A Maximum Principle Result for a General Fourth Order Semilinear Elliptic Equation

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.


Introduction
In [1], Payne obtains maximum principle results for the semilinear fourth order elliptic equation ( ) by proving that certain functionals defined on the solution of (1) are subharmonic.In this work, functionals containing the terms 2 , , i i u u u ∇ − ∆ are utilized and apriori bounds on the integral of the square of the second gradient and on the square of the gradient of the solution are deduced.Since then, many authors [2]- [11] and references therein have used this technique to obtain maximum principle results for other fourth order elliptic differential equations whose principal part is the biharmonic operator.
Other works deal with the more general fourth order elliptic operator 2 L u , where , : ij ij Lu a u = and ij ji a a = .
In [12], Dunninger mentions that functionals containing the term ( ) Lu can be used to obtain maximum principle results for such linear equations as 2 0.

L u aLu bu + + =
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: Then we briefly indicate how these maximum principles can be used to obtain apriori bounds on a certain quantity of interest.

Results
Throughout this paper, the summation convention on repeated indices is used; commas denote partial differentiation.Let ( ) ij a x be a symmetric matrix.Moreover let , : ij ij Lu a u = , be a uniformly elliptic operator, i.e, the symmetric matrix ( ) ij a x is positive definite and satisfies the uniform ellipticity condition: , where Ω is a bounded domain in n R and 2 n ≥ .Let u be a 5  C solution to the equation ( ) where f is say, a 1 C function.Now we define the functional We show that ( ) L P is subharmonic and note that the constants 1 c and 2 c and any constraints on f are yet to be determined.
By a straight-forward calculation, we have . .
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 Lu can be rewritten using (3) in the form ( ) .
A. Mareno Using the identity above for , ij mij a u and the additional identity, , , , which can be obtained by computing ( ) , ni ij j a A , for the terms at the ends of lines 6 and 3 respectively, we obtain To show that ( ) Repeated use of ( 9) on terms in lines 2, 3, 4, 5 in (7) yields the following: We add ( 10)-( 21) and label the resulting inequality, for part of ( )      such that P cannot attain its maximum value in Ω unless it is a constant.
We note that the function ( ) ( ) f u u u = − + satisfies the conditions stated in Theorem 1 for a solution that is bounded above.

Bounds
Here we give a brief application of Theorem 1.
Suppose that ij a x is positive definite, for a sufficiently large value of 2 c , where 2 c depends on the coefficients ij a and their derivatives, and for a sufficiently large value of 1 c , say ( ) 1 > , where 1 c depends on the constants 2 c , γ , ij a , and various derivatives of ij a , ( ) L P can be made nonnegative as desired.Thus we have the following result.
Using integration by parts on the first two terms of P yields the identity where ij A denotes the matrix which is the inverse of the positive definite matrix ( ) P is nonnegative, we establish a series of inequalities based on the following one from