ROBIN BOUNDARY VALUE PROBLEMS ON ARBITRARY DOMAINS

Daniel Daners^{∗}

School of Mathematics and Statistics University of Sydney, NSW 2006 Australia

March 25, 1996

Abstract. We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular we establish L_{p}-L_{q}-estimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations.

1. Introduction

It is well known that the equation −Δu + u = f subject to homogeneous Dirichlet or Neumann boundary conditions can be considered on arbitrary bounded domains Ω in ℝ^{N}. The idea is to introduce a weak formulation and to choose the “right” Hilbert space incorporating the boundary conditions in a generalised sense. One feature of this approach is that if data and domain are smooth enough we are back to classical solutions satisfying the boundary conditions pointwise. Looking at boundary conditions of Robin type such as

\frac{∂}{∂ν} u + βu = 0  (1.1)

(ν being the outer unit normal on ∂Ω and β a constant) it is well known that Dirichlet and Neumann boundary conditions correspond to two extreme cases, namely β = ∞ and β = 0, respectively. However, although introduced quite some time ago by Maz’ja [16], [17], Section 4.11.6, it seems not very well known that there is a weak formulation on arbitrary domains if β ∈ (0, ∞). The difficulty is to give sense to the boundary

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 35J25, Secondary 35K20, 35D05.

∗ Supported by a grant of the Australian Research Council

integral appearing in the Dirichlet form as there are problems with traces of functions in Sobolev spaces.

We shall use Maz’ja’s approach and develop an L_{p}-theory for Robin boundary value problems on arbitrary domains and general (non selfadjoint) second order elliptic operators in divergence form with bounded and measurable coefficients. It is known that the Neumann problem does not have any smoothing properties for a general domain as this is equivalent to embedding theorems for Sobolev spaces (see [5], Corollary 3.4), and that the Dirichlet problem always has (e.g. [4], Lemma 1). It turns out that the smoothing properties we prove in this paper lie “half way” between the two, and that they are the best possible in that framework (see Proposition 5.6 below). In a forthcoming paper [6] it will be shown that the L_{p}-L_{q}-estimates we obtain can not be improved even if we restrict ourselves to arbitrarily smooth domains if we don’t want the constants involved to depend on the geometry of the underlying domain. This follows from domain perturbation results which we shall prove in the above mentioned paper. The whole theory developed in this paper is in fact motivated by applications to domain perturbation of linear and nonlinear elliptic equations subject to Robin boundary conditions. It provides a priori estimates for solutions not depending on the domain geometry, which allows to deal with very singular perturbations of the domains such as cutting holes or adding small pieces. The results also allow to establish a positive lower bound for the first eigenvalue of −Δφ = λφ subject to boundary conditions of the form (1.1) with β > 0, uniformly with respect to all domains of the same volume. It would be interesting to know whether a Faber-Krahn type inequality holds or not.

We further show that the theory carries over to the corresponding parabolic problem, and that the parabolic problem fits into the framework of semigroup theory and abstract parabolic equations on L_{p}-spaces for 1 ≤ p < ∞. We also get estimates for the semigroup kernel (heat kernel). It turns out that the usual methods to prove kernel estimates such as described in [27] and [8] either apply directly or can be adapted to our situation very easily.

The outline of the paper is as follows. In Section 2 we give the precise assumptions, introduce notation and state some of our main results. Section 3 deals with the L_{2}- theory of weak solutions on arbitrary domains as introduced by Maz’ja [16], [17]. We discuss some problems arising with this approach and give examples. In Section 4 we provide global bounds for weak solutions of Robin boundary value problems. It is a slight extension of the results in [5]. These results are the basis to establish an L_{p}- theory for Robin boundary value problems on arbitrary domains which we develop in Section 5. The final section is concerned with the corresponding parabolic problem.

The paper concludes with an appendix on some facts on forms, the associated operators induced on various Banach spaces and the smoothing properties of their resolvents.

2. Assumptions and Main Results

In this section we give the precise assumptions, fix notation and state some of our main results. For more detailed statements, proofs and further comments we refer to later sections. In particular, we refer to Section 5 for the L_{p}-theory of elliptic problems and to Section 6 for parabolic problems.

We shall be concerned with the elliptic boundary value problem

\begin{aligned}Au & = f & in Ω \\ Bu & = 0 & on ∂Ω \end{aligned}  (2.1)

on arbitrary bounded domains Ω ⊂ ℝ^{N}, that is, a bounded open and connected set in ℝ^{N}. We always assume that N ≥ 2. For N = 1 every bounded domain is smooth and we do not have to prove anything. The differential operators A and B are supposed to be of the form

Au := − ∑_{i=1}^{N} ∂_{i}(∑_{j=1}^{N} a_{ij}(x)∂_{j}u + a_{i}(x)u) + ∑_{i=1}^{N} b_{i}(x)∂_{i}u + c_{0}(x)u  (2.2)

and

Bu := ∑_{i=1}^{N}( ∑_{j=1}^{N} a_{ij}(x)∂_{j}u + a_{i}(x)u)ν_{i} + b_{0}(x)u ,  (2.3)

where ∂_{i} := \frac{∂}{∂x_{i}} and ν := (ν_{1}, …, ν_{N}) denotes the outer unit normal on the boundary ∂Ω of Ω. We also assume that A is uniformly strongly elliptic, that is, there exists α_{0} > 0 such that

∑_{i,j=1}^{N} a_{ij}(x)ξ_{i}ξ_{j} ≥ α_{0}|ξ|^{2}  (2.4)

for all x ∈ \overline{Ω} and ξ ∈ ℝ^{N}, and that

inf_{x∈∂Ω} b_{0}(x) ≥ β_{0}  (2.5)

for some β_{0} > 0 (take the essential infimum if b_{0} is only measurable). We throughout assume that the coefficients of (A, B) are real valued L_{∞}-functions.

We denote by L_{p}(X) the Lebesgue spaces on a measurable subspace of X ⊂ ℝ^{N} and by ∥·∥_{p,X} its norm. If no confusion seems likely we just write L_{p} and ∥·∥_{p}, respectively.

Further, we write L_{p,loc}(X) for the space of all functions u such that u|_{K} ∈ L_{p}(K) for all K ⊂ X compact. Moreover, W_{p}^{k}(Ω) is the Sobolev space, which consists of all u ∈ L_{p} such that all the distributional derivatives up to the order k lie in L_{p}. D(Ω) is the space of smooth functions with compact support in Ω and W̊_{2}^{1}(Ω) its closure in W_{2}^{1}(Ω). Finally C(Ω) and C^{k}(Ω) are the spaces of continuous and k-times differentiable functions, respectively. If E, F are Banach spaces, we write E ↪ F if E ⊂ F and the natural injection is continuous. We write E′ for the topological dual and if A: E → F is a densely defined operator we denote its dual by A′.

Usually, a function u ∈ W_{2}^{1}(Ω) is called a weak solution of (2.1) if

a(u, v) = ⟨f, v⟩  (2.6)

for all v ∈ W_{2}^{1}(Ω), where a(· , ·) is the (generalised) Dirichlet form corresponding to (A, B) which is given by

a(u, v) := ∫_{Ω} ∑_{i=1}^{N}(∑_{j=1}^{N} a_{ij}∂_{j}u + a_{i}u)∂_{i}v + ∑_{i=1}^{N} b_{i}v∂_{i}u + c_{0}uv dx + ∫_{∂Ω} b_{0}uv dσ . (2.7)

Here, σ is the (N − 1)-dimensional Hausdorff measure restricted to ∂Ω (which coincides with the surface measure if ∂Ω is smooth (e.g. [10], Theorem 3.2.3)). Further,

⟨f, u⟩ := ∫_{Ω} fu dx .  (2.8)

It is well known that if data and domain are smooth any sufficiently smooth weak solution satisfies (2.1) pointwise. As W_{2}^{1}(Ω)-functions do not have well defined traces on ∂Ω for an arbitrary domain, and if so, it is not clear whether the boundary integral appearing in (2.7) is finite, we cannot use the space W_{2}^{1}(Ω) when dealing with weak solutions of (2.1) on arbitrary domains. The precise definition of a weak solution will be given in Section 3. As in case of the Dirichlet problem it turns out that (2.8) makes only sense for f ∈ L_{p}(Ω) if p is larger than some p_{0} > 1. In order to define solutions for inhomogeneities in “lower” L_{p}-spaces we “extrapolate” the problem by means of duality. This allows to define generalised solutions of (2.1) for all f ∈ L_{p}(Ω) and p ∈ [1, ∞]. One of the main result of this paper is the following theorem.

2.1 Theorem. Let Ω ⊂ ℝ^{N} be an arbitrary bounded domain and suppose that (A, B) are as above and that c_{0} ≥ 0. Then, for any f ∈ L_{p}(Ω) (1 ≤ p ≤ ∞) equation (2.1) has a unique (generalised) solution u which belongs to L_{m(p)}(Ω), where m(p) := Np(N − p)^{−1} if p ∈ (1, N), m(p) = ∞ if p > N and m(p) < N(N − 1)^{−1} arbitrary if p = 1.

Moreover, there exists a constant c > 0 depending only on N, p and upper bounds for γ := max{α_{0}^{−1}, β_{0}^{−1}}, δ := α_{0}^{−1}(∥a∥_{∞}^{2} + ∥b∥_{∞}^{2}) and the volume |Ω| of Ω such that

∥u∥_{m(p)} ≤ c∥f∥_{p}  (2.9)

for all f ∈ L_{p}(Ω).

We also show that the eigenvalue problem

\begin{aligned}Aφ & = λφ & in Ω \\ Bφ & = 0 & on ∂Ω \end{aligned}  (2.10)

has discrete spectrum. As a consequence of the above theorem we get the following result.

2.2 Corollary. Under the assumptions of the above theorem there exists λ_{∗} > 0 depending only on N and upper bounds for γ, δ and |Ω| such that

Re λ ≥ λ_{∗}  (2.11)

for all eigenvalues λ of (2.10).

In Section 5 we shall rewrite the above results in an abstract form. Theorem 2.1 along with its corollary will be proved in Theorem 5.3. The discreteness of the spectrum of (2.10) is contained in Corollary 5.2.

2.3 Remarks. (a) The existence of a positive lower bound λ_{∗} for the first eigenvalue of (2.10) for all domains of a given volume was first observed in [5]. This even seems to be the case for the simplest selfadjoint problem. Here, we prove this for a slightly more general differential operator than in [5]. If A = −Δ and b_{0} constant it is an open problem whether a Faber-Krahn type inequality holds, that is, whether λ_{∗} can be chosen to be the first eigenvalue of (2.10) on a ball with volume |Ω|. For partial results on that problem see [23]–[25] or [3]. In all these references it is assumed that the domains satisfy certain geometric conditions such as convexity or restrictions on the curvature of the boundary.

(b) It turns out that the above theorem, its corollary as well as Theorem 2.4 below remain true if we have Dirichlet boundary conditions on a closed subset of ∂Ω (see Remark 4.3(b) below). □

We finally consider the parabolic problem

∂_{t}u + Au = f  in Ω × (0, ∞)
Bu = 0  on ∂Ω × (0, ∞)
u(0, ·) = u_{0}(·)  in Ω ,

(2.12)

where ∂_{t} := ∂/∂t, and we show that the following results hold.

2.4 Theorem. Suppose that the same assumptions as in Theorem 2.1 hold but allow c_{0} to change sign. Further, let p, q ∈ [1, ∞], such that ∞ > p ≤ q and

N(\frac{1}{p} − \frac{1}{q}) < 1  (2.13)

Then, for all u_{0} ∈ L_{p} and f ∈ L_{∞}((0, ∞), L_{q}), equation (2.9) has a unique generalised solution u ∈ C([0, T], L_{p}). If f ∈ C^{θ}([0, T], L_{q}) for some θ ∈ (0, 1) then, u ∈ C^{1}((0, T], L_{p}).

A proof of the above theorem as well as the precise definition of a generalised solution will be given at the end of Section 6. The idea is to reformulate (2.9) as an abstract parabolic equation in L_{p}-spaces, where all the standard theory applies. Of course, one can then use the same methods to prove existence and uniqueness of solutions of nonlinear problems. The smoothing properties which for the parabolic problem are reflected by condition (2.13) allows the nonlinearity to have arbitrary growth provided we only take initial values in L_{∞}.

3. The L_{2}-Theory

We start this section by recalling Maz’ja’s approach to (2.1) on general domains. To do so we define Ṽ to be the abstract completion of the space

V_{0} := V_{0}(Ω) := \{u ∈ W_{2}^{1}(Ω) ∩ C(\overline{Ω}) ∩ C^{∞}(Ω): ∥u∥_{V} < ∞\} ,  (3.1)

endowed with the norm ∥·∥_{V} given by

∥u∥_{V} := \sqrt{∥∇u∥_{2}^{2} + ∥u|_{∂Ω}∥_{2,∂Ω}^{2}} .  (3.2)

Here, ∥·∥_{2,∂Ω} is the norm in L_{2}(∂Ω) := L_{2}(∂Ω, σ), and σ is the restriction of the (N − 1)-dimensional Hausdorff measure to ∂Ω. (In Maz’ja’s notation this is the space W_{2,2}^{2}(Ω, ∂Ω)). The key to the whole theory is the following inequality due to Maz’ja ([15], see [17], Corollary 4.11.1/2). It says that for all u ∈ V_{0}

∥u∥_{\frac{2N}{N−1}} ≤ c(N, |Ω|)∥u∥_{V}  (3.3)

holds, where c(N, |Ω|) > 0 is a constant depending only on N and an upper bound for |Ω|. This inequality tells us that the natural embedding

j_{0} : V_{0} → L_{\frac{2N}{N−1}} (Ω)  (3.4)

is continuous with norm dominated by c(N, |Ω|). Hence j_{0} has a unique extension j ∈ L(Ṽ , L_{\frac{2N}{N−1}} ). Using inequality (3.3) it is easy to check that the form a(· , ·) is continuous on V_{0} × V_{0}. It has therefore a unique continuous extension to Ṽ × Ṽ, which we denote again by a(· , ·). Moreover, using (3.3) and a standard argument it is easy to see that there exist constant λ_{0} ∈ ℝ and α > 0 such that

a(u, u) + λ_{0}∥u∥_{2} ≥ α∥u∥_{V}^{2}  (3.5)

for all u ∈ V_{0} and hence for u ∈ Ṽ by continuity and density. As obviously, D(Ω) is a subspace of Ṽ the image of j is dense in L_{\frac{2N}{N−1}} . Therefore, the dual map j′ ∈ L(L_{\frac{2N}{N+1}} , Ṽ′) is an injection, which means that each element of L_{\frac{2N}{N+1}} can be identified in a unique way with an element of Ṽ′. Therefore, the following definition makes sense.

3.1 Definition. Let f ∈ L_{p}, with 2N(N + 1)^{−1} ≤ p ≤ ∞. Then, u ∈ Ṽ is said to be a weak solution of (2.1) if and only if

a(u, v) = ⟨f, j(v)⟩ (= ⟨j′(f), v⟩_{V})  (3.6)

for all v ∈ Ṽ (or a dense subset thereof), where ⟨· , ·⟩_{V} is the duality pairing between Ṽ and Ṽ′, and ⟨· , ·⟩ is defined by (2.8).

3.2 Remarks. (a) Note that if Ω is a Lipschitz domain then due to (3.3), the trace inequality ∥u∥_{W_{2}^{1}(Ω)} ≤ c∥u∥_{2,∂Ω} and the fact that C^{∞}(\overline{Ω}) is dense in W_{2}^{1}(Ω) it follows that Ṽ = W_{2}^{1}(Ω) up to an equivalent norm. Hence, if Ω is Lipschitz the definition of a weak solution is the same as in the classical theory.

(b) As D(Ω) ⊂ Ṽ it follows that a weak solution of (2.1) is a weak solution of the equation

Au = f in Ω

in the usual sense, that is, a_{0}(u, v) = ⟨f, v⟩ for all v ∈ D(Ω) where a_{0}(u, v) is defined by (2.7) but without the boundary integral. Hence, all results on local properties of weak solutions of elliptic equations such as for instance given in [11], Section 8 apply.

(c) If (A, B) is selfadjoint and c_{0} ≥ 0 we can choose λ_{0} = 0 in (3.5) and hence, by the Lax Milgram Theorem (2.1) has a unique weak solution. This is in fact Maz’ja’s result in [16], [17]. □

There is one difficulty with the space Ṽ, which is not mentioned in [16], [17]. By definition of the embedding j ∈ L(Ṽ , L_{\frac{2N}{N−1}} ) it is not clear whether it is an injection or not. Injectivity of j is equivalent to the fact that for any sequence (u_{n}) in V_{0} converging to u in Ṽ such that u_{n} → 0 in L_{\frac{2N}{N−1}} implies that u = 0 in Ṽ. Using (3.3) it is clear that u_{n} → 0 in W_{2}^{1}(Ω) as n tends to infinity, but the question which seems difficult to answer is whether this implies that u_{n}|_{∂Ω} → 0 in L_{2}(∂Ω) for a general domain Ω. The above arguments also show that u ∈ ker j if and only if there exists a sequence (u_{n}) in V_{0} such that

lim_{n→∞} u_{n} = 0 in W_{2}^{1}(Ω) and lim_{n→∞} u_{n}|_{∂Ω} = w in L_{2}(∂Ω).  (3.7)

We do not know any example of a domain such that j is not an injection and we suspect that it always is. However, we do not have a proof. For further comments concerning this questions we refer to Remark 3.5. There is a way to get rid of this difficulty by replacing Ṽ by a closed subspace of V. To see this observe that by (3.6) any weak solution of (2.1) is orthogonal to the kernel, ker j, of j with respect to the form a(· , ·) and, moreover, u is a weak solution of (2.1) if and only if (3.6) holds for v in

V_{a} := {u ∈ Ṽ : a(u, v) = 0 for all v ∈ ker j} .  (3.8)

If it happens that j is not injective we could replace Ṽ by V_{a} which a priori depends on the operators (A, B). By (3.7) and the definition of a(· , ·) u ∈ V_{a} if and only if

∫_{∂Ω} b_{0}uv dσ = 0  (3.9)

for all v ∈ ker j and so V_{a} is independent of the coefficients of A. It turns out that V_{a} is also independent of B. The precise statement is contained in the following proposition.

3.3 Proposition. For all (A, B) satisfying (2.2)–(2.5) we have that V_{a} = (ker j)^{⊥} =: V. Furthermore, ker j is isometric isomorphic to L_{2}(S), where S is a measurable subset of ∂Ω, and

V = (ker j)^{⊥} = {u ∈ Ṽ : u = 0 on S} .  (3.10)

Finally, the function u is a weak solution of (2.1) if and only if u ∈ V and (3.6) holds for all v ∈ V.

The proof makes use of the following lemma which we prove at the end of this section.

3.4 Lemma. If v ∈ ker j then, b_{0}v ∈ ker j for all b_{0} ∈ L_{∞}(∂Ω).

Note that by definition of ker j v = 0 in Ω, and thus it makes sense to multiply v by a function just defined on ∂Ω.

Proof of Proposition 3.3. To prove the first assertion we have to show that ∫_{∂Ω} b_{0}uv dσ = 0 for all v ∈ ker j if and only if ∫_{∂Ω} uv dσ = 0 for all v ∈ ker v. But this is easily seen from the above lemma and (2.5) since b_{0}^{−1}v and b_{0}v are in the kernel of j.

Let S be the union of all essential supports of v ∈ ker j. Then, it follows from the above lemma that all simple functions on S are in ker j, and hence, ker j = L_{2}(S). As a simple consequence of this we obtain (3.10). The last assertion is clear from the remarks before Proposition 3.3. □

Next we have some more remarks and examples concerning the space Ṽ.

3.5 Remarks. (a) Due to (3.3) it turns out that Ṽ is isomorphic to the closure of

{(u, u|_{∂Ω}): u ∈ V_{0}}

in W_{2}^{1}(Ω) × L_{2}(∂Ω). All elements of Ṽ have well defined traces on ∂Ω given by lim_{n→∞} u_{n}|_{∂Ω}, where u_{n} is a sequence in V_{0} converging to u in Ṽ. To keep notation as simple as possible we do not distinguish between u and its trace on ∂Ω in our notation. However, note that due to the same problem we had with the existence of a kernel for the embedding j we do not know how much the trace can “disconnect” from the function in Ω.

(b) The space

V_{1} := \{u ∈ C(\overline{Ω}) ∩ W_{2}^{1}(Ω): ∥u∥_{V} < ∞\}

is dense in Ṽ. Indeed, a careful analysis of the proof that W_{2}^{1}(Ω) ∩ C^{∞}(Ω) is dense in W_{2}^{1}(Ω) such as given in [1], Theorem 3.16 reveals that if u ∈ C(\overline{Ω}) ∩ W_{2}^{1}(Ω) the approximating function ϕ lies in C(\overline{Ω}) ∩ C^{∞}(Ω) ∩ W_{2}^{1}(Ω), and has the same boundary values as u. Therefore, V_{1} is dense in V_{0} and hence in Ṽ.

(c) The space Ṽ is a Banach lattice, that is, if u ∈ Ṽ then the absolute value |u| also lies in Ṽ and ∥|u|∥_{V} = ∥u∥_{V}. To see this note that by [11], Lemma 7.6 W_{2}^{1}(Ω)×L_{2}(∂Ω) is a Banach lattice. As we know from (a) that Ṽ is a closed subspace thereof it is sufficient to show that |u| ∈ Ṽ_{1} for all u ∈ V_{1}, where V_{1} is the space defined in (b). But this is clear as C(\overline{Ω}) is a lattice too.

(d) Suppose that Ω is a domain with a boundary whose (N −1)-dimensional Hausdorff measure is locally infinite, that is, for all x ∈ ∂Ω and ε > 0 we have that

σ(B_{ε}(x) ∩ ∂Ω) = ∞ .  (3.11)

Then, V = Ṽ = W̊_{2}^{1}(Ω) up to an equivalent norm. In this case our boundary value problem coincides with the Dirichlet problem. If (3.11) only holds on part of the boundary Dirichlet boundary conditions are satisfied on that part. Examples of such domains are domains with fractional boundaries such as for instance the interior of the well known “snowflake curve” (e.g. [9]).

To prove our claim we first show that V_{0} ⊂ W̊_{2}^{1}(Ω). To do so let u ∈ V ∩ C(\overline{Ω}) be arbitrary. As u is continuous on \overline{Ω} it follows from (3.11) that u|_{∂Ω} = 0. By splitting u in positive and negative part (which by (c) also belong to V ∩ C(\overline{Ω})) we can assume without loss of generality that u ≥ 0. By the uniform continuity of u on \overline{Ω} it is clear that u_{ε} := max{u − ε, 0} ∈ W̊_{2}^{1}(Ω) for all ε > 0. It is then easily seen that u_{ε} tends to u in V as ε goes to zero. Finally, note that ∥∇u∥_{2} is an equivalent norm on W̊_{2}^{1}(Ω), and thus u_{ε} also converges in W̊_{2}^{1}(Ω). Hence, it follows that V = W̊_{2}^{1}(Ω).

(e) The above remark together with Proposition 3.3 show that wherever the boundary is bad either in the sense that its measure is locally infinite or that it allows j to have a kernel the weak solutions of (2.1) satisfy Dirichlet boundary conditions in some sense. In case of a nontrivial kernel we are not sure whether we really have Dirichlet boundary conditions in the weak sense because we do not know whether we can approximate the solutions by elements of V_{0} being zero in a neighbourhood of that part of the boundary.

(f) Suppose that K ⊂ ∂Ω is a compact set of capacity zero and that ∂Ω \ K is locally Lipschitz in the sense that it is locally the graph of a Lipschitz function. In that case every u ∈ Ṽ has locally a trace and for all x ∈ ∂Ω \ K there exists a ball B_{ε}(x) in ℝ^{N} and a constant c(ε, x) such that

(∫_{∂Ω∩B_{ε}(x)} |u(y)|^{2} dσ)^{\frac{1}{2}} ≤ c(ε, x)∥u∥_{W_{2}^{1}(Ω)}  (3.12)

for all u ∈ V_{0}. This implies that the trace of u is in L_{2,loc}(∂Ω \ K). Then, it turns out that

Ṽ = V = {u ∈ W_{2}^{1}(Ω): γu ∈ L_{2}(∂Ω \ K)} .  (3.13)

This characterisation shows that in general the space Ṽ is much smaller than W_{2}^{1}(Ω) even if the measure of ∂Ω is finite and the domain is smooth except for one point. As an example consider a domain with an outward pointing cusp. Using (3.3) it is clear that Ṽ ↪ W_{2}^{1}(Ω). If we had equality the open mapping theorem would imply that the two spaces coincide up to an equivalent norm. This is not possible since Ṽ ↪ L_{\frac{2N}{N−1}} but W_{2}^{1}(Ω) ⊄ L_{p} for all p > 2 if the domain has an exponential cusp (see [1], Theorem 5.32).

To prove (3.13) it is sufficient to prove that

V_{1} := \{u ∈ W_{2}^{1}(Ω) ∩ C^{∞}(\overline{Ω}): u = 0 in a neighbourhood of K\}  (3.14)

is dense in W_{2}^{1}(Ω) as well as in V_{0}. It is well known that it is dense in W_{2}^{1}(Ω). As K has zero capacity there exists a sequence φ_{k} ∈ D(ℝ^{N}) with 0 ≤ φ_{k} ≤ 1, φ_{k} = 1 in a neighbourhood of K such that ∥φ_{k}∥_{W_{2}^{1}(Ω)} → 0 as k tends to infinity. Let u ∈ V_{0} be arbitrary and set u_{k} := (1 − φ_{k})u. Then obviously u_{k} ∈ V_{1} and ∇u_{k} → ∇u in L_{2} as k tends to infinity. As K has zero capacity we have that σ(K) = 0 (e.g. [28], Theorem 2.6.16) and thus φ_{k} converges to zero σ-almost everywhere on ∂Ω. As |φ_{k}u| ≤ |u| on ∂Ω, it follows from the dominated convergence theorem that u_{k} → u in L_{2}(∂Ω). This shows that V_{1} is dense in V_{0} and hence in V_{0} which proves our claim.

(g) A sufficient condition making sure that j is injective is that W_{2}^{1}(Ω) allows at least locally a trace operator. More precisely assume that for σ-almost all x ∈ ∂Ω (3.12) holds. Then, u_{n}|_{∂Ω} → 0 in L_{2}(∂Ω ∩ B_{ε}(0)) if u_{n} → 0 in W_{2}^{1}(Ω). From this we conclude that u = 0 σ-almost everywhere on ∂Ω, whence u = 0 ∈ Ṽ and our claim follows. □

We conclude this section by proving Lemma 3.4

Proof of Lemma 3.4. In a first step we assume that b_{0} ∈ C^{∞}(\overline{Ω}). If v ∈ ker j take v_{n} ∈ V_{0} such that v_{n} → v in Ṽ. Clearly b_{0}v_{n} ∈ V_{0} and by a simple calculation

∥b_{0}v_{n}∥_{W_{2}^{1}(Ω)} ≤ (∥b_{n}∥_{∞} + ∥∇b_{0}∥_{∞})∥v_{n}∥_{W_{2}^{1}(Ω)} .

Thus, as ∥v_{n}∥_{W_{2}^{1}(Ω)} → 0 by (3.7), we conclude that ∥b_{0}v_{n}∥_{W_{2}^{1}(Ω)} → 0. Moreover, ∥b_{0}(v_{n} − v)∥_{2,∂Ω} ≤ ∥b_{0}∥_{∞}∥v_{n} − v∥_{2,∂Ω} → 0 as n → ∞. By (3.7) this implies that b_{0}v ∈ ker j.

In a second step we assume that b_{0} ∈ C(\overline{Ω}). Then, by Tietze’s Theorem there exists a continuous extension of b_{0} to ℝ^{N} which we denote again by b_{0}. This continuous function can then be approximated by smooth functions b_{k} uniformly on \overline{Ω}. By the first step b_{k}v ∈ ker j for all k ∈ ℕ. Moreover, as

∥b_{k}v − b_{ℓ}v∥_{V}^{2} = ∫_{∂Ω} |b_{k} − b_{ℓ}|^{2}v^{2} dσ  (3.15)

for all k, ℓ ∈ ℕ it turns out that b_{k}v is a Cauchy sequence in Ṽ, which converges to b_{0}v. As ker j is closed it follows that b_{0}v ∈ ker j.

We next try to approximate b_{0} ∈ L_{∞}(∂Ω) by continuous functions in a suitable way. To do so first note that the restriction of the (N − 1)-dimensional Hausdorff measure σ to ∂Ω is an inner regular Borel measure. Fix v ∈ ker j and set for k ∈ ℕ̇ := ℕ \ {0}

A_{k} := {x ∈ ∂Ω: |v(x)| ≥ k^{−1}}  and  A_{0} := {x ∈ ∂Ω: |v(x)| ≠ 0}.

Then, the union of the A_{k} is A_{0}. As v ∈ L_{2}(∂Ω) it follows that |A_{k}| := σ(A_{k}) < ∞ for all k ∈ ℕ̇. By Lusin’s Theorem (see [10], Theorem 2.3.5) we find for every k ∈ ℕ̇ a compact set C_{k} such that

|A_{k} \ C_{k}| < k^{−1}  (3.16)

and b_{0}|_{C_{k}} is continuous. By Tietze’s Theorem there exists b_{k} ∈ C(∂Ω) such that

∥b_{k}∥_{∞} ≤ ∥b_{0}∥_{∞}  (3.17)

and b_{k} = b_{0} on C_{k}. Without loss of generality we can choose C_{k} in such a way that C_{k+1} ⊃ C_{k} for all k ∈ ℕ̇. We prove now that b_{k} → b_{0} σ-almost everywhere on A_{0}. Suppose this is not the case. Then, there exists a set B ⊂ A_{0} with |B| > 0 such that b_{k} ↛ b_{0} on B. As the union of A_{k} is A_{0} there exists k_{0} ∈ ℕ and δ > 0 such that |B ∩ A_{k}| ≥ δ for all k ≥ k_{0}. On the other hand, |B ∩ A_{k}| = |B ∩ A_{k} \ C_{k}| + |B ∩ C_{k}| > δ for k ≥ k_{0}, whence |B ∩ C_{k}| > δ − k^{−1} > 0 by (3.16) for k large enough. As b_{0} = b_{k} on C_{k} this is a contradiction. This shows that b_{k}v → b_{0}v almost everywhere on ∂Ω. By step two, b_{k}v ∈ ker j for all k ∈ ℕ̇. Moreover, by (3.17) |b_{k} − b_{ℓ}|^{2}v^{2} ≤ 2∥b_{0}∥_{∞}v^{2}. Hence, it follows from (3.15) and the Dominated Convergence Theorem that b_{k}v is a Cauchy sequence in Ṽ with limit b_{0}v. As ker j is closed this concludes the proof of the lemma. □

4. Global Estimates for Weak Solutions

In this section we give global estimates for weak solutions of equation (2.1). The results here slightly generalise the results obtained in [5]. Before giving the details of the proof, which employs a version of Moser iteration techniques, we recall some definitions and state our results.

As usual u ∈ V is called a (weak) subsolution of (2.1) if

a(u, v) ≤ ⟨f, v⟩  (4.1)

for all nonnegative v ∈ V, where was defined in Proposition 3.3. If the reverse inequality holds u is called a (weak) supersolution. Further we denote by u^{+} and u^{−} the positive and negative part of u, respectively, that is, u^{±} := max{±u, 0}. We could also replace V by Ṽ in our definition, but then we have to replace v by j(v) on the right hand side of (4.1).

4.1 Theorem. Suppose (A, B) satisfy (2.2)–(2.5). Then, if p ∈ [2, N), there exists a constant c > 0 depending only on N, p and an upper bound for |Ω| such that for any weak solution of (2.1)

∥u∥_{\frac{Np}{N−p}} + ∥u∥_{\frac{(N−1)p}{N−p} ,∂Ω} ≤ cγ(∥f∥_{p} + γ^{μ}δ^{1+μ}∥u∥_{2}) ,  (4.2)

where

γ := max{α_{0}^{−1}, β_{0}^{−1}},  δ := α_{0}^{−1}(∥a∥_{∞} + ∥b∥_{∞})^{2} + ∥c_{0}^{−}∥_{∞}  (4.3)

and μ := N(p − 2)/2p. If p > N there exists a constant c > 0 depending on the same quantities as above such that

∥u∥_{∞} ≤ cγ(∥f∥_{p} + γ^{μ}δ^{1+μ}∥u∥_{2}) .  (4.4)

Moreover, ∥u∥_{∞,∂Ω} ≤ ∥u∥_{∞}. Furthermore, if u is a sub- or supersolution the above assertions hold for u replaced by u^{+} or u^{−}, respectively.

The assertion that ∥u∥_{∞,∂Ω} ≤ ∥u∥_{∞} is not completely obvious as we do not know how much the trace of a function u ∈ V can “disconnect” from the function in Ω as already mentioned in Remark 3.5(a). The next theorem contains an L_{∞}-estimate not containing the L_{2}-norm.

4.2 Theorem. Let (A, B) satisfy (2.2)–(2.5) with c_{0} ≥ 0. Then, if p > N there exists a constant c > 0 depending only on N, p and upper bounds for |Ω|, γ and δ such that for any weak solution u of (2.1)

∥u∥_{∞} ≤ c∥f∥_{p} .  (4.5)

If u is a sub- or a supersolution the inequality holds if we replace u by u^{+} and u^{−}, respectively.

Note that it is sufficient to give a proof of the above theorems for subsolutions. For supersolutions the assertion follows since −u is a subsolution and for solutions by combining the two inequalities.

Proof of Theorem 4.1. We define for every m > 0 and q ≥ 2 the functions G_{q,m}(ξ) := min\{ξ^{q−1}, m^{q−1}\} and H_{q,m}(ξ) := min\{ξ^{\frac{q}{2}} , m^{\frac{q}{2}} \} if ξ ≥ 0 and G_{q,m}(ξ) = H_{q,m}(ξ) := 0 for ξ ≤ 0. It is easy to see that the substitution operators induced by G_{q,m} and H_{q,m} are continuous on W_{2}^{1}(Ω) and on L_{2}(∂Ω) (see also [14]). Hence, it is continuous on V_{0} and therefore on V. To simplify notation we set v := G_{q,m}(u) and w := H_{q,m}(u). Then, taking into account [11], Lemma 7.6 we get that

∂_{i}w∂_{j}w = \frac{q^{2}}{4(q − 1)}∂_{i}u∂_{j}v, w∂_{i}w = \frac{q}{2}v∂_{i}u = \frac{q}{2(q − 1)}u∂_{i}v and w^{2} = uv .

Using this as well as (2.4) and Hölder’s inequality and we get that

\begin{aligned}α_{0}∥∇w∥_{2}^{2} & ≤ \frac{q^{2}}{4(q − 1)}(a_{0}(u, v) − \frac{2(q − 1)}{q} ∫_{Ω} a_{i}w∂_{i}w − \frac{2}{q} ∫_{Ω} b_{i}w∂_{i}w − ∫_{Ω} c_{0}w^{2}) \\ & ≤ q(a_{0}(u, v) + 2(∥a∥_{∞} + ∥b∥_{∞})∥w∥_{2}∥∇w∥_{2} + ∥c_{0}^{−}∥_{∞}∥w∥_{2}^{2}) , \end{aligned}

where a_{0}(· , ·) is defined as (2.7) but without the boundary term. By a standard trick we find that

α_{0}∥∇w∥_{2} ≤ q(a_{0}(u, v) + δq∥w∥_{2}^{2}) .  (4.8)

Obviously, we have that

β_{0}∥w∥_{2,∂Ω} ≤ q ∫_{∂Ω} b_{0}uv dσ .  (4.9)

Combining the two estimates and taking into account (3.3) we get that

∥w∥2_{\frac{2d}{d−2}} ≤ c(N, |Ω|)^{2}∥w∥_{V}^{2} ≤ c(N, |Ω|)^{2}γq(a(u, v) + δq∥w∥_{2}^{2})  (4.10)

where d := 2N. Using Hölder’s inequality, (3.3) and the fact that u is a subsolution of (2.1) we have that

∥w∥2_{\frac{2d}{d−2}} ≤ c(N, |Ω|)^{2}∥w∥_{V}^{2} ≤ c(N, |Ω|)^{2}γq^{2}(∥f∥_{p}∥v∥_{p′} + δ∥w∥_{2}^{2}) .  (4.11)

Furthermore, if p ≥ 2 then r := 4p(3p − 2)^{−1} ≤ 2, and by a well known interpolation inequality we obtain

∥w∥_{2}^{2} ≤ ∥w∥2(1−θ)_{\frac{2d}{d−2}}∥w∥_{r}^{2θ}

with θ satisfying 2^{−1} = θr^{−1} + (1 − θ)(d − 2)(2d)^{−1}. Applying Young’s inequality (see [11], Section 7.1) and (3.3) this implies that for all ε > 0

∥w∥_{2}^{2} ≤ εc(N, |Ω|)^{2}∥w∥_{V}^{2} + ε^{−μ}∥w∥_{r}^{2}

with μ := d(p − 2)(4p)^{−1}. Furthermore, by Hölder’s inequality

∥w∥_{r}^{2} = ∥w^{\frac{2}{q} + \frac{2}{q} (q−1)}∥\frac{r}{2} ≤ ∥u^{+}∥_{2}∥w^{\frac{2}{q} (q−1)}∥_{p′} = ∥u^{+}∥_{2}∥w^{\frac{2}{q}} ∥_{p′(q−1)}^{q−1}.

Note also that v = w^{\frac{2}{q} (q−1)}. Combining all this and setting ε := (2c(N, |Ω|)^{2}γδq^{2})^{−1} and c̄:= 2c(N, |Ω|)^{2}γ we conclude from (4.11) that

∥w^{\frac{2}{q}} ∥q_{\frac{dq}{d−2}} ≤ c(N, |Ω|)^{2}∥w∥_{V}^{2} ≤ c̄q^{2(1+μ)}(∥f∥_{p} + c̄^{μ}δ^{1+μ}∥u^{+}∥_{2})∥w^{\frac{q}{2}} ∥_{p′(q−1)}^{q−1} . (4.12)

In case p ∈ [2, d/2) we set q := p(d − 2)(d − 2p)^{−1} and divide inequality (4.12) by

∥u∥q−1_{\frac{dp}{d−2p}}. As w^{\frac{2}{q}} = min\{u^{+}, m\} we obtain

∥u^{+}∥_{\frac{dp}{d−2p}} ≤ Cc̄(∥f∥_{p} + c̄^{μ}δ^{1+μ}∥u^{+}∥_{2})

by letting m to infinity. Note that C just depends on d and p. To show that the L_{∞}- estimate holds for p > N = d/2 we set q_{0} := 2 and q_{n+1} := 1 + ηq_{n} for all n ∈ ℕ, where

η := \frac{d(p − 1)}{p(d − 2)} .

Moreover, set ū := u^{+}M^{−1} with M := c̄(∥f∥_{p} + c̄^{μ}δ^{1+μ}∥u∥_{2}). Then (4.12) yields

∥ū∥_{\frac{dq_{n+1}}{d−2}} ≤ q_{n+1}^{2 \frac{1+μ}{q_{n+1}}} ∥ū∥1− \frac{1}{q_{n+1}}_{\frac{dqn}{d−2}} ,

from which we get by induction that

∥ū∥_{\frac{dq_{n+1}}{d−2}} ≤ q_{n+1}^{2 \frac{1+μ}{q_{n+1}}} ∏_{k=1}^{n} q_{k}^{2 \frac{1+μ}{q_{k}} ∏_{i=k+1}^{n}(1− \frac{1}{q_{i}})}∥ū∥∏_{k=1}^{n+1}(1− \frac{1}{q_{k}} )_{\frac{2d}{d−2}}  (4.13)

Observe now that η ≤ q_{n+1}/q_{n} ≤ 2η and therefore, η^{n} ≤ q_{n} ≤ (2η)^{n}. Together with (4.13) we see that

∥ū∥_{\frac{dq_{n+1}}{d−2}} ≤ (2η)^{2 \frac{1+μ}{q_{n+1}} ∑_{k=0}^{∞} kα^{−1}}∥ū∥∏_{k=1}^{n+1}(1− \frac{1}{(2η)^{k}} )_{\frac{2d}{d−2}} .

As η > 1 for the range of p under consideration we can let n to infinity to get

∥ū∥_{∞} ≤ C∥ū∥θ_{\frac{2d}{d−2}}  (4.14)

for some θ ∈ (0, 1) depending only on d and p. As |Ω| this implies that

∥ū∥_{∞} ≤ (C|Ω|^{θ( \frac{1}{2} − \frac{1}{d})})^{1−θ}  (4.15)

which implies (4.4).

It remains to prove the estimate on the boundary. By definition of ∥·∥_{V} and setting q := p(N − 1)(N − p)^{−1} it follows from (4.12) that ∥ū∥_{\frac{p(N−1)}{N−p} ,∂Ω}^{q} ≤ ∥ū∥q−1_{\frac{Np}{N−p}}. Using the estimate for ∥ū∥_{\frac{dp}{d−2p}} we conclude that (4.2) holds. To prove that ∥u∥_{∞,∂Ω} ≤ ∥u∥_{∞} note

that as a consequence of (4.12) we have that ∥ū∥_{q,∂Ω} ≤ (q^{\frac{1}{q}} )^{2(1+μ)}∥ū∥_{∞}^{1− \frac{1}{q}} for all q ≥ 2 and our assertion follows by letting q to infinity as the constants in front of ∥ū∥_{∞} tend to one. This proves Theorem 4.1. □

Note that in the proof of Theorem 4.1 in case of a subsolution we just needed that (4.1) holds for all v ∈ V such that uv ≥ 0. This is essential in the proof of Theorem 4.2 which follows.

Proof of Theorem 4.2. To prove (4.5) we use a modification of a test function argument due to Trudinger [26] (see [11], proof of Theorem 8.6). We set

v := \frac{u^{+}}{m + k − u^{+}}  and  w := log \frac{m + k}{m + k − u^{+}} ,

where ∥u^{+}∥_{∞} and k := ∥f∥_{p}. We already know that u ∈ L_{∞} so the above definition makes sense. To estimate the boundary integral ∫_{∂Ω} w^{2} dσ we need the following inequality, which holds for all 0 ≤ s ≤ r:

(log \frac{r}{r − s})^{2} ≤ \frac{s^{2}}{r(r − s)} .  (4.16)

To prove the inequality we set t := s/r. It is then easy to see that (4.16) is equivalent to the fact that

(log \frac{1}{1 − t})^{2} ≤ \frac{t^{2}}{1 − t}

for all t ∈ (0, 1), which in turn is equivalent to g(t) := t^{−2}(1 − t)( log(1 − t))^{2} ≤ 1 for all t ∈ (0, 1). Using de l’Hôpital’s rule we find that lim_{t→0} g(t) = 1. As g ≥ 0 it is now sufficient to show that g′(t) < 0 for all t ∈ (0, 1). As

g′(t) = −t^{−2}(2 − t^{−1}(t − 2) log(1 − t)) log(1 − t) ,

it remains to show that t^{−1}(t − 2) log(1 − t) > 2 for all t ∈ (0, 1). Using the Taylor expansion of log(1 − t) we obtain

t^{−1}(t − 2) log(1 − t) = 2 + ∑_{k=2}^{∞}(\frac{2}{k + 1} − \frac{1}{k} )t^{k} > 2

for all t ∈ (0, 1) completing the proof of (4.16). Hence, setting r := m + k we conclude that

∫_{∂Ω} w^{2} dσ ≤ \frac{1}{m + k} ∫_{∂Ω} \frac{(u^{+})^{2}}{m + k − u^{+}} dσ .  (4.17)

Now we estimate ∥w∥_{2} in the same way as in the proof of [11] Theorem 8.16 using (4.17) for the boundary integral. This yields

∥w∥_{2} ≤ C ,  (4.18)

where C just depends on N and upper bounds for δ, γ and |Ω|. Set now v := φ(m + k − u^{+})^{−1}. Again using the same arguments as in the proof of [11] Theorem 8.16 and estimating the boundary integral in an obvious way we see that w satisfies

a(w, φ) ≤ a(u, v) ≤ ⟨f, v⟩ ≤ ⟨fk^{−1}, φ⟩

for all nonnegative φ ∈ V satisfying uφ ≥ 0. As wφ ≥ 0 if uφ ≥ 0, we may apply Theorem 4.1 to estimate ∥w∥_{∞} in terms of ∥k^{−1}f∥_{p} and ∥w∥_{2}. Using (4.18) and taking exponentials we finally obtain (4.5). □

4.3 Remarks. (a) Inequality (4.2) tells us that the solutions of (2.1) with f ∈ L_{p} lie in L_{\frac{Np}{N−p}} (Ω) × L_{\frac{p(N−1)}{N−p}} (∂Ω), that is, they lie in the Lebesgue spaces the Sobolev spaces

W_{p}^{1}(Ω) and its trace space W_{p}^{1− \frac{1}{p}} (∂Ω) inject into. Note that even not in case of a polygonal domain and smooth f we can expect the solution of the Robin problem to ly in W_{p}^{1}(Ω) for all p > 2. Concerning this, see [12], Theorem 4.4.3.7, where the S_{j,m} turn out to be not in W_{p}^{1}(Ω) for p large enough.

(b) The above results remain true if we have Dirichlet boundary conditions on a closed subset Γ_{0} of ∂Ω. To see that we replace the space V_{0} introduced in Section 3 by

V_{0,Γ_{0}} := \{u ∈ V_{0} : supp u ∈ \overline{Ω} \ Γ_{0}\}.

Then, (3.3) still holds for all u ∈ V_{0,Γ_{0}} and hence, for all u in the completion V_{Γ_{0}} of V_{0,Γ_{0}} with respect to the Norm ∥·∥_{V}. As the proof of the above theorems are based on this inequality everything remains true.

(c) If we restrict the class of domains we can get better estimates than those in Theorem 4.1. For instance if we take domains in the class K_{α,1}, α < 1, introduced by Maz’ja (see [17], Section 3.6.1), which are determined by the same constant E it turns out that V ↪ L_{\frac{2d}{d−2}} for d := 2(1 − α)^{−1} (replace u by u^{2} in [17], Theorem 3.6.3). In the above proofs we just have to replace d = 2N by the new value of d and in the assertion N by d/2. The classes of domains mentioned above allow to deal with nonsmooth domains having cusps of restricted sharpness. The sharpness of the cusp determines the class K_{α,1} ([17], §3.6). Above we only considered the class K_{1− \frac{1}{N} ,1} where all bounded domains belong to with the same constant E which is given by the isoperimetric constant ([17], Example 3.6.2/1).

(d) By using (3.3) we obviously can weaken the assumptions on the coefficients a_{i}, b_{i} and c_{0}. It is sufficient to assume that a_{i}, b_{i} ∈ L_{r} for some r > 2N (r may vary from one coefficient to another) eand c_{0} ∈ L_{s} for some s > N. We just have to replace the L_{∞}-norms in δ by the corresponding weaker norms. □

5. The L_{p}-Theory

In this section we prove smoothing properties of problem (2.1) making use of the a priori estimates established in the previous section, the L_{2}-theory in Section 2 and the abstract results in the Appendix. In particular we give a proof of Theorem 2.1.

Suppose that V is as in Proposition 3.3 and a(· , ·) is the form defined by (2.7). We already proved in Section 2 that a(· , ·) extends to a bounded form on V, and that it satisfies (3.5) for some α > 0 and λ_{0} ∈ ℝ. As D(Ω) ⊂ V clearly V ∩L_{p} is dense in L_{p}(Ω) for all p ∈ [1, ∞). If we put H := L_{2}(Ω) we are therefore in a setting which allows us to apply the abstract results in the Appendix. Define the adjoint form a^{♯}(· , ·) by (A.11). Note that this form corresponds to the formal adjoint boundary value problem (A^{♯}, B^{♯}) to (A, B) given by

A^{♯}u := − ∑_{i=1}^{N} ∂_{i}(∑_{j=1}^{N} a_{ji}(x)∂_{j}u + b_{i}(x)u) + ∑_{i=1}^{N} a_{i}(x)∂_{i}u + c_{0}(x)u  (5.1)

and

B^{♯}u := ∑_{i=1}^{N}( ∑_{j=1}^{N} a_{ji}(x)∂_{j}u + b_{i}(x)u)ν_{i} + b_{0}(x)u .  (5.2)

Denote by A and A^{♯} the operators induced on V′ by the corresponding forms a(· , ·) and a^{♯}(· , ·), respectively. Note also that (A^{♯}, B^{♯}) has the same structure as (A, B). Hence, A and A^{♯} satisfy (A.6) with the same constants (see proof of Theorem 4.1). Suppose that A_{p} and A_{p}^{♯} are defined as in Part III of the Appendix.

5.1 Theorem. All assertion of Theorem A.9 hold with d = 2N. In particular,

(λ + A)^{−1}(L_{p}) ⊂ L_{m(p)}  (5.3)

for all λ ∈ ϱ(−A), and m(p) is as in Theorem 2.1. Moreover, A_{p} has compact resolvent for all p ∈ [1, ∞] and for all λ ≥ λ_{0} (λ + A)^{−1} is irreducible.

Proof. Due to the results in Section 4 we can apply Theorem A.9 with d = 2N. Compactness of the resolvent follows since V is compactly embedded into L_{2} ([17], Corollary 4.11.1/3). Positivity of (λ + A)^{−1} follows by a simple test function argument (take the negative part of u := (λ + A)^{−1}f as a test function and conclude that it is zero). By a density argument it turns out that (λ + A_{p})^{−1} is positive for all p ∈ [0, ∞]. To show irreducibility it is sufficient to prove that there exists k ∈ ℕ such that for all nonnegative f ∈ L_{p} with f ≠ 0 (λ + A_{p})^{−k}f is strictly positive almost everywhere in Ω, that is, for any such f, (λ + A_{p})^{−k}f is in the quasi interior of the positive cone in L_{p} (see [22], Section III.8). Using the smoothing properties of the resolvent we find n ∈ ℕ such that (λ+A_{p})^{−n}(L_{1}) ⊂ L_{∞}. Clearly, for all 0 ≠ f ∈ L_{p} nonnegative 0 ≠ v := (λ+A_{p})^{−n}f ≥ 0. As v ∈ L_{∞} ∩ W_{2}^{1}(Ω) is a strict supersolution of (λ + A)w = 0 in Ω in the usual sense the Harnack inequality (cf. [11], Theorem 8.18) implies that (λ + A_{p})^{−1}v > 0 almost everywhere. Setting k = n + 1 our claim follows. □

5.2 Corollary. The spectrum of A_{p} consists of eigenvalues of finite algebraic multiplicity and the eigenfunctions lie in L_{∞}. Moreover, the spectral bound spb(A_{p}) = inf{Re λ: λ ∈ σ(A_{p})} is independent of p ∈ [1, ∞]. It is an algebraically simple eigenvalue and the only one having positive eigenfunction.

Proof. That the spectrum consists of eigenvalues of finite algebraic multiplicity follows since A_{p} has compact resolvent. The assertion on the spectral bound is a consequence of the abstract theory on compact irreducible operators (e.g. [22], Section V.5). □

Clearly, a solution u of the equation

A_{p}u = f  (5.4)

in L_{p}(Ω) is a weak solution of (2.1) and vice versa whenever p ≥ 2N(N + 1)^{−1}. For smaller p this is not the case in general. In either case we call a solution u of (5.4) a generalised solution of (2.1). We next establish solvability of (5.4) under the additional assumption that the zero order coefficient c_{0} of A is nonnegative. The following theorem is a reformulation of Theorem 2.1

5.3 Theorem. Let c_{0} ≥ 0 in (2.2). Then, 0 ∈ ϱ(A) and there exists a constant c > 0 depending only on the quantities listed in Theorem 2.1 such that

∥A_{p}^{−1}∥_{p,m(p)} ≤ c  (5.5)

for all p ∈ [1, ∞], where m(p) is as in Theorem 2.1. Moreover, there exists λ_{∗} > 0 depending only on N and upper bounds for γ, δ and |Ω| such that λ_{1} ≥ λ_{∗}, where λ_{1} is the first eigenvalue of Aφ = λφ.

Proof. The first assertion follows from the results in Section 4 and Theorem A.10. For the remaining assertion observe that the spectral radius of A_{p}^{−1} is given by λ_{1}^{−1} whence, λ_{1}^{−1} ≤ ∥A_{p}^{−1}∥_{L(L_{p})} ≤ c|Ω|^{\frac{1}{p} − \frac{1}{m(p)}} . Taking the minimum over all p we get the existence of λ_{∗} ≥ 0. □

5.4 Remarks. (a) Note that one could get an estimate of the form (5.5) from (4.2) and by using the fact that 0 ∈ ϱ(A_{2}). However, using this argument we have no control over the constant c. To have more information we used the bound for p > N, duality and interpolation. We also mention that in case of the Dirichlet problem considered in [11], Section 8 the corresponding estimate for small p is not included. By the same proof but this time with d = N one can get the corresponding estimates for the Dirichlet problem as well.

(b) Obviously, Remark 4.3(b)–(d) apply here as well. □

The next proposition will be used to show that without restrictions on the domains the mapping property (5.3) cannot be improved, that is, m(p) is the largest possible exponent in general. 5.5 Proposition. Let p ∈ (1, N) and suppose that (λ + A)^{−1}(L_{p}) ⊂ L_{q} for some q < Np(N − 2p)^{−1} if p < N/2 and q ≤ ∞ if p > N/2. Then,

(λ + A_{p})^{−1} ∈ L(L_{p}, L_{r})  (5.6)

is compact for all r ∈ [1, q). Proof. Note first that by our hypotheses D(A_{p}) ⊂ L_{q}. As L_{q} ↪ L_{p} it follows from the closed graph theorem that D(A_{p}) ↪ L_{q} and hence

(λ + A_{p})^{−1} ∈ L(L_{p}, L_{q}) .  (5.7)

Note also that we can assume without loss of generality that p ≥ 2N(N + 1)^{−1} as in the other case we consider the formal adjoint which has compact resolvent if and only if its dual has compact resolvent. To prove the assertion of the proposition it is sufficient to show that if f_{n} ⇀ 0 weakly in L_{p} then, u_{n} := (λ + A_{p})^{−1}f_{n} → 0 strongly in L_{r} for r ∈ (1, q). As bounded operators are weakly continuous we also have that u_{n} ⇀ 0 in L_{q}. Recall that by Remark 3.2(b) u_{n} is a weak solution of

λu_{n} + Au_{n} = f_{n}

in Ω in the usual sense. If we fix Ω′ ⊂⊂ Ω (that is, the closure of Ω′ is contained in the interior of Ω) by interior regularity of weak solutions (e.g. [11], Theorem 9.11) there exists a constant c depending on Ω′ such that

∥u_{n}∥_{W_{p}^{2}(Ω′)} ≤ c(∥u_{n}∥_{p} + ∥f_{n}∥_{p})

for all n ∈ ℕ. As weakly convergent sequences are bounded it follows from (5.7) that u_{n} is a bounded sequence in L_{q}. Hence, the above inequality shows that u_{n} is bounded in W_{p}^{2}(Ω′). By a well known embedding theorem we have that W_{p}^{2}(Ω) ↪ L_{q} compactly if we choose Ω′ sufficiently smooth. As we already know that u_{n} ⇀ 0 in L_{q}(Ω) it follows that u_{n} → 0 in L_{q}(Ω′) strongly. Given ε > 0 we choose Ω′ ⊂⊂ Ω sufficiently smooth such that |Ω \ Ω′| < ε. By Hölder’s inequality we can write

∥u_{n}∥_{r} ≤ |Ω \ Ω′|^{\frac{1}{r} − \frac{1}{q}} ∥u_{n}∥_{q} + ∥u_{n}∥_{q,Ω′} ≤ ε^{\frac{1}{r} − \frac{1}{q}} ∥u_{n}∥_{q} + ∥u_{n}∥_{q,Ω′} .

By the above considerations the last term tends to zero as n goes to infinity. As ∥u_{n}∥_{q} is uniformly bounded and ε was arbitrary this shows that u_{n} tends to zero in L_{r} as n goes to infinity. This completes the proof of the proposition. □

The following proposition shows that (5.3) is optimal.

5.6 Proposition. There exists a bounded domain Ω with smooth boundary except for one point and (5.3) fails for any q > m(p), where p ∈ (1, N).

Proof. First note that there exists a domain Ω such that the embedding

V ↪ L_{\frac{2N}{N−1}} .  (5.8)

is not compact (see [17], pages 259/60). This domain can easily be modified in such a way that its boundary is of class C^{∞} except for one point. The proof of the proposition is complete if we can show that (5.8) is compact if (5.7) holds for some p ∈ (1, p) and q > m(p). In case p = 2N(N + 1)^{−1} we conclude from Proposition 5.5 that (λ + A)^{−1} ∈ L(L_{\frac{2N}{N+1}} , L_{\frac{2N}{(N−1)}} ) is compact. Setting E := L_{\frac{2N}{N−1}} it follows from Proposition A.5 that (5.8) is a compact embedding. If p > (<) 2N(N +1)^{−1} we choose r > (<) 2N(N +1)^{−1}. By Theorem 5.3 we have that (λ + A_{r})^{−1} ∈ L(L_{r}, L_{m(r)}) and hence by interpolation (λ + A)^{−1} ∈ L(L_{\frac{2N}{N+1}} , L_{s}) for some s > 2N(N − 1)^{−1} and we are back to the situation before. This concludes the proof of the proposition. □

5.7 Remark. Together with the domain perturbation results in [6] the above proposition can be used to show that the estimate (5.5) can not be improved even if we restrict ourselves to smooth domains without making the upper bound c dependent on the geometry rather than the volume of the domain. □

6. Parabolic Problems

In this section we shall briefly discuss some consequences on parabolic problems of the form (2.9). We start off with the homogeneous abstract Cauchy problem

\begin{aligned}u̇ + A_{p}u & = 0  in (0, ∞) \\ u(0) & = u_{0} , \end{aligned}  (6.1)

where u̇ := du/dt. By well known abstract results −A_{2} generates a strongly continuous analytic semigroup T_{2}(t) := e^{−tA_{2}} on L_{2}(Ω) (see e.g. [7], Proposition XVII.6/3) and hence, (6.1) has a unique solution for all u_{0} ∈ L_{2} which is given by u(t) = e^{−tA_{2}}u_{0}. We shall show that T_{2}(t) acts on all the L_{p}-spaces for all t > 0. To do so note that by a well known smoothing property of analytic semigroups

T_{2}(t)(L_{2}) ⊂ ⋂_{k=0}^{∞} D(A_{2}^{k})  (6.2)

for all t > 0. It follows from (5.3) that ⋂_{k=1}^{∞} D(A_{2}^{k}) ↪ L_{∞} for k large enough and hence, therefore,

T_{2}(t) ∈ L(L_{2}, L_{∞})  (6.3)

for all t > 0. In particular this shows that T_{2} acts on L_{p} for all p ∈ [2, ∞] which allows us to define T_{p}(t) := T_{2}(t)|_{L_{p}} for that range of p. Obviously, we can do the same thing with the semigroup T_{2}^{♯} generated by −A_{2}^{♯}. Moreover, as A′_{2} = A_{2}^{♯}, we have that T′_{2} = T_{2}^{♯} (see e.g. [20], Corollary 1.10.6). Hence, it makes sense to set

T_{p}(t) := (T_{p′}^{♯}(t))′  (6.4)

for all p ∈ (1, 2], where p^{−1} + (p′)^{−1}. Due to the above remarks we have for all u, v ∈ L_{2} and t > 0

|⟨T_{2}(t)u, v⟩| = |⟨u, T_{2}^{♯}(t)v⟩| ≤ ∥u∥_{1}∥v∥_{2}∥T_{2}^{♯}(t)∥_{2,∞} ,

whence,

∥T_{2}(t)u∥_{2} ≤ ∥T_{2}^{♯}(t)∥_{2,∞}∥u∥_{1} .

For this reason, T_{2}(t) has a unique continuous extension to L_{1} which we denote by T_{1}(t). A simple density argument shows that T_{1}(·) is a semigroup on L_{1}. The main result of this section is the following theorem. The proof works much as in the case of the Dirichlet problem but with other exponents.

6.1 Theorem. For 1 ≤ p ≤ ∞ T_{p}(·) is a positive irreducible analytic semigroup of compact operators on L_{p}. For 1 ≤ p < ∞ it is strongly continuous at zero and its generator is −A_{p}. Further, T_{∞}(·) is weak^{∗} continuous at zero. Moreover, for 1 ≤ p ≤ ∞

∥T_{p}(t)∥_{p,p} ≤ e^{δt}  (6.5)

and

∥T_{1}(t)∥_{1,∞} ≤ c(1 + δt)^{−N}t^{−N}e^{δt} ,  (6.6)

where δ is as in (4.3) and c > 0 a constant just depending on N and upper bounds for γ and |Ω|. Finally, (T_{p}(t))′ = T_{p′}^{♯}(t) for all t ≥ 0 and 1 ≤ p < ∞.

Proof. We first prove (6.5) and (6.6). To do so we set for all q ≥ 2 u_{q} := sign u|u|^{q−1}. Applying (4.10) to u and −u, using the definition of A and letting m to infinity we get that

∥|u|^{\frac{q}{2}} ∥q_{\frac{qN}{N−1}} ≤ 2c(N, Ω|)^{2}qγ⟨Au + qδu, u_{q}⟩  (6.7)

holds whenever u ∈ V and the right hand side of (6.7) is finite. We are now in a position to apply Theorem II.3.5 in [27] with n = 2N, which implies that

∥T_{2}(t)∥_{2,∞} ≤ c(1 + δt)^{\frac{N}{2}} t^{− \frac{N}{2}} e^{δt}  (6.8)

with a constant c > 0 depending only on N and upper bounds for γ and |Ω|. As the same holds for the formal adjoint problem we obtain the same estimate for ∥T_{1}(t)∥_{1,2} by duality. Writing T_{1}(t) = T_{2}(t/2)T_{1}(t/2) and combining the two estimates (6.6) follows. For p ≥ 2 (6.5) is a simple consequence of an intermediate step in the proof of Theorem II.3.5 in [27], and for p ≤ 2 it follows by duality since the same estimate holds for the formal adjoint problem.

That (T_{p}(t))′ = T_{p′}(t) holds for p ∈ (1, ∞) is just a reformulation of (6.4). For p = 1 the assertion follows by a density argument by using that T_{1}(t) is a bounded extension of T_{p}(t) for all t > 0.

The remaining assertions are clear for p = 2. In particular, compactness and irreducibility of T_{2}(t) for t > 0 follows since the resolvent of its generator −A_{2} has the same property by Theorem 5.1 (see [18], Section C-III.3 for the irreducibility). For p ≠ 2 the assertion follows from the above by writing for ε > 0 arbitrary and t > ε

T_{p}(t) = T_{2}(ε/2)T_{2}(t − ε)T_{p}(ε/2)

and taking into account the smoothing properties of T_{2} and the embeddings for L_{p}- spaces.

Strong continuity of T_{p}(·) in zero for 1 ≤ p ≤ 2 follows since T_{2}(·) has that property, L_{2} is dense in L_{p} and (6.5). Next we show that −A_{p} is the generator of T_{p} for 1 ≤ p ≤ 2. For p = 2 the result is well known. As by the results Theorem 5.1 A_{p} is the closure of A_{2} it is sufficient to show that D(A_{2}) is a core for the generator −B_{p} of T_{p}. If so, A_{p} and B_{p} are closures of A_{∞} and hence must coincide. By a well known theorem (e.g. [18], Proposition I.1.9) we have to show that D(A_{2}) is dense in L_{p}, that it is invariant under T_{p}(t), and that A_{2} is a restriction of B_{p}. Density of D(A_{2}) in L_{p} follows from Theorem 5.1, invariance from (6.6) and (6.2), and A_{2} ⊂ A_{1} since A_{2} is the generator on L_{2}, the density of L_{2} in L_{1} and the bound (6.5). Hence, −A_{p} is the generator of T_{p} for p ∈ [1, 2). The same is true for the formal adjoint problem and hence, by Theorem 5.1 and [20], Theorem 1.10.6 for ∈ (2, ∞). This shows that −A_{p} is the generator of T_{p} for all p ∈ [1, ∞). □

6.2 Remarks. (a) It is easy to see that a solution u of (6.1) is a weak solution of the parabolic problem

∂_{t}u + Au = 0

in Ω×(0, ∞) in the usual sense. Hence, all the “interior regularity” results for parabolic equations apply to our situation.

(b) The comments in Remark 4.3(b)–(d) apply to the parabolic situation as well.

(c) As T_{1}(t) ∈ L(L_{1}, L_{∞}) it has a representation of the form

T_{1}(t)u(x) = ∫_{Ω} k_{t}(x, y)u(y) dy

where k_{t}(· , ·) ∈ L_{∞}(Ω) ⊗ L_{1}(Ω) is called the heat kernel. It is also well known that the estimate (6.6) leads to an estimate of the kernel of the form

|k_{t}(x, y)| ≤ c(1 + δt)^{N}t^{−N}e^{δt} .

Moreover, k_{t}(x, y) → δ(x − y) weakly as t → 0. (see e.g. [21], Appendix C.1.) By the positivity of the semigroup we have that k(x, y, t) > 0 for all x, y ∈ Ω and t > 0. As we have the same representation formula for all p ∈ [1, ∞] we denote the semigroup simply by T (·). □

For selfadjoint operators we can prove that the heat kernel has upper Gaussian estimates with N replaced by 2N. More precisely, we have the following results, whose proof is analogous to that given in [8] for the Dirichlet or Neumann problem.

6.3 Theorem. Suppose that (A, B) is selfadjoint with c_{0} ≥ 0, that is, a_{ij} = a_{ji} and a_{i} = b_{i} = 0 for i, j = 1, …, N. Then, for all x, y ∈ Ω and t > 0

0 ≤ k_{t}(x, t) ≤ Ct^{−N}e^{− \frac{|x−y|^{2}}{Ct}} ,  (6.9)

where C depends on the same quantities as c in (6.6).

Proof. The proof is a simple modification of the proof for the Dirichlet problem. First of all note that by (6.6) (with δ = 0) and the abstract Lemma 2.2.3 in [8] a logarithmic Sobolev inequality with β(ε) = c − (N/2) log ε and a constant c depending only on the quantities listed in the theorem holds for all 0 ≤ u ∈ V ∩L_{∞} ∩L_{1}. Note also that V_{0} ∩V is a form core for a(· , ·). Now we can repeat the calculations in [8], Section 3.3 with minor modifications including the boundary term of the form a(· , ·). □

6.4 Remarks. (a) With the aid of (6.9) one can determine the angle of the semigroups T_{p}. It turns out to be π/2. The proof is a modification of the proof in Ouhabaz [19]. Indeed, note that by (6.9) and [8], Theorem 3.4.8 we get an estimate for the complex heat kernel with the only difference that we have to replace N by 2N. Then, (2.2) in [19] holds, also with N replaced by 2N (Note that we only integrate over a set of finite measure). Hence, the proof of Theorem 2.4 in the above mentioned paper works in our case.

(b) By using the Laplace transform we get from (6.9) the estimate

0 ≤ g(x, y) ≤ \frac{C^{N}Γ(N − 1)}{|x − y|^{2(N−1)}}

for the kernel g(· , ·) of A^{−1}, where C is the same as in (6.9) and Γ(·) the Gamma function. □

Using the above results it is easy to deal with the abstract parabolic equation

\begin{aligned}u̇ + A_{p}u & = f(t)  in (0, ∞) \\ u(0) & = u_{0} . \end{aligned}  (6.10)

As usual we call

u(t) := T (t)u_{0} + ∫_{0}^{t} T (t − s)f(s) ds  (6.11)

a mild solution of (6.10) (e.g. [20]). If u is a mild solution of (6.10) we say that u is a generalised solution of (2.9). Using the results above it is now easy to prove Theorem 2.4.

Proof of Theorem 2.4. By (6.5), (6.6) and interpolation it follows that for all 1 ≤ q ≤ p ≤ ∞

∥T (t)∥_{q,p} ≤ c(1 + δt)^{N( \frac{1}{q} − \frac{1}{p})}t^{−N( \frac{1}{q} − \frac{1}{p})}e^{δt}

Using this, (6.11) obviously exists and hence (2.9) has a generalised solution. If f is Hölder continuous it easily follows that u is differentiable in t for t > 0. □

Appendix: Abstract Regularity Results

The purpose of this appendix is do provide some results on L_{p}-regularity of solutions to abstract linear variational problems. Suppose that (X, m) is a measure space and set H := L_{2} := L_{2}(X, m). Further, assume that V is a Hilbert space which is continuously and densely embedded in H. Identifying H with its dual we have that

V ↪^{d} H ↪^{d} V′.

The standard inner product in H is as usual given by

(u|v) := ∫_{X} uv dm .

We further suppose that ⟨· , ·⟩ is the duality pairing on V induced by (· | ·), that is, ⟨u, v⟩ = (u|v) holds for all u ∈ H ⊂ V′ and v ∈ V. The duality pairing ⟨· , ·⟩ is well defined as V is dense in H. Finally, we assume that

a(· , ·): V × V → ℝ  (A.1)

is a continuous sesquilinear form and that there exists α > 0 and λ_{0} ∈ ℝ such that

a(u, u) + λ_{0}∥u∥_{H}^{2} ≥ α∥u∥_{V} .  (A.2)

It is well known that a(· , ·) induces a an operator A ∈ L(V, V′) satisfying

a(u, v) = ⟨Au, v⟩  (A.3)

for all u, v ∈ V. We study properties of solutions of the equation

Au = f .  (A.4)

Clearly, u is a solution of (A.4) if and only if u satisfies

a(u, v) = ⟨f, v⟩  (A.5)

for all v ∈ V. We divide this appendix into three parts. In a first part we establish a priori estimates for solutions of (A.4). The method we use is an abstract version of the well known Moser iteration technique. The results are the “elliptic” counterpart to the results in the “parabolic” case given e.g. in [27], Theorem II.3.5. In a second part we look at properties of maximal restrictions of operators to Banach spaces and study duality. In the third part we apply and extend these results in case of L_{p}-spaces. Particularly in part three many results we prove are probably more or less folklore, but as we do not now of any explicit reference, we include the exact statements detailed proofs.

I. A priori estimates. In this section we prove a priori estimates for solutions of (A.4). The idea is to provide a general L_{p}-regularity result which bases on an inequality which is not hard to verify for concrete problems such as elliptic boundary value problems. We assume that there exist constants d > 2, c̄, δ ≥ 0 and a nondecreasing functions c_{1}, c_{2}: [2, ∞) → [1, ∞) of polynomial growth such that

∥|u|^{\frac{q}{2}} ∥2_{\frac{2d}{d−2}} ≤ c̄c_{1}(q)(⟨Au, sign u|u|^{q−1}⟩ + δc_{2}(q)∥|u|^{\frac{q}{2}} ∥_{2}^{2}) ,  (A.6)

for all u ∈ D(A) and q ≥ 2 for which the right hand side makes sense.

A.1 Theorem. Suppose that m(X) < ∞ and that u is the solution of the equation Au = f with f ∈ L_{p} and that (A.6) holds. Then, if p ∈ [2, d/2)

∥u∥_{\frac{dp}{d−2p}} ≤ Cc̄(∥f∥_{p} + c̄^{μ}δ^{1+μ}∥u∥_{2}) ,  (A.7)

where C > 0 only depends on d and p. If p > d/2

∥u∥_{∞} ≤ Cc̄(∥f∥_{p} + c̄^{μ}δ^{1+μ}∥u∥_{2}) ,  (A.8)

where C > 0 depends on d, p and an upper bound for m(X). Proof. The proof of the above theorem is almost identical to part of the proof of Theorem 4.1. In this case we set v := sign u|u|^{q−1} and w := |u|^{\frac{q}{2}} . Then, if f is a solution of (A.4) we get the estimates (4.11) and (4.12) for ∥w∥2_{\frac{2d}{d−2}} with q^{2} replaced by c_{3}(q) := max{c_{1}(q), c_{2}(q)}. If p ≥ d/2 we proceed exactly as in the proof of Theorem 4.1. In case p ∈ [2, d/2) we have to be a little more careful as we do not know a priori that w ∈ L_{p′(q−1)} and hence we can not just set q := q̄:= p(d − 2)(d − 2p)^{−1} and divide inequality (4.12) by ∥u∥q̄−1_{\frac{dp}{d−2p}} to get the result. The main problem therefore is to show that u has a finite L_{\frac{dp}{d−2p}} -norm. To do so we define q_{n} and η as in case of the L_{∞}-estimate. As for that range of p q_{n} < q̄ and η < 1 we have that

lim_{n→∞} q_{n} = lim_{n→∞} η^{n} + ∑_{k=0}^{n} η^{k} = (1 − η)^{−1} = q̄.

Using these facts we deduce from (4.13) that

∥w∥_{\frac{dq_{n+1}}{d−2}} ≤ c_{3}(q̄)^{\frac{1+μ}{2} ∑_{k=0}^{∞}(1− \frac{1}{q̄})^{k}}∥w∥(1− \frac{1}{q̄})^{n}_{\frac{2d}{d−2}} ≤ c_{3}(q̄)^{^{\frac{1+μ}{2}} q̄}∥w∥(1− \frac{1}{q̄})^{n}_{\frac{2d}{d−2}}

for all n ∈ ℕ. Letting n to infinity we obtain

∥w∥_{\frac{dp}{d−2p}} = ∥w∥_{\frac{dq̄}{d−2}} ≤ c_{3}(q̄)^{^{\frac{1+μ}{2}} q̄}

This completes the proof in case p ∈ [2, d/2). □ A.2 Remarks. With a slight modification of the proof the above theorem remains true if we omit the assumption that m(X) < ∞. In that case we have to assume that f ∈ L_{2} ∩ L_{p}, and we get that for p > d/2

∥u∥_{∞} ≤ Cc̄^{1+μ}(∥f∥_{p} + δ^{1+μ}∥u∥_{2}) + C∥u∥_{2} ,  (A.9)

where C > 0 just on d and p. To see this note first that the only place we used that the volume is finite is in (4.15). So we only have to prove (A.9). To do so note that (4.14) implies that ∥u∥_{∞} ≤ M^{1−θ}∥w∥θ_{\frac{2d}{d−2}} with M defined as in the proof of Theorem 4.1. By Young’s inequality we get that ∥u∥_{∞} ≤ CM + C∥u∥_{\frac{2d}{d−2}} . Finally, by a well known interpolation inequality we have that

∥u∥_{\frac{2d}{d−2}} ≤ 2^{−1}∥u∥_{∞} + 2^{\frac{2}{d−2}} ∥u∥_{2}

which completes the proof of (A.9). Note that (A.8) can not be true on unbounded domains. As a counterexample consider the equation −Δu = f in ℝ^{N}. In that case Δ = 0 and (A.8) would imply that (−δ)^{−1} is bounded from L_{p} to L_{∞}, which is not true. □

II. Maximal restrictions and duality. Suppose that V, H, the form a(· , ·) and the operator A are as before. Due to (A.2) and the Lax-Milgram Theorem λ + A has a bounded inverse for all λ ≥ λ_{0}. As V ↪^{d} V′ we may consider A as a densely defined operator on V′ with domain of definition V. It follows from the boundedness of a(· , ·) and (A.2) that the graph norm ∥u∥_{D(A)} = ∥Au∥_{V′} + ∥u∥_{V′} is equivalent to the norm on V. As A ∈ L(V, V′) it follows that A is a closed operator on V′. If F is a Banach space with F ↪ V′ the F -realisation A_{F} of A defined by

A_{F} u := Au  for all  u ∈ D(A_{F}) := {v ∈ V ∩ F: Av ∈ F }

is a closed operator on F. The operators A_{F} are also called the maximal restriction of A to F.

A.3 Lemma. Let F ↪^{d} V′ and (λ + A)^{−1}(F) ⊂ F for some λ ∈ ϱ(−A), then D(A_{F}) is dense in V.

Proof. First note that under the present assumptions ϱ(A) ⊂ ϱ(A_{F}) so in particular λ_{0} ∈ ϱ(−A_{F}). It is sufficient to show that the orthogonal complement of D(A_{F}) with respect to the bilinear form a(· , ·)+λ_{0}(· | ·)_{H} is zero. To see this note that 0 = a(u, v)+ λ_{0}(u|v)_{H} = ⟨(λ_{0} +A_{F})u, v⟩ for all u ∈ D(A_{F}) implies that v = 0 as the range of λ_{0} +A_{F} is F which is dense in V′. □

The following proposition which turns out to be useful later is a consequence of a result due to Arendt [2].

A.4 Proposition. Suppose that F ↪ V′, that (λ+A)^{−1}(F) ⊂ F and (λ+A)^{−k}(V′) ⊂ F for some λ ∈ ϱ(−A) and k ∈ ℕ. Then, ϱ(A) = ϱ(A_{F}) and (λ + A_{F})^{−1} = (λ + A)^{−1}|_{F} for all λ ∈ ϱ(−A).

A.5 Proposition. Suppose that E is a Banach space with V ↪^{d} E. Then, the embedding V ↪ E is compact if and only if (λ_{0} + A_{E′})^{−1} ∈ L(E′, E) is compact.

Proof. That the condition is necessary is clear. To prove the reverse we can assume that λ_{0} = 0 in (5.2) by replacing A by λ_{0} + A. Doing so we get that for all u ∈ E′ ↪ V′

α∥A^{−1}u∥_{V}^{2} ≤ a(A^{−1}u, A^{−1}u) = ⟨u, A^{−1}u⟩ ≤ ∥u∥_{E′}∥A^{−1}u∥_{E} .  (A.9)

Suppose that B ⊂ E′ is bounded and that w_{n} is a sequence in A^{−1}(B), that is, w_{n} = A^{−1}u_{n} for some u_{n} ∈ B. As A^{−1} is compact as an operator to E by hypotheses it follows that w_{n} has a convergent subsequence in E. Applying (A.9) to u = u_{k} − u_{ℓ} we get that

∥w_{k} − w_{ℓ}∥_{V}^{2} ≤ α^{−1}∥u_{k} − u_{ℓ}∥_{E′}∥w_{k} − w_{ℓ}∥_{E}

for all k, ℓ ∈ ℕ. As B is bounded in E′ this shows that this subsequence also converges in V. Consequently, A^{−1}(B) is relatively compact in V, whence A^{−1} ∈ L(E′, V) is compact. Therefore, the embedding E′ ↪ V′ given by AA^{−1} is compact which is equivalent to V ↪ E being compact. This concludes the proof of the proposition. □

Next we define extensions of A to Banach spaces which are not necessarily subspaces of V′. To do this let us consider the dual form a^{♯}(· , ·): V × V → ℝ given by

a^{♯}(u, v) := a(v, u)  (A.11)

for all u, v ∈ V. Denote by A^{♯} the closed operator induced by a^{♯}(· , ·) on V′ and let F be a Banach space such that F′ ↪^{d} V′ and (λ + A^{♯})^{−1}(F′) ⊂ F′. Assume in addition that D(A_{F′}^{♯}) is dense in F′. Then, we can define

A_{F} := (A_{F′}^{♯})′,  (A.12)

where the domain of definition of A_{F} is given by

D(A_{F}) = {u ∈ F: v ↦ ⟨A_{F′}^{♯}v, u⟩ is continuous on D(A_{F′}^{♯}) ⊂ F′} .

Then the following proposition holds.

A.6 Proposition. Suppose F and A_{F} are defined as above. Moreover, let E be a Banach space such that E ↪ F, E ↪^{d} V′ and (λ + A)^{−1}(E) ⊂ E for some λ ∈ ϱ(−A). Then, (A_{F})_{E} = A_{E}, that is, the E-realisation of A_{F} coincides with A_{E}. In particular, A_{F} is an extension of A_{E}.

Proof. Let us assume without loss of generality that 0 ∈ ϱ(A) ∩ ϱ(A^{♯}) by replacing A abd A^{♯} by λ_{0} + A and λ_{0} + A^{♯}, respectively. By our assumptions it is clear that 0 ∈ ϱ(A_{E}) ∩ ϱ(A_{F′}^{♯}). Let u ∈ D(A_{E}). Then, for all v ∈ D(A_{F′}^{♯})

|⟨A_{F′}^{♯}v, u⟩| = |a^{♯}(v, u)| = |a(u, v)| = |⟨Au, v⟩| ≤ ∥Au∥_{F} ∥v∥_{F′} ≤ c∥A_{E}u∥_{E}∥v∥_{F′} ,

where we used that E ↪ F. Hence, u ∈ D(A_{F}). The above calculation also shows that ⟨A_{F} u, v⟩ = ⟨A_{E}u, v⟩ for all v ∈ D(A_{F′}^{♯}), and therefore by density of D(A_{F′}^{♯}) in F′ we conclude that A_{F} u = A_{E}u for all u ∈ D(A_{E}), that is, A_{F} is an extension of A_{E}. Next assume that u ∈ D((A_{F})_{E}), that is, u ∈ D(A_{F}) ∩ E such that A_{F} u ∈ E. As 0 ∈ ϱ(A_{E}) and A_{F} u ∈ E ↪ V′ there exists w ∈ D(A_{E}) ⊂ V such that A_{F} u = A_{E}w ∈ E. Hence, by the definition of A_{F}

⟨A_{F′}^{♯}v, u⟩ = ⟨A_{F} u, v⟩ = ⟨A_{E}w, v⟩ = a(w, v) = a^{♯}(v, w) = ⟨A_{F′}^{♯}v, w⟩

for all v ∈ D(A_{F′}^{♯}). As the range of A_{F′}^{♯} is F′ (recall that 0 ∈ ϱ(A_{F′}^{♯})) it follows that u = w ∈ V ∩ E and A_{F} u = A_{E}u ∈ E which completes the proof of the proposition. □

A.7 Remarks. (a) Taking E = F = V′ in the above proposition we get that (A^{♯})′ = A which is of course well known. In particular, we have that ϱ(A) = ϱ(A^{♯}).

(b) Let F be as in the above proposition, and suppose that (λ + A^{♯})^{−k}(V′) ⊂ F′ for some λ ∈ ϱ(−A^{♯}) and k ∈ ℕ. Then, by definition (A.12) and Proposition A.4 we get that ϱ(A_{F}) = ϱ(A_{F′}^{♯}) = ϱ(A^{♯}) = ϱ(A). □

III. Smoothing properties of (λ + A)^{−1}. In what follows we throughout assume that m(X) < ∞ and that in addition to the assumptions given earlier

V ∩ L_{p} is dense in L_{p}  (A.13)

for all p ∈ [2, ∞). We further suppose that the operators A and A^{♯} induced by the forms a(· , ·) and a^{♯}(· , ·) satisfy (A.6) with the same constants. Observe that by (A.6) with q = 2, (A.3) and the continuity of a(· , ·) on V × V we get that ∥u∥_{\frac{2d}{d−2}} ≤ c_{1}∥u∥_{V}, that is,

V ↪^{d} L_{\frac{2d}{d−2}} .  (A.14)

Hence, by duality and the embeddings for L_{p}-spaces we conclude that

L_{p} ↪^{d} V′  (A.15)

for all p ∈ [2d(d + 2)^{−1}, ∞].

A.7 Lemma. For all p ∈ [2, ∞] we have that

(λ_{0} + A)^{−1} ∈ L(L_{p}, L_{m(p)}) ,  (A.16)

where m(p) = dp(d − 2p)^{−1} if p < d/2 and m(p) = ∞ if p > d/2, and λ_{0} from (A.2). Moreover, there exists k ∈ ℕ only depending on d such that

(λ_{0} + A)^{−k}(V′) ⊂ L_{∞} .  (A.17)

The assertion is the same if we replace A by A^{♯}.

Proof. Clearly λ_{0} ∈ ϱ(−A) = ϱ(−A^{♯}) and (λ_{0} + A)^{−1}(V′) ⊂ V ↪ L_{2}. Hence, there exists a constant c > 0 such that ∥u∥_{2} ≤ c∥f∥_{p} for p ≥ 2, where u := (λ_{0} + A)^{−1}f. Using (A.7) and (A.8) with A replaced by A + λ_{0} the first assertion follows. For the second assertion we again use that (λ_{0} +A)^{−1}(V′) ⊂ V ↪ L_{2} and iterate the first result a finite number of times depending on d only. □

For p ≥ 2 we denote by A_{p} and A_{p}^{♯} the L_{p}-realisation of A and A^{♯}, respectively. Recall that (L_{p})′ = L_{p′} with equal norms for all p ∈ [1, ∞), where as usual p^{−1} + (p′)^{−1} = 1. Thus, if p ∈ (1, 2], (L_{p})′ ↪^{d} V′, and by the above lemma (λ + A^{♯})^{−1}(Lp′) ⊂ L_{p′}. Hence, by assumption (A.13) and Lemma A.3 D(A_{p′}^{♯}) is dense in L_{p′}. Therefore, for all p ∈ (1, 2) we can define

A_{p} := (A_{p′}^{♯})′,  (A.18)

for p ∈ (1, 2]. As a consequence of Proposition A.6 A_{p} is an extension of A_{q} for all q > p. On the other hand, A_{p} is the closure of A_{q} for such p, q. To see this it is sufficient to show that D(A_{∞}) is dense in D(A_{p}) for all p > 1. Indeed, as L_{∞} is dense in L_{p} there exists a sequence v_{n} in L_{∞} approaching (λ_{0} + A)u ∈ L_{p}. Set u_{n} := (λ_{0} + A_{∞})^{−1}v_{n}. As (λ_{0} + A_{∞})^{−1}|_{L_{p}} = (λ_{0} + A_{p})^{−1} this shows the assertion.

Note that D(A_{∞}^{♯}) is not dense in L_{∞} in general, and that (L_{∞})′ ≠ L_{1}. For this reason we can not define A_{1} to be the dual of A_{∞}^{♯}. Instead, we define A_{1} to be the closure of A_{p} for some arbitrary p > 1. The next lemma shows that this is possible.

A.8 Lemma. For p > 1 A_{p} is closable in L_{1} and the closure is independent of p. If we denote its closure by A_{1} we have that (A_{1})′ = A_{∞}^{♯}.

Proof. Let p ∈ (1, ∞) be arbitrary and suppose that u_{n} ∈ L_{1} ∩D(A_{p}) is a sequence such that A_{p}u_{n} → g in L_{1} and u_{n} → 0 in L_{1} as n tends to infinity. For w ∈ D(A_{∞}^{♯}) ⊂ D(A_{p′}^{♯}) we have by (A.18) that ⟨A_{p}u_{n}, w⟩ = ⟨u_{n}, A_{p′}^{♯}w⟩. As A_{p′}^{♯}w ∈ L_{∞} it follows that the last term tends to zero, whereas the first tends to ⟨g, w⟩ as n tends to infinity. We know from (A.15) and Lemma A.3 that D(A_{∞}^{♯}) is dense in V and therefore dense in L_{q} for q ≤ 2d(d − 2)^{−1}. Hence, we can approximate every function v ∈ L_{∞} pointwise by a sequence w_{n} ∈ D(A_{∞}^{♯}) with ∥w_{n}∥_{∞} uniformly bounded and the Dominated Convergence Theorem yields 0 = ⟨g, w_{n}⟩ → ⟨g, w⟩ = 0, which implies that g = 0. This shows that A_{p} is closable. The closure is independent of p because A_{p} is the closure of A_{∞} for all p > 1.

It remains to show that (A_{1})′ = A_{∞}^{♯}. Let u ∈ D(A_{∞}^{♯}), that is, u ∈ V ∩ L_{∞} and A^{♯}u ∈ L_{∞}, and suppose that v ∈ D(A_{p}) ⊂ D(A_{1}) for some p ∈ (1, ∞). Using (A.18) we obtain

|⟨u, A_{1}v⟩| = |⟨u, A_{p}v⟩| = |⟨A_{p′}^{♯}u, v⟩| ≤ ∥A_{∞}^{♯}u∥_{∞}∥v∥_{1} .

As D(A_{p}) is dense in L_{1} this implies that u ∈ D((A_{1})′) and (A_{1})′u = A_{∞}^{♯}u. Suppose now that u ∈ D((A_{1})′). Therefore, for all v ∈ D(A_{p}) we have that

|⟨u, A_{p}v⟩| = |⟨u, A_{1}v⟩| ≤ c∥v∥_{1} ≤ c_{1}∥u∥_{p}

for some constants c, c_{1} > 0 independent of v ∈ D(A_{p}). Hence, u ∈ D((A_{p})′) = D(A_{p′}^{♯}). By (A.18) we conclude that (A_{1})′u = (A_{p})′u = A_{p′}^{♯}u = A^{♯}u. As u ∈ L_{∞} and (A_{1})′u ∈ L_{∞} this implies that u ∈ D(A_{∞}^{♯}). This concludes the proof of the lemma. □

A.9 Theorem. Let A_{p} and A_{p}^{♯} defined as above for all p ∈ [1, ∞]. Then, the following assertions hold.

(a) For p ∈ [1, ∞] A_{p} are closed. It is densely defined if p < ∞. Moreover, A_{p} is the closure of A_{q}, and A_{q} the L_{q}-realisation of A_{p} whenever 1 ≤ p ≤ q ≤ ∞.

(b) A′_{p} = A_{p′}^{♯} for all p ∈ [1, ∞).

(c) The spectrum of A_{p} is independent of p ∈ [1, ∞]. Moreover, if V is compactly embedded in L_{2}, then A_{p} has compact resolvent for all p ∈ [1, ∞], and if A_{p} has compact resolvent for some p ∈ [1, ∞] this is true for all and V ↪ L_{2} compactly.

(d) For p ∈ [1, ∞] and λ ∈ ϱ(−A) (A.16) holds, where m(p) = dp(d − 2p)^{−1} if p ∈ (1, d/2), m(p) = ∞ if p > d/2, and m(p) ∈ [1, d(d − 2)^{−1}) arbitrary if p = 1. Furthermore,

∥(λ + A_{p})^{−1}∥_{p,q} = ∥(λ + A_{q′}^{♯})^{−1}∥_{q′,p′}  (A.19)

whenever p, q ∈ [1, ∞] and one of the norms is finite.

Proof. (a) is clear from Lemma A.8 and the preceding discussion, and (b) follows from definition (A.18) and Lemma A.8. Independence of the spectrum of A_{p} from p ∈ (1, ∞] is a consequence of the definition of A_{p} and Remark A.7.(b). That ϱ(A_{1}) is the same follows by duality from (b). Compactness of the resolvent in case p = 2 is clear since V ↪ L_{2} compactly. Exactly in the same way as in the proof of Theorem 6.1 one can show that −A_{p} generates a C_{0}-semigroup of compact operators on L_{p} for all p ∈ [1, ∞). Hence, it follows from [20], Theorem 2.3.3 that A_{p} has compact resolvent for all p ∈ [1, ∞). By duality, also A_{∞} has compact resolvent. If A_{p} has compact resolvent for some p this implies that it is also true for p = 2 and by (b) for A_{2}^{♯}. Hence, it follows from Proposition A.5 that V ↪ L_{2} compactly. This proves (c). To prove (d) note that [(λ + A_{p})^{−1}]′ = (λ + A_{p′}^{♯})^{−1} for all p ∈ [1, ∞). As an operator and its dual have the same norm this implies (A.19). The remaining assertion follows from this together with (A.16). □

A.10 Theorem. Suppose that V ↪ L_{2} compactly and that for all p > d/2 there exists a constant M_{p} > 0 such that for the solution of (A.4)

∥u∥_{∞} ≤ M_{p}∥f∥_{p}  (A.20)

Further suppose that a similar statement holds for the solution of the adjoint problem with the same constants M_{p}. Then, 0 ∈ ϱ(A) and there exists a constant c > 0 depending only on c̄, d, p, δ and M_{d} such that

∥A_{p}^{−1}∥_{p,m(p)} ≤ c  (A.21)

for all p ∈ [1, ∞], where m(p) is as in Theorem A.9.

Proof. We first show that 0 ∈ ϱ(A). To do so note that u is a solution of (A.4) with f ∈ L_{p} if and only if u is a solution of

u − λ(λ + A_{p})^{−1}u = (λ + A_{p})^{−1}f  (A.22)

for some λ ∈ ϱ(−A). As A_{p} has compact resolvent by Theorem A.9 I −λ(λ + A_{p})^{−1} is Fredholm of index zero. On the other hand, we conclude from (A.20) that if p > d/2 (A.4) and hence (A.22) has at most one solution. By the Fredholm alternative it follows that (A.22) and hence (A.4) has a unique solution for all f ∈ L_{p}. It also follows from (A.20) that 0 ∈ ϱ(A_{p}) for this range of p and that (A.21) holds. By Theorem A.9 0 ∈ ϱ(A) = ϱ(A_{p}) for all p ∈ [1, ∞]. A similar statement with the same constants holds if we replace A by A^{♯}. It follows in particular that A^{−1}, (A^{♯})^{−1} ∈ L(L_{d}, L_{∞}). By Theorem A.9 A_{1}^{−1} ∈ L(L_{1}, L_{\frac{d}{d−1}} ) with norm equal to ∥(A^{♯})^{−1}∥_{d,∞}. Hence, by interpolation A^{−1} ∈ L(L_{\frac{2d}{d+2}} , L_{2}) with a norm depending on the same quantities. This means that the solution u of (A.4) with f ∈ L_{p} for p ≥ 2d(d + 2)^{−1} satisfies an a priori estimate ∥u∥_{2} ≤ c∥f∥_{p} with a constant c > 0 as mentioned above. If we plug this estimate into (A.7) we get that (A.21) holds for p ≥ 2. As the same assertion holds for the formal dual we get the remaining cases by duality using Theorem A.9. □

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