5.1. Stratifications and higher symbols for the Laplacian
A good example of boundary value problems is the mixed elliptic boundary value problems for elliptic differential operators in a domain M in Rn with smooth boundary ∂M = X. Here we focus on the Zaremba problem for the Laplace oper-ator where X is subdivided into sub manifolds X±with common smooth boundary Y of codimension 1, X = X−∪ X+and Y = X−∩ X+. On int X− we pose Dirich-let, on int X+ Neumann conditions. This problem has been investigated for a very long time, see Zaremba [59] and the subsequent development. For applications it is also interesting to admit domains with non-smooth boundary, e.g., polyhedral domains, and also interfaces with singularities.
Mixed elliptic problems can be studied from the point of view of pseudo-differential analysis with symbolic structures reflecting not only the operators in the domain and the boundary conditions but also the role of the interface Y which is here interpreted as an edge embedded on the boundary X. With the Zaramba problem we associate a stratification
(5.1) s(M ) = (s0(M ), s1(M ), s2(M ))
of M , where M = s0(M ) ∪ s1(M ) ∪ s2(M ) for the strata s2(M ) := Z, s1(M ) :=
Y \ Z, s0(M ) := M \ Y . Those are smooth manifolds of different dimension.
According to the general philosophy of pseudo-differential operators A on such spaces, especially, differential operators, we observe a principal symbolic hierarchy (5.2) σ(A) := (σ0(A), σ1(A), σ2(A))
that determines the ellipticity. Generalities on operators on stratified spaces M of some singularity order k ∈ N are developed in [44], [46]. Here k = 0 indicates smoothness, k = 1 conical or edge singularities, k = 2 corners of second order, etc.. The components of (5.2) are associated with the strata in (5.1). In particular, σ0(A) is the standard homogeneous principal symbol of A over s0(M ), moreover, σ1(A) is the boundary symbol, in the present case referring to int Y±, and σ2(A) is the edge symbol. For instance, if A is the Laplacian Pn
j=1
∂2
∂x2j in Rn+ = {x = (x1, . . . , xn) ∈ Rn: xn> 0}, we have σ0(A) = −|ξ|2with ξ being the covariable of x,
(5.3) σ1(A)(η) = −|η|2+ ∂2
∂x2n
where η is the covariable of y = (x1, . . . , xn−1) ∈ Rn−1 when we represent the operator in local coordinates x = (y, xn) close to the boundary, with xn ∈ R+
being the local variable normal to the boundary. We see that (5.3) is operator valued, and we take it as a family of operators Hs(R+) → Hs−2(R+) parametrised by η 6= 0 (and s ∈ R not too small, see the explanations in [45] with respect to the interface). Finally, if we locally represent Z as the hyperplane defined by xn−1= 0,
xn= 0 in x = (z, xn−1, xn) ∈ Rn+with ζ ∈ Rn−2being the covariable of z ∈ Rn−2, then the symbol σ2(A)(ζ) is the family of operators
(5.4) σ2(A)(ζ) = −|ζ|2+ ∂2
∂x2n−1+ ∂2
∂x2n
operating as Hs(R2+) → Hs−2(R2+), parametrised by ζ 6= 0 (and s ∈ R again not too small), where
R2+= {(xn−1, xn) ∈ R2: xn−1∈ R, xn> 0}.
The idea of the analysis of elliptic boundary value problems is to add conditions (such as Dirichlet or Neumann conditions) over s1(M ) and to understand the nature of parametrices. As soon as we have no jump of the boundary conditions and impose the Shapiro-Lopatinskij condition then we are in the framework of
“smooth” elliptic boundary value problems. The boundary operators also have a symbolic structure, and there is a well-known pseudo-differential calculus of boundary value problems, cf. Boutet de Monvel [7] and the material of Section 4, that contains the elliptic elements themselves together with their parametrices.
In this case it is adequate to consider 2 × 2 block matrices (4.16) with A in the upper left corner, in general, together with so-called Green operators. Moreover, T represents the boundary (or trace) condition while an additional potential operator K appears, and Q is a pseudo-differential operator on the boundary. The boundary symbolic map σ1then also applies to the other entries in (4.16).
In the Subsection 5.2, we analyze the first order pseudo-differential operator R that is obtained by reducing the Neumann boundary condition to the boundary.
The operator R, often referred to as the Dirichlet-to-Neumann operator, has been widely studied by many authors. Here we look at X+ and realize R by a specific edge quantization as an element of the edge pseudo-differential calculus with Y as the edge, operating in weighted edge spaces.
Moreover, as we have seen in Chang, Habal, and Schulze [9], the Dirichlet-to-Neumann operator R has no the transmission property with respect to any interface on X. Therefore, we cannot expect that the truncated operator r+Re+ induces continuous maps Hs(int X+) → Hs−1(int X+) (say, for s > −1/2). Hence, this also provides us a good example of BVPs with no transmission property 5.2. The Dirichlet-to-Neumann operator for the Zaremba problem
Let M be the closure of a smooth bounded domain in Rn with smooth boundary
∂M = X. We reduce boundary problems in M to the boundary, first in terms of operators in Boutet de Monvel’s algebra [7] and then by using tools from the calcu-lus of BVPs on a manifold with conical or edge singularities. The ‘standard’ idea is as follows. Let Ai=t(A Ti), i = 0, 1, denote the row matrix operators represent-ing two elliptic BVPs for an elliptic operator A with trace (or boundary) operators representing boundary conditions satisfying the Shapiro-Lopatinskij condition. As-sume that A is a second order elliptic differential operator in Rn with smooth
coefficients, in the simplest case the Laplacian and T0u := u|X and T1u := ∂νu|X
are Dirichlet and Neumann conditions, respectively, with ∂ν being the derivative normal to the boundary. For convenience we start with the assumption that A0is invertible, say, as an operator
A0: C∞(M ) →
C∞(M )
⊕ C∞(X) (as for the Laplacian and in numerous other cases), and by
P0= (P0 K0)
we denote its inverse. In particular, we have AP0= 1, AK0= 0. Roughly speaking, P0is Green’s operator and K0is the Poisson kernel of this boundary value problem.
This yields
(5.5) A1P0=
1 0
T1P0 T1K0
where T1K0 is often called the Dirichlet-to-Neumann operator.
To illustrate phenomena we consider the case A = ∆. However, the ideas here can be applied to more general elliptic operators A. We have
(5.6) R := T1K0∈ L1cl(X), σψ(R)(η) = c|η|
for a constant c; here σψ(·) denotes the homogeneous principal symbol of the respective operator an η the covariable on X (the absolute value refers to a Rie-mannian metric on the boundary). The operator R is elliptic, as we see from (5.6).
In general the ellipticity of operators obtained by reducing an elliptic BVP to the boundary follows from the ellipticity of both factors in (5.5). Explicit computa-tions for other concrete BVPs reduced to the boundary may be found in [22, page 26].
In more complicated situations below, i.e., when we replace A1 by a mixed boundary problem with jumps of the conditions along an interface Y of codimen-sion 1 on the boundary, the intention will be (similarly as in the smooth case) to express parametrices within a controlled operator algebra with symbolic struc-ture. In the simplest case there is no jump at all, i.e., we have a reduction of A1
to the boundary by means of A0 in the form (5.5). The idea is then to construct a parametrix A(−1)1 =: P1 of A1 in terms of the known parametrix (or inverse) P0
of A0 and a parametrix R(−1) of R. First we obtain a parametrix of (5.5) as (A1P0)(−1)=
1 0
T1P0 T1K0
(−1)
=
1 0
−R(−1)T1P0 R(−1)
. Then P1 itself follows in the form
P1= P0(A1P0)(−1)= (P0 K0)
1 0
−R(−1)T1P0 R(−1)
= (P0− K0R(−1)T1P0 K0R(−1)) =: (P1 K1).
(5.7)
In the consideration below we interpret A0 and A1as continuous operators belong to Boutet de Monvel’s calculus of pseudo-differential BVPs with the trans-mission property at the boundary, more precisely, Pi = F + Gi with F = r+F e˜ + being the truncation of a fundamental solution (or a parametrix) ˜F of ∆ in Rn to int M . Here e+ is the operator of extension by zero from int M to Rn and r+ the operator of restriction of distributions to int M . Moreover, Gi is a Green operator and Ki a potential operator in Boutet de Monvel’s calculus. The advantage of this viewpoint is that we have the principal symbolic structure of such operators, more precisely, the pair σ = (σ0, σ1) of symbols where σ0 is the standard homo-geneous principal symbol of operators over M (smooth up to the boundary) while σ1 represents the so-called principal boundary symbol. Moreover, we can freely compose operators in Boutet de Monvel’s algebra; this was done in (5.5) as well as in (5.7), and the symbols are (componentwise) multiplicative. Concerning Ti and the other operators in lower left corners, those are trace operators in Boutet de Monvel’s calculus, and they have boundary symbols as well. Parametrices Pi of Ai belong to the inverted symbolic components, i.e., σ(Ai)−1 = σ(Pi), i = 0, 1, as a componentwise relation.
In the construction of R = T1K0, the reduction of the Neumann condition to the boundary by means of the solution of the Dirichlet problem, it is not essential that P0is the inverse of A0. We are mainly interested in Fredholm operators between the chosen Sobolev spaces, and it suffices to employ P0 as a parametrix of A0. By in-terchanging the role of A0and A1for a parametrix P1of the Neumann problem we obtain A0P1=
1 0
T0P1 T0K1
modulo a smoothing operator in Boutet de Mon-vel’s calculus. Let us ignore such remainders in the following compositions. Then the composition (A0P1)(A1P0) =
for D−u := (T0u)|int X−, N+u := (T1u)|int X+ where X := ∂M is subdivided into submanifolds X+, X−with common boundary Y = X+∩ X−, in the simplest case assumed to be smooth. Then by virtue of (5.5) together with A0P0 = idHs−2(G)
we obtain
D−K0= 1 on int X−, N+K0= R on int X+. (5.9)
Thus the reduction of the mixed condition gives rise to R on the manifold X+
with boundary ∂X+ =: Y and the identity on X−. The relation (5.9) shows that in the reduction of the mixed conditions T := t(D− N+) to the boundary the resulting operator on the boundary has a jump; it is equal to the Dirichlet-to-Neumann operator R on X+ and to the identity on X−. Apart from the identity one of the main tasks is to solve a boundary value problem for R on X+ with Y = ∂X+ as the boundary. However, R fails to have the transmission property at the boundary, it has in fact, the anti-transmission property, cf. [48] and it is hard to imagine that Boutet de Monvel’s calculus extends to this case. This problem has been studied by many mathematicians, cf. Vishik, Eskin [58] or Eskin [16], see also [22], [34], and [45]. In the present paper we develop an approach that is based on a Mellin quantisation, already indicated in Dines, Liu, and Schulze [14].
Here we employ this method for the construction of a parametrix of (5.8) within a variant of edge calculus where Y plays the role of an edge. For the complete solution of this problem, we refer to the paper [9].
5.3. The Dirichlet-to-Neumann operator for the ¯∂-Neumann problem
Let Ω be a bounded domain in Cn+1 with C∞ boundary, i.e., there exists a real-valued function ρ ∈ C∞( ¯Ω) such that
∂Ω = {z ∈ Cn+1: ρ(z) = 0}
with dρ(z) 6= 0, ∀ z ∈ ∂Ω.
One of the basic problem in several complex variables is to solve the inho-mogeneous Cauchy-Riemann equation
(5.10) ∂U = f¯ in Ω
with “good” bounds on Ω, where f is a given (0, 1)-form f = Pn+1
j=1fjd¯ωj. Obvi-ous, the right-hand side of (5.10) has n+1 data but the left-hand side of (5.10) has only one function. Therefore, the system (5.10) is over-determined when n ≥ 1.
It follows that the equation (5.10) is solvable only when f satisfies a consistence condition, i.e., ¯∂f = 0. Moreover, solution for the equation (5.10) is highly non-unique. Suppose U is a solution of (5.10), then U + F is also a solution whenever F ∈ H(Ω) where H(Ω) is the set of all holomorphic functions defined on Ω.
Denote A2(Ω) = L2(Ω) ∩ H(Ω) the Bergman space. Then we can fine a
“canonical solution” U by requiring that
U ⊥ H(Ω) in A2(Ω)
which minuses the L2-norm among all solutions. In order to find the canonical solution, let us consider a first-order differential operator D and φ, ψ ∈ C∞( ¯Ω), then the formal adjoint D∗ of D can be defined as follows
Z
Ω
(Dφ) ¯ψdV = Z
Ω
φ(D∗ψ)dV + Z
∂Ω
φ(A]ψ)dσ,
where A]is a 0th-order operator defined on ∂Ω. In our case D = ¯∂ is the Cauchy-Riemann operator. Hence,
dom( ¯∂∗) =ψ ∈ C∞( ¯Ω) : A]ψ = 0 on ∂Ω . Note that with U = ¯∂∗u, then for F ∈ H(Ω)
h ¯∂∗u, F i = hu, ¯∂F i = 0.
This means that is we solve the equation
(5.11) ∂ ¯¯∂∗u = f, u ∈ dom( ¯∂∗), then we solve (5.10) with a canonical solution.
In fact, problem (5.11) is equivalent to the case ¯∂f = 0 of the system
u = ¯∂ ¯∂∗+ ¯∂∗∂u = f,¯ u ∈ dom( ¯∂∗), ¯∂u ∈ dom( ¯∂∗).
(5.12)
To see that, 0 = ¯∂f = ¯∂ ¯∂ ¯∂∗+ ¯∂∗∂u = ¯¯ ∂ ¯∂∗∂u, and so¯
0 = h ¯∂u, ¯∂ ¯∂∗∂ui = h ¯¯ ∂∗∂u, ¯¯ ∂∗∂ui ⇒ ¯¯ ∂∗∂u = 0.¯
The system (5.12) is not over-determined. For general u ∈ B(0,q)(Ω), the sys-tem (5.12) is called that “ ¯∂-Neumann problem”. Here B(0,q)(Ω) is the collection of all (0, q) forms defined on Ω. The formalism of the ¯∂-Neumann problem was introducing by D.C. Spencer in the early 1950’s. Under certain assumptions on Ω, J.J. Kohn [26] obtained the first result in 1963 and 1964:
kukHs+1(Ω)≤ C kf kHs(Ω)+ kukL2(Ω), s ∈ R.
Moreover, the estimate is sharp in L2. Unlike the elliptic case, the solution u does not gain two in all directions. Therefore, the system (5.12) has great interests from the point of view of partial differential equations.
There are essentially three aspects to this problem:
•. Existence of solutions;
•. Find the solving operator N (and hence ¯∂∗N );
•. Sharp estimates for N and ¯∂∗N .
Let g be a smooth Hermitian metric on Cn+1. Then there is an open neigh-borhood V of ∂Ω such that if ρ denotes a signed geodesic distance in the metric g to ∂Ω, then
Ω+= Ω ∩ V = {z ∈ U : ρ(z) > 0};
5ρ(z) 6= 0 for all z ∈ ∂Ω ∩ V.
We choose a smooth orthogonal basis for (0, 1)-form on V , given by ¯ω1,...,¯ωn+1
are tangential to ∂Ω. If ∂ρ∂ is the vector field dual to the one form dρ, then Zn+1= 1 The Hermitian form (cjk) is called the Levi form.
The domain Ω is called pseudoconvex if each point of Ω has a neighborhood on which the vector field T can be chosen so that (cjk) ≥ 0. If the Levi form (cjk) > 0, then Ω is called strongly pseudoconvex.
To simplify our discussion, we just assume that Ω is strongly pseudoconvex in this subsection. In this case, Ω has a foliation. By a result of Chern and Moser [13], Ω and ∂Ω can be locally approximated by the “Siegel upper half-space” and the
“Heisenberg group” respectively in Cn+1. Readers can consult Stein’s book [56]
for background of Heisenberg group and its connection with analysis in several complex variables. The domain consists of all z ∈ Cn+1, n ≥ 1, so that
In this case, the vector fields Zj can be written as Zj = ∂
∂zj + i¯zj ∂
∂t, j = 1, . . . , n and the complex normal Zn+1is
Zn+1 = 1
Because the vector fields split into tangential and normal part, we may con-sider a (0, 1)-form u as follows:
u =
Then the ¯∂-Neumann problem is the following boundary value problem:
u = f in Ω
and the matrix S is defined by the equations
∂ ¯¯ω` = X
j<k
¯
s`j,kω¯j∧ ¯ωk.
Then the operator which is basically the complex Laplacian which can be written as where hn+1is a smooth function which comes from the volume element. ε(Z, ¯Z)u represents terms of first derivatives of u along horizontal directions and ε(u) rep-resents terms ofu multiplying by smooth functions. Here
` = −1
The “normal” component un+1is the solution of a Dirichlet problem for an elliptic operator:
n+1un+1 = fn+1 in Ω un+1 = 0 on ∂Ω
From previous discussion, the structure of this solution is well understood.
We note that a parametric for the (n + 1)-component Nn+1of the Neumann operator N is given by
Now we are left with solving the following nonelliptic boundary problem:
Given f on Ω, find a function u on ¯Ω such that
In order to solve the problem (5.13), we may assume the solution u is given by
u(x, t; ρ) = G(f ) + P (ub)
where G is the Green’s function for the Dirichlet problem and P is the Poisson operator. Here ub is the “boundary value” of u which we need to determine.
On the other hand, u satisfies the ¯∂-Neumann boundary conditions, i.e., R ¯Zn+1(u) = 0, j = 1, . . . , n where R is the restriction operator to ∂Ω. Therefore,
0 = R ¯Zn+1(u) = R ¯Zn+1G(f ) + R ¯Zn+1P (ub) i.e.,
+(ub) = R ¯Zn+1P (ub) = −R ¯Zn+1G(f ).
The operator +is called the Dirichlet-to-Neumann operator associated to the ¯ ∂-Neumann problem. This is a 1st order pseudo-differential operator defined on ∂Ω.
Hence, in order to solve the ¯∂-Neumann problems reduces to invert the operator
+.
After detailed calculation, one can see that he principal symbol of the
Obviously, + is elliptic when τ < 0 but doubly characteristic on half of the line bundle Σ+= {(z, t; ξ, τ ) : τ > ∆}
on the cotangent bundle T∗(∂Ω).
On the other hand, we may also construct the Dirichlet-to-Neumann operator − of the ¯∂-Neumann problem on
Ω¯−= {z ∈ U : ρ(z) ≤ 0}.
The principal symbol of −is σ − = 1
It is a 1st order pseudo-differential operator doubly characteristic on the half of the line bundle
Σ−= {(z, t; ξ, τ ) : τ < −∆}
but elliptic on the characteristics of +. An important phenomena is
+◦ −= −b+ zero order terms and
−◦ += −b+ zero order terms.
Here b is the sub-Laplacian on (0, 1)-forms defined on the boundary ∂Ω. More precisely,
In order to find a full parametrix for the ¯∂-Neumann problem, one needs to invert bon
∂Ω.
When n > 1, the sub-Laplacian bhas an inverse K by a result of Folland-Stein [18]
(see also Beals and Greiner [5] and Berenstein-Chang-Tie [6]) and then the ¯∂-Neumann problem has a parametrix
Nj = G + P (−K−R ¯Zn+1G) + S−∞, j = 1, . . . , n where S−∞is a smoothing operator.
When n = 1, we know that
(ϕ) = − 1 2
Z1Z¯1+ ¯Z1Z1
− iT (ϕ) + ε Z1, ¯Z1, ϕ
= − Z1Z¯1(ϕ) + ε Z1, ¯Z1, ϕ
In this case, b is not invertible since it is intimately connected with the non-solvability of the Lewy equation. However, by a result of Greiner and Stein [20], one can construct an operator eK, so that
Ke b = bK = I −e Cb
where Cb is the Cauchy-Szeg¨o projection.
Let us consider 0th-order pseudo-differential operators: E+ and E−with symbols in the class S1,00 such that the principal symbol of E+ equals to 1 on the set
n
∆ < 1 4τ
o and whose principal symbol equals to 0 on the set
n
∆ > 1 2τo The important fact is that
E+◦ ¯Cb = ¯Cb◦ E+ = 0, approximately,
and hence the projection ¯Cb is subordinate to projection E− = I − E+; moreover + is elliptic away from its characteristic variety. Thus there exists a pseudo-differential operator of order −1, QE−, so that
QE−◦ + = E− approximately.
It is then easy to see that −E+◦ ¯K−+ QE− is an approximate inverse to +, i.e.,
− E+◦ ¯K−+ QE−+ = − E+K¯ −++ QE−+ = −E+K(−¯ b) + E−+ T−1
= − E+ − I + ¯Cb + E−+ T−1
= E++ E−+ T−1− E+C¯b
= I − E+C¯b+ T−1= I + T−1.
Hence, the ¯∂-Neumann problem has a parametrix (see Chang-Nagel-Stein[12]):
N1 = G + P (−K+−R ¯Z2G) + S−∞, j = 1, . . . , n where S−∞is a smoothing operator and
(5.14) K+ = −QE−+ E+K ◦ ¯ −
where QE− denotes the parametric for + in the support of E− = I − E+, i.e., QE−◦ + = E−+ T−∞.
Finally, under the hypothesis that Ω is a smoothly bounded, strongly pseudoconvex domain, it can be proved that the Neumann operator satisfies
•. N : Hs(Ω) → Hs+1(Ω);
•. P(Z, ¯Z)N : Hs(Ω) → Hs(Ω). Here P(Z, ¯Z) is any quadratic monomial in tangential vector fields.
As a consequence of the above subelliptic estimates, one can show that the solving operator ¯∂∗N of the canonical solution of the inhomogeneous Cauchy-Riemann equation satisfies the following estimates:
•. ¯∂∗N : Hs(Ω) → Hs+12(Ω);
•. X ¯∂∗N : Hs(Ω) → Hs(Ω) for any tangential tangential vector fields X.
The above results had been generalized to many other situations. We will not discuss here.
Acknowledgment. This research project is partially supported by an NSF grant DMS-1203845, Multi-Year Research Grant MYRG115(Y1-L4)-FST13-QT at University of Macau and Hong Kong RGC competitive earmarked research grant #601410. The paper is based on the first part of a series of lectures which were given by the authors at the National Center for Theoretical Sciences, Hsinchu, Taiwan during May-July, 2014.
They would like to express their profound gratitude to the Director of NCTS, Professor Winnie Li for her invitation and for the warm hospitality extended to them during their stay in Taiwan.
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