TIME–DEPENDENT PARABOLIC PROBLEMS ON NON–CYLINDRICAL DOMAINS WITH INHOMOGENEOUS BOUNDARY CONDITIONS

GÜNTER LUMER AND ROLAND SCHNAUBELT

To the memory of Ralph Phillips, inspiring mathematician and human being, collaborator and friend

Abstract. We study the relationship between the Dirichlet problem and the Cauchy problem with inhomogeneous boundary conditions for local operators. Our results are applied to non–autonomous parabolic problems on non–cylindrical domains.

1. Introduction

In this paper we continue the investigation of non–autonomous parabolic problems on non–cylindrical domains begun in [14]. In that work we have characterized the wellposedness of the problem with homogeneous Dirichlet boundary conditions in terms of a barrier condition using the concepts of semigroups and of local operators, cf. [10], [14]. Our main tool from semigroup theory was the Lumer–Phillips theorem, [12]. We now want to treat inhomogeneous boundary conditions in the same framework.

In Section 2 we first work in the following more general setting. Let V be a bounded open subset of a locally compact metric space Ω and A be a linear operator on C(V). Following [11], we let A_{0} be the part of A in C_{0}(V) and define on C(\overline{V}) the operator A_{1} with Ventcel boundary conditions, cf. (2.1). Assuming closedness and dissipativity of A, we prove that the Dirichlet problem

Au = 0 on V,  u = φ on ∂V,  (1.1)

has a unique solution u ∈ D(A)∩C(\overline{V}) for each φ ∈ C(∂V) provided that A_{0} is invertible on C_{0}(V) and A_{1} is densely defined on C(\overline{V}), see Corollary 2.7. In fact, these properties characterize the solvability of (1.1) (under a mild extra assumption) by Theorem 2.13. Then A_{1} generates a mean ergodic semigroup and the corresponding ergodic projection yields the solution operator of (1.1). We point out that there are degenerate problems such that A_{0} is a generator but A_{1} is not densely defined and (1.1) has no solution for certain φ, see Example 2.8. In the regular case this does not happen as shown in [4, Thm.4.1] (together with [4, Thm.4.10]) for uniformly elliptic operators A and in Theorem 3.5 for parabolic problems on non–cylindrical domains.

Combining these results with the theory developed in [14], we investigate in Section 3 the parabolic problem

Lu = F on \underline{V}_{0},  u = f on Γ ∪ V (0),  (1.2)

1991 Mathematics Subject Classification. 35J25, 35K20, 47D06.

Key words and phrases. Dirichlet problem, inhomogeneous initial boundary value problem, semigroup, mean ergodic theorem, local operator, Cauchy barrier, variable space propagator, non-cylindrical domain.

on a non–cylindrical domain \underline{V}_{0} ⊆ (0, T] × Ω with lateral boundary Γ and bottom V (0), where L is a parabolic differential operator with continuous coefficients and F, f are continuous functions. This setting covers diffusion processes on open subsets of (0, T]×ℝ^{N} as well as on networks, see [14, §6]. In particular, we show in the non–degenerate case that (1.2) is solvable if and only if the corresponding homogeneous problem is well–posed, Theorem 3.5. Moreover, a problem of singular multiplicative perturbation is treated. We also study regularity properties of the variable space propagator solving the homogeneous problem in Theorem 3.6.

Different semigroup approaches to the Dirichlet problem and inhomogeneous initial boundary value problem are developed in e.g. [3], [4], [8]. These works, however, do not cover non–autonomous problems. We refer to [14] for further literature concerning local operator methods and parabolic problems on non–cylindrical domains, see also [1].

2. Dirichlet and Cauchy problems

Let Ω be a locally compact Hausdorff space having a countable base, Ω^{∗} be its one– point compactification, and V ≠ ∅ be a fixed open subset of Ω. We denote the closure and boundary of V in Ω^{∗} by \overline{V^{∗}} and ∂V^{∗}, respectively. Let C(V) be endowed with the topology of uniform convergence on compact subsets of V. We also use the Banach space C(\overline{V^{∗}}) equipped with the sup–norm ∥ · ∥ and its closed subspace C_{0}(V) of functions vanishing on ∂V^{∗}. We identify C(\overline{V^{∗}}) with a subspace of C(V) by restriction. Let A : D(A) ⊆ C(V) → C(V) be a linear operator. Our basic assumption reads as follows. (H1) A is closed in C(V). If x ∈ V and f ∈ D(A) with |f(x)| = ∥f∥, then

Re[(Af)(x) \overline{f(x)}] ≤ 0. This hypothesis will lead to closedness and disspativity of the operators

D(A_{0}) = {f ∈ D(A) ∩ C_{0}(V) : Af ∈ C_{0}(V)}, A_{0}f = Af,
D(A_{1}) = \{f ∈ D(A) ∩ C(\overline{V^{∗}}) : Af ∈ C_{0}(V)\}, A_{1}f = Af.  (2.1)

Throughout A_{0} and A_{1} are considered as operators on C_{0}(V) and C(\overline{V^{∗}}), respectively.

Lemma 2.1. Let (H1) hold. Then A_{0} and A_{1} are closed and dissipative in C_{0}(V) and C(\overline{V^{∗}}), respectively.

Proof. We only consider A_{1}, the operator A_{0} can be treated in the same way. Let D(A_{1}) ∋ f_{n} → f and A_{1}f_{n} → g in C(\overline{V^{∗}}). Since A is closed in C(V), f belongs to D(A) and Af = g ∈ C_{0}(V) so that A_{1} is closed in C(\overline{V^{∗}}). Given f ∈ D(A_{1}), there is x ∈ \overline{V^{∗}} such that ∥f∥ = |f(x)|. Set ϕ = \overline{f(x)} δ_{x} ∈ C(\overline{V^{∗}})′. Then ∥ϕ∥^{2} = ∥f∥^{2} = ϕ(f) and the quantity

Re⟨A_{1}f, ϕ⟩ = \begin{cases}0 & if x ∈ ∂V^{∗}, \\ Re[Af(x) \overline{f(x)}] & if x ∈ V, \end{cases}

is negative due to (H1). Thus A_{1} is dissipative.

The resolvent set of a linear operator B is denoted by ρ(B) and its resolvent by R(λ, B).

Lemma 2.2. Let (H1) hold and 0 < λ ∈ ρ(A_{1}). Then λ ∈ ρ(A_{0}) and R(λ, A_{1})f = R(λ, A_{0})f for f ∈ C_{0}(V).

Proof. Given g ∈ C_{0}(V), set f = R(λ, A_{1})g ∈ D(A_{1}). Then λf = A_{1}f + g ∈ C_{0}(V) so that f ∈ D(A_{0}) and (λ − A_{0})f = g. Lemma 2.1 now yields the assertion.

The above results allow to relate the generation properties of A_{0} and A_{1}. Example 2.8 below shows that the density of D(A_{1}) cannot be omitted in assertions (a) and (c) of the next theorem, in general.

Theorem 2.3. If (H1) holds, the following assertions are equivalent.

(a) (λ − A_{0})D(A_{0}) is dense in C_{0}(V) for some λ > 0 and D(A_{1}) is dense in C(\overline{V^{∗}}).

(b) A_{1} generates a C_{0}–semigroup T_{1}(·) on C(\overline{V^{∗}}).

(c) A_{0} generates a C_{0}–semigroup T_{0}(·) on C_{0}(V) and D(A_{1}) is dense in C(\overline{V^{∗}}). If this is the case, T_{0}(t) and T_{1}(t) are contractions, T_{1}(t)f = T_{0}(t)f for f ∈ C_{0}(V) and (T_{1}(t)f)(x) = f(x) for x ∈ ∂V^{∗}, f ∈ C(\overline{V^{∗}}), and t ≥ 0.

Proof. (a)⇒(b): If A_{1} were not a generator, then there would exist a finite Borel measure μ ≠ 0 on \overline{V^{∗}} with

∫\overline{V^{∗}}(λf − A_{1}f) dμ = 0 for all f ∈ D(A_{1})

because of Lemma 2.1 and the Lumer–Phillips theorem. Using (a), we see that μ must vanish on V. Thus

0 = ∫_{∂V^{∗}}(λf − A_{1}f) dμ = ∫_{∂V^{∗}} λf dμ

for all f ∈ D(A_{1}). The second condition in (a) now implies μ = 0 so that (b) is verified.

(b)⇒(c): In view of Lemma 2.1 and 2.2 and the Lumer–Phillips theorem, one only has to show that D(A_{0}) is dense in C_{0}(V). This fact follows from

λR(λ, A_{0})f = λR(λ, A_{1})f → f

as λ → ∞ for f ∈ C_{0}(V).

The implication ‘(c)⇒(a)’ is trivial. By Lemma 2.1, T_{0}(t) and T_{1}(t) are contractions, and by Lemma 2.2 both coincide on C_{0}(V). The identity

T_{1}(t)f − f = A_{1} ∫_{0}^{t} T_{1}(s)f ds ∈ C_{0}(V), f ∈ C(\overline{V^{∗}}), t ≥ 0,

establishes the last assertion.

As an immediate application we can solve the Cauchy problem for A with inhomogeneous boundary conditions. In [3, Thm.6.5] the case of uniformly elliptic partial differential operators A in divergence form was treated allowing for time depending boundary values. Non–autonomous extensions of the next result are given in Theorem 3.3 and 3.5.

Corollary 2.4. Let (H1) hold and f ∈ D(A_{1}). If (a)–(c) of Theorem 2.3 hold, then there is a unique u ∈ C^{1}(ℝ_{+}, C(\overline{V^{∗}})) such that u(t) ∈ D(A_{1}) for t ≥ 0 and

\frac{d}{dt} u(t, ·) = Au(t, ·),  t ≥ 0, on V,
u(t, x) = f(x), t ≥ 0, x ∈ ∂V^{∗}, (2.2)
u(0, x) = f(x),  x ∈ V.

Proof. The function u(t, x) = (T_{1}(t)f)(x) on ℝ_{+} × \overline{V^{∗}} solves (2.2) due to Theorem 2.3. The solution is unique since A_{0} is a generator.

We strengthen (H1) in order to obtain positive solutions: (H2) A is closed in C(V) and real (i.e., if f ∈ D(A), then f̄ ∈ D(A) and Af̄ = \overline{Af}). If

x ∈ V and f ∈ D(A) is real–valued with 0 < f(x) = sup_{V} f, then (Af)(x) ≤ 0. As in [14, Cor.2.10] one proves that (H2) implies (H1). The converse can be shown if A is given by a real local operator that satisfies (H1) on open subsets of V, see [14, Cor.2.10] and the definitions given in the next section. The following result can easily be deduced from R.S. Phillips’ characterization of generators of positive contraction semigroups [17, Thm.2.1] (compare the proof of Lemma 2.1).

Proposition 2.5. If (H2) holds and A_{1} is a generator, then T_{1}(·) is positive. Thus the solution of (2.2) is positive if f ≥ 0.

The results established so far are essentially stated in [11] without proof. In this work as well as in [3] further results were derived assuming that the Dirichlet problem (1.1) is solvable. Here we are looking for conditions on A_{0} and A_{1} characterizing the solvability of (1.1). Our main tool is the mean ergodic theorem which we apply to T_{1}(·).

Theorem 2.6. Assume that (H1) holds, A_{0} is invertible on C_{0}(V), and A_{1} is densely defined on C(\overline{V^{∗}}). Then T_{1}(·) is mean ergodic with projection P onto ker A_{1} = fix T_{1}(·). Moreover, ∥P ∥ ≤ 1 and P f(x) = f(x) for x ∈ ∂V^{∗} and f ∈ C(\overline{V^{∗}}). If in addition (H2) holds, then P is positive and ∥T_{1}(t) − P ∥ ≤ Me^{−εt} for t ≥ 0 and constants M, ε > 0.

Proof. By Theorem 2.3 the operators A_{k} generate the contraction semigroups T_{k}(·), k = 0, 1. For λ > 0 and f ∈ D(A_{1}), we obtain

λR(λ, A_{1})f = R(λ, A_{1})A_{1}f + f = R(λ, A_{0})A_{1}f + f
→ −A_{0}^{−1}A_{1}f + f  (2.3)

as λ → 0 due to Lemma 2.2. The mean ergodicity of T_{1}(·) follows from (2.3) and [5, Thm.5.1]. The other assertions except for the convergence of T_{1}(·) are straightforward consequences of Theorem 2.3 and Proposition 2.5. Let (H2) hold. Note that T_{0}(·) is positive and thus uniformly exponentially stable by [15, Thm.B-IV.1.4]. We now deduce

∥T_{1}(t)f − P f∥ = ∥T_{0}(t)(f − P f)∥ ≤ 2Ne^{−εt} ∥f∥

from T_{1}(t)P f = P f, for t ≥ 0, f ∈ C(\overline{V^{∗}}), and some constants N, ε > 0.

Observe that P f = f − A_{0}^{−1}A_{1}f for f ∈ D(A_{1}) due to (2.3).

Corollary 2.7. Assume that (H1) holds, A_{0} is invertible, and A_{1} is densely defined. Then for each φ ∈ C(∂V^{∗}) there exists a unique solution u ∈ C(\overline{V^{∗}}) ∩ D(A) of

Au = 0 on V,  u|∂V^{∗} = φ.  (2.4)

If (H2) holds and φ ≥ 0, then u ≥ 0.

Proof. The function u = P f solves (2.4), where P is given by Theorem 2.6 and f is any extension of φ to C(\overline{V^{∗}}). If u satisfies (2.4) for φ = 0, then u ∈ ker A_{0} so that u = 0.

We refer to [7] for a detailed account on the Dirichlet problem for elliptic operators. The following two examples show that Theorem 2.6 and Corollary 2.7 are not valid if one only assumes that A_{0} is an invertible generator or that A_{1} is a generator and A_{0} is injective.

Example 2.8. For V = (0, 1)^{2} and Ω = ℝ^{2} define

Af(x, y) = ∂_{xx} f(x, y) + y^{2} ∂_{yy} f(x, y) − f(x, y)

with D(A) = {f ∈ C(V) ∩ W_{p,loc}^{2}(V) : Af ∈ C(V)} for one (and hence all) p > 2. It can be checked that (H2) holds, cf. [16]. Clearly, A_{0} is densely defined. Further, the function

h : V → ℝ_{+}; h(x, y) = x(1 − x)(1 − y)\sqrt{y}

is a Cauchy barrier in the sense of (CB) below. Therefore A_{0} generates a contraction semigroup on C_{0}(V) by [16, Thm.1], see also [10, Thm.5.4] or [14, Thm.3.27]. By the same reasons A_{0} + 1 is also a dissipative generator so that A_{0} is invertible. However, following the arguments of [7, §6.6, p.116] one shows that the Dirichlet problem corresponding to A cannot be solved for φ ∈ C(∂V) vanishing for x = 0, 1 but not for y = 0. Therefore the operator A_{1} is not densely defined on C(\overline{V}) due to the above corollary.

Example 2.9. Let Ω = (0, ∞), V = (1, ∞), and Af = f′′ with D(A) = C^{2}(V). Then (H2) holds and A_{0} is an injective generator on C_{0}(V). Further, D(A_{1}) is dense in C([1, ∞]). Hence, A_{1} is a generator by Theorem 2.3. However, (2.4) has no solution for φ with φ(0) ≠ φ(∞), and so T_{1}(·) is not mean ergodic in view of the proof of Corollary 2.7.

Concluding this section we want to characterize the Dirichlet regularity of V in terms of A_{0} and A_{1}. Here we use the following notion, cf. [10].

Definition 2.10. The set V is called weakly (A–)Dirichlet regular if (2.4) has a solution for each φ ∈ C(∂V^{∗}) and (A–)Dirichlet regular if, in addition, the solution is unique.

Lemma 2.11. If V is weakly Dirichlet regular and D(A_{0}) is dense in C_{0}(V), then D(A_{1}) is dense in C(\overline{V^{∗}}).

Proof. For f ∈ C(\overline{V^{∗}}) there is v ∈ C(\overline{V^{∗}}) ∩ D(A) such that Av = 0 on V and v = f on ∂V^{∗}. By assumption, there exist u_{n} ∈ D(A_{0}) converging to f − v in C_{0}(V). Hence, u_{n} + v ∈ D(A_{1}) tends to f in C(\overline{V^{∗}}).

Lemma 2.12. If V is weakly Dirichlet regular and C_{0}(V) ⊆ A[D(A) ∩ C(\overline{V^{∗}})], then A_{0} is surjective.

Proof. For g ∈ C_{0}(V) there is u ∈ D(A) ∩ C(\overline{V^{∗}}) with Au = g. By assumption, there exists v ∈ D(A) ∩ C(\overline{V^{∗}}) such that Av = 0 and u = v on ∂V^{∗}. Then f = u − v belongs to D(A_{0}) and Af = Au = g.

The range condition used above holds if A is a local operator on Ω such that A is invertible on C_{0}(Ω), cf. [14, Prop.3.4] and the next section.

Theorem 2.13. Assume that (H1) holds, D(A_{0}) is dense in C_{0}(V), and C_{0}(V) ⊆ A[D(A) ∩ C(\overline{V^{∗}})]. Then the following assertions are equivalent.

(a) V is A–Dirichlet regular.

(b) A_{0} is invertible on C_{0}(V) and D(A_{1}) is dense in C(\overline{V^{∗}}).

(c) A_{1} generates a mean ergodic semigroup on C(\overline{V^{∗}}).

Proof. Assertion (b) follows from (a) in view of Lemma 2.1, 2.11, 2.12 and the uniqueness of (2.4). The implication ‘(b)⇒(c)’ was shown in Theorem 2.6. If (c) holds, then V is weakly Dirichlet regular due to the proof of Corollary 2.7. The operator A_{0} is thus surjective by Lemma 2.12. This implies the injectivity of A_{0} since T_{0}(·) is also mean ergodic, cf. [5, Thm.5.1]. Therefore solutions of (2.4) are unique.

3. Parabolic problems on non–cylindrical domains

In this section we make use of the theory of local operators, see [10], [14], and the references therein. Unexplained notation and concepts can be found in [14], but for the convenience of the reader we recall several basic definitions. A local operator A is a collection of linear operators A^{V} : D(A, V) ⊆ C(V) → C(V) which respect restriction to subsets, where V belongs to the set O(Ω) of non–empty open subsets of Ω. Usually we omit the superscript V. We introduce further notions, where V ∈ O(Ω).

(a) A is locally closed u.c. if, for f, g ∈ C(V), the existence of O(Ω) ∋ V_{n} ↑ V and f_{n} ∈ D(A, V_{n}) such that f_{n} → f and Af_{n} → g uniformly on compact subsets of V implies that f ∈ D(A, V) and A^{V} f = g.

(b) A is locally dissipative if, for all compact K ⊂ V and f ∈ D(A, V) with sup_{V \K} |f| < sup_{V} |f|, there is x_{0} ∈ V such that |f(x_{0})| = sup_{V} |f| and Re[(Af)(x_{0}) \overline{f(x_{0})}] ≤ 0 .

(c) A is real if f ∈ D(A, V) implies that \overline{f} ∈ D(A, V) and A\overline{f} = \overline{Af} on V. If we assume that (a)–(c) hold and that (S) for V contained in a base of O(Ω) the part A_{V} of A^{V} in C_{0}(V) is densely defined, then Lemma 2.9 and 2.10 of [14] show that A = A^{V} satisfies (H2) (and (H1) if A is not supposed to be real). In view of Section 2 we define the operator

Ã_{V} f = \begin{cases}Af & on V, \\ 0 & on ∂V^{∗}, \end{cases}  D(Ã_{V}) = \{f ∈ C(\overline{V^{∗}}) ∩ D(A, V) : Af ∈ C_{0}(V_{0})\},

on C(\overline{V^{∗}}). We set A_{0} = A_{V} and A_{1} = Ã_{V} if there is no danger of confusion.

The results in the previous section allow to treat a problem of degenerate multiplicative perturbation. For a local operator A and p ∈ C(Ω) we define the local operator (pA)f = pAf with domain D(pA, V) = D(A, V), cf. [10, §6].

Proposition 3.1. Assume that A is a locally dissipative, locally closed u.c., local operator satisfying (S) on O(Ω). Let V ∈ O(Ω) and p ∈ C(Ω) be bounded and positive such that W = V \ {x ∈ V : p(x) = 0} is dense in V. Further suppose that Ã_{V} is densely defined on C(\overline{V^{∗}}) and (pA)_{W} is a generator on C_{0}(W). Then D((p̃A)_{W}) is dense in C(\overline{W^{∗}}) and thus there exist unique solutions of (2.2) for pA on W.

Proof. Note that (pA)^{W} satisfies (H1) on W. For f ∈ C(\overline{W^{∗}}) = C(\overline{V^{∗}}) take f_{n} ∈ D(Ã_{V}) converging to f in sup–norm. Since pAf_{n} ∈ C_{0}(W), the functions f_{n}|W belong to D((p̃A)_{W}). Corollary 2.4 now implies the second assertion.

The above result can be applied to the following situation. (One can deal with more general examples in view of [10, Thm.6.4], see also [13].)

Example 3.2. Let V = B_{1} be the open unit ball in ℝ^{N}, W = B_{1} \ {0}, A = Δ with maximal domain, and p(x) = |x|^{2}. Then (pA)_{W} is a generator on C_{0}(W) by e.g. the remarks after [10, Thm.6.10]. D(Ã_{V}) is dense in C(\overline{B_{1}}) because of standard existence results for the Dirichlet problem for the Laplacian and Lemma 2.11. Thus we can solve (2.2) for r^{2}Δ on B_{1} \ {0}.

Turning to the parabolic case, we let \underline{Ω} = [S, T] × Ω and, for \underline{V} ∈ O(\underline{Ω}) and t ∈ [S, T],

V (t) = \{x ∈ Ω : (t, x) ∈ \underline{V} \}, \underline{V} (t) = \{t\} × V (t), \underline{V}_{t} = \{(s, x) ∈ \underline{V} : s > t\},
\underline{V}_{0} = \underline{V}_{S}, \underline{V} ′_{t} = \{(s, x) ∈ \underline{V} : s ≥ t\}, I\underline{V} = \{t ∈ [S, T] : V (t) ≠ ∅\}.

Further, \overline{\underline{V}^{∗}} and ∂\underline{V}^{∗} designate the closure and boundary of \underline{V} in \underline{Ω}^{∗} = [S, T] × Ω^{∗}, respectively. Observe that ∂\underline{V} _{0}^{∗} = ∂\underline{V}^{∗} ∪ \underline{V} (S), [14, Prop.4.1]. We write \underline{x} = (t, x) for a generic point of \underline{Ω} or \underline{Ω}^{∗} and consider C(\overline{\underline{V}^{∗}}) and C(\underline{V}_{0}) as subspaces of C(\underline{V}).

A local operator L defined on O(\underline{Ω}) is called parabolic if, for F ∈ D(L, \underline{V}), \underline{V} ∈ O(\underline{Ω}), and φ ∈ C^{1}(I\underline{V}), we have φF ∈ D(L, \underline{V}) and L(φF) = φ LF − φ′F.

Let \underline{V} ∈ O(\underline{Ω}) satisfy V (t) ≠ ∅ for t ∈ [S, T]. Take F ∈ C(\overline{\underline{V}^{∗}}), f ∈ C(\overline{V (S)^{∗}}), and g ∈ C(∂\underline{V}^{∗}) with g(S, x) = f(x) for x ∈ ∂V (S)^{∗}. We are looking for solutions u ∈ C(\overline{\underline{V}^{∗}}) ∩ D(L, \underline{V}_{0}) of the problem

Lu = F  on \underline{V}_{0},
u = g on ∂\underline{V}^{∗},  (3.1)
u(S) = f.

The above setting covers the parabolic differential operator

L(\underline{x}, D) = ∑_{k,l=1}^{N} a_{kl}(t, x)∂_{kl} + ∑_{k=1}^{N} b_{k}(t, x)∂_{k} + c(t, x) − ∂_{t}  (3.2)

on an open subset Ω of ℝ^{N}, where ∂_{t} = \frac{∂}{∂t} and ∂_{k} = \frac{∂}{∂x_{k}}. The coefficients a_{kl}, b_{k}, and c ≤ 0 are supposed to be real-valued and continuous and to satisfy

∑_{k,l=1}^{N} a_{kl}(t, x) y_{k} y_{l} > 0  (3.3)

for all y ∈ ℝ^{N} and (t, x) ∈ \underline{Ω}. The coefficients may be singular or degenerate at the boundary. (See e.g. [6] for the case of uniformly elliptic and Hölder continuous coefficients on regular domains \underline{V}.) Set \underline{Ω}_{00} = (S, T) × Ω and \underline{W}_{00} = \underline{W} ∩ \underline{Ω}_{00} for \underline{W} ∈ O(\underline{Ω}). As in [14, §6.1], we define

D(L, \underline{W}_{0}) = \{F ∈ C(\underline{W}_{0}) ∩ W_{p,loc}^{1,2}(\underline{W}_{0}) : L(\underline{x}, D)u = G a.e. on \underline{W}_{00} for G ∈ C(\underline{W}_{0})\},
LF = G on \underline{W}_{0}.  (3.4)

Here, \underline{W} ∈ O(\underline{Ω}), p > N + 2 is fixed, and W_{p,loc}^{1,2}(\underline{W}_{0}) (W_{p,loc}^{1,2}(\underline{W})) denotes the space of functions F such that F |\underline{U}_{00} belongs to the Sobolev space

W_{p}^{1,2}(\underline{U}_{00}) = \{F ∈ L^{p}(\underline{U}_{00}) : ∂_{k}F, ∂_{kl}F, ∂_{t}F ∈ L^{p}(\underline{U}_{00}), k, l = 1, …, N\}

for all relatively compact open subsets \underline{U} of \underline{W}_{0} (of \underline{W}). In [14, §6.1] it is shown that D(L, \underline{W}_{0}) does not depend on the choice of p > N +2 and that L satisfies the assumptions listed above. Analogous operators on networks were studied in [14, §6.2].

We come back to the general situation of a real, locally dissipative, locally closed u.c., parabolic, local operator L satisfying (S). Then (3.1) with g = 0, F = 0 on ∂\underline{V}^{∗}, and S replaced by s ∈ (S, T) can be solved if the part L_{\underline{V}_{0}} of L in C_{0}(\underline{V}_{0}) is a generator, [14, Thm.4.14]. Under some mild additional assumptions L_{\underline{V}_{0}} is a generator if and only D(L_{\underline{V}_{0}}) is dense in C_{0}(\underline{V}_{0}) and \underline{V}_{0} has a Cauchy barrier in the sense that (CB) there exists a compact subset \underline{K} of \underline{V}_{0} and a function h ∈ D(L, \underline{V}_{0} \ \underline{K}) such that

h > 0 and (L − λ)h ≤ 0 on \underline{V}_{0} \ \underline{K} for some λ ≥ 0, and for all ε > 0 there is a

compact set \underline{K}_{ε} with \underline{K} ⊆ \underline{K}_{ε} ⊆ \underline{V}_{0} and 0 ≤ h ≤ ε on \underline{V}_{0} \ \underline{K}_{ε},

see [14, Thm.3.25, 3.27] and [10, Thm.5.4]. Cauchy barriers are discussed in [14, §3.1] where further literature can be found. Examples for Cauchy barriers are given in [14, §6.1]. The next result is a straightforward consequence of the theory developed in the previous section.

Theorem 3.3. Let L be a locally dissipative, locally closed u.c., parabolic, local operator satisfying (S) and defined on O(\underline{Ω}), let \underline{V} ∈ O(\underline{Ω}) satisfy V (t) ≠ ∅ for t ∈ [S, T], and let F ∈ C(\overline{\underline{V}^{∗}}), f ∈ C(\overline{V (S)^{∗}}), and g ∈ C(∂\underline{V}^{∗}) with g(S, x) = f(x) for x ∈ ∂V (S)^{∗}. Assume that L_{\underline{V}_{0}} is a generator on C_{0}(\underline{V}_{0}) and that D(L̃_{\underline{V}_{0}}) is dense in C(\overline{\underline{V}^{∗}}). Then there is a unique solution u ∈ C(\overline{\underline{V}^{∗}}) ∩ D(L, \underline{V}_{0}) of (3.1). If L is real and −F, g, f ≥ 0, then u ≥ 0.

Proof. Uniqueness and positivity of solutions follow from the parabolic maximum principle [14, Thm.2.29, 2.30]. By Theorem 2.3, L_{1} generates a contraction semigroup on C(\overline{\underline{V}^{∗}}). Since e^{tL_{0}} = 0 for t ≥ T −S, see [14, Thm.4.9], Theorem 2.6 provides us with a projection P = P\underline{V} mapping C(\overline{\underline{V}^{∗}}) onto ker L_{1} such that P w(\underline{x}) = w(\underline{x}) for \underline{x} ∈ ∂\underline{V} _{0}^{∗} and w ∈ C(\overline{\underline{V}^{∗}}). Set F̂(t, x) = e^{−t} F (t, x), v̂ = (L_{1} − 1)^{−1}F̂, and v(t, x) = e^{t} v̂(t, x) on \overline{\underline{V}^{∗}}. Then v ∈ D(L, \underline{V}_{0}) ∩ C(\overline{\underline{V}^{∗}}) and

Lv = −e^{t} v̂ + e^{t} Lv̂ = −e^{t} v̂ + e^{t} (v̂ + F̂) = F

on \underline{V}_{0} since L is parabolic. Take G ∈ C(\overline{\underline{V}^{∗}}) with G(S) = f −v(S) on \underline{V} (S) and G = g−v on ∂\underline{V}^{∗}. The function u = v + P G clearly solves (3.1).

We now give a sufficient condition for the density of D(L_{1}) formulated for a general local operator.

Proposition 3.4. Let μ ≥ 0 be a finite regular Borel measure on Ω and V ∈ O(Ω) be relatively compact with μ(∂V) = 0. Let A be a local operator on Ω such that A_{Ω} is densely defined and invertible on C_{0}(Ω) and ∥A_{Ω}^{−1}f∥_{C_{0}(Ω)} ≤ c∥f∥_{L^{p}(Ω,μ)} for f ∈ C_{0}(Ω) and some p ∈ [1, ∞). Then Ã_{V} is densely defined on C(\overline{V}).

Proof. Note that there are relatively compact sets W_{n} ∈ O(Ω) such that ∂V ⊂ W_{n+1} ⊂ \overline{W_{n+1}} ⊂ W_{n} and μ(W_{n}) ≤ 1/n for n ∈ ℕ. We extend a given f ∈ C(\overline{V}) to a function f̃ ∈ C_{0}(Ω). For ε > 0 there is ũ ∈ D(A_{Ω}) such that ∥f̃− ũ∥_{∞} ≤ ε. Let g̃ = Aũ and take continuous functions α_{n} : Ω → [0, 1] vanishing on W_{n+1} and being equal to 1 off W_{n}. Set g̃_{n} = α_{n}g̃ and v_{n} = (A_{Ω}^{−1}g̃_{n})|\overline{V}. Then v_{n} ∈ D(Ã_{V}), g̃_{n} → g̃ in p–norm, and

∥f − v_{n}∥_{C(\overline{V})} ≤ ∥f̃− ũ∥_{C_{0}(Ω)} + ∥A_{Ω}^{−1}(g̃ − g̃_{n})∥_{C_{0}(Ω)} ≤ ε + c ∥g̃ − g̃_{n}∥_{L^{p}(Ω,μ)} ≤ 2ε

for large n.

In the remainder of this section we treat the case of the operator L given by (3.2) and (3.4). We first extend [4, Thm.4.1] (combined with [4, Thm.4.10]) to the parabolic case.

Theorem 3.5. Let L be given by (3.2) and (3.4), where Ω is a bounded open subset of ℝ^{N} with boundary of class C^{2} and the coefficients belong to C(\overline{\underline{Ω}}, ℝ), c ≤ 0, and (3.3) holds on \overline{\underline{Ω}} = [−1, T] × \overline{Ω}. Take a relatively compact subset \underline{V} ∈ O([0, T] × Ω) such that V (t) ≠ ∅ for 0 ≤ t ≤ T and ∂\underline{V} has Lebesgue measure 0 in \underline{Ω}. Then the following assertions are equivalent.

(a) L̃_{\underline{V}_{0}} is a generator in C(\overline{\underline{V}}).

(b) L_{\underline{V}_{0}} is a generator in C_{0}(\underline{V}_{0}).

(c) \underline{V}_{0} has a Cauchy barrier for L. In this case the conclusions of Theorem 3.3 are true.

Proof. Let p > N + 1. Using [9, Thm.IV.9.1] we see that L_{\underline{Ω}_{0}} is invertible (and densely defined) on C_{0}(\underline{Ω}_{0}) and its inverse maps (C_{0}(\underline{Ω}_{0}), ∥ · ∥_{p}) continuously into W_{p,0}^{2,1}(\underline{Ω}_{0}) ↪ C_{0}(\underline{Ω}_{0}). So the theorem follows from Proposition 3.4, Theorem 2.3, [14, Thm.4.19] (see the remarks before [14, Thm.6.1]), and Theorem 3.3.

We briefly indicate another method to verify the density of D(L_{1}) using L–harmonic functions. Consider A = Δ, L = Δ − \frac{d}{dt}, and \underline{V} = [0, T] × (0, 1)^{2}. Assume that L_{1} is not densely defined in C(\overline{\underline{V}}). Then there is a regular Borel measure μ ≠ 0 such that

∫\overline{\underline{V}} u dμ = 0 for all u ∈ D(L_{1}).

Since L_{0} is densely defined on C_{0}(\underline{V}_{0}), the support of μ must be contained in ∂\underline{V}_{0}. The function u(t, x, y) = (T_{0}(t)f)(x, y) belongs to C_{0}(\underline{V}) and to the kernel of L, where f ∈ D(A_{0}) and T_{0}(·) is the semigroup generated by the Dirichlet Laplacian A_{0} on C_{0}(V). This implies that suppμ ⊂ [0, T] × ∂V. Further let α ∈ C_{c}^{1}(0, 1), k ∈ ℕ_{0}, and g(t, x, y) be equal to e^{−kt}α(x) for y = 1 and (t, x) ∈ [0, T] × [0, 1] and be equal to 0 on the other faces of the lateral boundary. Extend α to a periodic odd function on ℝ with Fourier coefficients a_{n}. Define

u_{m}(t, x, y) = e^{−kt} ∑_{n=1}^{m} \frac{a_{n}}{sin \sqrt{k − 4π^{2}n^{2}}} sin(2πnx) sin(y\sqrt{k − 4π^{2}n^{2}})

for t ∈ [0, T], x, y ∈ ℝ, and m ∈ ℕ. Then Lu_{m} = 0, u_{m} = 0 on the faces x = 0, 1 and y = 0, and u_{m}(t, x, 1) → e^{−kt}β(x) uniformly for t and x. Thus ∫ g dμ = 0. By an approximation argument we now deduce that μ vanishes on [0, T] × (0, 1) × {1}. The other faces can be treated in the same way so that μ is supported in the lateral edges. In a last step one sees in a similar manner that this is impossible. Therefore μ = 0 and D(L_{1}) is dense.

We finally want to study the regularity of the solutions to the homogeneous problem

Lu = 0 on \underline{V}_{s},
u = 0 on ∂\underline{V} _{s}^{∗} \ \underline{V} (s),  (3.5)
u(s) = f.

for s ∈ (S, T), f ∈ C_{0}(V (s)), and the operator L given by (3.2) and (3.4) supposing that L_{0} = L_{\underline{V}_{0}} is a generator on C_{0}(\underline{V}_{0}). In [14, Thm.4.13, 6.1] we proved that the solution u ∈ C_{0}(\underline{V} ′_{s})∩⋂_{p>1} W_{p,loc}^{1,2}(\underline{V}_{s}) of (3.5) is given by u(t, ·) = U(t, s)f, t ∈ [s, T], for a variable space propagator U(t, s) : C_{0}(V (s)) → C_{0}(V (t)), S < s ≤ t ≤ T. This means that

U(s, s) = I_{C_{0}(V (s))} ,  U(t, r)U(r, s) = U(t, s), and  (3.6)
(t, s) ↦ (U(t, s)F (s))^{#} is continuous in C_{0}(Ω)  (3.7)

for S < s ≤ r ≤ t ≤ T and F ∈ C_{0}(\underline{V}_{0}), where g^{#} denotes the extension of g ∈ C_{0}(V) by 0 to C_{0}(Ω), cf. [14, Def.4.8]. Moreover, U(t, s) is a positive contraction.

In the following we consider initial functions f belonging locally to Slobodeckij spaces W_{p}^{α}(U), where U is a bounded open subset of ℝ^{N} with a smooth boundary, α ∈ [0, 2], and 1 < p < ∞. These spaces can be defined, e.g., by Hölder estimates of L^{p}–type, see [18, §4.4]. Real interpolation of W_{p}^{α}(U) yields

(W_{p}^{α}(U), W_{p}^{β}(U))_{θ,p} = W_{p}^{γ}(U)  (3.8)

if γ = α(1 − θ) + βθ ∉ ℕ and θ ∈ (0, 1) due to Theorem 4.3.1.1 and formula (2.4.2.16) of [18]. Taking α = 0 and β = 2 in (3.8), [2, Thm.III.4.10.2] leads to

W_{p}^{1,2}([a, b] × U) = L^{p}([a, b], W_{p}^{2}(U)) ∩ W_{p}^{1}([a, b], L^{p}(U)) ↪ C([a, b], W_{p}^{2− \frac{2}{p}} (U)) (3.9)

if p ≠ 2. Finally, from [18, Thm.4.6.1, 4.6.2] we deduce

W_{q}^{2− \frac{2}{q}} (U) ↪ W_{p}^{2− \frac{2}{r}} (U) ↪ W_{p}^{2− \frac{2}{p}} (U)  (3.10)

if 1 < p < r < q < ∞. The following result yields in the interior of \underline{V} ′_{s} the regularity at t = s which we can expect in view of the well known results in [9, §IV.9] for regular problems.

Theorem 3.6. Let L be given by (3.2) and (3.4), where Ω is an open subset of ℝ^{N} and the coefficients belong to C(\underline{Ω}, ℝ), c ≤ 0, and (3.3) holds. Moreover, let \underline{V} ∈ O(\underline{Ω}) satisfy V (t) ≠ ∅ for S ≤ t ≤ T. Assume that L_{\underline{V}_{0}} is a generator on C_{0}(\underline{V}_{0}). Given s ∈ (S, T)

and f ∈ C_{0}(V (s)) ∩ ⋂_{p>1} W_{p,loc}^{2− \frac{2}{p}} (V (s)), we set u(t, x) = (U(t, s)f)(x) for (t, x) ∈ \underline{V} ′_{s}. Then u ∈ C_{0}(\underline{V} ′_{s}) ∩ ⋂_{p>1} W_{p,loc}^{1,2}(\underline{V} ′_{s}).

Proof. Fix 2 < p < r < q < ∞, δ > 0, and open, relatively compact subsets U, W, W′ of V (s) with smooth boundaries such that \overline{U} ⊆ W ⊆ \overline{W} ⊆ W′ and [s − δ, s + δ] × \overline{W′} ⊆ \underline{V}_{0}. Choose F ∈ C_{0}(\underline{V}_{0}) with ∥F ∥_{∞} ≤ ∥f∥_{∞}, F (s) = f, F (t, x) = f(x) for x ∈ \overline{W′} and |t − s| ≤ δ. We proceed in two steps.

(1) Take τ ∈ (s − δ, s) and set \underline{W} = [τ, τ + δ] × W and \underline{W}′ = [τ, τ + δ] × W′. Using (3.8), (3.10), (3.9), and the interior apriori estimate [9, (IV.10.12)], we calculate

∥U(s, τ)F (τ) − f∥_{W_{p}^{2− \frac{2}{p}} (W)}
≤ c ∥U(s, τ)F (τ) − f∥_{L^{p}(W)}^{1−θ} ∥U(s, τ)F (τ) − f∥_{W_{p}^{2− \frac{2}{r}} (W)}^{θ}

≤ c ∥(U(s, τ)F (τ))^{#} − F (s)^{#}∥_{C_{0}(Ω)}^{1−θ} (∥f∥_{W_{q}^{2− \frac{2}{q}} (W)} + ∥U(s, τ)F (τ)∥_{W_{q}^{2− \frac{2}{q}} (W)})^{θ}

≤ c ∥(U(s, τ)F (τ))^{#} − F (s)^{#}∥_{∞}^{1−θ} (∥f∥_{W_{q}^{2− \frac{2}{q}} (W)} + ∥U(·, τ)F (τ)∥_{W_{q}^{1,2}(\underline{W})})^{θ}
≤ c ∥(U(s, τ)F (τ))^{#} − F (s)^{#}∥_{∞}^{1−θ} (∥f∥_{W_{q}^{2− \frac{2}{q}} (W′)} + ∥U(·, τ)F (τ)∥_{C(\overline{\underline{W}′})})^{θ}
≤ c ∥(U(s, τ)F (τ))^{#} − F (s)^{#}∥_{∞}^{1−θ}  (3.11)

where θ = (1 − \frac{1}{p})(1 − \frac{1}{r})^{−1} and the constants c do not depend on τ,

(2) Now let τ_{n} = s − \frac{1}{n} for large n, u_{n} = U(·, τ_{n})F (τ_{n}) on \underline{V} ′_{s}, \underline{U} = [s, s + δ] × U, \underline{W} = [s, s + δ] × W, and \underline{W}′ = [s, s + δ] × W′. By [14, Thm.4.13, 6.3] we have u_{n} ∈ ⋂_{p>1} W_{p,loc}^{1,2}(\underline{V} ′_{s}) since u_{n} solves (3.5) with initial value F (τ_{n}) at time τ_{n}. Further, u_{n} → u in C_{0}(\underline{V} ′_{s}) due to (3.7). An application of (3.6), [9, (IV.10.12)], and (3.11) yields

∥u_{n} − u_{m}∥_{W_{p}^{1,2}(\underline{U})} = ∥U(·, s) [U(s, τ_{n})F (τ_{n}) − U(s, τ_{m})F (τ_{m})]∥_{W_{p}^{1,2}(\underline{U})}

≤ c (∥u_{n} − u_{m}∥_{C(\overline{\underline{W}})} + ∥U(s, τ_{n})F (τ_{n}) − U(s, τ_{m})F (τ_{m})∥_{W_{p}^{2− \frac{2}{p}} (W)})

≤ c (∥u_{n} − u_{m}∥_{∞} + ∥(U(s, τ_{n})F (τ_{n}))^{#} − F (s)^{#}∥_{∞}^{1−θ} + ∥(U(s, τ_{m})F (τ_{m}))^{#} − F (s)^{#}∥_{∞}^{1−θ}).

Thus (u_{n}) is a Cauchy sequence in W_{p}^{1,2}(\underline{U}) by the strong continuity of U(t, s). As a result, u_{n} → u in W_{p}^{1,2}(\underline{U}), and the assertion follows.

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G. Lumer, Université de Mons–Hainaut, Institut de Mathématique et d’Informatique, Mons, and Solvay Institutes for Physics and Chemistry, Brussels. Postal address: Institut de Mathématique, Université de Mons–Hainaut, Place du Parc 20, 7000 Mons, Belgium.

  E-mail address: dufour@umh.ac.be

R. Schnaubelt, FB Mathematik und Informatik, Martin–Luther–Universität, Theodor– Lieser–Str. 5, 06120 Halle, Germany.

  E-mail address: roland@euler.mathematik.uni-halle.de