This a question where I have thought quite long about:

The eigenfunctions (or also normal modes) of an dry Euler beam subject to free-free boundary conditions are given by

$$ \frac{\partial^4\psi}{\partial x^4}=\lambda_k^4\psi\qquad(0\le x\le1)\,,$$ $$\frac{\partial^2\psi}{\partial x^2}=\frac{\partial^3\psi}{\partial x^3}=0\qquad(x=0\text{ and }x=1)\,, $$

The solution of this problem is given by $$\psi_0(x)=1\qquad\text{and}\qquad\psi_1(x)=\sqrt3(2x-1),$$ $$ \psi_k(x)=\frac{\cosh((\frac12-x)\lambda_k)}{\cosh(\frac12\lambda_k)}+\frac{\cos((\frac12-x)\lambda_k)}{\cos(\frac12\lambda_k)}\qquad (k\ge 2 \text{ even}),$$ $$ \psi_k(x)=\frac{\sinh((\frac12-x)\lambda_k)}{\sinh(\frac12\lambda_k)}+\frac{\sin((\frac12-x)\lambda_k)}{\sin(\frac12\lambda_k)}\qquad (k\ge 2\text{ odd}),$$ where $$ \cosh(\lambda_k)\cos(\lambda_k)=1\qquad (k\ge2)\,,$$ This is a complete set of eigenfunctions. These eigenfunctions are orthogonal (it can be shown that they are even orthonormal) in terms of the standard scalar product, since the fourth order derivative is a self-adjoint operator subject to the given boundary conditions and a standard smoothness condition for $\psi$. I would like to know if this set of eigenfunctions is also an orthonormal basis in $L_2([0,1])$. There seems to be a Sturm-Liouville theory for fourth order differential operators. Is there any standard book which discusses such a problem?


2 Answers 2


You need to specify $\lambda_k$ more clearly. More precisely, what is the spectrum of this operator? The equation

$$ \cos x\cosh x =1 $$

seems to have a unique solution $\mu_k$ on any interval of the form $(k\pi/2, k\pi/2+\pi)$, $k\in\mathbb{Z}$, so I assume the spectrum might be $\mu_k^4$, $k\in\mathbb{Z}$?!? (Please edit you question to remove this ambiguity.) In any case if the boundary value problem is elliptic (please check that) then the spectrum is discrete. In particular it can be determined by finding the eigenfunctions which means solving some ode's. My guess is that you found all the eigenfunctions, i.e., the system you found is complete.

Update. You need to check two things: 1) the boundary value problem is elliptic 2) it is symmetric. I'll deal with the 2nd issue first because it is faster. Denote by $A$ the operator

$$A=\frac{d^4}{dx^4}. $$

A simple integration by parts shows that for any $u,v\in C^4([0,1])$ we have

$$ \int_0^1 \bigl(\; v(Au) -u(Av)\;\bigr) dx=\sum_{j=0}^3(-1)^j\bigl( u^{(3-j)}(1)v^{(j)}(1)- u^{(3-j)}(0) v^{(j)}(0)\;\bigr). $$

If the function $u$ satisfies your boundary conditions $u^{(k)}(x)=0$ for $k=2,3$, $x=0,1$ the above equality simplifies a bit

$$ \int_0^1 \bigl(\; v(Au) -u(Av)\;\bigr) dx= \sum_{j=2}^3(-1)^j\bigl( u^{(3-j)}(1)v^{(j)}(1)- u^{(3-j)}(0) v^{(j)}(0)\;\bigr). $$

If the function $v$ satisfies the same boundary conditions as $u$, then the last equality takes the very simple form

$$ \int_0^1 \bigl(\; v(Au) -u(Av)\;\bigr) dx= 0. $$

This says that the boundary value problem is symmetric, or formally selfadjoint.

The ellipticity of this problem is another issue. The most readable account I could find is in Chap. 20 vol.3 of the book The Analysis of Linear Partial Differential Operators by the late great Lars Hormander.

The ellipticity of the boundary value problem requires that the symbol of your operator $A$ be elliptic (which it is) and that the boundary value conditions should satisfy the so called Lopatinskii-Schapiro conditions.

In your case they are trivially satisfied because you work on a one-dimensional space $[0,1]$. The upshot is that in your case the boundary conditions are elliptic. We can form the unbounded operator $\newcommand{\bD}{\boldsymbol{D}}$

$$ \hat{A}: \bD(\hat{A})\subset L^2(0,1)\to L^2(0,1), u\mapsto \frac{d^4 u}{dx^4} $$

Where the domain $\bD(\hat{A})$ of $\hat{A}$ consists of functions in the Sobolev space $L^{4,2}(0,1)$ (four weak derivatives in $L^2$) such that $u^{(j)}(x)=0$ for $x=0,1$, $j=2,3$.

The results in the above monograph show that $\hat{A}$ viewed as an unbounded operator on the Hilbert space $L^2(0,1)$, is closed, densely defined, selfadjoint and has compact resolvent. This is all you need. Arguably, the above argument is a bit heavy, and it feels like hunting a mosquito using a bazooka.

There is a direct, more elementary approach to proving that $\hat{A}$ has compact resolvent. Observe first that the above integration by parts formulae show that the operator $\hat{A}+1$ is positive, i.e.,

$$ (\hat{A}u,u)_2+(u,u)_2>0,\;\;\forall u\in \bD(\hat{A})\setminus 0, $$

where $(-,-)_2$ denotes the $L^2$-inner product. Hence $\hat{A}+1$ is injective. Then follow the strategy in the proof of Theorem 8.22 in Brezis' book Functional Analysis, Sobolev Spaces and Partial Differential Equations to prove that $\hat{A}+1$ is invertible and it's inverse is compact as an operator $L^2(0,1)\to L^2(0,1)$.

  • $\begingroup$ Thanks, I have edited the question. What do you mean with the problem is elliptic? I know only that the operator is self-adjoint and non-negative. However, the spectrum is discrete. And the solution displayed is also complete. $\endgroup$ Apr 17, 2013 at 18:02
  • $\begingroup$ Because of lack of comment space I'll give my reply as an update to my original answer. $\endgroup$ Apr 17, 2013 at 18:54
  • $\begingroup$ I know the term self-adjoint. $\endgroup$ Apr 17, 2013 at 19:24
  • $\begingroup$ Thank you! I do not mind to shoot with a bazooka on a mosquito. I will study the chapter you have cited. $\endgroup$ Apr 19, 2013 at 21:35

Choose some $\lambda \ne \lambda_k$ for all $k$. Consider the resolvent $R_\lambda = (\lambda-\partial_x^4)^{-1}$. This is a compact operator on the Hilbert space $H = \{ \psi \in L^2([0,1]) \; | \; \psi''(0) = \psi''(1) = \psi'''(0) = \psi'''(1) = 0\}$. Thus $H$ has a basis of eigenvectors of $R_\lambda$. But $R_\lambda$ and $\partial_x^4$ share eigenvectors, thus $H$ has a basis of eigenvectors of $\partial_x^4$ as desired. The key bit is using compactness to show that a maximizing sequence for the Rayleigh quotient converges to an eigenvector. The argument can be found in Lax's Functional Analysis Ch 28 Thm 3.

  • $\begingroup$ How do you know that the operator $R_\lambda=(\lambda^4-\partial_x^4)^{-1}$ is compact? If I know compactness, then Hilbert-Schmidt theorem directly implies that my set of eigenfunctions is an orthonormal basis. How can I show compactness of $R_\lambda$ or is there a reference? I do not understand your last two sentences $\endgroup$ Apr 17, 2013 at 18:38
  • $\begingroup$ Furthermore, your space $H$ does not make sense. $\endgroup$ Apr 17, 2013 at 19:06
  • 1
    $\begingroup$ (1) You are absolutely right about the space. $H$ makes no sense. Instead of $L^2$ I should have $H^4$. (2) I was confused about where you were stuck. The last two sentences are probably not relevant. (3) Perhaps I'm being quite sloppy here, but: given $R_\lambda f = g$ we have $\|g\|_{L^2} \le C\|f\|_{L^2}$ and $\|g''''\|_{L^2} \le \lambda\|g\|_{L^2} + \|f\|_{L^2}$ which implies \|g\|_{H^4} \le C'\|f\|_2$. Since bounded sets in $H^4([0,1])$ are compact in $L^2([0,1])$, the resolvent is compact. $\endgroup$ Apr 18, 2013 at 14:42

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