On Maximum Principles for 𝑚-Metaharmonic Equations

Abstract

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.

1. Introduction

The 𝑃-function technique for obtaining subharmonic functions defined on the solution of certain partial differential equations of order ≥4 is well established. In [1] Miranda shows that the functional 𝑃=𝑢_,𝑖𝑢_,𝑖−𝑢Δ𝑢 is subharmonic where 𝑢 is a solution to the biharmonic equation Δ²𝑢=0. Later, in [2], Payne uses functionals containing the square of the second gradient of the solution to semilinear equations of the formΔ²𝑢=𝑓(𝑢)(1.1) to deduce integral bounds on (Δ𝑢)².

Other works such as [3, 4] develop maximum principle results for the sixth-order equations of the formΔ³𝑢−𝑎Δ²𝑢+𝑏Δ𝑢−𝑐𝑢=0(1.2) with constant coefficients. In [5, 6], 𝑃-functions containing the squares of terms of the form Δ^𝑖𝑢 are used to obtain a priori that bounds for solutions to the constant coefficient 𝑚-metaharmonic equationΔ^𝑚𝑢+𝑎_𝑚−1Δ^𝑚−1𝑢+⋅⋅⋅+𝑎₀𝑢=0.(1.3)

Most recently the authors in [7] obtain maximum principles results for the more general variable coefficient 𝑚-metaharmonic equationΔ^𝑚𝑢−𝑎_𝑚−1(𝑥)Δ^𝑚−1𝑢+𝑎_𝑚−2(𝑥)Δ^𝑚−2𝑢−⋅⋅⋅+(−1)^𝑚𝑎₀(𝑥)𝑢=0(1.4) using 𝑃-functions containing terms of the form (Δ^𝑖𝑢)². For three special cases, namely, when 𝑚=3, 𝑚=4, and when (1.4) reduces to the equation Δ^𝑚𝑢=0 for any integer 𝑚≥2, a more complicated class of 𝑃-functions containing the squares of certain gradient terms is used. An open question arising from [7] is whether maximum principle results for (1.4) can be obtained for say any integer 𝑚>4 for the latter class of 𝑃-functions. In this work, we establish such results by requiring that certain bounds and differential inequalities for the the coefficient functions 𝑎₀(𝑥),…,𝑎_𝑚−2(𝑥) hold. Then we obtain integral bounds on various gradient terms.

2. Assumptions and Results

Throughout this work we assume that Ω is a bounded domain in 𝐑^𝐧 whose boundary 𝜕Ω is sufficiently smooth, that the integer 𝑛≥2 is even (without loss of generality), and that the integer 𝑚>4. We identify the products of the first, second, and third gradients of the functions 𝑣 and 𝑤 as follows:𝑣_,𝑖𝑤_,𝑖≡∇𝑣⋅∇𝑤,𝑣_,𝑖𝑗𝑤_,𝑖𝑗≡∇²𝑣∶∇²𝑤,𝑣_{,𝑖𝑗𝑘}𝑤_{,𝑖𝑗𝑘}≡∇³𝑣⋮∇³𝑤,(2.1) where the commas denote partial differentiation.

For functions 𝑎_𝑖(𝑥)∈𝐶²(Ω) we impose the boundedness conditions^𝑚⁻³_𝑖=0𝑎^2𝑖≤𝛽,^𝑚−1_𝑖=0∇𝑎_𝑖⋅∇𝑎_𝑖≤𝛾,(2.2) for constants 𝛾≥0, 𝛽>0.

Finally, Δ denotes the Laplace operator, Δ^𝑚≡Δ(Δ^𝑚−1), and Δ⁰≡𝐼.

Now we derive two general maximum principle results for (1.4).

Theorem 2.1. Suppose that 𝑢∈𝐶^2𝑚+1(Ω)∩𝐶^2𝑚−1(Ω) is a solution of (1.4). Furthermore for 𝑛>4 one defines 𝑃=∇+²𝑎Δ^𝑚−2_𝑚−1𝑢⋅∇²Δ^𝑚−2𝑢−∇Δ^𝑚−2𝑢⋅∇Δ^m−1𝑢2∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+𝑎^𝑚−32∇Δ^𝑚−3+_𝑚−2𝑢⋅∇Δ^𝑚−3𝑢_𝑖=0𝜙_𝑖(Δ^𝑖𝑢)²+𝑎_𝑚−2𝑛−4𝑛+2(Δ^𝑚−2𝑢)²−(4−𝑛)2(𝑛+2)(Δ^𝑚−1𝑢)²,(2.3) where the functions 𝜙₀(𝑥),…,𝜙_𝑚−2(𝑥)∈𝐶²(Ω)∩𝐶⁰(Ω) satisfy ∑^{𝑚−2𝑖=0}𝜙^2𝑖+1≤𝛼 for some positive constant 𝛼.
Additionally, one imposes the conditions 𝜙_𝑖≥𝛽𝑎2(𝑖=0,…,𝑚−2),_𝑚−2≥1,𝑎_𝑚−1−1Δ𝑎_𝑖2≥𝛾2𝑛𝑛−+42,2−∇𝑎^𝑚−𝑖⋅∇𝑎_𝑚−𝑖𝑎_𝑚−𝑖Δ𝑎_𝑚−2≥0(𝑖=1,3),−4∇𝑎^𝑚−2⋅∇𝑎_𝑚−2𝑎_𝑚−2Δ𝜙_𝑖≥0,3≥max𝛽2𝑛𝑛−+42+𝛾2,𝛼,4∇𝜙^𝑖⋅∇𝜙_𝑖𝜙_𝑖(𝑖=0,…,𝑚−2).(2.4)
Then, 𝑃 is subharmonic in Ω.

Proof. A straightforward calculation yields Δ𝑃=2∇³Δ^𝑚−2𝑢⋮∇³Δ^𝑚−2𝑢−6𝑛+2∇Δ^𝑚−1−∇Δ_𝑚−2𝑢⋅∇Δ^𝑚−1𝑢𝑢⋅∇(Δ^𝑚𝑢)−4−𝑛𝑛+2Δ^𝑚−1+𝑎_𝑚−1𝑢Δ^𝑚𝑢+𝑎_𝑚−3∇²Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+𝑎_𝑚−1∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−1𝑢+_𝑚−2∇²Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢+𝑎_𝑚−3∇Δ^𝑚−3𝑢⋅∇Δ^𝑚−2𝑢_𝑖=0Δ𝜙^𝑖Δ^𝑖+𝑎_𝑚−2𝑢²+4∇𝜙_𝑖⋅∇Δ^𝑖𝑢Δ^𝑖𝑢+2𝜙_𝑖∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢+𝑛−4𝑛𝑛+−42∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+Δ^𝑚−2𝑢Δ^𝑚−1𝑢𝑛+2Δ𝑎^𝑚−2(Δ^𝑚−2𝑢)²+4𝑛−4𝑛+2∇𝑎^𝑚−2+1⋅∇Δ^𝑚−2𝑢Δ^𝑚−2𝑢2Δ𝑎^𝑚−1∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+12Δ𝑎^𝑚−3+2∇𝑎_𝑚−1∇Δ^𝑚−3𝑢⋅∇Δ^𝑚−3𝑢+_𝑚−2∇Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+2∇𝑎_𝑚−3∇Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢_𝑖=02𝜙_𝑖Δ^𝑖𝑢Δ^𝑖+1𝑢.(2.5) Using the well-known inequality (see [2]) ∇³𝑤⋮∇³𝑤≥3𝑛+2∇(Δ𝑤)⋅∇(Δ𝑤),forallfunctions𝑤∈𝐶³(Ω),(2.6) and (1.4) we deduce the inequality Δ𝑃≥−∇Δ^𝑚−2𝑢⋅^𝑚−1_𝑖=0(−1)^𝑖+1∇𝑎_𝑖Δ^𝑖𝑢+^𝑚−1_𝑖=0+𝑛−4(−1)^𝑖+1𝑎_𝑖∇Δ^𝑖𝑢𝑛+2Δ^𝑚−1𝑢^𝑚−1_𝑖=0+𝑎_𝑚−1(−1)^𝑖+1𝑎_𝑖Δ^𝑖𝑢+𝑎_𝑚−3∇²Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+𝑎_𝑚−1∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−1𝑢+_𝑚−2∇²Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢+𝑎_𝑚−3∇Δ^𝑚−3𝑢⋅∇Δ^𝑚−2𝑢_𝑖=0Δ𝜙^𝑖+𝑎_𝑚−2(Δ^𝑖𝑢)²+4∇𝜙_𝑖⋅∇Δ^𝑖𝑢Δ^𝑖𝑢+2𝜙_𝑖∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢+𝑛−4𝑛𝑛+−42∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+Δ^𝑚−2𝑢Δ^𝑚−1𝑢𝑛+2Δ𝑎^𝑚−2(Δ^𝑚−2𝑢)²+4𝑛−4𝑛+2∇𝑎^𝑚−2+1⋅∇Δ^𝑚−2𝑢Δ^𝑚−2𝑢2Δ𝑎^𝑚−1∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+12Δ𝑎^𝑚−3+2∇𝑎_𝑚−1∇Δ^𝑚−3𝑢⋅∇Δ^𝑚−3𝑢+_𝑚−2∇Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+2∇𝑎_𝑚−3∇Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢_𝑖=02𝜙_𝑖Δ^𝑖𝑢Δ^𝑖+1𝑢.(2.7) The right side of (2.7) is =𝑎_𝑚−21+𝑛−4𝑛+2∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+𝑎_𝑚−1𝑛−4𝑛+2Δ^𝑚−1𝑢²+_𝑚−2+𝑎_𝑚−1∇²Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+𝑎_𝑚−3∇²Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢_𝑖=0Δ𝜙^𝑖Δ^𝑖+𝑛−4𝑢²+4∇𝜙_𝑖⋅∇Δ^𝑖𝑢Δ^𝑖𝑢+2𝜙_𝑖∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢𝑛+2Δ^𝑚−1𝑢^𝑚−3_𝑖=0(−1)^𝑖+1𝑎_𝑖Δ^𝑖𝑢−∇Δ^𝑚−2𝑢⋅^𝑚−1_𝑖=0+𝑛−4(−1)^𝑖+1∇𝑎_𝑖Δ^𝑖𝑢𝑛+2Δ𝑎^𝑚−2(Δ^𝑚−2𝑢)²+4𝑛−4𝑛+2∇𝑎^𝑚−2+1⋅∇Δ^𝑚−2𝑢Δ^𝑚−2𝑢2Δ𝑎^𝑚−1∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+12Δ𝑎^𝑚−3+2∇𝑎_𝑚−1∇Δ^𝑚−3𝑢⋅∇Δ^𝑚−3𝑢−∇Δ_𝑚−2∇Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+2∇𝑎_𝑚−3∇Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢𝑢⋅^𝑚−4_𝑖=0(−1)^𝑖+1𝑎_𝑖∇Δ^𝑖𝑢+^𝑚−2_𝑖=02𝜙_𝑖Δ^𝑖𝑢Δ^𝑖+1𝑢.(2.8) Subsequently, we obtain Δ𝑃≥𝑎_𝑚−21+𝑛−4𝑛+2−1∇Δ^𝑚−2+𝑎_𝑚−1𝑢⋅∇Δ^𝑚−2𝑢+𝑎∇²_𝑚−1Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+𝑎_𝑚−3∇²Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢−12𝑛𝑛+−42(Δ^𝑚−1𝑢)²−12^𝑚−1_𝑖=0(−1)^𝑖+1∇𝑎_𝑖Δ^𝑖𝑢²−12𝑛𝑛−+24^𝑚−3_𝑖=0(−1)^𝑖+1𝑎_𝑖Δ^𝑖𝑢²−12^𝑚−4_𝑖=0(−1)^𝑖+1𝑎_𝑖∇Δ^𝑖𝑢²+^𝑚−2_𝑖=0+𝑛−Δ𝜙4_𝑖Δ^𝑖𝑢²+4∇𝜙_𝑖⋅∇Δ^𝑖𝑢Δ^𝑖𝑢+2𝜙_𝑖∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢𝑛+2Δ𝑎^𝑚−2(Δ^𝑚−2𝑢)²+4𝑛−4𝑛+2∇𝑎^𝑚−2+1⋅∇Δ^𝑚−2𝑢Δ^𝑚−2𝑢2Δ𝑎^𝑚−1∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+12Δ𝑎^𝑚−3+2∇𝑎_𝑚−1∇Δ^𝑚−3𝑢⋅∇Δ^𝑚−3𝑢+_𝑚−2∇Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢+2∇𝑎_𝑚−3∇Δ^𝑚−3𝑢∶∇²Δ^𝑚−3𝑢_𝑖=02𝜙_𝑖Δ^𝑖𝑢Δ^𝑖+1𝑢.(2.9) We now state a series of inequalities to demonstrate that Δ𝑃≥0. First we note ^𝑚⁻²_𝑖=02𝜙_𝑖Δ^𝑖𝑢Δ^𝑖+1𝑢≥−^𝑚−2_𝑖=0𝜙_𝑖²+1^𝑚−2_𝑖=0(Δ^𝑖𝑢)².(2.10) Similarly, we obtain −12^𝑚−1_𝑖=0(−1)^𝑖+1∇𝑎_𝑖Δ^𝑖𝑢²≥−12^𝑚−1_𝑖=0∇𝑎_𝑖⋅∇𝑎_𝑖(Δ^𝑚−1𝑢)²−12^𝑚−1_𝑖=0∇𝑎_𝑖⋅∇𝑎_𝑖^𝑚−2_𝑖=0Δ^𝑖𝑢Δ^𝑖𝑢,(2.11)−12𝑛𝑛+−42^𝑚−3_𝑖=0(−1)^𝑖+1a_𝑖Δ^𝑖𝑢²≥−12𝑛𝑛−+24^𝑚−3_𝑖=0𝑎^{2𝑖𝑚−3}_𝑖=0(Δ^𝑖𝑢)²,(2.12)−12^𝑚−4_𝑖=0(−1)^𝑖+1𝑎_𝑖∇Δ^𝑖𝑢²≥−12^𝑚−4_𝑖=0𝑎^{2𝑖𝑚−4}_𝑖=0∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢,(2.13)+^𝑚−2_𝑖=0Δ𝜙_𝑖(Δ^𝑖≥_𝑚−2𝑢)²+4∇𝜙_𝑖⋅∇Δ^𝑖𝑢Δ^𝑖𝑢+𝜙_𝑖∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢_𝑖=0(Δ^𝑖𝑢)²Δ3𝜙^𝑖−4∇𝜙^𝑖⋅∇𝜙_𝑖𝜙_𝑖.(2.14) We also have that 12Δ𝑎^𝑚−1+𝑎∇_𝑚−1Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢+2∇𝑎_𝑚−1∇Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢≥∇Δ∇²_𝑚−2Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢𝑢⋅∇Δ^𝑚−2𝑢Δ𝑎^𝑚−12−∇𝑎^𝑚−1⋅∇𝑎_𝑚−1𝑎_𝑚−1.(2.15) Additional inequalities analogous to (2.15) can be developed for the remaining terms in (2.9) involving Δ𝑎_𝑚−2 and Δ𝑎_𝑚−3.
From (2.2), (2.10)–(2.15) and upon inclusion of the aforementioned additional inequalities we deduce thatΔ𝑃≥𝑎_𝑚−1−12𝑛𝑛+−42−21^𝑚−1_𝑖=0∇𝑎_𝑖⋅∇𝑎_𝑖Δ^𝑚−1𝑢²+^𝑚−2_𝑖=0Δ3𝜙^𝑖−𝛽2𝑛𝑛−+42−𝛾2Δ^𝑖𝑢²+^𝑚−2_𝑖=0𝜙^𝑖−𝛽2Δ^𝑖𝑢²+^𝑚−2_𝑖=0Δ3𝜙^𝑖−𝛼Δ^𝑖𝑢²+^𝑚−2_𝑖=0Δ^𝑖𝑢²Δ3𝜙^𝑖−4∇𝜙^𝑖⋅∇𝜙_𝑖+𝑎_𝑚−2𝜙_𝑖+∇Δ−_𝑚−21∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢𝑢⋅∇Δ^𝑚−2𝑢Δ𝑎^𝑚−12−∇𝑎^𝑚−1⋅∇𝑎_𝑚−1+∇Δ_𝑚−3𝑎_𝑚−1𝑢⋅∇Δ^𝑚−3𝑢Δ𝑎^𝑚−32−∇𝑎^𝑚−3⋅∇𝑎_𝑚−3+𝑛−4𝑎_𝑚−3𝑛+2(Δ^𝑚−2𝑢)²Δ𝑎^𝑚−2−4∇𝑎^𝑚−2⋅∇𝑎_𝑚−2𝑎_𝑚−2.(2.16) Utilizing (2.4) the conclusion follows.

Now we handle the case where 𝑛≤4 in the following theorem.

Theorem 2.2. Suppose that 𝑢∈𝐶^2𝑚+1(Ω)∩𝐶^2𝑚−1(Ω) is a solution of (1.4) where 𝑛≤4. For 𝜙₀(𝑥),…,𝜙_𝑚−1(𝑥)∈𝐶²(Ω)∩𝐶⁰(Ω)satisfying the inequality ∑^{𝑚−2𝑖=0}𝜙^2𝑖+1≤𝛼 for some positive constant 𝛼, one defines 𝑃=∇²Δ^𝑚−2𝑢∶∇²Δ^𝑚−2𝑢−∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−1𝑢+^𝑚−1_𝑖=0𝜙_𝑖Δ^𝑖𝑢²+12Δ^𝑚−2𝑢².(2.17) Then if 𝜙_𝑖≥𝛼2,Δ𝜙_𝑚−𝑖for𝑖=0,…,𝑚−1,3−21≥0,Δ𝜙_𝑖for𝑖=1,2,3−4∇𝜙^𝑖⋅∇𝜙_𝑖𝜙_𝑖≥0,for𝑖=0,…,𝑚−1,(2.18)𝑃is subharmonic in Ω.

Proof. A calculation similar to that of Theorem 2.1 yields Δ𝑃=−2∇∇³ΔΔ^𝑚−2_𝑚−2𝑢⋮∇³Δ^𝑚−2𝑢−∇Δ^𝑚−1𝑢⋅∇Δ^𝑚−1𝑢𝑢⋅^𝑚−1_𝑖=0−∇Δ_𝑚−2(−1)^𝑖+1∇𝑎_𝑖Δ^𝑖𝑢+∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢𝑢⋅^𝑚−1_𝑖=0(−1)^𝑖+1+_𝑚−1𝑎_𝑖∇Δ^𝑖𝑢+Δ^𝑚−2𝑢Δ^𝑚−1𝑢_𝑖=0+_𝑚−1Δ𝜙_𝑖(Δ^𝑖𝑢)²+4∇𝜙_𝑖⋅∇Δ^𝑖𝑢Δ^𝑖𝑢_𝑖=02𝜙_𝑖∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢+2𝜙_𝑖Δ^𝑖𝑢Δ^𝑖+1𝑢.(2.19) Using (2.6), (2.10), (2.11), (2.13), and (2.14) we obtain Δ𝑃≥^𝑚−1_𝑖=0Δ^𝑖𝑢²Δ3𝜙^𝑖−4∇𝜙^𝑖⋅∇𝜙_𝑖𝜙_𝑖+^𝑚−1_𝑖=02Δ𝜙_𝑖3Δ^𝑖𝑢²+^𝑚−1_𝑖=0−1𝜙_𝑖∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢−∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢2Δ^𝑚−2𝑢²−12Δ^𝑚−1−1𝑢²+∇Δ^𝑚−2𝑢⋅∇Δ^𝑚−2𝑢2^𝑚−1_𝑖=0∇𝑎_𝑖⋅∇𝑎_𝑖^𝑚−1_𝑖=0Δ^𝑖𝑢²−12^𝑚−1_𝑖=0𝑎^2𝑖^𝑚−1_𝑖=0≥_𝑚−1∇Δ^𝑖𝑢⋅∇Δ^𝑖𝑢_𝑖=0Δ^𝑖𝑢²Δ3𝜙^𝑖−4∇𝜙^𝑖⋅∇𝜙_𝑖𝜙_𝑖+^𝑚−1_𝑖=0𝜙^𝑖−𝛼2∇(Δ𝑢)⋅∇(Δ𝑢)+^𝑚−1_𝑖=0Δ3𝜙^𝑖−𝛾2Δ^𝑖𝑢²+Δ𝜙^𝑚−23−21Δ^𝑚−2𝑢²+Δ𝜙^𝑚−13−12Δ^𝑚−1𝑢².(2.20) Hence by (2.18), we conclude that Δ𝑃≥0.

3. Applications

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 Δ^𝑚−2𝑢. suppose that the hypotheses of theorem 1 are satisfied and that the 𝑚 conditionsΔΔ^𝑖𝑢=0,for𝑖=0,…,𝑚−5,_𝑚−2𝑢=Δ^𝑚−2𝑢𝜕𝑛=0,Δ^𝑚−3𝑢=Δ^𝑚−3𝑢𝜕𝑛=0(3.1) hold on 𝜕Ω. Let 𝐴 denote the area of Ω. As a consequence of integration by parts, Theorem 2.1 implies that_Ω||∇²Δ^𝑚−2𝑢||²𝑑𝑥≤𝐴2max^𝜕Ω||∇²Δ^𝑚−2𝑢||²+𝜙_𝑚−4(𝑥)Δ^𝑚−4𝑢²+𝑛−42(n+2)ΔΔ^𝑚−2𝑢².(3.2)

Now, if the hypotheses of Theorem 2.2 hold and if we impose the 𝑚 conditionsΔ^𝑖𝑢=Δ0,for𝑖=0,…,𝑚−3,_𝑚−2𝑢=Δ^𝑚−2𝑢𝜕𝑛=0(3.3) on 𝜕Ω we can deduce the inequality_Ω||∇²Δ^𝑚−2𝑢||²𝑑𝑥≤𝐴2max^𝜕Ω||∇²Δ^𝑚−2𝑢||²+𝜙_𝑚−1(𝑥)ΔΔ^𝑚−2𝑢²,(3.4) from Theorem 2.2.