Singularities of the Fundamental Solution of the Kirchhoff Operator

Miguel Angel Moreles^{∗}

Abstract

We construct the fundamental solution of the Kirchhoff Operator as a Fourier Integral Operator with non-homogeneous phase function and find the singular support. We conclude with some applications in Boundary Controllability.

1 Introduction

Let us recall the Kirchhoff operator, namely

L = ρh∂_{t}^{2} − \frac{ρh^{3}}{12} Δ∂_{t}^{2} + DΔ^{2}  (1)

Here Δ is the Laplacian Operator, and acts in the space variables, i.e.

Δ = ∑_{i=1}^{n} ∂_{x_{i}}^{2}

When n = 2 , the equation Lw = 0 in a suitable domain with appropiate initial and boundary data, models vibrations of an elastic plate. In such a case w(x, t) represents the vertical displacement of the middle surface. The physical constants are: ρ ≡ density, h ≡ plate thickness, D ≡ flexural rigidity. For simplicity we assume D = 1 . See Lagnese [3] and Lagnese & Lions [4] for a complete discussion on elastic plates.

∗Centro de Investigación en Matemáticas (CIMAT); Apdo. Postal 402; Guanajuato, GTO 36000; MEXICO. (moreles@fractal.cimat.mx)

In the sense of Ortner & Wagner [8, 9] the Kirchhoff Operator is quasihyperbolic with principal part

L_{4} = \frac{ρh^{3}}{12} Δ (∂_{t}^{2} − \frac{12}{ρh^{3}} Δ) .

Notice the wave operator ∂_{t}^{2} − \frac{12}{ρh^{3}} Δ whose characteristic cone is

Γ = \{(x, t) ∈ ℝ^{n+1} : t^{2} = \frac{12}{ρh^{3}} |x|^{2} , t ≥ 0\} .

We shall see that this hyperbolic part of L_{4} determines the singularities of the fundamental solution of L.

The motivation of this work is to solve a problem in Boundary Controllability for the Kirchhoff Operator using the method in Littman [5, 6]. An important step in Littman’s method is the construction of a fundamental solution which is smooth away from zero. In this work we carry out this step and construct a fundamental solution G(x, t) for the Kirchhoff operator and determine its singular support, that is the complement of the set where G(x, t) is not C^{∞} .

The outline is as follows. In Section 2, following Hörmander [1], we recall the notion of Fourier Integral Operator (FIO) and the results that apply here.

In Section 3 we construct G(x, t) as a FIO and prove that

sing supp G = Γ^{0}

where Γ^{0} is the dual cone of Γ , i.e.,

Γ^{0} = \{(x, t) ∈ ℝ^{n+1} : t^{2} = \frac{ρh^{3}}{12} |x|^{2} , t ≥ 0\} .  (2)

Section 4 relates our work to Boundary Controllability and other problems of interest. In particular we consider a perturbation of the Kirchhoff Operator, the Mindlin-Timoshenko Operator

L_{K} = ρh∂_{t}^{2} − \frac{ρh^{3}}{12} Δ∂_{t}^{2} + Δ^{2} + \frac{ρh}{K} (\frac{ρh^{3}}{12} ∂_{t}^{4} − Δ∂_{t}^{2})

This operator is hyperbolic with principal part

L_{K4} = \frac{ρ^{2}h^{4}}{12K} (∂_{t}^{2} − \frac{K}{ρh}Δ) (∂_{t}^{2} − \frac{12}{ρh^{3}} Δ) .

It has two characteristic cones one of which coincides with Γ . We discuss the behaviour of this operator and the corresponding equation when K → ∞ .

Throughout we shall assume the Theory of Distributions of L. Schwartz. A reference is Hörmander [2].

2 Fourier Integral Operators

The purpose of this section is to consider oscillatory integrals of the form

P u(x) = \frac{1}{(2π)^{n}} ^{∫∫} exp (iφ(x, y, ξ)) p(x, ξ)u(y)dξdy  (3)

where, ξ ∈ ℝ^{n} , x ∈ Ω_{1} , y ∈ Ω_{2} , and u ∈ E′(Ω_{2}) . Here Ω_{j} is an open subset of ℝ^{n_{j}} , j = 1, 2 , To describe φ(x, y, ξ) and p(x, ξ) we introduce some definitions.

For s ∈ ℝ , let

Λ_{s}(ξ) = (1 + |ξ|^{2})^{s/2} .

Definition 1 Let Ω be an open subset of ℝ^{n} . Let m be a real number. The class S^{m}(Ω, ℝ^{N}) consists of functions p(x, ξ) ∈ C^{∞}(Ω, ℝ^{N}) such that for any multi-index α and any compact set K ⊂ Ω a constant C_{α,K} exists for which

|∂_{x}^{β}∂_{ξ}^{α}p(x, ξ)| ≤ C_{α,β,K}Λ_{m−|α|}; x ∈ K , ξ ∈ ℝ^{N} .

The elements of S^{m} are called symbols of order m .

Definition 2 Let Ω_{j} be an open set in ℝ^{n_{j}} , j = 1, 2 . A real valued function φ ∈ S^{1}(Ω_{1} × Ω_{2}, ℝ^{N}) is called a phase function if for every compact set K ⊂ Ω_{1} × Ω_{2} there is a constant C such that

\begin{aligned}|ξ|^{2} & ≤ & C (|∇φ_{x}|^{2} + |ξ|^{2} |∇φ_{ξ}|^{2}) ; & |ξ| > C , (x, y) ∈ K; \\ |ξ|^{2} & ≤ & C (|∇φ_{y}|^{2} + |ξ|^{2} |∇φ_{ξ}|^{2}) ; & |ξ| > C , (x, y) ∈ K . \end{aligned}

Theorem 3 For any m there is a unique bilinear map

S^{m}(Ω_{1} × Ω_{2}, ℝ^{N}) × E′(Ω_{2}) → D′(Ω_{1}) , (p, u) ⟼ P u

Recall that P u is given by (3) if u is a function and the double integral is absolutely convergent. When u is a distribution P u is defined as follows. Let χ ∈ D with χ = 1 in a neighborhood of the origin. Let

ψ_{ε}(y) = ∫ exp (iφ(x, y, ξ)) χ(εξ) p(x, ξ)dξ

for 0 < ε < 1 , notice that ψ_{ε} ∈ C^{∞} . Consequently P u is given by

P u = lim_{ε→0} ⟨u, ψ_{ε}⟩  (4)

In any case we write P u as in (3) as customary.

A Fourier Integral Operator (FIO) is an oscillatory integral of the form (3) with phase function φ(x, y, ξ) and symbol p(x, ξ) .

Next we consider the singularities of P u . Let Ω_{φ} be the set of all (x, y) ∈ Ω_{1} × Ω_{2} such that for some constant C , depending on (x, y) ,

1 ≤ C |∇φ_{ξ}|^{2} , |ξ| > C .

The set Ω_{φ} is always open. Let F_{φ} be the complement of Ω_{φ} hence closed. For subsets K of Ω_{2} we write

F_{φ} K = {x ∈ Ω_{1} : (x, y) ∈ F_{φ} for some y ∈ K} .

Theorem 4 If u ∈ E′(Ω_{2}) then

sing supp P u ⊂ F_{φ} sing supp u

Our interest is u = δ, the Dirac’s Delta distribution. In this case

sing supp δ = {0} .  (5)

3 The Fundamental Solution

In this section we use the notation

D_{x_{j}} ≡ −i∂_{x_{j}} ; D_{t} ≡ −i∂_{t} ; \overline{Δ} = −Δ

The Kirchhoff Operator is written in the form

P (D_{x_{1}}, …, D_{x_{n}}, D_{t}) = (1 + \frac{h^{2}}{12}\overline{Δ}) D_{t}^{2} − \frac{1}{ρh}\overline{Δ}^{2}  (6)

We shall construct a distribution G(x, t) supported in the half space {(x, t) ∈ ℝ^{n+1} : t ≥ 0} such that

[(1 + \frac{h^{2}}{12}\overline{Δ}) D_{t}^{2} − \frac{1}{ρh}\overline{Δ}^{2}] G(x, t) = δ(x, t) .

Let

\begin{aligned}S(ξ, t) & = & \frac{1}{2λ(ξ)} (e^{iλ(ξ)t} − e^{−iλ(ξ)t}) \\ & = & i\frac{sin λ(ξ)t}{λ(ξ)} \end{aligned}

where

λ(ξ) = \begin{matrix}\sqrt{\frac{1}{ρh}} & |ξ| & 2 \\ √ & \overline{1 + \frac{h^{2}}{12} |ξ|^{2}} \end{matrix} .

it satisfies

\begin{matrix}[(1 + \frac{h^{2}}{12} |ξ|^{2}) D_{t}^{2} − \frac{1}{ρh} |ξ|^{4}] S(ξ, t) = 0 \\ S(ξ, 0) = 0 , D_{t}S(ξ, 0) = 1 \end{matrix}  (7)

observe that

P (ξ, D_{t}) = (1 + \frac{h^{2}}{12} |ξ|^{2}) D_{t}^{2} − \frac{1}{ρh} |ξ|^{4}

is obtained from (6) after Fourier Transform with respect to x .

Choose χ(ξ) ∈ D such that χ(ξ) = 1 in a neighborhood of 0 . Let

\begin{aligned}P_{0}(x, t) & = & \frac{1}{(2π)^{n}} ^{∫∫} e^{i(x−y)·ξ}_{\frac{1}{1+ \frac{h^{2}}{12} |ξ|^{2}}} S(ξ, t)χ(ξ)u(y)dξdy \\ Au(x, t) & = & \frac{1}{(2π)^{n}} ^{∫∫} e^{iφ(x,y,ξ)}a(ξ)u(y)dξdy \\ Bu(x, t) & = & \frac{1}{(2π)^{n}} ^{∫∫} e^{iψ(x,y,ξ)}a(ξ)u(y)dξdy \end{aligned}  (8)

where

a(ξ) = \frac{1}{2λ(ξ)} · \frac{1}{1 + \frac{h^{2}}{12} |ξ|^{2}} · (1 − χ(ξ))  (9)

\begin{aligned}φ(x, y, ξ) & = & (x − y) · ξ + λ(ξ)t \\ ψ(x, y, ξ) & = & (x − y) · ξ − λ(ξ)t \end{aligned}

Since \frac{∂φ}{∂x_{i}} = ξ_{i} it follows that

|∇_{x}φ|^{2} + |ξ|^{2} |∇_{ξ}φ|^{2} ≥ |ξ|^{2} .

Moreover, λ(ξ) ∈ S^{1} and so is φ(x, y, ξ) . Therefore, according to Definition 2, φ is a phase function . Similarly for ψ . Also, it is readily seen that a(ξ) in (9) is in S^{−3} . Thus A and B are FIO with symbol a(ξ) and phase functions φ(x, y, ξ) , ψ(x, y, ξ) respectively.

Our claim about the fundamental solution will be a consequence of the following theorem.

Theorem 5 Let Γ^{0} be the cone in (2) . Let u = δ in (8) , define

Eδ(x, t) = P_{0}(x, t) + Aδ(x, t) − Bδ(x, t)

Then Eδ is a distribution such that sing supp Eδ = Γ^{0}

Proof.

We prove that in the complement of Γ^{0} the distribution Eδ(x, t) is smooth. Observe that P_{0}(x, t) is C^{∞} in ℝ^{n+1} , hence we need to consider only Aδ(x, t) and Bδ(x, t) . In light of Theorem 4, it suffices to show that if |x|^{2} ≠ \frac{12}{ρh^{3}} t^{2} for t ≥ 0 then (x, t) ∉ F_{φ} sing supp δ and (x, t) ∉ F_{ψ} sing supp δ. We prove the assertion for φ. For ψ the proof is similar.

From (5) (x, t) ∉ F_{φ} sing supp δ is equivalent to ((x, t), 0) ∉ F_{φ} that is ((x, t), 0) ∈ Ω_{φ} .

We are led to show that there exists a constan C depending on ((x, t), 0) such that

1 ≤ C |∇_{ξ}φ((x, t), 0, ξ)| |ξ| > C

Easily

\frac{∂φ}{∂ξ_{i}} = x_{i} + \frac{\frac{1}{\sqrt{ρh}}t}{(1 + \frac{h^{2}}{12} |ξ|^{2})^{3/2}} (2 + \frac{h^{2}}{12} |ξ|^{2}) ξ_{i}

thus

∇_{ξ}φ((x, t), 0, ξ) = x + \frac{\frac{1}{\sqrt{ρh}}t}{(1 + \frac{h^{2}}{12} |ξ|^{2})^{3/2}} (2 + \frac{h^{2}}{12} |ξ|^{2}) ξ

and

|∇_{ξ}φ((x, t), 0, ξ)| ≥ ||x| − \frac{\frac{1}{\sqrt{ρh}}t}{(1 + \frac{h^{2}}{12} |ξ|^{2})^{3/2}} (2 + \frac{h^{2}}{12} |ξ|^{2}) |ξ||

Consider the function

f(r) = \frac{2 + \frac{h^{2}}{12}r^{2}}{(1 + \frac{h^{2}}{12}r^{2})^{3/2}} r;  r ≥ 0 ,  r = |ξ|

Then f(r) ↘ \frac{\sqrt{12}}{h} when r → ∞ and f attains its maximum value at r = \frac{\sqrt{24}}{h} , f (\frac{\sqrt{24}}{h}) = \frac{8}{3\sqrt{6}} \frac{\sqrt{12}}{h} .

If |x| < \sqrt{\frac{12}{ρh^{3}}} t then |x| = (1 − ε) \sqrt{\frac{12}{ρh^{3}}} t for some ε , 0 < ε < 1 , thus

|∇_{ξ}φ((x, t), 0, ξ)| ≥ \frac{t}{ρh}f(r) − |x|

but f(r) ≥ \frac{\sqrt{12}}{h} for r ≥ \frac{\sqrt{24}}{h} , so

|∇_{ξ}φ((x, t), 0, ξ)| ≥ ε\sqrt{\frac{12}{ρh^{3}}} t

or

|∇_{ξ}φ((x, t), 0, ξ)| C ≥ 1 , if |ξ| > C

with C = max \{_{\frac{1}{ε\sqrt{\frac{12}{ρh^{3}}} t}}, \frac{\sqrt{24}}{h} \} , i.e. (x, t) ∈ Ω_{φ} .

For the case |x| > \sqrt{\frac{12}{ρh^{3}}} t we have |x| = (1 + ε) \sqrt{\frac{12}{ρh^{3}}} t for some ε > 0 , thus

|∇_{ξ}φ((x, t), 0, ξ)| ≥ |x| − \frac{t}{ρh}f(r)

Choose ε̃ small enough so that

1 + ε − ε̃\frac{8}{3\sqrt{6}} > 0

there is r_{ε,_{ε}} ≥ \frac{\sqrt{24}}{h} so that, if r > r_{ε,_{ε}} then f(r) ≤ ε̃\frac{8}{3\sqrt{6}} \frac{\sqrt{12}}{h} . We have

|∇_{ξ}φ((x, t), 0, ξ)| ≥ \sqrt{\frac{12}{ρh^{3}}} t (1 + ε − ε̃\frac{8}{3\sqrt{6}})

Again

|∇_{ξ}φ((x, t), 0, ξ)| C ≥ 1 , if |ξ| > C

with C = max \{(\sqrt{\frac{12}{ρh^{3}}} t (1 + ε − ε̃\frac{8}{3\sqrt{6}}))^{−1} , r_{ε,_{ε}}\} . Therefore (x, t) ∈ Ω_{φ} .

Finally we show that the singular support is exactly the cone Γ^{0} . Now we use the phase function ψ. We have

∇_{ξ}ψ((x, t), 0, ξ) = x − \frac{\frac{t}{ρh}}{(1 + \frac{h^{2}}{12} |ξ|^{2})^{3/2}} (2 + \frac{h^{2}}{12} |ξ|^{2}) ξ

In the cone Γ^{0}

|x| = \sqrt{\frac{12}{ρh^{3}}} t

so

∇_{ξ}ψ((x, t), 0, ξ) = x − \frac{h |x|}{\sqrt{12}} \frac{f(r)}{r}ξ

let ξ = αx , α to be chosen later. Then

|∇_{ξ}ψ((x, t), 0, ξ)| = |1 − \frac{h}{\sqrt{12}}f (α |x|)| |x|

Since f(r) ↘ \frac{\sqrt{12}}{h} when r ↗ ∞ it follows that for any C > 0 we may choose α >> 0 so that

|1 − \frac{h}{\sqrt{12}}f (α |x|)| |x| < \frac{1}{C}

that is if |x| = \sqrt{\frac{12}{ρh^{3}}} t |x| then (x, t) ∉ Ω_{ψ} , i.e., (x, t) is in the singular support.

Let H(t) be the Heaviside function. From (7) we see that

[(1 + \frac{h^{2}}{12} |ξ|^{2}) D_{t}^{2} − \frac{1}{ρh} |ξ|^{4}] (\frac{i}{1 + \frac{h^{2}}{12} |ξ|^{2}} S(ξ, t)H(t)) = δ(t)

This fact together with the previous theorem give us

Corollary 6 If G(x, t) = iEδ(x, t)H(t) , then G(x, t) is a fundamental solution of the Kirchhoff Operator supported in the half space t ≥ 0 . Moreover sing supp G = Γ^{0} .

4 Boundary Controllability

We solve the following problem in boundary controllability: The Null Boundary Controllability Problem (NBCP): Suppose we have a well-posed initial-boundary value problem for an evolution equation Lw = 0 in a cylindrical domain Q = Ω × [0, +∞), where Ω is a bounded domain in ℝ^{n} . The NBCP deals with the following question: Given initial data in Ω at t = 0, can this data be supplemented with appropiate inhomogeneous time dependent boundary data (boundary controls), prescribed on the lateral boundary of Q, such that the solution of the initial-boundary value problem will vanish for t ≥ T_{0}?

Consider the Kirchhoff Operator in the form (1) . Let s = x · ω with |ω| = 1 . Then

L(∂_{x_{1}}, …, ∂_{x_{n}}, ∂_{t})U(x · ω, t) = [ρh∂_{t}^{2} − \frac{ρh^{3}}{12} ∂_{s}^{2}∂_{t}^{2} + ∂_{s}^{4}] U(s, t)

We obtain the Rayleigh beam operator

L_{1}(∂_{s}, ∂_{t}) = ρh∂_{t}^{2} − \frac{ρh^{3}}{12} ∂_{s}^{2}∂_{t}^{2} + ∂_{s}^{4}

which is hyperbolic in the s-direction. There are two fundamental solutions supported respectively in the cones

\begin{matrix}Γ_{+}^{0} = \{(s, t) ∈ ℝ^{2} : t^{2} = \frac{ρh^{3}}{12} |s|^{2} , s ≥ 0\} \\ Γ_{−}^{0} = \{(s, t) ∈ ℝ^{2} : t^{2} = \frac{ρh^{3}}{12} |s|^{2} , s ≤ 0\} \end{matrix}

This allows to state the analogue to Theorem 2 in Littman [5]. The proof is almost identical.

Theorem 7 Let T_{0} = \sqrt{\frac{ρh^{3}}{12}} d, where d is the diameter of Ω. Let w_{0}(x), w_{1}(x) be C^{∞} functions in \overline{Ω} . Then for each T_{1} > T_{0} there exists a solution w to

\begin{matrix}Lw = 0 , in \overline{Ω} × \{t ≥ 0\}; \\ w(x, 0) = w_{0}(x) , w_{t}(x, 0) = w_{1}(x) ; in \overline{Ω} \end{matrix}

and vanishing in \overline{Ω} × \{t ≥ T_{1}\} .

Some final remarks:

(i) To solve the NBCP just read off lateral boundary values. Littman’s method is constructive and independent of the boundary conditions. We just need those that make a well posed Initial-Boundary Value Problem. (ii) The NBCP for the Mindlin-Timoshenko Operator is solved as a direct application of Theorem 2 in Littman [5]. It is natural to ask if controllability is uniform, that is, Does the solution to the NBCP for the Mindlin-Timoshenko equation converge to that of the Kirchhoff Operator when K → ∞ ? For the corresponding beam equations strong convergence is established in Moreles [7]. See the remarks and references therein regarding weaker results for higher dimensional problems.

References

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[3] Lagnese, J.: Boundary Stabilization of Thin Plates; SIAM studies in applied mathematics, 10. SIAM Publications; Philadelphia. (1989)

[4] Lagnese, J. & Lions, J.L. : Modelling, Analysis and Control of Thin Plates; Masson; Paris. (1988)

[5] Littman, W. : Boundary Control Theory for Hyperbolic and Parabolic Partial Differential Equations with Constant Coefficients; Annali Scuola Normale Superiore; 567-580; Pisa. (1978)

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[7] Moreles, M.A. : A Classical Approach to Uniform Null Boundary Controllability for Elastic Beams; To appear

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