On Bounded Solutions of Elliptic Partial Differential
Equations of the Second Order
By Takasi KUSANO
(Chuo University)
Introduction
This paper deals with linear and nonlinear elliptic partial differential equations of the types
$¥sum_{i,j=1}^{n}a_{ij}(x)¥frac{¥theta^{2}u}{¥partial x_{i}¥partial x_{j}}+¥sum_{i=1}^{n}b_{i}(x)¥frac{¥partial u}{¥partial x_{i}}+c(x)u=f(x)$
$¥sum_{i.j=1}^{n}a_{j}i,(x)¥frac{¥theta^{2}u}{¥partial x_{i}¥partial x_{j}}+¥sum_{¥iota=1}^{n}b_{i}(x)¥frac{¥partial u}{¥theta x_{i}}=f(x, u)$
,
,
and the main concern is with the existence and unicity of their solutions which
in the whole Euclidean -space. No other restricare bounded and of class
tions but boundedness will be imposed on the behaviour of solutions at distant
points in the space.
It should be mentioned that our work is suggested by an interesting paper
the
of M. M. Belova [1], where for ordinary differential equations
$¥
infty<x<
¥
infty$
in
an
has been established
unique existence of bounded solutions in
original way.
The paper consists of two sections. In Section 1 we shall establish several
space, based on a variant of
simple properties of bounded solutions in the
the maximum principle (Theorem 1). Section2 develops the existence theory;
we first discuss the case of linear equations, and then turn to nonlinear, or
rather semilinear equations and show that the bounded solutions may be derived
In doing so, we shall utilize two theorems on the
with the aid of iterations.
interior estimates of solutions of linear elliptic differential equations of the second order : one of them is the classical theorem of J. Schauder, the other being
a modern theorem of S. N. Kruzhkov?an extension of the famous theorem of
Nash.
It is probable that our results may naturally be generalized to the more
general nonlinear elliptic equations. We announce here that this investigation
is now in progress and the results will shortly be published elsewhere.
In terminating the introduction the author wishes to express his most profound gratitude to Professor Masuo HUKUHARA for his constant encouragement
and invaluable criticisms during the preparation of this paper.
$C^{2}$
$¥mathrm{n}$
$y^{¥prime¥prime}=f(x, y, y^{¥prime})$
$¥mathrm{w}¥mathrm{h}¥mathrm{o}¥mathrm{l}¥dot{¥mathrm{e}}$
T. KUSANO
$¥mathrm{Z}$
1.
I.
Properties of bounded solutions
The linear equation $Lu=f(x)$ .
(1)
Consider the linear equation
$Lu¥equiv¥sum_{i,j=1}^{n}a_{ij}(x)¥frac{¥partial^{2}u}{¥theta x_{i}¥partial x_{j}}+¥sum_{i=1}^{n}b_{i}(x)¥frac{¥partial u}{¥partial x_{i}}+c(x)u=f(x)$
with the following hypotheses:
I. At every point in
and for all real
$E^{n}$
$x$
$¥sum_{i,j=1}^{n}a_{ij}(x)¥xi_{i}¥xi_{j}¥geqq a_{0}¥sum_{i=1}^{n}¥xi_{i}^{2}$
,
$n-$
tupfes
$(a_{0}>0)$
$¥xi=(¥xi_{1^{ }},¥cdots, ¥xi_{n})$
,
;
The coefficients $a_{ij}(x)$ ,
, and $c(x)$ and
function $f(x)$ are defined and bounded in
;
$c(x)
¥
leqq-m^{2}<0$
.
for all in .
We begin with a theorem which may be regarded as a variant of the maximum principle and will be of great use in what follows.
Theorem 1. Let a function $v(x)¥in C^{2}$ be bounded above (below) and satisfy
the differential inequality $Lv$ $¥geqq 0(¥leqq 0)$ in
. Then, $v(x)¥leqq 0(¥geqq 0)$ in
.
$Lv$
$v(x)
¥
leqq
0$
Proof. Assuming
we shall prove that
. To this end, $¥sup-$
pose for contradiction that $v(x)$ may be positive at some point .
Fixing ,
we define
$¥mathrm{I}¥mathrm{I}$
.
$b_{i}(x)$
$th¥dot{e}$
$E^{n}$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$E^{n}$
$x$
$E^{n}$
$E^{n}$
$¥geqq 0$
$x^{0}$
(2)
$w(x)=(v(x^{¥alpha})-¥epsilon)¥prod_{i=1}^{n}¥cosh k(x_{i}-x_{i}^{0})$
,
$x^{0}$
$*)$
$(x=(x_{1^{ }},¥cdots,x_{n}), x^{¥Phi}=(x_{1}^{0_{ }},¥cdots, x_{n}^{0}))$
,
where and $¥epsilon(0<¥epsilon<v(x^{l}))$ are positive constants.
An easy computation then yields
$k$
(3)
$Lw=(v(x^{0})-¥mathrm{e})i=n1¥mathrm{I}¥mathrm{I}¥cosh k(x_{i}-x_{i}^{0})[k^{2}i-1¥sum_{-}^{n}a_{ii}(x)+$
$+k^{2}¥sum_{i¥neq j}a_{ij}(x)¥tanh k(x_{i}-x_{i}^{0})¥tanh k(xj-x_{j}^{0})+$
$+k¥sum_{i=1}^{n}b_{i}(x)¥tanh k(x_{i}-x_{i}^{0})+c(x)]¥equiv¥Phi(x;x^{0} ; k)$
.
It is readily verified that in view of the assumptions , , and
we can
select a nonzero value of such that $¥Phi(x;x^{0} ; k)¥leqq 0$ for all in
.
Now the function $v(x)-w(x)$ has the property $L(v-w)¥geqq 0$ in
, which
implies that $v(x)-w(x)$ cannot take on a positive maximum at a finite point of
the space. On the other hand, since $w(x)$ grows, by definition, towards infinity
$¥mathrm{I}$
$k$
$¥mathrm{I}¥mathrm{I}$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$E^{n}$
$x$
$E^{n}$
It must be mentioned that the function (2) is essentially the same
employed in his illuminating paper: Sur
problen ι e de Dirichlet pour l’equation line’aire du type elliptique dans un do maine non borne, Rend. Accad.
Naz. Lincei., 4 (1948), pp. 408-416. (See especially pp. 413-414.)
$*Added$ in proof:
as the one M.
$¥mathrm{K}¥mathrm{r}¥mathrm{z}¥mathrm{y}¥dot{¥mathrm{z}}¥mathrm{a}¥mathrm{n}¥mathrm{s}¥mathrm{k}¥mathrm{i}$
$¥mathfrak{l}e$
On Bounded Solutions of Elliptic Partial Differential Equations of the Second Order
, while $v(x)(v(x^{0})>w(x^{0}))$ is bounded above by hypothesis, the funas
ction $v(x)-w(x)$ necessarily becomes negative in some neighbourhood of infinity.
Thus, $v(x)-w(x)$ is shown to attain a positive maximum at some finite point,
contrary to what was just remarked. Hence the proof is complete.
Theorem 2. (Uniqueness theorem) There cannot be more than one solution
in
.
of equation (1) which is bounded and of class
and
are bounded solution of equation (1), then their
Proof. If
difference $v(x)=u_{1}(x)-¥mathrm{u}_{2}(x)$ is also bounded and satisfies the homogeneous equ¥
in
.
ation $Lv=0$ . By Theorem 1 we conclude that $v(x)¥equiv 0$ , or
$u(x)
¥
in
C^{2}(E^{n})$
equation
(1), there
Theorem 3. For a bounded solution
of
the
estimate:
holds
$|x|¥rightarrow¥infty$
$E^{n}$
$C^{2}$
$u_{2}(x)$
$u_{1}(x)$
$u_{1}(x) equiv u_{2}(x)$
(4)
$E^{n}$
$¥sup_{x¥in E^{¥dot{n}}}|u(x)|¥leqq¥frac{M}{m^{2}}$
where
$M=¥sup_{x¥in E^{¥mathfrak{n}}}|f(x)|$
Proof.
quality
.
The functions
$v_{¥pm}(x)=M/m^{2}¥pm u(x)$
both satisfy the diffential ine-
.
, or equivalently,
$-M/m^{2}¥leqq u(x)¥leqq M/m^{2}$ in
.
Theorem 4. The bounded solution of equation (1) depends continuousl on
the coefficient $c(x)$ and the function $f(x)$ . If, moreover, the solution of equation
(1) possesses bounded derivatives up to the second order, then it continuously
, and $c(x)$ and the function $f(x)$ .
depends on the coefficients $a_{ij}(x)$ ,
Proof. For the proof we adopt the method due to O. A. Oleinik [8]. Consider, together with (1), the equation
$Lv_{¥pm}=c(x)M/m^{2}¥pm f(x)¥leqq-M¥pm f(x)¥leqq 0$
Theorem 1 then implies that
$v_{¥pm}(x)¥geqq 0$
$E^{n}$
$¥mathrm{y}$
$b_{i}(x)$
(1’)
$L^{¥prime}u^{¥prime}¥equiv¥sum_{i,j=1}^{n}a_{ij}^{¥prime}(x)¥frac{¥partial^{2}u^{¥prime}}{¥partial x_{i}¥partial x_{j}}+¥sum_{i=1}^{n}b_{i^{J}}(x)¥frac{¥partial}{¥partial}¥frac{u^{¥prime}}{x_{i}}+c^{¥prime}(x)u^{¥prime}=f^{¥prime}(x)$
. Let $u(x)$ and
equipped with the same restrictions , , and
. Then, obviously,
respective bounded solutions of (1) and
$¥mathrm{I}$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$¥mathrm{I}¥mathrm{I}$
$u^{¥prime}(x)$
be the
$(1^{¥prime})$
$L(u-u^{¥prime})=-[(.f^{¥prime}-f)+(c-c^{¥prime})u^{¥prime}+¥sum_{i=1}^{n}(b_{i}-b_{i^{J}})-¥partial¥theta¥frac{u^{¥prime}}{x_{i}}$
$+¥sum_{i,j=1}^{n}$
$(a_{ij}-a_{ij}^{¥prime})¥frac{¥partial^{2}}{¥partial x_{i}}¥overline{¥partial}u^{¥prime}x_{j}]¥equiv F(x)$
.
By Theorem 3 it follows that
$¥sup_{x¥in E^{n}}|u(x)-u^{¥prime}(x)|¥leqq¥frac{N}{m^{2}}$
,
$N=¥sup_{x¥in E^{n}}|F(x)|$
,
This establishes the
has bounded derivatives up to the second order.
if
desired continuous dependence of the solution, thereby proving the second part
of the theorem. The first part is easier.
$u^{¥prime}(x)$
T. KUSANO
Remark 1. It is to be regretted that our Theorem 2 is weaker than the
result of S. Simoda and M. Nagumo [11]. In so far as our theory is concerned, however, those somewhat stronger hypotheses adopted in this paper do not
seem so unnatural and would be enough for the time being.
Remark 2. Note that the assumption in the second part of Theorem 4 is
automatically fulfilled if we add the assumption:
. The functions $a_{ij}(x)$ ,
, $c(x)$ , and $f(x)$ are uniformly Holder conti; which will always be assumed
nuous {exponent ) on any compact subsets in
theory.
I
2,
existence
Section
and
the
(See
in
)
. The nonlinear equation $¥Lambda u=f(x, u)$ . In this paragraph we shall be
concerned with the nonlinear elliptic equation of the form
$b_{i}(x)$
$¥mathrm{I}¥mathrm{V}$
$E^{n}$
$¥alpha$
$¥mathrm{I}¥mathrm{I}$
$¥mathrm{I}¥mathrm{I}$
(5)
$¥Lambda u¥equiv¥sum_{i,j=1}^{n}a_{ij}(x)¥frac{¥partial^{2}}{¥partial x_{i}}¥partial u¥partial¥overline{x_{j}}¥partial+¥sum_{i=1}^{n}b_{i}(x)-¥frac{u}{x_{i}}=f(x, u)$
.
Concerning equation (5) we make the following assumptions:
. At every point in
and for every real vector ,
$¥mathrm{I}^{*}$
$x$
$E^{n}$
$¥xi$
$¥sum_{i,j=1}^{n}a_{ij}(x)¥xi_{i}¥xi_{j}¥geqq a_{0}¥sum_{i=1}^{n}¥xi_{i}^{2}$
.
,
$(a_{0}>0)$
;
The coefficients $a_{ij}(x)$ and
are bounded in
$funcri
¥
sigma
nf(x,u)$
The
is
in
.
defined the region
and is subject to the conditions:
;
1) $f(x,0)$ is bounded in
$¥mathrm{I}¥mathrm{I}^{*}$
$b_{i}(x)$
$E^{n}$
;
$¥mathfrak{D}=¥{x¥in E^{n}, -¥infty<u<_{¥backslash }¥infty¥}$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}^{*}$
$E^{n}$
2)
$¥frac{¥partial f(x,u)}{¥partial u}¥geqq m^{2}>0$
for
all
$(x, u)¥in ¥mathfrak{D}$
.
Observing that by the mean value theorem we may rewrite equation (5) in
the form
(5’)
$¥mathfrak{L}$
$u¥equiv¥Lambda u-¥frac{¥theta f(x,¥tilde{¥mathrm{u}})}{¥partial u}u=f(x, 0)$
,
lies between 0 and $u(x)$ , we shall be able to verify the following
where
theorems marked with asterisks in exactly the same manner as in the preceding
paragraph.
in
,
a bounded solution $u(x)¥in C^{2}$
Theorem 1*. If $f(x, 0)¥geqq 0$
$u(x)
¥
leqq
0$
in
.
of (5) has the property
Theorem 2*. There cannot be more than one solution of equation (5) wh.ich
is bounded and of class
in the whole of
.
Theorem 3*. For a bounded solution $u(x)¥in C^{2}$ of equation (5), we have the
estimate:
$¥tilde{¥mathrm{u}}(x)$
$C^{2}$
$E^{n}$
$(¥geqq 0)$
$E^{n}$
$th¥dot{e}n$
$E^{n}$
$¥sup_{x¥in E^{n}}|u(x)|¥leqq¥frac{M}{m^{2}}$
$(4^{*})$
where
$(¥leqq 0)$
$M=¥sup_{x¥in E^{n}}|f(x, 0)|$
.
On Bounded Solutions of Elliptic Partial Differential Equations of the Second Order
Theorem 4*.
The bounded solution of equation (5) depends continuously on
the function
. If, moreover, the bounded solution has bounded derivatives
up to second order, then it continuously depends on $a_{ij}(x)$ ,
, and $f(x, 0)$ .
Theorem 5. Let $u(x)$ be a bounded solution of equation (5) in
and let
satisfying in
and
the differential
be bounded functions of class
inequalities
,
and
respectively. Then, we have in
$f(x, 0)$
$b_{i}(x)$
$E^{n}$
$¥overline{¥omega}(x)$
$E^{n}$
$C^{2}$
$¥underline{¥omega}(x)$
$¥Lambda¥Phi¥leqq f(x,¥overline{¥omega})$
$¥Lambda¥underline{¥omega}¥geqq f(x,¥underline{¥omega})$
$E^{n}$
(6)
$E^{n}$
$¥underline{¥omega}(x)¥leqq u(x)¥leqq¥overline{¥omega}(x)$
Proof. By hypotheses,
So that,
.
$¥underline{¥omega}(x)$
satisfies
.
$¥Lambda¥underline{¥omega}=f(x,¥underline{¥omega})+¥varphi(x)$
with
$¥Lambda(¥underline{¥omega}-u)=f(x,¥underline{¥omega})-f(x, u)+¥varphi(x)=_{¥partial}^{¥underline{¥partial f(}}¥frac{x,¥tilde{u})}{u}$ $(¥underline{¥omega}-u)+¥varphi(x)$
$¥varphi(x)¥geqq 0$
,
By Theorem 1 we have
and $u(x)$ .
since
. The remaining inequality can be proved analogously.
Theorem 6. Let there be given the differential equations
$¥Lambda u=f(x, u)$ ,
(7)
lying between
$¥tilde{¥mathrm{u}}(x)$
in
$¥underline{¥omega}(x)-u(x)¥leqq 0$
$¥underline{¥omega}(x)$
,
$¥varphi(x)¥geqq 0$
(8)
$¥Lambda v=g(x, v)$
where both
$f(x, u)$
and
$g(x, v)$
,
satisfy the assumption
$s¥mathrm{I}^{*}$
,
$¥mathrm{I}¥mathrm{I}^{*}$
, and
$¥mathrm{I}¥mathrm{I}¥mathrm{I}^{*}$
. Let
further
(9)
for
$f(x, u)¥geqq g(x, u)$
$(f(x, u)¥leqq g(x, u))$
.
Let finally $u(x)$ and $v(x)$ in
tions of (7) and (8).
Then, the following inequality holds in
$(x, u)¥in ¥mathfrak{D}$
$C^{2}$
$E^{n}$
(10)
$u(x)¥leqq v(x)$
Proof. The difference
(11)
be the respective bounded solu-
:
$(u(x)¥geq--v(x))$
$u(x)-v(x)$
.
satisfies
$¥Lambda(u-v)=f(x,u)$ $-f(x, v)+f(x, v)$ $-g(x,v)$
$=_{¥partial^{¥frac{x,¥tilde{¥mathrm{u}})}{u}}}^{¥underline{¥partial}¥underline{f}(}$
$(u-v)+f(x, v)$ $-g(x, v)$ ,
lying between $u(x)$ and $v(x)$ , whence the desired inequality (10) is derived
in virtue of Theorem 1 and (9).
Theorem 7. Let $u(x)$ and $v(x)$ in
be the bounded solutions of (7). and
(8), respectively, and let the inequality
$¥tilde{u}(x)$
$C^{2}$
(12)
hold
$|f(x, u)$ $-g(x, u)|¥leqq N$
for
all
$(x, u)¥in ¥mathfrak{D}$
.
Then, we have
(13)
where
$|u(x)-v(x)|¥leqq¥frac{N}{k}$
$k=¥max¥{¥inf¥frac{¥partial f(x,u)}{¥partial u}$
,
$¥inf¥frac{¥partial g(x,v)}{¥partial v}¥}$
.
Proof. From Theorem 3 in view of (11) it follows that
T. KUSANO
$|u(x)-v(x)|¥leqq¥frac{N}{¥inf^{¥underline{¥partial}¥underline{f}(}¥partial^{¥frac{x,u)}{u}}}$
Analogously, replacing
$f(x, u)$
by
$g(x, v)$
, we obtain
$|u(x)-v(x)|¥leqq¥frac{N}{¥inf¥frac{¥partial g(x,v)}{¥partial v}}$
,
thus completing the proof.
Theorem 8. In this theorem we consider the equation
$¥Lambda u=f(u)$
(14)
under the following assumptions
i) The coefficients of the operator
are subject to the conditions
and
as before ;
The function $f(u)$ , defined for $-¥infty<u<¥infty$ , has the property
$¥geqq m^{2}>0$ .
Then, this equation has a unique solution
, bounded and of class
$I^{*}$
$¥Lambda$
$¥mathrm{I}¥mathrm{I}^{*}$
$f^{¥prime}(u)$
$¥mathrm{i}¥mathrm{i})$
$ u(x)¥equiv¥alpha$
in
$E^{n}$
, where
$¥alpha$
is a constant such that
$f(¥alpha)=0$
and
$C^{2}$
$|¥alpha|¥leqq¥frac{|f(0)|}{m^{2}}$
.
Proof. By the mean value theorem:
$f(¥mathrm{u})=f(0)+f^{¥prime}(¥theta u)u$
it is clear that
,
$0<¥theta<1$ ,
and $f(u)<0$ providing
$u<-|f(0)|/m^{2}$ .
Hence there exists a constant
such that $f(¥alpha)=0$ and
$|¥alpha|¥leqq|f(0)|/m^{2}$ .
That
is a solution of (14) is evident. We shall show
that this is the only solution, bounded and of class
in
.
In fact, let
$v(x)¥in C^{2}$ be a bounded solution of (14).
$
v(x)¥
alpha$
Then,
satisfies the equation
$¥Lambda(v-¥alpha)=f(v)-f(¥alpha)=f^{¥prime}(¥alpha+¥theta^{¥prime}(v-¥alpha))(v-¥alpha)$,
where
, from which, by Theorem 1, the assertion that
follows.
Theorem 9. Consider an elliptic equation
$f(u)>0¥mathrm{p}¥mathrm{r}¥mathrm{o}¥mathrm{v}¥mathrm{i}¥mathrm{d}¥tilde{¥mathrm{i}}¥mathrm{n}¥mathrm{g}$
$u>|f(0)|/m^{2}$
$ u¥equiv¥alpha$
$ u(x)¥equiv¥alpha$
$E^{n}$
$C^{2}$
$0<¥theta^{¥prime}<1$
$ v(x)¥equiv¥alpha$
(15)
with constant
$¥Lambda u=f(x, u)$
coefficients. If $f(x, u)$ satisfies the
(16)
,
$|_{¥partial}^{¥underline{¥theta}¥underline{f}(}¥frac{x,u)}{x_{i}}|¥leqq L$
as well as condition
$¥mathrm{I}¥mathrm{I}¥mathrm{I}^{*}$
$i=1$
inequalities
,
$¥cdots$
, then a bounded solution
, ,
$n$
’
$ u(x)¥in C¥cdot$
of (15) has the pro-
perty
(17)
$|_{¥theta}^{¥partial}-¥frac{u}{x_{i}}|¥leqq¥frac{L}{m^{2}}$
If, on the other hand,
(18)
for
$(x, u)¥in ¥mathfrak{D}$
$f(x, u)$
,
$i=1$
,
satisfies the
$|¥frac{¥partial f(x,u)}{¥theta x_{i}}|¥leqq K¥underline{¥partial}¥frac{f(x,u)}{¥partial u}$
as well as condition
$¥mathrm{I}¥mathrm{I}¥mathrm{I}^{*}$
$¥cdots$
, .
$n$
inequalities
,
$i=1$ ,
$¥cdots$
, ,
$n$
, then a bounded solution
$u(x)¥in C^{2}$
has
On Bounded Solutions of Elliptic Partial
Differential
of the Second Order
Equations
the property
,
(19)
$i=1$
$|¥frac{¥partial u}{¥partial x_{i}}|¥leqq K$
Proof. Clearly, each
$v=¥partial u/¥partial x_{i}$
,
$¥cdots$
, .
$n$
satisfies
$¥Lambda v=¥frac{¥partial f}{¥partial x_{i}}+¥frac{¥theta}{¥theta}¥frac{f}{u}v$
.
By Theorem 3 with the aid of (16) we see that (17) is valid.
On the other hand, the function $v-K$ satisfies
$¥Lambda(v-K)=_{X_{i}}^{¥underline{f}}¥frac{¥partial}{¥partial}+K¥frac{¥partial f}{¥partial u}+¥frac{¥partial f}{¥partial u}(v-K)$
According to Theorem 1 and (18), it follows that
ing inequality $v¥geqq-K$ may be derived similarly.
.
$v-K¥leqq 0$
in
$E^{n}$
. The remain-
Existence of bounded solutions
2.
I. Preliminaries. We begin by formulating two theorems concerning the
interior estimates of solutions of second order linear elliptic equations, which
will be of central importance in our existence proofs.
We shall introduce the
A) J. Schauder’s interior estimates ([10], [3]).
$D$
with
Douglis-Nirenberg norms as in [6].
Let
be a bounded domain in
$E^{n}$
boundary
and set
for
functions
define
$¥dot{D}$
$¥mathrm{d}_{¥mathrm{x}}=¥mathrm{d}¥mathrm{i}¥mathrm{s}¥mathrm{t}$
.
$(x,¥dot{D})$
and
$dxy^{=¥min}(d_{x}, d_{y})$
for
$x$
,
$y¥in D$
.
We
$w(x)¥in C^{i}(D)$
,
(1)
$|w|_{p,i}^{D}=¥sum_{j=0}^{i}¥sup_{x¥in D}d_{x}^{p+j}|D^{j}w(x)|$
(2)
$|w|_{pi+a}^{D},=|w|_{p,i}^{D}+¥sup_{x,y¥in D}d_{xy}^{p+i+¥mathrm{a}}¥frac{|D^{i}w(x)-D^{i}w(y)|}{|x-y|^{oe}}$
,
where $0<¥alpha<1$ and stands for an arbitrary real number and the suprema in
order, respectively,
and
(1) and (2) are taken over all derivatives of the
as well as over the domain.
J. Schauder’s interior estimates are then expressed in terms of these norms
as follows.
Theorem A. Concerning the elliptic equation
$p$
$i^{¥mathrm{t}¥mathrm{h}}$
$j^{¥mathrm{t}¥mathrm{h}}$
(3)
$Lu¥equiv¥sum_{i,j=1}^{n}a_{ij}(x)¥frac{¥partial^{2}u}{¥partial x_{i}¥partial x_{j}}+¥sum_{i=1}^{n}b_{i}(x)¥frac{¥partial u}{¥partial x_{i}}+c(x)u=f(x)$
we make the following assumptions
i) At all points in $D$ and for all real vectors ,
$¥xi$
$x$
$¥sum_{i,j=1}^{n}a_{ij}(x)¥xi_{i}¥xi_{j}¥geqq K_{1}¥sum_{i=1}^{n}¥xi_{i}^{2}$
$¥mathrm{i}¥mathrm{i})$
The
$a_{ij}(x)$
,
$b_{i}(x)$
,
$c(x)$
, and
$|a_{i_{j}}|_{0i+¥alpha}^{D},¥leqq K_{2}$
,
$f(x)$
are
,
$(K_{1}>0)$
defined
$|b_{i}|_{1i+¥alpha}^{D},¥leqq K_{2}$
,
in
$D$
;
and satisfy
$|c|_{2i+¥mathrm{a}}^{D},¥leqq K_{2}$
.
T. KUSANO
Then,
for any
solution
(4)
$u(x)¥in C^{i+2+¥alpha}(D)$
of equation (3) we have
$|u|_{0i+2+¥alpha}^{D},¥leqq K(_{1}^{1}u|_{00}^{D},+|f^{D}|_{2i+¥alpha},)$
,
,
, , , and .
is a constant depending only on
B) S. N. Kruzhkov’s interior estimates ([4], [9]). We shall explain a
theorem due to S. N. Kruzhkov which establishes the interior Holder continuity of bounded solutions of linear uniformy elliptic equations with bounded
where
$K$
$K_{1}$
$K_{2}$
$i$
$n$
$¥alpha$
$¥mathrm{m}¥mathrm{o}¥mathrm{d}-$
$¥mathrm{e}¥mathrm{r}¥mathrm{n}$
coefficients. It is a generalization of the well-known Nash’s estimation theorem.
Theorem B. Let there be given an elliptic equation of the form
(5)
$¥sum_{j=1}^{n}¥frac{¥partial}{¥partial x_{i}}(a_{ij}(x)¥overline{¥partial}¥frac{¥partial u}{x_{j}})+¥sum_{i=1}^{n}b_{i}(x)¥frac{¥partial u}{¥partial x_{i}}=f(x)$
$i$
,
with the following hypotheses
i) At each point in $D$ (bounded or unbounded) and for each real vec$tor$
,
$x$
$¥xi$
$¥mu_{1}¥sum_{i=1}^{n}¥xi_{i}^{2}¥leqq¥sum_{i,j=1}^{n}a_{ij}(x)¥xi_{i}¥xi_{j}¥leqq¥mu_{2}¥sum_{i=1}^{n}¥xi_{i}^{2}$
The
$¥mathrm{i}¥mathrm{i})$
,
$(0<¥mu_{1}¥leqq¥mu_{2})$
;
and $f(x)$ are bounded in $D$ :
$|b_{i}(x)|¥leqq B$ , $i=1$ ,
; $|f(x)|¥leqq N$.
functions
$b_{i}(x)$
$¥cdots,n$
denote the maximal subdomain of $D$ such that dist.
Let
.
$u(x):|u(x)|
¥
leqq
M$
equation
solution
any
bounded
Then,
(5) has the property
of
, $0<¥delta<1$ ,
that for ,
$(D^{¥delta},¥dot{D})=¥delta$
$D^{¥delta}(¥delta>0)$
$x^{1}$
(6)
$x^{2}¥in D^{¥delta}$
$|u(x^{1})-u(x^{2})|¥leqq A¥max[¥frac{M+N}{¥delta^{ae}},$
$(M+N)B^{¥mathrm{o}¥mathrm{e}}]|x^{1}-x^{2}|^{a}$
,
are constants depending only on
and .
Remark. This remark is important. With the aid of S. N. Kruzhkov’s
device (see, [4], [5], [9]) the interior Holder continuity of bounded solutions
may also be established concerning uniformly elliptic equations of the form,
where
$A$
and
$¥alpha(0<¥alpha<1/2)$
(7)
$¥mu_{1)}¥mu_{2}$
$n$
$¥sum_{i,j=1}^{n}¥frac{¥theta}{¥partial x_{i}}(a_{ij}(x)-¥partial¥partial¥overline{¥partial}¥partial x_{j}-^{¥underline{u}})+¥sum_{i=1}^{n}b_{i}(x)¥frac{u}{x_{i}}+c(x)u=f(x)$
with bounded coefficients. The method consists in introducing a new independent variable and considering the function $v=yu(x)$ as a solution of equation
of the form (6)
$y$
(8)
$¥sum_{i=1}^{n}¥frac{¥theta}{¥partial x_{i}}(a_{ij}(x)¥frac{¥partial v}{¥partial x_{j}})+¥sum_{i=1}^{n}¥frac{¥partial}{¥theta y_{i}}(yb_{i}(x)¥frac{¥partial v}{¥partial x_{i}})^{X}+¥frac{¥partial}{¥partial y}(K¥frac{¥partial v}{¥theta y})$
?
$¥sum_{i=1}^{n}b_{i}(x)-¥partial¥partial¥frac{v}{x_{i}}+yc(x)¥frac{¥partial v}{¥partial y}=yf(x)$
.
Here the constant $K>0$ has to be chosen so large that the
$¥sum_{i=1}^{n}yb_{i}¥xi_{i}¥eta+K¥eta^{2}$
$¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{m}¥sum_{i.j=1}^{n}a_{ij}¥xi_{i}¥xi_{j}+$
associated with (8) becomes positive definite in a domain
$fx¥in D$ ,
On Bounded Solutions of Elliptic Partial Differential Equations of the Second Order
$0<y<s¥}$ ,
$¥mathrm{I}¥mathrm{I}$
being a constant.
The linear equation $Lu=f(x)$ .
.
$s$
Theorem 10.
(Existence theorem) There exists a unique solution, bounded
class
in
, of equation (3), provided that the conditions , ,
,
of
stated in Section 1 are satisfied.
Proof. Since the unicity of the solution is guaranteed by Theorem 2, it is
enough to construct a solution having the demanded properties. This construction is carried out by the method usually employed for unbounded domains;
namely, the solution will be found by taking limits of functions constructed as
solutions of the Dirichlet problem for a sequence of domains with growing diameters.
Let indeed
, $k=1,2$ , be a sequence of open balls centered at the ori-
and
and
$E^{n}$
$C^{2}$
$¥mathrm{I}$
$¥mathrm{I}¥mathrm{I}$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$¥mathrm{I}¥mathrm{V}$
$¥{B_{k}¥}$
$¥cdots$
gin, with radii
tending monotonically to infinity and with boundaries
.
For each we set the Dirichlet problem for
with the homogeneous boundary condition
$Lu_{k}=f(x)$
(9)
in
,
; $k=1,2$ , .
By J. Schauder’s existence theorem each of these problems is uniquely solvable,
thus a sequence of solutions
is obtained.
From the maximum principle
and Theorem A applied to the individual solutions of (9) it follows that the
sequence
is locally uniformly bounded and locally equicontinuous. In fact,
the maximum principle yields
$¥{r_{h}¥}$
$¥{¥dot{B}_{k}¥}$
$k$
$B_{k}$
$¥mathrm{s}$
$B_{k}$
$¥cdots$
$u_{k}|_{¥dot{B}_{k}}=0$
$¥{u_{k}¥}$
$¥{u_{h}¥}$
$|u_{k}(x)|¥leqq¥frac{M}{m^{2}}$
,
in
, for all , which implies that
Theorem A we have in a fixed ball
$k$
$B_{k}$
$M=¥sup_{x¥in E^{n}}|f(x)|$
$¥{u_{k}¥}$
is uniformly bounded.
$l$
$>k$ ,
where
$K$
According to
$B_{k}$
$|u_{l}|_{02+a}^{B_{k}},¥leqq K(|u_{l_{00}}^{1^{B_{k}}},+_{¥mathrm{I}}^{1}|f|_{2a}^{B_{k}},)$
for all
,
does not depend on /.
,
Hence the quantities
$|u_{l}|_{02+¥alpha}^{B_{k}}$
,
are bounded by a constant independent of $l>k$ .
This shows the local equicontinuity of the sequence. By Ascoli-Arzela theorem and the usual diagonalization process we can extract from
a subsequence
that converges
(together with derivatives up to second order) on every closed ball
to a bounded function of class
(and its corresponding derivatives). This is nothing
else but the desired solutio of equation (3).
. The nonlinear equations $Lu=f(x, u)$ and $¥Lambda u=f(x, u)$ .
We are now in a position to construct bounded solutions for nonlinear elliptic equations. By using the method of successive approximations (see [2]), we
shall first of all find bounded solutions of the equation of the form
$¥{u_{k}¥}$
$¥{u_{k¥nu}¥}$
$¥overline{B}_{k}$
$u$
$C^{2}$
$¥mathrm{n}¥sim$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
T. KUSANO
10
(10)
,
$Lu¥equiv¥sum_{i.i=1}^{n}a_{ij}(x)¥frac{¥partial^{2}u}{¥partial x_{i}¥partial x_{j}}+¥sum_{i=1}^{n}b_{i}(x)-¥frac{¥partial u}{¥partial x_{i}}+c(x)u=f(x, u)$
in case $f(x, u)$ is bounded. The case where
and the equation of the form
(11)
$f(x, u)$
is of more general structure
$¥Lambda u¥equiv¥sum_{i.j=1}^{n}a_{ij}(x)¥frac{¥partial^{2}u}{¥partial x_{i}¥partial x_{j}}+¥sum_{i=1}^{n}b_{i}(x)¥frac{¥partial u}{¥partial x_{i}}=f(x, u)$
.
will be discussed afterwards.
A) Concerning equation (10) we make the following assumptions:
At each point
I.
$x$
in
$E^{n}$
and
for
all real
$¥mu_{1}¥sum_{i-1}^{n}-¥xi_{i}^{2}¥leqq¥sum_{i.j=1}^{n}a_{ij}(x)¥xi_{i}¥xi_{j}¥leqq¥mu_{2}¥sum_{i=1}^{n}¥xi_{i}^{2}$
$n-$
,
tuples
$¥xi$
,
$(0<¥mu_{1}¥leqq¥mu_{2})$
;
The functions $a_{ij}(x)$ ,
,
, and $c(x)$ are bounded in
and
uniformly Holder continuous (exponent ) on any compact subsets of
;
$c(x)
¥
leqq-m^{2}<0$
.
;
for in
. The function $f(x, u)$ is bounded : $|f(x, u)|¥leqq N$, and continuous with its
¥
in
derivative ¥
, $-¥infty<u<¥infty$ }. Moreover, $f(x, u)$ is uniformly Holder continuous (exponent ) on any compact subsets of
for each
fixed value of .
We observe that in this case equation (10) can be written in the form
$¥mathrm{I}¥mathrm{I}$
.
$¥partial a_{ij}/¥theta x_{i}$
$E^{n}$
$b_{i}(x)$
$E^{n}$
$¥alpha$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$E^{n}$
$x$
$¥mathrm{I}¥mathrm{V}$
$ partial f(x,u)/ theta u$
$¥mathfrak{D}=¥langle x¥in E^{n}$
$E^{n}$
$¥alpha$
$u$
.
$(10^{¥prime})$
$¥sum_{i.j=1}^{n}¥frac{¥theta}{¥partial x_{i}}(a_{ij}(x)¥frac{¥theta¥alpha}{¥partial x_{j}})+¥sum_{¥iota=1}^{n}b_{i^{¥prime}}(x)¥frac{¥partial u}{¥theta x_{i}}+c(x)u=f(x)$
Now let
be a bounded solution of the equation
$LU=-N$,
$N$
where
is a constant bounding $f(x,u)$ .
According to Theorem 10, Theorem
1 and Theorem A it follows that such a function $U(x)$ exists, is unique, is nonnegative and is uniformly Holder continuous on any compact subsets of
.
Consider the sequence of functions
defined by the equations
$¥mathfrak{L}u_{k}¥equiv Lu_{k}-hu_{k}=f(x, u_{k-1})-hu_{k-1}$ ,
(12)
$u_{0}=U$ ;
$k=1,2$ ,
,
where we have set $h=¥sup(¥partial f(x, u)/¥partial u)$ for all in
and $-¥sup U¥leqq u¥leqq¥sup$
$U(x)¥in C^{2}$
$E^{n}$
$¥{u_{k}¥}$
$¥cdots$
$x$
$E^{n}$
.
$U$
If
is bounded and uniformly Holder continuous (exponent ) on any
compact subsets of
, then so is the right-hand side of (12).
By our Theorem10, these functions
do exist. It is to be shown that the required solution may be obtained as the limit of the sequence
. To this end, note first
that
$¥alpha$
$u_{k-1}$
$E^{n}$
$u_{k}$
$¥{u_{k}¥}$
$¥mathfrak{L}u_{1}=f(x, U)$
from which the inequality
$u_{1}¥leqq U$
$-hU¥geqq-N-hU=¥mathfrak{L}U$,
follows in virtue of Theorem 1.
$Lu_{1}=f(x, U)+h(u_{1}-U)¥leqq N=L(-U)$ ,
Consequently
On Bounded Solutions of Elliptic Partial Differential Equations of the Second Order 11
so that, $u_{1}¥geqq-7$, by Theorem 1.
Suppose that the relation
(13)
holds in
?
$E^{n}$
for some .
$k$
$U(x)¥leqq u_{k}(x)¥leqq u_{k-1}(x)¥leqq U(x)$
Then,
,
$¥mathfrak{L}(u_{k+1}-u_{k})=f(x, u_{k})-f(x, u_{k-1})-h(u_{k}-u_{k-1})¥geqq 0$
by the definition of A.
Hence,
$u_{k+1}¥leqq u_{k}$
, by Theorem 1.
From the relation
$Lu_{k+1}=f(x, u_{k})+h(u_{k+1}-u_{k})¥leqq N=L(-U)$ ,
it follows immediately that
$u_{k+1}¥geqq-U$
in
$E^{n}$
.
the relation (8) holds for all
, the seqence
monotonically nonincreasing in
.
It remains to prove that the limit
$k;¥mathrm{i}.¥mathrm{e}.$
Thus, an induction shows that
$¥{u_{k}¥}$
is uniformly bounded and
$E^{n}$
$u(x)=¥lim_{k¥rightarrow¥infty}u_{k}(x)$
is the solution in question.
Noting that the functions
are uniformly bounded and applying S. N. Kruzhkov’s interior estimates (Theorem and Remark), we see that the
uniformly
Holder continuous in
with exponent and constant independent of $m$ .
Consequently, by J. Schauder’ interior estimates (Theorem A), we verify that the
second derivatives of the
are uniformly bounded in absolute value and equicontinuous on every compact subset of
By Ascoli-Arzela’s theorem, the
.
diagonalization process and the monotonicity of
, we may assert that the
entire sequence converges in
, together with their derivatives up to second
order, to the function $u(x)$ and its corresponding derivatives. Letting
tend
to infinity in (12) we finally conclude that $u(x)$ is a bounded solution of (10).
Thus, our existence-proof is complete.
Remark. The following example shows that the bounded solution of equation (10) need not be unique.
Example. Consider, for simplicity, the eqation
$u_{k}$
$¥mathrm{B}$
$u_{k}$
$E^{n}$
$¥mathrm{s}$
$u_{k}$
$E^{n}$
$¥{u_{k}¥}$
$E^{n}$
$k$
$Lu¥equiv u^{¥prime¥prime}-u=f(u)$
in one independent variable , where $f(u)$ is a bounded smooth fnuction which
equals ?2 for $-1¥leqq u¥leqq 1$ . The functions $u=k¥cos x$ , $-1¥leqq k¥leqq 1$ , are solutions of this equation.
The question naturally arises, do there exist the maximal and the minimal
solutions among all bounded solutions of (10) ?. The answer is positive and the
truth is that this problem has already been settled. For, the solution constructed just above is the maximal solution. In fact, if is an arbitrary bounded
solution of (10), then, by Theorem 1, $v¥leqq U$, and by induction,
, $k=0,1,2$ ,
In order to obtain the minimal solution it suffices to repeat our iteration
process, starting with $u_{0}=-U$.
B) Consider the equation (10) under the same assumptions as in A) except
$x$
$u$
$u$
$v$
$v¥leqq u_{k}$
$¥cdots$
T. KUSANO
12
that the assumption
$¥mathrm{I}¥mathrm{V}^{*}$
.
$¥mathrm{I}¥mathrm{V}$
is replaced by more general one
$f(x,u)=f_{1}(x, u)+f_{2}(x, u)$ ,
¥
¥
.
are bounded, and ¥
where $f_{1}(x, u)$ ,
for
Equation (10) with these assumptions is referred to as $(10^{*})$ .
We shall show that there exists at least one bounded solution of equation
$(10^{*})$ .
Let us observe that equation $(10^{*})$ can be rewritten in the form
$¥tilde{L}u¥equiv Lu-¥frac{¥partial f_{2}(x,¥tilde{¥mathrm{u}})}{¥partial u}u=f_{1}(x, u)+f_{2}(x,0)$
where
$¥tilde{¥mathrm{u}}(x)$
lies between 0 and
(14)
$(x, u)¥in ¥mathfrak{D}^{*)}$
$ partial f_{2}(x,u)/ theta u geqq 0$
$¥partial f_{1}(x,u)/¥partial u$
,
Therefore, by Theorem 3, we obtain
$u(x)$ .
$|u(¥backslash x)|¥leqq_{2}^{¥frac{1}{m}}-¥sup|f_{1}(x, u)+f_{2}(x, ¥mathrm{O})|¥equiv K$
,
.
for a bounded solution $u(x)$ of
continuously
Denoting by u(u) a
differentiable and monotonically increasing
function of for $-¥infty<u<¥infty$ , such that u-(u)=u for $|u|¥leqq K$, and |u(u)|≦ 2K
for $-¥infty<u<¥infty$ , we consider the equation
$(10^{*})$
$u$
$Lu=f_{1}(x, u)+f_{2}¥mathrm{A}¥mathrm{A}(x, u)¥equiv f(x, u)$
$(1¥hat{0})$
,
( , u(u)).
, a bounded solution
Since, obviously, $¥hat{f}_{2}(x,0)=f_{2}(x,0)$ and
is controlled by the estimate (14), so that, it is also a solution of equof
According to the conclusion of A) it follows that equation
.
ation
,
has at least one bounded solution of class
and hence,
, since
is bounded by some constant.
where we have set
$f_{2}¥mathrm{A}(x, u)=f_{2}$
$x$
$¥partial f_{2}¥mathrm{A}(x, u)/¥theta u¥geqq 0$
$(1¥hat{0})$
$(¥hat{1}0)$
$(10^{*})$
$C^{2}(E^{n})$
$(10^{*})$
$¥hat{f}(x, u)$
$|¥hat{f}(x, u)|¥leqq¥sup|f_{1}(x, u)+¥hat{f}_{2}(x,u)|¥leqq N$
. It goes without saying that equation $(10^{*})$ has the maximal and
for
minimal solutions.
C) The more general nonlinear equation (11) with $f(x, u)=f_{1}(x,u)+f_{2}(x, u)$
such that $f_{1}(x, u)$ is bounded and $¥theta f_{2}(x, u)/¥partial u¥geqq m^{2}>0$ , is equivalent to
$(x, u)¥in ¥mathfrak{D}$
$Lu-m^{2}u=f_{1}(x, u)+(f_{2}(x, u)-m^{2}u)$ ,
(15)
which is of the kind considered in ). Equation (11) has, therefore, bounded
.
solutions in
In the long run, we have demonstrated the following
Theorem 11. {Existence theorem) Consider equation (11) under the follo$¥mathrm{B}$
$E^{n}$
wing conditions:
i) At each point
$x$
in
$E^{n}$
and
for
all real
$¥mu_{1}¥sum_{i=1}^{n}¥xi_{i}^{2}¥leqq¥sum_{i.j=1}^{n}a_{ij}(x)¥xi_{i}¥xi_{j}¥leqq¥mu_{2}¥sum_{i=1}^{n}¥xi_{i}^{2}$
$¥#)$
Of course we tacitly assume that
for each fixed .
compact subset in
$E^{n}$
$u$
$f(x, u)$
$n-$
,
tuples
$¥xi$
,
$(0<¥mu_{1}¥leqq¥mu_{2})$
;
is uniformly Holder continuous on every
On Bounded Solutions of Elliptic Partial
Differential
Equations
of the Second Order 13
, and
The $a_{ij}(x)$ ,
are bounded and uniformly Holder
;
continuous {exponent ) on every compact subset of
$f(x,
u)$
is of the form
The function
$¥partial a_{ij}(x)/¥partial x_{i}$
$¥mathrm{i}¥mathrm{i})$
$b_{i}(x)$
$E^{n}$
$¥alpha$
$¥mathrm{i}¥mathrm{i}¥mathrm{i})$
$f(x, u)=f_{1}(x, u)+f_{2}(x, u)$
where 1) $f_{1}(x, u)$ , $f_{2}(x, u)$ are uniformly Holder continuous {exponent ) in
are bounded for $(x, u)$
for each fixed value of $u;2$ ) $f_{1}(x, u)$ and
$
¥
partial
f_{2}(x,
u)/
¥
partial
u
¥
geqq
m^{2}>0$
; and 3)
.
for
Under these assumptions, there exists at least one solution of equation (11)
which is bounded and of class
in
.
The bounded solution of (11) is not necessarily unique, and in case the uniqueness is violated, equation (15) possesses the maximal and the minimal bounded
solutions.
$¥alpha$
$E^{n}$
$¥partial f_{1}(x, u)/¥partial u$
$(x, u)¥in ¥mathfrak{D}$
$¥overline{¥in}¥mathfrak{D}$
$C^{2}$
$E^{n}$
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