Dirichlet to Neumann operators and the ∂ ¯ ¯ ¯ -Neumann problem
aa r X i v : . [ m a t h . C V ] N ov DIRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANNPROBLEM DARIUSH EHSANIA
BSTRACT . We study the Dirichlet to Neumann operator of the ¯ ∂ -Neumann prob-lem, and the relation between the ¯ ∂ -Neumann boundary conditions and the Dirich-let to Neumann operator.
1. I
NTRODUCTION
The ¯ ∂ -Neumann problem is an example of a boundary value problem with in-volving an elliptic operator but whose boundary conditions lead to non-ellipticequations. In order to conclude Sobolev estimates for the solution to the ¯ ∂ -Neumannproblem control (of L -norms) over derivatives in all directions must be obtained,but the boundary conditions of the problem disadvantage one direction. Theboundary conditions contain the boundary value operator, the Dirichlet to Neu-mann operator (DNO), giving the boundary values of the outward derivative ofthe solution to a homogeneous Dirichlet problem. In some cases (for example thecase of strictly pseudoconvex domains) the DNO allows for some control of thedisadvantaged direction, in other cases of weak pseudoconvexity, the situation ismore delicate. The purpose of this article therefore, is to study the DNO of relatedto the ¯ ∂ -Neumann problem with particular emphasis on the resulting boundaryequations.The DNO will be written as a pseudodifferential operator acting on a boundarydistribution, and our first results are a reworking of results of Chang, Nagel, andStein in [1]. It is well known that to highest order the DNO is given by the squareroot of the highest order tangential terms in the elliptic interior operator. The high-est two orders of the DNO are calculated, as in [1], and reduce to those results in aspecial case. The approach of [1] could be used here as well to calculate the DNO,but we take another approach outlined in [4] based on pseudodifferential opera-tors on domains with boundary, an approach which was useful in calculating thesymbol of the normal derivative to the Green’s operator, as well permitting similarcalculations and estimates in the situation of piecewise smooth domains [5]. Re-lations among operators comprising the DNO, as well as other derived boundary Mathematics Subject Classification. value operators in the boundary conditions are essential in the construction of asolution to the ¯ ∂ -problem if a solution operator to ¯ ∂ b is assumed in [3].We further demonstrate in this paper the persistence of the non-elliptic charac-ter of the ¯ ∂ -Neumann conditions. In Section 7, we examine what happens when aperturbation is made of the elliptic operator of the problem. This change naturallyalso leads to a different DNO, however as we shall see the associated boundarycondition is essentially the same (and non-elliptic!). The boundary operator canbe approximated by Kohn’s Laplacian, (cid:3) b . This suggests that the ¯ ∂ -Neumannproblem can be solved by inverting the (cid:3) b operator and that the ¯ ∂ -problem can besolved by using a solution operator for ¯ ∂ b . This approach to ¯ ∂ is taken up in [3].Most the work presented here was undertaken while the author was at the Uni-versity of Wuppertal and the hospitality of the University and its Complex Anal-ysis Working Group is sincerely appreciated. The author particularly thanks JeanRuppenthal for his warm and generous invitation to work with his group. A visitto the Oberwolfach Research Institute in 2013 as part of a Research in Pairs groupwas also helpful in the formation of this article, for which the author extends grat-itude to the Institute as well as to S ¨onmez S¸ ahuto ˘glu for helpful discussions.2. N OTATION AND BACKGROUND
We fix some notation used throughout the article. Our notation for derivativesis ∂ t : = ∂∂ t . We also use the index notation for derivatives: with α = ( α , . . . , α n ) amulti-index ∂ α x = ∂ α x · · · ∂ α n x n .Multiplication of derivatives with − i come in handy when dealing with symbolexpansions of pseudodifferential operators and we will use the notation D α x todenote − i ∂ α x .We let Ω ⊂ R n be a smoothly bounded domain and define pseudodifferentialoperators on Ω as in [12]: Definition 2.1.
We denote by S α ( Ω ) the space of symbols a ( x , ξ ) ∈ C ∞ ( Ω × R n ) which have the property that for any given compact set, K , and for any n - tuples k and k , there is a constant c k , k ( K ) > (cid:12)(cid:12)(cid:12) ∂ k ξ ∂ k x a ( x , ξ ) (cid:12)(cid:12)(cid:12) ≤ c k , k ( K ) ( + | ξ | ) α −| k | ∀ x ∈ K , ξ ∈ R n .Associated to the symbols in class S α ( Ω ) are the pseudodifferential operators,denoted by Ψ α ( Ω ) . If u ∈ E ′ ( Ω ) , we can define u ∈ E ′ ( R n ) by using an exten-sion by 0, and then define the Fourier Transform of the extended u . We denote thetransform of the extended distribution simply by b u ( ξ ) . The definition of pseudo-differential operators on a domain Ω is given by IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 3 Definition 2.2.
We say an operator A : E ′ ( Ω ) → D ′ ( Ω ) is in class Ψ α ( Ω ) if A canbe written as an integral operator with symbol a ( x , ξ ) ∈ S α ( Ω ) :(2.1) Au ( x ) = ( π ) n Z R n a ( x , ξ ) b u ( ξ ) e ix · ξ d ξ .In our applications in this article we will be dealing with operators defined onall of R n applied to functions defined on Ω (but which can be extended by 0 to thewhole space). The operators on Ω will thus be the composition of the restrictionto Ω operator with the pseudodifferential operators defined on R n .If we let χ j be such that { χ j ≡ } j is a covering of Ω , and let ϕ j be a partition ofunity subordinate to this covering, then locally, we describe a boundary operator A : E ′ ( Ω ) → D ′ ( Ω ) in terms of its symbol, a ( x , ξ ) according to Au = ( π ) n Z a ( x , ξ ) d χ j u ( ξ ) d ξ on supp ϕ j . Then we can describe the operator A globally on all of Ω by(2.2) Au = ( π ) n ∑ j ϕ j Z a ( x , ξ ) d χ j u ( ξ ) d ξ .The difference arising between the definitions in (2.1) and (2.2) is a smoothing term[12], which we write as Ψ − ∞ u , to use the notation of Definition 2.2.While Ψ α ( Ω ) will denote a class of operators, the use of Ψ α will be used to referto any operator in class Ψ α ( Ω ) . Furthermore, operators defined on the boundaryof a domain will be denoted with a subscript b . For instance, if A ∈ Ψ α ( ∂ Ω ) wewrite A = Ψ α b .In our use of Fourier transforms and equivalent symbols we use cutoffs in orderto make use of local coordinates, one of which being a defining function, denotedby ρ , for the domain. We use e to indicate transforms in tangential directions. Let p ∈ ∂ Ω and let ( x , . . . , x n − , ρ ) be local coordinates around p , ( ρ < ) . Let χ p ( x , ρ ) denote a cutoff which is ≡ p and vanishes outside a small neighborhood of p on which the local coordinates ( x , ρ ) are valid. Then with u ∈ L ( Ω ) we write d χ p u ( ξ , η ) = Z χ p u ( x , ρ ) e − ix ξ e − i ρη dxd ρ g χ p u ( ξ , ρ ) = Z χ p u ( x , ρ ) e − ix ξ dx .We also use the e notation when describing transforms of functions supported onthe boundary. With notation and coordinates as above, we let u b ( x ) ∈ L ( ∂ Ω ) andwrite ^ χ p ( x , 0 ) u b ( ξ ) = Z χ p ( x , 0 ) u b ( x ) e − ix ξ dx .We want to apply pseudodifferential operator techniques to vector fields on asmoothly bounded domain Ω ⊂ C n . Let ρ be a smooth defining function for Ω DARIUSH EHSANI ( Ω = { z ∈ C n : ρ ( z ) < } ), normalized so that |∇ ρ | = ∂ Ω . We choose anorthonormal basis of (
1, 0 ) forms, ω , . . . , ω n in which ω n = √ ∂ρ , and we denote L , . . . , L n the vector fields respectively dual to the ω j .We let T = i ( L n − L n ) , and T = T | ∂ Ω . If we choose a boundary point p wecan choose local coordinates, as above, in a neighborhood of p such that L n has theform L n = √ ∂∂ρ + iT = √ ∂∂ρ + iT + O ( ρ )= √ ∂∂ρ + i ∂∂ x n − + O ( ρ ) .Similarly, in a neighborhood of p , we can represent the L j vector fields as(2.3) L j = ∂∂ x j − − i ∂∂ x j ! + n − ∑ k = ℓ jk ( x − p ) ∂∂ x k + O ( ρ ) ,where ℓ jk ( x ) = O ( x ) .In Fourier space we use ξ to denote the dual variables to the x coordinates, ξ i corresponding to x i for i =
1, . . . , 2 n −
1, and η dual to ρ . To help distinguish thecomplex tangential behavior, we set ξ L to be given by ξ L = n − ∑ ξ j .We use the standard decomposition of the Fourier transform space to separatethree microlocal neighborhoods (see for instance [2, 7, 8, 10]). We let ψ + , ψ , and ψ − be a smooth partition of unity on the unit ball, | ξ | =
1. We choose the functionsso that ψ + has support in ξ n − > | ξ L | and ψ + ≡ ξ n − > | ξ L | .The function ψ − is defined symmetrically: ψ − has support in ξ n − < − | ξ L | and ψ − ≡ ξ n − < − | ξ L | . The function ψ has support in | ξ n − | < | ξ L | and satisfies ψ = − ψ + − ψ − . We extend the functions radially, so that, inparticular, they satisfy | ∂ k ξ ψ ∗ | . | ξ | − k outside of some compact neighborhood of ξ =
0. This last property ensures that ψ + , ψ , and ψ − are in the class of symbols, S ( R n − ) . Cutoffs are also introducedso that (using the same notation for the functions) ψ ≡ ξ = | ξ | <
1. The radial extensions from the unit circle togetherwith the support of ψ near 0 are then to form a partition of unity of the transformspace, i.e., ψ + + ψ + ψ − = ξ ∈ R n − . The operators corresponding tothe symbols, ψ + , ψ , and ψ − , will be denoted by Ψ ν + , Ψ ν , and Ψ ν − , respectively. IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 5 As mentioned above, we take an approach to calculating boundary value oper-ators based on a pseudodifferential calculus for domains with boundary workedout in [4]. In particular, we will make use of the results detailing the behaviorof functions which result from the application of pseudodifferential operators (in R n ) to distributions supported on the boundary of the domain, as well as certainoperators applied to distributions with support in the whole domain (which canbe thought of as a distribution on all of R n with an extension by 0). Let us recallhere a few results from [4]. The results in [4] were stated for half-planes and thesewill be applied to domains Ω ⊂ C n ≃ R n , using local coordinates ( x , ρ ) with ρ < Definition 2.3.
Let A ∈ Ψ − k ( R n ) for k ≥ N ∈ N ,it can be written in the form A = B + Ψ − N ,where B ∈ Ψ − k ( R n ) has symbol, σ ( B )( x , ρ , ξ , η ) , which is meromorphic (in η )with poles at η = q ( x , ρ , ξ ) , . . . , q k ( x , ρ , ξ ) with q i ( x , ρ , ξ ) themselves, as well as the imaginary parts, Im q i , symbols of pseu-dodifferential operators of order 1 (restricted to η =
0) such that for each ρ ,Res η = q i σ ( B ) ∈ S k + ( R n − ) with symbol estimates uniform in the ρ parameter.We call such an operator, A , a decomposable operator.The first theorem is taken from Theorems 2.2 and 2.4 of [4]. Theorem 2.4.
Let g ∈ D ( Ω ) of the form g ( x , ρ ) = g b ( x ) δ ( ρ ) for g b ∈ W s ( ∂ Ω ) . LetA ∈ Ψ k ( Ω ) , k ≤ − be a decomposable operator. Then for all s, k Ag k W s ( Ω ) . k g b k W s + k + ( ∂ Ω ) .Theorem 2.4 for instance is applicable for any term arising in the symbol expan-sion of the inverse to an elliptic differential operator. With A an elliptic differentialoperator of order k , we can write for any N ∈ N ,(2.4) A − = B − k + Ψ − N where B − k is a pseudodifferential operator of order − k and satisfies the conditionsof the B operator in Definition 2.3.Thus for instance if △ is a second order elliptic differential operator on R n ,and for some given s ≥ g b ∈ W s ( ∂ Ω ) , with g = g b × δ ( ρ ) as above. Then △ − DARIUSH EHSANI satisfies the hypothesis of Theorem 2.4 and we have k△ − g k W s + ( Ω ) . k g b k W s ( ∂ Ω ) .We also have the following useful Lemmas. Lemma 2.5.
Let g ∈ D ( Ω ) be of the form g ( x , ρ ) = g b ( x ) δ ( ρ ) for g b ∈ D ( ∂ Ω ) .Let A ∈ Ψ k ( Ω ) , be a pseudodifferential operator of order k. Let ρ denote the operator ofmultiplication with ρ . Then ρ ◦ A induces a pseudodifferential operator of order k − ong: ρ Ag ≡ Ψ k − g .Let R denote the restriction operator, R : D ( Ω ) → D ( ∂ Ω ) , given in local coor-dinates ( x , ρ ) by R φ = φ | ρ = . Lemma 2.6.
Let g ∈ D ( Ω ) be of the form g ( x , ρ ) = g b ( x ) δ ( ρ ) for g b ∈ D ( ∂ Ω ) . LetA ∈ Ψ k ( Ω ) , be an operator of order k, for k ≤ − . Then R ◦ A induces a pseudodifferen-tial operator in Ψ k + b ( ∂ Ω ) acting on g b viaR ◦ Ag ≡ Ψ k + b g b .3. ¯ ∂ -N EUMANN PROBLEM
We look more closely at the ¯ ∂ -Neumann problem, (cid:3) u = f , where (cid:3) = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ .For f a ( q ) -form, the equation (cid:3) u = f comprises a system of equations, andwe write our equations in matrix form. We use the convention of writing indiceswith increasing entries: a particular index of length q , J = ( j , . . . , j q ) , is orderedaccording to j l < j m for l < m . For the matrix we consider the ordering of twoindices, J = ( j , j , . . . , j q ) and J = ( j , j , . . . , j q ) , according to J < J if j k < j k for the first k such that j k = j k , and J = J if j k = j k for all k =
1, . . . q .The rows (and columns) of the matrix are in order of increasing indices. Thus, forinstance, if we denote J = (
1, 2, . . . , q ) , the (
1, 1 ) -entry of the matrix correspondsto the action on u J which results in a form whose component is ¯ ω J . Similarly,with J = (
1, 2, . . . , q − n ) , the ( n − q +
1, 1 ) -entry of the matrix corresponds tothe action on u J which results in a form whose component is ¯ ω J , etc.We want to calculate in general a J th row of the matrix of operators describing (cid:3) . We thus need to know which forms would result in a ¯ ω J term when some inputform is given into (cid:3) . Let J = ( j , . . . , j q ) with j m = k , where 1 ≤ k ≤ n , for some m .We use the notation J ˆ k to denote the index of length q − ( j , . . . , j m − , j m + , . . . , j q ) .We further use the set notation J ˆ k ∪ { l } to denote the index of length q (we assumethe case l = j i for 1 ≤ i ≤ q , i = m ) consisting of the set { j , . . . , j m − , j m + , . . . , j q , l } IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 7 in the appropriate order (recall a particular index K = ( k , . . . , k q ) is ordered if k r < k s for r < s ).Since ¯ ∂ increases the type of the form by 1 and ¯ ∂ ∗ decreases it by one, we see it iswhen (cid:3) operates on a form of the type u J ˆ k ∪{ l } ¯ ω J ˆ k ∪{ l } that a ¯ ω J term would result.And so we calculate (cid:3) u J ˆ k ∪{ l } ¯ ω J ˆ k ∪{ l } . With abuse of notation, for a prescribed J ,we write u kl in place of u J ˆ k ∪{ l } , assuming k = l , with the obvious simplification indimension 2.We use our notation in the case k = l , although the case k = l is also in-cluded in the same calculations. We start with the ¯ ω J ˆ k components resulting from¯ ∂ ∗ (cid:16) u kl ¯ ω J ˆ k ∪{ l } (cid:17) . We use the notation(3.1) c JJ ∪{ m } = ¯ ∂ ( ¯ ω J ) ⌋ ¯ ω J ∪{ m } ,and for j =
1, . . . , n , we define d j according to an integration by parts ( φ , L j ϕ ) = (cid:16) ( − L j + d j ) φ , ϕ (cid:17) for φ , ϕ ∈ C ∞ ( Ω ) with support in a coordinate patch such that so that L j can bewritten in terms of local coordinates as in (2.3).From (cid:16) ¯ ∂ ∗ u kl ¯ ω J ˆ k ∪{ l } , ϕ ¯ ω J ˆ k (cid:17) = (cid:16) u kl ¯ ω J ˆ k ∪{ l } , ¯ ∂ ( ϕ ¯ ω J ˆ k ) (cid:17) = (cid:16) u kl ¯ ω J ˆ k ∪{ l } , L l ϕ ¯ ω l ∧ ¯ ω J ˆ k + c J ˆ k J ˆ k ∪{ l } ϕ ¯ ω J ˆ k ∪{ l } (cid:17) = (cid:16) u kl , ε l J ˆ k J ˆ k ∪{ l } L l ϕ + c J ˆ k J ˆ k ∪{ l } ϕ (cid:17) ,where we write, for some index K = ( k , . . . k q − ) and mK = ( m , k , . . . k q − ) , ε mKK ∪{ m } = ¯ ω mK ⌋ ¯ ω K ∪{ m } ,we have(3.2) ¯ ∂ ∗ (cid:16) u kl ¯ ω J ˆ k ∪{ l } (cid:17) = (cid:16) ε l J ˆ k J ˆ k ∪{ l } ( − L l + d l ) u kl + c J ˆ k J ˆ k ∪{ l } u kl (cid:17) ¯ ω J ˆ k + · · · ,with ϕ some test function, where the · · · refers to terms whose contraction with¯ ω J ˆ k results in 0 (and which contain a ¯ ω l component). And thus¯ ∂ ¯ ∂ ∗ u kl ¯ ω J ˆ k ∪{ l } = (cid:16) ε l J ˆ k J ˆ k ∪{ l } L k ( − L l + d l ) u kl + c J ˆ k J ˆ k ∪{ l } L k u kl (cid:17) ¯ ω k ∧ ¯ ω J ˆ k − ε l J ˆ k J ˆ k ∪{ l } c J ˆ k J L l u kl ¯ ω J + · · · = (cid:16) − ε l J ˆ k J ˆ k ∪{ l } ε kJ ˆ k J L k L l u kl + (cid:0) ε l J ˆ k J ˆ k ∪{ l } ε kJ ˆ k J d l + ε kJ ˆ k J c J ˆ k J ˆ k ∪{ l } (cid:1) L k u kl − ε l J ˆ k J ˆ k ∪{ l } c J ˆ k J L l u kl (cid:17) ¯ ω J + · · · ,(3.3) DARIUSH EHSANI where here the · · · refers to terms which upon contraction with ¯ ω J result in 0 aswell as zero order terms.We note that the calculations also show(3.4) ¯ ∂ ¯ ∂ ∗ u J ¯ ω J = ∑ l ∈ J (cid:16) − L l L l u J + (cid:0) d l + ε l J ˆ l J c J ˆ l J (cid:1) L l u J − ε l J ˆ l J c J ˆ l J L l u J (cid:17) ¯ ω J + · · · .Similarly, to calculate ¯ ∂ ∗ ¯ ∂ u kl ¯ ω J ˆ k ∪{ l } we start with¯ ∂ u kl ¯ ω J ˆ k ∪{ l } = (cid:16) ε kJ ˆ k ∪{ l } J ∪{ l } L k u kl + c J ˆ k ∪{ l } J ∪{ l } u kl (cid:17) ¯ ω J ∪{ l } modulo terms whose contraction with ¯ ω J ∪{ l } result in 0. As in (3.2), we have¯ ∂ ∗ v ¯ ω J ∪{ l } = (cid:16) ε l JJ ∪{ l } ( − L l + d l ) v + c JJ ∪{ l } v (cid:17) ¯ ω J modulo terms whose contraction with ¯ ω J result in 0, which when applied to ¯ ∂ u kl ¯ ω J ˆ k ∪{ l } above, yields¯ ∂ ∗ ¯ ∂ u kl ¯ ω J ˆ k ∪{ l } = (cid:16) − ε l JJ ∪{ l } ε kJ ˆ k ∪{ l } J ∪{ l } L l L k u kl + ε l JJ ∪{ l } ε kJ ˆ k ∪{ l } J ∪{ l } d l L k u kl + ε kJ ˆ k ∪{ l } J ∪{ l } c JJ ∪{ l } L k u kl − ε l JJ ∪{ l } c J ˆ k ∪{ l } J ∪{ l } L l u kl (cid:17) ¯ ω J + · · · ,(3.5)where the . . . refers to terms whose contraction with ¯ ω J result in 0 as well as termsof order 0. And similarly,¯ ∂ ∗ ¯ ∂ u J ¯ ω J = ∑ l / ∈ J (cid:16) − L l L l u J + (cid:0) d l + ε l JJ ∪{ l } c JJ ∪{ l } (cid:1) L l u J − ε l JJ ∪{ l } c JJ ∪{ l } L l u J (cid:17) ¯ ω J + · · · .(3.6)Adding (3.4) and (3.6) yields (cid:3) u J ¯ ω J : (cid:3) u J ¯ ω J = − ∑ l ∈ J L l L l u J ¯ ω J − ∑ l / ∈ J L l L l u J ¯ ω J + ( − ) | J | (cid:16) − c J ˆ n J L n u J ¯ ω J + c J ˆ n J L n (cid:17) u J ¯ ω J + d n L n u J ¯ ω J + · · · n ∈ J ( − ) | J | (cid:16) c JJ ∪{ n } L n − c JJ ∪{ n } L n (cid:17) u J ¯ ω J + d n L n u J ¯ ω J + · · · n / ∈ J .We only collect the (complex) normal derivatives, as they are enough to determinethe behavior of the relevant boundary value operators in the direction of the field T (see the discussion at the end of Section 5). The terms included in the · · · thusrefer to terms which are orthogonal to ¯ ω J , of zero order, or involve only vectorfields orthogonal to L n or L n . IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 9 The two cases can be combined into the single expression (cid:3) u J ¯ ω J = − ∑ l ∈ J L l L l u J ¯ ω J − ∑ l / ∈ J L l L l u J ¯ ω J +( − ) | J ∪{ n }| (cid:16) c J \{ n } J \{ n }∪{ n } L n − c J \{ n } J \{ n }∪{ n } L n (cid:17) u J ¯ ω J + d n L n u J ¯ ω J + · · · .We now add (3.3) and (3.5) to obtain (cid:3) u kl ¯ ω J ˆ k ∪{ l } . To simplify the result we note(3.7) ε l JJ ∪{ l } ε kJ ˆ k ∪{ l } J ∪{ l } = − ε l J ˆ k J ˆ k ∪{ l } ε kJ ˆ k J for k = l , and as we are only interested in the complex normal derivatives, incomparing ε l JJ ∪{ l } c J ˆ k ∪{ l } J ∪{ l } and ε l J ˆ k J ˆ k ∪{ l } c J ˆ k J we look at the case l = n , for which we have ε nJJ ∪{ n } c J ˆ k ∪{ n } J ∪{ n } = − ε nJ ˆ k J ˆ k ∪{ n } c J ˆ k J .Similarly, in comparing ε kJ ˆ k ∪{ l } J ∪{ l } c JJ ∪{ l } and ε kJ ˆ k J c J ˆ k J ˆ k ∪{ l } we are only interested in thecase k = n for which ε nJ ˆ n ∪{ l } J ∪{ l } c JJ ∪{ l } = − ε nJ ˆ n J c J ˆ n J ˆ n ∪{ l } .We thus can write, by adding (3.3) and (3.5), (cid:3) u kl ¯ ω J ˆ k ∪{ l } = − ε l J ˆ k J ˆ k ∪{ l } ε kJ ˆ k J [ L k , L l ] u kl ¯ ω J for k = l , modulo terms with the vector fields L j or L j for j =
1, . . . n −
1, zeroorder terms, or forms orthogonal to ¯ ω J .We collect our results in the following proposition: Proposition 3.1.
Modulo the vector fields L j or L j acting on components of u for j =
1, . . . n − , zero order terms, or forms orthogonal to ¯ ω J , we havei ) (cid:3) (cid:0) u J ¯ ω J (cid:1) = − ∑ l ∈ J L l L l u J ¯ ω J − ∑ l / ∈ J L l L l u J ¯ ω J + ( − ) | J ∪{ n }| (cid:16) c J \{ n } J \{ n }∪{ n } L n − c J \{ n } J \{ n }∪{ n } L n (cid:17) u J ¯ ω J + d n L n u J ¯ ω J ii ) (cid:3) u J ˆ k l ¯ ω J ˆ k ∪{ l } = − ε l J ˆ k J ˆ k ∪{ l } ε kJ ˆ k J [ L k , L l ] u kl ¯ ω J .4. T HE D IRICHLET TO N EUMANN OPERATOR
The Dirichlet to Neumann operator (DNO) is the boundary value operator giv-ing the outward normal derivative of the solution to a Dirichlet problem. We lookat the DNO corresponding to the operator 2 (cid:3) . We study the solution, v , whichsolves 2 (cid:3) v = Ω v = g b on ∂ Ω , and we obtain an expression for ∂ v ∂ρ (modulo smooth terms) near a given point p ∈ ∂ Ω in terms of g b .If χ p is a smooth cutoff function with support in a small neighborhood of p and χ ′ p a smooth cutoff such that χ ′ p ≡ χ p , we have2 (cid:3) ( χ p v ) = Ψ ( χ ′ p v ) on Ω (4.1) χ p v = χ p g b on ∂ Ω .The term Ψ ( χ ′ p v ) above arises due to derivatives falling on the cutoff function χ p . The use of the cutoff function allows us to consider the equation locally, andis equivalent to using pseudodifferential operators with symbols defined in localcoordinate patches, with one of the coordinates given by ρ . We thus consider v tobe supported in a neighborhood of a given boundary point, p ∈ ∂ Ω .To study the operator (cid:3) and the associated boundary operators, we consider theequation (cid:3) v = ( q ) -form v . In a small neighborhood of a boundary point,which we take to be 0 ∈ ∂ Ω , we write the vector fields, L j in local coordinates asin (2.3):(4.2) L j = ∂∂ x j − − i ∂∂ x j ! + n − ∑ k = ℓ jk ( x ) ∂∂ x k + O ( ρ ) ,where ℓ jk ( x ) = O ( x ) . Also we recall from Section 2, the representation of L n :(4.3) L n = √ ∂∂ρ + i ∂∂ x n − + O ( ρ ) .We use the symbol notation σ ( ∂ ρ ) = i ησ ( ∂ x j ) = i ξ j j =
1, . . . , 2 n − (cid:3) from Proposition3.1 are given by − ∑ l ∈ J L l L l − ∑ l / ∈ J L l L l .Expanding this operator using (4.2) and (4.3), we write in local coordinates − ∑ l ∈ J L l L l − ∑ l / ∈ J L l L l = − ∂ ∂ρ + ∑ j ∂ ∂ x j + ∂ ∂ x n − + n − ∑ j , k = l jk ∂ ∂ x j ∂ x k ! + O ( ρ ) ,where l jk = O ( x ) , and modulo first order terms. We define the operator Γ to begiven by the terms without a ρ factor on the right hand side: Γ : = − ∂ ∂ρ + ∑ j ∂ ∂ x j + ∂ ∂ x n − + n − ∑ j , k = l jk ∂ ∂ x j ∂ x k ! . IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 11 We let L bj for j =
1, . . . , n − σ ( L bj ) = σ ( L j ) (cid:12)(cid:12) ρ = .Then we have σ ( (cid:3) ) = η + ξ n − + n − ∑ j = σ ( L bj ) σ ( L bj ) + O ( ρ ) O ( ξ )= η + ξ n − + n − ∑ j = ξ j + O ( x ) O ( ξ ) + O ( ρ ) O ( ξ )= σ ( Γ ) + O ( ρ ) O ( ξ ) .We use the Kohn-Nirenberg notation, σ j to denote the part of a symbol homoge-neous of degree j in ξ and η in its symbol expansion.Let us now denote Ξ ( x , ξ ) = ξ n − + n − ∑ j = σ ( L bj ) σ ( L bj )= ξ n − + n − ∑ j = ξ j + O ( x ) O ( ξ ) (4.4)so that we can write σ ( Γ ) = η + Ξ ( x , ξ ) .We now collect the second order O ( ρ ) terms from 2 (cid:3) in an operator, τ , i.e. σ ( τ ) (cid:12)(cid:12)(cid:12) ρ = = ∂∂ρ σ ( (cid:3) ) (cid:12)(cid:12)(cid:12) ρ = ,and all tangential first order operators in the expression of the operator 2 (cid:3) asin Proposition 3.1 into a pseudodifferential operator, denoted A . We also denoteby the operator S the zero order operator which is multiplication by the (matrix)coefficient of ∂∂ρ in the operator 2 (cid:3) .With the notation v | ρ = = g b ( x ) , the equation 2 (cid:3) v = Γ v + √ S (cid:18) ∂ v ∂ρ (cid:19) + Av + ρτ ( v ) = (cid:3) v =
0, with boundary values v = g b on ∂ Ω . Inthe equation (4.5) above, the DNO can be found by solving for ∂ ρ v (cid:12)(cid:12) ρ = . We rewrite (4.5) using Fourier Transforms, extending (4.5) to R n by 0. Let E denote the extension by 0. The term E ◦ Γ v can be written E ◦ Γ v = Γ ◦ Ev − ( π ) n Z (cid:18) ∂ ρ e v (cid:12)(cid:12)(cid:12) ρ = + i η e g b ( ξ ) (cid:19) e i ρη e ix · ξ d ξ d η .For ease of notation, we will disregard the extension operator, E , and instead usethe subscript int to signify an operator is to be applied to the extension by 0 to R n of a distribution defined in Ω . With this convention, we write Γ v = Γ int v − ( π ) n Z (cid:18) ∂ ρ e v (cid:12)(cid:12)(cid:12) ρ = + i η e g b ( ξ ) (cid:19) e i ρη e ix · ξ d ξ d η ,where Γ on the left-hand side is to be understood as an operator Γ : D ′ ( Ω ) → D ′ ( R n ) via (left-side) composition with E , and where Γ int : D ′ ( R n ) → D ′ ( R n ) has as symbol: σ ( Γ int ) = η + Ξ ( x , ξ ) .The term S (cid:16) ∂ v ∂ρ (cid:17) can be written S (cid:18) ∂ v ∂ρ (cid:19) = ( π ) n Z s ( x , ρ ) i η b v ( ξ , η ) e i ρη e ix · ξ d ξ d η + ( π ) n Z s ( x , 0 ) e g b ( ξ ) e ix · ξ d ξ : = S int v + S b g b ,where similarly the left-hand side is understood to be composed on the left by E ,and where S int : = S ◦ E . We have σ ( S int ) = s ( x , ρ ) i η , and S b ∈ Ψ ( ∂ Ω ) with σ ( S b ) = s ( x , 0 ) . From Proposition 3.1, s ( x , ρ ) is a diagonal matrix (of smoothfunctions).We now rewrite (4.5) as Γ int v = ( π ) n Z (cid:18) ∂ ρ e v (cid:12)(cid:12)(cid:12) ρ = + i η e g b ( ξ ) (cid:19) e i ρη e ix · ξ d ξ d η − √ S int v − √ S b g b − Av − ρτ v ,where v is understood to be extended by 0 to all of R n . Γ int is an elliptic operator on R n and so we can apply an inverse to Γ int : v = ( π ) n Γ − int ◦ Z (cid:18) ∂ ρ e v (cid:12)(cid:12)(cid:12) ρ = + i η e g b ( ξ ) (cid:19) e i ρη e ix · ξ d ξ d η − √ Γ − int ◦ S int v − √ Γ − int ◦ S b g b − Γ − int ◦ Av − Γ − int ◦ ( ρτ v ) (4.6)modulo smoothing terms. The idea behind our calculations of the DNO is to write ∂ ρ v (cid:12)(cid:12)(cid:12) ρ = = Λ b g b + Λ b g b + · · · , insert this expansion into the first integral on theright-hand side of (4.6), set ρ = Ξ ( x , ξ ) (see [1] for another approach). IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 13 We first prove a Proposition about the Poisson operator, giving the solution, v , above. In order to consolidate the various smoothing terms which arise, wewrite R − ∞ b to include the restriction to ρ = Ψ − ∞ ( Ω ) applied to v , or smoothing operators in Ψ − ∞ b ( ∂ Ω ) applied to theboundary values g b or ∂ ρ v | ρ = . We also write R − ∞ to include any sum of smooth-ing operators in Ψ − ∞ ( Ω ) applied to v , smoothing operators in Ψ − ∞ ( Ω ) appliedto g b × δ ( ρ ) or ∂ ρ v | ρ = × δ ( ρ ) , or decomposable operators in Ψ − k ( Ω ) for k ≥ R − ∞ b (such terms can thus be estimated in terms of smooth boundaryterms, see Theorem 2.4) as well as smoothing operators in Ψ − ∞ ( Ω ) applied to R − ∞ b . From the definitions we have R ( R − ∞ ) = R − ∞ b .Estimates for the Poisson operator corresponding to an elliptic operator wereworked out in [4]. In those results, the highest order term of the DNO was alsocalculated. The calculations here follow those in [4] to find the Poisson operatorcorresponding to (cid:3) . As the operator, (cid:3) , is slightly different than the operatorconsidered in the author’s earlier work (namely in the first order terms), and asthe Poisson operator will be used to obtain the lower order terms of the DNO, wego through the calculations in detail, obtaining first an expression for the Poissonoperator, and then calculating the DNO.We define the Poisson operator corresponding to (cid:3) as the solution operator, P ,mapping ( q ) -forms on ∂ Ω to ( q ) -forms on Ω , to(4.7) 2 (cid:3) ◦ P = R ◦ P = I .We assume the classical results guaranteeing existence and uniqueness of a solu-tion. Theorem 4.1.
Let g b be a ( q ) -form on ∂ Ω ; each component of g b is a distributionsupported on ∂ Ω . Let g = g b ( x ) × δ ( ρ ) in local coordinates. ThenPg = Ψ − g + R − ∞ . Proof.
From (4.6), we have modulo R − ∞ v = ( π ) n Z ∂ ρ e v ( ξ , 0 ) + i η e g b ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η − √ Γ − int ◦ S (cid:18) ∂ v ∂ρ (cid:19) − Γ − int ◦ A ( v ) − Γ − int ◦ ρτ ( v )+ Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − v ,(4.8)locally, in a small neighborhood of the origin; we recall, using the pseudodiffer-ential analysis, we consider v to have compact support in a neighborhood of a boundary point, which we take to be the origin, and the pseudodifferential opera-tors are also composed on the left with cutoffs with support in a neighborhood ofthe origin; see the discussion in Section 2 as well as the discussion following (4.1).For ease of notation, we omit the writing of the cutoffs. We will also omit mentionof the smooth R − ∞ terms, inserting them again at the end of the calculations.We note the terms Γ − int ◦ S (cid:16) ∂ v ∂ρ (cid:17) and Γ − int ◦ A ( v ) contribute terms Ψ − v and Ψ − g .To handle the term(4.9) Γ − int ◦ ρτ ( v ) ,we write the operator τ using the form of its symbol σ ( τ ) = n − ∑ j , k = τ jk ( x , ρ ) ξ j ξ k ,and we rearrange (4.8) as v = ( π ) n Z ∂ ρ e v ( ξ , 0 ) + i η e g b ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + ( π ) n Z ρ ∑ j , k τ jk ( x , ρ ) ξ j ξ k η + Ξ ( x , ξ ) b v ( ξ , η ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − v ,as the terms involving the operators S and A are included in the last two remainderterms. We then bring the second term on the right to the left-hand side:1 ( π ) n Z − ρ ∑ j , k τ jk ( x , ρ ) ξ j ξ k η + Ξ ( x , ξ ) !b v ( ξ , η ) e ix ξ e i ρη d ξ d η = ( π ) n Z ∂ ρ e v ( ξ , 0 ) + i η e g b ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − v .(4.10)For small enough ρ (which, without loss of generality, can be assumed by choos-ing the cutoffs defining the pseudodifferential operators appropriately small) thesymbol(4.11) 1 − ρ ∑ n − j , k = τ jk ( x , ρ ) ξ j ξ k η + Ξ ( x , ξ ) is non-zero, and so (shrinking the support of v if necessary) we can apply a parametrixof the operator with symbol (4.11) to both sides of (4.10). We note the symbol ofsuch an operator is of the form 1 + O ( ρ ) , where the second term is a symbol of IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 15 class S ( Ω ) , which is O ( ρ ) . We obtain v = ( π ) n Z ( + O ( ρ )) ∂ ρ e v ( ξ , 0 ) + i η e g b ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − v .(4.12)From Lemma 2.5 we have that1 ( π ) n Z O ( ρ ) ∂ ρ e v ( ξ , 0 ) + i η e g b ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η = Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g .Returning to (4.12) we write v = ( π ) n Z ∂ ρ e v ( ξ , 0 ) + i η e g b ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − v .(4.13)The expression above is locally confined to a neighborhood of the origin, but usingcoverings and a partition of unity (as in the explanation in (2.2)) we can obtain anexpression for v on all of Ω . Then inverting an operator of the form I − Ψ − givesan expression for v on Ω :(4.14) v = Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − ∞ v ,as a vector-valued relation, with matrix-valued pseudodifferential operators.Using the residue calculus, we can take an inverse transform in (4.13) with re-spect to η . For ρ → + , we have0 = ( π ) n π i (cid:18) Z i | Ξ ( x , ξ ) | ∂ ρ e v ( ξ , 0 ) e ix ξ d ξ − Z | Ξ ( x , ξ ) | i | Ξ ( x , ξ ) | e g b ( ξ ) e ix ξ d ξ (cid:19) + R ◦ Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + R ◦ Ψ − g + R ◦ Ψ − v = ( π ) n − Z (cid:18) | Ξ ( x , ξ ) | ∂ ρ e v ( ξ , 0 ) − e g b ( ξ ) (cid:19) e ix ξ d ξ + Ψ − b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = (cid:17) + Ψ − b g b + R ◦ Ψ − v ,where we apply Lemma 2.6 to the terms with operators R ◦ Ψ − and R ◦ Ψ − in thesecond step. We can now invert the operator with symbol 1/2 | Ξ ( x , ξ ) | and solvefor ∂ ρ v | ρ = :(4.15) ∂ v ∂ρ ( x , 0 ) = Z | Ξ ( x , ξ ) | e g b ( ξ ) e ix ξ d ξ + Ψ b g b + Ψ b ◦ R ◦ Ψ − v ,locally, in a small neighborhood of the origin. Alternatively, we could in a similarmanner use the residue calculus to take an inverse transform with respect to η in(4.13) and calculate for ρ → − with the same result. For the term Ψ b ◦ R ◦ Ψ − v we insert (4.14) in the argument: Ψ b ◦ R ◦ Ψ − v = Ψ b ◦ R ◦ Ψ − (cid:16) Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − ∞ v (cid:17) = Ψ b ◦ R ◦ Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ b ◦ R ◦ Ψ − g + Ψ b ◦ R ◦ Ψ − ∞ v = Ψ b ◦ Ψ − b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = (cid:17) + Ψ b ◦ Ψ − b g b + R ◦ Ψ − ∞ v = Ψ − b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = (cid:17) + Ψ b g b + R − ∞ b v .(4.15) above leads to the well-known result that the DNO is a first order operatoron the boundary data, with principal term | Ξ ( x , ξ ) | : ∂ v ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = = Z | Ξ ( x , ξ ) | e g b ( ξ ) e ix ξ d ξ + Ψ b g b + Ψ − b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = (cid:17) + R − ∞ b v .Again, using a covering and the local expressions to obtain a global relation,and solving for (the vector) ∂ ρ v (cid:12)(cid:12) ρ = , and absorbing extra Ψ − ∞ b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = (cid:17) termsinto the remainder term, R − ∞ b , leads to the expression:(4.16) ∂ ρ v (cid:12)(cid:12) ∂ Ω = | D | g b + Ψ b g b + R − ∞ b where | D | is defined as the first order operator with symbol locally given by σ ( | D | ) = | Ξ ( x , ξ ) | .We can now insert (4.16) in (4.13) and obtain in a small neighborhood of theorigin v = ( π ) n Z ( | Ξ ( x , ξ ) | + i η ) e g b ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − v + Ψ − R − ∞ b = i ( π ) n Z e g b ( ξ ) η + i | Ξ ( x , ξ ) | e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ( Ψ b g b + R − ∞ b ) × δ ( ρ ) (cid:17) + Ψ − g + Ψ − v + R − ∞ ,which we write as(4.17) v = Ψ − g + Ψ − v + R − ∞ .We thus obtain v = Ψ − g + R − ∞ ,on all of Ω . (cid:3) From the proof of the Theorem we also have the principal symbol of the opera-tor Ψ − acting on g b × δ ( ρ ) ; it is given locally by (the diagonal matrix)(4.18) i η + i | Ξ ( x , ξ ) | , IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 17 which we note for future reference. Using the representation as in (4.16),(4.19) ∂ ρ v (cid:12)(cid:12) ∂ Ω = Ψ b g b + R − ∞ b ,we can obtain with Lemma 2.4 estimates for the Poission operator.We first handle the smooth terms, R − ∞ and R − ∞ b : Lemma 4.2.
For R − ∞ and R − ∞ b , and g b , defined as above, we have for all s k R − ∞ k W s ( Ω ) . k g b k L ( ∂ Ω ) and k R − ∞ b k W s ( ∂ Ω ) . k g b k L ( ∂ Ω ) . Proof.
We note the L estimates for the Poisson operator, k P ( g ) k L ( Ω ) . k g b k L ( ∂ Ω ) (see for instance [9]).For R − ∞ , we have by definition k R − ∞ k W s ( Ω ) . k u k W − ∞ ( Ω ) + k g b k W − ∞ ( ∂ Ω ) + k ∂ ρ u (cid:12)(cid:12) ∂ Ω k W − ∞ ( ∂ Ω ) . k g b k L ( ∂ Ω ) + k ∂ ρ u (cid:12)(cid:12) ∂ Ω k W − ∞ ( ∂ Ω ) for any s ≥ ∂ ρ u (cid:12)(cid:12) ∂ Ω j by assuming support in aneighborhood of ∂ Ω intersected with Ω and writing ∂ ρ u (cid:12)(cid:12) ρ = = Z − ∞ ∂ ρ ud ρ = Z − ∞ D t ud ρ + Z − ∞ (cid:0) φ ∂ ρ u + φ u (cid:1) d ρ ,where D t is a second order tangential operator, and φ and φ are smooth withsupport in the interior of Ω . From interior regularity, we have k φ j u k W ( Ω ) . k g b k L ( ∂ Ω ) .Thus, applying a tangential smoothing operator to both sides and integratingyields k ∂ ρ u (cid:12)(cid:12) ∂ Ω k W − ∞ ( ∂ Ω ) . k g b k L ( ∂ Ω ) .Hence, k R − ∞ k W s ( Ω ) . k g b k L ( ∂ Ω ) .The estimates for R − ∞ b follow similarly. (cid:3) Theorem 4.3.
Let P be the Poisson operator on Ω for the system (4.7) . Then for s ≥ k P ( g ) k W s + ( Ω ) . k g b k W s ( ∂ Ω ) . Proof.
We use the representation P ( g ) = Ψ − g + R − ∞ as in Theorem 4.1, wherethe Ψ − operator is decomposable. The estimates then follow from Theorem 2.4and Lemma 4.2. (cid:3) Included in the proof of Theorem 4.1 is the calculation of the highest order termof the DNO; from (4.16) we have in particular the first component of the DNO,which we write as, N − (the − superscript to denote we compute the outwardpointing normal derivative): Theorem 4.4. (4.20) N − g = | D | g b + Ψ b ( g b ) + R − ∞ b .We now want to write out the highest order terms included in Ψ b ( g ) in (4.20).That is to say, writing ∂ ρ v (cid:12)(cid:12)(cid:12) ρ = = Λ b g b + Λ b g b + · · · , we have Λ b = | D | , and wewant to calculate an expression for the operator Λ b .Recall in (4.6) we had the relation v = ( π ) n Γ − int ◦ Z (cid:18) ∂ ρ e v (cid:12)(cid:12)(cid:12) ρ = + i η e g b ( ξ ) (cid:19) e i ρη e ix · ξ d ξ d η − √ Γ − int ◦ S int v − √ Γ − int ◦ S b g b − Γ − int ◦ Av − Γ − int ◦ ( ρτ v ) modulo smooth terms. With ∂ ρ v (cid:12)(cid:12)(cid:12) ρ = = Λ b g b + Λ b g b + · · · , and using1 ( π ) n Γ − int ◦ Z (cid:18) ∂ ρ e v (cid:12)(cid:12)(cid:12) ρ = + i η e g b ( ξ ) (cid:19) e i ρη e ix · ξ d ξ d η = Θ + g + Ψ − g + R − ∞ ,we can write the relation as v = Θ + g + Ψ − g + Γ − int ◦ Λ b g − √ Γ − int ◦ S int v − √ Γ − int ◦ S b g b − Γ − int ◦ Av − Γ − int ◦ ( ρτ v ) + R − ∞ ,(4.21)where Θ + is defined by σ ( Θ + ) = i η + i | Ξ ( x , ξ ) | .The pseudodifferential calculus also yields the principal term of the Ψ − operatorin (4.21). The operator arises in the expansion of the symbol for the inverse, Γ − int : σ (cid:16) Γ − int (cid:17) = η + Ξ ( x , ξ ) + ∂ ξ Ξ · D x Ξ ( η + Ξ ( x , ξ )) + · · · .And so the principal symbol of the Ψ − operator in (4.21) is given by(4.22) ∂ ξ Ξ · D x Ξ ( η + Ξ ( x , ξ )) ( | Ξ ( x , ξ ) | + i η ) = ∂ ξ Ξ · ∂ x Ξ ( η + Ξ ( x , ξ )) ( η + i | Ξ ( x , ξ ) | ) . IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 19 For the term Λ b , we set ρ = − Ξ ( x , ξ ) . The first term, Θ + leads to a term which is homogeneous of order 0 in | Ξ ( x , ξ ) | . We go through the other terms individually. For the operator with sym-bol as in (4.22) we calculate1 ( π ) n Z ∂ ξ Ξ · ∂ x Ξ ( η + Ξ ( x , ξ )) ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ξ x d η d ξ = − i ( π ) n − Z ∂ ξ Ξ · ∂ x Ξ Ξ ( x , ξ ) e g b ( ξ ) e i ξ x d ξ .Next, we have R ◦ Γ − int ◦ Λ b g = ( π ) n − Z ] Λ b g b ( ξ ) | Ξ ( x , ξ ) | e i ξ x d ξ + Ψ − b g .For terms involving v we use the expression(4.23) v = Θ + g + Ψ − g modulo smoothing terms as in Theorem 4.1.With (4.23), and s ( x ) : = s ( x , 0 ) , we thus have R ◦ Γ − int ◦ S int v = − ( π ) n Z s ( x ) ηη + Ξ ( x , ξ ) e g b ( ξ ) η + i | Ξ ( x , ξ ) | d η e ix · ξ d ξ = − ( π ) n − Z s ( x ) e g b ( ξ ) | Ξ ( x , ξ ) | e ix · ξ d ξ ,modulo lower order terms. Note that any O ( ρ ) terms from an expansion of s ( x , ρ ) = s ( x ) + O ( ρ ) lead to lower order terms by Lemma 2.5.Next, R ◦ Γ − int ◦ S b g b = ( π ) n Z s ( x ) e g b ( ξ ) η + Ξ ( x , ξ ) d η e ix · ξ d ξ = ( π ) n − Z s ( x ) e g b ( ξ ) | Ξ ( x , ξ ) | e ix · ξ d ξ ,modulo lower order terms.Similar to the calculation involving Γ − int ◦ S int v above, we have for Γ − int ◦ AvR ◦ Γ − int ◦ Av = ( π ) n − Z a ( x , ξ ) e g b ( ξ ) Ξ ( x , ξ ) e ix · ξ d ξ modulo lower order terms, where a ( x , ξ ) = σ ( A ) (cid:12)(cid:12) ρ = .For the term, Γ − int ◦ ( ρτ v ) , we use ρτ v = ρτ ◦ Θ + g + · · · , where the · · · means lower order terms or smoothing terms. Hence, modulo lowerorder terms, we have ρτ v = ρτ ◦ Θ + g = ρ i ( π ) n ∑ jk Z τ jk ( x ) ξ j ξ k η + i | Ξ ( x , ξ ) | e g b ( ξ ) e i ρη e ix ξ d η d ξ = ( π ) n ∑ jk Z τ jk ( x ) ξ j ξ k ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ρη e ix ξ d η d ξ ,where τ jk ( x ) : = τ jk ( x , 0 ) , and thus Γ − int ( ρτ v ) = ( π ) n ∑ jk Z η + Ξ ( x , ξ ) τ jk ( x ) ξ j ξ k ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ηρ e i ξ x d η d ξ = ( π ) n ∑ jk Z η − i | Ξ ( x , ξ ) | τ jk ( x ) ξ j ξ k ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ηρ e i ξ x d η d ξ ,again, modulo lower order terms. Integrating over η and setting ρ = R ◦ Γ − int ( ρτ v ) = − ( π ) n − ∑ jk Z τ jk ( x ) ξ j ξ k | Ξ ( x , ξ ) | e g b ( ξ ) e i ξ x d ξ ,modulo Ψ − b g b and smoothing terms.We can now read off the symbols homogeneous of degree -1 with respect to | ξ | in (4.21):0 = − ( π ) n − i Z ∂ ξ Ξ · ∂ x Ξ Ξ ( x , ξ ) e g b ( ξ ) e i ξ x d ξ + ( π ) n − Z g Λ b g b | Ξ ( x , ξ ) | e i ξ x d ξ − ( π ) n − √ Z s ( x ) e g b ( ξ ) | Ξ ( x , ξ ) | e i ξ x d ξ − ( π ) n − Z a ( x , ξ ) e g b ( ξ ) Ξ ( x , ξ ) e i ξ x d ξ + ( π ) n − ∑ jk Z τ jk ( x ) ξ j ξ k | Ξ ( x , ξ ) | e g b ( ξ ) e i ξ x d ξ .Solving for σ ( Λ b )( x , ξ ) yields the Proposition 4.5. σ ( Λ b ) = √ s ( x ) + a ( x , ξ ) | Ξ ( x , ξ ) | − ∑ jk τ jk ( x ) ξ j ξ k Ξ ( x , ξ ) + i ∂ ξ Ξ · ∂ x Ξ | Ξ ( x , ξ ) | .Finally, we can state the IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 21 Theorem 4.6.
Modulo pseudodifferential operators of order − , the symbol for N − isgiven by σ ( N − )( x , ξ ) = | Ξ ( x , ξ ) | + √ s ( x ) + a ( x , ξ ) | Ξ ( x , ξ ) | − ∑ jk τ jk ( x ) ξ j ξ k Ξ ( x , ξ ) + i ∂ ξ Ξ · ∂ x Ξ | Ξ ( x , ξ ) | .This is the same as Theorem 1.2 in [1].5. T HE ZERO ORDER TERM
In this section we will look at the zero order term of the DNO, and note itspossible vanishing under the hypothesis of a weakly pseudoconvex domain. Thevector field ( L n − L n ) /2 i will play a special role in the following sections and thebehavior of the boundary value operators in its direction will be studied now. Weuse the terminology transverse tangential to refer to a vector field which is tangen-tial and transverse to the complex tangent space (also called the vector field of the”missing direction” or the ”bad direction” in the literature).We start by recalling our notation used in writing N − . Let N − denote the op-erator which is given by the principal (first order) symbol of N − , homogeneous ofdegree 1 in | Ξ ( x , ξ ) | , where Ξ ( x , ξ ) is given in (4.4):(5.1) | Ξ ( x , ξ ) | = r ξ n − + ∑ k < n σ ( L bk ) σ ( L bk ) .For the zero operator, we write the symbol a ( x , ξ ) in Theorem 4.6 according to a ( x , ξ ) = n − ∑ j = α j ( x ) ξ j .From Theorem 4.6, the zero order operator, denoted by N − , has symbol given by σ ( N − ) = √ s ( x ) + ∑ n − j = α j ( x ) ξ j | Ξ ( x , ξ ) | − τ jk ( x ) ξ j ξ k Ξ ( x , ξ ) + i ∂ ξ Ξ · ∂ x Ξ | Ξ ( x , ξ ) | ,(5.2)in a neighborhood of a boundary point, which we assume to be 0 ∈ ∂ Ω . Recall thefunctions s , α j and τ jk as defined in Section 4.According to Proposition 3.1, s ( x ) is a diagonal matrix. (i.e. there are no nor-mal derivatives of u I for I = J which contribute to the term f J ¯ ω J on the right handside of (cid:3) u = f J ¯ ω J ). We have s J , the diagonal ( J , J ) -entry of the matrix s , to be ofthe form(5.3) s J = − i ( − ) | J | ℑ ( c JJn ) + d n for n / ∈ J , from Proposition 3.1 i ) . Now let A J denote the J th row of the matrix of first order operators, A . We need(the vector product) A J · u , so as to calculate the J th component of Au , and in par-ticular, we will need the operators with the transversal tangential, T , componentin the expression A J · u (in applying the results to the ¯ ∂ -Neumann condition inSection 6, we are interested in the behavior of the operators in a microlocal neigh-borhood determined by ψ − ).For the contribution of the sum − ∑ l / ∈ J L l L l − ∑ l ∈ J L l L l occurring in Proposition 3.1 i ) to the A operators, we handle the case l = n sepa-rately (again, assuming n / ∈ J ): − L n L n = − (cid:18) √ ∂∂ρ + iT (cid:19) (cid:18) √ ∂∂ρ − iT (cid:19) = − ∂ ∂ρ − ∂ ∂ x n − + i √ (cid:20) ∂∂ρ , T (cid:21) + O ( ρ ) Ψ = − ∂ ∂ρ − ∂ ∂ x n − + i √ T + O ( ρ ) Ψ ,where T is defined to be h ∂∂ρ , T i at ρ = ρ = − L n L n , that is, in(5.4) 1 | T | (cid:28) i √ T , T | T | (cid:29) .We use < · , · > to denote the interior product of two vector fields. To ease notationwe will also use the notation of the dot product to denote the interior product inwhat follows. Thus 1 | T | i √ T · T | T | : = | T | (cid:28) i √ T , T | T | (cid:29) We could calculate this term explicitly, but we will not need to; it will eventuallycancel out with another term in the DNO.For L k L k , k = n , we recall (4.2) and write L k (cid:12)(cid:12)(cid:12) ρ = = (cid:18) ∂∂ x k − + i ∂∂ x k (cid:19) + n − ∑ j = ℓ kj ( x ) ∂∂ x j ,where ℓ kj ( x ) = O ( x ) . We also use the representation L k = √ ∑ γ kj ∂∂ ¯ z j , IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 23 where the γ kj have the property that ∑ j γ kj γ lj = δ kl , 1 ≤ k ≤ n − δ kl = k = l and δ kl = k = l , and ∑ j γ kj ∂ρ∂ ¯ z j =
0, 1 ≤ k ≤ n − ∈ ∂ Ω , we can write L k (cid:12)(cid:12)(cid:12) p = = (cid:18) ∂∂ x k − + i ∂∂ x k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p = = √ ∑ γ kj ( ) ∂∂ ¯ z j ,for k ≤ n − L k · T = ℓ k n − ( x ) = − √ ∑ γ kj ( ) ∂∂ ¯ z j ! · T + O ( x )= − √ i ∑ γ kj ( ) ∂∂ ¯ z j ! · L n + O ( x )= i ∑ γ kj ( ) ∂ρ∂ ¯ z j ! + O ( x ) ,where we use T · ∂ x j = O ( x ) for j = n − T · T = and L n = √ ∑ ∂ρ∂ z j ∂∂ ¯ z j .Hence, ℓ k n − ( x ) = i ∑ γ kj ( ) ∂ρ∂ ¯ z j + O ( x ) .With L k = √ ∑ ¯ γ kl ∂∂ z l ,we can write the coefficient of the transverse tangential vector field, T , in the ex-pression from − L k L k as − L k ( ℓ k n − ( x )) = − i ∑ γ kj ( ) L k ∂ρ∂ ¯ z j ! + O ( x )= − i √ ∑ γ kj ( ) ¯ γ kl ∂ ρ∂ z l ¯ z j + O ( x )= − i √ | L k | L + O ( x ) ,(5.5) where | · | L refers to the length with respect to the Levi metric, which is define by ds = ∑ ∂ ρ∂ z l ¯ z j dz l d ¯ z j .That is, − L k L k = − ∂ ∂ x k − + ∂ ∂ x k ! − i √ | L k | L ∂∂ x n − + · · · where the · · · refer to second order terms with coefficients in O ( x ) or first orderterms which upon contraction with T result in O ( x ) functions. And similarly, − L k L k = − ∂ ∂ x k − + ∂ ∂ x k ! + i √ | L k | L ∂∂ x n − + · · · .Thus the transverse tangential component to be included in the operator A ofthe first order vector fields from − ∑ k / ∈ J L k L k − ∑ k ∈ J L k L k written in our localcoordinates is given by(5.6) i √ ∑ k ∈ J | L k | L − ∑ k / ∈ J | L k | L ! .From the first order operators in Proposition 3.1, we see there are also the T components to be included in the operator A given by(5.7) − ( − ) | J | i ℜ ( c JJn ) − id n .We now move to the operator τ . From Proposition 3.1, τ is a diagonal operator.Let us calculate the asymptotic behavior of the entries of the symbol of τ for large | ξ n − | . Recall that in the τ operator, we collected all the second order tangentialderivatives with coefficients which are O ( ρ ) .We expand(5.8) T = T + ρ T + ρ T + · · · and L j = L j + ρ L j + ρ L j + · · · for 1 ≤ j ≤ n − O ( ρ ) operators arise from − L n L n = − (cid:18) √ ∂∂ρ + i ( T + ρ T + · · · ) (cid:19) (cid:18) √ ∂∂ρ − i ( T + ρ T + · · · ) (cid:19) = · · · − ρ T T − ρ T T + · · · IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 25 as well as − L j L j = − (cid:16) L j + ρ L j + · · · (cid:17) (cid:16) L j + ρ L j + · · · (cid:17) = · · · − ρ L j L j − ρ L j L j + · · · .We specifically want, τ n − n − , the coefficient of ∂ ∂ x n − in (the diagonal com-ponents of) τ . Thus, for instance, from the L n L n , τ n − n − contains the coefficientof − (cid:20)(cid:18) T · T | T | (cid:19) T | T | (cid:21) T − T (cid:20)(cid:18) T · T | T | (cid:19) T | T | (cid:21) ,i.e.,(5.9) τ n − n − = − √ (cid:28) T , T | T | (cid:29) ,since there are no contributions from − L j L j , in the form of − L j L j and − L j L j ,due to the property that L j · T =
0. As we mentioned earlier, we will have no needto calculate explicitly the interior product.Furthermore, we can handle the last term in (5.2) by noting that ∑ ∂ x j Ξ ( x , ξ ) ∂ ξ j Ξ ( x , ξ ) | Ξ ( x , ξ ) | = (cid:0) O ( | ξ L || ξ n − | ) + O ( ξ L ) + O ( x ) O ( ξ n − ) (cid:1) · O ( ξ ) | Ξ ( x , ξ ) | = O (cid:18) | ξ L || Ξ ( x , ξ ) | (cid:19) + O ( x ) ,(5.10)for large ξ n − .Lastly, to handle the non-diagonal terms in σ ( N − ) we consider the transversetangential components of the terms in Proposition 3.1 ii ) . Noting that [ L j , L k ] · T give the entries for the Levi matrix, and assuming without loss of generality thatthe Levi matrix is diagonal (at the given point 0 ∈ ∂ Ω ), the contributions of suchcomponents in the transverse tangential direction are O ( x ) . The non-diagonalterms in σ ( N − ) are thus in O (cid:16) | ξ L || Ξ ( x , ξ ) (cid:17) + O ( x ) .In the expression for the zero order term of N − we write ( b J ) J to mean thediagonal matrix whose ( J , J ) th entry is given by b J . All terms in the expression for σ ( N − ) , with the exception of error terms, will be diagonal matrices. Using (5.3),(5.4), (5.6), (5.7), and (5.9) in the expression for σ ( N − ) in (5.2), and restricting tothe boundary, we have Proposition 5.1.
Let σ ( N − ) be the zero order symbol in the expansion of the DNO as-sociated with the (cid:3) operator. Then in a microlocal neighborhood of the boundary point ∈ ∂ Ω for large | ξ n − | we have σ ( N − ) = √ (cid:16) − i ( − ) | J | ℑ ( c JJn ) + d n (cid:17) J + (cid:18) ( − ) | J | ℜ ( c JJn ) + d n − (cid:28) T , T | T | (cid:29)(cid:19) J ξ n − | Ξ ( x , ξ ) |− √ ∑ k ∈ J | L bk | L − ∑ k / ∈ J | L bk | L ! J ξ n − | Ξ ( x , ξ ) |− √ (cid:18)(cid:28) T , T | T | (cid:29)(cid:19) J ξ n − Ξ ( x , ξ ) + O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) + O ( x ) .6. B OUNDARY EQUATION
The ¯ ∂ -Neumann problem for a ( q ) -form f ∈ L ( q ) ( Ω ) is to find a solution u ∈ L ( q ) ( Ω ) to (cid:3) u = f . As the (cid:3) operator consists of ¯ ∂ ∗ operators, boundaryconditions arise on u so as to fulfill conditions regarding its inclusion in the do-main of ¯ ∂ ∗ . The ¯ ∂ -Neumann is the boundary value problem (cid:3) u = f in Ω with boundary conditions u ⌋ ¯ ∂ρ = ∂ u ⌋ ¯ ∂ρ = ∂ Ω .The first boundary condition u ⌋ ¯ ∂ρ = u J = ∂ Ω for any J such that n ∈ J . For the second condition involving ¯ ∂ u , we note¯ ∂ u = ∑ J n (cid:16) ( − ) | J | L n u J + c JJn u J + ε kJ ˆ kn Jn L k u J ˆ k n (cid:17) ¯ ω Jn + · · · ,where · · · refers to terms with no ¯ ω n component.Assuming the boundary condition u ⌋ ¯ ∂ρ =
0, we have ¯ ∂ u ⌋ ¯ ∂ρ = L n u J + ( − ) | J | c JJn u J = ∂ Ω for J such that n / ∈ J .We write the solution u in terms of a Green’s solution and Poisson solution: u = G ( f ) + P ( u b ) IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 27 where the operators G and P satisfy(6.2) 2 (cid:3) ◦ G = IR ◦ G = (cid:3) ◦ P = R ◦ P = I ,respectively.The J th component will be written u J = G J ( f ) + P J ( u b ) . L n u J can now bewritten on ∂ Ω as R ◦ L n u J = √ R ◦ ∂ ρ ◦ G J ( f ) + (cid:18) √ N − − iT (cid:19) u b , J ,where u b , J is the J th component of u b .From [4] (Theorem 3.3), we use the the property that R ◦ ∂ ρ ◦ G J ≡ R ◦ Ψ − modulo smoothing terms. The boundary condition (6.1) for J n can therefore bewritten as (cid:18) √ N − − iT (cid:19) u b , J + ( − ) | J | c JJn u b , J + √ (cid:0) N − u b (cid:1) J = R ◦ Ψ − f ,modulo lower order terms in u b . As mention above in Section 5, we will concen-trate on the microlocal region determined by the symbol, ψ − , that is, the regionin which (in local coordinates) ξ n − is large and negative. The reason is thatin the other regions, estimates for u b can be obtained by inverting the operator,1/ √ N − − iT .We will need the behavior of the operators in the microlocal neighborhood ofa boundary point, 0 ∈ ∂ Ω and with support in the support of the symbol ψ − . Tothis end, we first consider the limit of N − as ξ n − → − ∞ .Let us write (cid:0) N − u b (cid:1) J = N − J u b , J + N − JX u b ,where N − J is the ( J , J ) entry in the matrix N − and N − J X is the matrix consisting ofthe J th row of N − , with the ( J , J ) entry replaced with 0, and zeros elsewhere.From Proposition 5.1, σ (cid:16) N − J X (cid:17) = O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) + O ( x ) , and as ξ n − → − ∞ , we see σ ( N − J ) → √ (cid:16) − i ( − ) | J | ℑ ( c JJn ) + d n (cid:17) − √ (cid:18) ( − ) | J | ℜ ( c JJn ) + d n − (cid:28) T , T | T | (cid:29)(cid:19) + ∑ k ∈ J | L bk | L − ∑ k / ∈ J | L bk | L − √ (cid:28) T , T | T | (cid:29) + O ( x ) + O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) = − ( − ) | J | √ c JJn + ∑ k ∈ J | L bk | L − ∑ k / ∈ J | L bk | L + O ( x ) + O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) .(6.4)We could also at this point proceed to calculate each of the c JJn , but as we willsee, these will also cancel in what follows. We will denote the zero order operator ( − ) | J | c JJn + √ N − (with c JJn referring to the operator with a single diagonal entry)by Υ J . From above we have σ (cid:16) Υ J (cid:17) → ( − ) | J | c JJn + √ − ( − ) | J | √ c JJn + ∑ k ∈ J | L bk | L − ∑ k / ∈ J | L bk | L ! + O ( x ) + O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) = √ ∑ k ∈ J | L bk | L − ∑ k / ∈ J | L bk | L ! + O ( x ) + O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) ,as ξ n − → − ∞ , recalling that N − J X = O ( x ) + O (cid:16) | ξ L || Ξ ( x , ξ ) (cid:17) .We collect our results in the following Proposition Proposition 6.1.
The boundary equation for the ¯ ∂ -Neumann problem has the form (6.5) (cid:18) √ N − − iT (cid:19) u b , J + Υ J u b = R ◦ Ψ − f , where (6.6) Υ J u b = Υ J , J u b , J + ∑ K = J Υ J , K u b , K , and Υ J , J is a psedodifferential operator of order 0, whose symbol has the property (6.7) σ ( Υ J , J ) = √ ∑ k ∈ J | L bk | L − ∑ k / ∈ J | L bk | L ! + O ( x ) + O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) and Υ J , K is a psedodifferential operator of order 0, whose symbol has the property (6.8) σ ( Υ J , K ) = O ( x ) + O (cid:18) | ξ L || Ξ ( x , ξ ) (cid:19) IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 29 for K = J. At this point, we take a moment to review how previous work on inverting theKohn Laplacian, (cid:3) b , defined on the boundary, could be useful in solving (6.5). Aninverse to (cid:3) b in the case of strictly pseudoconvexity was studied in [6], and we firstrelate our boundary equation (6.5) to that of [6]. We simplify our equation, throw-ing away the O ( x ) and O (cid:16) | ξ L || Ξ ( x , ξ ) (cid:17) terms (for the purpose of illustration only) andconsider (cid:18) √ N − − iT (cid:19) u b , J + Y J u b , J = R ◦ Ψ − f ,with σ ( Y J ) = √ ∑ k ∈ J | L bk | L − ∑ k / ∈ J | L bk | L ! .We now apply the operator √ N − + iT to both sides: (cid:18) ( N − ) + ( T ) (cid:19) u b , J + i √ [ T , N − ] u b , J + (cid:18) √ N − + iT (cid:19) ◦ Y J u b , J = R ◦ Ψ f .(6.9)We first note some properties of the operators involved. Consider the first orderoperator [ T , N − ] . By expanding the symbol for N − for large | ξ n − | , we see (forlarge | ξ n − | ) σ ([ T , N − ]) = ∂ x n − σ (cid:0) N − (cid:1) = O ( | ξ L | ) + O ( x ) O ( | ξ | ) modulo S − ∞ ( ∂ Ω ) . We also write the operator ( N − ) + ( T ) in terms of thevector fields L j and L j . The symbol of ( N − ) is given by σ (cid:16) ( N − ) (cid:17) = σ ( N − ) σ ( N − ) − i ∂ ξ σ ( N − ) · ∂ x σ ( N − ) + · · · = Ξ ( x , ξ ) + O ( | ξ L | ) + O ( x ) (6.10)modulo S − ( ∂ Ω ) . For the term Ξ ( x , ξ ) we have from (4.4) Ξ ( x , ξ ) = ξ n − + ∑ j σ ( L bj ) σ ( L bj ) and for σ ( L bj ) σ ( L bj ) we have the relations σ ( L bj L bj ) = σ ( L bj ) σ ( L bj ) − i ∑ j ∂ ξ σ ( L bj ) · ∂ x σ ( L bj )= σ ( L bj ) σ ( L bj ) + L bj ( ℓ j n − )( i ξ n − ) + O ( x ) O ( | ξ | )= σ ( L bj ) σ ( L bj ) + √ ξ n − | L bj | L + O ( x ) O ( | ξ | ) , modulo ξ L terms and symbols of class S , where we use (5.5) in the last line.Similarly, we have for σ ( L bj ) σ ( L bj ) σ ( L bj L bj ) = σ ( L bj ) σ ( L bj ) − √ ξ n − | L bj | L + O ( x ) O ( | ξ | ) + · · · .Then the expression for Ξ ( x , ξ ) gives Ξ ( x , ξ ) = ξ n − + ∑ k / ∈ J σ ( L bk L bk ) + ∑ k ∈ J σ ( L bk L bk ) − √ ξ n − ∑ k / ∈ J | L bk | L + √ ξ n − ∑ k ∈ J | L bk | L + O ( x ) O ( | ξ | ) + O ( | ξ L | ) ,which, combined with the expression in (6.10) above, yields (for the ( J , J ) -entry) σ (cid:18) ( N − ) + ( T ) (cid:19) = ∑ k / ∈ J σ ( L bk L bk ) + ∑ k ∈ J σ ( L bk L bk ) − √ ξ n − ∑ k / ∈ J | L bk | L + √ ξ n − ∑ k ∈ J | L bk | L + O ( x ) O ( | ξ | ) + O ( | ξ L | ) .Furthermore, σ (cid:18)(cid:18) √ N − + iT (cid:19) ◦ Y J (cid:19) = √ | ξ n − | ∑ k ∈ J | L k | L − ∑ k / ∈ J | L k | L ! + O ( | ξ L | ) + O ( x ) O ( | ξ | ) for | ξ L | ≪ | ξ n − | , and ξ n − <
0, modulo lower order symbols.(6.9) is thus reduced to studying ∑ k / ∈ J L bk L bk + ∑ k ∈ J L bk L bk modulo first order operators with symbols which can be made arbitrarily small ina microlocal neighborhood of the boundary point 0 ∈ ∂ Ω for | ξ L | ≪ ξ n − .In the highest order, this is just the Kohn Laplacian, (cid:3) b which, under the hy-pothesis of strict pseudoconvexity, can be inverted by analyzing the operator onthe Heisenberg group, as in [6], or in the case of finite type by considering relationsof commutators of the vector fields, L k and their conjugates, as in [11]. The prob-lem in the case of weak pseudoconvexity is that the means to control derivativesin the direction of T , namely through commutators of the vector fields, L j , withvector fields, L k , is no longer available.One of the immediate difficulties in using the method of applying the boundaryoperator √ N − + iT as above leading to (6.9) is that the resulting highest ordersymbol, σ (cid:18) ( N − ) + ( T ) (cid:19) IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 31 is not elliptic. It is missing estimates from below by the ξ n − transform variable.In other words, an estimate of the form σ (cid:18) ( N − ) + ( T ) (cid:19) & + | ξ n − | for | ξ L | ≫ σ (cid:20)(cid:18) √ N − + iT (cid:19) ◦ Υ J (cid:21) & + | ξ n − | and use the missing first order estimate as a (weaker) substitute for an ellipticsecond order estimate. This idea is used in [5] to obtain (weighted) estimates ofthe boundary solution.The aim of the next sections is to show how persistent the absence of ellipticityin the boundary equation is.7. V ARIATIONS OF THE (cid:3)
OPERATOR
In this section we consider operators obtained from the (cid:3) operator by addingadditional terms. In particular, we let φ be a function supported near the boundaryand with (cid:3) φ = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ ◦ ( + φ ) , we consider the boundary value problem: (cid:3) φ u = f with the boundary conditions,(7.1) u ⌋ ¯ ∂ρ = ∂ (cid:0) ( + φ ) u (cid:1) ⌋ ¯ ∂ρ = ∂ Ω . The first condition ensures u ∈ dom ( ¯ ∂ ∗ ) and the second that¯ ∂ (cid:0) ( + φ ) u (cid:1) ∈ dom ( ¯ ∂ ∗ ) .We first look at the case φ only depends on ρ : φ = φ ( ρ ) , and φ ( ) =
0, andwe use the notation from the previous sections. In this case the condition ¯ ∂ (cid:0) ( + φ ) u (cid:1) ⌋ ¯ ∂ρ = ∑ k ( + φ ) L k u J ˆ k n + ( + φ )( − ) | J | L n u J + ( − ) | J | ( L n φ ) u J + ( + φ ) c JJn u J = u ⌋ ¯ ∂ρ =
0, and recalling φ ( ) = L n u J + ( L n φ ) u J + ( − ) | J | c JJn u J = L n φ allows for a strictly positive (diagonal) addition to the Υ J operator. We repeat the steps of the previous sections to obtainan expression of (7.2) in terms of the complex tangential vector fields, L j ; as before,the main calculation concerns the DNO.To recall, we write u J as a sum of solutions to Dirichlet problems, the solutionswritten in terms of Green’s operator and a Poisson operator (for the analogues tothe systems, (6.2) and (6.3), with (cid:3) replaced by (cid:3) φ ): u = G φ ( f ) + P φ ( u b ) .Also, we have L n P φ ( u b ) (cid:12)(cid:12)(cid:12) ρ = = (cid:18) √ ∂ ρ − iT (cid:19) P φ ( u b ) (cid:12)(cid:12)(cid:12) ρ = = √ N φ , − u b − iT u b ,and L n G φ ( f ) (cid:12)(cid:12)(cid:12) ρ = = √ ∂ ρ G φ ( f ) (cid:12)(cid:12)(cid:12) ρ = = R ◦ Ψ − f .We can now rewrite (7.2) as(7.3) (cid:18) √ N φ , − − iT (cid:19) u bJ + (cid:18) √ φ ′ ( ) + ( − ) | J | c JJn (cid:19) u bJ = R ◦ Ψ − f .As in the case with φ ≡
0, the operator √ N φ , − − iT is of first order, but it isnot elliptic since its principal symbol,1 √ | Ξ ( x , ξ ) | + ξ n − tends to 0 as ξ n − → − ∞ . However, a non-vanishing zero order term in thesymbol expansion of N φ , − would, after composition with √ N φ , − + iT lead toa first order term whose symbol is non-vanishing in the support of ψ − . We thusexamine the term σ ( N φ , − ) .For φ =
0, we have Theorem 4.6. In the case φ = φ ( ρ ) = (cid:3) φ and (cid:3) are iden-tical, so we determine the operators, S φ , A φ , and τ φ , (and their correspondingsymbols) with which (cid:3) φ v can be written as in (4.5) in the interior of Ω as Γ v + √ S φ (cid:18) ∂ v ∂ρ (cid:19) + A φ v + ρτ φ ( v ) = (cid:3) φ in local coordinates. We have (cid:3) φ = (cid:3) + ¯ ∂ ∗ ¯ ∂φ . IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 33 In Proposition 3.1, we examined the forms which, upon action through the (cid:3) op-erator would result in terms with a certain ¯ ω J component. We follow this sameapproach here for the operator (cid:3) φ in order to obtain an expression for the DNOcorresponding to the (cid:3) φ operator.We examine the term ¯ ∂ ∗ ¯ ∂ ( φ u ) . We have(7.4) ¯ ∂ ( φ u J ¯ ω J ) = ( − ) | J | ( L n φ ) u J ¯ ω J ∧ ¯ ω n + φ ( ρ ) ∑ J ( − ) | J | ( L n u J ) ¯ ω J ∧ ¯ ω n + · · · ,where the · · · refer to forms with no ¯ ω n component, or, in the case n ∈ J involveonly the L j operators for j =
1, . . . , n − ∂ ∗ v ¯ ω J ∧ ¯ ω n = (cid:16) ( − ) | J | ( − L n + d n ) + c JJ ∪{ n } (cid:17) v ¯ ω J + · · · ,where here the · · · denote terms which are orthogonal to ¯ ω J , we get¯ ∂ ∗ ¯ ∂ ( φ u J ¯ ω J )= (cid:18) − φ ( ρ ) L n L n u J − ( L n φ ) L n u J − ( L n φ ) L n u J + φ ( ρ ) (cid:16) d n + ( − ) | J | c JJ ∪{ n } (cid:17) L n u J (cid:19) ¯ ω J + · · · = (cid:18) − φ ( ρ ) L n L n u J − φ ′ ( ρ ) ∂ ρ u J + φ ( ρ ) (cid:16) d n + ( − ) | J | c JJ ∪{ n } (cid:17) L n u J (cid:19) ¯ ω J + · · · .Again, for the error terms we include all 0 order terms, terms orthogonal to ¯ ω J ,and terms involving only L j and/or L j for j =
1, . . . , n − · · · .From (3.5), we have¯ ∂ ∗ ¯ ∂ ( φ ( ρ ) u kl ¯ ω J ˆ k ∪{ l } ) = − φ ( ρ ) ε l JJ ∪{ l } ε kJ ˆ k ∪{ l } J ∪{ l } L l L k u kl ¯ ω J + · · · for l = n and J n , where here the · · · refer to terms which are of the form O ( ρ ) L j + O ( ρ ) L j , or are of order 0, or are terms orthogonal to ¯ ω J . In the case l = n we have¯ ∂ ∗ ¯ ∂ ( φ u kn ¯ ω J ˆ k ∪{ n } ) = − ( − ) | J | ε kJ ˆ k J √ φ ′ ( ρ ) L k u kn ¯ ω J − ( − ) | J | ε kJ ˆ k J φ ( ρ ) L n L k u kn ¯ ω J + · · · . From these calculations, we see that in a small neighborhood of a boundarypoint p ∈ ∂ Ω , for which again we assume p =
0, the equation 2 (cid:3) φ v = v J = n ∈ J can be written − ∂ ∂ρ + ∑ j ∂ ∂ x j + ∂ ∂ x n − + n − ∑ j , k = l jk ∂ ∂ x j ∂ x k ! v + √ S φ (cid:18) ∂ v ∂ρ (cid:19) + A φ v + ρτ φ ( v ) = S φ = S − √ φ ′ ( ρ ) + O ( ρ ) , A φ = A + O ( ρ ) ,and τ φ = τ − φ ( ρ ) L n L n + · · · where S , A , and τ are the operators from Section 4, and the · · · in the expressionfor τ φ refer to second order terms which are O ( ρ ) , and are compositions with atleast one L k or L k for k ∈ {
1, . . . , n − } (this also holds true in the case n ∈ J ,although is not needed).We now examine the contributions from the φ function to the DNO. UsingLemma 2.5 the O ( ρ ) terms of the operator S φ and A φ above lead to operatorsof order − s ( x ) for the DNOcorresponding to 2 (cid:3) should be replaced with s ( x ) − √ φ ′ ( ) .For the contributions from the τ φ operator we expand φ ( ρ ) = φ ′ ( ) ρ + O ( ρ ) and look at the terms − φ ′ ( ) ρ (cid:18) ∂ ρ + T (cid:19) coming from − φ ( ρ ) L n L n in τ φ . A term ρ∂ ρ v can be written using transforms,assuming the support of v is contained in a small coordinate patch around 0 ∈ ∂ Ω ,as ρ ( π ) n Z (cid:16) − η b v ( ξ , η ) + ∂ ρ e v ( ξ , 0 ) + i η e g b ( ξ ) (cid:17) e ix ξ e i ρη d ξ d η .Since ρ · δ ( ρ ) ≡
0, we have ρ Z ∂ ρ e v ( ξ , 0 ) e i ρη e ix · ξ d ξ d η ≡ ρ∂ ρ v = ρ ( π ) n Z (cid:16) − η b v ( ξ , η ) + i η e g b ( ξ ) (cid:17) e ix ξ e i ρη d ξ d η . IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 35 We examine first the term ρ R η b ve ix ξ e i ρη d ξ d η , recalling that v can be written as v = Θ + g modulo lower order terms: ρ ( π ) n Z (cid:16) − η b v ( ξ , η ) (cid:17) e ix ξ e i ρη d ξ d η = − ρ i ( π ) n Z η η + i | Ξ ( x , ξ ) | e g b ( ξ ) e i ρη e ix ξ d η d ξ = − ( π ) n Z η ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ρη e ix ξ d η d ξ + ( π ) n Z ηη + i | Ξ ( x , ξ ) | e g b ( ξ ) e i ρη e ix ξ d η d ξ ,modulo lower order terms and smooth terms (of the form R − ∞ ). In the calculationof the DNO, the above term contributes(7.6) 2 | Ξ ( x , ξ ) | Γ − int ◦ φ ′ ( ) ρ ◦ F . T . − (cid:16) η b v (cid:17) (see the calculation preceding Proposition 4.5). We thus need φ ′ ( ) Γ − int ◦ ρ F . T . − (cid:16) η b v (cid:17) = φ ′ ( )( π ) n Z η + Ξ ( x , ξ ) η ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ηρ e i ξ x d η d ξ − φ ′ ( )( π ) n Z η + Ξ ( x , ξ ) ηη + i | Ξ ( x , ξ ) | e g b ( ξ ) e i ηρ e i ξ x d η d ξ = φ ′ ( )( π ) n Z η − i | Ξ ( x , ξ ) | η ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ηρ e i ξ x d η d ξ − φ ′ ( )( π ) n Z η − i | Ξ ( x , ξ ) | η ( η + i | Ξ ( x , ξ ) | ) e g b ( ξ ) e i ηρ e i ξ x d η d ξ ,again, modulo lower order terms and smooth terms. Integrating over η and setting ρ = φ ′ ( ) ( π ) n − Z e g b ( ξ ) | Ξ ( x , ξ ) | e i ξ x d ξ − φ ′ ( ) ( π ) n − Z e g b ( ξ ) | Ξ ( x , ξ ) | e i ξ x d ξ ,modulo Ψ − b g b and smoothing terms. When setting the terms of order − | Ξ ( x , ξ ) | factors equal as we did to show Proposition 4.5, we are led to the symbols φ ′ ( ) − φ ′ ( ) = − φ ′ ( ) .for the contribution of (7.6) in the DNO for the operator 2 (cid:3) φ .We further need the contribution of the boundary term, g b in (7.5) to the DNO.Similar to above, the contribution comes through − | Ξ ( x , ξ ) | φ ′ ( ) Γ − int ◦ ρ F . T . − (cid:0) i η e g b ( ξ ) (cid:1) for which we have φ ′ ( ) Γ − int ◦ ρ F . T . − (cid:0) i η e g b ( ξ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = = − φ ′ ( )( π ) n Z η ( η + Ξ ( x , ξ )) e g b ( ξ ) e i ξ x d η d ξ = − φ ′ ( ) ( π ) n − Z e g b ( ξ ) | Ξ ( x , ξ ) | e i ξ x d η d ξ ,modulo lower order and smoothing terms.As in the calculations of Theorem 4.6 the − φ ′ ( ) ρ T terms lead to a term withsymbol − φ ′ ( ) ξ n − Ξ ( x , ξ ) in the DNO.We note the O ( ρ ) second order terms with at least one of L k or L k with k ∈{
1, . . . , n − } lead to terms O (cid:16) | ξ L || Ξ ( x , ξ ) | (cid:17) . Therefore, the contributions from theoperator τ φ in addition to those from τ are given by adding − φ ′ ( ) + φ ′ ( ) − φ ′ ( ) ξ n − Ξ ( x , ξ ) to the DNO for 2 (cid:3) . Note this term tends to 0 as ξ n − → − ∞ .We thus have the following description of the DNO in a microlocal neighbor-hood in the support of ψ − : Proposition 7.1.
Modulo pseudodifferential operators of order − , the symbol for N φ , − is given by σ ( N φ , − )( x , ξ ) = σ ( N − )( x , ξ ) − φ ′ ( ) + O (cid:18) | ξ L || Ξ ( x , ξ ) | (cid:19) .Returning to the boundary conditions, we see how the additional terms fromthe DNO coming from the added φ ( ρ ) function affect the boundary equations(7.1). The first condition, u ⌋ ¯ ∂ρ remains the same, and is equivalent to u J = n ∈ J .We recall the second condition written as in (7.3): (cid:18) √ N φ , − − iT (cid:19) u bJ + (cid:18) √ φ ′ ( ) + ( − ) | J | c JJn (cid:19) u bJ = R ◦ Ψ − ( f ) .From Proposition 7.1 we can write σ ( N φ , − ) = σ ( N − ) − φ ′ ( ) + O (cid:16) | ξ L || Ξ ( x , ξ ) | (cid:17) , mod-ulo lower order symbols. In particular, the φ ′ ( ) term in the boundary equationcancels with that coming from the DNO. We can state the IRICHLET TO NEUMANN OPERATORS AND THE ¯ ∂ -NEUMANN PROBLEM 37 Theorem 7.2.
Let (cid:3) φ = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ ◦ ( + φ ) . Let φ = φ ( ρ ) be a smooth function whichdepends only on the defining function, with the property φ ( ρ ) = O ( ρ ) . The condition ¯ ∂ ◦ ( + φ ) u ∈ dom ( ¯ ∂ ∗ ) , equivalent to ¯ ∂ (cid:0) ( + φ ) u (cid:1) ⌋ ¯ ∂ρ = on ∂ Ω , has the form (cid:18) √ N − − iT (cid:19) u b , J + Υ J u b = R ◦ Ψ − fas in (6.5) of Proposition 6.1, with Υ J sharing the same properties as those of (6.6) , (6.7) and (6.8) . R EFERENCES[1] D. C. Chang, A. Nagel, and E. Stein. Estimates for the ¯ ∂ -Neumann problem in pseudoconvexdomains of finite type in C . Acta Math. , 169:153–228, 1992.[2] M. Christ. On the ¯ ∂ equation for three-dimensional CR manifolds. Proc. Sympos. Pure Math. ,52(3):63–82, 1991.[3] D. Ehsani. Exact regularity of the ¯ ∂ -problem with dependence on the ¯ ∂ b -problem on weakly pseu-doconvex domains in C . Preprint.[4] D. Ehsani. Pseudodifferential analysis on domains with boundary. Preprint.[5] D. Ehsani. Weighted estimates for the ¯ ∂ -neumann problem on intersection domains in C .Preprint.[6] G. Folland and E. Stein. Estimates for the ¯ ∂ b complex and analysis on the Heisenberg group. Comm.Pure Appl. Math. , 27:429–522, 1974.[7] J. Kohn and A. Nicoara. The ¯ ∂ b equation on weakly pseudoconvex CR manifolds of dimension 3. J. Funct. Anal. , 230(2):251–272, 2006.[8] J.J. Kohn. Estimates for ¯ ∂ b on pseudoconvex CR manifolds. Proc. Sympos. Pure Math. , 43:207–217,1985.[9] J.-L. Lions and E. Magenes.
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