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Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds

Journal of the Egyptian Mathematical Society volume 27, Article number: 24 (2019) Cite this article

Abstract

Consider the bi-harmonic differential expression of the form

\( A=\triangle _{M}^{2}+q\ \)

on a manifold of bounded geometry (M,g) with metric g, where △M is the scalar Laplacian on M and q≥0 is a locally integrable function on M.

In the terminology of Everitt and Giertz, the differential expression A is said to be separated in Lp(M), if for all u∈Lp(M) such that Au∈Lp(M), we have qu∈Lp(M). In this paper, we give sufficient conditions for A to be separated in Lp(M),where 1<p<∞.

Introduction

In the terminology of Everitt and Giertz, the concept of separation of differential operators was first introduced in [1]. Several results of the separation problem are given in a series of pioneering papers [2–5]. For more backgrounds concerning to our problem, see [6–8]. Atia et al. [9] have studied the separation property of the bi-harmonic differential expression \(A=\triangle _{M}^{2}+q\ \), on a Riemannian manifold (M,g) without boundary in L2(M), where △M is the Laplacian on M and \(0\leq q\in L_{loc}^{2}\left (M\right) \ \) is a real-valued function.

Recently, Atia [10] has studied the sufficient conditions for the magnetic bi-harmonic differential operator B of the form \(B=\triangle _{E}^{2}+q\ \) to be separated in L2(M), on a complete Riemannian manifold (M,g) with metric g, where △E is the magnetic Laplacian on M and q≥0 is a locally square integrable function on M. In [11], Milatovic has studied the separation property for the Schrodinger-type expression of the form L=△M+q, on non-compact manifolds in Lp(M). Let (M,g) be a Riemannian manifold without boundary, with metric g (i.e., M is a C∞−manifold without boundary and g=(gjk) is a Riemannian metric on M) and dimM=n. We will assume that M is connected. We will also assume that we are given a positive smooth measure dμ, i.e., in any local coordinates x1,x2,…,xn, there exists a strictly positive C∞−density ρ(x) such that dμ=ρ(x)dx1dx2…dxn. In the sequel, L2(M) is the space of complex-valued square integrable functions on M with the inner product:

$$ \left(u,v\right) =\int\limits_{M}\left(uv^{-}\right) d\mu, $$
(1)

and ∥.∥ is the norm in L2(M) corresponding to the inner product (1). We use the notation L2(Λ1T∗M) for the space of complex-valued square integrable 1-forms on M with the inner product:

$$ \left(W,\Psi \right)_{L^{2}\left(\Lambda^{1}T^{\ast }M\right) }=\int\limits_{M}\left\langle W,\overline{\Psi }\right\rangle d\mu, $$
(2)

where for 1-forms W=Wjdxj and Ψ=Ψkdxk, we define 〈W,Ψ〉=gjkWjΨk, where (gjk) is the inverse matrix to (gjk), and \(\overline {\Psi }=\overline {\Psi _{k}}dx^{k}\ \)(above, we use the standard Einstein summation convention).

The notation \(\left \Vert.\right \Vert _{L^{2}\left (\Lambda ^{1}T^{\ast }M\right) }\) stands for the norm in L2(Λ1T∗M) corresponding to the inner product (2). To simplify notations, we will denote the inner products (1) and (2) by (.,.). In the sequel, for 1≤p<∞,Lp(M) is the space of complex-valued p-integrable functions on M with the norm:

$$ \left\Vert u\right\Vert_{p}=\left(\int\limits_{M}\left\vert u\right\vert^{p}~d\mu \right)^{\frac{1}{p}}, $$
(3)

In what follows, by C1(M), we denote the space of continuously differentiable complex-valued functions on M, and by C∞(M), we denote the space of smooth complex-valued functions on M, by \(C_{c}^{\infty }\left (M\right) -\)the space of smooth compactly supported complex-valued functions on M, by Ω1(M)− the space of smooth 1-forms on M, and by \(\Omega _{c}^{1}\left (M\right)-\) the space of smooth compactly supported 1-forms on M. In the sequel, the operator d:C∞(M)→Ω1(M) is the standard differential and d∗:Ω1(M)→C∞(M) is the formal adjoint of d defined by the identity: \(\left (du,v\right)_{L^{2}\left (\Lambda ^{1}T^{\ast }M\right) }=\left (u,d^{\ast }v\right),\ \ \ \ \ \ \ \ u\in C_{c}^{\infty }\left (M\right),~v\in \Omega ^{1}\left (M\right).\)By ΔM=d∗d, we will denote the scalar Laplacian on M. We will use the product rule for d∗ as follows:

$$ d^{\ast }(uv)=ud^{\ast }v-\left\langle du,v\right\rangle,~u\in C^{1}\left(M\right),~v\in \Omega^{1}\left(M\right). $$
(4)

We consider the bi-harmonic differential expression:

$$ A=\triangle_{M}^{2}+q,\ $$
(5)

where q≥0 is a locally integrable function on M.

Definition 1

The set Dp:

Let A be as in (5), we will use the notation

$$ D_{p}=\{u\in L^{p}\left(M\right) :Au\in L^{p}\left(M\right) \}. $$
(6)

Remark 1

In general, it is not true that for all u∈Dp, we have \(\triangle _{M}^{2}u\in L^{p}\left (M\right) \) and qu∈Lp(M) separately. Using the terminology of Everitt and Giertz, we will say that the differential expression \(A=\triangle _{M}^{2}+q\) is separated in Lp(M) when the following statement holds true: for all u∈Dp, we have qu∈Lp(M).

We will give sufficient conditions for A to be separated in Lp(M). Assume that the manifold (M,g) has bounded geometry, that is

(a) infx∈Mrinj(x)>0, where rinj(x) is the injectivity radius of (M,g),

(b) all covariant derivatives ∇jR of the Riemann curvature tensor R are bounded: |∇jR|≤Kj, j=0,1,2,...,where Kj are constants.

Let (M,g) be a manifold of bounded geometry. Then, there exists a sequence of functions (called cut-off functions) {ϕj} in \( C_{c}^{\infty }\left (M\right) \) such that for all j=1,2,3...,

(i) 0≤ϕj≤1;

(ii) ϕj≤ϕj+1;

(iii) for every compact set S⊂M, there exists j such that ϕj|S=1;

(iv) supx∈M|dϕj|≤C1, supx∈M|△Mϕj|≤C1, and \(\sup _{x\in M}\left \vert \triangle _{M}^{2}\phi _{j}\right \vert \leq C_{1},\) where C1>0 is a constant independent of j. For the construction of ϕj satisfying the above properties, see [12].

Preliminary lemma

In the following, we introduce a preliminary lemma which will be used in the sequel.

Lemma 1

Assume that (M,g)is a connected C∞−Riemannian manifold without boundary, with metric g and has bounded geometry. Assume that there exist a constant γ such that 0<γ≤q∈C1(M), and

$$ \left\vert \triangle_{M}q(x)\right\vert \leq \sigma q^{\frac{3}{2} }(x),~for~all~x\in M, $$
(7)

where \(0<\sigma <\frac {2}{\sqrt {p-1}},~1< p<\infty \), and |△Mq(x)| denotes the norm of \(\triangle _{M}q(x)\in T_{x}^{\ast }M\) with respect to the inner product in \(T_{x}^{\ast }M\) induced by the metric g. Assume that f∈Lp(M) and that u∈Lp(M)∩C1(M) is a solution of the equation

$$ \triangle_{M}^{2}u+qu=f. $$
(8)

Additionally assume that for all \(k\in \left [ -\frac {1}{2},p-1\right ],\)

$$ \left\vert u\right\vert^{p}q^{k+\frac{1}{2}}\in L^{1}\left(M\right) \text{ and }{\lim}_{j\rightarrow \infty }\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) =0. $$
(9)

Then, the following properties hold:

$$ {\lim}_{j\rightarrow \infty }\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) =0, $$
(10)

and

$$ q^{k+1}\left\vert u\right\vert^{p}\in L^{1}\left(M\right),\text{ and } \int_{M}q^{k+1}\left\vert u\right\vert^{p}~~d\mu \leq C_{1}\left\Vert f\right\Vert_{p}^{p}, $$
(11)

for all \(k\in \left [ -\frac {1}{2},p-1\right ] \), where {ϕj} is as in (i-iv) and C1≥0 is a constant independent of u.

Proof

We first prove (10): Since u∈Lp(M)∩C1(M), using integration by parts, product rule of d, the definition of ΔM=d∗d, and the formula \(d(u_{\epsilon })=\frac {udu}{ u_{\epsilon }},\) we have

$$\begin{array}{@{}rcl@{}} \left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,d\left(q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(dudq^{k},u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(duq^{k}du,(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &&+(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(duq^{k}du,u^{2}(u_{\epsilon })^{p-4}\triangle_{M}\phi_{j}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,q^{k}u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,dq^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right)\\&&\quad +{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,q^{k}u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &&+(p-1)\left(du,q^{k}du\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,d^{\ast }\left(dq^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,d^{\ast }\left(q^{k}u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(u,d^{\ast }\left(q^{k}du(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \right), \end{array} $$

using the product rule (4) of d∗, we get

$$\begin{array}{@{}rcl@{}} \left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right),dq^{k}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}q^{k}\right) \\ &&-(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(q^{k}(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right),du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}u\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(q^{k}u(u_{\epsilon })^{p-2}\right),d(\triangle_{M}\phi_{j})\right) \\&&\quad+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}^{2}\phi_{j}\right), \end{array} $$

using the product rule of d again, we get

$$\begin{array}{@{}rcl@{}} \left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(\triangle_{M}\phi_{j}\right),u(u_{\epsilon })^{p-2}dq^{k}\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(u\triangle_{M}\phi_{j}du,(u_{\epsilon })^{p-2}dq^{k}\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(u\triangle_{M}\phi_{j}du,u^{2}(u_{\epsilon })^{p-4}dq^{k}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}q^{k}\right) \\&&\quad+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}^{2}\phi_{j}\right) \end{array} $$
$$\begin{array}{@{}rcl@{}} &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k},(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(uq^{k}d\left(\triangle_{M}\phi_{j}\right),(u_{\epsilon })^{p-2}du\right) \\ &&-(p-1)(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(uq^{k}du,u(u_{\epsilon })^{p-4}\triangle_{M}\phi_{j}du\right) \\ &&-(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}u\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k},u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k}du,(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k}du,u^{2}(u_{\epsilon })^{p-4}d(\triangle_{M}\phi_{j})\right). \end{array} $$

Hence, we obtain

$$\begin{array}{@{}rcl@{}} p\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&\left(u\triangle_{M}q^{k},u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\left(u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}^{2}\phi_{j}\right) \\ &&-(p-1)(p-2)\left(uq^{k}du,u\left\vert u\right\vert^{p-4}\triangle_{M}\phi_{j}du\right). \end{array} $$

Taking the limit as j→∞, we get

$$\begin{array}{@{}rcl@{}} p{\lim}_{j\rightarrow \infty }\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&{\lim}_{j\rightarrow \infty }\left(u\triangle_{M}q^{k},u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) \\&&\quad+{\lim}_{j\rightarrow \infty }\left(u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}^{2}\phi_{j}\right) \\ &&-(p-1)(p-2){\lim}_{j\rightarrow \infty }\left(uq^{k}du,u\left\vert u\right\vert^{p-4}\triangle_{M}\phi_{j}du\right). \end{array} $$

By properties of {ϕj}, it follows that for all x∈M,ϕj(x)→1, dϕj(x)→0, △Mϕj(x)→0 and \(\triangle _{M}^{2}\phi _{j}(x)\rightarrow 0\) as j→∞, we apply dominated convergence theorem by using the assumption (7), the assumption \(\left \vert u\right \vert ^{p}q^{k+\frac {1}{2}}\in L^{1}\left (M\right) \) and the condition (iv), we obtain (10).

We now prove (11): Since u∈Lp(M)∩C1(M), using (8), integration by parts, product rule of d, the definition of ΔM=d∗d, and the formula \(d(u_{\epsilon })= \frac {udu}{u_{\epsilon }}\), we have

$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &=&\left(\triangle_{M}^{2}u,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}^{2}u,q^{k}u(u_{\epsilon })^{p-2}\phi_{j}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),d\left(q^{k}u(u_{\epsilon })^{p-2}\phi_{j}\right) \right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \end{array} $$
$$\begin{array}{@{}rcl@{}} &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),q^{k}u(u_{\epsilon })^{p-2}d\phi_{j}\right) +{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),q^{k}(u_{\epsilon })^{p-2}\phi_{j}du\right) \\ &&+(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),q^{k}u^{2}(u_{\epsilon })^{p-4}\phi_{j}du\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),u(u_{\epsilon })^{p-2}\phi_{j}dq^{k}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&\left(d\left(\triangle_{M}u\right),q^{k}u\left\vert u\right\vert^{p-2}d\phi_{j}\right) +\left(d\left(\triangle_{M}u\right),u\left\vert u\right\vert^{p-2}\phi_{j}dq^{k}\right) \\ &&+(p-1)\left(d\left(\triangle_{M}u\right),q^{k}\left\vert u\right\vert^{p-2}\phi_{j}du\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,d^{\ast }\left(q^{k}u(u_{\epsilon })^{p-2}d\phi_{j}\right) \right) +(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,d^{\ast }\left(q^{k}(u_{\epsilon })^{p-2}\phi_{j}du\right) \right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,d^{\ast }\left(u(u_{\epsilon })^{p-2}\phi_{j}dq^{k}\right) \right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right), \end{array} $$

using the product rule (4) of d∗, we get

$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &\,=\,&-\!{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}ud\left(q^{k}u(u_{\epsilon })^{p-2}\right),d\phi_{j}\right) \,+\,{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &&\,-\,{\lim}_{\epsilon \rightarrow 0^{+}}\left(\! \triangle_{M}ud\left(u(u_{\epsilon })^{p-2}\phi_{j}\right),dq^{k}\right) \,+\,{\lim}_{\epsilon \rightarrow 0^{+}}\left((\triangle_{M}u)u(u_{\epsilon })^{p-2}\phi_{j},\triangle_{M}q^{k}\!\right) \\ &&-(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}ud\left(q^{k}(u_{\epsilon })^{p-2}\phi_{j}\right),du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}(u_{\epsilon })^{p-2}\phi_{j},\triangle_{M}u\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right), \end{array} $$

using the product rule of d again, we get

$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &=&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}udq^{k},u(u_{\epsilon })^{p-2}d\phi_{j}\right) -{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}du,(u_{\epsilon })^{p-2}d\phi_{j}\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}du,u^{2}(u_{\epsilon })^{p-4}d\phi_{j}\right) \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}ud\phi_{j},u(u_{\epsilon })^{p-2}dq^{k}\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u\phi_{j}du,(u_{\epsilon })^{p-2}dq^{k}\right) +\left((\triangle_{M}u)u\left\vert u\right\vert^{p-2}\phi_{j},\triangle_{M}q^{k}\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u\phi_{j}du,u^{2}(u_{\epsilon })^{p-4}dq^{k}\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}udq^{k},(u_{\epsilon })^{p-2}\phi_{j}du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k},(u_{\epsilon })^{p-2}d\phi_{j}du\right) \\ &&-(p-1)(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}du,u(u_{\epsilon })^{p-4}\phi_{j}du\right) \\ &&+(p-1)\left(\triangle_{M}uq^{k}\left\vert u\right\vert^{p-2}\phi_{j},\triangle_{M}u\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right). \end{array} $$

Hence, we obtain

$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &=&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) \notag \\ &&+\left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right). \end{array} $$
(12)

We now estimate the term (△Mu,u|u|p−2ϕj△Mqk).

Using the assumption (7), we get

$$ \left\vert \triangle_{M}q^{k}\right\vert \leq \sigma \left\vert k\right\vert q^{k+\frac{1}{2}}. $$
(13)

Using (13) and the inequality \(ab\leq (p-1)a^{2}+\frac {b^{2}}{4(p-1)},\) for all 0≤a,b∈R, we have

$$\left\vert \left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) \right\vert \leq \int\limits_{M}\left\vert \triangle_{M}u\right\vert \left\vert \triangle_{M}q^{k}\right\vert \left\vert u\right\vert^{p-1}\phi_{j}~d\mu $$
$$\begin{array}{@{}rcl@{}} &\leq &\int\limits_{M}\sigma \left\vert \triangle_{M}u\right\vert \left\vert k\right\vert q^{k+\frac{1}{2}}\left\vert u\right\vert^{p-1}\phi_{j}~d\mu \notag \\ &=&\int\limits_{M}\left(\left\vert \triangle_{M}u\right\vert \left\vert u\right\vert^{\frac{p-2}{2}}\phi_{j}^{\frac{1}{2}}q^{\frac{k}{2}}\right) \left(\sigma \left\vert k\right\vert q^{\frac{k+1}{2}}\phi_{j}^{\frac{1}{2} }\left\vert u\right\vert^{\frac{p}{2}}\right) ~d\mu \notag \\ &\leq &(p-1)\int\limits_{M}\left\vert \triangle_{M}u\right\vert^{2}\left\vert u\right\vert^{p-2}\phi_{j}q^{k}~d\mu +\frac{\sigma^{2}k^{2} }{4(p-1)}\int\limits_{M}q^{k+1}\phi_{j}\left\vert u\right\vert^{p}~d\mu \notag \\ &=&(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) +\frac{\sigma^{2}k^{2}}{4(p-1)}\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \notag \\ &=&(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) +(1-\alpha)\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right), \end{array} $$
(14)

where \(\alpha =1-\frac {\sigma ^{2}k^{2}}{4(p-1)}\), and α∈(0,1].

From (14), we get

$$\left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) \geq -\left\vert \left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) \right\vert $$
$$ \geq (1-p)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) +(\alpha -1)\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right). $$
(15)

From (15) into (12), we obtain

$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &\geq &-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) \notag \\ &&+(1-p)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) \notag \\ &&+(\alpha -1)\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \notag \\ &=&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\alpha \left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right). \end{array} $$
(16)

Now, we use the inequality:

$$ \left\vert ab\right\vert \leq \frac{\left\vert a\right\vert^{p}}{\lambda^{p}}+\lambda \left\vert b\right\vert^{t},\text{ } $$
(17)

where \(\frac {1}{p}+\frac {1}{t}=1, a,~b\in R\), and λ∈(0,1). Since ϕj≤1 and \(t=\frac {p}{p-1}>1,\) this implies (ϕj)t≤ϕj.

Using this and (17), we have

$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &\leq &\left\vert \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \right\vert \notag \\ &\leq &\frac{1}{\lambda^{p}}\int\limits_{M}\left\vert f\right\vert^{p}d\mu +\lambda \int\limits_{M}(\phi_{j})^{t}q^{kt}\left\vert u\right\vert^{t}\left\vert u\right\vert^{(p-2)t}~d\mu \notag \\ &\leq &\lambda^{-p}\left\Vert f\right\Vert_{p}^{p}+\lambda \int\limits_{M}\phi_{j}q^{kt}\left\vert u\right\vert^{t}\left\vert u\right\vert^{(p-2)t}~d\mu \notag \\ &=&\lambda^{-p}\left\Vert f\right\Vert_{p}^{p}+\lambda \left(q^{\frac{kp}{ p-1}}\left\vert u\right\vert,\phi_{j}\left\vert u\right\vert^{p-1}\right). \end{array} $$
(18)

From (18) into (16), we get

$$\begin{array}{@{}rcl@{}} &&\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\alpha \left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \leq \lambda^{-p}\left\Vert f\right\Vert_{p}^{p}+\lambda \left(q^{\frac{kp}{p-1}}\left\vert u\right\vert,\phi_{j}\left\vert u\right\vert^{p-1}\right). \end{array} $$

Since k≤p−1 and λ∈(0,1) is arbitrary, we can choose a sufficiently small λ>0 such that

$$\begin{array}{@{}rcl@{}} &&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\frac{\alpha }{2}\left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \leq \lambda^{-p}\left\Vert f\right\Vert_{p}^{p}. \end{array} $$
(19)

By Fatou’s lemma, we have

$$ \int\limits_{M}q^{k+1}\left\vert u\right\vert^{p}~d\mu \leq {\lim}_{j\rightarrow \infty }\inf \left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right). $$
(20)

Combining (19) and (20) and using (9) and (10), we obtain \(\int \limits _{M}q^{k+1} \left \vert u\right \vert ^{p}~d\mu \leq C_{1}\left \Vert f\right \Vert _{p}^{p}, \) where C1≥0 is a constant independent of u, which is the proof of (11) and the lemma. □

Preparatory result

The following proposition is the most important result of this section.

Proposition 1

Assume that (M,g)is a connected C∞−Riemannian manifold without boundary, with metric g and has bounded geometry. Assume that the hypotheses (7), (8), and (9) of the Lemma 1 are satisfied. Then

$$ \left\Vert qu\right\Vert_{p}\leq C\left\Vert f\right\Vert_{p}, $$
(21)

where C≥0 is a constant independent of u.

Proof

Let m be an integer such that \(\frac {m}{2}< p\leq \frac {m+1}{2}.\) By the result (11) in Lemma 1 with \(k=-\frac {1}{2},~0,~\frac {1}{2},~1,~\frac {3}{2},...,\frac {m}{2},\) we get \(q^{\frac {1}{2}}\left \vert u\right \vert ^{p}\in L^{1}\left (M\right),~q\left \vert u\right \vert ^{p}\in L^{1}\left (M\right),...,q^{\frac {m}{2}+1}\left \vert u\right \vert ^{p}\in L^{1}\left (M\right).\) Since q(x)≥γ>0, thus \(\left \vert u\right \vert ^{p}q^{p-\frac {1}{2 }}=\left \vert u\right \vert ^{p}q^{\frac {m}{2}+1}q^{\beta }\leq \left \vert u\right \vert ^{p}q^{\frac {m}{2}+1}\gamma ^{\beta },\)where \(\beta =p-\frac {m+1 }{2}\leq 0.\) This implies \(\left \vert u\right \vert ^{p}q^{(p-1)+\frac {1}{2} }\in L^{1}\left (M\right),\) so by (11) (for k=p−1), we obtain qp|u|p∈L1(M) and \( \int \limits _{M}q^{p}\left \vert u\right \vert ^{p}~d\mu \leq C_{1}\left \Vert f\right \Vert _{p}^{p},\) which implies \(\left \Vert qu\right \Vert _{p}^{p}\leq C_{1}\left \Vert f\right \Vert _{p}^{p},\) that is ∥qu∥p≤C∥f∥p,where C≥0 is a constant independent of u. Hence, the proof of the proposition. □

Lemma 2

Let (M,g) be a Remannian manifold, and let \(u\in L_{loc}^{1}\left (M\right),~\triangle _{M}u\in L_{loc}^{1}\left (M\right).\) Then, \(\triangle _{M}^{2}\left \vert u\right \vert \leq \text {Re}\left ((\triangle _{M}^{2}u)sign\overline {u}\right),\) where \(signu(x)=\left \{\begin {array}{ll}\frac {u(x)}{\left \vert u(x)\right \vert } & if~u(x)\neq 0\\ 0 & otherwise \end {array}\right.\). See [13].

Distributional inequality For 1<p<∞ and λ>0, we consider the inequality, \(\left (\triangle _{M}^{2}+\lambda \right) u=v\geq 0,~u\in L^{p}\left (M\right),\)where v≥0 means that v is a positive distribution, i.e., 〈v,ϕ〉≥0 for every \( 0\leq \phi \in C_{c}^{\infty }\left (M\right).\) See [14].

Lemma 3

Let (M,g) be a manifold of bounded geometry and let 1<p<∞. If u∈Lp(M) satisfies the distributional inequality: \(\left (\triangle _{M}^{2}+\lambda \right) u\geq 0,\)then u≥0 (almost every where or, equivalently, as a distribution). See [15].

Lemma 4

If u∈Lp(M) satisfies the equation \(\triangle _{M}^{2}u+qu=0,\) (which is understood in distributional sense), then u=0.

Proof

Since \(q\in C^{1}\left (M\right) \subset L_{loc}^{\infty }\left (M\right),\) it follows that \(qu\in L_{loc}^{1}\left (M\right).\) Since we have \( \triangle _{M}^{2}u+qu=0,\) it follows that \(\triangle _{M}^{2}u=-qu\in L_{loc}^{1}\left (M\right).\) From Lemma 2 and the assumption q≥γ>0, we get

$$\triangle_{M}^{2}\left\vert u\right\vert \leq \text{Re}\left((\triangle_{M}^{2}u)sign\overline{u}\right) =-\text{Re}\left((qu)sign\overline{u} \right) =-qu\frac{\overline{u}}{\left\vert \overline{u}\right\vert }=-q\frac{ \left\vert u\right\vert^{2}}{\left\vert u\right\vert }=-q\left\vert u\right\vert \leq -\gamma \left\vert u\right\vert, $$

which implies \(\left (\triangle _{M}^{2}+\gamma \right) \left \vert u\right \vert \leq 0.\) From Lemma 3, we get |u|≤0. This implies u=0, hence the proof. □

The Main result

We now introduce our main result of this paper.

Theorem 1

Assume that (M,g)is a connected C∞−Riemannian manifold without boundary, with metric g and has bounded geometry. Assume that the assumption (7) of the Lemma 1 is satisfied. Then

$$ \left\Vert qu\right\Vert_{p}\leq C\left\Vert Au\right\Vert_{p},~\text{for all }u\in D_{p}, $$
(22)

where C≥0 is a constant independent of u.

Proof

Let u∈Dp and

$$ \left(\triangle_{M}^{2}+q\right) u=f, $$
(23)

so f∈Lp(M). Thus, there exist a sequence (fj) in \(C_{c}^{\infty }\left (M\right) \) such that fj→f in Lp(M) as j→∞. Let T be the closure of \(\left (\triangle _{M}^{2}+q\right) |_{C_{c}^{\infty }\left (M\right) }\) in Lp(M). By [15], it follows that:

(i) Dom(T)=Dp, and \(Tu=\left (\triangle _{M}^{2}+q\right) u\) for all u∈Dp.

(ii) The operator T is invertible, and T−1:Lp(M)→Lp(M) is a bounded linear operator.

Consider the sequence T−1fj=wj, since T−1:Lp(M)→Lp(M) is a bounded linear operator, so wj→T−1f in Lp(M) as j→∞. Let w=T−1f. Using the property (i) of T, we get

$$ \left(\triangle_{M}^{2}+q\right) w=f. $$
(24)

From (23) and (24), we get \(\left (\triangle _{M}^{2}+q\right) (u-w)=0.\) By Lemma 4, we obtain u=w. Since T−1fj=wj, it follows that wj∈Dp, and by the property (i) of T, we get

$$ \left(\triangle_{M}^{2}+q\right) w_{j}=f_{j}. $$
(25)

In (25), we have q∈C1(M) and \(f_{j}\in C_{c}^{\infty }\left (M\right),\) so by elliptic regularity, we get \(w_{j}\in W_{loc}^{2,p}\left (M\right).\) By Sobolev embedding theorem [16], we get \(w_{j}\in W_{loc}^{2,p}\left (M\right) \subset L_{loc}^{t}\left (M\right),\) where \( \frac {1}{t}=\frac {1}{p}-\frac {2}{m}.\) Hence, \(qw_{j}\in L_{loc}^{t}\left (M\right).\) Using elliptic regularity again, we get \(w_{j}\in W_{loc}^{2,t}\left (M\right) \) with t>p. Applying the same procedure, we will obtain wj∈C1(M). Thus, wj∈C1(M)∩Lp(M) satisfies the conditions of Proposition 1. From (25) for j, r=1,2,..., we get \(\left (\triangle _{M}^{2}+q\right) (w_{j}-w_{r})=f_{j}-f_{r}.\) Also, from (21), we get

$$ \left\Vert q(w_{j}-w_{r})\right\Vert_{p}\leq C\left\Vert f_{j}-f_{r}\right\Vert_{p}. $$
(26)

Since (fj) is a cauchy sequence in Lp(M), from (26), it follows that (qwj) is also a cauchy sequence in Lp(M), which implies (qwj) converges to s∈Lp(M). Let \(\Psi \in C_{c}^{\infty }\left (M\right),\) then 0=(qwj,Ψ)−(wj,qΨ)→(s,Ψ)−(w,qΨ)=(s−qw,Ψ). So qw=s (because \(C_{c}^{\infty }\left (M\right) \) is dense in Lp(M)). Hence, qwj→qw in Lp(M) as j→∞. But, we have u=w, so qu=qw. Since we have ∥qwj∥p≤C∥fj∥p, by taking the limit as j→∞, we obtain ∥qu∥p≤C∥f∥p=C∥Au∥p, where C≥0 is a constant independent of u. This concludes the proof of the Theorem. □

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    H. A. Atia

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Atia, H.A. Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds. J Egypt Math Soc 27, 24 (2019). https://doi.org/10.1186/s42787-019-0029-6

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Keywords

  • Separation problem
  • Bi-harmonic differential operator
  • Manifold

AMS Subject Classification

  • 47F05
  • 58J99