We study homogeneous linear elliptic partial differential equations of even order. Several maximum principle results are deduced for such equations as well as a priori bounds for certain boundary value problems.
The -function technique for obtaining subharmonic functions defined on the solution of certain partial differential equations of order is well established. In  Miranda shows that the functional is subharmonic where is a solution to the biharmonic equation . Later, in , Payne uses functionals containing the square of the second gradient of the solution to semilinear equations of the form
to deduce integral bounds on .
Other works such as [3, 4] develop maximum principle results for the sixth-order equations of the form
with constant coefficients. In [5, 6], -functions containing the squares of terms of the form are used to obtain a priori that bounds for solutions to the constant coefficient -metaharmonic equation
Most recently the authors in  obtain maximum principles results for the more general variable coefficient -metaharmonic equation
using -functions containing terms of the form . For three special cases, namely, when , , and when (1.4) reduces to the equation for any integer , a more complicated class of -functions containing the squares of certain gradient terms is used. An open question arising from  is whether maximum principle results for (1.4) can be obtained for say any integer for the latter class of -functions. In this work, we establish such results by requiring that certain bounds and differential inequalities for the the coefficient functions hold. Then we obtain integral bounds on various gradient terms.
2. Assumptions and Results
Throughout this work we assume that is a bounded domain in whose boundary is sufficiently smooth, that the integer is even (without loss of generality), and that the integer . We identify the products of the first, second, and third gradients of the functions and as follows:
where the commas denote partial differentiation.
For functions we impose the boundedness conditions
for constants , .
Finally, denotes the Laplace operator, , and .
Now we derive two general maximum principle results for (1.4).
Theorem 2.1. Suppose that is a solution of (1.4). Furthermore for one defines
where the functions satisfy for some positive constant . Additionally, one imposes the conditions
Then, is subharmonic in .
Proof. A straightforward calculation yields
Using the well-known inequality (see )
and (1.4) we deduce the inequality
The right side of (2.7) is
Subsequently, we obtain
We now state a series of inequalities to demonstrate that . First we note
Similarly, we obtain
We also have that
Additional inequalities analogous to (2.15) can be developed for the remaining terms in (2.9) involving and . From (2.2), (2.10)–(2.15) and upon inclusion of the aforementioned additional inequalities we deduce that
Utilizing (2.4) the conclusion follows.
Now we handle the case where in the following theorem.
Theorem 2.2. Suppose that is a solution of (1.4) where . For satisfying the inequality for some positive constant , one defines
is subharmonic in .
Proof. A calculation similar to that of Theorem 2.1 yields
Using (2.6), (2.10), (2.11), (2.13), and (2.14) we obtain
Hence by (2.18), we conclude that .
Here we briefly indicate how theorem 1 and theorem 2 can be used to obtain integral bounds on the square of the second gradient of . suppose that the hypotheses of theorem 1 are satisfied and that the conditions
hold on . Let denote the area of . As a consequence of integration by parts, Theorem 2.1 implies that
Now, if the hypotheses of Theorem 2.2 hold and if we impose the conditions
on we can deduce the inequality
from Theorem 2.2.
C. Miranda, “Formule di maggiorazione e teorema di esistenza per le funzioni biarmoniche de due variabili,” Giornale di Matematiche di Battaglini, vol. 2, no. 78, pp. 97–118, 1948.