Wentzell boundary conditions in the context of Dirichlet forms

Hendrik Vogt^{∗} and Jürgen Voigt^{∗}

Fachrichtung Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany

dedicated to the memory of Klaus Floret (1941 – 2002)

whose untimely death we deeply mourn

Abstract

Let X be a locally compact space, m a Radon measure on X, 𝔥 a regular Dirichlet form in L_{2}(X, m). For a Radon measure μ we interpret 𝔥 as a regular Dirichlet form τ in L_{2}(m + μ). We show that μ decomposes as μ_{r} + μ_{s} where μ_{r} is coupled to 𝔥 and μ_{s} decouples from 𝔥. Additionally to this ‘space perturbation’, a second perturbation is introduced by a measure ν describing absorption.

The main object of the paper is to apply this setting to a study of the Wentzell boundary condition

−αAu + n · a∇u + γu = 0 on ∂Ω

for an elliptic operator A = −∇ · (a∇), where Ω ⊆ ℝ^{d} is open, n the outward normal, and α, γ are suitable functions. It turns out that the previous setting can be applied with μ = αdS, ν = γdS, under suitable conditions. Besides the description of the d-dimensional case we give a more detailed analysis of the one-dimensional case.

As a further topic in the general setting we study the question whether mass conservation carries over from the unperturbed form 𝔥 to the space perturbed form τ. In an appendix we extend a known closability criterion from the minimal to the maximal form.

2000 MSC: 47D06, 31C25, 35J25, 34B05 Key words: perturbation of Dirichlet forms, initial boundary value problems, elliptic operators, mass conservation, closability of forms

∗partly supported by the DFG vogt@math.tu-dresden.de, voigt@math.tu-dresden.de

1 Introduction

The object of this paper is to present the treatment of Wentzell boundary conditions in the context of Dirichlet forms.

Let Ω ⊆ ℝ^{d} be open, and A = −∇ · (a∇) an elliptic operator in divergence form. (For precise conditions we refer to Section 3.) Then the Wentzell boundary condition is given by

−αAu + n · a∇u + γu = 0

on (part of) the boundary of Ω, where α, γ are non-negative functions, and n is the outward normal. (See, e.g., [23; formula (3)]; there even more general boundary conditions including diffusion on the boundary and jumps are considered.) In [9], the operator A subject to the Wentzell boundary condition was studied in the L_{p}- context, where the Lebesgue measure on Ω is augmented by a suitable measure on the boundary. It is the purpose of the present paper to continue and extend this investigation in various directions. The essential new ingredient in our study is putting the problem into the context of Dirichlet forms. This makes it possible to obtain results under rather general hypotheses.

In most of the previous papers, Wentzell boundary conditions have been studied in spaces of continuous functions and with operators that are not in divergence form; see, e.g., [10], [23], [6], [5]. It is only in [9] that the measure space (Ω ∪ Γ, m + μ)—where m is the Lebesgue measure on Ω, Γ is part of the boundary of Ω, and μ is a suitable measure on Γ—is proposed as the adequate measure space for studying Wentzell boundary conditions in the L_{p}-framework. In the one-dimensional case, a space with an additional measure on the boundary was already used in [Feller; Sections 18 and 22] for the L_{1}-treatment. In [2], more special boundary conditions have been studied thoroughly in the L_{p}-framework.

The time evolution generated by the operator A presented above describes particles which diffuse in Ω and which may stick (for infinitesimally short times) to the boundary. This behaviour is in contrast to the totally absorbing Dirichlet boundary condition, the totally reflecting Neumann boundary condition, or the partially absorbing, partially reflecting Robin boundary condition. One may think of the Wentzell boundary condition as describing a situation like the Robin boundary condition, with the additional feature that the boundary has the capacity for storing (but not conducting) heat.

Probabilistically speaking, the L_{1}-treatment means that one wants to describe the time development of measures which are given as densities with respect to a reference measure. As a consequence, if one wants to describe particles which may temporarily concentrate on the boundary, the reference measure should offer this possibility—meaning that the boundary should not be a null set. This aspect is not apparent as long as one studies the problem in spaces of continuous functions.

The paper is organised as follows. In Section 2 we present our abstract version of the Wentzell boundary condition in the context of Dirichlet forms. We start with a Dirichlet form 𝔥 which corresponds to the basic diffusion of particles in a locally compact space X, endowed with a Radon measure m. As a first perturbation we introduce a part of X (described by a measure μ) where additionally particles should stick. The state space X itself is not changed, but the reference measure m is augmented by μ. The interpretation of this perturbation is that the diffusion is slowed down on the support of μ (see, e.g., [11; Sec. 6.2]). A second perturbation is then added as an absorption by a measure ν.

In Section 3 we apply the setting of Section 2 to the situation sketched at the beginning of the introduction. The conclusion of this application is formulated in Theorem 3.4—the main result of the paper. In the first part of the section we give a motivation for our interpretation of the Wentzell boundary condition in the context of Dirichlet forms. As the state space we take X = Ω ∪ Σ, with Lebesgue measure m, where Σ is the part of ∂Ω on which we want to pose the Wentzell boundary condition. The form 𝔥 is associated with the differential expression A = −∇ · (a∇). The measures μ and ν are defined as μ = αdS and ν = γdS, respectively, where dS denotes the surface measure on Σ. The main part of Section 3 is devoted to the investigation of conditions on the coefficients guaranteeing that the results of Section 2 can be applied to the problem under consideration.

In Section 4 we present a more detailed analysis of the one-dimensional case Ω = (0, 1), Σ = {0, 1}. We give an explicit description of the L_{2}-operator associated with the form, and we discuss the various possibilities subsumed in our general approach.

In Section 5 we return to the general context of Section 2 and discuss the question of mass conservation, assuming that the unperturbed situation produces a mass conserving semigroup. Under the additional assumption that m and μ are finite measures it is easy to show that then the semigroup that is space perturbed by μ also is mass conserving. The conjecture that this statement should be true without the additional assumption is answered in the positive only in the case that μ is absolutely continuous with respect to m; see Theorem 5.5. (Unfortunately, the latter result is not applicable in the context of Section 3.) As an auxiliary result we present a partial description of the L_{p}-generator of the semigroup associated with a Dirichlet form (Proposition 5.2).

In the appendix we show that the maximal form associated with −∇· (a∇) is closable if a ∈ W_{2,loc}^{1}(Ω). This result (which is well-known for the minimal form) is of independent interest and serves to provide information in the context of Section 3.

2 Space perturbation of Dirichlet forms

In this section, let X be a locally compact space. Let m and μ be two Radon measures on X (i.e., locally finite, inner regular Borel measures), and assume that supp m = X. Moreover, let 𝔥 be a regular symmetric Dirichlet form in L_{2}(X, m); cf. [11], [14]. We recall that ‘regular’ means that C_{c}(X) ∩ D(𝔥) is dense in D(𝔥) with respect to the 𝔥-norm and dense in C_{0}(X) with respect to the sup-norm. In fact, this implies that C_{c}(X) ∩ D(𝔥) also is dense in C_{c}(X) with respect to the inductive limit topology of C_{c}(X), i.e., for each φ ∈ C_{c}(X) there exists a sequence (φ_{n}) ⊆ C_{c}(X) ∩ D(𝔥) such that ⋃_{n∈ℕ} supp φ_{n} is relatively compact, and φ_{n} → φ in the sup-norm.

We do not assume countable compactness, metrisability, or separability of X since the properties we are going to use do not depend on these assumptions. In the applications in the two subsequent sections, however, X will be a subset of ℝ^{d}.

Our aim is to interpret the form 𝔥 as a regular Dirichlet form in the space L_{2}(m + μ). As a first step we define a Dirichlet form τ̂ in L_{2}(m + μ) by

D(τ̂) := {u ∈ L_{2}(m + μ); u ∈ D(𝔥)},  τ̂(u, v) := 𝔥(u, v) (u, v ∈ D(τ̂)).

(Saying that u ∈ L_{2}(m + μ) belongs to D(𝔥) means that u should also be considered as a function in L_{2}(m). The observation that the mapping u ↦ u: L_{2}(m + μ) → L_{2}(m) is continuous immediately implies that τ̂ is a closed symmetric form in L_{2}(m + μ). Also, τ̂ is densely defined since C_{c}(X) ∩ D(𝔥) is dense in C_{c}(X) with respect to the inductive limit topology, and C_{c}(X) is dense in L_{2}(m + μ).)

We now couple μ to the form 𝔥 by restricting the domain of τ̂. Namely, we define

τ := \overline{τ̂↾_{C_{c}(X)∩D(𝔥)}}.

Then, by its very definition, τ is a regular symmetric Dirichlet form in L_{2}(m+μ).

Before analysing τ in more detail we finish the definition of the form we present as the generalisation of the Wentzell boundary condition. For this purpose let ν be a (positive) Borel measure on X which is absolutely continuous with respect to τ-capacity. Then τ + ν, defined by

D(τ + ν) := {u ∈ D(τ); ∫ |ũ|^{2} dν < ∞},
(τ + ν)(u, v) := τ(u, v) + ∫ ũ\overline{ṽ} dν (u, v ∈ D(τ + ν))

(where ũ denotes a τ-quasi-continuous version of u) is a symmetric (not necessarily regular) Dirichlet form in \overline{D(τ + ν)}^{L_{2}(m+μ)}. We refer to [21] for this statement and for further information.

Let 𝔅_{σ} be the σ-ring on X generated by the compact sets. Because of the inner regularity of m, the values of m on 𝔅_{σ} determine m, and the same is true for μ.

We recall from [21; Prop. 1.1(a)] that μ can be decomposed as

μ = μ_{r} + μ_{s},

where μ_{r}—the ‘regular’ part—is a measure which is absolutely continuous with respect to 𝔥-capacity, and μ_{s}—the ‘singular’ part—has the property that for each compact set K ⊆ X there exists a Borel set N ⊆ K of zero 𝔥-capacity such that μ_{s}(K \ N) = 0.

Lemma 2.1. There exists a local Borel set M such that 𝔥-cap(M ∩ K) = 0 for all compact K ⊆ X, and

m + μ_{r} = 11_{X\M}(m + μ), μ_{s} = 11_{M}(m + μ) on 𝔅_{σ}.

(In case X is σ-compact, the local Borel set M is automatically a Borel set, and M has zero 𝔥-capacity.)

As consequences, M is locally (m + μ_{r})-negligible,

μ_{r} = 11_{X\M}μ, μ_{s} = 11_{M}μ on 𝔅_{σ},

and the decomposition of m+μ given above induces a decomposition of L_{2}(m+μ) as an orthogonal sum of L_{2}(m + μ_{r}) and L_{2}(μ_{s}).

Proof. There exists a locally countable disjoint family (K_{ι})_{ι∈I} of compact subsets of X such that X \ ⋃_{ι∈I} K_{ι} is a local (m+μ)-null set and supp(11_{K_{ι}}(m+μ)) = K_{ι} for all ι ∈ I (cf. [4; § 1, n^{o} 4, Prop. 4]; this property expresses the strict localisability of the measure m+μ). Defining N_{ι} corresponding to K_{ι} as in the paragraph before the lemma, we obtain that the set M := ⋃_{ι∈I} N_{ι} has the asserted properties.

The main result of this section is the following description of τ.

Theorem 2.2. With the above notation,

D(τ) = {u ∈ L_{2}(m + μ) ∩ D(𝔥); u = ũ μ_{r}-a.e.}

(where ũ denotes an 𝔥-quasi-continuous version of u). The form τ decomposes as τ = τ_{r} ⊕ τ_{s}, where τ_{r} is the form in L_{2}(m + μ_{r}) defined by τ_{r}(u, v) = 𝔥(u, v) on

D(τ_{r}) = {u ∈ L_{2}(m + μ_{r}) ∩ D(𝔥); u = ũ μ_{r}-a.e.},

and τ_{s} = 0 in L_{2}(μ_{s}).

Remark 2.3. (a) The condition ‘u = ũ μ_{r}-a.e.’ is the coupling between 𝔥 and μ announced above. It may be considered as a kind of (interior) boundary condition.

(b) Theorem 2.2 states that the form τ decomposes into the interesting part τ_{r} and the trivial part τ_{s}. In the application in Sections 3 and 4, it will be one of the issues to indicate conditions excluding the trivial part τ_{s}.

(c) It is a well-established procedure to associate with a given Dirichlet form 𝔥 in L_{2}(m) a process on a space L_{2}(m̃) with a new measure m̃. We refer to [11; Sec. 6.2] for the description of the ‘time change’ of the process associated with the additive linear functionals generated my m̃. In this case, the space X is assumed to be metric and separable, and m̃ is absolutely continuous with respect to capacity. In our setting we have m̃ = m + μ ⩾ m; this is not assumed in [11].

Assume that X satisfies the additional properties mentioned in the previous paragraph. Having established the decomposition of Theorem 2.2 it is clear that the results of [11] apply to the measure m̃ = m + μ_{r}.

In the proofs of the next lemmas we will make use of the following fact. Let u ∈ D(𝔥), (u_{k}) ⊆ D(𝔥) a || · ||_{∞}-bounded sequence with u_{k} → u in D(𝔥), and ψ ∈ C_{c}(X) ∩ D(𝔥). Then there exists a sequence (v_{k}) of convex combinations of {u_{j}; j ⩾ k} such that ψv_{k} → ψu in D(𝔥) and in L_{2}(μ_{r}). Indeed, 𝔥(ψu_{k})^{1/2} ⩽ ||ψ||_{∞}𝔥(u_{k})^{1/2} + ||u_{k}||_{∞}𝔥(ψ)^{1/2} for all k ∈ ℕ; cf. [11; Thm. 1.4.2(ii)]. Thus, ψu_{k} → ψu weakly in D(𝔥). Also, without restriction ψu_{k} → ψu 𝔥-q.e., hence μ_{r}-a.e. This implies the assertion.

Lemma 2.4. Let u ∈ L_{2}(m + μ)∩D(𝔥), u = ũ μ_{r}-a.e. Then there exists (φ_{k}) ⊆ C_{c}(X) ∩ D(𝔥) such that φ_{k} → u in D(𝔥) and in L_{2}(m + μ_{r}).

Proof. Without restriction u ⩾ 0 and u bounded. By the regularity of 𝔥 there exists (u_{k}) ∈ C_{c}(X) ∩ D(𝔥) with u_{k} → u in D(𝔥); without restriction u_{k} ⩾ 0 (k ∈ ℕ) and u_{k} → u 𝔥-q.e. This implies u_{k} ∧ u → u in D(𝔥) and in L_{2}(m + μ_{r}). Therefore it suffices to approximate each u_{k} ∧ u or, expressed differently, we may assume supp u compact to start with.

As before, there exists a || · ||_{∞}-bounded sequence (u_{k}) ∈ C_{c}(X) ∩ D(𝔥) with u_{k} → u in D(𝔥). There exists ψ ∈ C_{c}(X) ∩ D(𝔥) with 0 ⩽ ψ ⩽ 1, ψ = 1 on supp u. By the remark before the lemma we may assume without restriction that φ_{k} := ψu_{k} → ψu = u in D(𝔥) and in L_{2}(μ_{r}), which concludes the proof.

Lemma 2.5. Let u ∈ L_{2}(μ_{s}), ε > 0. Then there exists φ ∈ C_{c}(X) ∩ D(𝔥) with ||u − φ||_{L_{2}(μ_{s})} ⩽ ε, 𝔥(φ) + ||φ||_{L_{2}(m+μ_{r})}^{2} ⩽ ε.

Proof. Since C_{c}(X) ∩ D(𝔥) is dense in L_{2}(m + μ), we obtain ψ ∈ C_{c}(X) ∩ D(𝔥) such that ||u − ψ||_{L_{2}(μ_{s})} ⩽ \frac{ε}{2}. There exists a Borel set N ⊆ K := supp ψ with 𝔥-cap(N) = 0, μ_{s}(K \ N) = 0. Since μ_{s} is inner regular, there exists a compact set K_{0} ⊆ N such that ||11_{N\K_{0}}ψ||_{L_{2}(μ_{s})} ⩽ \frac{ε}{2}. For the compact set K_{0} we obtain

0 = 𝔥-cap(K_{0}) = inf{𝔥(φ) + ||φ||_{L_{2}(m)}^{2}; φ ∈ C_{c}(X) ∩ D(𝔥), φ ⩾ 1 on K_{0}};

cf. [11; Lemma 2.2.7, Problem 1.4.1]. Thus, there exists a sequence (φ_{k}) ⊆ C_{c}(X) ∩ D(𝔥) with 0 ⩽ φ_{k} ⩽ 1, φ_{k} = 1 on K_{0}, φ_{k} → 0 in D(𝔥). As above we may assume without restriction that φ_{k}ψ → 0 in D(𝔥) and in L_{2}(μ_{r}). Also,

||u − φ_{k}ψ||_{L_{2}(μ_{s})} ⩽ ||u − ψ|| + ||ψ − φ_{k}ψ|| ⩽ \frac{ε}{2} + ||11_{N\K_{0}}ψ|| ⩽ ε (k ∈ ℕ).

Thus, taking φ := φ_{k}ψ for suitably large k, we obtain φ as asserted.

Proof of Theorem 2.2. First we show D(τ) ⊆ {u ∈ L_{2}(m + μ) ∩ D(𝔥); u = ũ μ_{r}-a.e.}. Let u ∈ D(τ). By definition there exists (φ_{n}) ⊆ C_{c}(X) ∩ D(𝔥) with φ_{n} → u in the 𝔥-norm and in L_{2}(m+μ). Without restriction φ_{n} → ũ 𝔥-q.e. This implies φ_{n} → ũ μ_{r}-a.e., so u = ũ μ_{r}-a.e.

Conversely, let u ∈ L_{2}(m + μ) ∩ D(𝔥), u = ũ μ_{r}-a.e. Let (φ_{n}) ∈ C_{c}(X) ∩ D(𝔥) be the sequence obtained in Lemma 2.4. By Lemma 2.5 there exists (ψ_{n}) ⊆ C_{c}(X) ∩ D(𝔥) with

||(u − φ_{n}) − ψ_{n}||_{L_{2}(μ_{s})} ⩽ \frac{1}{n}, 𝔥(ψ_{n}) + ||ψ_{n}||_{L_{2}(m+μ_{r})}^{2} ⩽ \frac{1}{n} (n ∈ ℕ).

This implies that the sequence (φ_{n} +ψ_{n}) ⊆ C_{c}(X)∩D(τ) converges to u in D(τ):

||u − (φ_{n} + ψ_{n})||_{τ}^{2} ⩽ (𝔥(u − φ_{n})^{\frac{1}{2}} + 𝔥(ψ_{n})^{\frac{1}{2}} )^{2}
+ (||u − φ_{n}||_{L_{2}(m+μ_{r})} + ||ψ_{n}||_{L_{2}(m+μ_{r})})^{2} + ||u − φ_{n} − ψ_{n}||_{L_{2}(μ_{s})}^{2} → 0.

It remains to show that τ decomposes as indicated. For u ∈ D(τ) we have u = ũ μ_{r}-a.e., thus u ∈ D(τ_{r}) and

τ_{r}(u, v) = 𝔥(u, v) = τ(u, v) (u, v ∈ D(τ)).

Conversely, let u_{r} ∈ D(τ_{r}), u_{s} ∈ L_{2}(μ_{s}). Then there exists u ∈ L_{2}(m + μ) with u_{r} = u (m + μ_{r})-a.e., u_{s} = u μ_{s}-a.e. From ũ = ũ_{r} 𝔥-q.e. we obtain u ∈ D(τ).

In order to show that the measure ν introduced above decomposes analogously to τ we need the following fact on capacities.

Lemma 2.6. Let N ∈ 𝔅_{σ}. Then τ-cap(N) = 0 if and only if 𝔥-cap(N) = 0 and μ_{s}(N) = 0.

Proof. Necessity is obvious. For sufficiency, we first note that we may assume that N is contained in a compact set. Let ε > 0. There exists ψ ∈ C_{c}(X) ∩ D(𝔥) with 0 ⩽ ψ ⩽ 1, ψ = 1 in a neighbourhood of N, ∫ ψ^{2} dμ_{s} ⩽ ε. Moreover, there exists a sequence (φ_{k}) ⊆ D(𝔥) with 0 ⩽ φ_{k} ⩽ 1, φ_{k} = 1 in a neighbourhood of N (k ∈ ℕ), φ_{k} → 0 in D(𝔥) as k → ∞. As above we may assume that φ_{k}ψ → 0 in D(𝔥) and in L_{2}(m + μ_{r}) as k → ∞. Since φ_{k}ψ = 1 in a neighbourhood of N (k ∈ ℕ), we obtain

τ-cap(N) ⩽ lim inf_{k→∞} (𝔥(φ_{k}ψ) + ∫ (φ_{k}ψ)^{2} d(m + μ)) ⩽ ε.

This proves τ-cap(N) = 0.

As before, let ν be a Borel measure on X which is absolutely continuous with respect to τ-capacity. We note that only the restriction of ν to 𝔅_{σ} is relevant for the form τ + ν. This is due to the fact that, for u ∈ D(τ), there exists a quasi-continuous version ũ such that [ũ ≠ 0] belongs to 𝔅_{σ}. Indeed, there exists a sequence (u_{k}) ⊆ C_{c}(X) ∩ D(τ) such that u_{k} → ũ τ-q.e., and we may assume [ũ ≠ 0] ⊆ ⋃_{k∈ℕ} supp u_{k}.

Proposition 2.7. There exists a decomposition ν = ν_{r} +ν_{s} on 𝔅_{σ} such that ν_{r} is absolutely continuous with respect to 𝔥-capacity and ν_{s} is μ_{s}-absolutely continuous. As a consequence, τ + ν decomposes as τ + ν = (τ_{r} + ν_{r}) ⊕ ν_{s} in L_{2}(m + μ) = L_{2}(m + μ_{r}) ⊕ L_{2}(μ_{s}).

Proof. We use the local Borel set M obtained in Lemma 2.1 and claim that ν_{r} := 11_{X\M}ν, ν_{s} := 11_{M}ν constitutes the asserted decomposition.

Let N ∈ 𝔅_{σ} be a set of 𝔥-capacity zero. Then μ(N \ M) = μ_{r}(N) = 0. Lemma 2.6 implies that τ-cap(N \M) = 0. Since ν is absolutely continuous with respect to τ-capacity, we obtain ν_{r}(N) = ν(N \ M) = 0.

On the other hand, let N ∈ 𝔅_{σ} be a μ_{s}-null set. Then μ(N ∩ M) = μ_{s}(N) = 0, and we have 𝔥-cap(N ∩ M) = 0 by Lemma 2.1, so Lemma 2.6 implies τcap(M ∩ N) = 0. Thus ν_{s}(N) = ν(M ∩ N) = 0.

The decomposition of τ +ν now follows from the decomposition of τ obtained in Theorem 2.2.

In part (a) of the following remark we recall some of the far-reaching consequences of the results obtained so far.

Remarks 2.8. (a) By [21; Thm. 4.1(a)] there exists a locally measurable set X_{0} ⊆ X such that L_{2}(X_{0}, m + μ) = \overline{D(τ + ν)}^{L_{2}(m+μ)}. Therefore, the form τ + ν is associated with a selfadjoint operator A in L_{2}(X_{0}, m + μ). Additionally, the form τ + ν is a (symmetric) Dirichlet form which means that it satisfies the Beurling-Deny criteria; we refer to [7; Sec. 1.3], [14; Ch. I, Sec. 4], [21; Sec. 4] for these notions and their consequences. In particular, the C_{0}-semigroup (e^{−tA}; t ⩾ 0) extends to a positive contractive C_{0}-semigroup T_{p} on L_{p}(X_{0}, m + μ), for all p ∈ [1, ∞). As a side remark we mention that, for p ∈ (1, ∞), the semigroup T_{p} is analytic of angle arccos |1 − \frac{2}{p}|, by [13; Cor. 3.2] (see also [17]).

(b) If μ has a non-trivial singular part μ_{s} then the form τ + ν decomposes as described in Proposition 2.7. The singular part ν_{s} is μ_{s}-absolutely continuous, thus ν_{s} = ρ_{s}μ_{s} for a suitable locally measurable function ρ_{s} (Indeed, this follows from the usual Radon-Nikodym theorem together with the strict localisability of (X, μ_{s}) described in the proof of Lemma 2.1). Since τ vanishes on L_{2}(μ_{s}), the part of the form τ +ν in L_{2}(μ_{s}) corresponds to the maximal operator of multiplication by ρ_{s}. Thus, the semigroup T_{p} acts on L_{p}(X_{0}, μ_{s}) = L_{p}(X_{0} ∩ M, m + μ) as multiplication by e^{−tρ_{s}}.

3 Second order elliptic operators

Let Ω ⊆ ℝ^{d} be open. Let Σ ⊆ ∂Ω be relatively open in the C^{1}-part of ∂Ω. Let a: Ω → ℝ^{d×d} be measurable, a(x) symmetric and positive semi-definite for all x ∈ Ω.

We want to use the setting of Section 2 in order to obtain an operator realisation of A = −∇ · (a∇), with zero Dirichlet boundary condition on ∂Ω \ Σ and Wentzell boundary condition

−αAu + n · a∇u + γu = 0 on Σ.  (3.1)

Here we assume γ: Σ → [0, ∞] measurable, 0 ⩽ α ∈ L_{1,loc}(Σ, dS), where dS denotes the (d−1)-dimensional surface measure on Σ, and n denotes the outward normal on Σ.

In the following we motivate our formulation of the Wentzell boundary condition. For an analogous motivation in terms of resolvents we refer to [9; Sec. 2].

We want to define A = −∇ · (a∇) by the form method, with boundary conditions as above. Taking u satisfying the above boundary conditions and Au ∈ L_{2}(Ω), moreover v vanishing on ∂Ω \ Σ in a suitable sense, and applying the divergence theorem purely formally, we obtain

∫_{Ω} Au \overline{v} dx = − ∫_{Ω} ∇· (a∇u)\overline{v} dx = ∫_{Ω} a∇u · ∇\overline{v} dx − ∫_{Σ} n · a∇u \overline{v} dS
= ∫_{Ω} a∇u · ∇\overline{v} dx − ∫_{Σ} Au \overline{v} αdS + ∫_{Σ} u\overline{v} γdS.

Looking at the right hand side one realises that the part containing Au should rather appear on the left hand side. The only way how to achieve this is including the measure μ := αdS in the measure for the L_{2}-space in which A should act, for then the above formula can be written as

(Au|v)_{L_{2}(m+μ)} = ∫_{Ω} a∇u · ∇\overline{v} dx + ∫_{Σ} u\overline{v} γdS.  (3.2)

The right hand side defines a symmetric form with a domain which will be specified subsequently. So A should be the operator in L_{2}(m + μ) associated with this form.

Remark 3.1. (a) On those parts where one does not want to pose zero Dirichlet boundary conditions, the coefficient of n · a∇u in (3.1) should be different from zero; otherwise one could not replace this term in the above computation. This assumption also appears in [9; Sec. 2]. We simply choose the corresponding coefficient as 1.

(b) In [9], the special case Σ = ∂Ω and a = ã id, with a scalar function ã is studied. In that paper, the Wentzell boundary condition is formulated as −Au+β̃n·∇u+γ̃u = 0, with β̃ strictly positive. Rewriting this in our notation we obtain α = \frac{ã}{β̃}, γ = \frac{ãγ̃}{β̃}. Thus, implicitly, α = γ = 0 on [a = 0] is assumed in [9]. This amounts to posing a Neumann boundary condition on [a = 0]. It turns out, however, that even on this set a Wentzell boundary condition—formulated in our way—may be meaningful if a does not vanish too strongly; see Propositions 3.7 and 4.2 below.

We define the symmetric form 𝔥̃ in L_{2}(Ω) by

D(𝔥̃) := \{u ∈ L_{2}(Ω); ∇u ∈ L_{1,loc}(Ω), ∫_{Ω} a(x)∇u(x) · ∇\overline{u}(x) dx < ∞\},
𝔥̃(u, v) := ∫_{Ω} a(x)∇u(x) · ∇\overline{v}(x) dx (u, v ∈ D(𝔥̃)).

We assume that

(H1) 𝔥̃ is densely defined and closable.

Then the closure of 𝔥̃ is a symmetric Dirichlet form.

Remark 3.2. Obviously, 𝔥̃ is densely defined if

a ∈ L_{1,loc}(Ω).

A sufficient condition for 𝔥̃ to be closed is given by

a a.e. invertible, a^{−1} ∈ L_{1,loc}(Ω)

(cf. [20; Thm. 3.2]).

In the appendix we show that (H1) also holds if a ∈ W_{2,loc}^{1}(Ω).

Moreover we assume that

(H2) C_{c}(Ω ∪ Σ) ∩ D(𝔥̃) is dense in C_{0}(Ω ∪ Σ) with respect to the sup-norm.

Remark 3.3. Assumption (H2) is satisfied, e.g., if

a ∈ L_{1,loc}(Ω ∪ Σ).

Indeed, then C_{c}^{1}(Ω∪Σ) is contained in D(𝔥̃), and it is easy to show that C_{c}^{1}(Ω∪Σ) is dense in C_{c}(Ω ∪ Σ) and hence in C_{0}(Ω ∪ Σ).

In the one-dimensional case, however, (H2) is satisfied if only a ∈ L_{1,loc}(Ω); see Section 4.

If (H1) and (H2) hold then

𝔥 := \overline{𝔥̃↾_{C_{c}(Ω∪Σ)∩D(𝔥̃)}}

is a regular symmetric Dirichlet form in L_{2}(X, m), with X = Ω ∪ Σ and m the Lebesgue measure on X. The operator associated with this form should be considered as the realisation of −∇· (a∇) with zero Dirichlet boundary condition on ∂Ω \ Σ and Neumann boundary condition on Σ.

The hypothesis α ∈ L_{1,loc}(Σ, dS) implies that μ = αdS is a Radon measure on X. Starting from the form 𝔥 above, we thus can define the form τ in L_{2}(X, m+μ) as in Section 2. As indicated by formula (3.2), our realisation of A = −∇· (a∇), with boundary condition (3.1), will be the operator associated with the form τ +ν in L_{2}(m + μ), with ν := γdS. For the form τ + ν to be well-defined, we have to assume that

(H3) ν = γdS is absolutely continuous with respect to τ-capacity.

Theorem 3.4. Let X = Ω ∪ Σ, a, α, and γ be as above. Assume that (H1) and (H2) are satisfied. Let 𝔥 and τ be as above, and assume that (H3) is satisfied.

Then the form τ + ν is associated with a selfadjoint operator A ⩾ 0 in L_{2}(X_{0}, m + μ) = \overline{D(τ + ν)}^{L_{2}(m+μ)} (see Remark 2.8(a) for the meaning of X_{0}). The C_{0}-semigroup (e^{−tA}; t ⩾ 0) on L_{2}(X_{0}, m + μ) is positivity preserving and contractive in L_{p}(X_{0}, m + μ), for all 1 ⩽ p ⩽ ∞.

The proof of this theorem is contained in the explanations in Remark 2.8(a).

Remark 3.5. The preceding result summarises our procedure to associate a Dirichlet form with the differential operator −∇·(a∇) subject to Wentzell boundary conditions (3.1), under rather general assumptions. Having achieved this goal one knows that this Dirichlet form gives rise to a C_{0}-semigroup on L_{2} which in turn acts as a positive contractive C_{0}-semigroup on L_{p}, for all p ∈ [1, ∞). The negative generator A_{p} of this C_{0}-semigroup on L_{p} duly should be considered as the L_{p}-realisation of the differential operator subject to the boundary condition.

It is then a separate (difficult) task to obtain an explicit description of the operator A_{p}. In [9], a different approach is chosen: an L_{p}-operator is defined, and the authors endeavour to show that the closure of this operator generates a contractive C_{0}-semigroup.

In the remainder of this section we will discuss properties of μ = αdS and ν = γdS; in particular we will indicate conditions implying condition (H3).

Remarks 3.6. (a) If γ = 0 dS-a.e. on [α = 0] then ν is μ-absolutely continuous, so ν is absolutely continuous with respect to τ-capacity.

(b) We do not exclude α to be zero on part of Σ. In these points (3.1) becomes

n · a∇u + γu = 0,

i.e., a Robin boundary condition. In this way, the Robin boundary conditions can be realised as a measure perturbation of the Dirichlet form for the Neumann boundary condition (or rather ‘intermediate Neumann form’ because of our definition of 𝔥); see also [16; § 4.11.6].

Posing the Robin boundary condition is only possible on those parts of Σ where γdS is absolutely continuous with respect to 𝔥-capacity.

(c) Assume that μ = αdS is not absolutely continuous with respect to 𝔥capacity, i.e., μ as non-trivial singular part μ_{s}. From Lemma 2.1 we know that there exists a Borel set Σ_{0} ⊆ Σ such that μ_{s} = 11_{Σ_{0}}αdS. Then the density of ν_{s} with respect to μ_{s} is given by \frac{γ}{α}, and the semigroup generated by −A acts as (e^{−t \frac{γ}{α}}; t ⩾ 0) on L_{2}(X_{0}, μ_{s}) = L_{2}(X_{0} ∩ Σ_{0}, αdS), by Remark 2.8(b).

The following statement supplies a condition guaranteeing that μ and ν are absolutely continuous with respect to 𝔥-capacity. Then μ has no singular part μ_{s}, and ν is absolutely continuous with respect to τ-capacity.

Proposition 3.7. (a) In addition to the hypotheses (H1), (H2) assume that for all x ∈ Σ there exist an open neighbourhood U and vector fields Y, Z: U ∩X → ℝ^{d}, Y continuous, satisfying

Y = a^{1/2}Z, ∇· Y ∈ L_{2}(U ∩ X), Z ∈ L_{2}(U ∩ X)^{d}, n · Y ⩾ 1 on U ∩ Σ.

Then the surface measure dS on Σ is absolutely continuous with respect to 𝔥capacity. As a consequence, (H3) is satisfied.

(b) The additional hypothesis formulated in (a) is satisfied if a is a.e. invertible and

a^{−1} ∈ L_{1,loc}(Ω ∪ Σ).

For the proof we need the divergence theorem in the following general form.

Lemma 3.8. Let w ∈ C_{c}(X)^{d} with ∇ · w ∈ L_{1}(Ω) (where ∇ · w is taken in the sense of distributions on Ω). Then

∫_{Ω} ∇· w(x) dx = ∫_{Σ} w(x) · n(x) dS(x),

where n is the outward normal on Σ.

Proof. Using a partition of unity one can approximate w by functions having a continuous extension to ℝ^{d}, with divergence in L_{1}(ℝ^{d}). For such functions, the assertion follows by approximation by smooth functions.

Proof of Proposition 3.7. (a)(i) In the first step we show that functions in D(𝔥) have L_{1,loc}-trace on Σ.

Let x ∈ Σ, and U, Y, Z as in the hypothesis. Let φ ∈ C_{c}^{1}(ℝ^{d}), supp φ ⊆ U, 0 ⩽ φ ⩽ 1, φ = 1 in a neighbourhood of x. For u ∈ C_{c}(X) ∩ D(𝔥̃) (⊆ W_{1}^{1}(Ω)), Lemma 3.8 is applicable to the function φ|u|Y on X, and we obtain

∫_{Σ} φ|u| dS ⩽ ∫_{Σ} φ|u|Y · n dS = ∫_{Ω} ∇· (φ|u|Y) dx
= ∫_{Ω} ∇φ · Y |u| dx + ∫_{Ω} φY · ∇|u| dx + ∫_{Ω} φ|u|∇· Y dx
⩽ ||∇φ · Y ||_{2}||u||_{2} + ||φZ · a^{1/2}∇u||_{1} + ||φ∇· Y ||_{2}||u||_{2}.

The second term on the right hand side can be estimated by ||φZ||_{2}𝔥(u)^{\frac{1}{2}} . Thus, the trace mapping C_{c}(X) ∩ D(𝔥̃) ∋ u ↦ u↾_{Σ} ∈ L_{1,loc}(Σ) is continuous with respect to the 𝔥-norm on C_{c}(X) ∩ D(𝔥̃). Since C_{c}(X) ∩ D(𝔥̃) is dense in D(𝔥) (by definition), we obtain the assertion.

(ii) Let N ⊆ Σ be a set of 𝔥-capacity zero, N relatively compact in Σ. Then there exists a sequence (u_{n}) ⊆ D(𝔥)_{+} with ⋃_{n} supp u_{n} relatively compact in X, u_{n} ⩾ 1 in a neighbourhood of N (n ∈ ℕ), and 𝔥(u_{n}) → 0 as n → ∞. By (i) this implies ||11_{N}||_{L_{1}(Σ)} ⩽ ||u_{n}||_{L_{1}(Σ)} → 0 as n → ∞, i.e., N is a dS-null set.

This shows that each set of 𝔥-capacity zero is a dS-null set.

(b) Let x ∈ Σ, and let n_{x} be the outward normal in x. Then, for any c > 1, the hypothesis of (a) holds for the constant vector field Y := cn_{x}, Z := a^{−1/2}Y, and a suitably small neighbourhood U of x.

4 The one-dimensional case

In this section we discuss the example d = 1, Ω = (0, 1), Σ = {0, 1} in more detail, where the notation is as in Section 3. Then X = [0, 1], so C_{c}(X) = C[0, 1]. Throughout we assume that a, a^{−1} ∈ L_{1,loc}(0, 1), so (H1) holds (and 𝔥̃ is closed), by Remark 3.2. A standard hypothesis in papers dealing with the present context is that a is continuous.

In view of the abundant literature for the one-dimensional case, most of the facts presented in this section are likely to be known. The purpose of the following discussion is presenting these facts under the point of view of Sections 2 and 3.

First of all we show that 𝔥̃ is a regular Dirichlet form; in particular, 𝔥 = 𝔥̃, and assumption (H2) is automatically fulfilled.

Proposition 4.1. (a) D(𝔥̃) ⊆ C(0, 1).

(b) Let

D_{0} := {u ∈ D(𝔥̃); u↾_{[0,ε]}, u↾_{[1−ε,1]} are constant for some ε > 0}.

Then D_{0} ⊆ C[0, 1] ∩ D(𝔥̃), and D_{0} is dense in D(𝔥̃) as well as in C[0, 1]. In particular, 𝔥̃ is a regular Dirichlet form.

Proof. (a) is clear since D(𝔥̃) ⊆ W_{1,loc}^{1}(0, 1).

(b) The first assertion follows from (a). Let u ∈ D(𝔥̃). Then

D_{0} ∋ u_{ε} := 11_{(0,ε)}u(ε) + 11_{[ε,1−ε]}u + 11_{(1−ε,1)}u(1 − ε) ⟶ u

in D(𝔥̃) as ε → 0. On the other hand,

{u ∈ C^{1}[0, 1]; u↾_{[0,ε]}, u↾_{[1−ε,1]} are constant for some ε > 0} ⊆ D_{0}

is dense in C[0, 1].

As an immediate consequence of D(𝔥) = D(𝔥̃) ⊆ C(0, 1), for all x ∈ (0, 1) the set {x} has positive 𝔥-capacity. In the following we characterise when the endpoints of the interval have positive 𝔥-capacity.

Proposition 4.2. (cf. Proposition 3.7) The set {0} has positive 𝔥-capacity if and only if a^{−1}↾_{(0, \frac{1}{2} )} ∈ L_{1}. Analogously, 𝔥-cap(\{1\}) > 0 if and only if a^{−1}↾_{( \frac{1}{2} ,1)} ∈ L_{1}.

Proof. If a^{−1}↾_{(0, \frac{1}{2} )} ∈ L_{1} then u′ = a^{− \frac{1}{2}} a^{\frac{1}{2}} u′ ∈ L_{1,loc}[0, 1) for all u ∈ D(𝔥̃) and hence D(𝔥̃) ⊆ C[0, 1). Thus, {0} has positive 𝔥-capacity.

Assume now that a^{−1}↾_{(0, \frac{1}{2} )} ∉ L_{1}. Define the function f ∈ W_{1,loc}^{1}(0, 1) by

f(x) := 1 + ∫x^{\frac{1}{2}} \frac{dt}{a(t)} (x ⩽ \frac{1}{2}), f(x) := 1 (x > \frac{1}{2}).

For u := ln f we then have u′ = \frac{f′}{f} = −\frac{1}{af} on (0, \frac{1}{2}), and hence

a|u′|^{2} = au′ · u′ = −\frac{1}{f} · \frac{f′}{f} = (\frac{1}{f} )′.

Since f(x) → ∞ as x → 0, this yields ∫_{0}^{1} a|u′|^{2} = 1.

Finally, let u_{n} := (\frac{u}{n}) ∧ 1. Then u_{n} ∈ C[0, 1] ∩ D(𝔥), u_{n}(0) = 1, 𝔥(u_{n}) ⩽ \frac{1}{n^{2}} for all n ∈ ℕ, u_{n} → 0 in L_{2}(0, 1), and therefore 𝔥-cap({0}) = 0.

Let α, γ, μ, ν be as in the previous section. Then μ = α(0)δ_{0} + α(1)δ_{1}, ν = γ(0)δ_{0}+γ(1)δ_{1}, where δ_{j} is the Dirac measure in {j}, for j = 0, 1. Assume that, for j = 0, 1, α(j) = γ(j) = 0 if 𝔥-cap({j}) = 0. Then μ, ν are absolutely continuous with respect to 𝔥-capacity. Moreover, let γ(0), γ(1) < ∞ (see Remark 4.4(b) concerning the case that this is not satisfied). Then the form τ + ν is densely defined. Let A be the selfadjoint operator in L_{2}(m + μ) and T_{p} the C_{0}-semigroup on L_{p}(m + μ) associated with τ + ν.

Proposition 4.3. (a) The operator A is given by

D(A) = D_{A} := {u ∈ D(τ); au′ absolutely continuous, (au′)′ ∈ L_{2}(0, 1),

(−1)^{j+1}(au′)(j) + (γu)(j) = 0 if α(j) = 0 (j = 0, 1)}

(where (γu)(j) = 0 if γ(j) = 𝔥-cap({j}) = 0; note that u may have no trace at j if 𝔥-cap({j}) = 0),

Au = −(au′)′ m-a.e.,
(Au)(j) = \frac{1}{α(j)}((−1)^{j+1}(au′)(j) + (γu)(j)) if α(j) > 0 (j = 0, 1).

(b) If \frac{1}{a} ∈ L_{1}(0, 1) then T_{1}(t): L_{1}(m + μ) → C[0, 1] for all t > 0,

||T_{1}(t): L_{1}(m + μ) → C[0, 1]|| ⩽ ct^{−1} (0 < t ⩽ 1)

for some c > 0, and (T_{1}(t)↾_{C[0,1]}; t ⩾ 0) is a C_{0}-semigroup on C[0, 1]. In particular, T_{1} has the strong Feller property (in the sense of [18; Sec. 2]).

Proof. (a) Observe that D(τ + ν) = D(τ) = D(𝔥̃) with equivalence of the norms. Thus, the set D_{0} defined in Proposition 4.1(b) is dense in D(τ +ν). Let u ∈ D(A), f := Au. Then u ∈ D(τ) and (f|v)_{L_{2}(m+μ)} = (τ + ν)(u, v), i.e.,

∫_{0}^{1} fv dm + ∑_{j=0}^{1}(αfv)(j) = ∫_{0}^{1} au′v′ dm + ∑_{j=0}^{1}(γuv)(j) (v ∈ D_{0})  (4.1)

(where (αfv)(j) = 0 if α(j) = 0; note that f(j) is not defined and v may have no trace at j in the case α(j) = 0). For v ∈ C_{c}^{∞}(0, 1) ⊆ D_{0} the second term on both sides vanishes. We obtain (au′)′ = −f↾_{(0,1)} m-a.e., so au′ is absolutely continuous and Au = −(au′)′ m-a.e. We infer that

∫_{0}^{1} fv dm = − ∫_{0}^{1}(au′)′v dm = − lim_{ε→0} au′v|_{ε}^{1−ε} + ∫_{0}^{1} au′v′ dm (v ∈ D_{0}) (4.2)

and hence, by (4.1),

∑_{j=0}^{1}(αfv)(j) = lim_{ε→0} au′v|_{ε}^{1−ε} + ∑_{j=0}^{1}(γuv)(j) (v ∈ D_{0}).  (4.3)

This shows (αAu)(j) = (αf)(j) = (−1)^{j+1}(au′)(j) + (γu)(j) for j = 0, 1. Thus u ∈ D_{A}, and the representation for Au is valid.

Conversely, let u ∈ D_{A}. Define the function f ∈ L_{2}(m + μ) by

f = −(au′)′ m-a.e., f(j) = \frac{1}{α(j)}((−1)^{j+1}(au′)(j) + (γu)(j)) if α(j) > 0

for j = 0, 1. Then (4.2) and (4.3) are easily verified, so (4.1) holds. Since D_{0} is dense in D(τ + ν), this shows u ∈ D(A), Au = f.

(b) As in the beginning of the proof of Proposition 4.2 we obtain D(τ + ν) ↪ C[0, 1]. Since T_{2} is an analytic semigroup on L_{2}(m + μ), we have T_{2}(t): L_{2}(m + μ) → D(τ +ν) (⊆ C[0, 1]), thus also T_{1}(t): L_{1}(m+μ) → L_{2}(m+μ), by symmetry and duality. This shows the first assertion. Since ||u||_{2}^{4} ⩽ ||u||_{∞}^{2}||u||_{1}^{2} ⩽ c((τ + ν + 1)(u))||u||_{1}^{2} for all u ∈ D(τ + ν), the norm estimate is an easy consequence of [7; Cor. 2.4.7, (ii) ⟹ (i)], applied with μ = 2.

By Proposition 4.1(b), D(τ + ν) ⊇ D_{0} is dense in C[0, 1]. Moreover, (T_{2}(t)↾_{D(τ+ν)}; t ⩾ 0) is a C_{0}-semigroup on D(τ +ν). This implies the last assertion since T_{2} is L_{∞}-bounded.

Remarks 4.4. (a) The assumption that α(j) = γ(j) = 0 if 𝔥-cap({j}) = 0 was made for simplicity. It suffices to assume γ(j) = 0 if α(j) = 𝔥-cap({j}) = 0, for j = 0, 1. Then ν is still absolutely continuous with respect to τ-capacity. But, as discussed in Remark 2.8(b), T_{2} acts trivially on L_{2}({j ∈ {0, 1}; α(j) > 0, 𝔥cap({j}) = 0}, αdS).

As a consequence, if \frac{1}{a} ∉ L_{1}(0, \frac{1}{2}), α(0) ≠ 0 (or \frac{1}{a} ∉ L_{1}(\frac{1}{2}, 1), α(1) ≠ 0) then T_{2} in general does not act on C[0, 1].

(b) We could also allow γ to assume the value ∞. This would lead to Dirichlet boundary conditions on [γ = ∞]. If Σ_{0} := [γ = ∞] ∩ [α > 0] ≠ ∅ then τ + ν is not densely defined, and \overline{D(τ + ν)}^{L_{2}(m+μ)} = L_{2}(m + μ↾_{[0,1]\Σ_{0}}).

(c) It follows from Remark 5.1(b) below that T_{1} is stochastic if γ = 0.

(d) The representation of the operator A given in Proposition 4.3(a) illustrates that the Wentzell boundary condition in L_{p} actually does not constitute a boundary condition but determines A on ∂Ω. Only in C[0, 1] it really is a boundary condition.

5 Conservation of mass

Let 𝔥, μ, τ be as in Section 2, T the C_{0}-semigroup on L_{2}(m) associated with 𝔥, and T_{μ} the C_{0}-semigroup on L_{2}(m + μ) associated with τ. The interpretation of T_{μ} is that the ‘diffusion’ described by T is slowed down corresponding to μ. It is thus natural to conjecture that T_{μ} is stochastic whenever T is stochastic. It is easy to see that this is true if m and μ are finite (see Remark 5.1(b) below).

For general m and μ we cannot answer the question, but in case μ is absolutely continuous with respect to m the answer is positive; see Theorem 5.5. We prove this in the following more general setting.

Let (X, m) be a measure space, 𝔥 a (not necessarily symmetric) Dirichlet form in L_{2}(m). Let −A_{p} denote the generator of the associated C_{0}-semigroup T_{p} on L_{p}(m), for 1 ⩽ p < ∞.

Remark 5.1. (a) It is easy to see that T_{1} is stochastic (i.e., T_{1} positive and ||T_{1}(t)f||_{1} = ||f||_{1} for all f ∈ L_{1}(m)_{+}) if and only if

∫ A_{1}u dm = 0 (u ∈ D(A_{1})).

(b) If m is finite then (a) implies that T_{1} is stochastic if and only if

1 ∈ D(𝔥) and 𝔥(u, 1) = 0 (u ∈ D(𝔥)).

(For necessity we note that T_{1}(t)11 = 11 since T_{1}(t)11 ⩽ 11 and ∫ T_{1}(t)11 dm = ∫ 11 dm; for sufficiency recall that D(A_{1}) ∩ D(𝔥) is a core for A_{1}.)

The following proposition is an auxiliary result describing A_{p}↾_{D(A_{p})∩D(𝔥)} in terms of the form 𝔥.

Proposition 5.2. Let p ∈ [1, ∞), \frac{1}{p} + \frac{1}{p′} = 1, D ⊆ D(𝔥) ∩ L_{p′}(m) a sublattice, D dense in D(𝔥). Let u ∈ L_{p}(m) ∩ D(𝔥), v ∈ L_{p}(m). Then the following are equivalent:

(i) u ∈ D(A_{p}), A_{p}u = v; (ii) 𝔥(u, φ) = ∫ v\overline{φ} dm for all φ ∈ D(𝔥) ∩ L_{p′}(m); (iii) 𝔥(u, φ) = ∫ v\overline{φ} dm for all φ ∈ D.

For the proof of the implication ‘(iii) ⟹ (ii)’ we need the density argument stated in the following lemma. The proof of this result is adapted from [15; Kor. 11.2.5].

Lemma 5.3. Let η: X → [0, ∞] be measurable. Let 1 ⩽ p ⩽ ∞, D ⊆ D(𝔥) ∩ L_{p}(ηm) a sublattice, D dense in D(𝔥).

(a) If 1 ⩽ p < ∞ then D is dense in D(𝔥) ∩ L_{p}(ηm).

(b) D ∩ B_{L_{∞}(ηm)} is dense in D(𝔥) ∩ B_{L_{∞}(ηm)} with respect to the topology of D(𝔥). (Here, B_{L_{∞}(ηm)} denotes the closed unit ball of L_{∞}(ηm)).

Proof. (a) Let u ∈ D(𝔥) ∩ L_{p}(ηm); without restriction u ⩾ 0.

In the first step we assume additionally that there exists ψ ∈ D such that u ⩽ ψ. There exists (ũ_{k}) ⊆ D, ũ_{k} → u in D(𝔥); without restriction ũ_{k} → u m-a.e. Then D ∋ u_{k} := (Re ũ_{k})^{+} ∧ ψ → u in D(𝔥) ∩ L_{p}(ηm).

For the general case let (ũ_{k}) ⊆ D, ũ_{k} → u in D(𝔥); without restriction ũ_{k} → u m-a.e. Then u_{k} := (Re ũ_{k})^{+} ∧ u is in the closure of D in D(𝔥) ∩ L_{p}(ηm), by the first step, and u_{k} → u in D(𝔥) ∩ L_{p}(ηm).

(b) The proof is analogous to the proof of (a).

Proof of Proposition 5.2. (i) ⟹ (ii),(iii) Firstly, \frac{1}{t}(u−T_{p}(t)u) → v in L_{p}(m) as t → 0. Secondly, u ∈ D(𝔥) implies

(\frac{1}{t}(u − T_{p}(t)u)|φ) → 𝔥(u, φ) (φ ∈ D(𝔥))

(see, e.g., [19; Lemma 1.1]). This implies the assertion.

(iii) ⟹ (ii) For 1 < p < ∞ this is immediate from Lemma 5.3(a); for p = 1 it follows from Lemma 5.3(b) and the dominated convergence theorem.

(ii) ⟹ (i) First observe that the complex case is an easy consequence of the real case, so we assume 𝕂 = ℝ. Let Ǎ_{p} := A_{p}↾_{D(A_{p})∩D(A_{2})}. It is easy to see that D(A_{p}) ∩ D(A_{2}) is a core for A_{p}, so we obtain that R(1 + Ǎ_{p}) is dense in L_{p}(m). Let Â_{p} be defined by

D(Â_{p}) := {u ∈ D(𝔥) ∩ L_{p}(m); ∃ v =: Â_{p}u ∈ L_{p}(m) ∀ φ ∈ D(𝔥) ∩ L_{p′}(m) :
𝔥(u, φ) = ∫ vφ dm}.

Then Ǎ_{p} ⊆ Â_{p}, by ‘(i) ⟹ (ii)’. In order to show the assertion it now suffices to show that Â_{p} is accretive (i.e., −Â_{p} is dissipative). (Then (1+Ǎ_{p})^{−1} ⊆ (1+Â_{p})^{−1} have unique continuous extensions to L_{p}(m) coinciding with (1 + A_{p})^{−1}; thus Â_{p} ⊆ A_{p}.)

Let φ: ℝ → ℝ, φ(t) := |t|^{p−1} sgn t (= sgn t if p = 1). Let u ∈ D(Â_{p}). Then we have to show

∫ Â_{p}u(φ ∘ u) dm ⩾ 0.  (5.1)

For k ∈ ℕ we define φ_{k}(t) := (|t|^{p−1} ∧ (k|t|)) sgn t. Then |φ_{k}(t)| ⩽ |φ(t)|, φ_{k}(0) = 0, φ_{k} monotone increasing and Lipschitz continuous. This implies φ_{k} ∘ u ∈ D(𝔥) ∩ L_{p′}(m), and by [3; Cor. 9] we obtain

∫ Â_{p}u(φ_{k} ∘ u) dm = 𝔥(u, φ_{k} ∘ u) ⩾ 0.

Finally, φ_{k} → φ pointwise, so (5.1) follows by the dominated convergence theorem.

Let now η: X → [1, ∞] be measurable, and assume that D(𝔥) ∩ L_{2}(ηm) is dense in D(𝔥) as well as in L_{2}(ηm). Similarly to the setup of Section 2, with the measure μ := (η − 1)m (here absolutely continuous with respect to m), we define the Dirichlet form τ in L_{2}(ηm) by

τ(u, v) := 𝔥(u, v) on D(τ) := D(𝔥) ∩ L_{2}(ηm).

Remarks 5.4. (a) Let D be a sublattice of D(𝔥) ∩ L_{2}(ηm), D dense in D(𝔥). Then D is dense in D(𝔥) ∩ L_{2}(ηm) = D(τ), by Lemma 5.3(a). Thus, \overline{τ↾_{D}} = τ.

(b) For the application to Section 2 we obtain the following. Assume that μ is absolutely continuous with respect to m, μ = (η−1)m. Then D := C_{c}(X)∩D(𝔥) is a sublattice of D(𝔥) ∩ L_{2}(ηm) = D(𝔥) ∩ L_{2}(m + μ) since m + μ is a Radon measure. Part (a) shows that the form τ defined above coincides with the form τ introduced in Section 2.

A version of part (b) of the following theorem, in a different setting, is contained in [8; Thm. 1].

For 1 ⩽ p < ∞ we denote by T_{η,p} the C_{0}-semigroup on L_{p}(ηm) associated with τ, and by −A_{η,p} its generator.

Theorem 5.5. (a) Let 1 ⩽ p < ∞, u ∈ D(A_{η,p}) ∩ D(𝔥), ηA_{η,p}u ∈ L_{p}(m). Then u ∈ D(A_{p}), ηA_{η,p}u = A_{p}u.

(b) The operator A_{η,1} is given by

D(A_{η,1}) = D(A_{1}) ∩ L_{1}(ηm), ηA_{η,1}u = A_{1}u (u ∈ D(A_{η,1})).

(c) Assume that T_{1} is stochastic. Then T_{η,1} is stochastic.

Proof. (a) For φ ∈ D(τ) ∩ L_{p′}(ηm) =: D, Proposition 5.2, (i) ⟹ (ii), implies

𝔥(u, φ) = τ(u, φ) = ∫ A_{η,p}u \overline{φ} ηdm = ∫ ηA_{η,p}u \overline{φ} dm.

Recall that D(A_{η,2}) ∩ L_{p′}(ηm) ⊆ D is a core for A_{η,2} and hence for τ. Thus, D is dense in D(𝔥), and D is a sublattice of D(𝔥) ∩ L_{p′}(ηm). By Proposition 5.2, (iii) ⟹ (i), we obtain the assertion.

(b) Let u ∈ D(A_{η,1}). Since D(A_{η,1}) ∩ D(A_{η,2}) is a core for A_{η,1}, there exists a sequence (u_{k}) ⊆ D(A_{η,1}) ∩ D(A_{η,2}) with u_{k} → u, A_{η,1}u_{k} → A_{η,1}u in L_{1}(ηm). Then ηA_{η,1}u_{k} ∈ L_{1}(m), and by part (a) we obtain A_{1}u_{k} = ηA_{η,1}u_{k} → ηA_{η,1}u in L_{1}(m). Therefore u ∈ D(A_{1}), A_{1}u = ηA_{η,1}u.

So we have shown A_{η,1} ⊆ \frac{1}{η} Ǎ_{1}, where Ǎ_{1} := A_{1}↾_{D(A_{1})∩L_{1}(ηm)}. In the relation

1 + A_{η,1} ⊆ 1 + \frac{1}{η} Ǎ_{1} = \frac{1}{η}(Ǎ_{1} + η)  (5.2)

we know that 1 + A_{η,1} is surjective since A_{η,1} generates a contractive C_{0}-semi-group. On the other hand, A_{1} + (η − 1) is an accretive operator in L_{1}(m), and thus Ǎ_{1} + η is injective. Moreover, \frac{1}{η} Ǎ_{1} is an operator in L_{1}(ηm). This implies equality in (5.2).

(c) For u ∈ D(A_{η,1}) = D(A_{1}) ∩ L_{1}(ηm) we have ∫ A_{η,1}u ηdm = ∫ A_{1}u dm. Thus, the assertion follows from Remark 5.1(a).

Appendix

Let 𝔥̃ be the maximal form defined in Section 3. It is well-know that the restriction of 𝔥̃ to C_{c}^{∞}(Ω) is closable if a ∈ W_{2,loc}^{1}(Ω) (see, e.g., [7; Thm. 1.2.5]). We show that even 𝔥̃ is closable in this case.

Proposition A.1. Assume that a ∈ W_{2,loc}^{1}(Ω). Then 𝔥̃ is closable. Thus, assumption (H1) of Section 3 holds.

In [15; Prop. 14.2.7] this result is shown under the additional assumption a ∈ L_{∞,loc}(Ω). In order to remove this restriction, we need the following product rule for weak derivatives. The crucial point in the lemma is that only the scalar product ∇u · v is assumed to be locally integrable, not the single components v_{j}∂_{j}u.

Lemma A.2. Let 𝕂 = ℝ, and u ∈ W_{1,loc}^{1}(Ω), v ∈ W_{1,loc}^{1}(Ω)^{d} with

uv_{j}, u∂_{j}v_{k}, ∇u · v ∈ L_{1,loc}(Ω) (j, k = 1, …, d).

Then uv ∈ W_{1,loc}^{1}(Ω)^{d}, ∇· (uv) = ∇u · v + u∇· v.

Proof. If u and v are bounded then the assertion follows by a standard convolution argument. We deduce the general case from the case of bounded functions.

For n ∈ ℕ let u_{n} := (u ∧ n) ∨ (−n), w_{n} := (1 + \frac{|v|}{n})^{−1}, and v_{n} := w_{n}v. Then u_{n} ∈ W_{1,loc}^{1}(Ω) and ∇u_{n} = 11_{[|u|<n]}∇u. We claim that v_{n} ∈ W_{1,loc}^{1}(Ω)^{d} and

∇· v_{n} = w_{n}∇· v − w_{n} \frac{v}{|v|} · ∑_{j=1}^{d} \frac{v_{j}}{|v| + n}∂_{j}v.  (A.3)

Indeed, let φ(x) := (1+ \frac{|x|}{n})^{−1}x and j ∈ \{1, …, d\}. Then v_{n} = φ∘v, φ_{j} ∈ C_{b}^{1}(ℝ^{d}), ∇φ_{j}(x) = (1 + \frac{|x|}{n})^{−1}e_{j} − \frac{1}{n}(1 + \frac{|x|}{n})^{−2} \frac{x}{|x|}x_{j}. Thus, φ_{j} ∘ v ∈ W_{1,loc}^{1}(Ω) and

∂_{j}(φ_{j} ∘ v) = (∇φ_{j}) ∘ v · ∂_{j}v = w_{n}∂_{j}v_{j} − (n + |v|)^{−1}w_{n} \frac{v}{|v|}v_{j} · ∂_{j}v.

This shows (A.3).

Since u_{n}, v_{n} are bounded, we have

∇· (u_{n}v_{n}) = ∇u_{n} · v_{n} + u_{n}∇· v_{n}.

Now the assertion follows since u_{n}v_{n} → uv in L_{1,loc}(Ω)^{d} and

∇u_{n} · v_{n} = 11_{[|u|<n]}w_{n}∇u · v → ∇u · v,

u_{n}∇· v_{n} = w_{n}u_{n}∇· v − w_{n} \frac{v}{|v|} · ∑_{j=1}^{d} \frac{v_{j}}{|v| + n}u_{n}∂_{j}v → u∇· v

in L_{1,loc}(Ω) as n → ∞, by the dominated convergence theorem.

Proof of Proposition A.1. Since 𝔥̃ is real (i.e., 𝔥̃(u) = 𝔥̃(Re u) + 𝔥̃(Im u) for all u ∈ D(𝔥̃)), we can assume 𝕂 = ℝ. We have to show that for all (u_{n}) ⊆ D(𝔥̃) with 𝔥̃(u_{n} − u_{m}) → 0, ||u_{n}||_{2} → 0 as n, m → ∞ one has 𝔥̃(u_{n}) → 0 as n → ∞. It is easy to see that it suffices to show 𝔥̃(u_{n}, φ) → 0 for all φ ∈ D(𝔥̃).

Let W_{a} := {f ∈ L_{1,loc}(Ω)^{d}; af · f ∈ L_{1}(Ω)}. Then ⟨f, g⟩_{a} := ∫ af · g dx defines a semi-scalar product on W_{a}. It is readily seen that C_{c}^{∞}(Ω)^{d} is dense in W_{a} with respect to this semi-scalar product. Moreover, D(𝔥̃) = {u ∈ L_{2}(Ω); ∇u ∈ W_{a}}. We have to show

𝔥̃(u_{n}, φ) = ⟨∇u_{n}, ∇φ⟩_{a} → 0 (φ ∈ D(𝔥̃)).

The sequence (∇u_{n}) is bounded in W_{a} since ||∇u_{n} − ∇u_{m}||_{W_{a}}^{2} = 𝔥̃(u_{n} − u_{m}) → 0 as n, m → ∞, so it suffices to show

⟨∇u_{n}, Φ⟩_{a} → 0 for all Φ ∈ C_{c}^{∞}(Ω)^{d}.

From u_{n} ∈ L_{2}(Ω), aΦ ∈ W_{2,c}^{1}(Ω)^{d}, and ∇u_{n}, Φ ∈ W_{a} it follows that

u_{n}aΦ ∈ L_{1}(Ω)^{d}, u_{n}∂_{j}(aΦ)_{k} ∈ L_{1}(Ω) (j, k = 1, …, d), ∇u_{n} · aΦ ∈ L_{1}(Ω).

By Lemma A.2 we thus obtain u_{n}aΦ ∈ W_{1,c}^{1}(Ω)^{d} and

∇· (u_{n}aΦ) = ∇u_{n} · aΦ + u_{n}∇· (aΦ).

Since ∫ ∇· f dx = 0 for all f ∈ W_{1,c}^{1}(Ω)^{d}, u_{n} → 0 in L_{2}(Ω), and ∇· (aΦ) ∈ L_{2}(Ω), we conclude that

⟨∇u_{n}, Φ⟩_{a} = − ∫ u_{n}∇· (aΦ) dx → 0 (n → ∞).

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