Two maximum principles for a nonlinear fourth order equation from thin plate theory

We develop two maximum principles for a nonlinear equation of fourth order that arises in thin plate theory. As a consequence, we obtain uniqueness results for the corresponding fourth order boundary value problem under the boundary conditions w = ∆w = 0, as well as some bounds of interest.


Introduction
In the pioneering work [9], Payne introduced a technique, which utilizes a maximum principle for a function defined on solutions to an elliptic differential equation, in order to obtain bounds for the gradient of the solution of the relevant differential equation.Several authors have contributed to the growing literature developing this technique (see the references cited here, especially [23], and the references therein).
This paper employs Payne's technique to treat the following equation that arises in the thin plate theory where Ω is a bounded domain, D(x) > 0 is the flexural rigidity of the plate, [u, v] = u xx v yy − 2u xy v xy + v xx u yy , and 0 < ν < 1 2 is the elastic constant (Poisson ratio) and is defined by ν = λ/2(λ + µ) with material depending constants λ and µ, the so-called Lamé constants.Usually λ and µ > 0 and hence 0 < ν < 1 2 .For metals the value ν is about 0.3.Some exotic materials have a negative Poisson ratio.We have denoted partial derivatives by a subscript and will use the summation convention on repeated indices.
In Section 2, we establish two maximum principles for an auxiliary P function containing the terms w, |∇w| 2 , (∆w) 2 .We note that Mareno [5,8] was the first to prove a maximum principle for the equation (1.1).
Email: cristiandanet@yahoo.com C.-P. Danet Finally, in Section 3 we use these results to prove uniqueness results for classical solutions C 4 (Ω) ∩ C 2 (Ω) and some bounds.

Maximum principles
The following maximum principle for second order operators will be useful ( [2]).
Suppose that Ω lies in the strip of width d, 0 < x i < d, for some i ∈ {1, . . ., n} and that (2.1) Then the function u/ϕ satisfies a generalized maximum principle in Ω, i.e., there exists a constant k ∈ IR such that u/ϕ ≡ k in Ω or u/ϕ does not attain a nonnegative maximum in Ω. Here where ε > 0 is small.Similarly, if we replace (2.1) by then u/ψ satisfies a generalized maximum principle in Ω. Here We define the function where F(s) = s 0 f (t) dt, C > 0 is a constant and prove the following maximum principle.
Theorem 2.2.Let w ∈ C 4 (Ω) be solution of (1.1) and let c, D ∈ C 2 (Ω), f ∈ C 1 (IR).Suppose that the following requirements are satisfied Then the function P/ϕ satisfies a generalized maximum principle in Ω. Here then the function P/ψ satisfies a generalized maximum principle in Ω.Here ψ Proof.From equation (1.1) we get and hence A computation shows that Adding and using (a 2 ) we get We observe that Consequently adding and subtracting 2D 2 ij w 2 ij in order to complete the square of the first two terms and using the fact that Completing the square of the first two terms see that C.-P. Danet Using the first inequality in (a 4 ), adding and subtracting ( f 2 c i c i )/(c f + C/α − C 2 /D) to the previous inequality we are left with by the second inequality in (a 4 ).
The desired proof follows from the generalized maximum principle (Theorem 2.1).
Now we assume that C ≤ D/α and state a similar result.
Theorem 2.3.Let w ∈ C 4 (Ω) be solution of (1.1) and let c, D ∈ C 2 (Ω), f ∈ C 1 (IR).Suppose that the following requirements are satisfied Then the function P/ϕ satisfies a generalized maximum principle in Ω, where Proof.Since C ≤ D/α inequality (2.3) reduces to Adding and subtracting ( f 2 c i c i )/(c f ) to the previous inequality we get By (b 1 ) we get c∆c/c i c i ≥ 2 and hence and the proof follows.
Here (Theorem 2.3, case α = 1) we imposed a geometric restriction on Ω that allowed us to drop the restriction (c 4 ) imposed by Mareno [5].Moreover, Theorem 2.2 works without any sign restriction for f and ∆c.

Uniqueness results and bounds
With the aid of the above theorem we can establish the uniqueness results.where k is the curvature of ∂Ω.
Then w ≡ 0 is the only solution of the boundary value problem Proof.According to Theorem 2.2, [5] the function R attains its maximum value on ∂Ω, at a point x 0 .From Hopf's lemma it follows that ∂R ∂n > 0 at x 0 .A computation shows that By introducing normal coordinates in the neighborhood of the boundary, we can write (see [23, p. 46, relation 4.3]) where ∂w ∂s denotes the tangential derivative of w.

Applications.
(a) From Theorem 3.1 we obtain a uniqueness result for convex domains (k ≥ 0) under the hypothesis ∂D/∂n ≤ 0 on ∂Ω.
(b) Suppose that the plate has the shape of the ellipse ∂Ω : We see that relation (3.7) is fulfilled.
In order to get a uniqueness result, it remains to check the validity of (
Theorem 3.2.Suppose that we are under the hypotheses of Theorem 2.3.We also assume that ∂Ω ∈ C 2+ε , D ∈ C 2 (Ω), k ≥ 0 and Then w ≡ 0 is the only solution of the boundary value problem (3.2).A similar uniqueness result holds if we replace ϕ by ψ in (3.6) and (3.7).