Wiener Regularity and Heat Semigroups on Spaces of Continuous Functions

by

Wolfgang Arendt and Philippe Bénilan

November 3, 1998

Dedicated to Professor Herbert Amann on the occasion of his sixtieth birthday.

Abstract. Let Ω ⊂ IR^{N} be open. It is shown that the Dirichlet Laplacian generates a (holomorphic) C_{0} −semigroup on C_{0}(Ω) if and only if Ω is regular in the sense of Wiener. The same result remains true for elliptic operators in divergence form.

AMS classification: 47D06, 35 K50

0 Introduction

Let Ω be an open subset of IR^{N} . By C_{0}(Ω) we denote the space of all continuous scalar valued functions f on Ω̄ which are 0 on the boundary ∂Ω of Ω and 0 at infinity; i.e., C_{0}(Ω) = {f : Ω → lC continuous: lim_{x→z} f(x) = 0 for all z ∈ ∂Ω and lim_{{|x|→∞ \atop x∈Ω}} f(x) = 0\} . Then C_{0}(Ω) is a Banach space for the supremum norm

∥f∥_{∞} = sup_{x∈Ω} |f(x)| ,

and a Banach lattice for pointwise ordering. We consider the Laplacian Δ_{0} on C_{0}(Ω) with maximal distributional domain; i.e.,

D(Δ_{0}) = {f ∈ C_{0}(Ω) : Δf ∈ C_{0}(Ω)} Δ_{0}f = Δf .

It is a closed, dissipative operator. Our aim is to characterize those open sets Ω for which Δ_{0} is generator of a C_{0} − semigroup on C_{0}(Ω) . We say that Ω is regular (in the sense of Wiener) if at each point z ∈ ∂Ω there exists a barrier (see Definition 3.1). If Ω is bounded then, by classical potential theory (see e.g. [DL], [GT]), Ω is regular if and only if the Dirichlet problem

D(φ) \{ \begin{matrix}u ∈ C(Ω̄) , u_{|∂Ω} = φ \\ Δu = 0 in D(Ω)′ \end{matrix}

has a solution for all φ ∈ C(∂Ω) . The purpose of this paper is to establish the following result:

Theorem 0.1 Let Ω ⊂ IR^{N} be open. The following assertions are equivalent:

(i) Ω is regular;

(ii) the resolvent set of Δ_{0} is non-empty;

(iii) Δ_{0} generates a holomorphic C_{0} − semigroup on C_{0}(Ω) .

The proof of Theorem 0.1 is based on classical potential theory and the methods are well-known. Still it seems that the result is nowhere given explicitely in the literature. On the other hand, it is certainly of importance. For example, treating the semilinear heat equation and associated dynamical systems, the realization of the Laplacian on the space C_{0}(Ω) is precisely what is needed, (cf. Cazanave-Haraux [CH], where, even though merely the case of Lipschitz boundary is treated, only vague references and indications for the proof are given; see [CH], Lemma 2.6.5 and Remark 2.6, in particular). In this paper we give complete proofs and also describe in detail properties of the Laplacian we use.

The paper is organized as follows. Section 1 contains prerequisites concerning the heat semigroups on L^{p}(Ω) . Then Theorem 0.1 is proved for bounded Ω in Section 3 by establishing the equivalence of (ii) and (iii) to well-posedness of the Dirichlet problem. In Section 4 we prove the result for arbitrary open sets establishing equivalence to regularity in the sense that there is a barrier at each point. These two sections are completely elementary and self-contained. In Section 4 we prove the corresponding result for elliptic operators. Here we use the fundamental result of De Giorgi and Nash on (Hölder-) continuity of weak solutions and further regularity results by Stampacchia. We also show several spectral and regularity results for the elliptic operators on C_{0}(Ω) . In particular, we prove holomorphy of the semigroup on C_{0}(Ω) using that the adjoint semigroup is holomorphic on L^{1}(Ω) (see [AB], [AE], [Ou2]). For previous results exploiting this duality see Amann [Am] and Amann and Escher [AmE].

1 The Dirichlet Laplacian on L^{p}(Ω)

In this section we put together some known results on the Laplacian with Dirichlet boundary conditions. Let Ω ⊂ IR^{N} be an open set. By D(Ω) we denote the space of all test functions and by D(Ω)′ the space of all distributions. The first Sobolev space is denoted by

H^{1}(Ω) := {u ∈ L^{2}(Ω) : D_{j}u ∈ L^{2}(Ω) , j = 1 … N}

where D_{j} = \frac{∂}{∂x_{j}} . Moreover, we let

H_{0}^{1}(Ω) = \overline{D(Ω)}^{H^{1}(Ω)} .

Recall that for real-valued u ∈ H^{1}(Ω) one has u^{+}, u^{−}, (u − 1)^{+} ∈ H^{1}(Ω) and

D_{j}u^{+} = 1_{[u>0]}D_{j}u , D_{j}(u − 1)^{+} = 1_{[u>1]}D_{j}u . (1.1)

Moreover, u^{+}, (u − 1)^{+} ∈ H_{0}^{1}(Ω) if u ∈ H_{0}^{1}(Ω) . Note also that the mappings u ↦ u^{+} and u ↦ (u − 1)^{+} at continuous from H^{1}(Ω) into H^{1}(Ω) . See [GT], Chapter 7, for these simple facts. By Δ_{2}^{Ω} we denote the Laplacian with Dirichlet boundary conditions on L^{2}(Ω) , also called the Dirichlet Laplacian for short; i.e.

D(Δ_{2}^{Ω}) = {u ∈ H_{0}^{1}(Ω) : Δu ∈ L^{2}(Ω)} , Δ_{2}^{Ω}u = Δu .

This is the operator associated with the form

a(u, v) = ∫_{Ω} ∇u∇u

on H_{0}^{1}(Ω) ; i.e., for u ∈ H_{0}^{1}(Ω), v ∈ L^{2}(Ω) we have

u ∈ D(Δ_{2}^{Ω}) , Δ_{2}^{Ω}u = v ⇔ − ∫ ∇u∇φ = ∫ vφ ∀ φ ∈ H_{0}^{1}(Ω) .

The operator Δ_{2}^{Ω} is self-adjoint and generates a positive contraction C_{0}−semigroup (e^{tΔ_{2}^{Ω}})_{t≥0} on L^{2}(Ω) . Moreover, 0 ∈ ϱ(Δ_{2}^{Ω}) (the resolvent set of Δ_{2}^{Ω}) if Ω is bounded. If Ω = IR^{N} we write Δ_{2} = Δ_{2}^{IR^{N}} . Then (e^{tΔ_{2}})_{t≥0} is the Gaussian semigroup which we denote by G_{2} ; i.e.,

(G_{2}(t)f)(x) = (4πt)^{−N/2} ∫_{IR^{N}} f(y)e^{−(x−y)^{2}/4t}dy

for all f ∈ L^{2}(IR^{N}) . We denote by ϱ(A) the resolvent set of an operator A , and by R(λ, A) = (λ−A)^{−1} its resolvent (λ ∈ ϱ(A)) . The following domination property is well-known (cf. [Da], Theorem 2.1.6).

For convenience of the reader we give a simple direct proof.

Proposition 1.1 Let Ω_{1}, Ω_{2} be open subsets of IR^{N} such that Ω_{1} ⊂ Ω_{2} . Then

0 ≤ e^{tΔ_{2}^{Ω_{1}}}f  ≤ e^{tΔ_{2}^{Ω_{2}}}f ,
0 ≤ R(λ, Δ_{2}^{Ω_{1}})f ≤ R(λ, Δ_{2}^{Ω_{2}})f

for f ∈ L^{2}(Ω)_{+} , λ > 0 , t ≥ 0 .

Here we identify L^{2}(Ω_{1}) with a subspace of L^{2}(Ω_{2}) extending functions by 0 . We let L^{2}(Ω)_{+} = {f ∈ L^{2}(Ω) : f(x) ≥ 0 a.e.} , H^{1}(Ω)_{+} = L^{2}(Ω)_{+} ∩H^{1}(Ω) , H_{0}^{1}(Ω) = H_{0}^{1}(Ω) ∩ L^{2}(Ω)_{+} , D(Ω)_{+} = {φ ∈ D(Ω) : φ(x) ≥ 0 for all x ∈ Ω} . For w_{1}, w_{2} ∈ D(Ω)′ we define

w_{1} ≤ w_{2} :⇔ < w_{1}, φ > ≤ < w_{2}, φ > for all φ ∈ D(Ω)_{+} .

Note that D(Ω)_{+} is dense in H_{0}^{1}(Ω)_{+} . Moreover, we recall: If u ∈ H^{1}(Ω) has compact support, then u ∈ H_{0}^{1}(Ω) (see [B], IX.5 p. 171). The proof of Proposition 1.1 is based on the following.

Lemma 1.2 Let Ω ⊂ IR^{N} be open, λ > 0 . Let u ∈ H_{0}^{1}(Ω) , v ∈ H^{1}(Ω) such that v ≥ 0 and

λu − Δu ≤ λv − Δv in D(Ω)′ .

Then u(x) ≤ v(x) a.e.. Taking in particular v = 0 , we obtain for u ∈ H_{0}^{1}(Ω)

λu − Δu ≤ 0 ⇒ u ≤ 0 . (1.2)

Proof. By hypothesis we have

λ ∫ (u − v)φ + ∫ ∇(u − v)∇φ ≤ 0 (1.3)

for all φ ∈ D(Ω)_{+} . It follows by density that (1.3) remains true for all φ ∈ H_{0}^{1}(Ω)_{+} . Note that (u−v)^{+} ∈ H_{0}^{1}(Ω) . In fact, let u_{n} ∈ D(Ω) , u_{n} → u in H^{1}(Ω) .

Then (u_{n} − v)^{+} ∈ H^{1}(Ω) has compact support and thus (u_{n} − v)^{+} ∈ H_{0}^{1}(Ω) . Since (u_{n} − v)^{+} → (u − v)^{+} (n → ∞) in H^{1}(Ω) , the claim follows. Now (1.3) for φ = (u − v)^{+} yields

λ ∫ ((u − v)^{+})^{2} + ∫ |∇(u − v)^{+}|^{2} ≤ 0 .

Hence (u − v)^{+} = 0 .

Proof of Proposition 1.1 a) Let λ > 0 , f ∈ L^{2}(Ω_{1})_{+} , u = R(λ, Δ_{2}^{Ω_{1}})f , v = R(λ, Δ_{2}^{Ω_{2}})f . We show that 0 ≤ u ≤ v . In fact, u ∈ H_{0}^{1}(Ω_{1}) , λu − Δu = f in D(Ω_{1})′ , v ∈ H_{0}^{1}(Ω_{2}) , λv −Δv = f in D(Ω_{2})′ . It follows from (1.2) that v(x) ≥ 0 and u(x) ≥ 0 a.e.. Since λu − Δu = λv − Δv in D(Ω_{2})′ , it follows from Lemma 1.2 that u ≤ v a.e..

b) Let 0 ≤ f ∈ L^{2}(Ω_{1}) . Then it follows from a) that (I−tΔ_{2}^{Ω_{1}})^{−1}f ≤ (I−tΔ_{2}^{Ω_{2}})^{−1}f

for all t > 0 . Hence e^{tΔ_{2}^{Ω_{1}}}f = lim_{n→∞}(I − \frac{t}{n}Δ_{2}^{Ω_{1}})^{−n}f ≤ lim_{n→∞}(I − \frac{t}{n}Δ_{2}^{Ω_{2}})^{−n}f = e^{tΔ_{2}^{Ω_{2}}}f .

It follows from Proposition 1.1 that

0 ≤ e^{tΔ_{2}^{Ω}}f ≤ G(t)f (t ≥ 0) , (1.4)

0 ≤ R(λ, Δ_{2}^{Ω})f ≤ R(λ, Δ_{2})f (λ > 0) (1.5)

for all f ∈ L^{2}(Ω)_{+} . Since the Gaussian semigroup is a contractive C_{0} − semigroup on L^{p}(IR^{N}) , it follows that there exist positive contraction C_{0}−semigroups (e^{tΔ_{p}^{Ω}})_{t≥0} on L^{p}(Ω) such that

e^{tΔ_{p}^{Ω}}f = e^{tΔ_{p}^{Ω}}f (t ≥ 0)

for all 1 ≤ p , q < ∞ , f ∈ L^{p}(Ω) ∩ L^{q}(Ω) . Moreover,

0 ≤ R(λ, Δ_{p}^{Ω})f ≤ R(λ, Δ_{p})f (1.6)

for all f ∈ L^{p}(Ω)_{+} , λ > 0 , p ∈ [1, ∞) . Finally, defining Δ_{∞}^{Ω} as the adjoint of Δ_{1}^{Ω} , we have consistency in the sense that

R(λ, Δ_{p}^{Ω})f = R(λ, Δ_{q}^{Ω})f (f ∈ L^{p}(Ω) ∩ L^{q}(Ω))

for λ > 0 , 1 ≤ p, q ≤ ∞ . Moreover, for each p ∈ [1, ∞] , one has Δ_{p}^{Ω}f = Δf in D(Ω)′ . It has been proved by Ouhabaz [Ou1] that Gaussian estimates (and in particular (1.4)) imply that the semigroup generated by Δ_{1} is holomorphic of angle \frac{π}{2} (see also [AE]). Denoting by Δ_{∞} the adjoint of Δ_{1} we obtain in particular that σ(Δ_{∞}) ⊂ (−∞, 0] and a bound

∥λR(λ, Δ_{∞})∥ ≤ M_{θ} (1.7)

for all λ ∈ Σ(θ) = {re^{iα} , r > 0 , |α| < θ} whenever θ ∈ [0, π) . This estimate will be used later.

Next we establish some spectral properties. Let E, F be Banach spaces such that F ↪ E (i.e. F is continuously injected into E) . Let A be an operator on E . The part B of A in F is defined by

D(B) = {x ∈ D(A) ∩ F : Ax ∈ F } Bx = Ax .

This notation is motivated by the following observation. Let λ ∈ ϱ(A) . Then

λ ∈ ϱ(B) ⇔ R(λ, A)F ⊂ F , (1.8)

and in this case, R(λ, B) = R(λ, A)_{|F} . A proof of the following easy result can be found in [ANS].

Proposition 1.3 Assume that ϱ(A) ≠ ∅ . If there exists k ∈ IN such that D(A^{k}) ⊂ F , then σ(A) = σ(B) .

Now by (1.6) we have 0 ≤ R(1, Δ_{p}^{Ω})^{k} ≤ R(1, Δ_{p})^{k} (1.9)

for 1 ≤ p < ∞ , k ∈ IN . Since D(Δ_{2}^{k}) = H^{2k}(IR^{N}) ⊂ L^{∞}(IR^{N}) for k > \frac{N}{4} we conclude that

D((Δ_{p}^{Ω})^{k}) ⊂ L^{∞}(Ω) for k > \frac{N}{4} , 2 ≤ p ≤ ∞ . (1.10)

If Ω has bounded Lebesgue measure, it follows from Proposition 1.3 that σ(Δ_{p}^{Ω}) = σ(Δ_{∞}^{Ω}) for p ∈ [2, ∞] . By duality and selfadjointness we conclude

Proposition 1.4 Let Ω ⊂ IR^{N} be open and of finite measure. Then

σ(Δ_{p}^{Ω}) = σ(Δ_{2}^{Ω}) (1 ≤ p ≤ ∞) . (1.11)

Moreover, R(λ, Δ_{p}^{Ω}) = R(λ, Δ_{q}^{Ω})_{|L^{q}} for all λ ∈ σ(Δ_{p}^{Ω}) , ∞ ≥ p ≥ q ≥ 1 .

If Ω is arbitrary, then more elaborate arguments are needed, but (1.11) still remains true (see [Ar]). We also note that

R(1, Δ_{p})L^{p}(IR^{N}) ⊂ L^{∞}(IR^{N})

for p > \frac{N}{2} . Hence

D(Δ_{p}^{Ω}) ⊂ L^{∞}(Ω) (1.12)

for \frac{N}{2} < p ≤ ∞ . Finally we recall the following result by de Pagter [dP].

Proposition 1.5 Let A be the generator of a C_{0} − semigroup on a Banach space E . Assume that R(λ, A)^{k} is compact for some k ∈ IN , λ ∈ ϱ(A) . Then A has compact resolvent.

2 Bounded domains

Let Ω ⊂ IR^{N} be a bounded, open, nonempty set and C_{0}(Ω) = {u ∈ C(Ω̄) : u_{|∂Ω} = 0} . By Δ_{0}^{Ω} we denote the Laplacian on C_{0}(Ω) with maximal domain; i.e.,

D(Δ_{0}^{Ω}) = {u ∈ C_{0}(Ω) : Δu ∈ C_{0}(Ω)} Δ_{0}^{Ω}u = Δu .

Here, for u ∈ L_{loc}^{1}(Ω) we denote by Δu ∈ D(Ω)′ the distributional Laplacian of u . Note that D(Δ_{0}^{Ω}) ⊄ C^{2}(Ω) as is well-known (cf. [DL], II. § 3 Remark 5). We recall the following easy local properties of the Laplacian (see e.g., [DL], II. § 3 Prop. 6 p. 336).

Lemma 2.1 Let u ∈ D(Ω)′ .

a) If Δu ∈ L_{loc}^{p}(Ω) , p > \frac{N}{2} , then u ∈ C(Ω) ;

b) if Δu ∈ L_{loc}^{p}(Ω) , p > N , then u ∈ C^{1}(Ω) .

In particular,

D(Δ_{0}^{Ω}) ⊂ C^{1}(Ω) . (2.1)

Next we show that Δ_{0}^{Ω} is the part of Δ_{2}^{Ω} in C_{0}(Ω) .

Lemma 2.2 a) Let u ∈ C_{0}(Ω) such that Δu ∈ L^{p}(Ω) where p > N . Then u ∈ H_{0}^{1}(Ω) .

b) In particular, the operator Δ_{0}^{Ω} is the part of Δ_{2}^{Ω} in C_{0}(Ω) .

Proof. Let u ∈ C_{0}(Ω) such that Δu = f ∈ L^{p}(Ω) where p > N . We can assume that u is real-valued. Let ε > 0 . Then (u − ε)^{+} has compact support. Let ω ⊂ Ω be open such that ω̄ ⊂ Ω and supp(u − ε)^{+} ⊂ ω . Since u ∈ C^{1}(Ω) (by (2.1)), we have (u − ε)^{+} ∈ H_{0}^{1}(ω) . By hypothesis, we have

∫ ∇u∇φ = ∫ fφ for all φ ∈ D(Ω) ,

hence also for all φ ∈ H_{0}^{1}(ω) by density. Taking φ = (u − ε)^{+} we obtain

∫ (∇(u − ε)^{+})^{2} = ∫ ∇u∇(u − ε)^{+} = ∫ f(u − ε)^{+}
≤ ∥f∥_{p}∥u∥_{q} , where \frac{1}{p} + \frac{1}{q} = 1 .

Thus {(u − ε)^{+} : 0 < ε ≤ 1} is bounded in H_{0}^{1}(Ω) . Since H_{0}^{1}(Ω) is reflexive, we find a sequence ε_{n} ↓ 0 (n → ∞) such that (u−ε_{n})^{+} converges weakly in H_{0}^{1}(Ω) , to v ∈ H_{0}^{1}(Ω) , say. Then (u − ε_{n})^{+} converges weakly to v in L^{2}(Ω) (n → ∞) .

Since (u − ε_{n})^{+} → u^{+} in L^{2}(Ω) , it follows that u^{+} = v ∈ H_{0}^{1}(Ω) . In the same way one sees that u^{−} ∈ H_{0}^{1}(Ω) .

We say that the Dirichlet problem is well-posed in Ω if for every φ ∈ C(∂Ω) there exists a solution u of (D(φ)) (see Introduction). It is known that the solution is unique and in C^{∞}(Ω) . Moreover, the Dirichlet problem is well-posed whenever D(φ) has a solution for all φ in a dense subspace of C(∂Ω) .

Theorem 2.3 Let Ω be a bounded open set in IR^{N} such the Dirichlet problem is well-posed. Assume that Ω is regular. Then Δ_{0}^{Ω} generates a holomorphic C_{0} − semigroup T_{0} = (T_{0}(t)_{t≥0} of angle \frac{π}{2} on C_{0}(Ω) . Moreover, T_{0}(t) is compact for all t > 0 and σ(Δ_{0}^{Ω}) = σ(Δ_{2}^{Ω}) .

Proof. a) We show that R(0, Δ_{2}^{Ω})L^{∞}(Ω) ⊂ C_{0}(Ω) . Let f ∈ L^{∞}(Ω) . Let v = E_{N} ∗ f ∈ C(IR^{N}) , where E_{N} is the Newtonian potential. Then Δv = f in D(Ω)′ . Let φ = v_{|∂Ω} . By hypothesis there exists w ∈ C(Ω̄) such that w_{|∂Ω} = φ and Δw = 0 in D(Ω)′ . Thus u = w − v ∈ C_{0}(Ω) and −Δu = f in D(Ω)′ . It follows from Lemma 2.2 that u ∈ H_{0}^{1}(Ω) . Thus R(0, Δ_{2}^{Ω})f = u ∈ C_{0}(Ω) .

b) It follows from (1.8) that 0 ∈ ϱ(Δ_{0}^{Ω}) . Moreover, ϱ(Δ_{0}^{Ω}) = ϱ(Δ_{∞}^{Ω}) by Proposition 1.3. By (1.8) again, we have R(λ, Δ_{0}^{Ω}) = R(λ, Δ_{∞}^{Ω})_{|C_{0}(Ω)} for all λ ∈ ϱ(Δ_{∞}^{Ω}) . Now it follows from (1.7) that Δ_{0}^{Ω} generates a bounded holomorphic C_{0} − semigroup . Note that D(Δ_{0}^{Ω}) is dense in C_{0}(Ω) since D(Ω) ⊂ D(Δ_{0}^{Ω}) .

c) We show that Δ_{0} has compact resolvent. By (1.10) we have R(0, Δ_{2})^{k}L^{2}(Ω) ⊂ L^{∞}(Ω) where k > \frac{N}{4} . Thus by a) R(0, Δ_{2})^{k+1}L^{2}(Ω) ⊂ C_{0}(Ω) . Note that R(0, Δ_{2}) ∈ L(L^{2}(Ω)) is compact. Thus we can write R(0, Δ_{0})^{k+2} = R(0, Δ_{2})^{k+1} ∘ R(0, Δ_{2}) ∘ j where j : C_{0}(Ω) → L^{2}(Ω) is the canonial embedding. It follows that R(0, Δ_{0})^{k+2} is compact. Consequently, R(0, Δ_{0}) is compact by Proposition 1.5.

d) Finally, from Proposition 1.3 one sees that σ(Δ_{0}^{Ω}) = σ(Δ_{2}^{Ω}) .

Let C^{b}(Ω) be the space of all bounded continuous scalar-valued functions on Ω with supremum norm. Denote by Δ_{b} the part of Δ_{2} in C^{b}(Ω) . Since R(0, Δ_{2})C^{b}(Ω) ⊂ C^{b}(Ω) is follows from Proposition 1.3 and 1.4 that

σ(Δ_{b}) = σ(Δ_{2}) = σ(Δ_{p}) (2.2)

for 1 ≤ p ≤ ∞ . Moreover, R(λ, Δ_{b}) = R(λ, Δ_{2})_{|C^{b}(Ω)} for all λ ∈ ϱ(Δ_{2}) .

We now consider necessary conditions for well-posedness of the Dirichlet problem.

Theorem 2.4 Let Ω ⊂ IR^{N} be bounded and open. The following are equivalent:

(i) the Dirichlet problem is well-posed;

(ii) ϱ(Δ_{0}^{Ω}) ≠ ∅ ; (iii) D(Δ_{p}^{Ω}) ⊂ C_{0}(Ω) for all p ∈ (\frac{N}{2} , ∞] ;

(iv) there exist \frac{N}{2} < p ≤ ∞ , λ ∈ ϱ(Δ_{p}^{Ω}) and f ∈ L^{p}(Ω) such that f(x) > 0 a.e. and R(λ, Δ_{p}^{Ω})f ∈ C_{0}(Ω) .

Proof. (i) ⇒ (ii) follows from Theorem 2.3. (ii) ⇒ (iii) Since ϱ(Δ_{b}^{Ω}) has non-empty interior, it follows from the hypothesis that U := ϱ(Δ_{0}^{Ω})∩ϱ(Δ_{b}^{Ω}) ≠ ∅ . The set U is open and relatively closed in ϱ(Δ_{b}^{Ω}) . In fact, let λ ∈ ϱ(Δ_{b}^{Ω}) such that λ = lim_{n→∞} λ_{n} ∈ U . Then (R(λ_{n}, Δ_{b}^{Ω}))_{n∈IN} is bounded in L(C_{b}(Ω)) , thus also (R(λ_{n}, Δ_{0}^{Ω}))_{n∈IN} is bounded in L(C_{0}(Ω)) . This implies that λ ∈ ϱ(Δ_{0}^{Ω}) . Since ϱ(Δ_{b}^{Ω}) = ϱ(Δ_{2}^{Ω}) is connected, it follows that ϱ(Δ_{b}^{Ω}) ⊂ ϱ(Δ_{0}^{Ω}) . In particular, 0 ∈ ϱ(Δ_{0}^{Ω}) . It follows from (1.8) that R(0, Δ_{b}^{Ω})C_{0}(Ω) ⊂ C_{0}(Ω) . Let \frac{N}{2} < p ≤ ∞ . Then R(0, Δ_{p}^{Ω}) is a bounded operator from L^{p}(Ω) into L^{∞}(Ω) (by (1.12)). Since R(0, Δ_{p}^{Ω})C_{0}(Ω) ⊂ C_{0}(Ω) and since C_{0}(Ω) is closed in L^{∞}(Ω) , it follows that R(0, Δ_{p}^{Ω})L^{p}(Ω) ⊂ C_{0}(Ω) . (iii) ⇒ (i) We show that the Dirichlet problem has a solution for all φ ∈ C(∂Ω) . At first, assume that φ = w_{|∂Ω} for some w ∈ C^{2}(Ω̄) . Let v = −R(0, Δ_{p}^{Ω})Δw . Then v ∈ C_{0}(Ω) and Δv = Δw in D(Ω)′ . Thus u = w − v ∈ C(Ω̄) , u_{|∂Ω} = φ and Δu = 0 in D(Ω)′ ; i.e. u is a solution of D(φ) . Since the set {w_{|∂Ω} : w ∈ C^{2}(Ω̄)} is dense in C(∂Ω) , it follows that D(φ) has a (unique) solution for all φ ∈ C(∂Ω) . (iii) ⇒ (iv) is obvious. (iv) ⇒ (ii) We can assume that N/2 < p < ∞ (since L^{∞}(Ω) ⊂ L^{p}(Ω)) . Note that by (1.12) and Lemma 2.1, R(λ, Δ_{p}^{Ω}) is a bounded operator from L^{p}(Ω) into C^{b}(Ω) . Let F = {g ∈ L^{p}(Ω) : ∃ c ≥ 0 , |g| ≤ cf} . Then the hypothesis implies that R(λ, Δ_{p}^{Ω})F ⊂ C_{0}(Ω) . Since F is dense in L^{p}(Ω) , it follows that R(λ, Δ_{p}^{Ω})L^{p}(Ω) ⊂ C_{0}(Ω) . In particular, R(λ, Δ_{p}^{Ω})C_{0}(Ω) ⊂ C_{0}(Ω) . Now it follows from (1.8) that λ ∈ ϱ(Δ_{0}) . Since Δ_{0} is the part of Δ_{p} in C_{0}(Ω) .

Corollary 2.5 Let Ω be a bounded, open, non-empty, connected subset of IR^{N} . Then Ω is regular if and only if the eigenfunction u_{1} associated with the first eigenvalue of Δ_{2}^{Ω} is in C_{0}(Ω) .

We conclude this section by a remark concerning the realization of Dirichlet boundary conditions in L^{2}(Ω) . There is another choice, namely to replace H_{0}^{1}(Ω) by

H̃_{0}^{1}(Ω) := {u ∈ L^{2}(Ω) , ũ ∈ H^{1}(IR^{N})}

where

ũ(x) = \{ \begin{matrix}u(x) & x ∈ Ω \\ 0 & x ∉ Ω . \end{matrix}

The operator associated with the form

a(u, v) = ∫_{Ω} ∇u∇v

on H̃_{0}^{1}(Ω) is called the pseudo-Dirichlet Laplacian in [AB] and is denoted by Δ̃_{Ω} . Thus D(Δ̃_{Ω}) = {u ∈ H̃_{0}^{1}(Ω) : ∃ v ∈ L^{2}(Ω) such that ∫_{Ω} ∇u∇φ =

∫ vφ ∀ φ ∈ H̃_{0}^{1}(Ω)} and Δ̃_{Ω}u = v . Equivalently, D(Δ̃_{Ω}) = {u ∈ H̃_{0}^{1}(Ω) : Δu ∈ L^{2}(Ω)} , Δ̃_{Ω}u = Δu . For example, if N = 1 , Ω = (0, 1) ∪ (1, 2) , then Δ̃_{Ω} ≠ Δ_{2}^{Ω} , but if Ω has Lipschitz boundary then Δ̃_{Ω} = Δ_{2}^{Ω} . Now assume that the semigroup (e^{tΔ̃_{Ω}})_{t≥0} generated by Δ̃_{Ω} on L^{2}(Ω) leaves invariant C_{0}(Ω) . Hence R(1, Δ̃_{Ω})C_{0}(Ω) ⊂ C_{0}(Ω) . But then Lemma 2.2 implies that R(1, Δ̃_{Ω})C_{0}(Ω) ⊂ H_{0}^{1}(Ω) . Thus R(1, Δ̃_{Ω})f = R(1, Δ_{2}^{Ω})f for f ∈ C_{0}(Ω) . It follows that Δ̃_{Ω} = Δ_{2}^{Ω} ; i.e. H_{0}^{1}(Ω) = H̃_{0}^{1}(Ω) . Concluding we have the following result.

Proposition 2.6 Let Ω ⊂ IR^{N} be a bounded open set. Assume that

e^{tΔ̃_{Ω}}C_{0}(Ω) ⊂ C_{0}(Ω) (t > 0) .

Then Ω is regular and Δ̃_{Ω} = Δ_{2}^{Ω} .

3 Unbounded open sets

Whereas in the preceding paragraph, for bounded open sets, we established an equivalence with well-posedness of the Dirichlet problem, in this section it will be more convenient to consider barriers. Let Ω ⊂ IR^{N} be open.

Definition 3.1 a) Let z ∈ ∂Ω . A barrier is a function w ∈ C(\overline{Ω ∩ B}) such that

Δw ≤ 0 in D(Ω ∩ B)′ and w(z) = 0 , w(x) > 0 for x ∈ (Ω ∩ B) \ {z}

where B = B(z, r) is a ball centered at z . We call w an H^{1}− barrier if in addition w ∈ H^{1}(Ω ∩ B) .

b) We say that Ω is regular, if at each boundary point z ∈ ∂Ω there exists a barrier.

It is well-known that a bounded open set Ω is regular if and only if the Dirichlet problem is well-posed (see [GT], § 2.8 or [DL], II). For unbounded Ω the situation is more complicated since the behaviour at infinity has to be taken into account (see [DL], II § 4). Still, the existence of a barrier at each boundary point is the right regularity property in order that Δ_{0}^{Ω} be a generator. This is the assertion of Theorem 0.1 which will be proved below. We first show that Δ_{0}^{Ω} is dispersive. This is not new (cf. [LP1], [LP2]). In the bounded case, it follows immediately from our arguments since Δ_{0}^{Ω} is the part of Δ_{∞}^{Ω} in C_{0}(Ω) . We include a proof in the unbounded case for convenience of the reader. Recall that an operator A defined on the real space C_{0}(Ω) is called dispersive if for every u ∈ D(A) such that u^{+} ≠ 0 there exists x_{0} ∈ Ω such that

u^{+}(x_{0}) = ∥u^{+}∥_{∞} and (Au)(x_{0}) ≤ 0 . (3.1)

Dispersiveness on C_{0}(Ω) implies dissipativity (as is easy to see). In particular, (λ − A) is injective for λ > 0 whenever A is dispersive. This will be used in Theorem 3.7 below. More generally, a densely defined operator A generates a positive contractive C_{0} − semigroup if and only if A is dispersive and I − A is surjective (see [N], C-II.Theorem 1.2).

Proposition 3.2 The operator Δ_{0} is closed and dispersive.

Proof. It is obvious that Δ_{0} is closed. Let u ∈ D(Δ_{0}) , such that u^{+} ≠ 0 . Let (ϱ_{n})_{n∈IN} ⊂ C^{∞}(IR^{N}) be an approximate unit; i.e. ϱ_{n} ≥ 0 , ∫ ϱ_{n}(x)dx =

1 , supp ϱ_{n} ⊂ B(0, \frac{1}{n}) . Let u_{n} = ϱ_{n} ∗ u . Then u_{n} ∈ C_{0}(IR^{N}) and lim_{n→∞} u_{n} = u

in C_{0}(Ω) . Let n_{0} ∈ IN such that for all n ≥ n_{0} , ∥u_{n} − u∥_{∞} ≤ \frac{δ}{4} , δ = ∥u^{+}∥_{∞} . The set K = \{x ∈ Ω : u(x) ≥ \frac{δ}{2}\} is compact. Let x_{n} ∈ Ω such that u_{n}(x_{n}) = max_{y∈Ω} u_{n}(y) . Then u_{n}(x_{n}) ≥ \frac{3}{4}δ and so u(x_{n}) ≥ \frac{δ}{2} for n ≥ n_{0} . Hence

x_{n} ∈ K (n ≥ n_{0}) . Since K is compact, we can assume that x_{0} = lim_{n→∞} x_{n} exists. Then u(x_{0}) = ∥u^{+}∥_{∞} . Since u_{n} ∈ C^{∞}(Ω) we have (ϱ_{n}∗Δu)(x_{n}) = Δu_{n}(x_{n}) ≤ 0 . Since ϱ_{n} ∗ Δu converges to u uniformly, it follows that (Δu)(x_{0}) ≤ 0 .

Remark 3.3 More generally the following maximum principle holds, which clearly implies dispersiveness: If u ∈ C(Ω) has a local maximum at x ∈ Ω and Δu ∈ C(Ω) , then (Δu)(x) ≤ 0 .

We deduce from Proposition 3.2 that the operator Δ_{0}^{Ω} is the only possible realization of the Laplacian in C_{0}(Ω) which might generate a semigroup.

Corollary 3.4 Let A ⊂ Δ_{0}^{Ω} be the generator of a C_{0} − semigroup on C_{0}(Ω) . Then A = Δ_{0}^{Ω} .

Proof. There exists λ > 0 such that (λ − A) is surjective. Let u ∈ D(Δ_{0}^{Ω}) . There exists v ∈ D(A) such that (λ − A)v = (λ − Δ_{0}^{Ω})u . Since Av = Δ_{0}^{Ω}v it follows that u = v ∈ D(A) .

For the proof of Theorem 0.1 in the unbounded case we need some further preparation.

Lemma 3.5 Let Ω_{1}, Ω_{2} be regular, open subsets of IR^{N} . Then Ω_{1}∩Ω_{2} is regular.

Proof. Let Ω = Ω_{1} ∩ Ω_{2} . Then ∂Ω ⊂ ∂Ω_{1} ∪ ∂Ω_{2} . Let z ∈ ∂Ω . Suppose that z ∈ ∂Ω_{1} . Then there exists a barrier w ∈ C(\overline{Ω_{1} ∩ B}) where B = B(z, r) . Then w_{|\overline{Ω∩B}} is clearly a barrier on \overline{Ω ∩ B} at z .

We deduce from Lemma 3.5 the following

Lemma 3.6 Let Ω be a regular open subset of IR^{N} . Then for each z ∈ ∂Ω there exists an H^{1}− barrier at z .

Proof. It follows from Lemma 3.5 that Ω∩B is regular, where B = B(z, r) , z ∈ ∂Ω , r > 0 . Thus the Dirichlet problem is well-posed on Ω ∩ B . Let φ(x) = |x − z|^{2} . Then there exists v ∈ C(\overline{Ω ∩ B}) such that v(x) = φ(x) for all x ∈ ∂(Ω ∩ B) and Δv = 0 in D(Ω ∩ B)′ . It follows from the maximum principle that v(x) > 0 for all x ∈ B ∩ Ω . We show that v ∈ H^{1}(Ω ∩ B) . Note that φ ∈ C^{2}(\overline{Ω ∩ B}) and Δφ = const . Let u = v − φ . Then u ∈ C_{0}(Ω ∩ B) and Δu = −Δφ in D(Ω ∩ B)′ . It follows from Lemma 2.2 that u ∈ H_{0}^{1}(Ω ∩ B) . Thus v = u + φ ∈ H^{1}(Ω ∩ B) .

We recall the following simple fact which is easy to proof (cf. [B], IX).

Lemma 3.7 Let u ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω) . Then there exist u_{n} ∈ D(Ω) such that

∥u_{n}∥_{∞} ≤ ∥u∥_{∞} (n ∈ IN) and lim_{n→∞} u_{n} = u in H_{0}^{1}(Ω) .

Moreover, we recall that for a bounded open set Ω_{1} ⊂ IR^{N} one has

C_{0}(Ω_{1}) ∩ H^{1}(Ω_{1}) ⊂ H_{0}^{1}(Ω_{1}) (3.2)

(see [B], Remark 20, p. 172 or [Da], Theorem 1.5.7).

Theorem 3.8 Let Ω ⊂ IR^{N} be a regular, open set. Then Δ_{0}^{Ω} generates a holomorphic C_{0}−semigroup on C_{0}(Ω) . Moreover, Δ_{0}^{Ω} is the part of Δ_{∞}^{Ω} in C_{0}(Ω) and σ(Δ_{p}^{Ω}) = σ(Δ_{0}^{Ω}) for 1 ≤ p ≤ ∞ .

Proof. a) Let f ∈ C_{0}(Ω) ∩ L^{2}(Ω) , u = R(1, Δ_{2}^{Ω})f , i.e. u ∈ H_{0}^{1}(Ω) , u − Δu = f in D(Ω)′ . We show that u ∈ C_{0}(Ω) . It follows from local regularity that u is continuous and bounded.

1. Let z ∈ ∂Ω . We show that lim_{x→z} u(x) = 0 . Let ε > 0 . Choose w ∈ C^{1}(IR^{N}) such that Δw = u − f in D(Ω)′ and w(z) = ε (one can take w = E_{N} ∗ (u − f) + const) . Let v ∈ C(\overline{B ∩ Ω}) ∩ H^{1}(Ω ∩ B) be an H^{1}− barrier where B = B(z, r) is so small that w ≥ 0 on B̄ . Multiplying v with a positive constant if necessary, we can assume that v(x) > ∥u∥_{∞} on ∂B ∩ ∂(Ω ∩ B) . Then u − v − w ∈ H^{1}(Ω ∩ B) and (u − v − w)^{+} ∈ H_{0}^{1}(Ω ∩ B) . To see the last point, choose u_{n} ∈ D(Ω) such that ∥u_{n}∥_{∞} ≤ ∥u∥_{∞} and u_{n} → u in H^{1}(Ω) (by Lemma 3.7). Then (u_{n} − v − w)^{+} → (u − v − w)^{+} in H^{1}(Ω ∩ B) . Note that (u_{n}−v−w)^{+} ∈ C(\overline{Ω ∩ B}) vanishes on ∂(Ω∩B) = (∂(Ω∩B)∩∂Ω)∪(∂(Ω∩B)∩∂B) . Thus (u_{n} − v − w)^{+} ∈ H_{0}^{1}(Ω ∩ B) by (3.2). Thus (u − v − w)^{+} ∈ H_{0}^{1}(Ω ∩ B) and the claim is proved.

Now Δ(u − v − w) = −Δv ≥ 0 in D(Ω ∩ B)′ . Since u − v − w ∈ H^{1}(Ω) , it follows that ∫ ∇(u − v − w)∇φ ≤ 0 for all φ ∈ D(Ω ∩ B)_{+} and hence for all φ ∈ H_{0}^{1}(Ω∩B)_{+} . Taking φ = (u−v−w)^{+} , we deduce that ∫ |∇(u−v−w)^{+}|^{2} ≤ 0 . Hence (u − v − w)^{+} = 0 ; i.e., u ≤ w + v . Thus \overline{lim_{x→z}}u(x) ≤ \overline{lim_{x→z}}(v(x) + w(x)) = ε . Since ε > 0 is arbitrary, it follows that \overline{lim_{x→z}} u(x) ≤ 0 . Replacing u by −u we obtain lim_{{x→z \atop x∈∂Ω}} u(x) = 0 .

2. Since |u| ≤ R(1, Δ_{2})|f| (by (1.5)) and R(1, Δ_{2})C_{0}(IR^{N}) ⊂ C_{0}(IR^{N}) , it follows that lim_{|x|→∞} |u(x)| = 0 . Thus a) is proved.

b) Recall that R(1, Δ_{2}^{Ω}) and R(1, Δ_{∞}^{Ω}) are consistent. It follows from a) and by density that R(1, Δ_{∞}^{Ω})C_{0}(Ω) ⊂ C_{0}(Ω) . We show that 1 ∈ ϱ(Δ_{0}^{Ω}) and R(1, Δ_{0}^{Ω}) = R(1, Δ_{∞}^{Ω})_{|C_{0}(Ω)} . Let f ∈ C_{0}(Ω) . Then u = R(1, Δ_{∞}^{Ω})f ∈ C_{0}(Ω) and u − Δu = f in D(Ω)′ . (In fact, for φ ∈ D(Ω) we have < u − Δu , φ > = < u , φ − Δφ > = < R(1, Δ_{∞}^{Ω})f , φ − Δφ > = < f , R(1, Δ_{1}^{Ω})(φ − Δφ) > = < f, φ >) . Thus u ∈ D(Δ_{0}) and u − Δ_{0}u = f . Conversely, let u ∈ D(Δ_{0}^{Ω}) and f = u − Δu . Let v = R(1, Δ_{∞}^{Ω})f . Then w = u − v ∈ D(Δ_{0}) and w − Δ_{0}w = 0 . Since Δ_{0} is dissipative, it follows that w = 0 . We have shown that 1 ∈ ϱ(Δ_{0}^{Ω}) and R(1, Δ_{0}^{Ω}) = R(1, Δ_{∞}^{Ω})_{|C_{0}(Ω)} . Thus Δ_{0}^{Ω} is a generator and Δ_{0}^{Ω} is the part of Δ_{∞}^{Ω} in C_{0}(Ω) .

c) It is clear from analyticity of the resolvent, that the set U = {λ ∈ ϱ(Δ_{∞}^{Ω}) : R(λ, Δ_{∞}^{Ω})C_{0}(Ω) ⊂ C_{0}(Ω)} is open and closed in ϱ(Δ_{∞}^{Ω}) . Since ϱ(Δ_{∞}^{Ω}) is connected it follows that U = ϱ(Δ_{∞}^{Ω}) ; i.e. ϱ(Δ_{∞}^{Ω}) ⊂ ϱ(Δ_{0}^{Ω}) . We have shown that σ(Δ_{0}^{Ω}) ⊂ σ(Δ_{∞}^{Ω}) = σ(Δ_{2}^{Ω}) . In order to show the converse denote by B the adjoint of Δ_{0}^{Ω} on C_{0}(Ω)′ . Let λ > 0 , f ∈ L^{1}(Ω) ∩ L^{2}(Ω) . Then for φ ∈ D(Ω) , < φ , R(λ, B)f > = <R(λ, Δ_{0}^{Ω})φ , f > = <φ , R(λ, Δ_{2}^{Ω})f > = <φ , R(λ, Δ_{1}^{Ω})f > . Hence R(λ, B)L^{1}(Ω) ⊂ L^{1}(Ω) and R(λ, B)_{|L^{1}(Ω)} = R(λ, Δ_{1}^{Ω}) . As before, it follows that ϱ(Δ_{0}^{Ω}) = ϱ(B) ⊂ ϱ(Δ_{1}^{Ω}) = ϱ(Δ_{2}^{Ω}) . Thus σ(Δ_{2}^{Ω}) ⊂ σ(Δ_{0}^{Ω}) .

Of course, if Ω is regular, the semigroup (e^{tΔ_{0}^{Ω}})_{t≥0} generated by Δ_{0}^{Ω} is consistent with (e^{tΔ_{p}^{Ω}})_{t≥0} i.e.

e^{tΔ_{0}^{Ω}}f = e^{tΔ_{p}^{Ω}}f (f ∈ C_{0}(Ω) ∩ L^{p}(Ω) , t ≥ 0 , 1 ≤ p ≤ ∞) . (3.3)

This follows from consistency of the resolvents.

Next we show that Δ_{0}^{Ω} has compact resolvent whenever Ω is regular and has finite Lebesgue measure. This follows from the following more general result.

Proposition 3.9 Let Ω be a regular open subset of IR^{N} with finite Lebesgue measure. Then e^{tΔ_{0}^{Ω}} is a compact operator on C_{0}(Ω) for all t > 0 .

Proof. Let t > 0 . Then e^{tΔ_{2}^{Ω}} is a bounded operator from L^{2}(Ω) into L^{∞}(Ω) . We have seen that e^{tΔ_{2}^{Ω}}C_{0}(Ω) ⊂ C_{0}(Ω) . Hence by density e^{tΔ_{2}^{Ω}}L^{2}(Ω) ⊂ C_{0}(Ω) . Denote by j : C_{0}(Ω) → L^{2}(Ω) the canonical injection. Then e^{tΔ_{0}^{Ω}} can be written as

e^{\frac{t}{2} Δ_{2}^{Ω}} ∘ e^{\frac{t}{2} Δ_{2}^{Ω}} ∘ j : C_{0}(Ω) → L^{2}(Ω) → L^{2}(Ω) → C_{0}(Ω) .

Since e^{\frac{t}{2} Δ_{2}^{Ω}} is a compact operator on L^{2}(Ω) , the claim follows.

In order to prove the converse of Theorem 3.1 we establish a spectral characterization of Δ_{0}^{Ω} being a generator.

Proposition 3.10 Let Ω ⊂ IR^{N} be an open set. The following are equivalent:

(i) Δ_{0}^{Ω} is a generator;

(ii) ϱ(Δ_{0}^{Ω}) ≠ ∅ ;

(iii) R(λ, Δ_{2}^{Ω})(L^{2}(Ω) ∩ C_{0}(Ω)) ⊂ C_{0}(Ω) .

Proof. (i) ⇒ (ii) This is clear. (ii) ⇒ (iii) If ϱ(Δ_{0}^{Ω}) ≠ ∅ , since σ(Δ_{2}^{Ω}) ⊂ IR , there exists λ ∈ ϱ(Δ_{0}^{Ω}) ∩ ϱ(Δ_{2}^{Ω}) . Let B be the adjoint of Δ_{0}^{Ω} . We show that R(λ, B)L^{1}(Ω) ⊂ L^{1}(Ω) and R(λ, B)_{|L^{1}} = R(λ, Δ_{1}^{Ω}) . Let g ∈ L^{1}(Ω) ∩ L^{2}(Ω) , u = R(λ, B)g . Let φ ∈ D(Ω) . v = R(λ, Δ_{0}^{Ω})φ . Then v ∈ C_{0}(Ω) and λv − Δv = φ in D(Ω)′ . As in the proof of Lemma 2.2 one sees that v ∈ H_{0}^{1}(Ω) . Thus v = R(λ, Δ_{2}^{Ω})φ . Consequently, < φ, R(λ, B)g > = < R(λ, Δ_{0}^{Ω})φ, g > = < R(λ, Δ_{2}^{Ω})φ, g > =< φ, R(λ, Δ_{2}^{Ω})g > = < φ, R(λ, Δ_{1}^{Ω})g > . We have shown that R(λ, Δ_{1}^{Ω})g = R(λ, B)g for all g ∈ L^{1}(Ω) ∩ L^{2}(Ω) , hence for all g ∈ L^{1}(Ω) . As in the proof of Theorem 3.8 we deduce that ϱ(Δ_{1}^{Ω}) ⊂ ϱ(B) = ϱ(Δ_{0}^{Ω}) and R(λ, B)_{|_{L^{1}(Ω)}} = R(λ, Δ_{1}^{Ω}) for all λ ∈ ϱ(Δ_{1}^{Ω}) . Let f ∈ C_{0}(Ω) ∩ L^{2}(Ω) . Then for g ∈ L^{1}(Ω) ∩ L^{2}(Ω) , < R(λ, Δ_{0}^{Ω})f, g > = < f, R(λ, B)g > = < f, R(λ, Δ_{2}^{Ω})g > = < R(λ, Δ_{2}^{Ω})f, g > . Hence R(λ, Δ_{0}^{Ω})f = R(λ, Δ_{2}^{Ω})f . Thus (iii) is proved. (iii) ⇒ (i) Let f ∈ L^{2}(Ω)∩C_{0}(Ω) . Then by hypothesis u = R(λ, Δ_{2}^{Ω})f ∈ C_{0}(Ω) . Thus u ∈ D(Δ_{0}^{Ω}) and u − Δ_{0}u = f . We have shown that λ − Δ_{0}^{Ω} has dense image. Since Δ_{0}^{Ω} is closed and dissipative, it follows that Δ_{0}^{Ω} is a generator.

Corollary 3.11 Let Ω_{1}, Ω_{2} ⊂ IR^{N} be open. If Δ_{0}^{Ω_{1}} and Δ_{0}^{Ω_{2}} are generators, then Δ_{0}^{Ω_{1}∩Ω_{2}} is a generator.

Proof. Let 0 ≤ f ∈ C_{0}(Ω) ∩ L^{2}(Ω) , Ω = Ω_{1} ∩ Ω_{2} . Since by Proposition 1.1 R(λ, Δ_{2}^{Ω})f ≤ R(λ, Δ_{2}^{Ω_{1}})f and R(λ, Δ_{2}^{Ω})f ≤ R(λ, Δ_{2}^{Ω_{2}})f and ∂Ω ⊂ ∂Ω_{1} ∪ ∂Ω_{2} , it follows that R(λ, Δ_{2}^{Ω})f ∈ C_{0}(Ω) .

Corollary 3.12 Assume that Ω ⊂ IR^{N} is open. If Δ_{0}^{Ω} is a generator, then at each z ∈ ∂Ω there exists an H^{1}− barrier. In particular, Ω is regular.

Proof. Let z ∈ ∂Ω , B = B(z, r) , r > 0 . By Corollary 3.11, the operator Δ_{0}^{Ω∩B} is a generator on C_{0}(B ∩Ω) . By Theorem 2.4 this implies that the Dirichlet problem on B ∩ Ω is well-posed. Taking φ(x) = |x − z|^{2} on the boundary, the solution of the Dirichlet problem on B ∩ Ω gives an H^{1}− barrier.

Concluding, we mention that the proofs given in Section 2 and 3 are almost selfcontained. In particular, for a bounded open set Ω in IR^{N} , we gave a complete proof of the equivalence of the following three assertions

(a) the Dirichlet problem on Ω is well-posed;

(b) at each z ∈ ∂Ω there exists an H^{1}− barrier;

(c) Δ_{0}^{Ω} is a generator.

For the fact that condition (b) can be replaces by the more general condition that Ω is regular we refer to classical Potential Theory. Concerning holomorphy of the semigroup generated by Δ_{0} the following should be added: The case where Ω is bounded and of class C^{∞} is due to Stewart [Stu]. Lumer and Paquet [LP1] (see also [LP2]) proved by a beautiful dissipativity argument that the semigroup (e^{tΔ_{0}})_{t≥0} is holomorphic whenever Δ_{0} is a generator. The duality argument we give here has the advantage to give the optimal angle. It was first used by Ouhabaz [Ou2].

4 Elliptic Operators on C_{0}(Ω)

Whereas Section 2 and 3 were self-contained, using only elementary results of potential theory, the following investigation of parabolic equations on C_{0}(Ω) depends on results by Stampacchia [Sta] among others who studied boundary behavior for solutions of elliptic equations. We consider merely bounded open sets generalizing the approach of Section 2.

Let Ω ⊂ IR^{N} be a bounded open set. We introduce elliptic operators using the notation of Gilbarg-Trudinger [GT], Chapter 8. Let a_{ij} ∈ L^{∞}(Ω) (i, j = 1, …, N) be real functions such that

∑_{i,j=1}^{N} a_{ij}(x)ξ_{i}ξ_{j} ≥ α|ξ|^{2}

for all ξ ∈ IR^{N} , x− a.e., where α > 0 , and let d, b_{j}, c_{j} ∈ L^{∞}(Ω) be real coefficients, j = 1, … N . We consider the elliptic operator L , formally given by

Lu = ∑_{i=1}^{N} D_{i}(∑_{j=1}^{N} a_{ij}D_{j}u + b_{i}u) + ∑_{i=1}^{N} c_{i}D_{i}u + du .

Defining the form

a(u, v) = ∫_{Ω} { ∑_{i,j=1}^{N} a_{ij}D_{j}uD_{i}v + ∑_{i=1}^{N}(b_{i}uD_{i}v − c_{i}D_{i}uv) − duv}dx (4.1)

for u ∈ H_{loc}^{1}(Ω) , v ∈ D(Ω) , we can realize L as an operator. L : H_{loc}^{1}(Ω) → D(Ω)′ given by

< Lu , v > = −a(u, v) (u ∈ H_{loc}^{1}(Ω) , v ∈ D(Ω)) . (4.2)

We observe furthermore that

(4.3) Lu ∈ (H_{0}^{1}(Ω))′ whenever u ∈ H^{1}(Ω) .

We define the operator A_{0} on C_{0}(Ω) as the part of L in C_{0}(Ω) ; i.e.

D(A_{0}) = {u ∈ C_{0}(Ω) ∩ H_{loc}^{1}(Ω) : Lu ∈ C_{0}(Ω)} A_{0}u = Lu .

In the following we will assume throughout that

∑_{j=1}^{N} D_{j}b_{j} + d ≤ 0 in D(Ω)′ . (4.4)

Then the following is the main result of this section.

Theorem 4.1 Assuming (4.4), the following are equivalent:

(i) Ω is regular;

(ii) ϱ(A_{0}) ≠ ∅ ;

(iii) A_{0} generates a positive, contractive C_{0} − semigroup on C_{0}(Ω) .

By A_{2} we denote the realization of L in L^{2}(Ω) with Dirichlet boundary conditions; i.e.

D(A_{2}) = {u ∈ H_{0}^{1}(Ω) : Lu ∈ L^{2}(Ω)} A_{2}u = Lu .

Note that −A_{2} is associated with the form a on the form domain H_{0}^{1}(Ω) ; i.e., for u, v ∈ L^{2}(Ω) one has

u ∈ D(A_{2}) , A_{2}u = v ⇔ u ∈ H_{0}^{1}(Ω) and a(u, φ) = − ∫_{Ω} vφ for all φ ∈ H_{0}^{1}(Ω) .

The form a is elliptic; i.e. for some β > 0 , w ≥ 0 one has

a(u, u) + w(u|u)_{L^{2}} ≥ β∥u∥_{H^{1}}^{2} (4.5)

for all u ∈ H_{0}^{1}(Ω) . Thus A_{2} generates a holomorphic C_{0} − semigroup T_{2} on L^{2}(Ω) . It follows from the first Beurling-Deny criterion that T_{2}(t) ≥ 0 (t ≥ 0) (see [Ou1]). In virtue of (4.4), by the second Beurling-Deny criterion one has for f ∈ L^{2}(Ω) ∩ L^{∞}(Ω) ,

∥λR(λ, A_{2})f∥_{∞} ≤ ∥f∥_{∞} (λ > 0) (4.6)

or equivalenty, ∥T_{2}(t)f∥_{∞} ≤ ∥f∥_{∞} (t > 0) (4.7)

(see [Ou1]). Actually (4.6) (and (4.7)) are equivalent to (4.4) (see [ABBO]). It follows from (4.6) that there are operators A_{p} on L^{p}(Ω) , 2 ≤ p ≤ ∞ , such that (0, ∞) ⊂ ϱ(A_{p}) and R(λ, A_{p}) = R(λ, A_{2})_{|_{L^{p}(Ω)}} . Moreover, for 2 < p < ∞ , by (4.7) the restriction of T_{2} to L^{p} is a C_{0} − semigroup T_{p} on L^{p}(Ω) whose generator is A_{p} .

Lemma 4.2 Let u ∈ H_{loc}^{1}(Ω) ∩ C_{0}(Ω) such that Lu ∈ L^{2}(Ω) . Then u ∈ H_{0}^{1}(Ω) .

Proof. Let v = Lu . Then

a(u, φ) = − ∫ vφ for all φ ∈ D(Ω) .

Let ε > 0 . Then (u − ε)^{+} ∈ H_{0}^{1}(Ω) . Let ω be open such that ω̄ ⊂ Ω . Since u ∈ H^{1}(ω) , (u − ε)^{+} ∈ H_{0}^{1}(Ω) , we have

a(u, (u − ε)^{+}) = − ∫_{Ω} v(u − ε)^{+}dx ≤ ∥v∥_{L^{2}}∥u∥_{L^{2}} .

Since D_{j}(u − ε)^{+} = 1_{{u>ε}}D_{j}u we have

a(u, (u − ε)^{+}) =

a((u − ε)^{+} , (u − ε)^{+}) + ε ∫_{Ω} ∑_{i=1}^{N} b_{i}D_{i}(u − ε)^{+}dx − ε ∫_{Ω} d(u − ε)^{+}dx

≥ a((u − ε)^{+} , (u − ε)^{+}) (by (4.4)) ≥ β∥(u − ε)^{+}∥_{H^{1}}^{2} − ω∥(u − ε)^{+}∥_{L^{2}}^{2} .

Thus β∥(u − ε)^{+}∥_{H^{1}}^{2} ≤ a(u , (u − ε)^{+}) + ω∥u∥_{L^{2}}^{2} ≤ ∥v∥_{L^{2}}∥u∥_{L^{2}} + ω∥u∥_{L^{2}}^{2} . As in the proof of Lemma 2.2 we deduce from this that u^{+} ∈ H_{0}^{1}(Ω) . Similarly, u^{−} ∈ H_{0}^{1}(Ω) .

Corollary 4.3 The operator A_{0} is the part of A_{2} in C_{0}(Ω) . In particular, A_{0} is closed and dissipative.

Proof. It follows from Lemma 4.2 that A_{0} is the part of A_{2} in C_{0}(Ω) . Since A_{2} is closed, A_{0} is closed as well. In order to show dissipativity, let λ > 0 , u ∈ D(A_{0}) , λu − A_{0}u = f . Since A_{0} is the part of A_{2} in C_{0}(Ω) , it follows that u = R(λ, A_{2})f . Now by (4.6), ∥λu∥_{∞} ≤ ∥f∥_{∞} .

Theorem 4.4 Assume that Ω is regular, and that (4.4) is satisfied. Then A_{0} generates a positive contractive C_{0} − semigroup T_{0} on C_{0}(Ω) . Moreover, T_{0}(t) = T_{2}(t)_{|_{C_{0}(Ω)}} (t ≥ 0) .

Proof. 1. We extend the coefficients to IR^{N} by setting

a_{ij}(x) = \{ \begin{matrix}1 & if & i = j \\ 0 & if & i ≠ j \end{matrix}

for x ∈ IR^{N} \ Ω and b_{i}(x) = c_{i}(x) = d(x) = 0 for x ∈ IR^{N} \ Ω . Denote by L̃ : H_{loc}^{1}(IR^{N}) → D(IR^{N})′ the corresponding elliptic operator and by Ã_{2} its realization in L^{2}(IR^{N}) . Let λ > max{0, ω(Ã_{2})} , where ω(Ã_{2}) is the type of Ã_{2} . We show that λ − A_{0} is surjective, which proves our claim in view of Corollary 4.3. Let g ∈ C_{0}(Ω) , v = R(λ, Ã_{2})g . Then v ∈ H^{1}(IR^{N}) and v − Lv = g in D(Ω)′ . By the famous result of De Georgi and Nash [GT], Theorem 8.22, the function v is continuous on IR^{N} . Let φ = v_{|_{∂Ω}} . By [GT], Theorem 8.31, (in the case where Ω is connected, see [Sta], Section 10 for the general case), there exists w ∈ H_{loc}^{1}(Ω) ∩ C(Ω̄) such that w − Lw = 0 in D(Ω)′ and ω_{|∂Ω} = φ . Now let u = v−w . Then u ∈ C_{0}(Ω)∩H_{loc}^{1}(Ω) and u−Lu = g in D(Ω)′ . Thus u ∈ D(A_{0}) and u−A_{0}u = g . We have shown that A_{0} generates a contractive C_{0}−semigroup . Since A_{0} is the part of A_{2} , it follows that R(λ, A_{2})_{|_{C_{0}(Ω)}} = R(λ, A_{0}) (λ > 0) which implies the last claim in the proposition. We have shown that (I − A_{0}) is surjective. Since A_{0} is dissipative, it follows that (0, ∞) ⊂ ϱ(A_{0}) and ∥λR(λ, A_{0})∥ ≤ 1 (λ > 0) . Moreover, A_{0} being the part of A_{2} in C_{0}(Ω) , we have R(λ, A_{0}) = R(λ, A_{2})_{|_{C_{0}(Ω)}} for λ > 0 .

2. We show that D(A_{0}) is dense in C_{0}(Ω) . For that we have to show that R(λ, A_{0})′ ∈ L(C_{0}(Ω)′) is injective. Let μ ∈ C_{0}(Ω)′ such that R(λ, A_{0})′μ = 0 . Take μ_{n} ∈ L^{1}(Ω)∩L^{2}(Ω) such that sup_{n∈IN} ∥μ_{n}∥ < ∞ and lim_{n→∞} μ_{n} = μ for the topol-

ogy σ(C_{0}(Ω)′ , C_{0}(Ω)) . Let v_{n} = R(λ, A_{0})′μ_{n} . Let 1 < q < \frac{N}{N−1} . By a result of Stampacchia [Sta], Théorème 4.4, R(λ, A_{0})′ is a bounded operator from L^{1}(Ω) into W_{0}^{1,q}(Ω) . By reflexivity, we can assume that (v_{n})_{n∈IN} converges weakly to v in W_{0}^{1,q}(Ω) (choosing a subsequence otherwise). Since w^{∗}− lim_{n→∞} v_{n} = R(λ, A_{0})′μ = 0 , we have v = 0 . Since R(λ, A_{0}) = R(λ, A_{2})_{|_{C_{0}(Ω)}} one has v_{n} = R(λ, A_{2})′μ_{n} . Hence for φ ∈ D(Ω) ,

∫ φdμ = lim_{n→∞} ∫ μ_{n}φ =

lim_{n→∞}((I − A_{2})′v_{n}|φ)_{L^{2}} =

lim_{n→∞}{∫ v_{n}φ + a(φ, v_{n})} = 0 .

It follows that μ = 0 .

3. We have shown that A_{0} generates a contractive C_{0} − semigroup T_{0} on C_{0}(Ω) . Since R(λ, A_{0}) = R(λ, A_{2})_{|_{C_{0}(Ω)}} (λ > 0) , it follows that

T_{0}(t)f = lim_{n→∞}(I − \frac{t}{n}A_{0})^{−n}f = lim_{n→∞}(I − \frac{t}{n}A_{2})^{−n}f = T_{2}(t)f (f ∈ C_{0}(Ω)) .

We introduce the following notation. Let S ∈ L(L^{p}(Ω)) . Then

∥S∥_{L(L^{q},L^{r})} = sup{∥Sf∥_{r} : f ∈ L^{q} ∩ L^{p} , ∥f∥_{q} ≤ 1}

where 1 ≤ p , q , r ≤ ∞ . We recall the following result on ultracontractivity (see [V], théorème 2 or [C], Lemma 1).

Theorem 4.5 Let T be a C_{0} − semigroup on L^{2}(Ω) such that

∥T (t)f∥_{∞} ≤ M_{∞}∥f∥_{∞} (f ∈ L^{2} ∩ L^{∞} , 0 < t ≤ 1) (4.8)

and

∥T (t)f∥_{q} ≤ M_{q}t^{−α}∥f∥_{2} (f ∈ L^{2} ∩ L^{q} , 0 < t ≤ 1) (4.9)

where 2 < q < ∞ , α > 0 , M_{q} ≥ 0 , M_{∞} ≥ 0 . Then there exists a constant c > 0 such that

∥T (t)f∥_{∞} ≤ ct^{−β}∥f∥_{2} (0 < t ≤ 1) (4.10)

where β = \frac{α}{1− \frac{2}{q}} .

Corollary 4.6 Assume that Ω is a bounded regular open set. Then the operators T_{0}(t) (t > 0) are compact. Moreover,

σ(A_{0}) = σ(A_{2}) .

Proof. Since T_{2} is holomorphic, one has sup_{0<t≤1} ∥tA_{2}T_{2}(t)∥_{L(L^{2})} < ∞ . Since

H_{0}^{1}(Ω) ↪ L^{2N/N−2} , it follows that

sup_{0<t≤1} ∥tT_{2}(t)∥_{L(L^{2},L^{2N/N−2})} < ∞ .

Now it follows from Theorem 4.5 that

∥T_{2}(t)∥_{L(L^{2},L^{∞})} ≤ ct^{−N/2} (0 < t ≤ 1) . (4.11)

Since the injection H_{0}^{1}(Ω) ↪ L^{2}(Ω) is compact, it follows as in the proof of Proposition 3.8 that T_{0}(t) is compact (t > 0) . In particular, A_{0} has compact resolvent. Now it follows from [Ar], Proposition 2.6 that σ(A_{0}) = σ(A_{2}) .

Corollary 4.7 Assume that b_{i}, c_{i} ∈ W^{1,∞}(Ω) i = 1, …, N . If Ω is a bounded, regular, open set, then A_{0} generates a holomorphic C_{0} − semigroup on C_{0}(Ω) .

Proof. It follows from [Ou1] that there exists a C_{0} − semigroup T_{1} on L^{1}(Ω) such that T_{1}(t)_{|_{L^{2}(Ω)}} = T_{2}(t) (t ≥ 0) . Denote the generator of T_{1} by A_{1} . It follows from [AE] that T_{1} is holomorphic. By duality

∥λR(λ, A_{∞})∥ ≤ M (Re λ > ω)

for some ω, M ≥ 0 . This implies that ∥λR(λ, A_{0})∥ ≤ M (Re λ > ω) , so A_{0} generates a holomorphic C_{0} − semigroup .

Remark 4.8 In Corollary 4.7 the hypothesis (4.4) is not needed a priori: it is satisfied if we replace L by L − ω for a suitable ω .

Lemma 4.9 Let ∞ > p > N . Then

D(A_{p}) ⊂ L^{∞}(Ω) .

Moreover,

D(A_{2}^{k}) ⊂ L^{∞}(Ω)

for k > max\{\frac{N}{2} , 1\} .

Proof. By (4.11) we have

∥T_{2}(t)∥_{L(L^{2},L^{∞})} ≤ ct^{−N/2} (0 < t ≤ 1) . (4.12)

By interpolation we deduce from this that

∥T_{p}(t)∥_{L(L^{p},L^{∞})} ≤ c_{p}t^{−N/p} (0 < t ≤ 1) (4.13)

for some c_{p} ≥ 0 , 2 ≤ p < ∞ . Let p > N . Let M ≥ 0 , ω ≥ 0 such that ∥T_{p}(t)∥_{L(L^{p})} ≤ Me^{ωt} . Then for λ > ω ,

∥R(λ, A_{p})f∥_{∞} ≤ ∫_{0}^{1} e^{−λt}∥T (t)f∥_{∞} dt + ∫_{0}^{∞} e^{−λ(t+1)}∥T (1)T (t)f∥_{∞} dt
≤ c_{p} ∫_{0}^{1} t^{−N/p} dt ∥f∥_{p} + c_{p} ∫_{0}^{∞} e^{−λ(t+1)}∥T (t)f∥_{p} dt
≤ c_{p}{∫_{0}^{1} t^{−N/p} dt + ∫_{0}^{∞} e^{−λ(t+1)}Me^{ωt} dt}∥f∥_{p}
≤ const. · ∥f∥_{p} .

Similarly, for λ sufficiently large, since

R(λ, A_{2})^{k} = (k − 1)! ∫_{0}^{∞} e^{−λt}t^{k−1}T_{2}(t)dt

one obtains by (4.13) that ∥R(λ, A_{2})^{k}∥_{L(L^{2},L^{∞})} < ∞ if k > \frac{N}{2} .

In order to prove the remaining implication (ii) ⇒ (i) of Theorem 4.1 we consider the Dirichlet problem with respect to L−I . For this, fix Ω̃ a large ball containing Ω̄ . Let h ∈ C(∂Ω) be the trace of a function w ∈ H^{1}(Ω̃) ∩ C(Ω̃) ; i.e. h = w_{|_{∂Ω}} . Then there exists a unique function u ∈ H^{1}(Ω) ∩ C^{b}(Ω) such that

Lu − u = 0 in D(Ω)′ and u − w ∈ H_{0}^{1}(Ω) (4.14)

(see [Sta], Théorème 10.1 and [GT], 8.22). We interprete (4.14) as a weak form of u_{|_{∂Ω}} = h . Then by the maximum principle (see [Sta], § 10 or [GT], 8.1),

∥u∥_{L^{∞}(Ω)} ≤ ∥h∥_{C(∂Ω)} . (4.15)

We set u = Bh . Since by the Stone-Weierstraß theorem traces of functions w ∈ C(Ω̃) ∩ H^{1}(Ω̃) are dense in C(∂Ω) , we find a unique linear extension

B : C(∂Ω) → H^{1}(Ω) ∩ C^{b}(Ω)

such that ∥Bh∥_{L^{∞}(Ω)} ≤ ∥h∥_{C(∂Ω)} . (4.16)

Recall that L : H^{1}(Ω) → D(Ω)′ is a continuous linear mapping. Let h ∈ C(∂Ω) , then u = Bh is the solution of the Dirichlet problem

\{ \begin{matrix}u ∈ H^{1}(Ω) \\ Lu − u = 0 in D(Ω)′ \\ u_{|_{∂Ω}} = h \end{matrix} (4.17)

where the last identity has to be understood in the sense of the construction.

It turns out that Ω is regular (which means by our definition regular with respect to the Laplacian) if and only if Ω is regular with respect to L . More precisely, the following remarkable result is due to Stampacchia [Sta], § 10 and Littmann, Stampacchia, Weinberger [LSW].

Theorem 4.10 The following are equivalent:

(i) Ω is regular;

(ii) lim_{{x→z \atop x∈Ω}}(Bh)(x) = h(z) and all z ∈ ∂Ω for all h ∈ C(∂Ω) .

Recall that here L is a fixed elliptic operator satisfying (4.4).

On the basis of this result we now deduce the following.

Proposition 4.11 Assume (4.4). If ϱ(A_{0}) ≠ ∅ , then Ω is regular.

Proof. 1. It follows from Lemma 4.8 and Proposition 1.3 that σ(A_{2}) = σ(A_{∞}) . Since by hypothesis ϱ(A_{0}) ≠ ∅ and since σ(A_{∞}) ⊂ IR , it follows that ϱ(A_{0}) ∩ ϱ(A_{∞}) ≠ ∅ . As in the proof of Theorem 4.4 we conclude that λ ∈ ϱ(A_{0}) . Let p > N . Then by Lemma 4.8, R(λ, A_{p})L^{p}(Ω) ⊂ L^{∞}(Ω) . Since R(λ, A_{p})C_{0}(Ω) = R(λ, A_{0})C_{0}(Ω) ⊂ C_{0}(Ω) , it follows by density that R(λ, A_{p})L^{p}(Ω) ⊂ C_{0}(Ω) .

2. Let Ω̃ be a ball containing Ω̄ . Note that Ω̃ is regular. Consider the elliptic operator L̃ : H_{loc}^{1} ˜(Ω) → D(Ω̃)′ one obtains by extending the coefficients as in the proof of Theorem 4.4 and let A_{0} be its realization in C_{0}(Ω̃) . Then 1 ∈ ϱ(Ã_{0}) by Theorem 4.4. Since Ã_{0} has dense domain in C_{0}(Ω̃) , and since each function in C(∂Ω) has an extension to a function in C_{0}(Ω̃) , the space

F = {h ∈ C(∂Ω) : ∃ g ∈ C_{0}(Ω̃) , h = (R(λ, Ã_{0})g)_{|_{∂Ω}}}

is dense in C(∂Ω) . Let h ∈ F , h = w_{|_{∂Ω}} , w = R(1, Ã_{0})g , g ∈ C_{0}(Ω̃) . Then g_{|_{Ω}} ∈ L^{p}(Ω) . Let v = R(λ, A_{p})(g_{|_{Ω}}) . Then v ∈ C_{0}(Ω) by 1 . Moreover, we have

v − Lv = g in D(Ω)′ and
w − Lw = g in D(Ω)′ .

Let u = (w − v)_{|_{Ω}} . Then u ∈ H^{1}(Ω) ∩ C(Ω̄) and u − Lu = 0 in D(Ω)′ and u_{|_{∂Ω}} = h . Thus (ii) of Theorem 4.9 is satisfied for all h ∈ F and hence for all h ∈ C(∂Ω) by density.

The proof of Theorem 4.1 is complete now.

Concluding Remark For elliptic operators we restricted ourselves to bounded open sets. In order to carry over the strategy used for the Laplacian in Section 3 further arguments are needed. One can actually show that L_{0} is dissipative under condition (4.4). However, one obtains barriers with respect to L merely in H_{loc}^{1} which presents an additional difficulty.

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W. Arendt  Ph. Bénilan
Universität Ulm  Université de Franche-Comté
Abteilung Mathematik V  Equipe de Mathématiques
D - 89069 Ulm, Germany  UMR CNRS 6623
arendt@mathematik.uni-ulm.de  F - 25030 Besançon, France

benilan@math.univ-fcomte.fr