Gradual Loss of Positivity and Hidden Invariant Cones in a Scalar Heat Equation

Patrick Guidotti^{∗} and Sandro Merino^{∗∗}

∗ Applied Mathematics, 217-50 Caltech, Pasadena, CA 91125 USA, gpatrick@cco.caltech.edu

∗∗ University of Strathclyde, Department of Mathematics, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK, merino@math.unizh.ch

Abstract. Invariance properties of a scalar, linear heat equation with nonlocal boundary conditions are discussed as a function of a real parameter appearing in the boundary conditions of the problem. The equation is a model for a thermostat with sensor and controller positioned at opposite ends of an interval, whence the non-locality. It is shown that the analytic semigroup associated with the evolution problem is positive if and only if the parameter is in (−∞, 0] . For the corresponding elliptic problem three maximum principles are proved which hold for different parameter ranges.

Key Words: Invariant Cones - Scalar Reaction Diffusion Equations - Nonlinear and Nonlocal Boundary Conditions - Maximum Principle.

AMS Subject Classifications: 35 K60, 35 B32, 35 B10.

1 Introduction

Consider the linear parabolic evolution problem

\begin{aligned}u_{t} − u_{xx} & = & 0 , & (x, t) ∈ (0, π) × (0, ∞) , \\ \frac{∂ u}{∂ ν}(0, t) + β u(π, t) & = & 0 , & t ∈ (0, ∞) , \\ \frac{∂ u}{∂ ν}(π, t) & = & 0 , & t ∈ (0, ∞) , \\ u(x, 0) & = & u_{0}(x) , & x ∈ (0, π) . \end{aligned}  (1)

The nonlocal boundary condition ∂_{ν} u(0, t) + β u(π, t) = 0 is a variation of the classical local Robin condition ∂_{ν} u(0, t) + β u(0, t) = 0 . The problem (1) can be interpreted as a model for a thermostat where the sensor measures the temperature at x = π and the controller releases or extracts heat at x = 0 proportionally to the temperature at x = π. It is intuitively clear that only positive values of β can have a stabilizing effect on the reference state u(x) = 0 . It may be less clear that stability is also lost when β is too large. In fact, as proved in [8], a Hopf bifurcation occurs at β_{0} ≈ 5.6655 and the results in [8] suggest that (1) might be termed “ the parabolic oscillator ”. In this paper we are interested in the invariance (positivity) properties of (1) . More precisely, we study the positivity of the analytic C_{0} -semigroup e^{−t A(β)} on L_{2}(0, π) that is canonically associated with (1) (see Section 2 for details). For β = 0 we are considering the heat equation with no flux boundary conditions and it is a consequence of the parabolic maximum principle that

u_{0} ≥ 0  ⟹  e^{−t A(β)}u_{0} ≥ 0 , t ≥ 0 ,  (2)

in the usual L_{2} sense. It is also a simple application of the parabolic maximum principle to see that (2) remains true for β ∈ (−∞, 0] . To give a concise formulation of our results we introduce the following closed and convex subsets (cones) of L_{2}(0, π) :

P := { u ∈ L_{2}(0, π) ; u ≥ 0 } ,
Q := { u ∈ L_{2}(0, π) ; P^{s}u ∈ P and P^{a}u ∈ P } ,
R := { u ∈ L_{2}(0, π) ; P^{s}u ∈ P } ,

where

P^{s}u(·) := \frac{u(·) + u(π − ·)}{2} and P^{a}u(·) := \frac{u(·) − u(π − ·)}{2}

are the symmetric and antisymmetric components of u, respectively. In Section 3 we characterize the positivity of the analytic semigroup associated with (1) :

Theorem e^{−t A(β)}(P) ⊂ P , t ≥ 0 , iff β ∈ ( −∞ , 0 ] .

The behaviour of the nonlocal problem is thus very different from its local counterpart with the classical Robin condition ∂_{ν} u(0, t) + β u(0, t) = 0 , where (2) holds for each β ∈ lR . For the elliptic problem A(β) u = f associated with (1) the main result is Theorem 5.6 where the following invariance properties (maximum principles) are proved: Theorem The following assertions hold:

a) A(β)^{−1}(P) ⊂ P for β ∈ (0, \frac{1}{π}] .

b) A(β)^{−1}(Q) ⊂ P for β ∈ (0, \frac{2}{π}] .

c) A(β)^{−1}(Q) ⊂ R for β ∈ (0, \frac{4}{π}] .

Observe that for β ∈ ( 0, \frac{1}{π} ] the second and third inclusions are trivial consequences of the first and for β ∈ ( 0, \frac{2}{π} ] the third assertion is a consequence of the second one. Thus in that sense, as β is increased, positivity is lost gradually and the nonstandard (hidden) cones Q and R come into play to formulate the weaker positivity persisting for β > \frac{1}{π} .

What led us to consider the projections P^{s} and P^{a} in the definition of Q and R is the observation that the adjoint problem to (1) can be obtained by the linear transformation on L_{2}(0, π)

T u(·) := u(π − ·) .

In Sections 4 and 5 we discuss the adjoint problem and the properties of the linear transformation above. In Section 6 we discuss the existence of positive principal eigenvalues. We find that in contrast to second order elliptic problems with local boundary conditions the existence of a positive principal eigenvalue is not related to the validity of a maximum principle. In Section 7 we briefly discuss how monotone iteration techniques can be applied to nonlinear problems when the parameter β is in the range (0, \frac{1}{π}] where the maximum principle holds. A problem closely related to (1) has been studied by A. Friedman and L. Jiang in [7]. A similar nonlocal boundary condition is considered together with a hysteresis functional. However, Friedman and Jiang do not discuss the spectrum of the linearization and therefore do not find a periodic solution as the result of a Hopf bifurcation. The periodic solution is constructed by a more direct approach applying Schauder’s fixed point theorem. Furthermore their problem is piecewise linear. We were not aware of the paper [7] when we submitted [8] and thank M. Grinfeld and J. Ockendon for pointing out that paper to us. In Section 8 we briefly give an intuitive interpretation of the functional analytic approach sketched in Section 2. For a more detailed introduction in extrapolation techniques the reader may also consult the introduction in [3]. Observations made in numerical experiments are also briefly reported. We conclude this introduction with the unavoidable:

A Misprint in [8]

Unfortunately, a misprint appears in [8]: Equation (7) should read

ω( ρ, β) = 2 iρ ( ρ sin(πρ) − β) .

The subsequent discussion of the zeros of ω is not affected by this misprint since the case ρ = 0 is easily discussed separately.

2 A Sketch of the Semigroup Approach

In this section we briefly discuss how the linear evolution equation (1) can be reformulated and solved in the context of semigroup theory on Banach scales. We rewrite (1) more concisely as

\begin{aligned}u_{t} + Au & = & 0 , & (x, t) ∈ Ω × (0, ∞) , \\ B(β)u & = & 0 , & (x, t) ∈ ∂Ω × (0, ∞) , \\ u(x, 0) & = & u_{0}(x) , & x ∈ Ω , \end{aligned}  (3)

where Ω := (0 , π) and

Au := −u_{xx} , B(β)u := (B_{1}(β)u , B_{2}u) := ( \frac{∂ u}{∂ ν}(0) + βu(π) , \frac{∂ u}{∂ ν}(π) ) .

We closely follow the approach described in [2] and [3] for solving the above evolution problem. For (s, p) ∈ lR × (1, ∞) we denote the Bessel potential spaces by H_{p}^{s}(Ω). For later convenience we also set

E_{1} := H_{2}^{2}(Ω) , E_{\frac{1}{2}} := H_{2}^{1}(Ω) , E_{0} := L_{2}(Ω) , E_{− \frac{1}{2}} := H_{2}^{1}(Ω)′ .

The evolution equation (1) can be reformulated as an abstract Cauchy problem in the Banach space E_{− \frac{1}{2}}

\begin{aligned}u̇ + A_{− \frac{1}{2}} u + βγ′_{0}γ_{π} u & = & 0 , t > 0 , \\ u(0) & = & u_{0} , \end{aligned}  (4)

where

γ_{x} ∈ L(E_{\frac{1}{2}} , lR) , γ_{x}u := u(x) ,

for x ∈ {0, π} and

γ′_{x} ∈ L(lR, E_{− \frac{1}{2}} )

denotes the dual operator to γ_{x}. The above abstract Cauchy problem is obtained by reinterpreting the weak formulation of (1):

\begin{aligned}∫_{0}^{π} u̇φ dx + ∫_{0}^{π} u_{x}φ_{x} dx + βu(π) φ(0) & = & 0 , t > 0 , \\ u(0) & = & u_{0} , \end{aligned}  (5)

where the test functions φ are chosen in E_{\frac{1}{2}} . We note that in (4)

A_{− \frac{1}{2}} ∈ L(E_{\frac{1}{2}} , E_{− \frac{1}{2}} )

is the linear operator canonically induced via the L_{2} duality pairing by the continuous bilinear form, the “ Dirichlet form ”

(u, v) ↦ ∫_{0}^{π} u_{x} v_{x} dx

on E_{\frac{1}{2}} × E_{\frac{1}{2}} . In the following we set

A_{− \frac{1}{2}} (β) := A_{− \frac{1}{2}} + βγ′_{0}γ_{π} .

Since (A, B(β)) is not a normally elliptic boundary value problem (see [2, Section 4]) we use a perturbation argument to show that −A_{− \frac{1}{2}} (β) is the generator of an analytic semigroup: following the notation used in [3] we denote the set of generators of analytic C_{0} -semigroups on E_{− \frac{1}{2}} with domain E_{\frac{1}{2}} by

H(E_{\frac{1}{2}} , E_{− \frac{1}{2}} ) .

Proposition 2.1 For β ∈ lR

A_{− \frac{1}{2}} (β) ∈ H(E_{\frac{1}{2}} , E_{− \frac{1}{2}} ) .

Furthermore

A_{− \frac{1}{2}} (β)^{−1} ∈ L(E_{− \frac{1}{2}} )

for β ≠ 0 .

Proof We verify that γ′_{0}γ_{π} ∈ L(E_{\frac{1}{2}} , E_{− \frac{1}{2}} ) is a compact operator. The first assertion then follows from a standard perturbation result from semigroup theory (see e.g. [3, Chapter 1]). It is clear that we only need to prove the compactness of γ_{π} ∈ L(E_{\frac{1}{2}} , lR) . Since E_{\frac{1}{2}} is compactly embedded in C(\overline{Ω}) the compactness of γ_{π} follows from the factorization

E_{\frac{1}{2}} ⟶ C(\overline{Ω}) ⟶^{γ_{π}} lR .

That A_{− \frac{1}{2}} (β) has a bounded inverse for β ≠ 0 is shown in [8] where the spectrum of A_{− \frac{1}{2}} (β) is discussed.

Based on the above generation result interpolation-extrapolation techniques can be applied to construct a whole scale of analytic C_{0}-semigroups. For our purposes it is sometimes convenient to restrict the semigroup on E_{− \frac{1}{2}} to E_{0} so that we can formulate our invariance results in terms of the standard positive cone in E_{0} . The situation is best described by the following commutative diagram:

E_{1} ⟶^{A_{0}(β)} E_{0}  ⟶^{e^{−tA_{0}(β)}} E_{0}
d ↓  d ↓  d ↓
E_{\frac{1}{2}} ⟶^{A_{− \frac{1}{2}} (β)} E_{− \frac{1}{2}} ⟶^{e^{−tA_{− \frac{1}{2}} (β)}} E_{− \frac{1}{2}}

We refer to [2, Section 6] or [3, Chapter 5] for a complete exposition of the theory.

3 Characterization of Positivity

In this section we prove our main result on the analytic C_{0} -semigroup on E_{−1/2} generated by −A_{−1/2}(β). Following [3, Sections 6.3 and 6.4] a cone C is defined as a nonempty subset of a vector space that satisfies lR^{+}C ⊂ C. A cone is proper if C ∩ (−C) = {0}. Let P^{#} be the cone in E_{−1/2} containing all positive functionals on E_{1/2}. It is given by

P^{#} = {u ∈ E_{−1/2} | ⟨u, ϕ⟩ ≥ 0 , 0 ≤ ϕ ∈ E_{1/2}}

and is a proper cone with nonempty interior. Then we have:

Theorem 3.1

e^{−t A_{−1/2}(β)}(P^{#}) ⊂ P^{#} , t ≥ 0 , iff β ∈ ( −∞ , 0 ] .

If β ≤ 0 and t > 0 then e^{−t A_{−1/2}(β)}(P^{#}) ⊂ P.

Proof That β ∈ (−∞, 0] is a sufficient condition is a consequence of the parabolic maximum principle: pick an arbitrary u_{0} ∈ P^{#} . Since the semigroup is analytic u(t, u_{0}) := e^{−tA_{0}(β)}(u) ⊂ E_{1/2} for t > 0 . In particular, u(t, u_{0}) is Hölder continuous for t > 0 . By classical regularity arguments we then obtain that u(t, u_{0}) ∈ BUC^{∞}(Ω) for t > 0 . Since u is a classical solution of

u_{t} − u_{xx} = 0 in Ω × (0, ∞) ,

we can apply the parabolic maximum principle as stated in [6, Propositions 13.1 and 13.2] to conclude that u(t, u_{0}) ∈ P for t > 0 . Our assumption β ≤ 0 together with [6, Proposition 13.2] is used to exclude that u changes sign at the boundary of Ω . To show that the condition is necessary we study the resolvent of the generator −A_{−1/2}(β) on E_{−1/2}: For λ > 0 consider

λ u + A_{−1/2}(β)u = δ_{π} , in E_{−1/2}  (6)

where β > 0 and δ_{π} ∈ E_{−1/2} denotes the Dirac measure supported at the right endpoint of the interval. The solution of (6) will turn out to be negative at x = 0 for all sufficiently large λ. In other words we will show that

(λ + A_{−1/2}(β))^{−1}(P^{#}) ⊄ P^{#}

for λ ∈ (λ_{0}, ∞) , λ_{0} > 0 , i.e. −A(β) is not resolvent positive. Consequently the semigroup e^{−t A_{−1/2}(β)} is not positive for β > 0 . If the righthand side of (6) is replaced by a function f ∈ E_{0} it is easy to see that a particular solution of (6) is given by convolution with the kernel \frac{sinh(π\sqrt{λ})}{\sqrt{λ}}. Thus its general solution is given through

u(x, λ) = ∫_{0}^{x} \frac{sinh(\sqrt{λ}(x − y)}{\sqrt{λ}}f(y) dy + A cosh(\sqrt{λ}(π − x)) + B sinh(\sqrt{λ}(π − x)) ,

where the constants A and B are fixed by imposing the boundary conditions. An easy computation leads to the following expressions for A and B:

B = − ∫_{0}^{π} \frac{cosh(\sqrt{λ}(π − y)}{\sqrt{λ}}f(y) dy ,  (7)
A = \frac{1}{β + \sqrt{λ} sinh(\sqrt{λ}π)}[β ∫_{0}^{x} \frac{sinh(\sqrt{λ}(x − y)}{\sqrt{λ}}f(y) dy − B\sqrt{λ} cosh(\sqrt{λ}π)] .  (8)

Thus we obtain

u(0, λ) = \frac{\frac{β}{\sqrt{λ}} cosh(\sqrt{λ}π)}{β + \sqrt{λ} sinh(\sqrt{λ} π)} ∫_{0}^{π} sinh(\sqrt{λ} (π − τ)) f(τ) dτ  (9)

+ \frac{1 − \frac{β}{\sqrt{λ}} sinh(\sqrt{λ}π)}{β + \sqrt{λ} sinh(\sqrt{λ} π)} ∫_{0}^{π} cosh(\sqrt{λ} (π − τ)) f(τ) dτ .

Now choosing an approximating sequence (f_{n})_{n≥0} of L_{2}-functions for the Dirac measure δ_{π}, which is possible since E_{0} is dense in E_{−1/2}, and using the continuity of the resolvent on E_{−1/2} we infer from (9) that

((λ + A_{−1/2}(β))^{−1}δ_{π})(0) = \frac{1}{β + \sqrt{λ} sinh(\sqrt{λ}π)}[1 − πβ \frac{sinh(\sqrt{λ}π)}{\sqrt{λ}π}]  (10)

since

(λ + A_{−1/2}(β)^{−1}f_{n} → (λ + A_{−1/2}(β)^{−1}δ_{π} in E_{1/2} ↪ C[0, 1]

and

∫_{0}^{π} sinh(\sqrt{λ} (π − τ)) f_{n}(τ) dτ → sinh(\sqrt{λ} π) , ∫_{0}^{π} cosh(\sqrt{λ} (π − τ)) f_{n}(τ) dτ → cosh(\sqrt{λ} π)

as n tends to infinity. We easily conclude that, whenever β > 0, a λ_{0} can be found which makes the righthand side of (10) negative whenever λ ≥ λ_{0}. It is interesting to observe that

lim_{λ→0}((λ + A_{−1/2}(β))^{−1}δ_{π})(0) = \frac{1}{π} − β .

It will in fact turn (see Section 5) that a maximum principle holds for A_{0}(β) as long as β < 1/π. We refer to that section for the details.

Remark 3.2 The above result implies that

e^{−t A_{0}(β)}(P) ⊂ P , t ≥ 0 , iff β ∈ ( −∞ , 0 ] .

In other words positivity is preserved along the Banach scale.

The Time Dependent Case

For the sake of completeness we briefly discuss the case where the parameter β depends on time. The abstract results in [3, Theorem II.6.3.3] can be applied to obtain invariance properties analogous to the autonomous case. The main assumption is that β(t) varies within the corresponding interval uniformly in time. We consider the time dependent problem:

\begin{aligned}u_{t} + Au & = & 0 , & (x, t) ∈ Ω × (0, ∞) , \\ B(β(t))u & = & 0 , & (x, t) ∈ ∂Ω × (0, ∞) , \\ u(x, 0) & = & u_{0}(x) , & x ∈ Ω , \end{aligned}  (11)

where

β ∈ C^{ρ}([0, ∞), lR) for some ρ ∈ (0, 1) .  (12)

The abstract theory in [3, Chapter II] can now be invoked to construct a parabolic evolution operator (fundamental solution, propagator) to the above evolution problem. Furthermore the evolution operator inherits the invariance properties shared by the family of analytic semigroups generated by −A_{0}(β(t)) , t > 0 . Theorem 3.3 The evolution problem (11) induces a parabolic evolution operator

U : J_{△} := { (t, s) ∈ [0, ∞) × [0, ∞) | t ≥ s } → L(E_{0}) .

If β([0, ∞)) ⊂ (−∞, 0 ] , then U(t, s)(P) ⊂ P for (t, s) ∈ J_{△}. Proof The result is an application of [3, Theorem II.6.3.4].

4 The Adjoint Evolution Problem

In this section we will define the formally adjoint problem to (1) in the usual way. The adjoint problem will reveal a symmetry which turns out to be the key for unveiling hidden invariant sets of the elliptic problem associated with (1).

Definition 4.1 We define the formally adjoint operators (A^{#}, B^{#}) to (A, B) by

A^{#}u := −u_{xx} = Au , B^{#}u := ( B_{1}^{#}u , B_{2}^{#}(β) u) := ( \frac{∂ u}{∂ ν}(0) , \frac{∂ u}{∂ ν}(π) + β u(0) ) .

The formally adjoint evolution problem to (3) is thus given by

\begin{aligned}u_{t} + A^{#}u & = & 0 , & (x, t) ∈ Ω × (0, ∞) , \\ B^{#}(β)u & = & 0 , & (x, t) ∈ ∂Ω × (0, ∞) , \\ u(x, 0) & = & u_{0}(x) , & x ∈ Ω . \end{aligned}  (13)

Hence the adjoint problem is obtained by the change of variables

x ↦ π − x .

5 Eigenspaces and Maximum Principles

In this section we collect some properties of the linear transformation

T ∈ L(E_{0}) , T u(·) := u(π − ·) .

We will then obtain various maximum principles that can be formulated in terms of the eigenspaces of T .

Lemma 5.1 The transformation T has the following properties

a) T ∈ Lis (E_{0}) , T^{−1} = T , T^{∗} = T .

b) σ(T) = {1, −1} , ker(1 − T) = E_{0}^{s} , ker(1 + T) = E_{0}^{a} , where

\begin{matrix}E_{0}^{s} := \{ u ∈ E_{0} | u(\frac{π}{2} − x) = u(\frac{π}{2} + x) for |x| ≤ \frac{π}{2} \} , \\ E_{0}^{a} := \{ u ∈ E_{0} | u(\frac{π}{2} − x) = −u(\frac{π}{2} + x) for |x| ≤ \frac{π}{2} \} . \end{matrix}

c) E_{0} is decomposed by the direct sum E_{0} = E_{0}^{s} ⊕ E_{0}^{a}. The corresponding continuous projections are given by

\begin{matrix}P^{s}u(·) := \frac{1}{2}( u(·) + u(π − ·) ) , \\ P^{a}u(·) := \frac{1}{2}( u(·) − u(π − ·) ) . \end{matrix}

Proof a) That T is a unitary, self-adjoint, isometric isomorphism on the Hilbert space E_{0} is easily verified.

b) To study the spectrum of T consider the equation

(λ − T)u = f

in E_{0} . By part a) this is equivalent to

λT u − u = T f .

This can be written as the system

[ \begin{matrix}λ & −1 \\ −1 & λ \end{matrix} ] [ \begin{matrix}u \\ T u \end{matrix} ] = [ \begin{matrix}f \\ T f \end{matrix} ] .

The spectrum of T is given by the spectrum of the above matrix and it is clear that the eigenspaces are given by the symmetric and antisymmetric functions in E_{0} .

c) P^{s} is a continuous projection on E_{0} . Hence

E_{0} = range(P^{s}) ⊕ ker(P^{s})

and clearly

range(P^{s}) = E_{0}^{s} , ker(P^{s}) = E_{0}^{a} .

We can now study the projection onto E_{0}^{s} and E_{0}^{a} of the following problem in E_{0}

A_{0}(β)u = f,

which is the abstract formulation of

\begin{aligned}Au & = & f , & x ∈ Ω , \\ B(β)u & = & 0 , & x ∈ ∂Ω . \end{aligned}  (14)

Proposition 5.2 For f ∈ E_{0} consider the solution u := A_{0}(β)^{−1}f ∈ E_{1} of (14). Then the symmetric and antisymmetric components of u

v := P^{s}u and w := P^{a}u ,

restricted to [0, \frac{π}{2}] , solve the linear system

\begin{aligned}−v_{xx} & = & f^{s} , & x ∈ (0, \frac{π}{2}) , \\ −w_{xx} & = & f^{a} , & x ∈ (0, \frac{π}{2}) , \\ v_{x}(0) & = & −\frac{β}{2} (v(0) − w(0)) , \\ w_{x}(0) & = & −\frac{β}{2} (v(0) − w(0)) , \\ v_{x}(\frac{π}{2}) & = & 0 , \\ w(\frac{π}{2}) & = & 0 . \end{aligned}  (15)

Proof Note that the boundary condition at x = \frac{π}{2} is a consequence of the symmetry and antisymmetry of v and w, respectively. We leave the simple verification to the reader.

We will now derive integral representations for the solutions of (14) and (15) .

Lemma 5.3 a) The solution of (14) is given by

u(x) = \frac{1}{β} ∫_{0}^{π} f(s)ds − ∫_{x}^{π} ds ∫_{s}^{π} f(r)dr .

b) The solution of (15) is given by

v(x) = (\frac{2}{β} − \frac{π}{2} ) ∫_{0}^{\frac{π}{2}} f^{s}(q)dq + ∫_{0}^{\frac{π}{2}} dq ∫_{0}^{q} f^{a}(r)dr + ∫_{0}^{x} dq ∫_{q}^{\frac{π}{2}} f^{s}(r)dr ,

w(x) = (x − \frac{π}{2} ) ∫_{0}^{\frac{π}{2}} f^{s}(q)dq + ∫_{x}^{\frac{π}{2}} dq ∫_{0}^{q} f^{a}(r)dr .

Proof

a) Integrating (14) over (x, π) and using the boundary condition at x = π we find

u_{x}(x) = ∫_{x}^{π} f(q)dq .

From the boundary condition we deduce

u(π) = \frac{1}{β} u_{x}(0) = \frac{1}{β} ∫_{0}^{π} f(q)dq .

A second integration over (x, π) yields

u(π) − u(x) = ∫_{x}^{π} dq ∫_{q}^{π} f(r)dr.

Inserting u(π) concludes the proof.

b) Integrating the first equation in (15) we obtain

∫_{x}^{\frac{π}{2}} v_{xx}(q)dq = v_{x}(x) − v_{x}(\frac{π}{2} ) = ∫_{x}^{\frac{π}{2}} f^{s}(q)dq.

In particular

v_{x}(0) = ∫_{x}^{\frac{π}{2}} f^{s}(r)dr .

Using the boundary condition v_{x}(\frac{π}{2}) = 0 and integrating once more over (0, x) we find

v(x) = v(0) + ∫_{0}^{x} dq ∫_{q}^{\frac{π}{2}} f^{s}(r)dr .  (16)

Analogously, integrating the second equation in (15) yields

w_{x}(x) = w_{x}(0) − ∫_{0}^{x} f^{a}(r)dr

and a second integration over (x, \frac{π}{2}) produces

w(x) = w_{x}(0)(x − \frac{π}{2} ) + ∫_{x}^{\frac{π}{2}} dq ∫_{0}^{q} f^{a}(r)dr ,  (17)

where we have made use of the boundary condition w(\frac{π}{2}) = 0 . Next we will use the boundary conditions in (15) at x = 0 , which can now be written as

\begin{aligned}∫_{0}^{\frac{π}{2}} f^{s}(r)dr & = & \frac{β}{2} (v(0) + \frac{π}{2} w_{x}(0) − ∫_{0}^{\frac{π}{2}} dq ∫_{0}^{q} f^{a}(r)dr) , \\ w_{x}(0) & = & \frac{β}{2} (v(0) + \frac{π}{2} w_{x}(0) − ∫_{0}^{\frac{π}{2}} dq ∫_{0}^{q} f^{a}(r)dr) . \end{aligned}

Interpreting the above equations as a linear system for the unknowns w_{x}(0) and v(0) we can easily solve it and find

\begin{aligned}w_{x}(0) & = & ∫_{0}^{\frac{π}{2}} f^{s}(r)dr , \\ v(0) & = & (\frac{2}{β} − \frac{π}{2} ) ∫_{0}^{\frac{π}{2}} f^{s}(r)dr + ∫_{0}^{\frac{π}{2}} dq ∫_{0}^{q} f^{a}(r)dr . \end{aligned}

Inserting this result into (16) and (17) proves assertion b).

Definition 5.4 We introduce the following subsets of E_{0} :

P := {u ∈ E_{0} ; u(x) ≥ 0 a.e. in Ω } ,
Q := {u ∈ E_{0} ; P^{s}u and P^{a}u ∈ P } ,
R := {u ∈ E_{0} ; P^{s}u ∈ P } .

We collect some easily verified properties of P, Q and R :

Remarks 5.5

a) P, Q and R are convex closed cones in E_{0}. The cones P and Q are proper while R is not.

The following inclusions hold

Q ⊂ P and Q ⊂ R .

b) P is the standard positive, proper, generating, normal cone in E_{0}.

The above representations of the solution of (14) can be used to prove the following “ Maximum Principles ” :

Theorem 5.6 (Maximum Principles) The following assertions hold

a) A_{0}(β)^{−1}(P) ⊂ P for β ∈ (0, \frac{1}{π}] .

b) A_{0}(β)^{−1}(Q) ⊂ P for β ∈ (0, \frac{2}{π}] .

c) A_{0}(β)^{−1}(Q) ⊂ R for β ∈ (0, \frac{4}{π}] .

Furthermore the upper bounds given for β are sharp.

Proof a) Lemma 5.3 a) shows that u(x) ≥ u(0) for f ≥ 0 and x ∈ [0, π] . From

u(0) = (\frac{1}{β} − π) ∫_{0}^{π} f(s)ds

we conclude that u(0) ≥ 0 for β ∈ (0, \frac{1}{π}] .

b) Lemma 5.3 b) yields

u(x) = v(x)+w(x) = (x+ \frac{2}{β} −π) ∫_{0}^{\frac{π}{2}} f(s)ds+ ∫_{0}^{x} dq ∫_{q}^{\frac{π}{2}} f^{s}(r)dr+ ∫_{0}^{x} dq ∫_{0}^{q} f^{a}(r)dr+2 ∫_{x}^{\frac{π}{2}} dq ∫_{0}^{q} f^{a}(r)dr .

The assumption then implies that u(x) ≥ 0 on [0, \frac{π}{2}] if (x + \frac{2}{β} − π) ≥ 0 , i.e. if β ∈ (0, \frac{2}{π}] . From assertion a) we see that

min_{x∈[0,π]} u(x) = u(0)

and consequently u ∈ P for β ∈ (0, \frac{2}{π}] .

c) By Lemma 5.3 b) we immediately see that v(x) ≥ 0 , x ∈ [0, \frac{π}{2}] , if (\frac{2}{β} − \frac{π}{2}) ≥ 0 , i.e., if β ∈ (0, \frac{4}{π}] . Hence v ∈ P by its symmetry.

We end this section with a “strong maximum principle” for Problem 14 with non homogeneous boundary conditions:

\begin{aligned}Au & = & f & , in Ω , \\ B(β)u & = & g & , on ∂Ω , \end{aligned}  (18)

where f ∈ E_{0} and g := (g_{1}, g_{2}) ∈ lR^{2} . We show that for β ∈ (0, \frac{1}{π}) a strong maximum principle holds for (18) .

Theorem 5.7 (Strong Maximum Principle) Assume β ∈ (0, \frac{1}{π}) and let u be the solution of (18) . Then f ≥ 0 and (g_{1}, g_{2}) ∈ lR_{+}^{2} imply u ≥ 0 . If in addition f(x) > 0 almost everywhere on an open subset of Ω or (g_{1}, g_{2}) ≠ (0, 0) , then u(x) > 0 on \overline{Ω} .

Proof The linearity of (18) allows to decompose its solution as

u = u_{f} + u_{g} ,

where

\begin{aligned}A u_{f} & = & f & , in Ω , \\ B(β) u_{f} & = & 0 & , on ∂Ω , \end{aligned}  (19)

and

\begin{aligned}A u_{g} & = & 0 & , in Ω , \\ B(β) u_{g} & = & g & , on ∂Ω . \end{aligned}  (20)

If (g_{1}, g_{2}) = 0 , then u = u_{f} and the assertions are a consequence of Lemma 5.3 a). If f ≡ 0 , then noting that u = u_{g} is an affine function we obtain by a straightforward calculation that

u_{g}(x) = g_{2} x + g_{2} (\frac{1}{β} − π) + \frac{g_{1}}{β} .

This concludes the proof.

6 Principal Eigenvalues and the Maximum Principle

In this section we discuss the existence of a positive principal eigenvalue in the spectrum of A_{0}(β) , or equivalently, the existence of positive principal eigenvalues of

\begin{aligned}Au & = & λ u , & x ∈ Ω , \\ B(β)u & = & 0 , & x ∈ ∂Ω , \end{aligned}  (21)

and the implications on the validity of a maximum principle for A_{0}(β)u = f . It is a well-known result that for classical local Robin or Dirichlet boundary conditions the existence of a positive principal eigenvalue for second-order elliptic equations is equivalent to the validity of the strong maximum principle. In particular, for the local counterpart of (21):

\begin{aligned}Au & = & λ u , & x ∈ Ω , \\ B1^{loc}(β)u := u_{x}(0) − β u(0) & = & 0 , \\ B2^{loc}u := u_{x}(π) & = & 0 , \end{aligned}  (22)

the following are equivalent (cf. [4, Theorem 2.4]):

• The least eigenvalue of (22) is positive.

• (A, B^{loc}) satisfies the strong maximum principle, i.e., (Au, B^{loc}u) > 0 ⟹ u > 0 .

Note that for each β ∈ lR the least eigenvalue of (22) is simple and its unique normalized principal eigenfunction is strictly positive. For our nonlocal problem (21) we give the following result

Theorem 6.1 For β ∈ (0, \frac{1}{2}) the spectral bound s(A_{0}(β)) := inf\{ Re(λ) | λ ∈ σ(A_{0}(β)) \} is a positive principal eigenvalue, i.e. a simple eigenvalue with a unique normalized eigenfunction which is strictly positive on \overline{Ω} .

Proof Note that the least eigenvalue of A_{0}(β = \frac{1}{2}) is given by λ_{1}(\frac{1}{2}) := ρ^{2} where ρ is the first positive root of

ρ sin(πρ) − \frac{1}{2}

(see [8, Section 3] for details). The associated normalized eigenfunction is

φ_{1}(β = \frac{1}{2})(x) := sin(\frac{x}{2}) , x ∈ (0, π) .

For β = 0 clearly the least eigenvalue is 0 and the associated normalized eigenfunction is

φ_{1}(β = 0)(x) := 1 , x ∈ (0, π) .

It is easily verified that the normalized first eigenfunctions φ_{1}(β) > 0 are strictly positive as β varies in [0, \frac{1}{2}) . The first eigenfunction picks up a nodal point as β crosses \frac{1}{2} .

Remarks 6.2

a) In Theorem 5.6 we have seen that a maximum principle (in the usual sense) is only valid for β ∈ (0, \frac{1}{π}] . Hence for β ∈ (\frac{1}{π}, \frac{1}{2}) no maximum principle is available despite the existence of a positive principal eigenvalue. This shows that the characterization of the maximum principle in [4] can not be extended to the nonlocal problem A_{0}(β)u = f .

b) The positivity of a semigroup implies properties of the spectrum of its generator. In particular, since P is a generating and normal proper cone, the invariance

e^{−tA_{0}(β)}(P) ⊂ P  (23)

implies that s(A_{0}(β)) ∈ σ(A_{0}(β)) . This is proved in [5, Theorem 7.4] as a consequence of the Pringsheim-Landau Lemma. The discussion of the spectrum of A_{0}(β) in [8, Section 3] shows that for β > β_{r} ≈ 0.5792 we have that s(A_{0}(β)) ∉ σ(A_{0}(β)) . Hence the general abstract result in [5] shows that (23) is not true for β > β_{r}. Using the structure of the problem we have found the sharp estimate β > 0 for the loss of the invariance (23) .

7 Monotone Iterations

Having established the validity of a strong maximum principle in Section 5 we will briefly discuss the application of the method of sub- and super-solutions to nonlinear problems of the form

\begin{aligned}Au & = & f(u) & , in Ω , \\ B(β)u & = & g(u) & , on ∂Ω . \end{aligned}  (24)

We will be rather brief in this section since it is quite clear that as long as a maximum principle is valid we only need to follow well-known standard arguments. We refer to the review article [1] for details. We impose the following conditions on f and g = (g_{1}, g_{2}):

f, g_{1}, g_{2} ∈ C^{1}(lR, lR) ,  (25)

and

f, g_{1}, g_{2} are strictly increasing.  (26)

We define sub- and supersolutions in the usual way: we call w ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}) a subsolution if

\begin{aligned}Aw & ≤ & f(w) & , in Ω , \\ B(β)w & ≤ & g(w) & , on ∂Ω . \end{aligned}  (27)

The sub-solution is called strict if one inequality is strict in some x ∈ \overline{Ω} . Super-solutions are defined by reversing the inequalities. In the next theorem we show that for β ∈ (0, \frac{1}{π}) solutions of (24) can be found by monotone iteration.

Theorem 7.1 Assume that β ∈ (0, \frac{1}{π}) and that f and g satisfy (25) and (26) . Then if (24) admits a pair (φ, ψ) of ordered sub- and super-solutions the problem (24) possesses a least and a greatest solution \underline{u} and \overline{u} in the order interval [φ, ψ] .

Proof We verify that the ideas in [1, Section 9] can be applied. Consider the solution operators K_{β} of

\begin{aligned}Au & = & h & , in Ω , \\ B(β)u & = & 0 & , on ∂Ω , \end{aligned}

and L_{β} of

\begin{aligned}Au & = & 0 & , in Ω , \\ B(β)u & = & h & , on ∂Ω . \end{aligned}

We can reformulate (24) as the following fixed point equation in a suitable ordered Banach space:

u = S_{β}(u) := K_{β}F (u) + L_{β}G(u) ,

where F and G are the superposition (Nemytskii) operators induced by f and g. By our assumption (26) these operators are increasing. The assertion now follows from the fact that S_{β} is increasing and completely continuous by applying a standard monotone iteration scheme.

We conclude this section with a remark on the boundary conditions in (24) .

Remark 7.2 The boundary conditions in (24) can be nonlocal, i.e. more explicitly they can chosen to be

\begin{aligned}B_{1}(β) u & = & \frac{∂u}{∂ν}(0) + β u(π) & = & g_{1}(u(ξ)) , \\ B_{2} u & = & \frac{∂u}{∂ν}(π) & = & g_{2}(u(η)) , \end{aligned}

with arbitrary ξ, η ∈ \overline{Ω} .

8 Final Remarks

The trace operator and its dual

The trace operator γ_{π} ∈ L(E_{\frac{1}{2}} , lR) and the dual γ′_{0} ∈ L(lR, E_{− \frac{1}{2}} ) of the trace operator γ_{0} appearing in the abstract formulation of (3) in E_{− \frac{1}{2}} :

\begin{aligned}u̇ + A_{− \frac{1}{2}} u & = & − β γ′_{0} γ_{π} u  , t > 0 , \\ u(0) & = & u_{0} , \end{aligned}  (28)

have a simple interpretation that we briefly discuss: for s ∈ lR

γ′_{0}(s) ∈ E_{− \frac{1}{2}} = L(E_{\frac{1}{2}} , lR) ,

and for u ∈ E_{\frac{1}{2}} duality yields

⟨γ′_{0}(s) , u⟩_{E_{− \frac{1}{2}} , E_{\frac{1}{2}}} = ⟨s , γ_{0}(u)⟩_{lR′ , lR} = s γ_{0}(u) = s u(0) .

Hence γ′_{0}(s) is a Dirac measure s δ_{0} of mass s concentrated at the left boundary point x = 0 . Note that the test functions D(Ω) are not dense in E_{\frac{1}{2}} and as a consequence E_{− \frac{1}{2}} ⊄ D′(Ω) . As an example γ′_{0}(s) ∈ E_{− \frac{1}{2}} but γ′_{0}(s) ∉ D′(Ω) . The right-hand side of the abstract formulation (28) is thus a reinterpretation of the boundary conditions in (3) in the sense of (generalized) distributions. At each time t a “heat pulse” −β u(π, t) δ_{0} is released at the left boundary point x = 0 . This interpretation of the boundary conditions as a source term makes the problem (3) accessible to semigroup techniques. Considering (28) it becomes evident that it can be interpreted as a feedback control equation of the form

u̇ + A_{− \frac{1}{2}} u = −β C S u , t > 0 ,

where

C := γ′_{0} ∈ L(lR, E_{− \frac{1}{2}} )

is the controller and

S := γ_{π} ∈ L(E_{\frac{1}{2}} , lR)

is the sensor. In fact, (28) is a model for a rudimentary thermostat studied in [8]. For more information on feedback control problems and the related questions of controllability and observability we refer to [9].

A particular initial condition

Consider the following initial value problem in E_{− \frac{1}{2}} :

\begin{aligned}u̇ + A_{− \frac{1}{2}} u & = & − β γ′_{0} γ_{π} u  , t > 0 , \\ u(0) & = & γ′_{π}(1) = δ_{π} . \end{aligned}  (29)

We have thus chosen a unit heat pulse at x = π as the initial condition. We proved that the solution of (29) looses positivity when β > 0 . The initial condition δ_{π} is extreme in the sense that it manipulates the controller of the thermostat to cool at x = 0 by heating the sensor at x = π, even though the temperature is zero almost everywhere. To observe the loss of positivity at β = 0 in numerical experiments it is thus advisable to start close to the initial condition δ_{π}.

Acknowledgements

The first author would like to aknowledge the support of the Swiss National Science Foundation. The second author is very grateful to Michael Grinfeld and Sean McKee for several motivating discussions and helpful remarks. The work of the second author was supported by the European Union TMR network “Differential Equations in Industry and Commerce”.

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