We obtain maximum principles for solutions of some general fourth order elliptic equations by modifying an auxiliary function introduced by L.E. Payne. We give a brief application of these maximum principles by deducing apriori bounds on a certain quantity of interest.
In [
by proving that certain functionals defined on the solution of (1) are subharmonic. In this work, functionals containing the terms
Other works deal with the more general fourth order elliptic operator
A similar approach is taken in [
In this paper, we modify the results in [
Then we briefly indicate how these maximum principles can be used to obtain apriori bounds on a certain quantity of interest.
Throughout this paper, the summation convention on repeated indices is used; commas denote partial differentiation. Let
Let u be a
where f is say, a
We show that
By a straight-forward calculation, we have
Now we write
By expanding out the derivative terms in parentheses, we see that
The terms in lines 2 and 3 above containing two or more derivatives of
Using the identity above for
To show that
One can deduce
Repeated use of (9) on terms in lines 2, 3, 4, 5 in (7) yields the following:
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):
We add (10)-(21) and label the resulting inequality, for part of
Now,
Since
Theorem 1. Suppose that
We note that the function
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
A. Mareno, (2016) A Maximum Principle Result for a General Fourth Order Semilinear Elliptic Equation. Journal of Applied Mathematics and Physics,04,1682-1686. doi: 10.4236/jamp.2016.48176