A FRACTIONAL DIRICHLET-TO-NEUMANN OPERATOR ON BOUNDED LIPSCHITZ DOMAINS

. Let Ω ⊂ R N be a bounded open set with Lipschitz continuous boundary ∂ Ω. We deﬁne a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on L 2 ( ∂ Ω) which can also be ultracontractive. We also use the fractional Dirichlet-to-Neumann operator to compare the eigenvalues of a realization in L 2 (Ω) of the fractional Laplace operator with Dirichlet boundary condition and the regional fractional Laplacian with the fractional Neumann boundary conditions.


Introduction.
Let Ω ⊂ R N be a bounded domain with Lipschitz continuous boundary ∂Ω. Given g ∈ L 2 (∂Ω), let u ∈ W 1,2 (Ω) be the solution of the Dirichlet problem ∆u = 0 in Ω, u = g on ∂Ω. (1.1) The operator D 1,0 defined on L 2 (∂Ω) by      D(D 1,0 ) = {g ∈ L 2 (∂Ω), ∃ u ∈ W 1,2 (Ω) solution of (1.1) and ∂ ν u exists in L 2 (∂Ω)}, is called the Dirichlet-to-Neumann operator. Here, ∂ ν u denotes the normal derivative of the function u in direction of the outer normal vector ν. The Dirichletto-Neumann operator is well-known and has been studied by several authors (see e.g. [3,4,5,12,16,26] and their references). Some properties of the operator D 1,0 have been used in [3] to give another proof on Lipschitz domains of the Friedlander's result [16] (initially proved on smooth domains) that is, λ N n+1 ≤ λ D n for all n ∈ N, where λ N n+1 is the (n + 1) th -eigenvalue of the Neumann Laplace operator and λ D n is the n th -eigenvalue of the Dirichlet Laplacian. The Dirichlet-to-Neumann operator has been also defined on very rough domains in [4] by using the method of bilinear forms, where the authors have shown that −D 1,0 generates a strongly continuous Markov semigroup on L 2 (∂Ω) and the asymptotic behavior of this semigroup is related to properties of the trace of functions in the first order Sobolev space W 1,2 (Ω). Other generation of semigroups results are contained in [3,12,17,26].
More precisely, heat kernel estimates for the classical Dirichlet-to-Neumann operator and fractional Laplacians (but not the fractional Dirichlet-to-Neumann) have been obtained in [17,26].
The main concerns of the present paper are the following: • Find a right definition of a fractional Dirichlet-to-Neumann type operator associated with the fractional Laplace operator, study its spectral properties and some generation of semigroup results. • Use the spectral properties of the fractional Dirichlet-to-Neumann operator to obtain the Friedlander's type result for the fractional Laplace operator on bounded domains in R N with Lipschitz continuous boundary. That is, for 0 < s < 1, we would like to show that λ N n+1,s ≤ λ D n,s for all n ∈ N, where λ N n+1,s is the (n + 1) th -eigenvalue of the realization in L 2 (Ω) of the fractional Laplace operator with fractional Neumann type boundary conditions and λ D n,s denotes the n th -eigenvalue of a realization in L 2 (Ω) of the fractional Laplace operator with the Dirichlet boundary condition. For the convenience of the reader and in order to make the paper as self-contained as possible, we start by introducing the fractional Laplace operator which is not so familiar as the well-known Laplace operator. For 0 < s < 1 and Ω ⊂ R N an arbitrary open set, we let  provided that the limit exists. Most recently, Caffarelli et al. [8,9] have deeply studied the operator (−∆) s on R N and on subsets of R N with the Dirichlet boundary conditions. Some fundamental and beautiful results have been obtained. Some semi-linear problems involving (−∆) s with Dirichlet boundary condition have been also investigated in [8,9,23] and the references therein. The fractional Laplacian and fractional derivative operators are commonly used to model anomalous diffusion. Physical phenomena exhibiting this property cannot be modeled accurately by the usual advection-dispersion equation; among others, we mention turbulent flows and chaotic dynamics of classical conservative systems. The operator (−∆) s is the most typical non-local operator. Further properties and applications of the fractional Laplace operator and more general non-local operators are contained in [15] and the references therein.
As we have mentioned above, the aim of the present paper is to find a right definition of a fractional Dirichlet-to-Neumann type operator. We have the following situation. Let g ∈ C(R N ) be a given function and Ω ⊂ R N a bounded domain with Lipschitz continuous boundary ∂Ω. It is well known (see e.g. [20] and their references) that the following Dirichlet type problem is not well-posed. The well-posed Dirichlet problem for the fractional Laplace operator is given by Since the problem (1.5) is not well-posed, compare with (1.1), the operator (−∆) s is not suitable to obtain a fractional Dirichlet-to-Neumann operator. We have to consider an operator that depends on the domain Ω. We proceed as follows. Let Ω ⊂ R N be an arbitrary open set. We restrict the integral kernel in (1.4) to the open set Ω. For u ∈ L 1 (Ω), x ∈ Ω and ε > 0, we let and we define the operator A s Ω as follows: provided that the limit exists. In [18,19] the operator A s Ω has been called the regional fractional Laplacian. We have the following. Let u ∈ D(Ω). Since u = 0 on R N \ Ω, a simple calculation gives, (1.6) where the potential V Ω is given for every x ∈ Ω by More details and the relation between A s Ω and (−∆) s , and some useful properties of the potential V Ω can be found in [6,14] and their references. Now, let Ω ⊂ R N be a bounded domain with Lipschitz continuous boundary ∂Ω. Given a function g ∈ C(∂Ω), it has been shown in [19] that the Dirichlet problem for the regional fractional Laplacian, is well-posed. Therefore, to define a fractional Dirichlet-to-Neumann type operator, it makes sense to consider the regional fractional Laplacian. Let A D and A N be the realization in L 2 (Ω) of the operator A s Ω with Dirichlet and fractional Neumann boundary conditions, respectively, that is, where N 2−2s u denotes the fractional normal derivative of the function u (see (2.11) and (2.13) below). We refer to Section 2.2 below for more details. We notice that if 0 < s ≤ 1 2 , then the operators A D and A N coincide. That is, if 0 < s ≤ 1 2 , then D(A D ) = D(A N ) and A D u = A N u for every u ∈ D(A D ) = D(A N ). Therefore without any restriction we may assume that 1 2 < s < 1. Let σ(A D ) denote the spectrum (which is discrete) of A D . Let λ ∈ R \ σ(A D ). Then using [19], we have that given g ∈ L 2 (∂Ω), there exists u ∈ W s,2 (Ω) solution of the Dirichlet problem (1.8) Next, let 1 2 < s < 1 and λ ∈ R \ σ(A D ). As in (1.2) for the Laplace operator, the fractional Dirichlet-to-Neumann operator D s,λ is defined on L 2 (∂Ω) by (1.8) and where C s is an explicit normalized constant (see (2.8) below). We show in Section 3 below that D s,λ is well defined, is associated with a bilinear symmetric, continuous and elliptic form and has also a compact resolvent. Further spectral properties of the operator D s,λ and some generation of semigroup results have been investigated. More precisely, we obtain that for every λ ∈ R\σ(A D ), the operator −D s,λ generates a strongly continuous analytic and compact semigroup on L 2 (∂Ω) satisfying the following: • If λ < λ D 1,s , then the semigroup is positive. • If λ ≤ 0, then the semigroup is also submarkovian and ultracontractive.
• If λ = 0, then the semigroup is in addition Markovian.
Using the spectral properties of D s,λ , we prove in Section 4 the Friedlander type result for the regional fractional Laplace operator, that is, λ N n+1,s ≤ λ D n,s for all n ∈ N.
The rest of the paper is organized as follows. In Section 2 we introduce the function spaces needed to investigate our problem and we prove some intermediate results on the regional fractional Laplace operator A s Ω as they are needed throughout the paper. The fractional Dirichlet-to-Neumann operator is introduced in Section 3 where we also show that it has a compact resolvent and generates a strongly continuous analytic semigroup which can also be ultracontractive. Finally in Section 4 we use the fractional Dirichlet-to-Neumann operator to compare the eigenvalues of the regional fractional Laplacian with Dirichlet boundary condition and the regional fractional Laplace operator with fractional Neumann boundary conditions. 2. Intermediate results. In this section we introduce the function spaces needed to investigate our problem and we prove some intermediate results that will be used to obtain our main results. Ω Ω |u(x) − u(y)| 2 |x − y| N +2s dx dy < ∞} the fractional order Sobolev space endowed with the norm We let .
We have the following result taken from [6, Corollary 2.8 and Remark 2.3].
Theorem 2.1. Let Ω ⊂ R N be a bounded domain with a Lipschitz continuous boundary. Then for every 0 < s ≤ 1 2 , the spaces W s,2 (Ω) and W s,2 0 (Ω) coincide with equivalent norms.
In view of Theorem 2.1, the characterization (2.2) and [27], we have that if 0 < s ≤ 1 2 and Ω has a Lipschitz continuous boundary, then every u ∈ W s,2 (Ω) is zero quasi-everywhere and σ-a.e. on ∂Ω. Therefore, to talk about traces (not necessarily zero) of functions in W s,2 (Ω), it is not a restriction to assume that 1 2 < s < 1. We have the following result taken from [10,14]. Theorem 2.2. Let 1 2 < s < 1 and Ω ⊂ R N a bounded domain with a Lipschitz continuous boundary ∂Ω. Then the following assertions hold.
The following theorem gives another equivalent norm for the space W s,2 (Ω). Theorem 2.3. Let Ω ⊂ R N be a bounded domain with a Lipschitz continuous boundary ∂Ω and 1 2 < s < 1. Then there exists a constant C = C(Ω, N, s) > 0 such that for every u ∈ W s,2 (Ω), Proof. Let 1 2 < s < 1. It suffices to show that defines an equivalent norm on W s,2 (Ω). Since 2 : To prove the converse inequality, we proceed by contradiction. Assume that for every n ∈ N, there exists a sequence u n ∈ W s,2 (Ω) such that By possibly dividing (2.6) by u n 2 L 2 (Ω) we may assume that u n 2 L 2 (Ω) = 1 for any n. Hence, by (2.6), u n is a bounded sequence in the Hilbert space W s,2 (Ω). Therefore, after passing to a subsequence, if necessary, we may assume that u n converges weakly to some u ∈ W s,2 (Ω) and strongly to u in L 2 (Ω) (since the embedding W s,2 (Ω) → L 2 (Ω) is compact by Theorem 2.2). Moreover, u n | ∂Ω converges strongly to u| ∂Ω in L 2 (∂Ω) (since the embedding W s,2 (Ω) → L 2 (∂Ω) is compact by Theorem 2.2). It follows from (2.6) that This implies that u n | ∂Ω converges strongly to 0 in L 2 (∂Ω) and (after passing to a subsequence, if necessary) Using (2.7) and the fact that (after passing to a subsequence, if necessary) u n converges a.e. to u in Ω, we get that lim n→∞ u n (x) = lim n→∞ u n (y) for a.e. x, y ∈ Ω. Hence, u n converges almost everywhere to some constant function c. The uniqueness of the limit and the uniqueness of the trace (since u n | ∂Ω converges to 0 on ∂Ω after passing to a subsequence, if necessary) imply that c = u = 0 a.e. on Ω. On the other hand, we have u 2 L 2 (Ω) = lim n→∞ u n 2 L 2 (Ω) = 1, and this is a contradiction. The proof is finished.
For more information on the fractional order Sobolev spaces we refer to [1,10,14,27] and their references. 2.2. The regional fractional Laplacian with various boundary conditions. Before we introduce various realizations of the regional fractional Laplace operator, we give a Green type formula. Let Ω ⊂ R N be a bounded domain of class C 1,1 with boundary ∂Ω. Let 1 2 < s < 1 and the constant where C 1,s is given by (1.3) with N = 1. Let the constant B N,s be such that Remark 1. We notice that a simple calculation gives that , (2.10) where B denotes the usual beta function. Replacing (2.10) into (2.9), we get that B N,s = C s , so that it is independent of N and depends on s only.
The following integration by parts formula has been recently obtained by Guan [18,Theorem 3.3].
Comparing (2.12) and the classical Green formula for the Laplace operator, we get that the function C s N 2−2s u plays the role (for the regional fractional Laplace operator) that the normal derivative ∂ ν u does for the Laplace operator.
Next, we introduce a weak formulation on non-smooth domains of a fractional normal derivative.
(a) Let u ∈ W s,2 (Ω). We say that A s for all v ∈ D(Ω) and hence, for all v ∈ W s,2 0 (Ω) by density. In that case we , hence for all v ∈ W s,2 (Ω) by density and by using (2.3). In that case, the function g is uniquely determined by (2.13), we write C s N 2−2s u = g and call g the fractional normal derivative of u.
Throughout the remainder of the article without any mention, Ω ⊂ R N denotes a bounded domain with Lipschitz continuous boundary ∂Ω. Let 1 2 < s < 1 and µ ∈ R. We define the bilinear symmetric form E µ with domain W s,2 (Ω) by We have the following result.
Then the bilinear symmetric form E µ is continuous and elliptic.
Proof. Let µ ∈ R. It is easy to see that the form E µ is continuous. We show that it is elliptic. We notice that (2.3) together with the fact the embedding W s,2 (Ω) → L 2 (Ω) is compact, imply that there exists a constant C > 0 such that for every u ∈ W s,2 (Ω), Therefore, for every u ∈ W s,2 (Ω), We have shown that E µ is elliptic and the proof is finished.
Next, let A µ be the closed linear self-adjoint operator on L 2 (Ω) associated with the form E µ in the sense that, The operator A µ is the realization in L 2 (Ω) of the regional fractional Laplace operator A s Ω with the fractional Robin type boundary conditions. More precisely, we have the following characterization. 16) where N 2−2s u is to be understood in the sense of (2.13) in Definition 2.5.
Proof. Let A µ be the operator on L 2 (Ω) defined in (2.15). Let D(A µ ) be given by (2.15) and set It follows from (2.17) that in particular, for every v ∈ D(Ω), By Definition 2.5, this implies that A s Ω u ∈ L 2 (Ω) and w = A s Ω u. It also follows from (2.17), the fact that A s Ω u = w and Definition 2.5 that u has a fractional normal derivative and C s N 2−2s u = µu in L 2 (∂Ω). Hence, u ∈ D and A µ u = A s Ω u.
To prove the converse inclusion, let u ∈ D. Then u ∈ W s,2 (Ω), w := A s Ω u ∈ L 2 (Ω) and C s N 2−2s u = µu in L 2 (∂Ω). It follows from the identity (2.14) that for every v ∈ W s,2 (Ω), Hence, u ∈ D(A µ ) and A µ u = A s Ω u. We have shown (2.16) and the proof is finished.
Since the embedding W s,2 (Ω) → L 2 (Ω) is compact, we have that A µ has a compact resolvent, hence, it has a discrete spectrum. For µ ∈ R and n ∈ N, we denote by λ n,s (µ) the n th -eigenvalue of A µ .
Remark 3. We have the following situation.
(a) If µ = 0, then we denote the operator A 0 by A N which is given by (2.18) The operator A N is the realization in L 2 (Ω) of the regional fractional Laplace operator A s Ω with fractional Neumann boundary condition. The spectrum of A N is an increasing sequence 0 = λ N 1,s < λ N 2,s < · · · < λ N n,s < · · · of real numbers such that lim n→∞ λ N n,s = ∞. (b) We extend E µ to µ = −∞. If µ = −∞, since ∞ · ∂Ω |u| 2 dσ must be finite for every u ∈ D(E −∞ ), then u = 0 a.e. on ∂Ω and hence, the form E −∞ has domain W s,2 0 (Ω) and is given for u, v ∈ W s,2 0 (Ω) by We denote the associated operator by A D which is the realization in L 2 (Ω) of A s Ω with the Dirichlet boundary condition. By [27], we have that Since the embedding W s,2 0 (Ω) → L 2 (Ω) is compact, then A D also has a compact resolvent. Its spectrum is an increasing sequence 0 < λ D 1,s < λ D 2,s < · · · < λ D n,s < · · · of real numbers and lim n→∞ λ D n,s = ∞. We have the following Poincaré type inequality. For every u ∈ W s,2 0 (Ω), In fact, (−∆) s Ω is the self-adjoint operator on L 2 (Ω) associated with the closed bilinear symmetric form 6). They coincide if and only if R N \ Ω has zero capacity and this cannot happen if Ω is bounded. The operator (−∆) s Ω also has a compact resolvent. Its spectrum is an increasing sequence 0 < λ 1,s,Ω < λ 2,s,Ω < · · · < λ n,s,Ω < · · · of real numbers and lim n→∞ λ n,s,Ω = ∞. The variational characterization and some basic properties of λ n,s,Ω have been investigated in [24, Proposition 9 and Appendix A]. It follows directly from the min-max variational formula for the eigenvalues that 0 < λ D n,s ≤ λ n,s,Ω , ∀ n ∈ N. (2.22) In [25] the authors have shown that λ 1,s,Ω < (λ D 1 ) s , where we recall that λ D 1 denotes the first eigenvalue of the Dirichlet Laplace operator. For more details on this topics we refer to [14,24,25,27] and their references.
3. The fractional Dirichlet-to-Neumann operator. Let σ(A D ) denote the spectrum of the operator A D introduced in (2.19). Recall that σ(A D ) is a discrete set and its elements form an increasing sequence 0 < λ D 1,s < λ D 2,s < · · · < λ D n,s < · · · of real numbers such that lim n→∞ λ D n,s = ∞.
for all v ∈ D(Ω), and hence, for all v ∈ W s,2 0 (Ω) by density. Proof. Let 1 2 < s < 1 and (W s,2 0 (Ω)) the dual of the Hilbert space W s,2 0 (Ω). Consider the operator B : W s,2 0 (Ω) → (W s,2 0 (Ω)) given by where ·, · is the duality map between (W s,2 0 (Ω)) and W s,2 0 (Ω). Note that B is a bounded operator. First, we claim that the operator B is bijective. In fact, if one defines E B with domain W s,2 0 (Ω) by we have that E B is a bilinear, symmetric and continuous form. Moreover, using the Poincaré inequality (2.20), we have that there exists a constant C > 0 such that E B (u, u) ≥ C u 2 W s,2 0 (Ω for every u ∈ W s,2 0 (Ω). Hence, the form E B is also coercive. Now it follows from the classical Lax-Milgram theorem (see e.g. [7, Corollary 5.8] that for every v ∈ (W s,2 0 (Ω)) there exists a unique u ∈ W s,2 0 (Ω) such that Bu, ϕ = E B (u, ϕ) = v, ϕ for every ϕ ∈ W s,2 0 (Ω) and this proves the claim.
• If λ < 0, then Hence, u − = 0 in Ω. • If λ = 0, then using also the Poincaré inequality (2.20), we get that Hence, u − = 0 in Ω. We have shown that in any case, u − = 0 in Ω and this implies that u = u + ≥ 0 on Ω. The proof is finished. Now, we are ready to introduce the fractional Dirichlet-to-Neumann operator. where N 2−2s u is to be understood in the sense of Definition 2.5 and C s is the constant given in (2.8).
We have the following result. Proof. Let 1 2 < s < 1 and λ ∈ R \ σ(A D ). We prove the proposition in several steps.
Step 1: Using the Hölder inequality and Remark 4 we get that there exists a constant C > 0 such that for all u, v ∈ H s,λ (Ω) with ϕ = u| ∂Ω and ψ = v| ∂Ω , we have Hence, the form F λ is continuous.
Step 4: The generation result of the strongly continuous analytic semigroup on L 2 (∂Ω) by the operator −D s,λ follows from the fact that F λ is continuous and elliptic, and that V s (∂Ω) is dense in L 2 (∂Ω). Since the embedding V s (∂Ω) → L 2 (∂Ω) is compact, we have that D s,λ has a compact resolvent and hence, has a discrete spectrum. The proof is finished.
, u ∈ W s,2 (Ω) and w ∈ L 2 (∂Ω). It follows from the proof of Proposition 3.1 that u| ∂Ω ∈ D(D s,λ ) and D s,λ (u| ∂Ω ) = w if and only if for all v ∈ W s,2 (Ω), The following theorem is the main result of this section.
The following result shows that the semigroup can also be ultracontractive.
Proof of Proposition 3.3. . Let λ ∈ R \ σ(A D ) and µ 1,s (λ) the first eigenvalue of D s,λ which is given by the infimum of the Rayleich coefficient, that is, .