Maximum principle for higher order operators in general domains

Daniele Cassani; Antonio Tarsia

doi:10.1515/anona-2021-0210

Open Access Published by De Gruyter November 20, 2021

Maximum principle for higher order operators in general domains

Daniele Cassani and Antonio Tarsia

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0210

Abstract

We first prove De Giorgi type level estimates for functions in W^1,t(Ω), Ω⊂RN , with t>N≥2 . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W^1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

Keywords: Harnack’s inequality; Higher order PDEs; Polyharmonic operators; Positivity preserving property

MSC 2010: 35J30; 35J48; 35B50

1 Introduction

One of the most powerful tools in the study of partial differential equations and nonlinear analysis is without any doubts the Maximum Principle (MP in the sequel). It turns out to be fundamental in obtaining existence, uniqueness and regularity results in the theory of linear elliptic equations, as well as to establish qualitative properties of solutions to nonlinear equations. We mainly refer to [22] for classical results and historical development, where suitable applications also to the parabolic and hyperbolic cases are discussed. Let us merely mention that the roots of MP date back two centuries in the work of Gauss on harmonic functions, up to the ultimate version of Hopf [16], and then further extended in the seminal work of Nirenberg [20], Alexandrov [2] and Serrin [24], within the foundations of modern theory of PDEs.

The underlying idea is simple: positivity of a suitable set of derivatives of a function induces positivity of the function itself. This is elementary true for real functions of one variable which vanish at the endpoints of an interval where −u″(x)=0 and the validity can be extended to second order uniformly elliptic operators for which a prototype is the Laplace operator:

(1.1) −Δu=f, in Ω⊂RN,N≥2u=0, on ∂Ω

for which we have

f≥0⇒u≥0 in Ω .

Surprisingly, this is no longer true when considering higher order elliptic operators such as the biharmonic operator Δ²:

(1.2) Δ2u=f, in Ωu=∂u∂ν=0, on ∂Ω .

Indeed, in this case in general one has

f ≥ 0 ⇏ u ≥ 0 in Ω .

This is a well known fact as long as the domain Ω is not a ball, for which the positive Green function was computed by Boggio [6] and which keeps on being positive for slight deformations of the ball [25]. As deeply investigated in [11] and references therein, the lack of the positivity preserving property is due to the appearance of sets carrying small Hausdorff measure (see [15]) where u<0 and apparently without robust physical motivations. Recently the loss of the MP has been established in [1] also in the case of higher order fractional Laplacians. This paper is a step forward a better understanding of this phenomena and at the same time gives some general principle in order to recover the validity of the MP in the higher order setting.

Let us briefly recall some physical interpretation of (1.1)–(1.2). Indeed, (1.1) is modeling, among many other things, a membrane whose profile is u which deflects under the charge load f and clamped along the boundary ∂Ω. This is the case in which tension forces prevale on bending forces which can be neglected because of the ‘‘thin’' membrane. However, the model does not suite the case of a ‘‘thick’' plate in which bending forces have to be taken into account. Here higher order derivatives come into play which yield (1.2). As one expects for (1.1), and there this is true by the MP, upwards pushing of a plate, clamped along the boundary, should yield upwards bending: this is false for (1.2) in contrast to some heuristic evidences in applications (see e.g. [17] and references therein).

Our point of view here, roughly speaking, is that approaching the boundary, where the bending energy carries some minor effect because of the clamping condition, tensional forces can not be neglected for which the contribute of lower order derivatives may restore the validity of the MP. As a reference example, consider the following simple model:

(1.3) Δ2u−γΔu=f, in Ω⊂RN,γ≥0u=∂u∂ν=0, on ∂Ω .

Clearly for γ=0 one has (1.2) whence formally as γ → ∞, in a sense one may expect that (1.3) inherits some properties of (1.1).

As we are going to see, this is the case and for the more accurate model (1.3) surprisingly the MP holds true, for fairly general domains, provided γ≥γ0>0 , which is essentially given in terms of Sobolev and Poincaré best constants. Let us state our main result in the case of (1.3) though it extends to cover the general case of uniformly elliptic operators of any even order, see Corollary 5.1.

Theorem 1.1

Let Ω⊂RN,N=2,3 , be an open connected and bounded set, with sufficiently smooth boundary and which satisfies the interior sphere condition. Let u∈H02(Ω) be a weak solution to (1.3), where f ∈ L²(Ω), f=0 in Ω and |{x:f(x)>0}|>0 . Then, there exists γ0>0 (which depends on the diameter of Ω, Sobolev and Poincaré best constants but does not depend on f), such that for γ>γ0 one has u>0 in Ω.

As a consequence of Theorem 1.1, the operator Δ²−γΔ, which in addition to (1.2) contains the contribute of lower order derivatives, turns out to be a more natural extension of (1.1) to the higher order setting.

Overview. In Section 2 we prove some preliminary estimates which will be the key ingredient to prove in Section 3 a new Harnack type inequality. Indeed, in the higher order case, it is well known how truncation methods fail [11]. Our approach here is to demand some extra integrability on the function entering the Harnack inequality in place of being solution to a PDE, which usually yields Caccioppoli’s inequality and the solution belongs to the corresponding De Giorgi class. In [10] the authors prove a Harnack type inequality just for functions with membership in some De Giorgi classes. Here we drop this assumption though we assume more regularity in terms of integrability which however enables us to prove De Giorgi type pointwise level estimates. In Section 5 we apply the results obtained to prove the strong maximum principle for polyharmonic operators of any order, which contain lower order derivatives, in sufficiently smooth bounded domains which enjoy the interior sphere condition. This is done by a limiting procedure starting from compactly supported functions and then extending the results and estimates to the solutions of higher order PDEs subject to Dirichlet boundary conditions. Those boundary conditions are in a sense the natural ones as the higher order operator in this case does not decouple into powers of a second order operator. In one hand the result we obtain is a first step towards the investigation of qualitative properties of higher order nonlinear PDEs, such as uniqueness, optimal regularity, symmetries and concentration phenomena [5, 7, 13, 18, 19 ,21 ,23 ,26]. On the other hand, we are confident the tools introduced here may reveal useful also in different higher order contexts, such as parabolic problems, in the study of the sign of solutions to quasilinear equations and in the higher order fractional Laplacian setting [4, 8, 14].

This research started in 2010 when Theorem 1.1 was settled by the first named author in the form of conjecture in a conference in Pisa. New advances towards the results in this paper have been made in 2014 during the first visit of Louis Nirenberg in Varese, then in New York 2015, Pisa 2016 and Varese again in 2017 (his last trip), occasions in which Louis has further stimulated this research during long discussions of which we keep nostalgic memories. Goodbye Louis!

Notation. In the sequel we will use the following basic definitions:

B(x₀, r) denotes the ball in RN of center at x₀ and radius r;
ω_N is the volume of the unit ball in ℝ^N;
d_Ω denotes the diameter of the bounded set Ω in RN ;
∣ · ∣ applied to sets denotes the Lebesgue measure in RN otherwise it is the Euclidean norm in RN with scalar product (· , ·);
A+(x0,k,r):={x:x∈B(x0,r),u(x)>k} ;
(u − k)⁺ ≔ max{u − k, 0};
{f>0} denotes the set {x∈Ω:f(x)>0} ;
Ω satisfies the interior sphere condition if for all x ∈ ∂Ω there exists y ∈ Ω and r0>0 such that B(y, r₀)⊂Ω and x ∈ ∂B(y, r₀);
c and C denote positive constants which may change from line to line and which do not depend on the other quantities involved unless explicitly emphasized;
W^{m, p}(Ω) is the standard Sobolev space endowed with the norm ∥⋅∥m,pp=∑0≤|α|≤m∥Dαu∥pp ;
W0m,p(Ω) is the completion of smooth compactly supported functions with respect to the norm ∣·∣_{m, p};
the critical Sobolev exponent p∗:=NpN−mp , 1<p<N/m .

2 Preliminaries

Let Ω⊂RN , N=2, be an open bounded set with sufficiently smooth boundary. The following holds true

Lemma 2.1

Let u ∈ W^1,t(Ω), t>N and 1<s<N . Then there exists c(s,t)>0 such that for all k∈R , x₀ ∈ Ω and ρ ∈ (0, r) where 0<r<dist(x0,∂Ω) the following holds

(2.1) ∫A+(x0,k,ρ)(u−k)s∗dx≤c(s,t,N)(r−ρ)s∗|A+(x0,k,r)|(1−st)s∗s∫A+(x0,k,r)(u−k)tdx+rt∫A+(x0,k,r)|∇u|tdxs∗t .

Proof. Consider a standard cut-off function Θ∈C0∞(RN) given by

(2.2) Θ(x)=1,x∈B(x0,ρ)0,x∉B(x0,r),

such that 0 ⩽ Θ(x) ⩽ 1 and |∇Θ(x)|≤cr−ρ .

As W^1,t(Ω)↪ W^1,s(Ω), one has from Sobolev’s emedding and Hölder’s inequality

□

Remark 2.1

The condition 0<r<dist(x0,∂Ω) , namely that x₀ lies in the interior of Ω, is crucial to extend to the whole RN the function (u − k)Θ. Therefore when x₀ approaches ∂Ω, necessarily r=r(x₀) tends to zero.

Lemma 2.2

Let u ∈ W^1,t(Ω), t>N≥2 and 1<s<N . Let l, k∈R such that l>k , x₀ ∈ Ω and r<dist(x0,∂Ω) . Then for all ρ ∈ (0, r) one has

(2.3) ∫A+(x0,l,ρ)(u−l)2dx≤c(t)|A+(x0,k,r)|β(r−ρ)2(p−1)p∫A+(x0,k,r)(u−k)2dx1p⋅∫A+(x0,k,r)(u−k)tdx+rt∫A+(x0,k,r)|∇u|tdx2(p−1)pt ,

where β=1−2q+1−sts∗s2p−qpq , s=2qN(p−1)N(2p−q)+2q(p−1) and 2<q<2p , p>1 .

Proof. Let x₀ ∈ Ω, r<dist(x0,∂Ω) and for simplicity let us write A⁺(k, r) in place of A⁺(x₀, k, r).

For l,k∈R and ρ ∈ (0, r), since A⁺(l, ρ)⊂ A⁺(k, ρ) we have

(2.4) ∫A+(l,ρ)(u−l)2dx≤∫A+(k,ρ)(u−k)2dx.

Let q > 2 for which one has

(2.5) ∫A+(k,ρ)(u−k)2dx≤|A+(k,ρ)|1−2q∫A+(k,ρ)(u−k)qdx2q .

Let now p > 1 and 2 < q < 2p and estimate by Hölder’s inequality

(2.6) ∫A+(k,ρ)(u−k)qdx ≤∫A+(k,ρ)(u−k)2dxq2p∫A+(k,ρ)(u−k)2q(p−1)(2p−q)dx2p−q2p≤c(p,q,t)(r−ρ)q(p−1)p|A+(k,r)|(1−st)q(p−1)sp∫A+(k,ρ)(u−k)2dxq2p⋅∫A+(k,r)(u−k)tdx+rt∫A+(k,r)|∇u|tdxq(p−1)tp ,

where in the last inequality we have used Lemma 2.1 with s∗=2q(p−1)2p−q . Combine (2.5) and (2.6) to get

(2.7) ∫A+(k,ρ)(u−k)2dx ≤c(p,q,t,N)(r−ρ)2p−1p|A+(k,r)|(1−2q)+(1−st)2(p−1)sp∫A+(k,ρ)(u−k)2dx1p⋅∫A+(k,r)(u−k)tdx+rt∫A+(k,r)|∇u|tdx2(p−1)tp .

In what follows we will use the following result from [3] in order to prove a version of the well known Poincaré inequality.

Theorem 2.1

(Theorem A. 28, p. 184 in [3). Let u ∈ W^1,1(B_r), such that u=0 and |{x:u(x)=0}|≥|Br|2 . Then

(2.8) ∫Bru1∗dx11∗≤c∫Br|∇u|dx,

where c=c(N) depends only on the dimension N.

Lemma 2.3

Let u ∈ W^1,p(B_r) be such that |{x:u(x)=0}|≥|Br|2 , with p≥NN−1 and N=2. Then, the following holds

(2.9) ∫Br|u|pdx1p≤cωN1NpN−1Nr∫Br|∇u|pdx1p,

where c=c(N) is the constant in (2.8).

Proof. Apply Theorem 2.1 to the function ∣u∣^p, taking p≥NN−1 and N=2, to get

∫Br|u|pdx=∫Br|u|pN−1NNN−1dx≤cNN−1∫Br|∇|u|pN−1N|dxNN−1=cNN−1∫BrpN−1N|u|pN−1N−1|∇u|dxNN−1=cNN−1pNN−1N−1NNN−1∫Br|u|pN−1N−1|∇u|dxNN−1=cNN−1pNN−1N−1NNN−1∫Br|u|p(N−1N−1p)(|∇u|p)1p11NdxNN−1≤cNN−1pNN−1N−1NNN−1∫Br|u|pdx(N−1N−1p)NN−1⋅∫Br|∇u|pdx1pNN−1|Br|1N−1.

Since

N−1N−1pNN−1=1−1pNN−1,

we have

∫Br|u|pdx1pNN−1≤cNN−1pNN−1N−1NNN−1ωN1N−1rNN−1∫Br|∇u|pdx1pNN−1.

□

3 A Harnack type inequality

Next we derive a De Giorgi type level estimate (see [3,12]) for functions u ∈ W^1,t, t>N≥2 which will be the key ingredient in establishing a new Harnack type inequality. Let us emphasize that in De Giorgi’s theorem [9], level estimates hold for u ∈ W^1,2 which is a solution to a uniformly elliptic second order equation with bounded and measurable coefficients. As a consequence, Caccioppoli’s inequality holds and u ∈ W^1,2 belongs to the corresponding so-called De Giorgi class. Later, Di Benedetto and Trudinger relaxed the framework and in [10] they merely assume u ∈ W^1,2 belonging to some De Giorgi class. Here, we further improve the setting, without requiring any of those previous assumptions, though demanding for some augmented integrability which turns out to be necessary, as it is well known, functions in W^1,N(Ω), Ω⊂RN , may not be bounded.

Theorem 3.1

Let u ∈ W^1,t(Ω), t>N≥2 , Ω⊂RN be open and bounded set with sufficiently smooth boundary ∂Ω. For all k∈R , y ∈ Ω, r>0 such that r<dist(y,∂Ω) , the following holds

(3.1) supB(y,r2)u≤k+d,

where

d=crξ(p−1)ηp∫A+(k,r)|u(x)−k|tdx+rt∫A+(k,r)|∇u(x)|tdxξ(p−1)tpη∫A+(k,r)|u(x)−k|2dxξ(θ−1)2η|A+(k,r)|θ−12 .

Remark 3.1

Here we write for simplicity A⁺(k, ρ) in place of A⁺(y, k, ρ) and c=c(t, p, ξ, η, θ) is a positive constant which depends on the parameters p > 1, t > N obtained in Lemma 2.2, while ξ > 0, η > ), θ > 1 are defined by suitable equations stated in the proof.

Proof of Theorem 3.1. Let us set:

I(l,ρ)=∫A+(l,ρ)|u−l|2dx ,M(r,k,t,p)=c(t)∫A+(k,r)|u−k|tdx+rt∫A+(k,r)|∇u|tdx2(p−1)pt,

where c(t) is the constant of (2.3). For all l,k∈R , such that l > k and for all ρ ∈ (0, r), one has

(3.2) |A+(l,ρ)|≤1(l−k)2I(k,ρ)

and clearly ∣A⁺(l, ρ)∣ ⩽ ∣A⁺(k, ρ)∣, for l > k.

Set

(3.3) Φ(l,ρ)=I(l,ρ)ξ|A+(l,ρ)|η,

then from (3.2) and (2.3) we have

(3.4) Φ(l,ρ)≤1(r−ρ)2ξp−1p(l−k)2ηΦ(k,r)θM(r,k,t,p)ξ,

where η,ξ,θ > 0 satisfy the following algebraic equations

(3.5) ξp+η=θξβξ=θη

from which we have θ² − θ/p − β=0 and we take θ=θ₁ given by

(3.6) θ1=1/p+1/p2+4β2 .

As one can easily check θ1>1 , for all 2<q<2p , t>N and 1<s<N .

From (3.4) we are done provided we prove that for all k∈R and r<dist(y,∂Ω) there exists d>0 satisfying

Φk+d,r2=0,

which in turn by (3.3) yields

A+k+d,r2=0 .

Next we proceed by using the iterative scheme from the proof of De Giorgi’s theorem. For m∈N set

rm=r2+r2m+1,km=k0+d−d2m,

where the parameter d>0 has to be chosen in the sequel and k₀=k. The idea is to exploit the inequality (3.4) with r=r_m and ρ=r_m+1 where the sequence {rm}m∈N is decreasing so that B(r_m+1)⊂ B(r_m). On the other hand {km}m∈N is increasing, and we set in (3.4) l=k_m+1 and k=k_m. With this choice we obtain from (3.4) the following inequality

(3.7) Φ(km+1,rm+1)≤22(p−1)p(m+2)ξ+2(m+1)ηr2(p−1)pξd2ηΦ(km,rm)θM(rm,km,t,p)ξ.

Now multiply (3.7) by 2^μ(m+1), μ>0 and set

(3.8) Ψm=2μmΦ(km,rm)

to obtain form (3.7)

(3.9) Ψm+1≤22(p−1)p(m+2)ξ+2(m+1)ηr2(p−1)pξd2η2μ[1+m(1−θ)]ΨmθM(rm,km,t,p)ξ.

Let us choose μ>0 to avoid the dependence on m in the first factor in the right hand side of (3.9), namely

(3.10) μ=2(p−1)pξ+2ηθ−1,

and thus (3.9) becomes

Ψm+1≤24(p−1)pξ+2η+μr2(p−1)pξd2ηΨmθM(rm,km,t,p)ξ≤24(p−1)pξ+2η+μr2(p−1)pξd2ηΨmθM(r,k0,t,p)ξ .

Set

A=24(p−1)pξ+2η+μr2(p−1)pξM(r,k0,t,p)ξ,

so that for all m∈N one has

Ψm+1≤Ad2ηΨmθ .

At this point we choose d>0 such that

(3.11) A d 2 η Ψ 0 θ − 1 = 1 ,

and by induction on m∈N we have

Ψm≤Ψ0,for allm∈N .

Finally by (3.8) we obtain

Φ(km,rm)≤12μmΦ(k0,r)

and the proof is complete by letting m → ∞. □

Remark 3.2

It is important to note that in Lemma 2.2 as t → N one has β → (p − 1)/p so that θ₁ → 1 in (3.6). As a consequence, from (3.10) one has μ → ∞ and this, as expected, prevents the result to hold.

Next we prove the following Harnack type inequality

Theorem 3.2

Let u ∈ W^1,t(Ω), t>N≥2 , and Ω⊂RN be an open bounded set with sufficiently smooth boundary ∂Ω. Let B(x₀, r)⊂Ω, then there exists a constant c>0 , which depends only on N, such that

(3.12) supB(x0,r2)u≤infB(x0,r)u+cr[(ξη+N2)(θ−1)]∫B(x0,r)|∇u|tdxξηp−1tp∫B(x0,r)|∇u|2dxξ(θ−1)2η .

Proof. Set M=supB(x0,r)u , m=minB(x0,r)u , and let

I 1 = k : k ∈ ( m , M ) : x : x ∈ B ( x 0 , r ) , u ( x ) > k < | B ( x 0 , r ) | 2 , I 2 = k : k ∈ ( m , M ) : x : x ∈ B ( x 0 , r ) , u ( x ) ≥ k ≥ | B ( x 0 , r ) | 2 .

If I₁≠∅ then we prove for all k ∈ I₁ the following

(3.13) supB(x0,r2)u≤k+cr[(ξη+N2)(θ−1)]∫B(x0,r)|∇u|tdxξηp−1tp∫B(x0,r)|∇u|2dxξ(θ−1)2η .

Indeed, by Theorem 3.1 we have for all k ∈ I₁

(3.14) supB(x0,r2)u≤k+cr−ξ(p−1)ηp|A+(k,r)|θ−12 ∫A+(k,r)|u−k|tdx+rt∫A+(k,r)|∇u|tdxξηp−1tp ∫A+(k,r)|u−k|2dxξ(θ−1)2η .

Since k ∈ I₁ one has

|{x:(u(x)−k)+=0}|≥|B(x0,r)|2,

and apply Lemma 2.3 to the function (u(x) − k)⁺ to get

∫A+(k,r)|u(x)−k|2dx12=∫B(x0,r)|(u(x)−k)+|2dx12≤c(N)r∫B(x0,r)|∇u|2dx12 ,

∫A+(k,r)|u(x)−k|tdx1t=∫B(x0,r)|(u(x)−k)+|tdx1t≤c(N)r∫B(x0,r)|∇u|tdx1t .

In the case I₂≠∅, for all k ∈ I₂ set h=−k and v(x)=−u(x). Thus h ∈ (−M, −m) and the following holds

x : x ∈ B ( x 0 , r ) : u ( x ) ≥ k = x : x ∈ B ( x 0 , r ) : − u ( x ) ≤ − k = x : x ∈ B ( x 0 , r ) : v ( x ) ≤ h ≥ | B ( x 0 , r ) | 2 .

Therefore, the function v enjoys (3.13), namely

(3.15) supB(x0,r2)v≤h+cr[(ξη+N2)(θ−1)]∫B(x0,r)|∇v|tdxξηp−1tp∫B(x0,r)|∇v|2dxξ(θ−1)2η.

From

supB(x0,r2)v=−infB(x0,r2)u

and (3.15) we have

−infB(x0,r2)u≤−k+cr[(ξη+N2)(θ−1)]∫B(x0,r)|∇u|tdxξηp−1tp∫B(x0,r)|∇u|2dxξ(θ−1)2η.

As a consequence, for all k ∈ I₂ we get

(3.16) k≤infB(x0,r2)u+cr[(ξη+N2)(θ−1)]∫B(x0,r)|∇u|tdxξηp−1tp∫B(x0,r)|∇u|2dxξ(θ−1)2η.

Next we distinguish three cases, precisely:

I₁≠∅ and I₂=∅. In this case any k ∈ (m, M) belongs to I₁, for which (3.13) which holds for all k ∈ I₁, it holds for k=m as well;
I₁=∅ and I₂≠∅. In this case any k ∈ (m, M) belongs to I₂, and thus (3.16) which holds for all k ∈ I₂, in particular holds for k=M;
I₁≠∅ and I₂≠∅. In this case we consider inf I₁ and sup I₂ and it is standard to prove there exists a unique k₀=inf I₁=sup I₂ which enjoys both (3.13) and (3.16) and the theorem follows.

□

In order to state the next result let us introduce the following

Definition 3.1

Let Ω be an open set in RN , N=2 with non-empty and sufficiently smooth boundary and which enjoys the interior sphere condition. Let x ∈ ∂Ω and consider balls of radius r, B_x(r)⊂Ω which are tangent in the interior to ∂Ω at point x and let δ(x)=sup r. We define the narrowness index of Ω as follows:

(3.17) δ=infx∈∂Ωδ(x) .

Theorem 3.3

Let Ω⊂RN , N=2 be open, connected, with sufficiently smooth boundary and which enjoys the interior sphere condition. Let δ be the narrowness index of Ω as in Definition 3.17. Let x_max and x_min be respectively a local maximum and local minimum for u ∈ W^1,t(Ω), t>N .

Then, there exists h∈N and r ∈ (0, δ), such that

(3.18) u(xmax)≤u(xmin)+chr(ξη+N2)(θ−1)∫Ω|∇u(x)|tdxξ(p−1)tpη∫Ω|∇u(x)|2dxξ(θ−1)2η ,

with c=c(N) provided by Thorem 3.2 and where in particular h depends only on dist (x_max, ∂Ω), dist (x_min, ∂Ω) and δ.

Proof. Let r>0 be such that:

for all x ∈ B(x_min, r)⊂Ω one has u(x)=u(x_min);
B(xmin,r)‾⊂Ω ;
B(xmax,r)‾⊂Ω .

Consider the arc g: [0, 1] → Ω such that g (0)=x_min and g (1)=x_max. Let t0=0<…th=1 be a partition of [0, 1] such that setting x_i=g(t_i) one has

(3.19) Bxi,r2∩Bxi+1,r2≠∅,i=0,…,h−1

and where r is such that B(x_i, r)⊂Ω.

By Theorem 3.2 we have

supB(x0,r2)u≤u(xmin)+cr(ξη+N2)(θ−1)∫B(x0,r)|∇u(x)|tdxξ(p−1)tpη∫B(x0,r)|∇u(x)|2dxξ(θ−1)2η,

which we rewrite in the following form

(3.20) ∀x∈Bx0,r2,u(x)≤u(xmin)+N0,

where we have set for i=0, ⋅, h

Ni:=c∫B(xi,r)|∇u(x)|tdxξ(p−1)tpη∫B(xi,r)|∇u(x)|2dxξ(θ−1)2η .

Now inequality (3.20) in particular holds for

x∈Bx1,r2∩Bx0,r2

and thus

(3.21) infBx1,r2u≤u(x)≤u(xmin)+N0 .

By applying iteratively Theorem 3.2 we end up with

supB(xh+1,r2)u≤u(xmin)+Nh+⋯+N1+N0 ,

which completes the proof. □

Remark 3.3

One may wonder what happens if in the construction of Theorem 3.3 we consider a sequence of balls with increasing radius and center converging to a point on the boundary of Ω. For this purpose consider {xn}n∈N⊂Ω converging to a point x_∞ ∈ ∂Ω. Consider balls of center x_n and radius r_n such that:

B(x_n, r_n)⊂Ω;
rn<dist(xn,∂Ω) =dist (x_n, x_∞);
B(xn,rn2)∩B(xn+1,rn+12)≠∅ .

Applying to this sequence the reasoning carried out in the proof of Theorem 3.3 where x₀=x_max, we get

u(xmax)≤u(x∞)+c∑n=0∞rn(ξη+N2)(θ−1)1γ∫Ω1f(x)dxξ2η(θ−1)u(xmax).

We would get a contradiction if the above series converge. Actually as we are going to see this is not the case. Consider B(xn,rn2) and B(xn+1,rn+12) and let C∈∂B(xn,rn2)∩∂B(xn+1,rn+12) and D its projection on the segment with endpoints A=x_n and B=x_n+1. Set AD=ρ_n, DB=ρ_n+1, so that considering the triangles ADC and CDB one has rn24−ρn2=rn+124−ρn+12 , and then

rn4+ρnrn+14+ρn+1=rn+14−ρn+1rn4−ρn.

We can apply Kummer’s test to the series with general terms an=rn4+ρna and bn=rn4−ρna , a>0 , from which since anan+1=bn+1bn , for all n∈N , and ∑n=0∞1bn=+∞ we obtain ∑n=0∞an=+∞ . From an<rna we have ∑n=0∞rna=+∞ .

4 Towards the Positivity Preserving Property

Next we apply the results so far obtained to prove the strong maximum principle for the biharmonic operator perturbed by the Laplacian for compactly supported data. As we are going to see, here it comes for the first time the restriction on the Euclidean dimension N<4 and the fact that we deal with the solution to a PDE. Precisely, this section is devoted to proving the following

Theorem 4.1

Let Ω⊂RN , N=2, 3 be an open and bounded set, with sufficiently smooth boundary and which enjoys the interior sphere condition. Let u∈W4,2(Ω)∩H02(Ω) be a solution to

(4.1) Δ2u(x)−γΔu(x)=f(x),x∈Ω,

where γ>0 , f ∈ L²(Ω), f=0 in Ω and |{x:f(x)>0}|>0 . Moreover, f(x)=0 on Ω∖Ω₁, with Ω₁ a bounded subset of Ω such that dist(∂Ω1,∂Ω)>0 . Then, there exists γ0>0 such that for all γ>γ0 the solution to (4.1) satisfies u(x)>0 , for all x ∈ Ω.

Assuming the hypotheses of Theorem 4.1 we have the following preliminary lemmas:

Lemma 4.1

The following holds true

(4.2) supΩ1u>0 .

Proof. By multiplying (4.1) by u and integrating by parts

(4.3) ∫Ω|Δu(x)|2dx+γ∫Ω|∇u(x)|2dx=∫Ωf(x)u(x)dx≤supΩ1u∫Ω1f(x)dx

In order to apply the Harnack inequality established in Section 3 we next estimate first order derivatives of the solution to (4.1). Though from one side elliptic regularity yields enough summability, on the other side we need estimates which are uniform with respect to the parameter γ, and for this reason we restrict ourself to dimensions N<4 .

Lemma 4.2

There exists a constant c=c(N)>0 which does not depend on γ in (29) such that

∥∇u∥Lt(Ω)≤cdΩ2t(3−N)∫Ωf(x)u(x)dx12,

for any t>2 when N=2 and for t=6 when N=3.

Proof. Since u=∇ u=0 on ∂Ω, one has

∫Ω|Δu(x)|2dx=∑i,j=1n∫Ω|Diju(x)|2dx=∫Ω∥D2u(x)∥2dx .

By Sobolev’s embedding, Poincaré inequality and from (4.3), when N=3 and t=6 we have,

∥∇u∥Lt(Ω)≤cSdΩ∥∇u∥L2(Ω)+cS∥D2u∥L2(Ω)≤c∥D2u∥L2(Ω)=c∥Δu∥L2(Ω)≤c∫Ωf(x)u(x)dx12 .

Similarly when N=2 and t=1 we obtain

∥∇u∥Lt(Ω)≤cSdΩ1−2t∥∇u∥L2(Ω)+cSdΩ2t∥D2u∥L2(Ω)≤cdΩ2t∥D2u∥L2(Ω)=cdΩ2t∥Δu∥L2(Ω)≤cdΩ2t∫Ωf(x)u(x)dx12 .

Proof of Theorem 4.1. Let x_max be an absolute maximum point for u in Ω1‾ and x_min a local minimum for u in Ω. Set

a=ξ(θ−1)2η+ξ(p−1)2ηp,b=ξη+N2(θ−1),c=ξηp−1p .

From (30) of Theorem 3.3, (36) and Lemma 4.2 we have

(4.4) u(xmax)≤u(xmin)+chrbdΩ[2t(3−N)]c∫Ωf(x)u(x)dxaγξ2η(θ−1)

where a=ξ(θ−1)2η+ξ(p−1)2ηp<1 . If supΩ1u≥1 then we have

u(xmax)≤u(xmin)+chrbdΩ[2t(3−N)]c∫Ωf(x)dxau(xmax)γξ2η(θ−1) .

The thesis follows as γ is large enough. If supΩ1u<1 , let k>0 be such that ksupΩ1u≥1 . Set w_k(x)≔ k u(x), which satisfies

(4.5) wk∈W4,2(Ω)∩H02(Ω)Δ2wk(x)−γΔwk(x)=kf(x),x∈Ω,

Peforming the change of variable x=sy, with s>0 , v_k(y)=w_k(sy), g(y)=f (sy), ymin=xmins , ymax=xmaxs , we obtain

(4.6) vk∈W4,2(Ωs)∩H02(Ωs)Δ2vk(y)−γs−2Δvk(y)=s−4kg(y),y∈Ωs,

where Ωs={y:y=x/s,x∈Ω} . Next apply (4.4) to the solution of (4.6) to get

(4.7) vkymax≤vkymin+chrsbdΩs2t(3−N)c∫Ωsg(x)dxavkymaxγξ2η(θ−1)kas3a.

With respect to the original variables it reads as follows

(4.8) uxmax≤uxmin+chrsbdΩs2t(3−N)c∫Ωf(x)dxauxmaxγξ2η(θ−1)kas(N+3)a.

Let us now observe that thanks to the interior sphere condition, the number h of balls covering the path from y_max to y_min does not depend on the parameter s. The same happens for the parameter k. Thus we choose the parameter s such that

(4.9) hkasb+[2t(3−N)]c+(N+2)a=1,

namely the thesis of the theorem follows for all

(4.10) γ>c2ηξ(θ−1)dΩ2ηξ(θ−1){[2t(3−N)]c+b}∫Ωf(x)dxa2ηξ(θ−1),

and thus γ₀ is the right hand side of (4.10) with optimal constant c. When γ=γ₀ we just get the weak inequality u=0. □

5 The validity of the strong maximum principle for higher order elliptic operators

In this section we first prove Theorem 1.1 for which we have to remove the restriction to compactly supported data of Theorem 4.1. Then, we will extend the result obtained to polyharmonic operators and to more general uniformly elliptic operators of any even order with constant coefficients.

Proof of Theorem 1.1. Consider the following family of sets {Ωm}m∈N such that for all m∈N satisfy:

Ω‾m⊂Ωm+1⊂Ω‾m+1⊂Ω ;
∪m=1∞Ωm=Ω ;
x∈Ω:|{f>0}|>0∩Ω1≠Ω1 ;
dist (∂Ω_m, ∂Ω) → 0 as m → ∞.

Let χ_m be the characteristic function of Ω_m

χm(x)=1,x∈Ωm,0,x∉Ωm.

and set

(5.1) gm(x):=1S(x)χm(x)m2f(x),x∈Ω,

where

S(x)=∑m=1+∞χm(x)m2,

converges pointwise on Ω. Moreover, notice that g_m ∈ L²(Ω).

Next consider the following problems

(5.2) um∈W4,2∩H02(Ω)Δ2um(x)−γΔum(x)=gm(x),x∈Ω,

where by construction g_m(x)=0 for x ∈ Ω∖Ω_m and thus by Theorem 4.1 there exists γm>0 such that for all γ>γm , one has um(x)>0 , for all x ∈ Ω, m∈N .

It is crucial here that by (4.9) and (4.10) the parameter γ_m does not depend on h, namely does not depend on the distance of the maximum point of u_m from the boundary (recall the proof of Theorem 3.3). Indeed, this prevents γ_m to blow up and actually remain bounded since from (4.10)

γm=c2ηξ(θ−1)dΩ2ηξ(θ−1){[2t(3−N)]c+b}∫Ωmgm(x)dxa2ηξ(θ−1)≤c2ηξ(θ−1)dΩ2ηξ(θ−1){[2t(3−N)]c+b}∫Ωf(x)dxa2ηξ(θ−1)=γ∞ .

Therefore, for all γ>γ∞ and for all m∈N one has

(5.3) um(x)>0,x∈Ω

Finally, we prove that the function

(5.4) v(x)=∑m=1∞um(x)

solves the following

(5.5) v∈W4,2∩H02(Ω)Δ2v(x)−γΔv(x)=f(x),x∈Ω

and thus by (5.3) we conclude that for all γ>γ∞ and for all x ∈ Ω one has

v(x)>0 .

By uniqueness of the solution to the Dirichlet problem (4.1) the Theorem follows. Hence, it remains to show that vm→v∈W4,2∩H02(Ω) which is a solution to (5.5).

Set

fm=∑i=1mgi,vm=∑i=1mui .

By Lebesgue’s dominated convergence f_m → f in L²(Ω) and notice that v_m solves the following

(5.6) vm∈W4,2∩H02(Ω)Δ2vm(x)−γΔvm(x)=fm(x),x∈Ω .

Thus for all m, l∈N we have

(5.7) vm−vl∈W4,2∩H02(Ω)Δ2vm(x)−vl(x)−γΔvm(x)−vl(x)=fm(x)−fl(x)

and multiplying by v_m−v_l and integrating by parts we get

∫Ω|Δ[vm(x)−vl(x)]|2dx≤∫Ω|fm(x)−fl(x)|2dx

which together with the equation (5.6) yields

∫Ω|Δ2[vm(x)−vl(x)]|2dx≤c(2γ2+2)∫Ω|fm(x)−fl(x)|2dx .

Thus {v_m} is a Cauchy sequence in W^4,2 (Ω) which converges to v ∈ W^4,2 (Ω), the solution to (5.5). □

Remark 5.1

Observe that in (5.5) the solution can be normalized dividing the equation by ∫Ωfdx>0 , so that the parameter γ₀ identified in (4.10) does not depend effectively on f.

What we have seen so far naturally extends to polyharmonic operators of any order and more in general to uniformly elliptic operators of any even order as established in the following

Corollary 5.1

Let u∈W2m,2∩W0m,2(Ω) , m=2 be the solution to the following equation

(5.8) (−1)mA2m(D)u(x)−γA2(x,D)u(x)=f(x),x∈Ω,

where f ∈ L²(Ω),

A2m(D)=∑|α|=|β|=maαβDα+β

and

A2(x,D)=∑i,j=1nDi[aij(x)Dj],

are uniformly elliptic operators on Ω, namely there exist νm>0 and ν1>0 such that for all ξ∈RN and x ∈ Ω

νm∥ξ∥2m≤∑|α|=|β|=maαβξα+β,ν1∥ξ∥2≤∑i,j=1naij(x)ξiξj,

with aαβ∈R and a_ij(x) ∈ L^∞(Ω). Then, there exists γ0>0 such that for all γ>γ0 one has u(x)>0 for all x ∈ Ω.

Proof. We have to estimate intermediate derivatives of suitable order avoiding the dependance on γ. Multiplying the equation (5.8) by u and integrating by parts we get

∫Ω∑|α|=|β|=maαβDαu(x)Dβu(x)dx+γ∫Ω∑i,j=1naij(x)Dju(x)Diu(x)dx≤∫Ωf(x)u(x)dx≤∥f∥L2(Ω)∥u∥W0m(Ω) .

By the ellipticity condition and Gårding’s inequality one has

vm∥u∥W0m(Ω)2+γv1∥u∥W01,2(Ω)2≤∫Ω|f(x)||u(x)|dx

together with Poincaré's inequality

νm∥u∥W0m(Ω)2≤c(N,Ω)∫Ω|f(x)|2dx .

We conclude by the Sobolev embedding theorem as follows:

• If N ⩽ 2(m − 1) one has ∇ u ∈ L^t(Ω), for all t=1 and in particular for t>N and

∥∇u∥Lt(Ω)≤c∥u∥Wm,2(Ω)≤c∥f∥L2(Ω) ;

• If N=2m − 1 one has ∇ u ∈ L^t(Ω) with t=4m − 2 and

∥∇u∥Lt(Ω)≤c∥u∥Wm,2(Ω)≤c∥f∥L2(Ω) ,

where the constant c does not depend on γ.

It is well known from [11] that the positivity preserving property of the ball for polyharmonic operators carries over to small deformations of the ball. Actually on those domains what we have proved yields the positivity preserving property of the γ-perturbed polyharmonic operator for all γ⩾0. For simplicity let us state the result in the case of the biharmonic operator:

Corollary 5.2

Let Ω⊂RN be an open bounded domain such that for all f ∈ L²(Ω) with f=0 and ∣{x ∈ Ω: f(x)=0}∣=0, the solution u∈W4,2∩W02,2(Ω) to

(5.9) Δ2u=f

enjoys u(x)>0 , a.e. in Ω. If there exists γ0>0 such that the solution v∈W4,2∩W02,2(Ω) to

(5.10) Δ2v−γ0Δv=f

enjoys v(x)>0 , a.e. in Ω, then for all γ ∈ [0, γ₀] the solution w∈W4,2∩W02,2(Ω) to Δ² w_γ − γ Δw_γ=f enjoys wγ(x)>0 , a.e. in Ω.

Proof. Set w_τ=τ v + (1−τ) u, γ_τ=τ γ₀, hence Δ² w_τ− γ_τ Δw_τ=f. For τ=0, w_τ is a solution to (5.9) whence for τ=1, w_τ enjoys (5.10) and then wτ>0 a.e. in Ω for all τ ∈ [0, 1]. By uniqueness of the Dirichlet problem w_τ=w_γ and the claim follows. □

From Corollary 5.2 we also have

Corollary 5.3

Let Ω⊂RN be an open bounded domain such that for all f ∈ L²(Ω) with f=0 and ∣{x ∈ Ω: f(x)=0}∣=0, the solution u∈W4,2∩W02,2(Ω) to Δ² u=f enjoys u(x)>0 , a.e. in Ω. Then, if N=2, 3, we have for all γ ∈ [0, +∞) that wγ>0 .

Conflict of interest

Author’s Statement

Conflict of interest: Authors state no conflict of interest.
Dedicated to In loving memory of Louis Nirenberg

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Received: 2021-06-06

Accepted: 2021-09-23

Published Online: 2021-11-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

Maximum principle for higher order operators in general domains

Abstract

1 Introduction

Theorem 1.1

2 Preliminaries

Lemma 2.1

Remark 2.1

Lemma 2.2

Theorem 2.1

Lemma 2.3

3 A Harnack type inequality

Theorem 3.1

Remark 3.1

Remark 3.2

Theorem 3.2

Definition 3.1

Theorem 3.3

Remark 3.3

4 Towards the Positivity Preserving Property

Theorem 4.1

Lemma 4.1

Lemma 4.2

5 The validity of the strong maximum principle for higher order elliptic operators

Remark 5.1

Corollary 5.1

Corollary 5.2

Corollary 5.3

References

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