On Sign Preservation for Clotheslines, Curtain Rods, Elastic Membranes and Thin Plates

Abstract

All problems mentioned in the title seem to have one thing in common. Whenever a force is applied in one direction, the object moves in that direction. At least this is what one might expect. The corresponding boundary value problems contain differential equations of second order for line and membrane, while rod, or beam, and plate are modeled through fourth order differential equations. How a positive source will give a positive solution crucially depends on this order and the boundary conditions. We will present a survey concerning the so-called positivity preserving property for these various models.

Appendices

Appendix A: 1d Proofs when $\lambda >0$

The result of Lemma 2.1 is standard. Lemma 2.2 and 2.3 can be found in [81] or [53]. The crucial idea to compute $\lambda _{c}$ as ‘anti-eigenvalue’ or switched eigenvalue goes back to Schröder in [73]. See also [79]. The direct approach is to compute the Green function and see for which $\lambda$ a sign change occurs. We choose this approach for the first cases, since it will supply us with a formula for the Green function, which has some independent interest. The calculations are tedious and maybe not so interesting, but since the existing results in the literature do not always agree, it might be useful to have them explicitely stated.

Proof of Lemma 2.1

Although we may restrict ourselves to the case $\lambda >0$ one directly computes the following explicit Green function for (1):

$$ g_{\lambda } ( x,y ) =\left \{ \textstyle\begin{array}{l@{\quad }l} \frac{\sinh ( \sqrt{\lambda }\min ( x,y ) ) \sinh ( \sqrt{\lambda } ( 1-\max ( x,y ) ) ) }{\sqrt{\lambda }\sinh ( \sqrt{\lambda } ) } & \text{for }\lambda >0, \\ \frac{\sin ( \sqrt{-\lambda }\min ( x,y ) ) \sin ( \sqrt{-\lambda } ( 1-\max ( x,y ) ) ) }{\sqrt{-\lambda }\sin ( \sqrt{-\lambda } ) } & \text{for }\lambda < 0,\end{array}\displaystyle \right . $$

(A.1)

when $\sqrt{-\lambda }$ is not a multiple of $\pi$. Consistent with (5) one finds

$$ g_{0} ( x,y ) =\lim_{\lambda \rightarrow 0}g_{\lambda } ( x,y ) =\min ( x,y ) \bigl( 1-\max ( x,y ) \bigr) . $$

The denominator in (A.1) turns 0 for $\lambda =-k^{2}\pi ^{2}$ with $k\in \mathbb{N}^{+}$. Since the numerators in (A.1) changes sign for some $x$ and $y$ if and only if $\lambda <-\pi ^{2}$, one finds that the Green function is positive if and only if $\lambda >-\pi ^{2}$. □

Proof of Lemma 2.2

Let $\lambda >0$ and write $\mu =\sqrt[4]{\lambda /4}$. The function

$$\begin{aligned} w ( \mu ,p ) =&\sin ( \mu p ) \cosh \bigl( \mu ( 2-p ) \bigr) +\cos ( \mu p ) \sinh \bigl( \mu ( 2-p ) \bigr) \\ &{}+\sinh ( \mu p ) \cos \bigl( \mu ( 2-p ) \bigr) +\cosh ( \mu p ) \sin \bigl( \mu ( 2-p ) \bigr) , \end{aligned}$$

(A.2)

satisfies $( ( \frac{d}{dx} ) ^{4}+\lambda ) w ( \mu ,x ) =0$. Then the Green function is

$$ g_{\lambda } ( x,y ) =\dfrac{w ( \mu ,\vert x-y\vert ) -w ( \mu ,x+y ) }{8\mu ^{3}(\cosh (2\mu )-\cos (2\mu ))}. $$

(A.3)

See Fig. 28 (right). Indeed, it satisfies the same differential equation as $w$ for $x\neq y$. One may check that

$$ w ( \mu ,p ) =w_{0} ( \mu ) +w_{2} ( \mu ) p^{2}+w_{3} ( \mu ) p^{3}+\mathcal{O} \bigl(p^{4} \bigr) $$

(A.4)

with

$$\begin{aligned} w_{0} ( \mu ) =&\sinh ( 2\mu ) +\sin ( 2\mu ) , \\ w_{2} ( \mu ) =&-\mu ^{2} \bigl( \sinh ( 2\mu ) -\sin ( 2 \mu ) \bigr) , \\ w_{3} ( \mu ) =&\tfrac{2}{3}\mu ^{3}\bigl(\cosh (2\mu )-\cos (2\mu )\bigr). \end{aligned}$$

The first odd power of $p$ that appears in (A.4) is 3 and with its coefficient this explains

$$ \biggl( \frac{d}{dx} \biggr) ^{4}g_{\lambda } ( x,y ) + \lambda g_{\lambda } ( x,y ) =\delta _{y} ( x ) . $$

The boundary conditions $g_{\lambda } ( 0,y ) =0$ and $( \frac{d}{dx} ) ^{2}g_{\lambda } ( 0,y ) =0$ follow from (A.3) and (A.2). The ones in $x=1$ from $w ( \mu ,p ) =w ( \mu ,2-p )$.

The first eigenvalue is reached for $\lambda =-\pi ^{4}$ and for $\lambda >-\pi ^{4}$ the Green function depends smoothly on $\lambda$. Where does positivity break down for $\lambda >0$? One may show that for $\lambda \mapsto g_{\lambda } ( \cdot ,\cdot )$ no interior zero can appear and hence that new ‘zeros come in through the boundary’ at $\lambda _{c}$ if $\lambda _{c}$ denotes the value for which positivity of the Green function breaks down. Schröder [73] studied the question and showed that for $\lambda =\lambda _{c}$ one encounters either a standard eigenvalue or a ‘switched’ eigenvalue. The standard eigenvalues are negative, so $\lambda _{c}$ is, according to Schröder, an eigenvalue with a positive eigenfunction of the ‘switched’ eigenvalue problem:

$$ \left \{ \textstyle\begin{array}{l} \varphi ^{\prime \prime \prime \prime }+ \lambda \varphi =0\quad \text{in } ( 0,1 ) , \\ \varphi ( 0 ) =\varphi ^{\prime } ( 0 ) =\varphi ^{\prime \prime } ( 0 ) =0, \\ \varphi ( 1 ) =0. \end{array}\displaystyle \right . $$

(A.5)

Switched, since $\varphi ^{\prime \prime } ( 1 ) =0$ is replaced by $\varphi ^{\prime } ( 0 ) =0$. One could also have replaced $\varphi ^{\prime \prime } ( 0 ) =0$ is replaced by $\varphi^{\prime } ( 1 ) =0$, but by symmetry one finds the same value $\lambda _{c}$. The first eigenfunction for (A.5) is

$$ \varphi ( x ) =\sin ( \mu _{c}x ) \cosh ( \mu _{c}x ) - \sinh ( \mu _{c}x ) \cos ( \mu _{c}x ) , $$

with $\mu _{c}$ the first positive solution of $\tan \mu _{c}=\tanh \mu _{c}$ and hence $\lambda _{c}=4\mu _{c}^{4}$. □

Proof of Lemma 2.3

The Green function here is

$$ gg ( \mu ,x,y ) =g ( \mu ,x,y ) -rl ( \mu ,x,y ) -rr ( \mu ,x,y ) , $$

with $g ( \mu ,x,y )$ from (A.3),

$$\begin{aligned} rl ( \mu ,x,y ) =&\frac{\sin ( \mu ( 2-y ) ) \sinh ( \mu y ) -\sin ( \mu y ) \sinh ( \mu ( 2-y ) ) }{\mu ^{3} ( \cosh ( 2\mu ) -\cos ( 2\mu ) ) ( \cosh ( 2\mu ) +\cos ( 2\mu ) -2 ) }\\ &{}\times \bigl( \sin ( \mu ) \sinh ( \mu x ) \sin \bigl( \mu ( 1-x ) \bigr) -\sinh ( \mu ) \sin ( \mu x ) \sinh \bigl( \mu ( 1-x ) \bigr) \bigr) \end{aligned}$$

and

$$ rr ( \mu ,x,y ) =rl ( \mu ,1-x,1-y ) . $$

See Fig. 28 (left). The value $\lambda _{1}=-\rho _{1}:=- ( 2\mu _{1} ) ^{4}$ corresponds to the first eigenvalue for the clamped problem

$$ \varphi _{1} ( x ) =\cos \bigl( 2\mu _{1} \bigl( x- \tfrac{1}{2} \bigr) \bigr) -\frac{\cos \mu _{1}}{\cosh \mu _{1}}\cosh \bigl( 2\mu _{1} \bigl( x-\tfrac{1}{2} \bigr) \bigr) $$

with $\mu _{1}$ the first positive solution of $\tan \mu _{1}+\tanh \mu _{1}=0$. To determine the value $\lambda _{c}$ one considers the switched eigenvalue problem where $\varphi ^{\prime } ( 1 ) =0$ is replaced by $\varphi ^{\prime \prime } ( 0 ) =0$, which is again problem (A.5), and hence one obtains the same value $\lambda _{c}$ as before. □

Remark A.1

It might appear surprising, that the value $\lambda _{c}$ from Lemma 2.2 for the supported beam and the $\lambda _{c}$ from Lemma 2.3 for the clamped beam are identical. The reason is that the corresponding switched eigenvalue problems in the sense of [73], which determine $\lambda _{c}$, are identical. Indeed, in both cases one arrives at (A.5).

Fig. 28

Sketches of the Green functions for the clamped (left) and for the hinged beam with $\lambda =5184>\lambda _{c}$. On the dark (red) part the Green function is negative. The line depicts the diagonal, where $x\mapsto ( \frac{d}{dx} ) ^{3}g_{\lambda } (x,y )$ has a jump

Proof of Lemma 2.4

The first eigenfunction that corresponds to Problem 5, is

$$ \varphi _{1} ( x ) =\sinh ( \nu x ) -\sin ( \nu x ) - \frac{\sin \nu +\sinh \nu }{\cos \nu +\cosh \nu } \bigl( \cosh ( \nu x ) -\cos ( \nu x ) \bigr) , $$

with eigenvalue $d_{1}=\nu _{1}^{4}$, where $\nu _{1}$ the first positive zero of $\cosh \nu \cos \nu +1=0$. According to [73] the critical value $d_{c}$ is determined by one of the following two switched eigenvalue problems, namely

$$ \left \{ \textstyle\begin{array}{l} \varphi ^{\prime \prime \prime \prime }+ \lambda \varphi =0\quad \text{in } ( 0,1 ) , \\ \varphi ( 0 ) =\varphi ^{\prime } ( 0 ) =\varphi ^{\prime \prime } ( 0 ) =0, \\ \varphi ^{\prime \prime } ( 1 ) =0,\end{array}\displaystyle \right . \quad \text{and}\quad \left \{ \textstyle\begin{array}{l} \varphi ^{\prime \prime \prime \prime }+ \lambda \varphi =0\quad \text{in } ( 0,1 ) , \\ \varphi ( 0 ) =0, \\ \varphi ( 1 ) =\varphi ^{\prime \prime } ( 1 ) =\varphi ^{\prime \prime \prime } ( 1 ) =0.\end{array}\displaystyle \right . $$

(A.6)

The corresponding first eigenfunctions of (A.6) are related through

$$ \varphi _{\text{left}} ( x ) =\varphi _{\text{right}}^{\prime \prime } ( 1-x ), $$

and hence give the same eigenvalue. Since

$$ \varphi _{\text{left}} ( x ) =\sinh ( \mu _{1}x ) \cos ( \mu _{1}x ) -\sin ( \mu _{1}x ) \cosh ( \mu _{1}x ), $$

where $\mu _{1}$ is the first positive number such that $\varphi _{\text{left}}^{\prime \prime } ( 1 ) =0$, the same $\mu _{1}$ as for Lemma 2.3 but here implying that $d_{c}=4\mu _{1}^{4}\approx 125.141$. □

Appendix B: A Sharper Version of Kreĭn–Rutman

In a more general setting such as for cooperative systems the strong positivity condition, which is used for the version of Kreĭn–Rutman above, i.e. Theorem 3.4, is too restrictive. Instead of strong positivity, some weaker positivity is sufficient, whenever one can show that the spectral radius is strictly positive. Showing that the spectral radius is positive, most of the time will need some tedious constructions in order to obtain a more or less explicit (super)solution. A breakthrough is a theorem of de Pagter in [66]. The slight disadvantage is that one needs the structure of Banach lattices, which may not be a common tool for the analyst working on partial differential equations. The spaces $C(\overline{\varOmega })$ and $\mathcal{L}^{p} ( \varOmega )$, home turf for analysts, with the natural ordering, however are Banach lattices. Let us briefly introduce the setting.

Definition B.1

The real vectorspace $X$ gives a vector lattice $( X,\leq )$, if:

• 1. ≤ is a partial ordering on $X$;

• 2. for all $u,v,w\in X$: $u\leq v$ implies $u+w\leq v+w$;

• 3. for all $u\in X$ and $t\in \mathbb{R}^{+}$: $0\leq u$ implies $0\leq tu$;

• 4. for all $u,v\in X$ also $\inf ( u,v ) ,\sup ( u,v ) \in X$, where $\inf ( u,v )$ is the largest lower bound and $\sup ( u,v )$ the least upper bound for $u,v$.

Note that $u\in C(\overline{\varOmega })$ implies $\vert u\vert \in C(\overline{\varOmega })$ and a similar result holds for $\mathcal{L}^{p} ( \varOmega )$ or $W^{1,2} ( \varOmega )$. For $C^{1}(\overline{\varOmega })$ and $W^{2,2} ( \varOmega )$ such a result is not true. One also needs a relation between the norm and the absolute value of a function, $\vert u\vert :=\sup ( u,-u )$.

Definition B.2

If $( X,\leq )$ is a vector lattice and $( X,\Vert \cdot \Vert )$ a Banach space, then $( X, \Vert \cdot \Vert ,\leq )$ is a Banach lattice if for all $u,v\in X$ one has: $\vert u\vert \leq \vert v\vert$ implies $\Vert u\Vert \leq \Vert v\Vert$.

The space $( C(\overline{\varOmega }),\Vert \cdot \Vert _{\infty },\le )$ is a Banach lattice and $( W^{1,2} ( \varOmega ) ,\Vert \cdot \Vert _{W^{1,2}},\le )$ is not.

Definition B.3

Let $( X,\Vert \cdot \Vert ,\leq )$ be a Banach lattice. A subspace $I\subset X$ is called a lattice ideal, if

• 1. for all $u\in I$ one has $\vert u\vert \in I$ and

• 2. for all $u\in I$ and $v\in X$ with $0\leq v\leq u$ one has $v\in I$.

Definition B.4

Let $( X,\Vert \cdot \Vert ,\leq )$ be a Banach lattice and $T\in \mathsf{L} ( X )$. $T$ is called irreducible if $\{ 0 \}$ and $X$ are the only closed $T$-invariant lattice ideals.

The properties of $g_{1}$, $g_{2}$, $g_{3}$ and $g_{5}$ in (5), (6), (10), (11), and for more general second order equations the strong maximum principle, imply that the corresponding $\mathcal{G}\in \mathsf{L} ( C(\overline{\varOmega }) )$ is irreducible, without assuming more on the boundary than that $\mathcal{G}$ indeed maps from $C(\overline{\varOmega })$ into $C(\overline{\varOmega })$.

Using the concept of Banach lattice one may formulate an optimal version of the Kreĭn–Rutman theorem.

Theorem B.5

(Kreĭn–Rutman–de Pagter)

Let $( X,\Vert \cdot \Vert ,\leq )$ be a Banach lattice with $\dim ( X ) >1$ and let $T\in \mathsf{L} ( X )$ be compact, positive and irreducible. Then:

• 1. the spectral radius $\nu ( T )$ is strictly positive;

• 2. $\nu ( T )$ is an eigenvalue for $T$ with algebraic multiplicity 1;

• 3. all other eigenvalues $\nu _{i}$ satisfy $\vert \nu _{i}\vert <\nu ( T )$;

• 4. the eigenfunction for $\nu ( T )$ is of fixed sign and (up to multiples) it is the only eigenfunction of $T$ with a fixed sign.

The value $\nu ( T )$ is called the principal eigenvalue of $T$ just as the corresponding eigenfunction is called the principal eigenfunction of $T$.

The theorem combines a classical result of Kreĭn and Rutman with a result of de Pagter [66]. See also [37] or [77].