Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space C(∂M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {C}(\partial M)$$\end{document} of continuous functions on the boundary ∂M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial M$$\end{document} of a compact manifold M¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{M}$$\end{document} with boundary. We prove that it generates an analytic semigroup of angle π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\pi }{2}$$\end{document}, generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\pi }{2}$$\end{document} on the space C(M¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {C}(\overline{M})$$\end{document}.


Introduction
Differential operators with dynamic boundary conditions on manifolds with boundary describe a system whose dynamics consisting of two parts: a dynamics on the manifold interacting with an additional dynamics on the boundary. This leads to differential operators with so called Wentzell boundary conditions, see [17,Sect. 2].
On spaces of continuous functions on domains in R n such operators have first been studied systematically by Wentzell [39] and Feller [26]. Later Arendt et al. [8] proved that the Laplace operator with Wentzell boundary conditions generates a positive, contractive C 0 -semigroup. Engel [20] improves this by showing that this semigroup is analytic with angle of analyticity π 2 . Later Engel and Fragnelli [17] generalize this result to uniformly elliptic operators, however without specifying the corresponding angle of analyticity. For related work see also [11][12][13][14]22,23,38,39] and the references therein. Our interest in this context is the generation of an analytic semigroup with the optimal angle of analyticity.
As shown in [9,17] this problem is closely connected to the generation of an analytic semigroup by the Dirichlet-to-Neumann operator on the boundary space. More precisely, based on the abstract theory for boundary perturbation problems developed by Greiner [25], it has been shown in [9,17] that the coupled dynamics can be decomposed into two independent parts: a dynamics on the interior and a dynamics on the boundary. The first one is described by the differential operator on the manifold with Dirichlet boundary conditions while the second is governed by the associated Dirichlet-to-Neumann operator.
On domains in R n the generator property of differential operators with Dirichlet boundary conditions is quite well understood, see [7,29,32]. On compact Riemannian manifolds with boundary it has been shown in [10] that strictly elliptic operators with Dirichlet boundary conditions are sectorial of angle π 2 and have compact resolvents on the space of continuous functions.
Dirichlet-to-Neumann operators have been studied e.g. by [30,32,37] and [36,App. C]. For the operator-theoretic context see, e.g., the work of Amann and Escher [4] and Arendt and ter Elst [5][6][7]. In particular, on domains in R n Escher [19] has shown that such Dirichlet-to-Neumann operators generate analytic semigroups on the space of continuous functions, however without specifying the corresponding angle of analyticity. Finally, ter Elst and Ouhabaz [16] proved that this angle is π 2 and extended the result of Escher [19] to differential operators with less regular coefficients.
In this paper we study such Dirichlet-to-Neumann operators on the space of continuous functions on Riemannian manifolds and show that they generate compact and analytic semigroups of angle π 2 on the continuous functions. We first explain our setting and terminology. Consider a strictly elliptic differential operator A m : D(A m ) ⊂ C(M) → C(M), as given in (4.3), on the space C(M) of continuous functions on a smooth, compact, orientable Riemannian manifold M with smooth boundary ∂ M. Moreover, let ∂ a ∂ν g : D( ∂ a ∂ν g ) ⊂ C(M) → C(∂ M) be the outer conormal derivative, β > 0 and γ ∈ C(∂ M). We consider B : (4.4), and define the operator A B f := A m f with Wentzell boundary conditions by requiring For a continuous function ϕ ∈ C(∂ M) on the boundary the corresponding Dirichlet problem That is, N ϕ is obtained by applying the Neumann boundary operator −β ∂ a ∂ν g to the solution f of the Dirichlet problem (1.2).
Our main results are the following. This extends the results from Escher [19] and Cor. 4.5] to elliptic operators on compact manifolds with boundaries and gives the maximal angle of analyticity π 2 in both cases. In the flat case the result for the Dirichlet-to-Neumann operator coincides with the result of ter Elst-Ouhabaz [16] in the smooth case. The techniques here are different and our proof is independent from theirs. This paper is organized as follows. In Sect. 2 below we recall the abstract setting from [9,17] needed for our approach. Based on [20,Sect. 2], we study in Sect. 3 the special case where A m is the Laplace-Beltrami operator and B the normal derivative. In Sect. 4 we then generalize these results to arbitrary strictly elliptic operators and their conormal derivatives. Moreover, we use this to obtain uniqueness, existence and estimates for the solutions of the Robin-Problem. Here the main idea is to introduce a new Riemannian metric induced by the coefficients of the second order part of the elliptic operator. Then the operator takes a simpler form: Up to a relatively bounded perturbation of bound 0, it coincides with a Laplace-Beltrami operator for the new metric. Regularity and perturbation theory for operator semigroups as in [9,Sect. 4] then yield the first part of the main theorem in its full generality. The second part follows from [17,Thm. 3.1] and [10, Thm. 1.1]. In the Appendix we collect some results about elliptic operators on manifolds with boundary.
In this paper the following notation is used. For a closed operator T : D(T ) ⊂ X → X on a Banach space X we denote by [D(T )] the Banach space D(T ) equipped with the graph norm • T := • X + T (•) X and indicate by → a continuous and by c → a compact embedding. Moreover, we use Einstein's notation of sums, i.e.,

The abstract setting
The starting point of our investigation is the abstract setting proposed first in this form by [25] and successfully used, e.g., in [11,12,17] for the study of boundary perturbations.

Abstract Setting 2.1 Consider
(i) two Banach spaces X and ∂ X , called state and boundary space, respectively; (ii) a densely defined maximal operator A m : D(A m ) ⊂ X → X ; (iii) a boundary (or trace) operator L ∈ L(X , ∂ X ); (iv) a feedback operator B : D(B) ⊆ X → ∂ X .
Using these spaces and operators we define the operator A B : D(A B ) ⊂ X → X with generalized Wentzell boundary conditions by For our purpose we need some more operators.

Notation 2.2
We denote the (closed) kernel of L by X 0 := ker(L) and consider the restriction A 0 of A m given by The abstract Dirichlet operator associated with A m is, if it exists, i.e., L A m 0 ϕ = f is equal to the solution of the abstract Dirichlet problem If it is clear which operator A m is meant, we simply write L 0 . Moreover for λ ∈ C we define the abstract Robin operator associated with (λ, A m , B) by i.e., R λ ϕ = f is equal to the solution of the abstract Robin problem If it is clear which operators A m and B are meant, we simply write R λ .

T. Binz
Furthermore, we introduce the abstract Dirichlet-to-Neumann operator associated with (A m , B) defined by If it is clear which operators A m and B are meant, we call N simply the (abstract) Dirichlet-to-Neumann operator. This Dirichlet-to-Neumann operator is an abstract version of the operators studied in many places, e.g., [19], [36,Sect. 7.11] and [35, Sect. II.5.1].
The Dirichlet-to-Neumann and the Robin operator are connected in the following way.
This again is equivalent to is an isomorphism, there exists a unique ϕ ∈ D(N ) for every ψ ∈ ∂ X . Moreover its given by ϕ = −L R λ,μ ψ and therefore the boundedness of the inverse follows from the boundedness of L and R λ . The formula for the resolvent of N follows, since L| ker(A m ) is an isomorphism with inverse L 0 and the image of R λ is contained in ker(A m ).

Boundary problems for the Laplace-Beltrami operator
In order to obtain a concrete realization of the above abstract objects we consider a smooth, compact, orientable Riemannian manifold (M, g) with smooth boundary ∂ M, where g denotes the Riemannian metric. Moreover, we take the Banach spaces X := C(M) and ∂ X = C(∂ M) and as the maximal operator the Laplace-Beltrami operator (3.1) As feedback operator we take the normal derivative denotes the gradient on M, which in local coordinates is given as Moreover, ν g is the outer normal on ∂ M given in local coordinates by ν l g = g kl ν k . Furthermore, we choose L as the trace operator, i.e., which is bounded with respect to the supremum norm. Later on we will also need the unique bounded extension of L to W 1,2 (M), denoted by L : W 1,2 (M) → L 2 (∂ M), and call it the (generalized) trace operator.

The Laplace-Beltrami operator with Robin boundary conditions
In this setting we consider the Laplace-Beltrami operator with Robin boundary conditions and prove existence, uniqueness and regularity for the solution of (2.3). Moreover, we show that this solution satisfies a maximum principle.
For this purpose we need the concept of a weak solution of (2 . This motivates the following definition. Next we prove the existence of such weak solutions. Proof We consider a and F as defined above. Obviously a is sesquilinear and F is linear. By the Cauchy-Schwarz Inequality we have for f , φ ∈ W 1,2 (M) that and remark that v k W 1,2 (M) = 1 and therefore Next we prove that every weak solution is even a strong solution.

Lemma 3.4 (Regularity of the Robin problem)
. Therefore, we obtain by the fundamental lemma of the calculus of variation that Moreover we need a maximum principle for the Robin problem.
Proof We consider a point p ∈ M, where | f | and therefore | f | 2 assumes its maximum. By the interior maximum principle (cf. Theorem A.1) it follows that q ∈ ∂ M. Hence, we have Since Re(λ) ≥ 0, this implies Summing up we obtain the following.

Generator property for the Dirichlet-to-Neumann operator
Now we are able to prove our main result: The Dirichlet-to-Neumann operator generates a contractive and analytic semigroup of angle π 2 on ∂ X = C(∂ M). To do so we represent the Dirichlet-to-Neumann operator as a relatively bounded perturbation of We first need the existence of the associated Dirichlet operator. Next we prove a first generation result for the Dirichlet-to-Neumann operator.
Since C 2 (∂ M) is dense in ∂ X , N is densely defined. By Lemma 2.3 and Corollary 3.6 it follows that the resolvent R(λ, N ) exists for all Re(λ) > 0. By the interior maximum principle L| ker(A m ) : ker(A m ) ⊂ X → ∂ X is an isometry. Therefore, Lemmas 2.3 and 3.5 imply generates an analytic semigroup of angle π 2 on ∂ X.
We proceed as in the proof of [20,Thm. 2.1]. Let N and W be the closure of N and W , respectively, in Y := L 2 (∂ M). Moreover we need results from the theory of pseudo differential operators. We use the notation from [35] and denote by OPS k (∂ M) the pseudo differential operators of order k ∈ Z on ∂ M.
Step 1 Then the part N | ∂ X coincides with N .
Proof By Proposition 3.8 the Dirichlet-to-Neumann operator N is densely defined and λ − N , considered as an operator on Y , has dense range rg(λ − N ) = ∂ X ⊂ Y for all λ > 0. By Green's Identity we have Hence, for f :   ) is a core for W . Hence, C ∞ (∂ M) is a core for W and since C ∞ (∂ M) ⊂ D(W ) we obtain that D(W ) is a core for W on Y . This implies that W is indeed the closure of W in Y . Moreover, we obtain where 1 − W is surjective and 1 − W is injective on ∂ X . This is possible only if for the domains we have for p > n−1 1−α , the claim follows.
Step 5 The difference P := N − W ∈ OPS 0 (∂ M) is a pseudo differential operator of order 0. Moreover, P considered as an operator on Y is bounded. Therefore, by Ehrling's lemma (cf. [34,Thm. 6.99]), for every ε > 0 there exists a constant C ε > 0 such that

Proof
for every ϕ ∈ D(W ), i.e. P is relatively W -bounded with bound 0.
Step 7 (Proof of Theorem 3.9) Proof First we note that by Step 5 we have and therefore using the Steps 1, 3, 6 it follows that On the other hand, by Steps 2, 6 and [18, Lem. III.2.6], W − P generates an analytic semigroup of angle π 2 on ∂ X . Moreover, λ ∈ ρ(N ) ∩ ρ(W − P) for λ large enough. This implies equality in (3.4) and hence the claim.

Remark 1
After we finished this paper, a different proof of Theorem 3.9 came to our mind, based on the work of ter Elst and Ouhabaz [15].

The Laplace-Beltrami operator with Wentzell boundary conditions
In this subsection we study the Laplace-Beltrami operator with Wentzell boundary conditions and prove that it generates an analytic semigroup of angle π 2 on X = C(M). To show this, we verify the assumptions of [ Therefore, by Ehrling's lemma (cf. [34,Thm. 6.99]), for every ε > 0 there exists a constant C ε > 0 such that 1) satisfying the strict ellipticity condition for all co-vectorfields X k , X l on M with (X 1 (q), . . . , X n (q)) = (0, . . . , 0). Further, denote |a| := det(a k j ). Then we define the maximal operator in divergence form as As feedback operator we takẽ The key idea is to reduce the strictly elliptic operator and the conormal derivative on M, equipped by g, to the Laplace-Beltrami operator and to the normal derivative on M, endowed by a new metricg.
For this purpose we consider a (2, 0)-tensorfield on M given bỹ Its inverseg is a (0, 2)-tensorfield on M, which is a Riemannian metric since a k j g jl is strictly elliptic on M. We denote M with the old metric by (M, g) and with the new metric by (M,g) and remark that (M,g) is a smooth, compact, orientable Riemannian manifold with smooth boundary ∂ M. Since the differentiable structures of (M, g) and Since b l ∈ C c (M) we obtain follows fromÂ m f =Ã m f − P f . By (4.6) we conclude from Ehrling's Lemma (see [34,Thm. 6.99]) that for f ∈ D(Â m ) and all ε > 0 and hence P 1 is relatively A m -bounded of bound 0. Finally, remark that is bounded and thatÃ Proof We have to verify this equality for all q ∈ M. Choose local coordinates around q ∈ M and calculateÂ

Lemma 4.3 The operators B andB differ only by a bounded perturbation.
Proof Note that the Sobolev spaces coincide. We have to verify this equality for all q ∈ M. Choose local coordinates around q ∈ M and computẽ Proof The claim follows by Theorem 3.9 and Lemma 4.4.

Remark 4
As in Remark 2 we can insert a strictly positive, continuous function β > 0 and the same result as Theorem 3.9 becomes true.
Remark 5 Theorem 4.5 improves and generalizes the main result in [19]. If we consider M = ⊂ R n equipped with the euclidean metric g = δ, we obtain the maximal angle π 2 of analyticity in this case. This is the main result in [16] for smooth coefficients.
Now we use Theorem 4.5 to obtain existence and uniqueness for the associated Robin problem (2.3). Moreover, we obtain a maximum principle for the solutions of these problems.
Proof The existence and uniqueness follows immediately by Theorem 4.5. The first inequality is the interior maximum principle. The second inequality is a direct consequence from Lemma 2.3 and Theorem 4.5.  Proof The claim follows by Theorem 3.12 and Lemma 4.7.

Remark 6
As in Remark 2 we can insert a strictly positive, continuous function β > 0 and the same result as Theorem 4.8 becomes true.
Remark 7 Theorem 4.8 improves and generalizes [17,Cor. 4.5]. If we consider M = ⊂ R n equipped with the euclidean metric g = δ, we obtain the maximal angle π for i ∈ {1, . . . , r }. Note that since φ i are smooth diffeomorphisms and the operators A i are strictly elliptic, we obtain that the operatorsÃ i are elliptic. Thus our strictly elliptic partial differential equation (A.3) is the collection of finitely many strictly elliptic differential equations (A.4) on bounded sets V i with piese-wise smooth boundary. Since every (second-countable) manifold (with boundary) admits an adequate atlas, we can even choose V i = B 1 (0) + := {x ∈ B 1 (0) : x n ≥ 0} ⊂ R n + . Further, let f + be the positive part of f . Since χ is positive, we obtain f + i = χ i · f + . The following weak maximum principle is a direct consequence of [ We conclude Using u = r i=1 f i and h = r i=1 f i the claim follows.
Since the proofs of the a priori bounds (cf. [24, Thm. 9.11, Thm. 9.13 & Lem. 9.16]) use localization techniques they can be easily generalized to manifolds with boundary. We obtain the following result. admits a unique solution f i =f i • φ ∈ W 2,q (U i ) and the claim follows from f = r i=1 f i .
Using the same approximation technique as [24, Section 9.6] we conclude the following statement. The differences ( f l − f k ) satisfies Now the a priori bounds (Theorem A.2) and the maximum principle (Theorem A.1) yields for all compact subsets K ⊂ M. Hence ( f l ) l∈N converges in C( ) ∩ W