A Maximum Principle Result for a General Fourth Order Semilinear Elliptic Equation ()
Received 2 June 2016; accepted 27 August 2016; published 30 August 2016
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1. Introduction
In [1] , Payne obtains maximum principle results for the semilinear fourth order elliptic equation
(1)
Other works deal with the more general fourth order elliptic operator
, where
and
. In [12] , Dunninger mentions that functionals containing the term
can be used to obtain maximum principle results for such linear equations as
![](//html.scirp.org/file/14-1720616x12.png)
A similar approach is taken in [13] for a class of nonlinear fourth order equations.
In this paper, we modify the results in [1] and a matrix result from [14] to deduce maximum principles defined on the solutions to semilinear fourth order elliptic equations of the form:
(2)
Then we briefly indicate how these maximum principles can be used to obtain apriori bounds on a certain quantity of interest.
2. Results
Throughout this paper, the summation convention on repeated indices is used; commas denote partial differentiation. Let
be a symmetric matrix. Moreover let
, be a uniformly elliptic operator, i.e, the symmetric matrix
is positive definite and satisfies the uniform ellipticity condition:
, where
is a bounded domain in
and
.
Let u be a
solution to the equation
(3)
where f is say, a
function. Now we define the functional
![]()
We show that
is subharmonic and note that the constants
and
and any constraints on f are yet to be determined.
By a straight-forward calculation, we have
![]()
Now we write
(4)
By expanding out the derivative terms in parentheses, we see that
is
(5)
The terms in lines 2 and 3 above containing two or more derivatives of
can be rewritten using (3) in the form
, where
denotes the matrix which is the inverse of the positive definite matrix
. Furthermore, we use the identity
to rewrite the last two terms in line 4. Hence,
(6)
Using the identity above for
and the additional identity,
, which can be obtained by computing
, for the terms at the ends of lines 6 and 3 respectively, we obtain
(7)
To show that
is nonnegative, we establish a series of inequalities based on the following one from [14] : Let
be any
matrix. From the inequality
(8)
One can deduce
(9)
Repeated use of (9) on terms in lines 2, 3, 4, 5 in (7) yields the following:
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
Furthermore, by completing the square, we obtain useful inequalities for the last two terms in line 1 and the third term in line 2 of (7):
(19)
(20)
(21)
We add (10)-(21) and label the resulting inequality, for part of
, as
![]()
Now,
![]()
Since
is positive definite, for a sufficiently large value of
, where
depends on the coefficients
and their derivatives, and for a sufficiently large value of
, say
, where
depends on the constants
,
,
, and various derivatives of
,
can be made nonnegative as desired. Thus we have the following result.
Theorem 1. Suppose that
is a solution of (2) and
. If
, where
,
,
is a nonnegative function such that
then there exists positive constants
and
sufficiently large
such that P cannot attain its maximum value in
unless it is a constant.
We note that the function
satisfies the conditions stated in Theorem 1 for a solution that is bounded above.
3. Bounds
Here we give a brief application of Theorem 1.
Suppose that
![]()
By Theorem 1,
![]()
Using integration by parts on the first two terms of P yields the identity
![]()
Upon integrating both sides of the previous inequality we deduce
(22)
(23)