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[ Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København, Denmark
Abstract

Consider the fractional powers and of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator on a smooth bounded subset of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970’s, we demonstrate how they imply regularity properties in full scales of -Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of are derived by use of recent results on -calculus. We also include an overview of the various Dirichlet- and Neumann-type boundary problems associated with the fractional Laplacian.

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theDOIsuffix \Volume248 \Month01 \Year2007 \pagespan1 \ReceiveddateXXXX \ReviseddateXXXX \AccepteddateXXXX \DatepostedXXXX \subjclass[msc2000]35P99, 35S15, 47G30

Regularity of spectral problems ]Regularity of spectral fractional Dirichlet and Neumann problems

G. Grubb]Gerd Grubb111Corresponding author E-mail: grubb@math.ku.dk, Phone +45  3532 0743 Fax +45  3532 0704 .

1 Introduction

There is currently a great interest in fractional powers of the Laplacian on , , and derived operators associated with a subset of . The fractional Laplacian can be described as the pseudodifferential operator

(1.1)

with symbol , see also (6.1) below. Let be a bounded -smooth subset of . Since is nonlocal, it is not obvious how to define boundary value problems for it on , and in fact there are several interesting choices.

One choice for a Dirichlet realization on is to take the power defined from the Dirichlet realization of by spectral theory in the Hilbert space ; let us call it “the spectral Dirichlet fractional Laplacian”, following a suggestion of Bonforte, Sire and Vazquez [8].

Another very natural choice is to take the Friedrichs extension of the operator (where denotes restriction to ); let us denote it and call it “the restricted Dirichlet fractional Laplacian”, following [8].

Both choices enter in nonlinear PDE; is moreover important in probability theory. The operator can be replaced by a variable-coefficient strongly elliptic second-order operator (not necessarily symmetric).

For the restricted Dirichlet fractional Laplacian, detailed regularity properties of solutions of in Hölder spaces and Sobolev spaces have just recently been shown, in Ros-Oton and Serra [36, 37, 38], Grubb [27, 26].

For the spectral Dirichlet fractional Laplacian, regularity properties in -spaces have been known for many years, as a consequence of Seeley’s work [41, 42]; we shall account for this below in Sections 2 and 3. Further results have recently been presented by Caffarelli and Stinga in [12], treating domains with limited smoothness and obtaining certain Hölder estimates of Schauder type. See also Cabré and Tan [9] Th. 1.9, for the case .

In Section 4 we show how similar regularity properties of the spectral Neumann fractional Laplacian follow from Seeley’s results. Also for this case, [12] has recently shown Hölder estimates of Schauder type under weaker smoothness hypotheses.

In Section 5, we first briefly discuss extensions to more general scales of function spaces. Next, for generalizations to nonsmooth domains, we show how a recent result of Denk, Dore, Hieber, Prüss and Venni [16], on the existence of -calculi for boundary problems, can be combined with more recent results of Yagi [48, 49], to extend the regularity properties of Sections 3 and 4 to suitable nonsmooth situations for small , leading to new results.

Finally, Section 6 gives a brief overview of the many kinds of boundary problems associated with , expanding the references given above. This includes several other Neumann-type problems.

A primary purpose of the present note is to put forward some direct consequences of Seeley [41, 42] for the spectral fractional Laplacians. One of the main results is that when is second-order strongly elliptic and stands for either a Dirichlet or a Neumann condition, and , then for solutions of

(1.2)

for an implies if and only if itself satisfies all those boundary conditions of the form () that have a meaning on . Consequences are also drawn for -solutions and for solutions where is in or a Hölder space. We think this is of interest not just as a demonstration of early results, but also in showing how far one can reach, as a model for less smooth situations.

Section 5 shows one such generalization to nonsmooth domains and coefficients.

2 Seeley’s results on complex interpolation

Let be a strongly elliptic second-order differential operator on with -coefficients. (The following theory extends readily to -order systems with normal boundary conditions as treated in Seeley [41, 42] and Grubb [24], but we restrict the attention to the second-order scalar case to keep notation and explanations simple.)

Let be a -smooth bounded open subset of , and let denote the realization of in with domain ; here stands for either the Dirichlet condition or a suitable Neumann-type boundary condition. In details,

(2.1)

here , and is a first-order differential operator on such that together form a strongly elliptic boundary value problem. Then is lower bounded with spectrum in a sectorial region . Our considerations in the following are formulated for the case where is bijective. Seeley’s papers also show how to handle a finite-dimensional 0-eigenspace.

The complex powers of can be defined by spectral theory in in the cases where is selfadjoint, but Seeley has shown in [41] how the powers can be defined more generally in a consistent way, acting in -based Sobolev spaces (), by a Cauchy integral of the resolvent around the spectrum

(2.2)

Here is the set of distributions (functions if ) such that , and (denoted in [26, 27]), where stands for restriction to . The general point of view is that the resolvent is constructed as an integral operator (found here by pseudodifferential methods) that can be applied to various function spaces, e.g. when varies. The different realizations coincide on their common domains, so the labels and are used without indication of the actual spaces, which are understood from the context (this is standard terminology).

The formula (2.2) has a good meaning for ; extensions to other values of are defined by compositions with integer powers of . As shown in [41, 42], one has in general that , and the operators consitute a holomorphic semigroup in for . This is based on the fundamental estimates of the resolvent shown in [40]. For , the define unbounded operators in , with domains . Note in particular that

(2.3)

We can of course not repeat the full analysis of Seeley here. An abstract framework for similar constructions of powers of operators in general Banach spaces is given in Amann [3, 4].

The domains in of the positive powers of will now be explained for the cases in (2.1).

The domain of the realization of in with boundary condition is

(2.4)

In [42], Seeley showed that for , the domain of (the range of applied to ) equals the complex interpolation space between and of the appropriate order. He showed moreover that this is the space of functions satisfying if , and the space of functions with no extra condition if . He gives the special description for the case :

(2.5)

one can say that vanishes at in a generalized sense. (It is also recalled in Triebel [T95], Th. 4.3.3.) We here use a notation of [30, 26, 27], where stands for the space of functions in with support in .

Let us define:

{definition}

The spaces are defined by:

(2.6)

where .

Note that in the first three statements, consists of the functions in satisfying those boundary conditions for which (i.e., those that are well-defined on ). The definition in the fourth statement, although slightly complicated, is included here primarily in order that we can use the notation freely without exceptional parameters.

The spaces were defined in Seeley [42] (in Grisvard [22] for ); we have added the definitions for (they can be called extrapolation spaces, as in [3, 4]). In the -case, the extra requirement in (2.5) can be replaced by , where is the distance from to .

With this notation, Seeley’s works show:

{theorem}

When , equals the space obtained by complex interpolation between and .

For all , .

Proof.

The first statement is a direct quotation from [42]. So is the second statement for , and it follows for , and , by using (2.3) with , . ∎

Observe the general homeomorphism property that follows from this theorem in view of formula (2.3):

{corollary}

For , defines homeomorphisms:

(2.7)

The characterization of the interpolation space was given (also for -order operators) by Grisvard in the case of scalar elliptic operators in Sobolev spaces in [22], in terms of real interpolation. Seeley’s result for is shown for general elliptic operators in vector bundles, with normal boundary conditions.

3 Consequences for the Dirichlet problem

Let , denoted for brevity. Corollary 2 already shows how the regularity of and are related, when the functions are known on beforehand to lie in the special spaces in (2.6). But we can also discuss cases where is just given in a general Sobolev space. Namely, we have as a generalization of the remarks at the end of [42]:

{theorem}

Let . Let for some , and assume that is a solution of

(3.1)

If , then .

Let . Then for all . Moreover, if and only if , and then in fact .

Proof.

. When , we can simply use that , where defines a homeomorphism from to in view of (2.7).

. We first note that since , all , the preceding result shows that for all .

Now if , then by (2.6). Hence since defines a homeomorphism from to according to (2.7).

Conversely, let . Then since we know already that , we see that (taking ). Then by (2.6), for ; such exist since . Hence with and therefore has . ∎

Point in the theorem shows that may have to be provided with a nontrivial boundary condition in order for the best possible regularity to hold for . This is in contrast to the case where , where it is known that for satisfying with , always implies .

The case can be included in when we use the generalized boundary condition in (2.4); details are given for the general case in Theorem 3.2 below.

The importance of a boundary condition on for optimal regularity of is also demonstrated in the results of Caffarelli and Stinga [12] (and Cabré and Tan [9]).

By induction, we can extend the result to higher :

{theorem}

Let . Let be the solution of (3.1) with for some . One has for any :

If , and for (i.e., ), then .

On the other hand, if , then necessarily for (and hence and ).

Let . If , then . On the other hand, if , then necessarily and .

Proof.

Statement was shown for in Theorem 3 . We proceed by induction: Assume that the statement holds for . Now show it for :

If for , then by (2.6). Hence since defines a homeomorphism from to according to (2.7).

Conversely, let . Note that since , all , the result for shows that for all . Then, taking , we see that for . Now in view of (2.6), for ; such exist since . Hence with ; therefore it has for .

The first part of statement follows immediately from (2.7). For the second part, let , . Since , we see by application of with that . For this shows that for . Now also lies in (since ) so in fact , and . ∎

Briefly expressed, the theorem shows that in order to have optimal regularity, namely the improvement from lying in an -space to lying in an -space, it is necessary and sufficient to impose all the boundary conditions for the space on .

In the following, we assume throughout that . (Results for higher can be deduced from the present results by use of elementary mapping properties for integer powers, and are left to the reader.) As a first corollary, we can describe -solutions. Define

(3.2)
{corollary}

The operator defines a homeomorphism of onto itself.

Moreover, if for some , then implies (and hence ).

Proof.

Fix . We first note that

(3.3)

Here the inclusion ’’ follows from the observation

by taking the intersection over all . The other inclusion follows from

by taking intersections for .

The fact that maps homeomorphically to for all now implies that maps to with inverse .

Next, let . If , then Theorem 3.2 can be applied with arbitrarily large , showing that , and hence .∎

{remark}

It follows that for each , the eigenfunctions of (with domain ) belong to ; they are the same for all . In particular, when is selfadjoint in , the eigenfunctions of defined by spectral theory (that are the same as those of ) are the eigenfunctions also in the -settings.

Finally, let us draw some conclusions for regularity properties when or is in a Hölder space. As in [27], we denote by the space of functions that are continuously differentiable up to order when , and are in the Hölder class when , and . Recall that the Hölder-Zygmund spaces , also denoted , coincide with when , and there is the Sobolev embedding property

(Embedding and trace mapping properties for Besov-Triebel-Lizorkin spaces and are compiled e.g. in Johnsen [32], Sect. 2.3, 2.6; note that .) Recall also that for . Here we use the notation (it is applied similarly to -spaces).

{corollary}

Let with . If , resp. , then the solution of (3.1) is in , resp. , with .

If , then the solution of (3.1) is in with .

Proof.

. When , then by Theorem 3.1 . Now when , Sobolev embedding gives that , except when , where it gives . Since à fortiori , we see from (2.6) that in , hence in resp. .

. When , then for all . Using and letting , we conclude that .∎

{corollary}

Let , and let . If with for , then the solution of (3.1) satisfies:

(3.4)
Proof.

When , then for all , all . For so small that , we see from (2.6) that since for , . Then it follows from (2.7) that .

If , we have for so small that , and then sufficiently small, that satisfies the boundary conditions for . For , this implies that satisfying these boundary conditions.

If , we have for in a small interval that , and then for all sufficiently small, that satisfies the boundary conditions for . For , this implies that satisfying those boundary conditions. ∎

The regularity results of Caffarelli and Stinga [12] are concerned with cases assuming much less smoothness of the domain and coefficients, getting results in Hölder spaces of low order (). See also Section 5.

The above results deduced from [42] explain the role of boundary conditions on . The results in Hölder spaces resemble the results of [12] for the values of considered there, however with a loss of sharpness (the ’’) in some of the estimates in Corollary 3.

4 Consequences for Neumann-type problems

The proofs are analogous for a Neumann-type boundary operator ( in (2.1)ff.).

{theorem}

Let . Let be the solution of

(4.1)

where for some .

If , then .

One has for any :

If , and for (i.e., ), then .

On the other hand, if , then necessarily for (and hence and ).

Let . If , then . On the other hand, if , then necessarily and .

Define

(4.2)
{corollary}

The operator defines a homeomorphism of onto itself.

Moreover, if for some , then implies (and hence ).

{corollary}

Let with . If , resp. , then the solution of (4.1) is in , resp. , with if .

If , then the solution of (4.1) is in , with precisely when .

Proof.

It is seen as in Corollary 3 that resp. . If , then à fortiori , and in ; this carries over to the space we embed in.

. When , then for all , so we have for all . Letting , we conclude that , and is assured if . When , then for all , so for all ; no boundary condition is imposed. ∎

{corollary}

Let , and let satisfy .

If with for , then the solution of (4.1) satisfies:

(4.3)

In the case of considered on a connected set , there is a one-dimensional nullspace consisting of the constants (that are of course in ). This case is included in the above results by a trick found in [41]: Replace by

(4.4)

note that is a projection onto the constants, orthogonal in (it is also a pseudodifferential operator of order ). Here and , where . With , equals and is invertible, and the above results apply to it and lead to similar regularity results for itself (note that ).

5 Further developments

5.1 More general function spaces

The above theorems in Sobolev spaces are likely to extend to a large number of other scales of function spaces. Notably, it seems possible to extend them to the scale of Besov spaces with , , since the decisive complex interpolation properties of domains of elliptic realizations have been shown by Guidetti in [G91].

It is not at the moment clear to the author whether the scale of Hölder-Zygmund spaces, or the scale of “small” Hölder-Zygmund spaces (obtained by closure in -spaces of the compactly supported smooth functions), cf. e.g. Escher and Seiler [19], can be or has been included for these boundary value problems. (It was possible to include in the regularity study for the restricted fractional Laplacian in [26] using Johnsen [32].) Such an extension would allow removing the ’’ in some formulas in Corollaries 3 and 4 above.

Let us mention for cases without boundary conditions, that the continuity of classical pseudodifferential operators on (such as and its parametrices) in Hölder-Zygmund spaces has been known for many years, cf. e.g. Yamazaki [50] for a more general result and references to earlier contributions. On this point, [12] refers to Caffarelli and Silvestre [10].

5.2 Nonsmooth situations

It is of great interest to treat the problems also when the set and the coefficients of have only limited smoothness. One of the common strategies is to transfer the results known for constant-coefficient operators on to to variable-coefficient operators by perturbation arguments, and to sets by local coordinates. (This strategy is used in [12].) The pseudodifferential theory in smooth cases is in fact set up to incorporate the perturbation arguments in a systematic and more informative calculus. For nonsmooth cases, we remark that there do exist pseudodifferential theories requiring only limited smoothness in , cf. [2] and other works of Abels listed there. Applications to the present problems await development.

Another point of view comes forward in the efforts to establish so-called maximal regularity, -calculus and -boundedness properties for operators generating semigroups; see e.g. Denk, Hieber and Prüss [17] for results, references to the vast literature, and an overview of the theory. Fractional powers of boundary problems entered in this theory at an early stage, starting with Seeley’s results, but are not so much in focus in the latest developments, that are primarily aimed towards solvability of parabolic problems.

However, there is an interesting result by Yagi [48] that is relevant for the present purposes. He considers an operator

(5.1)

real in , real bounded and , on a bounded -domain . Define