we have 
u,, = (u,), + Cud,, = j- R gx,b, Y)~,(Y) 4 + s, gx,(x, Y).L(Y) do. 
Now the proof goes on in much the same way as Theorem 2.2 using the 
bound for the gradient of the Green function given by Lemma 1.2 and the 
Holder estimate in [GW] Theorem 3.51. 
Similarly we have 
THEOREM 2.8. Under the same assumptions on Q and aii of the previous 
theorem, if f E L’*‘., n - 1 <A <n, we may conclude that IVul is locally 
in Cog’. 
POISSON EQUATIONS AND MORREY SPACES 167 
REFERENCES 
CA1 D. ADAMS, A note on Riesz Potential, Duke Math. J. 42, No. 4 (1975), 765-778. 
[CF] F. CHIARENZA AND M. FRASCA, Morrey spaces and Hardy-Littlewood maximal 
function, Rend. Mat. Roma Serie (VII) I, Nos. 3-4 (1987), 273-279. 
[CFG] F. CHIARENZA, E. FABES, AND N. GAROFALO, Harnack’s inequality for SchrGdinger 
operator and the continuity of solution, Proc. Amer. Math. Sot. 98 (1986). 415425. 
[Dl G. DI FAZIO, Hiilder continuity of the solution for some SchrGdinger equations. 
Rend. Sem. Mat. Unia. Padoua 79 (1988) 173-183. 
[GW] M. GROTER AND K. WIDMAN, The Green function for uniformly elliptic equations, 
Munuscripfa Muth. 37 (1982), 303-342. 
[LA] N. S. LANDKOF, “Foundations of Modern Potential Theory.” Springer-Verlag, 
Berlin/New York, 1972. 
[LSW] W. LITTMAN, G. STAMPACCHIA, AND H. WEINBERGER, Regular points for elliptic 
equations with discontinuous coefficients, Af~ln. Scuola Norm. Sup. Pi.w I IIIJ 17 
(1963), 45-79. 
ISI G. STAMPACCHIA, “Equations elliptiques du second ordre a coefficients discon- 
tinuous,” Les Presses de I’Universite de Montreal (1965). 


Poisson equations and Morrey spaces 

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Poisson equations and Morrey spaces

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