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Abstract

We show well-posedness for an evolution problem associated with the Dirichlet-to-Robin operator for certain Robin boundary data. Moreover, it turns out that the semigroup generated by the Dirichlet-to-Robin operator is closely related to a weighted semigroup of composition operators on an appropriate Banach space of analytic functions.

Dirichlet-to-Robin Operators]Dirichlet-to-Robin Operators via Composition Semigroups

L. Perlich]Lars Perlich

The author is supported by Sächsisches Landesstipendium..\par\@mkboth\shortauthors\shorttitle \par [\par

Mathematics Subject Classification (2010). 47B38, 47B33, 47D06.

Keywords. Composition operators, Spaces of holomorphic functions, Dirichlet-to-Neumann, Dirichlet-to-Robin..

\par\@xsect\par

In recent years, the Dirichlet-to-Neumann operator has been studied intensively. In the beginning of the 20th century, these operators were dealt with theoretically, while in the 1980s and 1990s they were used to analyze inverse problems to determine coefficients of a differential operator. These problems apply, e.g., to image techniques in medicine and also to find defects in materials.
According to Arendt and ter Elst, the Dirichlet-to-Neumann operator can be obtained as an example of an operator associated with -sectorial forms, see [3]. Using methods from function theory, our purpose is to give an alternative approach to Poincaré-Steklov operators and the related semigroups on boundary spaces of Banach spaces of analytic functions. It turns out, as pointed out by Lax [14], that there is a surprising connection between semigroups of composition operators on spaces of harmonic functions on the unit disk referring to a specific semiflow and the Dirichlet-to-Neumann operator. In fact, we can extend this observation to the Laplace equation with Robin boundary conditions on Jordan domains in . More precisely, we study the evolution problem \par

(1.1)

where is a Jordan domain and and are boundary values of appropriate holomorphic functions on . We prove well-posedness of (\@setrefeq:DTR) in various spaces of distributions on including the scale of -spaces. As mentioned above, our approach does not use form methods but the theory of (weighted) composition operators on spaces of holomorphic and harmonic functions (for the moment only) on planar domains. Our method appears to be restricted to problems involving the Laplace operator, while the variational approach to Dirichlet-to-Neumann and Dirichlet-to-Robin operators using the theory of forms is quite flexible with respect the choice of elliptic operators in the domain . However, there it seems difficult to handle coefficients in front of the associated Neumann derivative (at least, we do not see how to handle them). Here, we can allow a large class of coefficient functions and . In particular, it may happen that degenerates at one point on the boundary. Moreover, using our method, we can define Dirichlet-to-Neumann and Dirichlet-to-Robin operators on several spaces of distributions. \parThis article is organized as follows. In Section 2 we introduce the notion of admissible spaces which is eventually our tool to solve the above posed evolution problem. We discuss some examples of admissible spaces, and we investigate corresponding boundary spaces. Then, in Section 3, we examine the connection between certain Poincaré-Stecklov operators, namely Dirichlet-to-Neumann and Dirichlet-to-Robin operators, and weighted semigroups of composition operators, and prove our main theorem. \par\par\@xsect \parInitiated by the famous paper by Berkson and Porta [6], semigroups of composition operators were studied intensively by many authors on various spaces of holomorphic functions defined on the unit disk, see, for example, [2, 5, 13, 18, 17]. In our approach, we consider (weighted) semigroups of composition operators on spaces of harmonic and holomorphic functions which are defined on a simply connected domain bounded by a Jordan curve. To give the definition of such a semigroup, we need the notion of a semiflow of holomorphic functions. \par\par\@xsect

Definition 2.1.

Let be simply connected. Let be holomorphic (we write ) such that for every the fractional iterates are holomorphic selfmaps in . A family is called a semiflow of holomorphic functions if it satisfies the following properties:

  1. for all ,

  2. for all and ,

  3. as for all .

Given a semiflow we define its generator by

for every . \parSince is simply connected, by the Riemann mapping theorem there exists a conformal map , and thus every semiflow on can be written in terms of a semiflow on the unit disk. Let be a semiflow on . As a consequence of the chain rule, the generator of can be written in terms of the generator of . For all holomorphic selfmaps in the unit disk which are not automorphisms, the embeddability into a semiflow can be characterized in terms of the Denjoy-Wolff point of , see for instance [10]. The Denjoy-Wolff point is defined as the unique fixed point of a holomorphic selfmap in the unit disk which is not an automorphism in the unit disk. Such a point can be found in the interior of the unit disk as well as on the boundary. Thus we can use appropriate Möbius transforms to shift an interior Denjoy-Wolff point to zero and a Denjoy-Wolff point on the boundary to 1. In our case, the representation of on in terms of a semiflow on the unit disk gives also the unique fixed point of every as where is the Denjoy-Wolff point of . From the theory of differential equations, we obtain that is univalent for every , hence the same is true for .
Let be the Denjoy-Wolff point of a semiflow in . Then, by [6], the generator of is given by the Berkson and Porta formula

(2.1)

where is holomorphic and . It is also well known that is holomorphic in and that . In fact, if a holomorphic function extends continuously to and for every , then is the generator of a semiflow in , see [1, Thm 1]. Conversely, a generator of a semiflow need not extend continuously to the closure of On the other hand, note that, by Fatou’s theorem, a generator has radial limits almost everywhere since the function is the composition of a bounded holomorphic function and a Möbius transform. The angle condition at the boundary still holds.

Lemma 2.2.

Let be a semiflow in the unit disk and its generator. Then .

\par
Proof.

Let be the Denjoy-Wolff point of . Then, by [6], the generator is given by (\@setrefeq:BP) and radial limits exist almost everywhere. For we have \par

∎ The same result holds true for generators of semiflows on Jordan domains.

Lemma 2.3.

Let be a Jordan domain. Let be a semiflow in and its generator. Then , where is the normal vector at

\par
Proof.

Let be conformal. Therefore is a semiflow in the unit disk. Let be the generator of . Then .
For , we have

(2.2)

The function extends continuously to (see [16, Thm. 2.6]) and has non-vanishing angular derivative a.e. (see [16, Thm. 6.8]). Furthermore, for , we have . For every there exists a unique such that , so \par

\par

Next, we transfer the characterization of generators of semiflows in the unit disk given above to Jordan domains. \par

Proposition 2.4.

Let be holomorphic, where is simply connected.
(I) If is Dini-smooth and extends continuously to and for a.e. , then is the generator of a semiflow in .
(II) If for every conformal map there exists and a holomorphic function with positive real part such that

(2.3)

is the generator of a semiflow in . In this case, we say that admits a conformal Berkson and Porta representation.

\par
Proof.

(I) Let conformal. Define for . Then is a holomorphic function which admits a uniformly continuous extension to , by [16, Thm 3.5]. Moreover, for ,

So we can apply [1, Thm. 1] which shows that is the generator of a semiflow in , and by (\@setrefeq:konfGen) is the generator of the semiflow .
(II) The function is given by the Berkson and Porta formula, hence it is the generator of a semiflow in with Denjoy-Wolff point . The assertion follows again by (\@setrefeq:konfGen). ∎ \par\par\@xsect \parSemiflows of holomorphic mappings lead to semigroups of composition operators on spaces of holomorphic functions. Let be simply connected, and consider the Frechét space equipped with the topology of uniform convergence on compact subset of . Let be an increasing sequence of compact subsets of such that . We define a sequence of seminorms on as follows

and a metric induced by these seminorms by

For a given semiflow , we define a family of composition operators acting on as follows

(2.4)

By the definiton of semiflows, this family is an operator semigroup which is, in particular, strongly continuous since for all , we have

This defintion makes also sense when the space of harmonic functions is under consideration. Since, by the Cauchy-Riemann equations, for every function , we have .

Definition 2.5.

Let be a Banach space and a semiflow of holomorphic functions in generated by . The space is called -admissible if the family of operators defined by (\@setrefeq:CO) satisfies the following two conditions:

  • is invariant under i.e., for all .

  • is strongly continuous on .

\par\par

Given a semigroup of composition operators on a -admissible Banach space , the generator admits a special form:

Note that is a directional derivative. This is true for holomorphic functions and harmonic functions as well, but for convenience we write instead of for harmonic functions to distinguish products of complex numbers from inner products. \par\@xsect \parTypical choices for the space are the Bergman spaces

where denotes the normalized Lebesgue measure on , and the Hardy spaces

The invariance is a consequence of Littlewood’s subordination principle, and the strong continuity follows from the density of the polynomials and the dominated convergence theorem, see [17], which is also a comprehensive survey on semigroups of composition operators. \parIndeed, this result carries over to Bergman and Hardy spaces on simply connected domains. The Bergman spaces can be defined analogously to the Bergman spaces for functions in the unit disk. For the Hardy space, we can give at least two definitions for simply connected domains, see [9], either using harmonic majorants or via approximating the boundary of by rectifiable curves. Both definitions are equivalent when analytic Jordan domains are considered. We use the definition in terms of harmonic majorants.

Definition 2.6.

Let be simply connected. For , the Hardy space consists of those functions such that the subharmonic functions is dominated by a harmonic function .

\par

Equipped with the norm where is some fixed point and is the least harmonic majorant for , the Hardy space over is a Banach space. As in the unit disk, functions in admit non-tangential limits a.e. on and the boundary function is in . For more details about Hardy spaces over general domains, we refer to [9, Ch. 10]. \par\par

Proposition 2.7.

Let be simply connected. Let be a semiflow of holomorphic functions in generated by . The Hardy space () is -admissible.

\par
Proof.

Let be conformal. Then there exists a semiflow in such that . By [9, Cor. to Thm. 10.1], if and only if). This and Littlewood’s subordination principle gives invariance since

Without loss of generality, we assume that . Then, by [8, p. 168], we have

\par

\par
Remark 2.8.

If we were using the definition of Hardy spaces by approximating level curves (sometimes called Hardy-Smirnov spaces), the last proof would involve boundary values of conformal maps. This would have forced us to prescribe conditions concerning the boundary of . Therefore it seems more appropriate to define Hardy spaces via harmonic majorants.

Proposition 2.9.

Let be a Jordan domain. Let be a semiflow of holomorphic functions in generated by . The Bergman space () is -admissible.

\par
Proof.

Let be conformal. Then there exists a semiflow in such that . Thus, for ,

The derivative of is non-vanishing in , see [16, Thm. 6.8]. \parFor invariance, we only need to show that . Indeed,

Now Littlewood’s subordination principle yields invariance. \parBy the same calculation, we obtain strong continuity of on from strong continuity on . ∎

Further examples of holomorphic function spaces on the unit disk which appear in the literature concerning semigroups of composition operators are the Bloch space and the space BMOA as well as their subspaces and VMOA. On these spaces the question of strong continuity is much more delicate, and in fact there is no nontrivial strongly continuous semigroup on and BMOA. So in these cases, one is studying so-called maximal subspaces of strong continuity denoted by and such that a given semiflow defines a strongly continuous semigroup of composition operators on resp. . In [5] it has been shown that , and in the recent paper [2] the analogous result for BMOA has been obtained, that is, \parIt is also natural to consider weighted semigroups of composition operators. Let be simply connected. Let be holomorphic. For we define a weight as follows

(2.5)

For a family of composition operators on with semiflow , we define a family of weighted composition operators as follows \par

(2.6)

This is again an operator semigroup on and also on but the question of strong continuity is more difficult since it depends heavily on the choice of . \parSpecial weights we are interested in are so-called cocycles.

Definition 2.10.

Let be a semiflow in . A family of holomorphic functions is called cocycle if

  1. ,

  2. for all and ,

  3. is continuous for every .

\par\par

If there exists a holomorphic function such that then the family is called a coboundary of . \parIt is easy to see that a family of cocycle weighted composition operators is also an operator semigroup on . Moreover, given an arbitrary holomorphic function , we can easily construct a cocycle to a semiflow : for , \par

(2.7)

is a cocycle.

Definition 2.11.

Let be a weighted semigroup of composition operators on , cf. (\@setrefeq:WSG), with semiflow generated by the holomorphic function and cocycle weight in terms of a holomorphic function , see (\@setrefeq:cocy) . A Banach space is called -admissible if it satisfies the following two conditions:

  1. is invariant under i.e., for all .

  2. is strongly continuous on .

Let be -admissible. Then the generator of is given by

\par\par\@xsect\par

In [13, Theorem 2] it has been shown that for certain holomorphic functions and their associated cocycles as in (\@setrefeq:cocy), and a semiflow generated by ,the Hardy space is -admissible in the sense of Definition \@setrefdef:admsibble. By a slight adjustment of the arguments in Proposition \@setrefprop:SGHardy, we obtain the result for Hardy spaces over simply connected sets.

Lemma 2.12.

Let be simply connected. Let be a holomorphic function such that , and let be a semiflow in with generator . Then is admissible.

\par
Proof.

Invariance follows by boundedness of and Proposition \@setrefprop:SGHardy. To show strong continuity, we use the same technique as in Proposition \@setrefprop:SGHardy, too. Since the real part of is bounded as well, we obtain the assertion from [18, Theorem 1]. ∎ Indeed, the proof of [18, Theorem 1] works as well for a family of -weighted composition operators on the Bergman space , where is a Jordan domain. \par

Lemma 2.13.

Let be a Jordan domain. Let be a holomorphic function such that , and let be a semiflow in with generator . Then is -admissible.

\par
Proof.

It suffices to prove the statement for and then apply the same technique as in Proposition \@setrefprop:SGBergman. Due to Siskakis [18, Theorem 1], strong continuity for a weighted SGCO on is achieved if which is satisfied by our assumptions on , see [13, Lemma 3.1]. To prove the assertion, we can simply follow the steps in the proof of [18, Theorem 1].
For all we have . This and the cocycle properties yield that defines a family of bounded operators on . Let . By Littlewood’s subordination principle we get that

(2.8)

thus for all .
First, we prove strong continuity if . Let be a sequence such that . Then we have . Since is reflexive and by (\@setrefeq:LSP), after passing to a subsequence again denoted by , the sequence is weakly convergent. The weak limit is because for all . By lower-semicontinuity of the norm, , and thus . This yields the desired strong continuity.
To show strong continuity in the case we use that ( is dense in . Let . For every there exists such that . Moreover,

Since , for all there exists a sufficiently small such that Thus as .

Remark 2.14.

Several authors are especially interested in semigroups of composition operators weighted by the derivative of the semiflow with respect to the complex varibale, i.e.,

See for example the recent paper [4]. \parIndeed, this weight is a cocycle given by

\par\par\@xsect\par

Finding boundary values of holomorphic functions is a fundamental problem in function theory. Strong results concerning the boundary values of functions in Hardy spaces are Fatou’s theorem and the theorem by F. and M. Riesz. But, in many spaces of holomorphic functions, convergence to boundary values in a nontangential sense is a rather strong condition. Therefore we consider boundary values in a weaker sense, namly in the sense of distributions. \parLet be a Jordan domain. This restriction guarantees existence and nonvanishing of boundary values of derivatives of conformal maps defined on . Up to now, we are not sure if the established theory works for rectifiable boundaries as well. \parIn what follows, we are exploring boundary distributions of functions in Banach spaces . Our first aim is to define the boundary space of consisting of appropriately defined distributional boundary values of elements of .

Definition 2.15.

Let be a Jordan domain. Let be a Banach space. If for every there exists a uniquely defined boundary distribution in the following sense

for every , where , and is any conformal map, then we denote the set consisting of all such boundary values by . If there exists an isomorphism , then is called the boundary space corresponding to . Moreover, we define a norm on by for every .

\par\par\@xsect\par

A first (though artificial) example is the space where denotes the disk algebra. The restriction to the boundary is an isometric homomorphism from into . So is a Banach subalgebra of which is even maximal due to Wermer’s maximality theorem. Thus the boundary space can be defined as the space of continuous functions on which are holomorphically extendable to .
Let and define as the Hardy space . Then it is well known that every function in has nontangential limits a.e. and the boundary function is in . For a comprehensive overview, we refer especially to [9, Chapter 3]. These boundary functions form a closed subspace of ) which consists of those function in with vanishing negative Fourier coefficients. Note that this theory is almost applicable when the analogously defined Hardy space of harmonic functions is considered. However, the case appears to be different. The boundary space on consists of finite Borel measures on the unit circle. \parIn both examples, the boundary space inherits some properties of the underlying space of holomorphic functions. Moreover, by the Luzin-Privalov theorem, a holomorphic function is in either case identically zero if the boundary function vanishes on a set of positive measure. Given a function in one of the two boundary spaces from the examples above, we can recover the holomorphic function in via Cauchy’s integral formula and the Poisson integral as well which acts as an isometric isomorphism between and . \par\par\@xsect \parThe theory of boundary values for functions in Hardy spaces on the unit disc is well established. The question of boundary functions is much more complicated if one wishes to work on Bergman spaces. In fact, the Bergman spaces contain functions which do not admit nontangential or radial limits almost everywhere, such as the Lacunary series. So it seems more appropriate to define boundary values in the sense of distributions. To establish such distributional boundary values, we emphasize a connection between Hardy and Bergman spaces. For simplicity we use the notation and . The following theorem can be found in [8, Lem. 4].

Theorem 2.16.

If and is an antiderivative of , then .

\par

For , Theorem \@setrefthm:.p=00003D00003D1 can be generalized to in the following way.

Theorem 2.17.

Let and an antiderivative of . Then .

\par
Proof.

Let . Then we have

To estimate , we examine the following two integrals

For the first term we have

Without loss of generality, we assume . Thus we obtain for the second integral

Combining these results, we have

Letting , the right-hand side is still finite since can be chosen arbitrarily in . ∎

This theorem remains true if we replace by a Jordan domain .

Corollary 2.18.

Theorem \@setrefthm:anti remains true if is replaced by a Jordan domain . \par

Proof.

By [9, Cor. to Thm. 10.1], it is enough to show that ) for some conformal mapping . Therefore, one can mostly copy the proof of Theorem \@setrefthm:anti, noting that for one has . \parThe derivative of does not vanish in , and so we have

It remains to show that :

\par

∎ Now we can define distributional boundary values for Bergman functions.

Theorem 2.19.

Let . Every function admits a distributional boundary value in , the dual space of ), where is the usual conjugate exponent of \par

\par
Proof.

Let . We denote by the antiderivative of , so we obtain

This limit exists by using Theorem \@setrefthm:anti, Hölder’s inequality, and the dominated convergence theorem. ∎

Corollary 2.20.

Let be Jordan domain. Then every function admits a distributional boundary value in .

\par
Proof.

Let , and let be conformal. For we define as usual Thus as . Then

(2.9)

It is easy to show that and . Since is a conformal, we also have . So, by [16, Thm 6.8], we obtain convergence of the integral (\@setrefeq:dbvo) as . ∎ Distributional boundary values of harmonic and holomorphic functions defined on a simply connected domain with smooth boundary have been studied in [19]. There it has been shown that a holomorphic function admits a distributional boundary value if and only if it lies in the Sobolev space for some , see [19, Thm. 1.3]. Moreover, by [19, Cor. 1.7], for all the map defined by

where is the Poisson kernel for , is an isomorphism. The inverse is given by assigning the distributional boundary value to a given function. Thus, functions in are uniquely determined by their boundary distributions. Therefore, restricting the map to the boundary space for some , we can recover each function in using the Poisson operator. \par\par\@xsect \parIn this section we work out our main result, the connection between partial differential equations on the boundary associated with Poincaré-Steklov operators and semigroups of composition operators on Banach spaces of holomorphic functions. \par\par\@xsect \parLet be a ’nice’ function and consider the following elliptic equation \par

(3.1)

The Dirichlet-to-Neumann operator maps the function to the Neumann derivative of the solution of (\@setrefeq:DP.D) provided that a solution exists and is sufficiently regular. As it is shown by Lax [14], if or in