problems in

We develop maximum principles for several P functions which are defined on solutions to equations of fourth and sixth order (including a equation which arises in plate theory and bending of cylindrical shells). As a consequence, we obtain uniqueness results for fourth and sixth order boundary value problems in arbitrary n dimensional domains.


Introduction
This paper represents the n dimensional analogue of Schaefer's paper [9] and is concerned with uniqueness results for boundary value problems of fourth and sixth order.
Schaefer [9] investigated the uniqueness of the solution for the boundary value problems and where a, b, ≥ 0, c > 0 are constants, ϕ ≡ 0, ρ > 0 in the bounded domain Ω, n = 2 and the curvature of the boundary is strictly positive.Our aim here is to remove via the P function method dimension and geometry conditions (convexity and smoothness) with, of course, further conditions on the coefficients a, b, c and ρ.
Finally, we deal with a equation that arises in plate theory and in bending of cylindrical shells.We prove the uniqueness result for the corresponding homogeneous boundary value problem without the hypothesis that the plate has a convex shape.
A word on notations.For simplicity, we shall say that a function Φ satisfies a generalized maximum principle in Ω, if either there exists a constant k ∈ IR such that Φ ≡ k in Ω or Φ does not attain a nonnegative maximum in Ω.Throughout the paper Ω and diamΩ denote respectively a bounded domain in IR n , the diameter of Ω. EJQTDE, 2011 No. 54, p. 1 We first involve second order operators and establish some useful results.The results are not only useful for our purposes but also yielding other results for partial differential equations (see [3], [7]).
We consider the problem of determining a smooth function w (a positive supersolution), which satisfies 2) The problem of determining such a function is of interest only if γ takes positive values or both positive and negative values.If γ ≤ 0 in Ω then, the function w ≡ c, where c is a positive constant, satisfies (2.1) and (2.2).
Proof.By virtue of Jung's theorem (see [5] or [1], Theorem 11.5.8,p. 357), we may suppose without loss of generality that Ω is embedded in the ball (the smallest ball containing Ω): Ω ⊂ B (n/2n+2) }.We define the function where the positive constant α is to be determined.By calculations we get By choosing If Ω lies in a slab of width d, then the result follows from Lemma 21.11, p.158, [8].Here for some i ∈ {1, . . ., n}, where ε > 0 is small.
Then, the function u/w 1 satisfies a generalized maximum principle in Ω.
2. We may improve the constant For sufficiently small ε we have A method for determining a function having properties (2.1) and (2.2) was given in [7], p. 73-74 .The authors proved that if then there exists a function w 4 fulfilling (2.1) and (2.2).Here Lw ≡ ∆w + γ(x)w, γ ≥ 0 in Ω and Ω is supposed to lie in a strip of width d.Note that (2.6) is the inequality in the footnote of p.

74.
Of course, our Lemma is sharper that this result.For a more general result concerning the construction of supersolutions see [3].

Maximum principles and uniqueness results for sixth order equations
We now tackle the uniqueness for the boundary value problem (1.1).For the sake of simplicity we consider four cases.
We deal with classical solutions (i.e.
The uniqueness results can be inferred from the following maximum principles.
Lemma 3.1.Let u be a classical solution of (3.7).i).Suppose that holds, where a > 1, b, c are constants.We consider the function P 1 given by Then, the function P 1 /w 1 satisfies a generalized maximum principle in Ω.
ii).Suppose that then, the function P 2 /w 1 satisfies a generalized maximum principle in Ω.
If a = c in Ω then, P 2 attains its maximum value on ∂Ω (the restriction (3.9) is not needed).iii).Suppose that holds, where a > 0 in Ω, and c is of arbitrary sign in Ω.
then, the function P 3 /w 1 satisfies a generalized maximum principle in Ω.
EJQTDE, 2011 No. 54, p. 4 Proof.i).By computation and using equation (3.7) we have in Ω Hence P 1 satisfies the differential inequality Since (3.8) holds, we can use the maximum principle (Theorem 2.1) to obtain the desired result.
ii).A computation shows that By (3.10) and the arithmetic -geometric mean inequality we get Adding, we obtain that P 2 satisfies and the proof follows.
iii).The proof follows by similar reasoning.
We consider the function P 4 given by Then, the function P 4 /w 1 satisfies a generalized maximum principle in Ω.
We consider the function P 5 given by Then, the function P 5 /w 1 satisfies a generalized maximum principle in Ω.Then, the function P 3 /w 1 satisfies a generalized maximum principle in Ω.

Proof. i). It is easily verified that
The proof is achieved by arguing exactly as in Lemma 3.1.
(3.21) holds, then the function P 6 /w 1 satisfies a generalized maximum principle in Ω. Here then, the function P 7 /w 1 satisfies a generalized maximum principle in Ω.Here The proof is achieved by arguing exactly as in Lemma 3.1.
We now conclude the uniqueness result.By the boundary conditions (3.23) we have We can argue similarly if we are under the hypotheses of Lemma 3.2, Lemma 3.3 or Lemma 3.4.

Comments.
1. Our uniqueness results extend Theorem 1, [9] to the n dimensional and nonconstant coefficient case without the convexity restriction imposed to Ω.In our paper, the removal was achieved by using P functions without gradient terms.2. We could also derive P functions containing gradient terms.This kind of functions would have led us to weaker uniqueness results.
For example, the function satisfies the inequality Hence P 8 /w 1 attains its maximum value on ∂Ω (unless P 8 < 0 in Ω), if a=0 and We note that this maximum principle can be used to obtain gradient bounds for the solution of (3.7) (the method is similar to the method presented in Section 4 and hence will be omitted).3. We note that the case a, b, c > 0 and n arbitrary was also treated in [4].We see that our results cannot be deduced from results in [4].Moreover, we are able here to treat the cases a = 0 or b = 0 or a = b = 0. 4. If b = c and a > 1 (see relation (3.14)), then Lemma 3.1 holds without the assumption (3.8).This particular result can be deduced from Theorem 2, [4]. 5. Different uniqueness results for boundary value problems of sixth order have been obtained in [2].6.The sign condition on the coefficients a and b is needed.The following example shows that if a < 0, then the uniqueness result (Theorem 3.1) is violated.
The boundary value problem has (at least) the solutions u 1 (x, y) ≡ 0 and u 2 (x, y) = sin x sin y in Ω.
EJQTDE, 2011 No. 54, p. 8 A maximum principle and an uniqueness result for a fourth order equation Finally, we deal with the following equation where k 1 , k 2 > 0 are constants.The equation (4.26) arises in the plate theory and in the bending of cylindrical shells [10].
The next maximum principle will be used to obtain solution and gradient bounds for the equation (4.26).
Then the function attains its maximum value on ∂Ω.
and the proof follows by the classical maximum principle.max where n ≥ 2. b). max where n = 2.
Proof.a).Case a). is a simple consequence of Lemma 4.1.b).From Theorem 1, [10] we know that the the function |∇u| 2 − u∆u attains it maximum value on ∂Ω, which we may rewrite as The hypothesis that is assumed over and over again in plate theory is convexity.Under this assumption, Schaefer [10] proved the uniqueness for the solution of where Ω ⊂ IR 2 is a convex domain.An application of our Lemma 4.1 shows that the convexity assumption is redundant.Moreover, the uniqueness result for solutions of (4.32) holds for n > 2.
The result reads as follows: Theorem 4.2.Let u be a classical solution of (4.32), where Ω ⊂ IR n is an arbitrary domain.
1. Some maximum principles and their applications for general equations of fourth and six order have been given in [11] and [6].Unfortunately, it is difficult to apply their results in the study of uniqueness results.2. If n ≥ 3 we can still obtain gradient bounds for solutions of (4.26).We must use the function P 10 = 2(∆u)