_{1}

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