# On Maximum Principles form-Metaharmonic Equations

Authors: A. Mareno
Journal: ISRN Mathematical Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction

TheP-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 form(1.1)Δ2u=f(u)
to deduce integral bounds on (Δu)2.Other works such as [3, 4] develop maximum principle results for the sixth-order equations of the form(1.2)Δ3u-aΔ2u+bΔu-cu=0
with constant coefficients. In [5, 6], P-functions containing the squares of terms of the form Δiu are used to obtain a priori that bounds for solutions to the constant coefficient m-metaharmonic equation(1.3)Δmu+am-1Δm-1u+⋅⋅⋅+a0u=0.Most recently the authors in [7] obtain maximum principles results for the more general variable coefficient m-metaharmonic equation(1.4)Δmu-am-1(x)Δm-1u+am-2(x)Δm-2u-⋅⋅⋅+(-1)ma0(x)u=0
using P-functions containing terms of the form (Δiu)2. For three special cases, namely, when m=3, m=4, and when (1.4) reduces to the equation Δmu=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),…,am-2(x) hold. Then we obtain integral bounds on various gradient terms.


## 2. Assumptions and Results

Throughout this work we assume thatΩ is a bounded domain in Rn 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:(2.1)v,iw,i≡∇v⋅∇w,v,ijw,ij≡∇2v:∇2w,v,ijkw,ijk≡∇3v⋮∇3w,
where the commas denote partial differentiation.For functionsai(x)∈C2(Ω) we impose the boundedness conditions(2.2)∑i=0m-3ai2≤β,∑i=0m-1∇ai⋅∇ai≤γ,
for constants γ≥0, β>0.Finally,Δ denotes the Laplace operator, Δm≡Δ(Δm-1), and Δ0≡I.Now we derive two general maximum principle results for (1.4).Theorem 2.1.
Suppose thatu∈C2m+1(Ω)∩C2m-1(Ω̅) is a solution of (1.4). Furthermore for  n>4 one defines
(2.3)P=∇2(Δm-2u)⋅∇2(Δm-2u)-∇(Δm-2u)⋅∇(Δm-1u)+am-12∇(Δm-2u)⋅∇(Δm-2u)+am-32∇(Δm-3u)⋅∇(Δm-3u)+∑i=0m-2ϕi(Δiu)2+am-2(n-4n+2)(Δm-2u)2-(4-n)2(n+2)(Δm-1u)2,
where the functions ϕ0(x),…,ϕm-2(x)∈C2(Ω)∩C0(Ω̅) satisfy ∑i=0m-2ϕi2+1≤α for some positive constant α.
Additionally, one imposes the conditions(2.4)ϕi≥β2(i=0,…,m-2),am-2≥1,(am-1-12)≥γ2(n+2n-4),Δai2-∇am-i⋅∇am-iam-i≥0(i=1,3),Δam-2-4∇am-2⋅∇am-2am-2≥0,Δϕi3≥max⁡{β2(n-4n+2)+γ2,α,4∇ϕi⋅∇ϕiϕi}(i=0,…,m-2).
Then,P is subharmonic in Ω.Proof.
A straightforward calculation yields(2.5)ΔP=2∇3(Δm-2u)⋮∇3(Δm-2u)-(6n+2)∇(Δm-1u)⋅∇(Δm-1u)-∇(Δm-2u)⋅∇(Δmu)-(4-nn+2)Δm-1uΔmu+am-1∇2(Δm-2u):∇2(Δm-2u)+am-1∇(Δm-2u)⋅∇(Δm-1u)+am-3∇2(Δm-3u):∇2(Δm-3u)+am-3∇(Δm-3u)⋅∇(Δm-2u)+∑i=0m-2(Δϕi(Δiu)2+4∇(ϕi)⋅∇(Δiu)Δiu+2ϕi∇(Δiu)⋅∇(Δiu))+am-2(n-4n+2)[∇(Δm-2u)⋅∇(Δm-2u)+Δm-2uΔm-1u]+(n-4n+2)Δam-2(Δm-2u)2+4(n-4n+2)∇am-2⋅∇(Δm-2u)Δm-2u+12Δam-1∇(Δm-2u)⋅∇(Δm-2u)+12Δam-3∇(Δm-3u)⋅∇(Δm-3u)+2∇am-1∇(Δm-2u):∇2(Δm-2u)+2∇am-3∇(Δm-3u):∇2(Δm-3u)+∑i=0m-22ϕiΔiuΔi+1u.
Using the well-known inequality (see [2])
(2.6)∇3w⋮∇3w≥3n+2∇(Δw)⋅∇(Δw),forallfunctionsw∈C3(Ω),
and (1.4) we deduce the inequality
(2.7)ΔP≥-∇(Δm-2u)⋅[∑i=0m-1(-1)i+1∇(ai)Δiu+∑i=0m-1(-1)i+1ai∇(Δiu)]+(n-4n+2)Δm-1u[∑i=0m-1(-1)i+1aiΔiu]+am-1∇2(Δm-2u):∇2(Δm-2u)+am-1∇(Δm-2u)⋅∇(Δm-1u)+am-3∇2(Δm-3u):∇2(Δm-3u)+am-3∇(Δm-3u)⋅∇(Δm-2u)+∑i=0m-2(Δϕi(Δiu)2+4∇(ϕi)⋅∇(Δiu)Δiu+2ϕi∇(Δiu)⋅∇(Δiu))+am-2(n-4n+2)[∇(Δm-2u)⋅∇(Δm-2u)+Δm-2uΔm-1u]+(n-4n+2)Δam-2(Δm-2u)2+4(n-4n+2)∇am-2⋅∇(Δm-2u)Δm-2u+12Δam-1∇(Δm-2u)⋅∇(Δm-2u)+12Δam-3∇(Δm-3u)⋅∇(Δm-3u)+2∇am-1∇(Δm-2u):∇2(Δm-2u)+2∇am-3∇(Δm-3u):∇2(Δm-3u)+∑i=0m-22ϕiΔiuΔi+1u.
The right side of (2.7) is
(2.8)=am-2(1+n-4n+2)∇(Δm-2u)⋅∇(Δm-2u)+am-1(n-4n+2)(Δm-1u)2+am-1∇2(Δm-2u):∇2(Δm-2u)+am-3∇2(Δm-3u):∇2(Δm-3u)+∑i=0m-2(Δϕi(Δiu)2+4∇(ϕi)⋅∇(Δiu)Δiu+2ϕi∇(Δiu)⋅∇(Δiu))+(n-4n+2)Δm-1u∑i=0m-3(-1)i+1aiΔiu-∇(Δm-2u)⋅∑i=0m-1(-1)i+1∇(ai)Δiu+(n-4n+2)Δam-2(Δm-2u)2+4(n-4n+2)∇am-2⋅∇(Δm-2u)Δm-2u+12Δam-1∇(Δm-2u)⋅∇(Δm-2u)+12Δam-3∇(Δm-3u)⋅∇(Δm-3u)+2∇am-1∇(Δm-2u):∇2(Δm-2u)+2∇am-3∇(Δm-3u):∇2(Δm-3u)-∇(Δm-2u)⋅∑i=0m-4(-1)i+1ai∇(Δiu)+∑i=0m-22ϕiΔiuΔi+1u.
Subsequently, we obtain
(2.9)ΔP≥[am-2(1+n-4n+2)-1]∇(Δm-2u)⋅∇(Δm-2u)+am-1∇2(Δm-2u):∇2(Δm-2u)+am-3∇2(Δm-3u):∇2(Δm-3u)+[am-1-12](n-4n+2)(Δm-1u)2-12(∑i=0m-1(-1)i+1∇(ai)Δiu)2-12[n-4n+2](∑i=0m-3(-1)i+1ai(Δiu))2-12(∑i=0m-4(-1)i+1ai∇(Δiu))2+∑i=0m-2Δϕi(Δiu)2+4∇(ϕi)⋅∇(Δiu)Δiu+2ϕi∇(Δiu)⋅∇(Δiu)+(n-4n+2)Δam-2(Δm-2u)2+4(n-4n+2)∇am-2⋅∇(Δm-2u)Δm-2u+12Δam-1∇(Δm-2u)⋅∇(Δm-2u)+12Δam-3∇(Δm-3u)⋅∇(Δm-3u)+2∇am-1∇(Δm-2u):∇2(Δm-2u)+2∇am-3∇(Δm-3u):∇2(Δm-3u)+∑i=0m-22ϕiΔiuΔi+1u.
We now state a series of inequalities to demonstrate that ΔP≥0. First we note
(2.10)∑i=0m-22ϕiΔiuΔi+1u≥-(∑i=0m-2(ϕi)2+1)∑i=0m-2(Δiu)2.
Similarly, we obtain
(2.11)-12(∑i=0m-1(-1)i+1∇(ai)Δiu)2≥-12(∑i=0m-1[∇(ai)⋅∇(ai)])(Δm-1u)2-12(∑i=0m-1[∇(ai)⋅∇(ai)])∑i=0m-2ΔiuΔiu,(2.12)-12[n-4n+2](∑i=0m-3(-1)i+1aiΔiu)2≥-12[n-4n+2]∑i=0m-3ai2∑i=0m-3(Δiu)2,(2.13)-12(∑i=0m-4(-1)i+1ai∇(Δiu))2≥-12∑i=0m-4ai2∑i=0m-4∇(Δiu)⋅∇(Δiu),(2.14)+∑i=0m-2(Δϕi)(Δiu)2+4∇(ϕi)⋅∇(Δiu)Δiu+ϕi∇(Δiu)⋅∇(Δiu)≥∑i=0m-2(Δiu)2[Δϕi3-4∇(ϕi)⋅∇(ϕi)ϕi].
We also have that
(2.15)12Δam-1∇(Δm-2u)⋅∇(Δm-2u)+2∇am-1∇(Δm-2u):∇2(Δm-2u)+am-1∇2(Δm-2u):∇2(Δm-2u)≥∇(Δm-2u)⋅∇(Δm-2u)[Δam-12-∇am-1⋅∇am-1am-1].
Additional inequalities analogous to (2.15) can be developed for the remaining terms in (2.9) involving Δam-2 and Δam-3.
From (2.2), (2.10)–(2.15) and upon inclusion of the aforementioned additional inequalities we deduce that(2.16)ΔP≥([am-1-12](n-4n+2)-12∑i=0m-1∇(ai)⋅∇(ai))(Δm-1u)2+∑i=0m-2[Δϕi3-β2(n-4n+2)-γ2](Δiu)2+∑i=0m-2[ϕi-β2](Δiu)2+∑i=0m-2[Δϕi3-α](Δiu)2+∑i=0m-2(Δiu)2[Δϕi3-4∇(ϕi)⋅∇(ϕi)ϕi]+(am-2-1)∇(Δm-2u)⋅∇(Δm-2u)+∇(Δm-2u)⋅∇(Δm-2u)[Δam-12-∇am-1⋅∇am-1am-1]+∇(Δm-3u)⋅∇(Δm-3u)[Δam-32-∇am-3⋅∇am-3am-3]+(n-4n+2)(Δm-2u)2[Δam-2-4∇am-2⋅∇am-2am-2].
Utilizing (2.4) the conclusion follows.Now we handle the case wheren≤4 in the following theorem.Theorem 2.2.
Suppose thatu∈C2m+1(Ω)∩C2m-1(Ω̅) is a solution of (1.4) where  n≤4. For  ϕ0(x),…,ϕm-1(x)∈C2(Ω)∩C0(Ω̅)satisfying the inequality  ∑i=0m-2ϕi2+1≤α for some positive constant  α, one defines
(2.17)P=∇2(Δm-2u):∇2(Δm-2u)-∇(Δm-2u)⋅∇(Δm-1u)+∑i=0m-1ϕi(Δiu)2+12(Δm-2u)2.
Then if
(2.18)ϕi≥α2,fori=0,…,m-1,Δϕm-i3-12≥0,fori=1,2,Δϕi3-4∇ϕi⋅∇ϕiϕi≥0,fori=0,…,m-1,Pis subharmonic in Ω.Proof.
A calculation similar to that of Theorem2.1 yields
(2.19)ΔP=2∇3(Δm-2u)⋮∇3(Δm-2u)-∇(Δm-1u)⋅∇(Δm-1u)-∇(Δm-2u)⋅∑i=0m-1(-1)i+1∇aiΔiu+∇(Δm-2u)⋅∇(Δm-2u)-∇(Δm-2u)⋅∑i=0m-1[(-1)i+1ai∇(Δiu)]+Δm-2uΔm-1u+∑i=0m-1Δϕi(Δiu)2+4∇(ϕi)⋅∇(Δiu)Δiu+∑i=0m-12ϕi∇(Δiu)⋅∇(Δiu)+2ϕiΔiuΔi+1u.
Using (2.6), (2.10), (2.11), (2.13), and (2.14) we obtain
(2.20)ΔP≥∑i=0m-1(Δiu)2[Δϕi3-4∇ϕi⋅∇ϕiϕi]+∑i=0m-12Δϕi3(Δiu)2+∑i=0m-1ϕi∇(Δiu)⋅∇(Δiu)-∇(Δm-2u)⋅∇(Δm-2u)-12(Δm-2u)2-12(Δm-1u)2+∇(Δm-2u)⋅∇(Δm-2u)-12(∑i=0m-1∇ai⋅∇ai)∑i=0m-1(Δiu)2-12(∑i=0m-1ai2)∑i=0m-1∇(Δiu)⋅∇(Δiu)≥∑i=0m-1(Δiu)2[Δϕi3-4∇ϕi⋅∇ϕiϕi]+∑i=0m-1[ϕi-α2]∇(Δu)⋅∇(Δu)+∑i=0m-1[Δϕi3-γ2](Δiu)2+(Δϕm-23-12)(Δm-2u)2+(Δϕm-13-12)(Δm-1u)2.
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 ofΔm-2u. suppose that the hypotheses of theorem 1 are satisfied and that the m conditions(3.1)Δiu=0,fori=0,…,m-5,Δm-2u=Δm-2u∂n=0,Δm-3u=Δm-3u∂n=0
hold on ∂Ω. Let A denote the area of Ω. As a consequence of integration by parts, Theorem 2.1 implies that(3.2)∫Ω|∇2(Δm-2u)|2dx≤A2max⁡∂Ω[|∇2(Δm-2u)|2+ϕm-4(x)(Δm-4u)2+n-42(n+2)(Δ(Δm-2u))2].Now, if the hypotheses of Theorem2.2 hold and if we impose the m conditions(3.3)Δiu=0,fori=0,…,m-3,Δm-2u=Δm-2u∂n=0
on ∂Ω we can deduce the inequality(3.4)∫Ω|∇2(Δm-2u)|2dx≤A2max⁡∂Ω[|∇2(Δm-2u)|2+ϕm-1(x)(Δ(Δm-2u))2],
from Theorem 2.2.



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