On the generation of C_{0}-semigroups in L^{1}(I) ^{∗}

Antonio Attalienti ^{†}, Michele Campiti ^{‡}

Abstract

In this paper we characterize completely the existence of a C_{0}-semigroup in L^{1}(I) (I real interval) generated by a second-order differential operator when suitable boundary conditions at the endpoints are imposed. In spaces of continuous functions similar characterizations have been obtained by Timmermans [6] on the maximal domain, by Clément and Timmermans [2] in the case of Ventcel’s boundary conditions and by Campiti, Metafune and Pallara [1] in the case of Neumann’s boundary conditions.

1 Introduction and preliminaries

Let I =]r_{1}, r_{2}[ (−∞ ≤ r_{1} < r_{2} ≤ +∞) be a real interval and consider the second-order differential operator

Bu(x) = α(x)u′′(x) + β(x)u′(x) ,  x ∈ I ,  (1.1)

where α, β : I → R are continuous functions and α(x) > 0 for every x ∈ I.

In [2] Clément and Timmermans gave necessary and sufficient conditions in order for B to be the generator of a C_{0}-semigroup in C(\overline{I}) on the domain

D_{V}(B) := \{u ∈ C(\overline{I}) ∩ C^{2}(I) | lim_{x→r_{1},r_{2}} Bu(x) = 0\} ,  (1.2)

where the corresponding boundary conditions are known as Ventcel’s boundary conditions.

A further characterization for B to be a generator when considered as an operator acting on its maximal domain

D_{Max}(B) := \{u ∈ C(\overline{I}) ∩ C^{2}(I) | lim_{x→r_{1},r_{2}} Bu(x) exist finite\}  (1.3)
= \{u ∈ C(\overline{I}) ∩ C^{2}(I) | Bu ∈ C(\overline{I})\}

is contained in Timmermans [6].

More recently, the existence of a C_{0}-semigroup has been characterized in [1] also in the case of the following Neumann’s domain

D_{N}(B) := \{u ∈ D_{Max}(B) | lim_{x→r_{1},r_{2}} \frac{u′(x)}{W (x)} = 0\}.

∗Work performed under the auspices of G.N.A.F.A. (C.N.R.) and M.U.R.S.T.

†Institute of Financial Mathematics, University of Bari, Via C. Rosalba 53, 70124 Bari, Italy.

‡Department of Mathematics, Polytechnic of Bari, Via E. Orabona 4, 70125 Bari, Italy.

1E-mails: campiti@dm.uniba.it, attalienti@matfin.uniba.it.

AMS Classification: 47D05, 34B05.

Keywords: Generators of Semigroups – Second-Order Differential Operators – Boundary Conditions.

However, all the results obtained in [2], [6] and [1] are very closely related to the pioneer work by Feller [4], but they have the advantage of being very simple to handle in practical applications, when the matter is to state whether B, on a given domain, is a generator.

In order to state the main characterizations, we fix x_{0} ∈ I and define, for every x ∈ I,

W (x) := exp (− ∫_{x_{0}}^{x} \frac{β(t)}{α(t)} dt)  (1.4)

and

Q(x) := \frac{1}{α(x)W (x)} ∫_{x_{0}}^{x} W (s) ds , R(x) := W (x) ∫_{x_{0}}^{x} \frac{1}{α(s)W (s)} ds .  (1.5)

Incidentally observe that the operator (1.1) can be written in the form Bu = αW (u′/W)′ for every u ∈ D_{Max}(B).

For the sequel it is also useful to set

I_{1} :=]r_{1}, x_{0}] , I_{2} := [x_{0}, r_{2}[ .

We recall that, according to the classification introduced by Feller [4], the endpoint r_{i}, i = 1, 2 is said to be

a regular boundary  if Q ∈ L^{1}(I_{i}) , R ∈ L^{1}(I_{i}) ;
an exit boundary  if Q ∉ L^{1}(I_{i}) , R ∈ L^{1}(I_{i}) ;
an entrance boundary  if Q ∈ L^{1}(I_{i}) , R ∉ L^{1}(I_{i}) ;
a natural boundary  if Q ∉ L^{1}(I_{i}) , R ∉ L^{1}(I_{i}) .

(1.6)

In terms of the above classification, the main result in [2] states that (B, D_{V}(B)) generates a C_{0}-semigroup of positive contractions in C(\overline{I}) if and only if both the endpoints are not entrance boundaries. Analogously, according to [6] the operator (B, D_{Max}(B)) generates a C_{0}-semigroup of positive contractions in C(\overline{I}) if and only if both the endpoints are entrance or natural boundaries.

At last, as regards Neumann’s domain, in [1] it is proved that the operator (B, D_{N}(B)) generates a C_{0}-semigroup if and only if, for every i = 1, 2,

\frac{1}{αW} ∉ L^{1}(I_{i}) ⟹ W ∉ L^{1}(I_{i}) , sup_{x∈I_{i}} |∫_{x}^{σ(x)} \frac{1}{α(t)W (t)} dt| < +∞,

where for every x ∈ I_{i}, σ(x) ∈ I_{i} is defined by |∫_{x}^{σ(x)} W (t) dt| = 1.

The aim of this paper is to state the analogous of the above results in the space L^{1}(I). Also in this case, the paper by Feller [4] contains a general treatment of the problem, but it is mainly concerned with the most general admissible boundary conditions under which B generates a C_{0}-semigroup.

On the contrary, our aim is to fix some particular boundary conditions and give characterizations of the existence of semigroups on the corresponding domains. Of course, the arguments used in the paper of Feller are decisive for the sequel and will be intensively used.

We shall denote by L_{loc}^{1}(I) the space of all Lebesgue measurable functions on I which are integrable over compact subsets of I and by AC(I) (respectively, AC_{loc}(I)) the space of all continuous functions on I which are absolutely continuous on I (respectively, on compact subsets of I).

Henceforth we shall be concerned with the operator A adjoint of B, i.e., with the operator

Av(x) := \frac{d}{dx} ((αv)′(x) − β(x)v(x)) (= \frac{d}{dx} (\frac{(αW v)′(x)}{W (x)})) ,  (1.7)

which is meaningful for any function v ∈ L_{loc}^{1}(I) such that αv ∈ AC_{loc}(I) and (αv)′ − βv ∈ AC_{loc}(I); as before, α, β are continuous functions on I and α(x) > 0 for every x ∈ I. The last equality in (1.7) is a consequence of the differentiability of W.

In the sequel, we shall denote again by W, Q and R the functions defined in (1.4) and (1.5) and, according to [4], we use the same classification given in (1.6) for the boundary points r_{1} and r_{2} relative to A. Nevertheless, it is useful to observe that this classification may be different when considering a second-order differential operator in C(I) rather than in L_{loc}^{1}(I), as pointed out in the following remark.

Remark 1.1 Assume that α ∈ C^{1}(I). It is easy to verify that the function Q (respectively, R) relative to the operator A in L^{1}(I) coincides with the function R (respectively, Q) relative to the operator

Cu(x) = α(x)u′′(x) + (β(x) − α′(x))u′(x)

in C(I) (u ∈ C(\overline{I}) ∩ C^{2}(I)).

Hence, r_{1} or r_{2} is a regular or a natural boundary point for A if and only if it is a regular or a natural boundary point for C, as well as r_{1} or r_{2} is an exit (respectively, an entrance) boundary point for A if and only if it is an entrance (respectively, an exit) boundary point for C.

Let λ > 0; there is a strong link between the sets of all the solutions of the homogeneous equation Bu = λu (u ∈ C(I)) and

Av = λv ,  v ∈ L_{loc}^{1}(I) .  (1.8)

Really, if we denote these sets by U and V respectively, then the map ϕ : U → V defined by ϕ(u) := u/(αW) is a linear bijection (see [4, Lemma 10.4] for more details).

Moreover, observe that every function v ∈ V is also continuous with αv ∈ C^{1}(I), (αv)′ − βv ∈ C^{1}(I) and αW v ∈ C^{2}(I) (indeed, the first relation follows from (αv)′ = βv + λw, w being an antiderivative of v, and the remaining ones from (αv)′ − βv = λw and (aW v)′ = W ((αv)′ − βv)).

The maximal domain of A consists of all functions v ∈ L^{1}(I) such that Av ∈ L^{1}(I). Since the integrability of Av is equivalent to the existence in R of the limits

n_{i}(v) := lim_{x→r_{i}}((αv)′(x) − β(x)v(x)) (= lim_{x→r_{i}} \frac{(αW v)′(x)}{W (x)})  (1.9)

for i = 1, 2, we can write down the maximal domain in different ways as follows

D_{Max}(A) := {v ∈ L^{1}(I) | Av ∈ L^{1}(I)}  (1.10)
= {v ∈ L^{1}(I) | αv ∈ AC_{loc}(I), (αv)′ − βv ∈ AC(I)}
= {v ∈ L^{1}(I) | αv, (αv)′ − βv ∈ AC_{loc}(I),
the limits n_{1}(v), n_{2}(v) exist finite} .

Before imposing boundary conditions, we observe that for every u ∈ D_{Max}(B), v ∈ D_{Max}(A) and x, y ∈ I, integrating twice by parts gives

∫_{x}^{y} Av(t)u(t)dt = [((αv)′ − βv)(t)u(t)]_{x}^{y}

− [(αW v)(t)\frac{u′}{W} (t)]_{x}^{y} + ∫_{x}^{y} v(t)Bu(t)dt

and therefore at the endpoint r_{i} the adjoint Dirichlet’s (or, respectively, Neumann’s) boundary condition is obtained by setting the following limit

d_{i}(v) := lim_{x→r_{i}}(αW v)(x) ,  i = 1, 2  (1.11)

(or, respectively, the limit n_{i}(v)) equal to 0.

Summing up, besides the maximal domain, we shall consider in a natural way also the following different domains of A:

1) The Dirichlet’s adjoint domain

D_{D}(A) := {v ∈ D_{Max}(A) | d_{1} = d_{2} = 0} .  (1.12)

2) The Neumann’s adjoint domain

D_{N}(A) := {v ∈ L^{1}(I) | αv ∈ AC_{loc}(I), (αv)′ − βv ∈ AC_{0}(I)}
= {v ∈ D_{Max}(A) | n_{1}(v) = n_{2}(v) = 0} ,  (1.13)

where AC_{0}(I) denotes the space of all functions v ∈ AC(I) such that n_{1}(v) = n_{2}(v) = 0.

As already announced, our aim is to state necessary and sufficient conditions under which the operator A generates a C_{0}-semigroup on each of the above domains.

We begin our analysis by stating some preliminary general properties. Since the discussion about the invertibility of λ − A is based on the properties of the solutions of the homogeneous equation, we collect them in the following lemma, whose proof follows from the results in [4, Sects. 10–11].

Lemma 1.2 The following properties hold true:

1) The boundary r_{2} is regular if and only if there exist two independent positive continuous solutions v_{2} and v_{3} of (1.8) satisfying

v_{i} ∈ L^{1}(x_{0}, r_{2}) , αW v_{i} decreasing, αv_{i} ∈ C^{1}(I), i = 2, 3

and, in addition, d_{2}(v_{2}) = 0, n_{2}(v_{2}) = −1 and d_{2}(v_{3}) = 1, n_{2}(v_{3}) = 0. Moreover, there exists a positive continuous solution v_{1} of (1.8) such that αW v_{1} is increasing. Finally, any solution v of (1.8) is in C(I) ∩ L^{1}(x_{0}, r_{2}), αW v is bounded on [x_{0}, r_{2}[, αv ∈ C^{1}(I) and n_{2}(v) exists finite; for every positive solution v of (1.8) we have also n_{2}(v) ≤ 0 if and only if v = c_{2}v_{2} + c_{3}v_{3} with c_{2}, c_{3} ≥ 0.

2) The boundary r_{2} is exit if and only if there exists a positive continuous solutions v_{2} of (1.8) satisfying

v_{2} ∈ L^{1}(x_{0}, r_{2}) , αW v_{2} decreasing, αv_{2} ∈ C^{1}(I)

and, in addition, d_{2}(v_{2}) = 0, n_{2}(v_{2}) = −1. Moreover, there exists a positive continuous solution v_{1} of (1.8) which is not in L^{1}(x_{0}, r_{2}) and such that αW v_{1} is increasing and satisfies d_{2}(v_{1}) < +∞, n_{2}(v_{1}) = +∞. A solution v of (1.8) is in L^{1}(x_{0}, r_{2}) if and only if v is a multiple of v_{2}.

3) The boundary r_{2} is entrance if and only if there exists a positive continuous solution v_{2} of (1.8) satisfying

v_{2} ∈ L^{1}(x_{0}, r_{2}) , αW v_{2} decreasing, αv_{2} ∈ C^{1}(I)

and, in addition, d_{2}(v_{2}) = 1, n_{2}(v_{2}) = 0. Moreover, there exists a positive continuous solution v_{1} of (1.8) which is in L^{1}(x_{0}, r_{2}) and such that αW v_{1} is increasing and satisfies d_{2}(v_{1}) = +∞, n_{2}(v_{1}) < +∞. For any solution v of (1.8), we have v ∈ L^{1}(x_{0}, r_{2}) and n_{2}(v) exists finite.

4) The boundary r_{2} is natural if and only if there exists a positive continuous solution v_{2} of (1.8) satisfying

v_{2} ∈ L^{1}(x_{0}, r_{2}) , αW v_{2} decreasing, αv_{2} ∈ C^{1}(I)

and, in addition, d_{2}(v_{2}) = 0, n_{2}(v_{2}) = 0. Moreover, there exists a positive continuous solution v_{1} of (1.8) which is not in L^{1}(x_{0}, r_{2}) and such that αW v_{1} is increasing and satisfies d_{2}(v_{1}) = +∞, n_{2}(v_{1}) = +∞. A solution v of (1.8) is in L^{1}(x_{0}, r_{2}) if and only if v is a multiple of v_{2}.

Obviously, the properties at the endpoint r_{1} are at all similar (with increasing replaced by decreasing).

As a consequence of the above properties, for every positive continuous solution of (1.8) such that αW v is increasing, condition v ∈ L^{1}(x_{0}, r_{2}) also implies Q ∈ L^{1}(x_{0}, r_{2}).

By using the functions v_{1} and v_{2} in the previous lemma, we consider the Green function Γ : I × I → R defined by

Γ(x, s) := \{ \begin{matrix}W_{0} α(s)W (s) v_{1}(s)v_{2}(x) , & x ≤ s, \\ W_{0} α(s)W (s) v_{1}(x)v_{2}(s) , & x ≥ s, \end{matrix} x, s ∈ I,

where W_{0} := 1/[(α(v_{2} − v_{1}))′(x_{0}) − β(x_{0})(v_{2} − v_{1})(x_{0})]; as stated in [4, (12.3)], we have v_{1} ∈ L^{1}(r_{1}, x_{0}), v_{2} ∈ L^{1}(x_{0}, r_{2}) and

n_{1}(v_{1}) ≥ 0 , n_{2}(v_{2}) ≤ 0 .  (1.14)

Now we may consider the operator L_{λ} : L^{1}(I) → L^{1}(I) defined by

L_{λ}g(x) := ∫_{r_{1}}^{r_{2}} Γ(x, s)g(s) ds , g ∈ L^{1}(I), x ∈ I .

The main properties of L_{λ} are collected in the following proposition (see [4, Sect. 13]).

Proposition 1.3 The operator L_{λ} is positive linear and satisfies

∥L_{λ}g∥_{1} ≤ \frac{1}{λ}∥g∥_{1} , g ∈ L^{1}(I) .

Moreover, for every g ∈ L^{1}(I), L_{λ}g is a particular solution in L^{1}(I) of the inhomogeneous equation

(λ − A)v = g .  (1.15)

The general solution of (1.15) is given by

G(x) = L_{λ}g(x) + c_{1} v_{1}(x) + c_{2} v_{2}(x) ,  x ∈ I , c_{1}, c_{2} ∈ R .  (1.16)

Since for every g ∈ L^{1}(I) the function L_{λ}g is itself in L^{1}(I), from the equation A(L_{λ}g) = λL_{λ}g − g we deduce that A(L_{λ}g) ∈ L^{1}(I) and that the limits n_{i}(L_{λ}g), i = 1, 2, both exist and are finite.

We end this section with some properties of the above limits; for the sake of brevity we shall refer only to n_{2}(L_{λ}g), the case of n_{1}(L_{λ}g) being analogous.

Lemma 1.4 In each of the following cases

1) 1/(αW) ∈ L^{1}(x_{0}, r_{2}) and Q ∉ L^{1}(x_{0}, r_{2}),

2) 1/(αW) ∉ L^{1}(x_{0}, r_{2}) and W ∉ L^{1}(x_{0}, r_{2}),

3) R ∉ L^{1}(x_{0}, r_{2}) and W ∈ L^{1}(x_{0}, r_{2}),

we have n_{2}(L_{λ}g) = 0 whenever g ∈ L^{1}(I).

Proof Let g ∈ L^{1}(x_{0}, r_{2}). If n_{2}(L_{λ}g) ≠ 0, then (αW L_{λ}g)′(x) ∼ n_{2}(L_{λ}g)W (x) as x → r_{2}. Therefore, integrating between x_{0} and x gives (αW L_{λ}g)(x) ∼ c + n_{2}(L_{λ}g) ∫_{x_{0}}^{x} W (t)dt as x → r_{2} whence

∫_{x_{0}}^{x} L_{λ}g ∼ ∫_{x_{0}}^{x} \frac{1}{α(t)W (t)} (c + n_{2}(L_{λ}g) ∫_{x_{0}}^{t} W (s)ds) dt ,  (1.17)

where c := (αW L_{λ}g)(x_{0}). Under the assumptions in 1), the integrability of L_{λ}g yields a contradiction in (1.17) and therefore n_{2}(L_{λ}g) = 0.

The assumptions in 2) imply that the function Q is not integrable on I_{2}, as well; moreover, on account of (1.17), for every g ∈ L^{2}(I), since L_{λ}g is integrable, 1/(αW) and Q must satisfy

lim_{x→r_{2}} \frac{c ∫_{x_{0}}^{x} 1/(α(t)W (t))dt}{∫_{x_{0}}^{x} Q(t) dt} = −n_{2}(L_{λ}g) .

Applying l’Hôpital’s rule, we get

lim_{x→r_{2}} \frac{c/(α(x)W (x))}{1/(α(x)W (x)) ∫_{x_{0}}^{x} W (t) dt} = lim_{x→r_{2}} c/ ∫_{x_{0}}^{x} W (t) dt = −n_{2}(L_{λ}g) ,

and thus necessarily n_{2}(L_{λ}g) = 0 because W ∉ L^{1}(I_{2}).

As for the last case 3), observe that again 1/(αW) ∉ L^{1}(I_{2}) (otherwise R ∈ L^{1}(I_{2}) as well, because W ∈ L^{1}(I_{2})). If n_{2}(L_{λ}g) ≠ 0 for some g ∈ L^{1}(I), from

(αW L_{λ}g)′(x) ∼ n_{2}(L_{λ}g)W (x) ,  x → r_{2}  (1.18)

and the integrability of W we deduce that the limit d_{2}(L_{λ}g) exists and is finite. Moreover, integrating (1.18) between x and r_{2} gives

d_{2}(L_{λ}g) − (αW L_{λ}g)(x) ∼ n_{2}(L_{λ}g) ∫_{x}^{r_{2}} W (t) dt , x → r_{2} ,

i.e.,

L_{λ}g(x) ∼ \frac{1}{α(x)W (x)} (d_{2}(L_{λ}g) − n_{2}(L_{λ}g) ∫_{x}^{r_{2}} W (t) dt) x → r_{2} .  (1.19)

Since L_{λ}g ∈ L^{1}(x_{0}, r_{2}) and 1/(αW) ∉ L^{1}(x_{0}, r_{2}), from (1.19) it follows that

lim_{x→r_{2}} (d_{2}(L_{λ}g) − n_{2}(L_{λ}g) ∫_{x}^{r_{2}} W (t) dt) = 0

and therefore d_{2}(L_{λ}g) = 0, because W ∈ L^{1}(x_{0}, r_{2}).

Then (1.19) reduces to

L_{λ}g(x) ∼ −\frac{n_{2}(L_{λ}g)}{α(x)W (x)} ∫_{x}^{r_{2}} W (t) dt , x → r_{2},

which contradicts the integrability of L_{λ}g, because 1/(αW) ∉ L^{1}(I_{2}), as already observed. Therefore, also in this case, n_{2}(L_{λ}g) = 0 for any g ∈ L^{1}(I).

On account of the above lemma, if r_{2} is a natural boundary, then n_{2}(L_{λ}g) = 0 for every g ∈ L^{1}(I) (the proof shows, indeed, that n_{2}(v) = 0 for every v ∈ L^{1}(I) such that Av ∈ L^{1}(x_{0}, r_{2})).

2 Generation on the maximal domain

First we consider the case of the maximal domain for the operator A as defined in (1.10). We begin with the following preliminary properties of (A, D_{Max}(A)).

Lemma 2.1 The operator (A, D_{Max}(A)) is closed and densely defined on L^{1}(I).

Proof Since D_{Max}(A) contains the space C_{0}^{∞}(I) which is dense in L^{1}(I), it follows that (A, D_{Max}(A)) is densely defined on L^{1}(I).

Now, let (v_{n})_{n≥1} be a sequence of functions in D_{Max}(A) converging to a function v ∈ L^{1}(I) in the L^{1}-norm and such that the sequence (Av_{n})_{n≥1} converges to w ∈ L^{1}(I).

This last property implies that

lim_{n→+∞} (\frac{(αW v_{n})′(x)}{W (x)} − (αW v_{n})′(x_{0})) = ∫_{x_{0}}^{x} w(t) dt  (2.1)

uniformly with respect to x ∈ I. Next, we show that the limit of (αW v_{n})′(x_{0}) as n → +∞ exists and is finite. Otherwise, a constant ε_{0} > 0 and a subsequence (v_{k(n)})_{n≥1} can be found such that

|(αW v_{k(n+1)})′(x_{0}) − (αW v_{k(n)})′(x_{0})| ≥ ε_{0}

for every n ≥ 1; moreover, in view of (2.1) and taking into account that lim_{x→x_{0}} W (x) = 1, we can choose ε_{1} > 0, a point x_{1} ∈ I, with x_{0} < x_{1}, and an integer ν ≥ 1 satisfying

|(αW v_{k(n+1)})′(x) − (αW v_{k(n)})′(x)| ≥ ε_{1}

for every x ∈ [x_{0}, x_{1}] and n ≥ ν. Integrating the above last relation, since αW is bounded and strictly positive on [x_{0}, x_{1}], for a suitable constant ε_{2} > 0 we get |(v_{k(n+1)} − v_{k(n)})(x) − (v_{k(n+1)} − v_{k(n)})(x)| ≥ ε_{2}(x − x_{0}) for every x ∈ [x_{0}, x_{1}] and n ≥ ν. This contradicts the fact that (v_{n})_{n≥1} converges in L^{1}([x_{0}, x_{1}]). Thus, the limit ℓ_{0} := lim_{n→+∞}(αW v_{n})′(x_{0}) exists finite and, accordingly, formula (2.1) becomes

lim_{n→+∞}((αv_{n})′(x) − βv_{n})(x) = lim_{n→+∞} \frac{(αW v_{n})′(x)}{W (x)} = ℓ_{0} + ∫_{x_{0}}^{x} w(t) dt  (2.2)

uniformly with respect to x ∈ I.

On every compact subinterval J of ]r_{1}, r_{2}[, we have

lim_{n→+∞}(αW v_{n})′(x) = W (x) (ℓ_{0} + ∫_{x_{0}}^{x} w(t) dt)

uniformly in J and consequently αW v ∈ AC_{loc}(I); therefore we also have αv, (αv)′ − βv ∈ AC_{loc}(I). The relation Av = w easily follows from (2.2) as well as the existence and finiteness of the limits (1.9), since w ∈ L^{1}(I).

From the above lemma it follows that, in order for (A, D_{Max}(A)) to be the generator of a C_{0}-semigroup, we have only to check the dissipativity of A and the invertibility of λ − A for λ > 0.

This is characterized in the following main result of this section.

Theorem 2.2 The following statements are equivalent:

a) The operator (A, D_{Max}(A)) generates a C_{0}-semigroup in L^{1}(I);

b) The endpoints r_{1} and r_{2} are exit or natural boundary points, i.e., Q ∉ L^{1}(r_{1}, x_{0}) and Q ∉ L^{1}(x_{0}, r_{2}).

Moreover the semigroup is positive and contractive.

Proof If one of the endpoints, say r_{2}, is neither an exit nor a natural boundary point, from Lemma 1.2 we deduce that λ − A is not injective since v_{1} ∈ L^{1}(I) and therefore the inhomogeneous equation (λ − A)v = g has infinitely many solutions in L^{1}(I) given by v = L_{λ}g + c_{1}v_{1}, c_{1} ∈ R.

So, assume that both the endpoints are exit or natural boundaries. Lemma 1.2 ensures that v_{1} ∉ L^{1}(I_{2}), v_{2} ∉ L^{1}(I_{1}) and therefore, for every g ∈ L^{1}(I), the unique solution in L^{1}(I) of (λ − A)v = g is L_{λ}g; thus λ − A is invertible and its inverse is the operator L_{λ}. From Proposition 1.3, we deduce the dissipativity of A and hence, on account of the classical Lumer-Phillips generation theorem (see, e.g., [5, Theorem 4.3, p. 14]), we conclude that the operator (A, D_{Max}(A)) generates a C_{0}-semigroup in L^{1}(I).

Finally, for every λ > 0, the inverse L_{λ} of λ − A is positive and satisfies ∥L_{λ}∥ ≤ 1/λ (see Proposition 1.3); hence ∥((n/t)R(n/t, A))^{n}∥ ≤ 1 for every n ≥ 1; therefore the semigroup is contractive and positive because e^{tA} = lim_{n→+∞}((n/t)R(n/t, A))^{n}.

We observe that whenever (A, D_{Max}(A)) generates a C_{0}-semigroup on L^{1}(I), for every g ∈ L^{1}(I), the unique solution L_{λ}g of (1.15) can be also characterized as the only one for which the limits (1.9) both exist and are finite (see Lemma 1.2).

Finally, if α ∈ C^{1}(I) then, on account of Remark 1.1, the operator (A, D_{Max}(A)) generates a strongly continuous semigroup in L^{1}(I) if and only if the operator (C, D_{Max}(C)) generates a strongly continuous semigroup in C(\overline{I}) (where D_{Max}(C) is defined according to (1.3)).

3 Adjoint Neumann’s boundary conditions

In this section we consider the boundary conditions

n_{1}(v) = n_{2}(v) = 0 ,  v ∈ D_{Max}(A)  (3.1)

and the corresponding domain defined in (1.13).

Some preliminary properties of (A, D_{N}(A)) are indicated in the next Lemma.

Lemma 3.1 The operator (A, D_{N}(A)) is closed and densely defined on L^{1}(I). Moreover, for every v ∈ D_{N}(A) and λ > 0

λ∥v∥_{1} ≤ ∥λv − Av∥_{1}  (3.2)

and therefore (A, D_{N}(A)) is dissipative. Finally, if (λ − A)v is positive for some λ > 0, then v is positive.

Proof As before, D_{N}(A) contains the space C_{0}^{∞}(I) and therefore it is dense in L^{1}(I).

If (v_{n})_{n≥1} is a sequence of functions in D_{N}(A) converging to a function v ∈ L^{1}(I) in the L^{1}-norm and if (Av_{n})_{n≥1} converges to w ∈ L^{1}(I), on account of Lemma 2.1 we have v ∈ D_{Max}(A) and Av = w. Consequently, (αv)′ − βv is absolutely continuous.

In order to obtain v ∈ D_{N}(A), we have to show that v satisfies (3.1). Indeed, since lim_{n→+∞} Av_{n} = Av in L^{1}(I) and v_{n} ∈ D_{N}(A) for every n ≥ 1, we have

lim_{n→+∞} \frac{(αW v_{n})′}{W} (x) = \frac{(αW v)′}{W} (x) − n_{1}(v)

uniformly with respect to x ∈ I; letting x → r_{1} gives n_{1}(v) = 0; the proof at the endpoint r_{2} is similar.

Finally, we prove (3.2). Let v ∈ D_{N}(A) and λ > 0; since v is locally absolutely continuous, we can divide the interval I into at most a countable number of subintervals J_{n} and K_{n} with pairwise disjoint interiors such that v is positive on every J_{n}, negative on every K_{n} and vanishing at the endpoints of J_{n} and K_{n} different from r_{1} and r_{2}. If J_{n} = [a, b] ⊂]r_{1}, r_{2}[, the function αv is positive on J_{n} and (αv)(a) = (αv)(b) = 0; consequently, (αv)′(a) ≥ 0, (αv)′(b) ≤ 0 and hence ∫_{J_{n}} Av = (αv)′(b) − (αv)′(a) ≤ 0. It follows that

λ ∫_{J_{n}} |v| = ∫_{J_{n}} λv ≤ ∫_{J_{n}}(λv − Av) ≤ ∫_{J_{n}} |λv − Av|.

If K_{n} = [c, d] ⊂]r_{1}, r_{2}[, the function αv is negative on K_{n} and again (αv)(c) = (αv)(d) = 0; as before, we have (αv)′(c) ≤ 0, (αv)′(d) ≥ 0 and ∫_{J_{n}} Av = (αv)′(d) − (αv)′(c) ≥ 0. In this case

λ ∫_{K_{n}} |v| = − ∫_{K_{n}} λv ≤ − ∫_{K_{n}}(λv − Av) ≤ ∫_{K_{n}} |λv − Av|.

Taking into account the boundary conditions (3.1), a similar estimate is still true if one or both the endpoints of an interval J_{n} or K_{n} coincides with r_{1} or r_{2}. Thus, we conclude that

λ∥v∥_{1} = ∑_{n} ∫_{J_{n}} |λv| + ∑_{n} ∫_{K_{n}} |λv|

≤ ∑_{n} ∫_{J_{n}} |λv − Av| + ∑_{n} ∫_{K_{n}} |λv − Av| = ∥λv − Av∥_{1}

which proves (3.2).

Now assume that there exists v ∈ D_{N}(A) such that (λ − A)v is positive for some λ > 0 and, nevertheless, v is not positive. Since v is continuous, we can choose at least one of the above intervals K_{n} where ∫_{K_{n}} v < 0; hence, from the previous inequalities, ∫_{K_{n}}(λ − A)v ≤ ∫_{K_{n}} v < 0, which is a contradiction.

The proof of the dissipativity of (A, D_{N}(A)) also shows that, for every v ∈ D_{N}(A), ∫_{r_{1}}^{r_{2}} Av(t) dt = 0.

Since the dissipativity of (A, D_{N}(A)) has been already established, in this case the generation of a C_{0}-semigroup in L^{1}(I) depends only on the surjectivity of λ − A. This is investigated in the following main result.

Theorem 3.2 The following statements are equivalent:

a) The operator (A, D_{N}(A)) generates a C_{0}-semigroup in L^{1}(I);

b) The endpoints r_{1} and r_{2} are not exit boundary points, i.e.,

Q ∈ L^{1}(I_{i}) or R ∉ L^{1}(I_{i}) ,  i = 1, 2 .  (3.3)

Moreover, the semigroup is positive and contractive.

Proof If both the endpoints r_{1} and r_{2} are entrance or regular boundaries, Lemma 1.2 implies that the limits n_{i}(v_{j}), i, j = 1, 2, exist and are finite. If g ∈ L^{1}(I), we can also consider the limits n_{i}(L_{λ}g).

Note that a general solution v = L_{λ}g+c_{1}v_{1}+c_{2}v_{2} of (1.15) satisfies (3.1) (i.e., v ∈ D_{N}(A)) if and only if the linear system

\{ \begin{matrix}n_{1}(L_{λ}g) + c_{1}n_{1}(v_{1}) + c_{2}n_{1}(v_{2}) = 0 , \\ n_{2}(L_{λ}g) + c_{1}n_{2}(v_{1}) + c_{2}n_{2}(v_{2}) = 0 \end{matrix}

has at least a solution.

Really, we prove that the above system has exactly a unique solution by showing that det(n_{i}(v_{j}))_{i,j=1,2} ≠ 0. Indeed, integrating (1.8) gives n_{2}(v_{j}) − n_{1}(v_{j}) = λ ∫_{r_{1}}^{r_{2}} v_{j}, j = 1, 2 and therefore

| \begin{matrix}n_{1}(v_{1}) & n_{1}(v_{2}) \\ n_{2}(v_{1}) & n_{2}(v_{2}) \end{matrix} | = | \begin{matrix}n_{1}(v_{1}) & n_{2}(v_{2}) − λ ∫_{r_{1}}^{r_{2}} v_{2} \\ n_{1}(v_{1}) + λ ∫_{r_{1}}^{r_{2}} v_{1} & n_{2}(v_{2}) \end{matrix} |
= λn_{1}(v_{1}) ∫_{r_{1}}^{r_{2}} v_{2} − λ ∫_{r_{1}}^{r_{2}} v_{1} (n_{2}(v_{2}) − λ ∫_{r_{1}}^{r_{2}} v_{2}) .

Taking into account that (see (1.14))

n_{1}(v_{1}) ≥ 0 , n_{2}(v_{2}) ≤ 0 , ∫_{r_{1}}^{r_{2}} v_{1} > 0 , ∫_{r_{1}}^{r_{2}} v_{2} > 0 ,

we have

n_{1}(v_{1}) ∫_{r_{1}}^{r_{2}} v_{2} ≥ 0 , n_{2}(v_{2}) − λ ∫_{r_{1}}^{r_{2}} v_{2} < 0

and hence det(n_{i}(v_{j}))_{i,j=1,2} > 0.

Thus we conclude that in this case the operator λ − A is surjective.

If both the endpoints are natural, in view of Lemma 1.4, n_{1}(L_{λ}g) = n_{2}(L_{λ}g) = 0 for every g ∈ L^{1}(I) and therefore L_{λ}g ∈ D_{N}(A); since L_{λ}g is a solution of (1.15), the surjectivity of λ − A is true also in this case.

If the endpoint r_{1} is natural and r_{2} is entrance or regular, then for every g ∈ L^{1}(I) n_{1}(L_{λ}g) = 0 (see Lemma 1.4) and v_{1} ∈ L^{1}(I); consequently, there exist infinitely many solutions v ∈ L^{1}(I) of (1.15) given by v = L_{λ}g +c_{1}v_{1}, c_{1} ∈ R. After imposing n_{2}(v) = 0, we get n_{2}(L_{λ}g) + c_{1}n_{2}(v_{1}) = 0; now n_{1}(v_{1}) = 0 (see Lemma 1.2) and the function (αv_{1})′ − βv_{1} is strictly increasing (indeed, its derivative is λv_{1}); thus n_{2}(v_{1}) > 0 and a constant c_{1} may be found such that n_{2}(L_{λ}g) + c_{1}n_{2}(v_{1}) = 0; the corresponding function v = L_{λ}g + c_{1}v_{1} is in D_{N}(A) (in r_{1} the boundary condition is satisfied since n_{1}(v_{1}) = 0) and is a solution of (1.15).

The case r_{2} natural and r_{1} entrance or regular is analogous.

So, in any case, we have shown that condition b) implies the surjectivity of λ − A; since A is dissipative, applying the Lumer-Phillips theorem (see, e.g., [5, Theorem 4.3, p. 14]) we get a).

Conversely, assume that r_{2} is an exit boundary point. As a consequence of Lemma 1.2, v_{1} ∉ L^{1}(I). We show that 1 − A is not surjective. In view of (1.6), let us consider a function g ∈ L^{1}(I) which satisfies

g(x) := \frac{1}{α(x)W (x)} ∫_{x}^{r_{2}} W (t) dt , x ∈ I_{2}[

and coincides with a function w ∈ L^{1}(I) on I_{1}. Then Ag = 0 on I_{2} and hence g is a solution of v − Av = g on the same interval. Since g is integrable, we have L_{1}g = g and moreover n_{2}(L_{1}g) = −1. If r_{1} is exit or natural, L_{1}g is the unique solution in L^{1}(I) of (1.16) and is not in D_{N}(A). Thus, in this case 1 − A is not surjective.

If r_{1} is regular or entrance, then v_{2} ∈ L^{1}(I) with n_{1}(v_{2}) = 0 and all the solutions v ∈ L^{1}(I) of (1.15) are given by v = L_{1}g+c_{2}v_{2}; in addition, n_{1}(v) = n_{1}(L_{1}g), where n_{1}(L_{1}g) is not always equal to 0 for any w ∈ L^{1}(I) (for example, take w(x) := 1/(α(x)W (x)) ∫_{x_{0}}^{x} W).

If r_{1} is exit, the proof is similar.

The last part of the proof directly follows from Lemma 3.1.

Observe that in (3.3), the integrability of Q implies that 1/(αW) is integrable, too, and the converse is also true since if Q and R are both not integrable, then 1/(αW) is itself not integrable. Thus, condition (3.3) can be restated explicitly as follows, for i = 1, 2,

\frac{1}{αW} ∈ L^{1}(r_{i}, x_{0}) or ∫_{x_{0}}^{r_{i}} W (x) ∫_{x_{0}}^{x} \frac{1}{α(t)W (t)} dt dx = +∞.

Moreover, if α ∈ C^{1}(I) then, as a consequence of Remark 1.1, the operator (A, D_{N}(A)) generates a strongly continuous semigroup in L^{1}(I) if and only if the operator (C, D_{V}(C)) generates a strongly continuous semigroup in C(\overline{I}) (here D_{V}(C) is defined by (1.2)).

4 Adjoint Dirichlet’s boundary conditions

In this section we consider the adjoint in the space L^{1}(I) of the Dirichlet’s boundary conditions in C(I), namely we want to characterize the operator (A, D_{D}(A)) as a generator of a C_{0}-semigroup.

We begin with some preliminary properties of (A, D_{D}(A)).

Lemma 4.1 The operator (A, D_{D}(A)) is densely defined and dissipative. Moreover, if (λ − A)v is positive for some λ > 0, then v is positive.

Proof Obviously C_{0}^{∞}(I) ⊂ D_{D}(A) and therefore the operator A is densely defined.

Now, let v ∈ D_{D}(A); since v is locally absolutely continuous, we can divide the interval I into at most a countable number of intervals J_{n} and K_{n}, v being positive on every J_{n} and negative on every K_{n}. If a and b denote the endpoints of a general interval J_{n}, the function αW v is positive on J_{n} and vanishes at a, b (this is obvious if [a, b] ⊂ I, whereas it is a consequence of the Dirichlet’s boundary conditions if one or both the endpoints coincide with the endpoints of I). Hence, (αW v)′(a) ≥ 0, (αW v)′(b) ≤ 0 and therefore

∫_{J_{n}} Av = \frac{(αW v)′}{W} (b) − \frac{(αW v)′}{W} (a) ≤ 0 ,

which implies

λ ∫_{J_{n}} |v| = ∫_{J_{n}} λv ≤ ∫_{J_{n}}(λv − Av) ≤ ∫_{J_{n}} |λv − Av| .

A similar argument is true for the intervals K_{n}, too, and therefore we may conclude that

λ∥v∥_{1} = ∑_{n} ∫_{J_{n}} |λv| + ∑_{n} ∫_{K_{n}} |λv|

≤ ∑_{n} ∫_{J_{n}} |λv − Av| + ∑_{n} ∫_{K_{n}} |λv − Av|
= ∥λv − Av∥_{1}

which proves the dissipativity of (A, D_{D}(A)).

Finally, let v ∈ D_{D}(A) such that (λ − A)v is positive for some λ > 0. As in the proof of Lemma 3.1, if v is not positive, we get a contradiction by considering a maximal interval K where ∫_{K} v < 0 and ∫_{K}(λ − A)v ≤ ∫_{K} v < 0.

The following result states that the closure of (A, D_{D}(A)) is always a generator of a positive C_{0}-semigroup in L^{1}(I).

Proposition 4.2 The closure (\overline{A}, D(\overline{A})) of the operator (A, D_{D}(A)) generates a positive C_{0}-semigroup in L^{1}(I).

Proof On account of Lemma 4.1, we have only to show that the range of λ − A is dense in L^{1}(I) for every λ > 0. To this purpose, it is sufficient to fix a function g ∈ L^{1}(I) which vanishes on some neighborhoods of r_{1} and r_{2} and to find a function v ∈ D_{D}(A) satisfying (1.15). But for such a function g, the solution L_{λ}g of (1.15) is integrable and coincides with a solution of the homogeneous equation on some neighborhoods of r_{1} and r_{2}. Taking into account the integrability of the functions v_{1} and v_{2} (see Lemma 1.2), we deduce that on some neighborhood of r_{1} and r_{2}, we shall have L_{λ}g = k_{1}v_{1} + k_{2}v_{2}, if r_{1} and r_{2} are both regular or entrance boundaries, L_{λ}g = k_{1}v_{1} (respectively, L_{λ}g = k_{2}v_{2}) if r_{1} is an exit or a natural boundary (respectively, a regular or an entrance boundary) and r_{2} is a regular or an entrance boundary (respectively, an exit or a natural boundary) and finally L_{λ}g = 0 if r_{1} and r_{2} are both exit or natural boundaries. It follows that in any case we can find a solution of the form (1.16) which is integrable and satisfies the adjoint Dirichlet’s boundary condition for suitable constants c_{1} and c_{2}.

From the above proposition, it follows that (A, D_{D}(A)) is the generator of a C_{0}-semigroup in L^{1}(I) if and only if it is closed. This is not always true, as shown in the following lemma.

Lemma 4.3 Assume that W ∉ L^{1}(x_{0}, r_{2}). If v ∈ D_{Max}(A) and d_{2}(v) = 0, then n_{2}(v) = 0, too. Conversely, if v ∈ D_{Max}(A) and n_{2}(v) = 0, then there exists a sequence (v_{n})_{n≥1} in D_{Max}(A) converging to v in the graph norm of A and such that d_{2}(v_{n}) = 0 for every n ≥ 1.

Proof Let v ∈ D_{Max}(A) such that d_{2}(v) = 0; if n_{2}(v) ≠ 0, then (αW v)′(x) ∼ n_{2}(v)W (x) as x → r_{2}, which is not possible because (αW v)′ is integrable on some neighborhood on r_{2}, by virtue of the corresponding adjoint Dirichlet’s boundary condition, whereas W is not by assumption. Thus n_{2}(v) = 0.

Next, we show that the set E of all functions in D_{Max}(A) vanishing on a neighborhood of r_{2} is dense for the graph norm in the set F of all functions in D_{Max}(A) satisfying the adjoint Neumann’s boundary condition at r_{2}. Indeed, if v ∈ F and if (b_{n})_{n≥1} is an increasing sequence tending to r_{2}, from the integrability of v and Av it follows that lim_{n→+∞} ∫_{b_{n}}^{r_{2}} |v| = lim_{n→+∞} ∫_{b_{n}}^{r_{2}} |Av| = 0; therefore, we can consider a sequence (φ_{n})_{n≥1} of functions in C_{∞}^{0}(I) such that the support of every φ_{n} is contained in ]r_{1}, b_{n+1}] and φ_{n} = 1 on ]r_{1}, b_{n}]. The sequence (v · φ_{n})_{n≥1} of functions in E converges to v in the graph norm of A, as required.

We can now easily prove the following proposition concerning the case when one of the endpoints is an entrance boundary.

Proposition 4.4 Assume that one of the endpoints, say r_{i}, i = 1, 2, is an entrance boundary point. Then, the operator (A, D_{D}(A)) is not closed and every function in its closure satisfies the adjoint Neumann’s boundary condition at r_{i}. Consequently, (A, D_{D}(A)) cannot generate a C_{0}-semigroup in L^{1}(I).

Proof Arguing at r_{2}, we show that, if r_{2} is an entrance boundary, then 1 − A is not surjective. Consider a function w which coincides with 1/(αW) on I_{2} and is integrable on I_{1} (if r_{1} is a regular or an entrance boundary, we just take w := 1/(αW) on I). In I_{2} we have αW w = 1 and consequently Aw = 0. Then w satisfies the equation v − Av = w in I_{2} and therefore the general solution of this equation in I_{2} is v = w + c_{1}v_{1} + c_{2}v_{2}. Since d_{2}(v_{1}) = +∞ (see Lemma 1.2), the solutions v ∈ L^{1}(I) for which the limit d_{2}(v) is finite are given by v = w + c_{2}v_{2}. If r_{1} is a natural or an exit boundary, the function v_{2} is not integrable on I_{1} and therefore w is the unique integrable solution of v − Av = w in I_{2}, but it does not satisfy the adjoint Dirichlet’s boundary condition at the point r_{2}. If r_{1} is a regular or an entrance boundary, v_{2} is integrable and the unique solution of v − Av = w in I satisfying the adjoint Dirichlet’s boundary condition at r_{2} is v = w − v_{2}; in this case the adjoint Dirichlet’s boundary condition is not satisfied at the point r_{1} since αW v_{2} is decreasing and consequently d_{1}(w − v_{2}) = lim_{x→r_{1}}(αW (w − v_{2}))(x) = 1 − (αW v_{2})(r_{1}) < 0 (the last inequality may be justified by taking the function v_{2} equal to the same v_{2} of Lemma 1.2 if r_{1} is an entrance boundary, and equal to the function v_{3} of Lemma 1.2 if r_{1} is a regular boundary). Thus, in any case, no integrable solution of v − Av = w is in D_{D}(A) and this proves that 1 − A is not surjective. We can conclude that (A, D_{D}(A)) cannot generate a C_{0}-semigroup and therefore it is not closed. The last part follows from Theorem 3.2 and Proposition 4.2, since the adjoint Neumann’s boundary condition ensures the generation and hence the closedness of the operator (as far as concerns the corresponding endpoint).

Now, we can state the main theorem of this section; if W is not integrable at one of the endpoints, say r_{2}, the generation of a C_{0}-semigroup will be shown to depend on the following condition (D)_{2} If J = ⋃_{n=1}^{+∞} J_{n} is a countable union of pairwise disjoint intervals contained in [x_{0}, r_{2}[ and such that sup J_{n} < inf J_{n+1} for every n ≥ 1, then

lim_{n→+∞} ∫_{J_{n}} W = +∞ ⟹ ∫_{J} \frac{1}{αW} = +∞.

The corresponding condition (D)_{1} at the endpoint r_{1} may be stated in just the same way by requiring that the sequence (J_{n})_{n≥1} satisfies sup J_{n+1} < inf J_{n} for every n ≥ 1, every J_{n} being contained in ]r_{1}, x_{0}].

Remark 4.5 1) By considering a subsequence (if necessary), we may and do require that (J_{n})_{n≥1} in (D)_{i} satisfies sup_{n≥1} ∫_{J_{n}} W = +∞.

2) If a subset J satisfies the requirements in (D)_{i}, the endpoint r_{i} is a cluster point for J; indeed, the functions W and 1/(αW) are not integrable only on some neighborhoods of r_{i}.

3) We explicitly observe that condition (D)_{i} implies that 1/(αW) is not integrable on I_{i} because W is not integrable as well.

Theorem 4.6 The operator (A, D_{D}(A)) is the generator of a (positive) C_{0}-semigroup in L^{1}(I) if and only if at each endpoint r_{i}, i = 1, 2, we have

W ∉ L^{1}(I_{i}) ⟹ (D)_{i} is satisfied.  (4.1)

Proof For the sake of brevity, we argue only on a neighborhood of r_{2}; the complete proof can be therefore obtained by combining all the cases discussed at r_{2} with their analogous ones at the endpoint r_{1}.

First, we observe that if W ∈ L^{1}(I_{2}), then the limit d_{2}(v) exists in R for every v ∈ D_{Max}(A). Indeed, let v ∈ D_{Max}(A); since the limit n_{2}(v) does exist, we get (αW v)(x) ∼ (αW v)(x_{0}) + n_{2}(v) ∫_{x_{0}}^{x} W (t) dt which, together with the integrability of W near r_{2}, implies that the limit d_{2}(v) exists and is finite, as well. Moreover, if in addition 1/(αW) ∉ L^{1}(I_{2}), then necessarily d_{2}(v) = 0 for every v ∈ D_{Max}(A) (if, for example, d_{2}(v) > 0, we should have αW v ≥ d_{2}(v)/2 on a neighborhood of r_{2} and therefore v ≥ d_{2}(v)/(2αW) on the same neighborhood, which contradicts the integrability of v).

After these preliminaries, on account of Proposition 4.2, the proof is carried out into the following steps:

1) First, we show that if W ∈ L^{1}(I_{2}), then (A, D_{D}(A)) is closed.

2) Secondly, in the case W ∉ L^{1}(I_{2}), we show that (A, D_{D}(A)) is closed if and only if (D)_{2} holds true at r_{2}.

1) Let (v_{n})_{n≥1} be a sequence in D_{D}(A) converging to v ∈ L^{1}(I) and such that the sequence (Av_{n})_{n≥1} converges to a function w ∈ L^{1}(I) (in the L^{1}-norm). Since the operator (A, D_{Max}(A)) is closed (see Proposition 2.1), we already know that v ∈ D_{Max}(A) and Av = w; hence we have only to show that d_{2}(v) = 0. Taking into account that every v_{n} is in D_{Max}(A) and integrating (1.7) between x and r_{2} we obtain

(αW v_{n})′(x) = n_{2}(v_{n})W (x) − W (x) ∫_{x}^{r_{2}} Av_{n}(t) dt;

a further integration between x_{0} and x gives

(αW v_{n})(x) − (αW v_{n})(x_{0}) = n_{2}(v_{n}) ∫_{x_{0}}^{x} W (t) dt
− ∫_{x_{0}}^{x} W (t) ∫_{t}^{r_{2}} Av_{n}(s) ds dt .

A similar relation holds true for the function v, too; therefore, passing to the limit as x → r_{2}, an easy computation shows that

|d_{2}(v) − d_{2}(v_{n})| ≤ |(αW v)(x_{0}) − (αW v_{n})(x_{0})|
+|n_{2}(v) − n_{2}(v_{n})| ∫_{x_{0}}^{r_{2}} W (t) dt
+ ∫_{x_{0}}^{r_{2}} W (t) |∫_{t}^{r_{2}} w(s) ds − ∫_{t}^{r_{2}} Av_{n}(s) ds| dt.

Now observe that the first term in the above sum tends to 0 as n → +∞ since αW v_{n} is continuous and converges almost everywhere to αW v; the second one tends to 0, too, since D_{Max}(A) is closed and W is integrable. Finally, the last term tends to 0 as well, because of the integrability of W and the uniform convergence of ∫_{t}^{r_{2}} Av_{n}(s) ds to ∫_{t}^{r_{2}} w(s) ds. Thus d_{2}(v) = lim_{n→+∞} d_{2}(v_{n}) = 0, as required.

2) Since W ∉ L^{1}(I_{2}), the endpoint r_{2} must be of an entrance or a natural boundary and therefore the adjoint Neumann’s boundary condition ensures the generation of a C_{0}- semigroup (as far as concernes the endpoint r_{2}). Assume first that 1/(αW) ∈ L^{1}(I_{2}) (this holds true if r_{2} is an entrance boundary). In this case we can choose v ∈ D_{Max}(A) equal to 0 on a neighborhood of r_{1} and to 1/(αW) on a neighborhood of r_{2}. Then Av = 0 near r_{2} but d_{2}(v) = 1 and n_{2}(v) = 0, so that v ∈ D_{N}(A) \ D_{D}(A). Hence, (A, D_{D}(A)) is not closed and (D)_{2} does not hold at r_{2} (see Remark 4.5, 3)).

Now, assume W, 1/(αW) ∉ L^{1}(I_{2}), so that r_{2} is necessarily a natural boundary. Again, on account of Theorem 3.2, we have to show that condition (D)_{2} is equivalent to d_{2}(v) = 0 for every v ∈ D_{N}(A). First, let (D)_{2} be true at r_{2} and suppose that there exists v ∈ D_{Max}(A) such that d_{2}(v) ≠ 0. Considering −v instead of v, if necessary, we may find ε_{0} > 0 and a sequence (t_{m})_{m≥1} tending to r_{2} such that (αW v)(t_{m}) ≥ ε_{0} for every m ≥ 1. On a suitable neighborhood of every t_{m} we have v(x) ≥ ε/(2α(x)W (x)) which is not possible near r_{2} because v ∈ L^{1}(I_{2}) and 1/(αW) ∉ L^{1}(I_{2}). Consequently, we may construct a sequence (J_{n})_{n≥1}, J_{n} = [x_{n}, y_{n}] of pairwise disjoint bounded intervals such that sup J_{n} < inf J_{n+1}, (αW v)(y_{n}) ≥ ε_{0} and (αW v)(x) ≥ ε_{0}/2 for every x ∈ J_{n} and n ≥ 1.

Since αW v ∈ C^{1}(I), for every n ≥ 1 we have

∫_{J_{n}} \frac{(αW v)′}{W} (x) W (x) dx = (αW v)(y_{n}) − (αW v)(x_{n}) ≥ \frac{ε_{0}}{2} .  (4.2)

Set J := ⋃_{n≥1} J_{n}; the function v is integrable on J and therefore the inequalities ∫_{I} |v| ≥ ∫_{J} |v| ≥ (ε_{0}/2) ∫_{J} 1/(αW) yield the integrability of 1/(αW) on J too. From (D)_{2} and the subsequent Remark 4.5, 1), it follows that there exists M > 0 such that ∫_{J_{n}} W ≤ M for every n ≥ 1. Moreover, the function v is in the maximal domain and therefore n_{2}(v) = 0; this implies that ε_{n} := sup_{x∈J_{n}}(αW v)′(x)/W (x) → 0 as n → +∞. Thus

|∫_{J_{n}} \frac{(αW v)′}{W} (x) W (x) dx| ≤ ε_{n} ∫_{J_{n}} W (x) dx ≤ Mε_{n} → 0, n → +∞

in contradiction with (4.2).

Conversely, assume that (D)_{2} is not satisfied and let (J_{n})_{n≥1}, J_{n} = [x_{n}, y_{n}] be a sequence of pairwise disjoint intervals such that y_{n} < x_{n+1} for every n ≥ 1 and lim_{n→+∞} ∫_{J_{n}} W = +∞, ∫_{J} 1/(αW) < +∞, where J = ⋃_{n≥1} J_{n}. Fix a function φ ∈ C^{∞}(R) with support contained in [0, 1], satisfying 0 ≤ φ(x) ≤ 1 and φ(1/2) = 1. For every n ≥ 1, define the function

φ_{n}(x) := φ (∫_{x_{n}}^{x} W / ∫_{x_{n}}^{y_{n}} W ) , x ∈ I .

Then each φ_{n} is in C^{∞}(R) and its support is contained in J_{n}; moreover, each φ_{n} attains the value 1 in at least one point of J_{n}.

Finally, set ψ := ∑_{n≥1} φ_{n} and consider the function v = ψ/(αW) on I. Since the support of ψ is contained in J, we have

∫_{I} |v| = ∫_{J} \frac{ψ}{αW} ≤ ∫_{J} \frac{1}{αW} < +∞.

Moreover, for every n ≥ 1 and x ∈ J_{n},

ψ′(x) = φ′_{n}(x) = φ′ (∫_{x_{n}}^{x} W / ∫_{x_{n}}^{y_{n}} W ) W (x)/ ∫_{x_{n}}^{y_{n}} W

and

\frac{(αW v)′}{W} (x) = \frac{ψ′}{W} (x) = φ′ (∫_{x_{n}}^{x} W / ∫_{x_{n}}^{y_{n}} W ) / ∫_{x_{n}}^{y_{n}} W .

But φ′ is bounded and lim_{n→+∞} ∫_{J_{n}} W = +∞, so that n_{2}(v) = 0.

Thus the function v is in D_{Max}(A), but does not satisfy the adjoint Dirichlet’s boundary condition since αW v = ψ takes the values 0 and 1 on every neighborhood of r_{2}.

Condition (D)_{2} is not automatically satisfied if the functions W and 1/(αW) are not integrable. For example, as for the operator Lv(x) := (αv)′′(x) on the interval ]r_{1}, +∞[, we have W (x) = 1 and (D)_{2} is satisfied if α is bounded, whereas it is not satisfied whenener 1/α is integrable on the union J of a sequence (J_{n})_{n≥1} of pairwise disjoint intervals contained in ]x_{0}, +∞[ (x_{0} ∈]r_{1}, +∞[) and satisfying lim_{n→+∞} |J_{n}| = +∞, but it is not integrable on ]x_{0}, +∞[\J.

Finally, observe that whenever (A, D_{D}(A)) does not generate a C_{0}-semigroup, its closure is always obtained taking an adjoint Neumann’s boundary condition instead of the adjoint Dirichlet’s one at the endpoint(s) which are not regular, exit or natural boundaries (in the natural case, the adjoint Neumann’s boundary condition is, in turn, equivalent to the maximal boundary condition).

5 Conclusions and examples

From the analysis carried out in the last sections, it follows that there is a link between the boundary condition at one of the endpoints and the classification of that same endpoint; this suggests us to consider, again in analogy with the case of the space C(I), different combinations of the boundary conditions for the operator A; for example, we can consider the domain

D_{Max,N}(A) = {v ∈ L^{1}(I) | Av ∈ L^{1}(I), n_{2}(v) = 0} .

As one may expect, in this case (A, D_{Max,N}(A)) generates a C_{0}-semigroup in L^{1}(I) if and only if r_{1} is an exit or natural boundary point and r_{2} is not an exit boundary point. In a similar way, one can define the domains D_{Max,D}(A), D_{N,D}(A) and also D_{N,Max}(A), D_{D,Max}(A), D_{D,N}(A)) and characterize A as a generator on each of them all. The proofs resemble the ones already carried out, so we omit them.

It may be helpful to summarize all the above results in terms of Feller’s classification (1.6) of each endpoint; namely, in order to have the generation of a C_{0}-semigroup, at each boundary r_{i}, i = 1, 2, we may impose the boundary conditions indicated in the following table (maximal means that no boundary condition is imposed at the endpoint).

Boundary  Admissible adjoint boundary condition(s)
regular  Neumann Dirichlet
exit Maximal  Dirichlet
entrance  Neumann
natural Maximal Neumann Dirichlet (if (4.1) holds)

Finally, we consider some examples of particular interest in the applications.

Example 5.1 Let I =]0, 1[ and define φ(x) = x(1 − x) for every x ∈ [0, 1]. Let β ∈ C[0, 1] such that β(0) ≠ 0, β(1) ≠ 0 and consider the differential operator

Av = ((φv)′ − β v)′ , v ∈ D_{Max}(A) .

It is easy to see that W (x) ∼ cx^{−β(0)} at 0 and W (x) ∼ c(1 − x)^{β(1)} at 1; after some straightforward computations, we obtain that R is integrable on ]0, 1/2] if and only if β(0) > 0, as well as Q is integrable on ]0, 1/2] if and only if β(0) < 1. Consequently, 0 is an entrance boundary if β(0) ≤ 0, a regular boundary if 0 < β(0) < 1 and an exit boundary if β(0) ≥ 1. Similarly, 1 is an exit boundary if β(1) ≤ −1, a regular boundary if −1 < β(1) < 0 and an entrance boundary if β(1) ≥ 0.

Accordingly, the generation of a C_{0}-semigroup is ensured if and only if at the point 0 we impose the Neumann’s adjoint boundary condition in the case β(0) ≤ 0, the Neumann’s or Dirichlet’s adjoint boundary condition in the case 0 < β(0) < 1 and the Dirichlet’s adjoint boundary condition or no boundary condition in the case β(0) ≥ 1 and at the point 1 we impose the Dirichlet’s adjoint boundary condition or no boundary condition in the case β(1) ≤ −1, the Dirichlet’s or Neumann’s adjoint boundary condition in the case −1 < β(1) < 0 and the Neumann’s adjoint boundary condition in the case β(1) ≥ 0.

Example 5.2 Consider the adjoint of the Ornstein-Uhlenbeck operator (see, e.g., [3])

Av(x) = v′′(x) − (x v(x))′ , v ∈ D_{Max}(A) , x ∈ R .

Since W (x) = exp(−x^{2}/2) and ∫_{0}^{x} W ∼ − exp(−x^{2}/2)/x as x → ±∞ (as a consequence of the l’Hôpital’s rule), it follows that Q(x) ∼ −1/x and, similarly, R(x) ∼ 1/x as x → ±∞. Thus, ±∞ are both natural boundaries and therefore A is a generator of a C_{0}-semigroup in L^{1}(R) on D_{Max}(A) or on D_{N}(A). Finally, note that (4.1) is satisfied because W ∈ L^{1}(R) and consequently also the Dirichlet’s adjoint boundary conditions at ±∞ are allowed.

Acknowledgement The authors are indebted to Professors G. Metafune and D. Pallara for several discussions on the adjoint domains.

References

[1] M. Campiti, G. Metafune, D. Pallara, One-dimensional Feller semigroups with reflecting barriers, preprint (1998).

[2] Ph. Clément, C. A. Timmermans, On C_{0}-semigroups generated by differential operators satisfying Ventcel’s boundary conditions, Indag. Math. New Ser. 89 (1986), 379–387.

[3] G. Da Prato, A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995), 94–114.

[4] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. Math., II. Ser. 55 (1952), no. 2, 468–519.

[5] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.

[6] C. A. Timmermans, On C_{0}-semigroups in a space of bounded continuous functions in the case of entrance or natural boundary points, in: Approximation and Optimization (J. A. Gómez Fernández et al. eds.), Springer (1988), 209–216.