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A priori bounds for elliptic equations
NonlineorAnalysis,
Pergamon
Theory, Methods & Applications. Vol. 30, NO. 8, pp. 5457-5461, 1997 Pm. 2nd World Congress of Nonlinear Analysts 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
PII: SO362-546X(97)00144-2
A PRIORI BOUNDS FOR ELLIPTIC EQUATIONS
E. LEMUS (2), P. PADILLA (1) 1. IIMAS, UNAM, Circuito Escolar, Mexico, D.F. Mexico Department of Mathematics, IJAM, Iztapalapa
2.
Mexico
Key ecords end phrases:
Maximum Principle, exit times, probabilistic approach, elliptic oper-
ators, diffusion. ABSTRACT In this paper we study the relationship prtl eigcnvalue diffusions.
of second order elliptic operators
A probabilistic
sults established
existing
between
the maximum
and the expected
principle
and the princi-
exit times of the corresponding
approach is discussed that provides a good understanding
analytically.
We also obtain an Alexandrov-Bakelman-Pucci
of classical re-
type of estimate
using
similar methods.
1. INTRODUCTION In what follows we introduce operators.
We emphasize
and probabilistic differentiability
approach.
some notation
and mention
Therefore, we do not insist on minimal assumptions
of the coefficients of the operators
to an elliptic operator for solutions
and the principal of elliptic equations.
Let R be a given bounded
on the domain, the
and so on (see also the concluding
remarks).
In
between the exit times for the diffusion associated
eigenvalue
(section 2). In section 3 we prove an a priori
Finally, section 4 is devoted to some concluding
regular domain and an elliptic operator
LU = a;juziz, + bju,, + cu satisfying
elliptic
that our goal in this note is to present some links between the analytic
the rest of the paper we consider the relationship estimate
several well known facts about
remarks.
54%
Second World Congress of Nonlinear Analysts
and
2. EXIT
TIMES
It is known that if c 5 0 we can problem
corresponding
AND THE PRINCH’AL
construct
to the operator
EIGENVALUE
diffusion Xt which is the solution
a
L. The solution
to the Dirichlet
to the martingale
problem
for L is unique
whenever ET, <
DC,
for any z E $2, where r, is the exit time of Xt for the initial condition expectation
operator
(see [3]).
Bcrestycki, Nirenberg only the principal
and Varadhan
[l] established
eigenvalue X1 of L is positive.
principle implies uniqueness
that the refined maximum
relationship
Theorem
1 1j c 5 0 then
principle holds if and
On the other hand, as is well known, the maximum
of the Dirichlet problem.
between X1 and ET, is to be expected.
A clear-cut
Proof.
20 = z and E stands for the
Using the representation
In fact:
for the principal eigenvalue (see [3], chapter 14, theorem
10.1)
Xl = sup{ X > 0; sup EZeXTz < w} XEn
it easily follows that when c = 0, the boundedness
of E,r,
implies X1 > 0. Indeed, in this case,
the expected exit time is finite for all z E R because of the conditions lemma 7.4.) and applying When c 5 0, ET,
may
Chebychev’s inequality
be infinite,
on L (see also [4], chapter 5,
we get the result.
so we use instead
the killing time that is bounded
by the exit
time of the related diffusion with c z 0 and the result follows. The converse, i.e. X > 0 implies finite expected esit time is immediate It would be interesting but not necessarily
by using 1-t 2 5 e”.
to give a probabilistic
nonpositive.
argument
in the general case, i.e. when c is bounded,
Notice however, that from the analytic
t,heorem 1, one can proceed along the same lines as in [l] (for instance
point of view, once we have theorem 6.1).
5459
Second World Congress of Nonlinear Analysts 3. A PRIORI
Results
as the above motivate
the study
as the Alexandrov-Bakelman-Pucci
of the probabilistic
interpretation
of classical
a priori bounds
inequality:
(ABP inequality) If c 5 0, f E Ln(0),
Theorem
BOUNDS
u E W2:,
p 2 n and
i) ,l2& U(X) 5 0 ii) Lu > j,jcLn(Q),
then
whewB is (1constant
depending on the dimension of the domain n, the ellipticity constant CO of L,
the diameter d oj Q and b. Recently
the dependence
they show that a quantity
of B on the diameter
this dependence
R measuring
of R was improved
on the diameter
the narrowness
in [l] and [2]. Roughly
may be improved
of the domain
by IQ]‘/” or, more precisely,
(see the above references
approach
to prove an inequality
that
works.
the result
and proving
that, for c 5 0, the Feynman-Kac
it, notice
generalizes
for details).
we use a Probabilistic Before stating
speaking,
those presented
by Here
in the cited formula
for
1 1, c < 0 and u E WZJ’ for R bounded such that Lu = f
and
Lu = f u=
0
in
0
on
dS2
implies
for the diffusion Theorern lim,,ao
X1 corresponding
2 Let f
E La,0
to L with X0 = CC.
U(Z) 5 0, then sup lul I B~l~l-“*llf
[IL?
where
r = sup E,r, n Remark.
For Q = n and a = co and using
bounds
presented
Proof.
We may assume
in [2] (theorem that
the fact that
1.4. ) and [l] (Theorem
f is a simple
function
lu(~)l 5 Es 5 E,
Ezrz 5 l52/2/n(see [l]), we obtain 10.1. ) respectively.
and proceed
then by approximation.
6%
If(4ldt
lai IXn,(Xt)dt
Then
the
Second World Congress
5460
for a finite partition
of Nonlinear Analysts
{0,}i of R and f = Ca;xo,
This implies
where
T& =
JoTxni (-Jw
=
“sojourn time in” 0;.
Hence, using Hiilder’s inequality
Under the hypothesis
on L, we have T,5, =
7.c JJ 0
ni
d4 2, YFY&
where q is the Green’s function for the Cauchy problem (see [3] and [5] for the details on the existence of the Green’s function) of the Green’s function
Now we integrate
over Q and use both Fubini’s theorem and the definition
to obtain that the RHS of the above expression is less than or equal to
SUPM~>l< c+y+y =
fwI+llfllL~
4.
Through
this approach the relationship
the coefficients of the operator degenerate
x (~IlfllL.)
CONCLUSION
between the measure of the domain, the Green’s measure and
L is made apparent.
It seems these ideas can be generalized
elliptic case, or when the domain fl is a manifold.
in a subsequent
work.
to the
We intend to pursue these questions
5461
Second World Congress of Nonlinear Analysts REFERENCES
[l] Berectycki,
H., Nirenberg,
L., Varadhan,
The Principal
S.R.S.,
Eigenvalue and Maximum Principle for Second-Order Elliptic Opemtors in Geneml Domains. Comm. Pure
Appl.
[2] Cabre,
Math., X.,
Vol. XLVII,
47-92
(1994).
On the Alexandro&Bakelman-Pucci
Estimate and the Reversed Holder Inequality for
Solutions oj Elliptic and Pambolic Equations. Comm. [3] Friedman,
A., it Stochastic
[4] Karatzas,
I., Brownian
[5] Pinsky,
R.G.,Positive
Differential
Motion
Equations.
and Stochastic
Harmonic
Functions
Pure
Appl.
Vol I. Academic
Calculus.
Springer
Math.,
Vol. XLVIII,
539-570
Press. Verlag
and Diflusion. Cambridge
(1991). Univ.
Press.
(1995).
(1995).
Pergamon
Theory, Methods & Applications. Vol. 30, NO. 8, pp. 5457-5461, 1997 Pm. 2nd World Congress of Nonlinear Analysts 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
PII: SO362-546X(97)00144-2
A PRIORI BOUNDS FOR ELLIPTIC EQUATIONS
E. LEMUS (2), P. PADILLA (1) 1. IIMAS, UNAM, Circuito Escolar, Mexico, D.F. Mexico Department of Mathematics, IJAM, Iztapalapa
2.
Mexico
Key ecords end phrases:
Maximum Principle, exit times, probabilistic approach, elliptic oper-
ators, diffusion. ABSTRACT In this paper we study the relationship prtl eigcnvalue diffusions.
of second order elliptic operators
A probabilistic
sults established
existing
between
the maximum
and the expected
principle
and the princi-
exit times of the corresponding
approach is discussed that provides a good understanding
analytically.
We also obtain an Alexandrov-Bakelman-Pucci
of classical re-
type of estimate
using
similar methods.
1. INTRODUCTION In what follows we introduce operators.
We emphasize
and probabilistic differentiability
approach.
some notation
and mention
Therefore, we do not insist on minimal assumptions
of the coefficients of the operators
to an elliptic operator for solutions
and the principal of elliptic equations.
Let R be a given bounded
on the domain, the
and so on (see also the concluding
remarks).
In
between the exit times for the diffusion associated
eigenvalue
(section 2). In section 3 we prove an a priori
Finally, section 4 is devoted to some concluding
regular domain and an elliptic operator
LU = a;juziz, + bju,, + cu satisfying
elliptic
that our goal in this note is to present some links between the analytic
the rest of the paper we consider the relationship estimate
several well known facts about
remarks.
54%
Second World Congress of Nonlinear Analysts
and
2. EXIT
TIMES
It is known that if c 5 0 we can problem
corresponding
AND THE PRINCH’AL
construct
to the operator
EIGENVALUE
diffusion Xt which is the solution
a
L. The solution
to the Dirichlet
to the martingale
problem
for L is unique
whenever ET, <
DC,
for any z E $2, where r, is the exit time of Xt for the initial condition expectation
operator
(see [3]).
Bcrestycki, Nirenberg only the principal
and Varadhan
[l] established
eigenvalue X1 of L is positive.
principle implies uniqueness
that the refined maximum
relationship
Theorem
1 1j c 5 0 then
principle holds if and
On the other hand, as is well known, the maximum
of the Dirichlet problem.
between X1 and ET, is to be expected.
A clear-cut
Proof.
20 = z and E stands for the
Using the representation
In fact:
for the principal eigenvalue (see [3], chapter 14, theorem
10.1)
Xl = sup{ X > 0; sup EZeXTz < w} XEn
it easily follows that when c = 0, the boundedness
of E,r,
implies X1 > 0. Indeed, in this case,
the expected exit time is finite for all z E R because of the conditions lemma 7.4.) and applying When c 5 0, ET,
may
Chebychev’s inequality
be infinite,
on L (see also [4], chapter 5,
we get the result.
so we use instead
the killing time that is bounded
by the exit
time of the related diffusion with c z 0 and the result follows. The converse, i.e. X > 0 implies finite expected esit time is immediate It would be interesting but not necessarily
by using 1-t 2 5 e”.
to give a probabilistic
nonpositive.
argument
in the general case, i.e. when c is bounded,
Notice however, that from the analytic
t,heorem 1, one can proceed along the same lines as in [l] (for instance
point of view, once we have theorem 6.1).
5459
Second World Congress of Nonlinear Analysts 3. A PRIORI
Results
as the above motivate
the study
as the Alexandrov-Bakelman-Pucci
of the probabilistic
interpretation
of classical
a priori bounds
inequality:
(ABP inequality) If c 5 0, f E Ln(0),
Theorem
BOUNDS
u E W2:,
p 2 n and
i) ,l2& U(X) 5 0 ii) Lu > j,jcLn(Q),
then
whewB is (1constant
depending on the dimension of the domain n, the ellipticity constant CO of L,
the diameter d oj Q and b. Recently
the dependence
they show that a quantity
of B on the diameter
this dependence
R measuring
of R was improved
on the diameter
the narrowness
in [l] and [2]. Roughly
may be improved
of the domain
by IQ]‘/” or, more precisely,
(see the above references
approach
to prove an inequality
that
works.
the result
and proving
that, for c 5 0, the Feynman-Kac
it, notice
generalizes
for details).
we use a Probabilistic Before stating
speaking,
those presented
by Here
in the cited formula
for
1 1, c < 0 and u E WZJ’ for R bounded such that Lu = f
and
Lu = f u=
0
in
0
on
dS2
implies
for the diffusion Theorern lim,,ao
X1 corresponding
2 Let f
E La,0
to L with X0 = CC.
U(Z) 5 0, then sup lul I B~l~l-“*llf
[IL?
where
r = sup E,r, n Remark.
For Q = n and a = co and using
bounds
presented
Proof.
We may assume
in [2] (theorem that
the fact that
1.4. ) and [l] (Theorem
f is a simple
function
lu(~)l 5 Es 5 E,
Ezrz 5 l52/2/n(see [l]), we obtain 10.1. ) respectively.
and proceed
then by approximation.
6%
If(4ldt
lai IXn,(Xt)dt
Then
the
Second World Congress
5460
for a finite partition
of Nonlinear Analysts
{0,}i of R and f = Ca;xo,
This implies
where
T& =
JoTxni (-Jw
=
“sojourn time in” 0;.
Hence, using Hiilder’s inequality
Under the hypothesis
on L, we have T,5, =
7.c JJ 0
ni
d4 2, YFY&
where q is the Green’s function for the Cauchy problem (see [3] and [5] for the details on the existence of the Green’s function) of the Green’s function
Now we integrate
over Q and use both Fubini’s theorem and the definition
to obtain that the RHS of the above expression is less than or equal to
SUPM~>l< c+y+y =
fwI+llfllL~
4.
Through
this approach the relationship
the coefficients of the operator degenerate
x (~IlfllL.)
CONCLUSION
between the measure of the domain, the Green’s measure and
L is made apparent.
It seems these ideas can be generalized
elliptic case, or when the domain fl is a manifold.
in a subsequent
work.
to the
We intend to pursue these questions
5461
Second World Congress of Nonlinear Analysts REFERENCES
[l] Berectycki,
H., Nirenberg,
L., Varadhan,
The Principal
S.R.S.,
Eigenvalue and Maximum Principle for Second-Order Elliptic Opemtors in Geneml Domains. Comm. Pure
Appl.
[2] Cabre,
Math., X.,
Vol. XLVII,
47-92
(1994).
On the Alexandro&Bakelman-Pucci
Estimate and the Reversed Holder Inequality for
Solutions oj Elliptic and Pambolic Equations. Comm. [3] Friedman,
A., it Stochastic
[4] Karatzas,
I., Brownian
[5] Pinsky,
R.G.,Positive
Differential
Motion
Equations.
and Stochastic
Harmonic
Functions
Pure
Appl.
Vol I. Academic
Calculus.
Springer
Math.,
Vol. XLVIII,
539-570
Press. Verlag
and Diflusion. Cambridge
(1991). Univ.
Press.
(1995).
(1995).