One-dimensional Feller semigroups with reflecting barriers

M. Campiti, G. Metafune, D. Pallara

December 16, 1998

Abstract Given the second order ordinary differential operator Au = mu′′ + qu′ in the real interval I, possibly degenerate at the boundary, we define W (x) = exp(− ∫_{x_{0}}^{x} \frac{q(s)}{m(s)} ds) (where x_{0} ∈ I is arbitrarily fixed) and the domain

D_{N}(A) = \{u ∈ C(\overline{I}) ∩ C^{2}(I) : Au ∈ C(\overline{I}), lim_{x→∂I} \frac{u′(x)}{W (x)} = 0\}.

Given f ∈ D_{N}(A), we consider the initial value problem u_{t} = Au, u(0) = f and study under which conditions on the coefficients a classical solution exists, and when (A, D_{N}(A)) is the generator of a strongly continuous semigroup in the space C(\overline{I}).

1 Introduction

Let I = (r_{1}, r_{2}) be a (possibly unbounded) open interval and m, q continuous functions, with m positive in I; without any extra assumption on the behaviour of the coefficients at the boundary, consider the second order differential operator Au = mu′′ + qu′ in I and the abstract Cauchy problem in C(\overline{I})

u_{t} = Au  (t > 0)  (1)
u(0) = f  (2)

with f ∈ C(\overline{I}). A general question is: which is the relation between the behaviour of the coefficients m, q near the endpoints and the boundary conditions that can be imposed in order to have a unique solution of (1), (2) satisfying them, for any initial datum? This problem was posed by W. Feller whose main motivation was the probabilistic interest of equation (1) for the transition probabilities u: in fact, (1) is the backward equation coming from a one-dimensional diffusion process (for this reason, C(\overline{I}) is the natural space where the problem should be discussed, keeping its original flavour). Accordingly, many boundary conditions that have been joined to problem (1), (2) have a genuine probabilistic meaning and this reflects also in the terminology introduced by Feller. We refer to [5], [6] and [7] for a discussion of the probabilistic background and of the model, and for a detailed study of the evolution problem (1), (2) in many different situations. In terms of semigroups, assigning boundary conditions amounts to endowing the operator A with a domain D(A) in C(\overline{I}); then one investigates whether (A, D(A)) is a generator. To begin with, let us introduce the maximal domain of A,

D_{M}(A) = \{u ∈ C(\overline{I}) ∩ C^{2}(I) : Au ∈ C(\overline{I})\}.

Moreover, fix once and for all a point x_{0} ∈ I, define the Wronskian

W (x) = exp\{− ∫_{x_{0}}^{x} \frac{q(s)}{m(s)}ds\}

and consider the “reflecting barrier” boundary conditions

u(t) ∈ D_{N}(A)  t > 0,  (3)

where

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

We point out that the above boundary conditions reduce to the homogeneous Neumann conditions if, e.g., I is bounded and m, q are continuous up to the boundary, with inf_{I} m > 0. In order to provide a general setting to all the boundary value problems associated to (1), (2), Feller proposed a classification of the boundary points r_{1}, r_{2}, which comes from the possible behaviour of the solutions of the equation λu − Au = 0. In order to recall Feller’s classification of the boundaries we introduce the functions

Q(x) = \frac{1}{m(x)W (x)} ∫_{x_{0}}^{x} W (s)ds, R(x) = W (x) ∫_{x_{0}}^{x} \frac{1}{m(s)W (s)}ds; (4)

the endpoint r_{2} is said to be

regular if Q ∈ L^{1}(x_{0}, r_{2}), R ∈ L^{1}(x_{0}, r_{2});
exit  if Q ∉ L^{1}(x_{0}, r_{2}), R ∈ L^{1}(x_{0}, r_{2});
entrance if Q ∈ L^{1}(x_{0}, r_{2}), R ∉ L^{1}(x_{0}, r_{2});
natural if Q ∉ L^{1}(x_{0}, r_{2}), R ∉ L^{1}(x_{0}, r_{2});

of course, analogous definitions are understood for r_{1}. If r_{2} is regular or exit, then it is called accessible, because in terms of Markov processes, there is a positive probability that the particle does reach r_{2} in a finite time; otherwise, r_{2} is unaccessible, see [6]. From this point of view, assigning boundary conditions serves to select (if possible) one among the infinite solutions of (1), (2) only in the case of accessible boundaries. Otherwise, (1), (2) has a unique solution because (A, D_{M}(A)) generates a strongly continuous semigroup (see [5] or [2, Theorem 4.14]). If this is the case and boundary conditions are assigned, the question is whether the solution u enjoys the required properties, and reduces to a kind of regularity control.

In this paper we give necessary and sufficient conditions in order that problem (1), (2), (3), with f ∈ D_{N}(A), has a unique classical solution, i.e. a function u ∈ C([0, ∞), C(\overline{I})) ∩ C^{1}((0, ∞), C(\overline{I})) such that u(t) ∈ D_{N}(A) for every t > 0 and u solves (1), (2). Our main result is the following.

Theorem 1.1 The initial-boundary value problem (1), (2), (3) has a unique classical solution for every f ∈ D_{N}(A) if and only if neither r_{1} nor r_{2} are exit boundaries.

Theorem 1.1 cannot be reformulated by saying that (A, D_{N}(A)) generates a strongly continuous semigroup if and only if the boundaries are not of exit type: in fact, this last statement is not true. In the language of semigroups, the situation is the following: if the boundaries are of regular or entrance type, then (A, D_{N}(A)) is a generator and the solution of problem (1), (2), (3) is given by the generated semigroup. In the exit case, Theorem 1.1 clearly implies that (A, D_{N}(A)) does not generate; however its closure does, but boundary conditions (3) transforms into conditions of a different kind (see Proposition 3.6). In the natural case the problem is the closedness of (A, D_{N}(A)), that is the continuity of the map u ↦ u′/W with respect to the graph norm. We shall prove the following result, where we indicate by I_{i} the interval whose endpoints are x_{0} and r_{i}, for i = 1, 2.

Theorem 1.2 The operator (A, D_{N}(A)) generates a strongly continuous semigroup of positive contractions if and only if at each endpoint r_{i} one of the following conditions is fulfilled:

(i) (mW)^{−1} ∈ L^{1}(I_{i})

(ii) (mW)^{−1} ∉ L^{1}(I_{i}), W ∉ L^{1}(I_{i}) and

sup_{x∈I_{i}} |∫_{x}^{σ(x)} \frac{ds}{m(s)W (s)}| < +∞,  (5)

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

Notice that condition (ii) above is meaningful only for natural boundaries. In Section 3 we prove Theorem 1.2 and show that the closure of (A, D_{N}(A))

always generates a strongly continuous semigroup; Section 4 is devoted to characterize when D_{N}(A) is invariant under the semigroup generated by the closure of (A, D_{N}(A)), and this leads to the proof of Theorem 1.1. Finally, in Section 5 we give some applications to the generation and the regularity of semigroups, under more general boundary conditions, in the case of regular boundaries. Acknowledgement. We are deeply indebted to K. Taira and R. Nagel for many useful remarks and suggestions.

2 Preliminaries

In this section we collect some results on the behaviour of the solutions of the stationary equation

λu − Au = 0,  (6)

which depends on the type of boundary, i.e. on the summability properties of the functions Q, R in (4). All the results we need are proved in [5] and collected in [2, Theorem 4.13], except what is stated in the following lemmas, which are however strongly inspired by the techniques in [5]. Notice that the function W allows us to write A in the form

Au = mW (\frac{u′}{W} )′

and that the summability of the functions R, Q near e.g. r_{2} implies the summability of W and (mW)^{−1} respectively. We state and prove the lemmas for r_{2}; analogous statements hold, of course, for r_{1}.

Lemma 2.1 Let r_{2} be an exit boundary and u_{1}, u_{2} be positive solutions of (6) with u_{1} increasing and u_{2} decreasing. Then

lim_{x→r_{2}} \frac{u′_{1}(x)}{W (x)} = +∞,  lim_{x→r_{2}} \frac{u_{2}(x)u′_{1}(x)}{W (x)} = 0.

There is no non-zero solution u of (6) satisfying lim_{x→r_{2}} \frac{u′(x)}{W (x)} = 0.

Proof. Integrating (6) with u = u_{1} we obtain

\frac{u′_{1}(x)}{W (x)} = u′_{1}(x_{0}) + ∫_{x_{0}}^{x} \frac{λu_{1}(s)}{m(s)W (s)}ds  (7)

and the divergence of the last integral implies that u′_{1}(x)/W (x) → +∞ as x → r_{2}. To prove that (u_{2}u′_{1})/W is infinitesimal, notice that by (7)

\frac{1}{λ}[\frac{u′_{1}(x)}{W (x)} − u′_{1}(x_{0})] = ∫_{x_{0}}^{x} \frac{u_{1}(s)}{m(s)W (s)}ds

and that u_{1} is bounded and u_{2} is infinitesimal near r_{2} (see [2, Theorem 4.13]), so that it suffices to prove that

lim_{x→r_{2}} u_{2}(x) ∫_{x_{0}}^{x} \frac{1}{m(s)W (s)}ds = 0.

To this aim, observe that (7), with u_{2} in place of u_{1}, and u′_{2} ≤ 0 imply

∫_{x_{0}}^{x} \frac{λu_{2}(s)}{m(s)W (s)}ds ≤ |u′_{2}(x_{0})|,

and then the integral ∫_{x_{0}}^{r_{2}} \frac{u_{2}(s)}{m(s)W (s)}ds is convergent. For x > y > x_{0} we have

u_{2}(x) ∫_{x_{0}}^{x} \frac{1}{m(s)W (s)}ds ≤ u_{2}(x) ∫_{x_{0}}^{y} \frac{1}{m(s)W (s)}ds + ∫_{y}^{r_{2}} \frac{u_{2}(s)}{m(s)W (s)}ds;

since the last integral is convergent and u_{2} is infinitesimal, fixing ε > 0, it is possible to choose y such that both quantities in the right hand side are less than ε for x close to r_{2} and our claim is proved.

Finally, let w_{0} = u′_{1}(x_{0})u_{2}(x_{0}) − u_{1}(x_{0})u′_{2}(x_{0}) > 0 and observe that u′_{1}u_{2} − u_{1}u′_{2} = w_{0}W; we obtain lim_{x→r_{2}} \frac{u′_{2}(x)}{W (x)} = k < 0 and since lim_{x→r_{2}} \frac{u′_{1}(x)}{W (x)} = +∞,

we deduce that no non-zero solution u of (6) satisfies lim_{x→r_{2}} \frac{u′(x)}{W (x)} = 0.

Let us now look at the behaviour of u′/W when (mW)^{−1} is summable, an hypothesis that covers regular, entrance and some natural boundaries.

Lemma 2.2 Let (mW)^{−1} ∈ L^{1}(x_{0}, r_{2}), then for every u ∈ D_{M}(A) the limit lim_{x→r_{2}} \frac{u′(x)}{W (x)} = ℓ exists finite. If W ∉ L^{1}(x_{0}, r_{2}) then ℓ = 0. Moreover, the map u ↦ u′/W is continuous from D_{M}(A), endowed with the graph norm, into C(\overline{I}).

Proof. Let f = Au ∈ C(\overline{I}). Then we have

\frac{u′(x)}{W (x)} = u′(x_{0}) + ∫_{x_{0}}^{x} \frac{f(s)}{m(s)W (s)}ds  (8)

and hence lim_{x→r_{2}} \frac{u′(x)}{W (x)} = ℓ exists finite. If W ∉ L^{1}(x_{0}, r_{2}) then ℓ = 0 other-

wise, if e.g. ℓ > 0, the inequality \frac{u′(x)}{W (x)} ≥ ℓ/2 holds in a neighbourhood of r_{2} and implies that u is unbounded near r_{2}. Finally, (8) gives the continuity of the map u ↦ u′/W, as stated.

The preceding lemma shows, in particular, that D_{N}(A) = D_{M}(A) in case of entrance endpoints. In the natural case, the Corollary to Theorem 13.1 in [5] shows that the maximal domain D_{M}(A) coincides with the the Ventcel domain, corresponding to “adhesive boundary” conditions

D_{V} (A) = \{u ∈ C(\overline{I}) ∩ C^{2}(I) : lim_{x→r_{1},r_{2}} Au(x) = 0\};  (9)

we recall that (A, D_{V} (A)) is a generator if and only if both r_{1} and r_{2} are not entrance boundaries (see [3] or [2, Theorem 4.17]).

We suppose now that r_{1}, r_{2} are not exit boundaries and fix two positive monotonic solutions of (6) u_{1}, u_{2} such that

lim_{x→r_{1}} \frac{u′_{1}(x)}{W (x)} = lim_{x→r_{2}} \frac{u′_{2}(x)}{W (x)} = 0;

the existence of these solutions is proved in [5], see also [2, Theorem 4.13]. We construct the Green function

G(x, s) = \{ \begin{matrix}\frac{u_{1}(x)u_{2}(s)}{w_{0}W (s)m(s)}, & x ≤ s, \\ \frac{u_{1}(s)u_{2}(x)}{w_{0}W (s)m(s)}, & x ≥ s, \end{matrix}  x, s ∈ I,  (10)

where we recall that w_{0} = u′_{1}(x_{0})u_{2}(x_{0})−u_{1}(x_{0})u′_{2}(x_{0}) > 0 so that u′_{1}u_{2}−u_{1}u′_{2} = w_{0}W. The integral operator

T_{λ}f(x) = ∫_{r_{1}}^{r_{2}} G(x, s)f(s)ds,  f ∈ C(\overline{I}).  (11)

is bounded from C(\overline{I}) to C(\overline{I}) with ∥T_{λ}∥ ≤ 1/λ and, for every f ∈ C(\overline{I}), T_{λ}f ∈ D_{M}(A) is a solution of the equation λu − Au = f. Moreover, T_{λ}1 = 1/λ by the choice of u_{1}, u_{2} (see [2, Prop. 4.8]).

3 Generation in C(\overline{I})

In this section we prove Theorem 1.2. Let us first show that the operator (A, D_{N}(A)) is always dissipative, independently of the type of boundaries.

Lemma 3.1 The operator (A, D_{N}(A)) is densely defined and dissipative. Moreover we have for every u ∈ D_{N}(A) and λ > 0:

sup λu ≤ sup(λ − A)u inf λu ≥ inf(λ − A)u.

Proof. Let sup u = u(c) with c ∈ \overline{I}. If c ∈ I, then Au(c) ≤ 0 and sup λu = λu(c) ≤ (λ − A)u(c) ≤ sup(λ − A)u. If c = r_{i}, i = 1, 2, then Au(r_{i}) ≤ 0 and one concludes as before. Suppose, in fact, Au(r_{2}) > 0; then there are b < r_{2}, ℓ > 0 such that Au(x) ≥ ℓ in [b, r_{2}[. Let b < x < y < r_{2}, then

\frac{u′(y)}{W (y)} − \frac{u′(x)}{W (x)} ≥ ∫_{x}^{y} \frac{ℓ}{m(s)W (s)}ds;

letting y → r_{2} we obtain

−\frac{u′(x)}{W (x)} ≥ ∫_{x}^{r_{2}} \frac{ℓ}{m(s)W (s)}ds > 0

whence u′(x) < 0 in [b, r_{2}[ which is absurd.

Replacing u with −u we obtain inf λu ≥ inf(λ − A)u and the dissipativity follows.

Observe that for λ > 0, (λ − A)^{−1} is a positive operator on C(\overline{I}), when it exists.

Unlike the dissipativity, the generation of a strongly continuous semigroup depends on the type of boundaries. We start our investigation pointing out a simple necessary condition.

Proposition 3.2 Assume that (A, D_{N}(A)) generates a strongly continuous semigroup in C(\overline{I}). Then neither r_{1} nor r_{2} are exit boundaries.

Proof. We suppose that λ − A is surjective and show that r_{2} is not an exit boundary; to this aim, by Lemma 2.1, it suffices to show that a non-zero solution v of λu − Au = 0 exists with v′/W vanishing at r_{2}. Choose f ≥ 0, f ≢ 0, vanishing in [x_{1}, r_{2}[ and let u ∈ D_{N}(A) be the solution of λu − Au = f; then u ≥ 0 by Lemma 3.1 and u is a solution of λu − Au = 0 in ]x_{1}, r_{2}[. If we show that u ≢ 0 in ]x_{1}, r_{2}[, we can take as v the continuation of u to the whole of I and we are done. In fact, we prove that if J = {x ∈ I : u(x) = 0} ≠ ∅, then u ≡ 0: hence u never vanishes in I. Assume that u(x_{2}) = 0 and note that u′(x_{2}) = 0, since u does not change sign; from the equation we have Au ≤ λu, that is

(\frac{u′}{W} )′ ≤ \frac{λu}{mW}

and then for x ∈]x_{2}, r_{2}[

u′(x) ≤ λW (x) ∫_{x_{2}}^{x} \frac{u(s)}{m(s)W (s)}ds.

Choose δ < 1 such that λW (x) ∫_{x_{2}}^{x} \frac{1}{m(s)W (s)}ds < 1 for x ∈]x_{2}, x_{2} + δ[ and let M = sup{u(x) : x ∈]x_{2}, x_{2} + δ[}. Then u′(x) ≤ M, hence u(x) ≤ Mδ for x ∈]x_{2}, x_{2} + δ[ and M ≤ Mδ implies M = 0. A similar argument shows that u vanishes in a left neighbourhood of x_{2}; therefore J is open, and since it is obviously closed, J = I.

We consider now non-exit boundaries.

Lemma 3.3 If r_{1}, r_{2} are not exit boundaries, the closure (A, D) of (A, D_{N}(A)) generates a strongly continuous semigroup of positive contractions.

Proof. By Lemma 3.1 and the Lumer-Phillips theorem, it suffices to show that λ − A has dense range. Suppose that f vanishes in a neighbourhood of r_{2}; then the function T_{λ}f defined in (11) coincides with a multiple of u_{2} near r_{2} and hence satisfies the boundary condition at r_{2}. Since T_{λ}1 = 1/λ we deduce that T_{λ}f satisfies the boundary condition at r_{2} if f is constant in a neighbourhood of r_{2}. Repeating the argument for r_{1} we obtain that T_{λ}f ∈ D_{N}(A) if f is constant in neighbourhoods of the endpoints, hence the range contains these functions and is dense in C(\overline{I}).

Observe that we have shown that T_{λ} coincides with the resolvent operator of (A, D).

The generation problem is now reduced to characterize when (A, D_{N}(A)) is closed. According to Lemma 2.2, this always happens in the regular or entrance case, and, even more generally, if (mW)^{−1} is summable, a condition that holds also in some natural cases; hence, we now study the case when r_{2} is a natural boundary and (mW)^{−1} ∉ L^{1}(x_{0}, r_{2}). Let us set

D_{1} = \{u ∈ D_{M}(A) : lim_{x→r_{1}} \frac{u′(x)}{W (x)} = 0\};

arguing as in Lemma 3.1 (and using that Au(x) → 0 as x → r_{2} by [5], Corollary to Theorem 13.1) one sees that (A, D_{1}) is dissipative. Moreover the function u_{2} entering into the definition of T_{λ} vanishes at r_{2} (see again [2, Theorem 4.13]) and repeating the proof of Lemma 3.3 we obtain that the closure of (A, D_{1}) is a generator and hence coincides with (A, D). Of course (A, D_{1}) is closed if (mW)^{−1} ∈ L^{1}(r_{1}, x_{0}) (by Lemma 2.2); if this condition is not satisfied we can repeat the above argument and obtain that the closure of (A, D_{1}) is the maximal operator (A, D_{M}(A)). At this point, it is clear that (A, D_{N}(A)) is closed if and only if D_{N}(A) coincides with the closure of D_{1} in D_{M}(A). This remark will be used in the next two lemmas.

Lemma 3.4 Suppose that r_{2} is a natural boundary, that (mW)^{−1} ∉ L^{1}(x_{0}, r_{2}) and W ∈ L^{1}(x_{0}, r_{2}); then the operator (A, D_{N}(A)) is not closed.

Proof. Let u be a C^{2}-function vanishing in a neighbourhood of r_{1} and equal to ∫_{x_{0}}^{x} W (s)ds in a neighbourhood of r_{2}. Then Au = 0 and u′/W = 1 near r_{2} and hence u ∈ D_{1} \ D_{N}(A) and the thesis follows form the above discussion.

It remains to be considered the case when r_{2} is a natural boundary and both the functions W, (mW)^{−1} are not summable near r_{2}.

In the following lemma we restrict to the particular case Au = mu′′ with r_{2} = +∞ (in this case W ≡ 1) and characterize when the condition u′(x) → 0 as x → r_{2} is automatically satisfied for every u ∈ D_{M}(A); the general case will be reduced to this one through a change of variables.

Lemma 3.5 Let Au = mu′′ in (r_{1}, +∞); the equality lim_{x→+∞} u′(x) = 0 holds for every function u ∈ D_{M}(A) if and only if the condition

sup_{x>x_{0}} ∫_{x}^{x+1} \frac{ds}{m(s)} < +∞  (12)

is satisfied.

Proof. Suppose first that (12) is satisfied. By Taylor’s formula we have for every x > x_{0} every u ∈ D_{M}(A)

u(x + 1) − u(x) = u′(x) + ∫_{x}^{x+1} \frac{x + 1 − s}{m(s)}Au(s)ds.

The left hand side of the above equality tends to 0 as x → +∞ as well as Au(x) by [5], Corollary to Theorem 13.1; then the integral converges to 0 as x → +∞ by (12) and the same holds for u′.

Suppose now that (12) fails and take points x_{n} > x_{0} such that

x_{n+1} > x_{n} + 1  and  ∫_{x_{n}}^{x_{n}+1} \frac{ds}{m(s)} > 2^{n};

for large n, we divide the interval [x_{n}, x_{n} + 1] into n^{2} equal parts and consider a subinterval K_{n} where the integral of 1/m is bigger than 2n. Further, we split K_{n} into two closed intervals I_{n}, J_{n} such that ∫_{I_{n}} 1/m ≥ n, ∫_{J_{n}} 1/m ≥ n. Finally we consider continuous functions ϕ_{n} : [r_{1}, +∞[→ R such that

(i) ϕ_{n}(x) = 0 for x ∉ K_{n}, ϕ_{n} ≥ 0 in I_{n}, ϕ_{n} ≤ 0 in J_{n};

(ii) |ϕ_{n}(x)| ≤ \frac{1}{nm(x)};

(iii) ∫_{I_{n}} ϕ_{n} = − ∫_{J_{n}} ϕ_{n} = 1.

Put

ψ(x) = ∑_{n=1}^{∞} ∫_{r_{1}}^{x} ϕ_{n}(s)ds;

notice that ψ(x) = 0 if x ∉ ∪_{n}K_{n} and that in each interval K_{n} only one term in the sum is different from zero. Moreover, ψ ∈ C^{1}([r_{1}, +∞[) and ψ ∈ L^{1}(r_{1}, +∞) because | ∫_{r_{1}}^{x} ϕ_{n}| ≤ χ_{K_{n}}(x) for any x ≥ r_{1}, and the length of K_{n} is 1/n^{2}. Finally, since by (ii) mψ′ → 0 as x → +∞, the function u(x) = ∫_{x_{0}}^{x} ψ(s)ds belongs to D_{M}(A) but u′ does not have a limit as x → +∞.

We are now in a position to prove Theorem 1.2.

Proof of Theorem 1.2. If the boundaries are of exit type both (i) and (ii) fail and, by Proposition 3.2, (A, D_{N}(A)) is not a generator, hence we can exclude the case of exit boundaries.

By Lemma 3.3, we have only to characterize when (A, D_{N}(A)) is closed, i.e. when for every sequence (u_{n}) ⊂ D_{N}(A), if u_{n} → u with respect to the graph norm then u′(x)/W (x) → 0 as x → r_{i}, i = 1, 2. We discuss only the case of r_{2}.

If (mW)^{−1} ∈ L^{1}(x_{0}, r_{2}), Lemma 2.2 ensures that (A, D_{N}(A)) is closed; if (mW)^{−1} ∉ L^{1}(x_{0}, r_{2}) and W ∈ L^{1}(x_{0}, r_{2}), Lemma 3.4 implies that (A, D_{N}(A)) is not closed, and only the case (mW)^{−1} ∉ L^{1}(x_{0}, r_{2}), W ∉ L^{1}(x_{0}, r_{2}) remains to be treated. Assume first Au = mu′′ in (r_{1}, +∞); by Lemma 3.5 (see also the remarks preceding Lemma 3.4), the boundary condition at r_{2} is preserved in D_{N}(A) by passing to the limit with respect to the graph norm if and only if (12), which coincides with (5) in this particular case, holds.

Let now Au = mu′′ + qu′; the change of variables

s(x) = ∫_{x_{0}}^{x} W (ξ)dξ, u(x) = v(s)

maps (r_{1}, r_{2}) into (s(r_{1}), +∞), transforms the operator Au into mW^{2}v_{ss} and the condition u′/W → 0 at r_{2} into v_{s} → 0 at +∞. Changing variables in the integral, it is immediately seen that (5) translates into (12).

Finally, we come back to the case of exit boundaries and describe the closure of (A, D_{N}(A)). It turns out that the closure of (A, D_{N}(A)) still generates a strongly continuous semigroup, but the Neumann boundary conditions are not preserved, as we shall prove in the next section. For simplicity, we suppose that both r_{1} and r_{2} are exit boundaries.

Proposition 3.6 Suppose r_{1}, r_{2} be exit boundaries. Then (A, D_{N}(A)) is not closed; its closure is (A, D_{V} (A)) and generates a strongly continuous semigroup.

Proof. First, we show that D_{N}(A) ⊂ D_{V} (A). For, let u ∈ D_{N}(A) and f = Au. Then, (8) gives f(r_{i}) = 0, i = 1, 2, otherwise the non summability of (mW)^{−1} would imply |u′(x)/W (x)| → +∞ as x → r_{i}; hence u ∈ D_{V} (A).

Let now (A, D) be the closure of (A, D_{N}(A)) and note that D is the closure of D_{N}(A) in D_{M}(A) with respect to the graph norm; we show that D ⊃ D_{V} (A), proving that every u ∈ D_{V} (A) can be approximated by functions in D_{N}(A). To this aim, let u ∈ D_{V} (A) and f = Au. Integrating (8) we obtain

u(x) − u(x_{0}) = u′(x_{0}) ∫_{x_{0}}^{x} W (s)ds + ∫_{x_{0}}^{x} W (s)(∫_{x_{0}}^{s} \frac{f(ξ)}{m(ξ)W (ξ)}dξ)ds. (13)

Since f(r_{1}) = f(r_{2}) = 0 and the function (mW)^{−1} is not summable near r_{1} and r_{2}, we can find a > r_{1}, b < r_{2} and g ∈ C(\overline{I}) with the following properties:

(i) supp g ⊂ [a, b];

(ii) ∥g − f∥_{∞} ≤ ε ;

(iii) ∫_{x_{0}}^{a} \frac{g}{mW} = ∫_{x_{0}}^{b} \frac{g}{mW} = −u′(x_{0}).

Let v such that v(x_{0}) = u(x_{0}) and Av = g, i.e.

v′(x) = W (x)[u′(x_{0}) + ∫_{x_{0}}^{x} \frac{g(s)}{m(s)W (s)}].

Since g ≡ 0 in ]r_{1}, a[ and ]b, r_{2}[, v is constant near r_{1}, r_{2} and belongs to D_{N}(A); moreover, by (ii), ∥Au − Av∥_{∞} = ∥f − g∥_{∞} ≤ ε. The equality v′(x_{0}) = u′(x_{0}) and (13) imply that

u(x) − v(x) = ∫_{x_{0}}^{x} W (s)(∫_{x_{0}}^{s} \frac{f(ξ) − g(ξ)}{m(ξ)W (ξ)}dξ)ds

whence ∥u − v∥_{∞} ≤ Kε with K = ∫_{r_{1}}^{r_{2}} |R(x)|dx < +∞ and hence D ⊃ D_{V} (A).

Since, by [3], (A, D_{V} (A)) is a generator and, by Proposition 3.2, (A, D_{N}(A)) is not, we deduce that (A, D_{N}(A)) is not closed.

We remark that Proposition 3.6 is of local character; in fact, if r_{2} is an exit boundary and r_{1} arbitrary then each function u ∈ D_{N}(A) satisfies Au(x) → 0 as x → r_{2} and, conversely, each function u ∈ D_{M}(A), zero near r_{1} and with Au vanishing at r_{2} is limit (in the graph norm) of a sequence of functions in D_{N}(A). Using the same techniques as above one can see that the closure of (A, D_{N}(A)) coincides with (A, D_{NV} (A)) if r_{1} is regular and with (A, D_{MV} (A)) if r_{1} is entrance or natural where

D_{NV} (A) = \{u ∈ D_{M}(A) : lim_{x→r_{1}} \frac{u′(x)}{W (x)} = lim_{x→r_{2}} Au(x) = 0\},

D_{MV} (A) = {u ∈ D_{M}(A) : lim_{x→r_{2}} Au(x) = 0}.

Arguing as in Lemma 3.3 and using a Green function constructed with monotonic solutions u_{1}, u_{2} such that u_{i} satisfies the boundary condition at r_{i}, we can prove that (A, D_{NV} (A)) and (A, D_{MV} (A)) are generators in the cases indicated above (see also [2]). From these observations, Lemma 3.3 and Proposition 3.6, we obtain

Proposition 3.7 The closure of (A, D_{N}(A)) always generates a strongly continuous semigroup of positive contractions in C(\overline{I}).

Remark 3.8 In particular, if the boundaries are of exit type then the operator (A, D_{N}(A)) is not closed and the evolution leads effectively to its closure, i.e., by Proposition 3.6, to “adhesive boundary”, or Ventcel conditions, see (9); from the point of view of diffusion processes, reflecting barrier condition (3) are supposed to correspond to a finite number of reflections near the boundary, and adhesive conditions to an infinite number of reflections (in finite time), hence our result seems to support this interpretation, as far as it asserts the possibility of approximating the last condition by the preceding one.

4 The initial-boundary value problem

In this section we prove Theorem 1.1, splitting the proof in two parts: in the first one we assume that neither r_{1} nor r_{2} are of exit type, and prove that, if f ∈ D_{N}(A), a classical solution exists; in the second one we assume for simplicity that both r_{1} and r_{2} are of exit type and show that a classical solution does not exist whenever the initial datum f ≢ 0 is positive with compact support in I. Since the closure (A, D) of (A, D_{N}(A)) generates a strongly continuous semigroup (T (t))_{t≥0}, it is clear that problem (1), (2), (3) with f ∈ D_{N}(A) has a (unique) classical solution if and only if T (t)f ∈ D_{N}(A) for every t > 0. However, since D_{N}(A) is not closed (in general) neither in C(\overline{I}) nor in D, this is not equivalent to say that (λ − A)^{−1}f ∈ D_{N}(A). To overcome this problem, we use the auxiliary Banach space

X = \{u ∈ C(\overline{I}) ∩ C^{1}(I) : \frac{u′}{W} ∈ C(\overline{I})\},

endowed with the norm ∥u∥_{X} = ∥u∥_{∞} + ∥u′/W ∥_{∞}.

The operator norm of an operator T acting on a space Z will be denoted by ∥T ∥_{L(Z)}.

Since the proofs of the two parts of Theorem 1.1 are very similar, we give complete details only for Part 1 and indicate the changes needed for Part 2.

Proof of Theorem 1.1, Part 1. Assume that r_{1}, r_{2} are not of exit type and f ∈ D_{N}(A); then a (unique) classical solution of problem (1), (2), (3) exists. We write (λ − A)^{−1} for the resolvent of (A, D) and recall that it coincides with the operator T_{λ} defined in (11), as observed after Lemma 3.3. Step 1. We prove the following: let

X_{0} = \{u ∈ X : \frac{u′}{W} (r_{1}) = \frac{u′}{W} (r_{2}) = 0\},

then T_{λ}f ∈ X_{0} for every f ∈ X_{0} and ∥T_{λ}∥_{L(X_{0})} ≤ 1/λ. Let f ∈ X_{0}, and set F = T_{λ}f; then λF − AF = f, and we have to prove that F ∈ X_{0} and to estimate the norm ∥T_{λ}∥_{L(X_{0})}. First, we compute

w_{0}F′(x) = u′_{2}(x) ∫_{r_{1}}^{x} \frac{u_{1}(s)f(s)}{m(s)W (s)}ds + u′_{1}(x) ∫_{x}^{r_{2}} \frac{u_{2}(s)f(s)}{m(s)W (s)}ds;

since mW (u′_{i}/W)′ = λu_{i} and lim_{x→r_{i}} u′_{i}(x)/W (x) = 0, integrating by parts the boundary terms vanish and we obtain

\frac{F′(x)}{W (x)} = −\frac{1}{λw_{0}} [\frac{u′_{2}(x)}{W (x)} ∫_{r_{1}}^{x} u′_{1}(s)\frac{f′(s)}{W (s)}ds + \frac{u′_{1}(x)}{W (x)} ∫_{x}^{r_{2}} u′_{2}(s)\frac{f′(s)}{W (s)}ds] (14)

whence

|\frac{F′(x)}{W (x)}| ≤ \frac{1}{λw_{0}W (x)}[u′_{1}(x)(u_{2}(x) − u_{2}(r_{2})) − u′_{2}(x)(u_{1}(x) − u_{1}(r_{1}))]∥\frac{f′}{W} ∥_{∞}
≤ \frac{1}{λw_{0}W (x)}[u′_{1}(x)u_{2}(x) − u_{1}(x)u′_{2}(x)]∥\frac{f′}{W} ∥_{∞} = \frac{1}{λ}∥\frac{f′}{W} ∥_{∞}.

���� This estimate, in connection with ∥T_{λ}∥_{L(C(\overline{I}))} ≤ 1/λ, gives ∥T_{λ}∥_{L(X_{0})} ≤ 1/λ, once we prove that F ∈ X_{0}. Let us show that F′(x)/W (x) → 0 as x → r_{2}, the computation for x → r_{1} being analogous: to do that, we check that both addends in the right hand side of (14) are infinitesimal as x → r_{2}. For every y < x, taking into account that u′_{1} > 0, u′_{2} < 0 and u′_{1}u_{2}/W ≤ w_{0}, |u_{1}u′_{2}|/W ≤ w_{0}, we have

|\frac{u′_{2}(x)}{W (x)} ∫_{r_{1}}^{x} u′_{1}(s)\frac{f′(s)}{W (s)}ds| ≤ |\frac{u′_{2}(x)}{W (x)} ∫_{r_{1}}^{y} u′_{1}(s)\frac{f′(s)}{W (s)}ds|

+\frac{|u_{1}(x)u′_{2}(x)|}{W (x)} sup_{y≤s<r_{2}} |\frac{f′(s)}{W (s)}|

≤ |\frac{u′_{2}(x)}{W (x)} ∫_{r_{1}}^{y} u′_{1}(s)\frac{f′(s)}{W (s)}ds| + w_{0} sup_{y≤s<r_{2}} |\frac{f′(s)}{W (s)}|;

the second addend can be made arbitrarily small by choosing y close enough to r_{2}, while the first, with y fixed, tends to 0 as x → r_{2} since u′_{2}(x)/W (x) → 0 as x → r_{2}. As regards the second term in (14), we have, arguing as above

|\frac{u′_{1}(x)}{W (x)} ∫_{x}^{r_{2}} u′_{2}(s)\frac{f′(s)}{W (s)}ds| ≤ \frac{u′_{1}(x)u_{2}(x)}{W (x)} sup_{x≤s<r_{2}} |\frac{f′(s)}{W (s)}|
≤ w_{0} sup_{x≤s<r_{2}} |\frac{f′(s)}{W (s)}|,

which goes to 0 as x → r_{2}. Step 2. (Conclusion). Let (T (t))_{t≥0} be the semigroup generated by (A, D), and introduce the operator (A_{0}, D_{0}) by setting

D_{0} = {u ∈ D ∩ X_{0} : Au ∈ X_{0}}, A_{0}u = Au for all u ∈ D_{0}.

Since (λ − A_{0})^{−1} = (λ − A)^{−1}|_{X_{0}} = T_{λ}|_{X_{0}} is a bounded operator and D_{0} = (λ − A)^{−1}X_{0}, the operator (A_{0}, D_{0}) is closed; moreover, as we shall show in Step 3 below, D_{0} is dense in X_{0} (for the norm of X_{0}) hence, by Step 1, we can apply the Hille-Yosida theorem and conclude that (A_{0}, D_{0}) generates a strongly continuous semigroup T_{0}(t) in X_{0}. Moreover, by the coincidence of the resolvent operators, T_{0}(t)f = T (t)f for every f ∈ X_{0}. Since D_{N}(A) = D ∩ X_{0}, we deduce that D_{N}(A) is invariant under T (t), and the first part of Theorem 1.1 will follow after the next step. Step 3. (Proof of the density of D_{0} in X_{0}). In order to prove that D_{0} is dense in X_{0} we consider the dense subspace Y ⊂ X_{0} whose elements are the C^{∞}- functions that are constant in some neighbourhoods of r_{1}, r_{2}. If the coefficients m, q are C^{1}(I), then trivially Y ⊂ D_{0}, hence the reader who has in mind regular coefficients may skip the rest of the proof. If m, q are assumed only to be continuous, we can prove the density of D_{0} in X_{0} through a rather indirect argument: given v ∈ Y and ε > 0, we shall find u ∈ D_{0} such that ∥u − v∥_{X_{0}} is

13 of order ε. Let v be locally constant out of a compact interval J = [s_{1}, s_{2}] ⊂ I, let g ∈ C_{0}^{∞}(J) be such that ∥Av − g∥_{∞} < ε and

∫_{s_{1}}^{s_{2}} \frac{u_{i}(s)Av(s)}{m(s)V (s)}ds = ∫_{s_{1}}^{s_{2}} \frac{u_{i}(s)g(s)}{m(s)V (s)}ds,  i = 1, 2  (15)

where u_{1}, u_{2} are two solutions of Au = 0 verifying the conditions u_{i}(s_{i}) = 0, u′_{i}(s_{i}) = (−1)^{i−1}, and V (x) = u′_{1}(x)u_{2}(x) − u_{1}(x)u′_{2}(x) is their Wronskian. Consider the following regular boundary value problem in J:

\{ \begin{matrix}Au = g & in [s_{1}, s_{2}] \\ u(s_{i}) = v(s_{i}), & i = 1, 2 \end{matrix}  (16)

whose solution is given by

u(x) = −[u_{2}(x) ∫_{s_{1}}^{x} \frac{u_{1}(s)g(s)}{m(s)V (s)}ds + u_{1}(x) ∫_{x}^{s_{2}} \frac{u_{2}(s)g(s)}{m(s)V (s)}ds]
+\frac{v(s_{2})}{u_{1}(s_{2})}u_{1}(x) + \frac{v(s_{1})}{u_{2}(s_{1})}u_{2}(x).  (17)

Notice now that v′(s_{1}) = v′(s_{2}) = 0, and a straightforward computation shows that also u′(s_{1}) = u′(s_{2}) = 0 holds by virtue of (15); using the equation Au = g we deduce that u′′(s_{1}) = u′′(s_{2}) = 0 as well, whence u, continued as a constant in (r_{1}, s_{1}) and in (s_{2}, r_{2}), belongs to C^{2}(\overline{I}). Moreover, Au ∈ C_{0}^{∞}(I) and Au = 0 in I \ J imply that u ∈ D_{0}. Finally, ∥Au − Av∥_{∞} < ε and the continuity of A^{−1} : C([s_{1}, s_{2}]) ⟶ C^{2}([s_{1}, s_{2}]) yield ∥u − v∥_{C^{2}([s_{1},s_{2}])} ≤ Cε and hence ∥u − v∥_{X_{0}} ≤ Cε, as claimed. Proof of Theorem 1.1, Part 2. Assume that r_{1}, r_{2} are both of exit type (for simplicity) and f ∈ D_{N}(A), f ≥ 0, f ≢ 0 with compact support; then a classical solution of problem (1), (2), (3) does not exist. Step 1 Let (T (t))_{t≥0} be the semigroup generated by (A, D_{V} (A)) (which is the closure of (A, D_{N}(A)) by Proposition 3.6) and let (λ − A)^{−1} its resolvent operator. Fixing two positive monotonic solutions u_{1}, u_{2} of (6) such that u_{i}(r_{i}) = 0, i = 1, 2, we define the Green function as in (10) and obtain

(λ − A)^{−1}f(x) = ∫_{r_{1}}^{r_{2}} G(x, s)f(s)ds, f ∈ C(\overline{I})

(see [2, Theorem 4.17] and [5, Lemma 14.1]). We define

X_{1} = {u ∈ X : u(r_{1}) = u(r_{2}) = 0}

and prove that (λ − A)^{−1}X_{1} ⊂ X_{1} and that ∥(λ − A)^{−1}∥_{L(X_{1})} ≤ 1/λ. In fact, with f ∈ X_{1} and F = (λ − A)^{−1}f, formula (14) still holds since f vanishes at the boundary and gives ∥(λ − A)^{−1}∥_{L(X_{1})} ≤ 1/λ once we prove that F ∈ X_{1}. This fact can be verified as in the proof of Part 1 (Step 1), using Lemma 2.1. Step 2 Introduce the operator (A_{1}, D_{1}) by setting

D_{1} = {u ∈ D_{V} (A) ∩ X_{1} : Au ∈ X_{1}}, A_{1}u = Au for all u ∈ D_{1}.

The domain D_{1} is dense in X_{1} (see Step 3 below) and hence, by Step 1, arguing as in the proof of Part 1 (Step 2), we conclude that (A_{1}, D_{1}) generates a strongly continuous semigroup (T_{1}(t))_{t≥0} in X_{1} such that T_{1}(t)f = T (t)f for every f ∈ X_{1}.

Let f ∈ D_{N}(A), f ≥ 0, f ≢ 0 with compact support and suppose that problem (1), (2), (3) has a classical solution, that is T (t)f ∈ D_{N}(A) for t ≥ 0. Putting

X_{2} = \{u ∈ X_{1} : \frac{u′}{W} (r_{i}) = 0, i = 1, 2\},

we deduce that T_{1}(t)f ∈ X_{2} for t ≥ 0 and hence (λ − A)^{−1}f ∈ X_{2} for λ > 0 since X_{2} is closed in X_{1}. We have therefore obtained a solution u = (λ − A)^{−1}f of the equation λu − Au = f that belongs to D_{M}(A) ∩ X_{2} ⊂ D_{N}(A). Arguing as in the proof of Proposition 3.2, we can then construct a non-zero solution v of λv − Av = 0 such that v′(x)/W (X) → 0 as x → r_{2}; this is in contrast with Lemma 2.1. Step 3 The proof of the density of D_{1} in X_{1} is similar to that of D_{0} in X_{0}; let Y_{1} ⊂ X_{1} be the dense subspace consisting of the C^{2}-functions that are multiples of ∫_{r_{i}}^{x} W (ξ)dξ in some neighbourhoods of r_{i}, i = 1, 2. As above, if the coefficients are regular, then Y_{1} ⊂ D_{1} and the density follows. Otherwise, we can argue as in Part 1, Step 3, choosing v ∈ Y_{1}, g as above, and solving again problem (16), where the s_{i} have the obvious meaning. The solution u is given by (17) and conditions (15) ensure that u′(s_{i}) = v′(s_{i}), and the same estimates hold; we omit the details.

Examples. Let us present some concrete examples of application of the results of Sections 3 and 4.

1. Let I = (0, 1) and Au = (mu′)′; then (mW)^{−1} = 1, hence (A, D_{N}(A)) generates a semigroup in C(\overline{I}) (this case has been treated also in L^{p}(I) in [1]).

2. Let I = (0, +∞), Au = mu′′ and assume that 0 is a regular point. If 1/m ∈ L^{1}(I), condition (i) of Theorem 1.2 holds and, if inf_{I} m > 0, condition (ii) of Theorem 1.2 holds, hence in these cases (A, D_{N}(A)) generates a semigroup. If m(x) → 0 as x → +∞, both conditions (i) and (ii) above are not satisfied and (A, D_{N}(A)) is not a generator; however +∞ is a natural boundary and problem (1), (2), (3) has a classical solution for any f ∈ D_{N}(A).

3. Let I = (0, 1), Au = mu′′ and assume that 1 is a regular point. If 1/m ∈ L^{1}(I), condition (i) of Theorem 1.2 holds; if 1/m ∉ L^{1}(I) and ∫_{1/2}^{x} 1/m ∈ L^{1}(I), 0 is an exit boundary and problem (1), (2), (3) has no solution in general; if ∫_{1/2}^{x} 1/m ∉ L^{1}(I), 0 is a natural boundary, (A, D_{N}(A)) is not a generator because W ∈ L^{1}(I), but problem (1), (2), (3) has a classical solution. In particular, for m(x) = x^{α} (with α ∈ R), the operator (A, D_{N}(A)) generates a semigroup if and only if α < 1, and problem (1), (2), (3) has a classical solution if and only if α < 1 or α ≥ 2. Notice that for α < 1 the point 0 is regular, whereas for α ≥ 2 the point 0 is natural, hence unaccessible. For 1 ≤ α < 2 the point 0 is exit, hence it is accessible as in the regular case, but condition u′(x) → 0 as x → 0 is not preserved in general.

4. Let I = (0, 1), Au = x^{k+1}u′′ + bx^{k}u′, with k, b ∈ R. Obviously, 1 is a regular point; as regards the point 0, if b ≤ k < 1 it is an exit boundary; in all the other cases, a classical solution of problem (1), (2), (3) exists for every f ∈ D_{N}(A). Moreover, condition (i) of Theorem 1.2 holds if and only if b > k while condition (ii) holds if and only if 1 ≤ b ≤ k ≤ 2b − 1: in fact, in this last case, 0 is a natural boundary and it is easy to check condition (12) after the change of variable s(x) = ∫_{0}^{x} W (ξ)dξ, as in the proof of Theorem 1.2.

5. Let I = (0, +∞) and Au = u′′ + bx^{α}u′, with α > 0. If b < 0, then condition (i) of Theorem 1.2 holds.

If b > 0, then W (x) = exp\{−b\frac{x^{α+1}}{α+1} \} belongs to L^{1}(I) and the function R(x) = W (x) ∫_{0}^{x} 1/W (ξ)dξ belongs to L^{1}(I) if and only if α > 1; in fact,

∫_{0}^{x} exp\{b\frac{ξ^{α+1}}{α + 1}\}dξ ≈ C exp\{b\frac{x^{α+1}}{α + 1}\}x^{−α}, as x → +∞

for a suitable C > 0. Therefore, for b > 0, problem (1), (2), (3) has a classical solution if and only if α ≤ 1, but the operator (A, D_{N}(A)) is never a generator.

5 Regular boundaries

Let r_{1}, r_{2} be regular boundaries; fixing real numbers α_{i}, β_{i} such that α_{i}^{2}+β_{i}^{2} > 0, i = 1, 2, we consider the so-called “elastic barrier” conditions, i.e. we put

D(A) = \{u ∈ D_{M}(A) : α_{i}u(r_{i}) + β_{i} lim_{x→r_{i}} \frac{u′(x)}{W (x)} = 0, i = 1, 2\}.

If β_{1} = β_{2} = 0 we are imposing Dirichlet (or “absorbing barrier”) conditions, while if α_{1} = α_{2} = 0 then we are imposing the “reflecting barrier” conditions treated in the previous section. We now show how it is possible to deduce from Theorem 1.2 that (A, D(A)) generates a strongly continuous semigroup and study its regularity properties. If the sign condition (−1)^{i}α_{i}β_{i} ≥ 0 holds, the operator (A, D(A)) is dissipative and generates a strongly continuous semigroup of positive contractions, as it is shown in [5] (see also [2]). If β_{1}β_{2} ≠ 0 we have a dense domain; in the other case, one still obtains a semigroup in the closure of D(A), as it can be easily verified. For example, if β_{1} = β_{2} = 0, then \overline{D(A)} = C_{0}(\overline{I}) and D(A) = D_{M}(A) ∩ C_{0}(\overline{I}).

Proposition 5.1 The operator (A, D(A)) generates a strongly continuous semigroup.

Proof. Assume β_{1}β_{2} ≠ 0. We perform two changes of variables, in the independent and in the dependent variable, in order to apply Theorem 1.2. First, set s(x) = ∫_{r_{1}}^{x} W (ξ)dξ, 0 ≤ s ≤ ℓ := s(r_{2}) < +∞, so that the operator (A, D(A)) transforms into (B, D(B)), where Bv = pv_{ss}, with p = mW^{2}, 1/p ∈ L^{1}(0, ℓ), and

D(B) = {v ∈ C^{1}([0, ℓ]) ∩ C^{2}(]0, ℓ[) : Bv ∈ C([0, ℓ]),
α_{1}v(0) + β_{1}v′(0) = α_{2}v(ℓ) + β_{2}v′(ℓ) = 0}.

Then, let ϕ ∈ C^{2}([0, ℓ]) be a strictly positive function such that ϕ(0) = ϕ(ℓ) = 1, ϕ′(0) = −α_{1}/β_{1}, ϕ′(ℓ) = −α_{2}/β_{2} with ϕ linear in neighbourhoods of 0, ℓ. The map T : C([0, ℓ]) → C([0, ℓ]) given by T v = v/ϕ is a linear isomorphism of C([0, ℓ]) onto itself and gives Bv = T^{−1}B_{1}T v, for v ∈ D(B), where

B_{1}z = p[z_{ss} + 2\frac{ϕ_{s}}{ϕ} z_{s} + \frac{ϕ_{ss}}{ϕ} z]

for z ∈ D(B_{1}) = {z ∈ C^{1}([0, ℓ])∩C^{2}(]0, ℓ[) : B_{1}z ∈ C([0, ℓ]), z_{s}(0) = z_{s}(ℓ) = 0}. Observe that 0, ℓ are regular boundaries for B_{1}, hence, in particular, condition (i) of Theorem 1.2 is satisfied. Moreover, the term (pϕ_{ss}/ϕ)z is a bounded perturbation because ϕ_{ss} = 0 near the endpoints, hence the operator (B_{1}, D(B_{1})) generates a strongly continuous semigroup, and also (A, D(A)) by similarity.

We now prove that the semigroup generated by (A, D(A)) is differentiable in C(\overline{I}). To this aim we introduce the auxiliary Hilbert space

H = \{u : I → C : ∫_{r_{1}}^{r_{2}} \frac{|u|^{2}}{mW} < +∞\},

endowed with the obvious inner product, and the operator (A_{1}, D(A_{1})), where A_{1}u = mu′′ + qu′ for u in

D(A_{1}) = \{u ∈ H : u ∈ H_{loc}^{2}(I), A_{1}u ∈ H, α_{i}u(r_{i}) + β_{i} lim_{x→r_{i}} \frac{u′(x)}{W (x)} = 0, i = 1, 2\}.

By [8, Sections 17.3 and 19.4] the operator (A_{1}, D(A_{1})) is self-adjoint and bounded from above in H, hence it generates an analytic semigroup in H.

We are now in a position to prove the announced regularity result.

Theorem 5.2 The semigroup generated by (A, D(A)) is compact and differentiable in C(\overline{I}).

Proof. Let (T (t))_{t≥0}, (T_{1}(t))_{t≥0} be the semigroups generated by the operators (A, D(A)) and (A_{1}, D(A_{1})) respectively and observe that C(\overline{I}) ⊂ H, D(A) ⊂ D(A_{1}) and Af = A_{1}f if f ∈ D(A). If f ∈ D(A), the uniqueness of the solution of the Cauchy problem u_{t} = A_{1}u, u(0) = f implies T (t)f = T_{1}(t)f for t ≥ 0 and hence, by density, T (t)f = T_{1}(t)f for every f ∈ C(\overline{I}), t ≥ 0.

If u ∈ D(A_{1}), by (8), we deduce |u′(x)| ≤ CW (x) and u ∈ C(\overline{I}). By the closed graph thorem, the inclusion D(A_{1}) ↪ C(\overline{I}) is continuous. Since (T_{1}(t))_{t≥0} is analytic, for every t > 0 the map AT_{1}(t) is bounded from H into D(A_{1}), hence AT (t) is bounded from C(\overline{I}) into itself and the differentiability of (T (t))_{t≥0} follows.

To show the compactness of (T (t))_{t≥0} it is sufficient to prove that the inclusion D(A) ↪ C(\overline{I}) is compact. For, using (8) again and observing that there is C > 0 such that both |u′(x_{0})|, ∥Au∥_{∞} are majorized by C∥u∥_{D(A)}, we obtain

|u′(x)| ≤ C∥u∥_{D(A)}[W (x) + |R(x)|];

since both W, R are summable, by Ascoli’s theorem we deduce the compactness of the above embedding.

References

[1] M. Campiti, G. Metafune, D. Pallara, Degenerate Self-adjoint Evolution Equations in the Unit Interval, Semigroup Forum, 57 (1998), 1-36.

[2] M. Campiti, G. Metafune, D. Pallara, S. Romanelli, Semigroups for ordinary differential operators, in [4].

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

[4] K. Engel, R. Nagel, One–parameter semigroups for linear evolution equations, Springer Graduate Texts in Math., to appear.

[5] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. 55 (1952), 468–519.

[6] W. Feller, Diffusion processes in one dimension, Trans. Am. Math Soc. 97 (1954), 1–31.

[7] P. Mandl, Analytical treatment of one–dimensional Markov processes, Springer 1968.

[8] M.A. Naimark, Linear differential operators II, Ungar, 1968.

M. Campiti  G. Metafune, D. Pallara
Dipartimento di Matematica  Dipartimento di Matematica
Politecnico di Bari  Università di Lecce
Via E. Orabona, 4, I-70125, Bari  P.O.B. 193, I-73100, Lecce

e-mail: campiti@dm.uniba.it               metafune@le.infn.it, pallara@le.infn.it