R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI
PADOVA
G IUSEPPE D I FAZIO
Hölder-continuity of solutions for some
Schrödinger equations
Rendiconti del Seminario Matematico della Università di Padova,
tome 79 (1988), p. 173-183
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Hölder-Continuity of Solutions
for Some Schrödinger Equations.
GIUSEPPE DI FAZIO
(*)
0. Introduction.
Recently the local regularity properties for
equations of the form
solutions of
Schr6dinger
many authors (see e.g. [A-S], [D-M], [C-F-G],
a very singular potential, precisely
[C-F-Z])
the Stummel-Kato class (see definition 1.1 ) .
Under this assumption in [C-F-G] was established a Harnack inequality and proved a local continuity result for solutions of ( ~ ).
It is easy to see that if SZ is an open bounded set in Rn then
for p &#x3E; n/2 ; hence the result in [C-F-G] generalizes the well
known H61der estimates by Stampacchia [ST], Ladizhenskaia [L-U] etc.
We stress that high integrability of V does not play an essential
role.
In fact also the Morrey space
is contained in S for ~ &#x3E; n - 2
in
and being
L1 ~~(SZ), for any 0 C ~ C n, does not imply any extra
integrability (see e.g. the examples in [P2]).
In this paper we assume V in .L1 ~~(,S~) (A &#x3E; n - 2) and prove local
holder-continuity for solutions of (*) hence, in this special situation,
we improve the continuity result in [C-F-G].
have been studied
by
allowing V to be
(*) Indirizzo
(Catania), Italy.
dell’A.: Via del Canalicchio 9 - 95030 Tremestieri Etneo
174
Our technique is very close to the one in [C-F-G] heavily relying
the exploitation of well known estimates for the Green function of L.
There is however a technical difficulty.
It is impossible to use the usual C°°-approximation for .L and V
(as in [C-F-G]) because functions in Morrey spaces are not close, in
general, to bounded functions in
(see [P1] p. 22 for an example
of an
function with distance from L°°(SZ) equal to 1). We
overcame this difficulty by developping a representation formula for
solutions of (*) that extends classical results on the Green function
(see e.g. [ST]).
on
1. Some function spaces.
Let ,S~ be an open bounded set of Rn (n &#x3E; 3).
We will need some mild regularity assumption to be satisfies
8Q e.g.
where
r:
0
diam (Q)
r
DEFINITION 1.1
belongs to the
functions q(r) &#x3E;
by
(1).
( Stummel-.gato clas s ) . W e sa y that
class S iff there exists a non decreasing
0 with
limq(r)
r-+0
=
0 such that
Obviously S C
DEFINITION 1.2 (Morrey spaces).
of functions f E Ll(92) such that
(1) JEJ denotes
the
Lebesgue
measure
(0
of
a
A
n) is the space
measurable subset L~’ of
175
LEMMA 1.1.
to the
If
u
belongs
to
L1’;’(Q) (n - 2
Â
n)
then
u
belongs
Stummel-Kato class and
where C
depends only
on
2 and
n.
Indeedy
REMARK 1.1:
is an immediate consequence of
Indeed the inclusion
Lemma 1.1 and the other inclusion is obvious.
We now recall the definitions of the Sobolev spaces
and
DEFINITION 1.3.
p
+ °°) iff
.gl~~(S~)
is
a
We say that u
belongs
to
’U,
Banach space under the
the closure
is the dual space o f
of
norm
with respect to the
norm;
1. We have T e
where lip + 11q
=
H-1,p (Q)
176
2. Green’s function and
a
representation
formula.
In the following sections we will consider the operator L - V
where L is the divergence form elliptic operator
satisfying
and V is
of
a
function
DEFINITION 2.1.
the equation
We say that
is
u E
a
local weak solution
iff
c S and [S]
meaningful by the inclusion
p. 138-140.
We recall that under the weaker hypothesis V E S the following
regularity result for weak solutions was proven in [C-F-G].
Definition 2.1 is
There exist two positive constants C
C(v, n), ro
from definition 1.1),.and a non decreasing function
0
0 such that, for any local weak solution of Lu + Vu
THEOREM 2.1.
=
ro(v, n, r~) (r¡
lim co(r)
r-+0
Q and for every ball
=
in
=
=
=
Q (0
r c ro )
we
have :
177
We
now
define
a
different class of solutions :
(2.1 ) holds, let It be
DEFINITION 2.2. Let L be such that
variation
measure
yYe say that
a
bounded
in Q and
u E
is
a
very weak solution
the
of
equation
i f and only if
for every V E B’o~2(S~) n CO(D) such that
way as in [ST] it is possible to show
LEMMA 2.1.
Assume p, is
a
bounded variation
I f u E H¿,2(Q)
then u is the very weak solution
of
Ly E
the
is
a
same
In much the
measure
weak solution
of
same
and T
the
=
equation
equation.
The proof is an easy consequence of the definitions above. We
recall the definition of fundamental solution.
Let y E SZ and 6, the Dirac mass at y.
Consider the equation
now
We call its (very weak) solution the Green’s function relative to
the operator L with pole at y and we denote it by
y).
of Lq
y~,
By the definition above the solution 99 C
=
178
is
where y E
given by
the formula
Consider:
bounded variation measure,y
where p
is
have the
following
a
THEOREM 2.2:
is the very weak solution
see
PROOF. We consider
[ST] Th. 8.3 p. 227).
We will show that
satisfies:
such that
Let
Then
We observe that
of (2.8).
only
the case p
=
0
(for
the
case
T
=
0
179
Indeed
Then
By
we
have :
(see [ST]
p. 220
Tonelli and Fubini’s theorems
REMARK 2.1.
integral; i.e.
In
(8.6))
fact,
In the
for every
proof
we
above
we
have:
we
may differentiate under the
have, using Fubini’s theorem:
180
sd
3.
Hölder-continuity
We
now
of local solutions.
state the main result of this paper
THEOREM 3.1.
of
=
PROOF.
i.e.
~c E
numbers ro
A, n)
ro(v, I
A) such that for any local solution u
CQ, 0 rro
any ball Br (xo ) , with
There exist
a(v, n), C C(v, n,
Vu in Q and f
Lu
have
we
a
=
=
I
or
Let V E
such that :
positive
and u
It is easy to
see
a
=
local weak solution of Lu
that uq is such that
holds.
Therefore, by
Lemma
2.1,
is
a
very weak solution of
=
Vu
181
By
Theorem 2.2
we
have
1 in
such that 0 c g~ c 1,
determined
is
0
where
and
ro
supp (g~) ~ B2r(’ x0 ) ~I
the local boundedness theorem 1.4 in [C-F-G].
we have:
Obviously, for every
Now
We
=
we
begin estimating
by
I.
Where N is a positive number to be fixed later.
To estimate A we use the inequality (see [G-T] p. 200 Th. 8.22
and Harnack’s Theorem)
182
hence
and
by
Lemma 1.1
To estimate B
we use
Lemma 1.1 and the
following
proven in [L-S-W].
We obtain:
and therefore
Now,
if
we
Estimating
choose
II and III
we
as
in
[C-F-G]
we
obtain
obtain
bound
183
and
The theorem
now
follows.
REFERENCES
[A-S]
M. AIZENMAN - B.
SIMON, Brownian motion and Harnack’s inequality
Comm. Pure and Appl. Math., 35 (1982),
for Schrödinger operators,
[C-F-G]
[C-F-Z]
[D-M]
[G-T]
[L-S-W]
[L-U]
[P1]
[P2]
[S]
pp. 209-271.
F. CHIARENZA - E. FABES - N. GAROFALO, Harnack’s inequality for
Schrödinger operators and the continuity of solutions, Proc. A.M.S.,
98 (1986), pp. 415-425.
M. CRANSTON - E. FABES - Z. ZHAO, Potential theory for the Schrödinger equation, Bull. A.M.S., 15 (1986), pp. 213-216.
G. DAL MASO - U. MOSCO, Wiener criteria and energy decay for relaxed Dirichlet problems, preprint.
D. GILBARG - N. TRUDINGER, Elliptic Partial Differential Equation
of Second Order (2nd edition), Springer-Verlag (1983).
W. LITTMAN - G. STAMPACCHIA - H. F. WEINBERGER, Regular points
for elliptic equations with discontinuous coefficients, Ann. Scuola Norm.
Sup. di Pisa, S. III, 17 (1963), pp. 45-79.
O. A. LADIZHENSKAIA - N. URALTZEVA, Linear and quasilinear elliptic equations, Academic Press (1968).
L. C. PICCININI, Proprietà di inclusione e interpolazione tra spazi di
Morrey e loro generalizzazioni (Tesi di perfezionamento), Pisa (1969).
L. C. PICCININI, Inclusioni tra spazi di Morrey, Boll. U.M.I., 2 (1969),
pp. 95-99.
M. SCHECHTER, Spectra of partial differential operators, North-Holland
(1971).
[ST]
G. STAMPACCHIA, Équations elliptiques du second ordre à coefficients
discontinus, Les Presses de l’Université de Montréal (1965).
Manoscritto pervenuto in redazione il 3
aprile
1987.

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