The Oblique Derivative Problem

Pengfei Guan^{∗} Department of Mathematics McMaster University Hamilton, Ontario L8S 4K1 Canada

Eric Sawyer^{†} Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 CANADA

1 Introduction

This is a report of our work ([GSI], [GSII]) on the oblique derivative problem:

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \end{matrix}  (1)

where P is an elliptic second order differential operator on R^{n}, Ω is a bounded domain in R^{n} and ℓ^{→} is a unit vector field on R^{n}. Here all above are assumed with some limited smoothness.

The problem was first proposed by Poincaré at the beginning of this century. If ℓ^{→} is everywhere transversal to the boundary, then the problem is an elliptic boundary value problem. (see [ADN]). When ℓ^{→} is permitted to become tangent to ∂Ω, the situation is significantly different. The existence and regularity of the problem are closely related to the behavior of the integral curves of the vector field near the boundary. (e.g. [Hor], [EK], and references in [GS]) In the case the coefficients of P, ∂Ω and ℓ^{→} are smooth, the sharp regularity results in L^{p} Sobolev spaces were obtained in the paper [GS]. However, in dealing with nonlinear versions of this problem, it is necesssary to consider the case where P and ℓ^{→} in (1) have limited smoothness.

In the case of limited smoothness, the techniques in [GS] cannot be used for the reduction to the boundary nor for the pseudodifferential calculus. One of the key steps in our new approach is the construction of a nice approximate Poisson operator using the solution to the Neumann problem. Which is then used in the reduction of the problem to a pseudodifferential operator on the boundary with the help of symbol splitting technique of pseudodifferential operators with limited smoothness.

∗Supported in part by NSERC grant OGP0046732 and a Sloan Fellowship.

†Research supported in part by NSERC grant OGP0005149

In this article, we only state Hölder regularity of the problem. We refer to [GSI] for a complete results in L^{p} and Hölder spaces.

To state our results,let Λ^{s} be the Hölder spaces. Let Ω be a connected open set in R^{n+2} with C^{λ+3} boundary ∂ Ω = Σ. Let

P (x, D) = ∑_{i,j=1}^{n+2} a_{ij}(x)\frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} + ∑_{i=1}^{n} b_{i}(x)\frac{∂}{∂x_{i}} + c(x)

be an elliptic second order real differential operator on R^{n+2} with C^{λ} coefficients, i.e. there are C, δ > 0 such that

C|ξ|^{2} ≥ ∑_{i,j=1}^{n+2} a_{ij}(x)ξ_{i}ξ_{j} ≥ δ|ξ|^{2}, x, ξ ϵ R^{n+2}

where a_{ij}, b_{i} and c are C^{λ} on R^{n+2}. Let ℓ⃗(x) be a real C^{λ+2} unit vector field in R^{n+2} with

ℓ⃗|_{Σ} = T⃗ + a(x)n⃗,

where n⃗ is the unit outward normal to Σ at x and T⃗ is tangential to Σ. Finally let N = {x ∈ Σ : a(x) = 0} ≠ ϕ. We assume throughout the following restriction (R_{int}) on T⃗ :

No integral curve of T⃗ in Σ lies in N for an infinite interval of time.  (2)

¿From the necessary conditions on local solvability needed for pseudodifferential operators (see [NT]), in order to obtain the existence results for the oblique derivative problem, we obtain corresponding necessary conditions on ℓ⃗. This leads us to consider following two cases:

Case (I) a(x) ≥ 0 for all x ∈ Σ or a(x) ≤ 0 for all x ∈ Σ.

Case (II) a(x) takes on both positive and negative values and there is an n-dimensional C^{λ+2} submanifold Γ (possibly with several components) contained in N and open subsets Σ^{+} and Σ^{−} of Σ such that Σ is a disjoint union of Σ^{+}, Σ^{−} and Γ and

(i) a(x) ≥ 0 for x ∈ Σ^{+}

(ii) a(x) ≤ 0 for x ∈ Σ^{−}

(iii) ∂Σ^{+} = ∂Σ^{−} = Γ

(iv) ℓ⃗ is transversal to Γ and points in the direction of Σ^{+}.

For case (I) we consider the problem

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \end{matrix} ,  (3)

and for case (II) the problem is

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \\ u & = & h & on Γ \end{matrix} .  (4)

We can now state our main theorem. Fix 0 < μ < min \{\frac{λ}{λ+1}, \frac{λ+2}{2(λ+3)}\}.

Theorem 1 Suppose that P, Ω and ℓ⃗ are as at the beginning of this section with λ > 0.

(i) If ℓ⃗ satisfies case (I), then for all 2 < s ≤ λ + 2, there is a subspace F of finite codimension in Λ^{s−2}(Ω)× Λ^{s}(∂Ω) such that for (f, g) ∈ F, there exists u satisfying (3) and u ∈ Λ^{s}(Ω). Moreover, if u ∈ Λ^{s′}(Ω) for some s′ > 1 satisfies (3) with f, g as above, then u ∈ Λ^{s}(Ω), and there is C_{s} such that

∥u∥_{Λ^{s}(Ω)} ≤ C_{s} (∥f∥_{Λ^{s−2}(Ω)} + ∥g∥_{Λ^{s}(∂Ω)} + ∥u∥_{Λ^{s−μ}(Ω)}) .

Furthermore, if the zero order term c (x) in the operator P is negative, then for every (f, g) ∈ Λ^{s−2}(Ω)× Λ^{s}(∂Ω), there is a unique solution u ∈ Λ^{s}(Ω). (ii) If ℓ⃗ satisfies case (II), then for all 2 < s ≤ λ + 2, there is a subspace F of finite codimension in Λ^{s−2}(Ω)× Λ^{s}(∂Ω)× Λ^{s}(Γ) such that for (f, g, h) ∈ F, there exists u satisfying (4) and u ∈ Λ^{s}(Ω). Moreover, if u ∈ Λ^{s′}(Ω) for some s′ > 1 satisfies (4) with f, g, h as above, then u ∈ Λ^{s}(Ω), and there is C_{s} such that

∥u∥_{Λ^{s}(Ω)} ≤ C_{s} (∥f∥_{Λ^{s−2}(Ω)} + ∥g∥_{Λ^{s}(∂Ω)} + ∥h∥_{Λ^{s}(Γ)} + ∥u∥_{Λ^{s−μ}(Ω)}) .

Furthermore, if the zero order term c (x) in the operator P is nonpositive, then for every (f, g, h) ∈ Λ^{s−2}(Ω)× Λ^{s}(∂Ω)× Λ^{s}(Γ), there is a unique solution u ∈ Λ^{s}(Ω).

Followings are two immediate colloraries of Theorem 1 to the nonlinear problems.

Theorem 2 Let P be a uniformly elliptic, quasi-linear, second order differential operator, with coefficients in C^{λ}, for some λ > 0. Suppose further that u ∈ Λ^{s} (\overline{Ω}), s > 1.

(i) If ℓ⃗ satisfies case (I) above, and if in addition u satisfies

\{ \begin{matrix}P (x, u, ∇u, ∇^{2}u) & = & 0 & in Ω \\ ℓ^{→} u & = & 0 & on ∂Ω \end{matrix} ,

then u ∈ Λ^{λ+2} (\overline{Ω}).

(ii) If ℓ⃗ satisfies case (II) above, and if in addition u satisfies

\{ \begin{matrix}P (x, u, ∇u, ∇^{2}u) & = & 0 & in Ω \\ ℓ^{→} u & = & 0 & on ∂Ω \\ u & = & 0 & on Γ \end{matrix} ,

then u ∈ Λ^{λ+2} (\overline{Ω}).

In particular, if the functions P, and the manifolds Γ, ∂Ω and the vector field ℓ⃗ are all smooth, and if u is a Λ^{s} solution to (i) or (ii) as above with s > 1, then u is also smooth.

Using the implicit function theorem, we have

Theorem 3 Let Ω and ℓ^{→} be as in Theorem 1. Suppose that λ > 0 is nonintegral and that

\begin{matrix}P (x, u, ξ, ζ) ∈ C^{λ} (\overline{Ω} × R × R^{n+2} × R^{(n+2)^{2}}) , \\ ∂_{u}P, ∂_{ξ}P, ∂_{ζ}P ∈ C^{λ} (\overline{Ω} × R × R^{n+2} × R^{(n+2)^{2}}) . \end{matrix}

Moreover, suppose P (x, 0, 0, 0) = 0 and that there is C > 0 such that

\begin{matrix}C|η|^{2} ≥ ∑_{i,j=1}^{n+2} \frac{∂}{∂ζ_{ij}} P (x, 0, 0, 0) η_{i}η_{j} ≥ C^{−1}|η|^{2}, \\ −C ≤ ∂_{u}P (x, 0, 0, 0) ≤ −C^{−1}, \end{matrix} for all x ∈ \overline{Ω}.

(i) If ℓ^{→} satisfies case (I), then there is c_{0} > 0 such that if ∥f∥_{Λ^{λ}(Ω)} and ∥g∥_{Λ^{λ+2}(∂Ω)} ≤ c_{0}, then there is u ∈ Λ^{λ+2} (Ω) satisfying

\{ \begin{matrix}P (x, u, ∇u, ∇^{2}u) & = & f (x) & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \end{matrix} .

(ii) If ℓ^{→} satisfies case (II), then there is c_{0} > 0 such that if ∥f∥_{Λ^{λ}(Ω)}, ∥g∥_{Λ^{λ+2}(∂Ω)} and ∥h∥_{Λ^{λ+2}(Γ)} ≤ c_{0}, then there is u ∈ Λ^{λ+2} (Ω) satisfying

\{ \begin{matrix}P (x, u, ∇u, ∇^{2}u) & = & f (x) & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \\ u & = & h & on Γ \end{matrix} .

2 Sketch of the proof of theorem 1

We sketch here the main ideas involved in the proof. Since the mapping properties of operators we deal with follow essentially from [Gu] and [GS], we won’t repeat their proofs here. For a complete proof, we refer to [GSI] and [GSII].

2.1 Reductions

First we try to reduce the oblique derivative problem to a pseudodifferential equation on the boundary. In an earlier paper (see section 3 of [GS]), we dealt with this problem in the case of smooth data, following the line of argument in [BFG]. Now since we are no longer in smooth case, the argument in [BFG] can’t apply anymore. This is one of the difficulties we have to overcome. Our reduction will rely on the elliptic theory of the Neumann problem and construction of a ”good” approximate Poisson operator (using the symbol splitting technique of pseudodifferential operators with limited smoothness). In order to simplify the computation of ℓ^{→} on the boundary, we pick a constant c such that the Neumann problem

\{ \begin{matrix}P F + cF = & f & on Ω \\ n^{→} F = & 0 & on ∂Ω \end{matrix}

has a unique solution F, which we denote by F = Nf. Then, by the theory of elliptic boundary value problems, N : Λ^{s}(Ω) → Λ^{s+2}(Ω) is bounded for 0 ≤ s ≤ λ. With v = u − Nf, the oblique derivative problem reduces to

\{ \begin{matrix}P v = & cNf & on Ω \\ ℓ^{→} v = & g− T^{→} Nf & on ∂Ω \end{matrix}  (5)

or

\{ \begin{matrix}P v = & cNf & on Ω \\ ℓ^{→} v = & g− T^{→} Nf & on ∂Ω \\ v = & h − Nf & on Γ. \end{matrix}  (6)

Since N : Λ^{s}(Ω) → Λ^{s}(Ω) is compact, our main theorem will follow if we can solve the following equations with the appropriate estimates:

\{ \begin{matrix}P v = & 0 & on Ω \\ ℓ^{→} v = & g− T^{→} Nf & on ∂Ω \end{matrix}  (7)

and

\{ \begin{matrix}P v = & 0 & on Ω \\ ℓ^{→} v = & g− T^{→} Nf & on ∂Ω \\ v = & h − Nf & on Γ \end{matrix} .  (8)

Using ODE theory, we may choose coordinate charts such that a(x, t) ≠ 0 on the top and bottom of the t-interval (where \frac{∂}{∂t} = T^{→}) to ensure that singularities won’t propagate out of the local charts. More specifically, for every p ∈ ∂Ω, if a(p) = 0, we can find a neighborhood U of p in the boundary which has the following property (*):

\begin{matrix}U = I^{n} × I, I = (−1, 1), | a(x, −1) |≠ 0, | a(x, 1) |≠ 0, ∀x ∈ I^{n,}, \\ either U ∩ Γ = ∅ or U ∩ Γ = \{t = 0\}. \end{matrix}

In the above coordinate chart, ℓ^{→}= \frac{∂}{∂t} −a\frac{∂}{∂r} . Furthermore, the coefficients in P and ℓ^{→} have the same degree of smoothness (C^{λ} and C^{λ+2} respectively) as those in the original variables by the theory of ODE. By shrinking U if necessary, we may assume that a is as small as we wish on U. For points p ∈ ∂Ω, a(p) ≠ 0, we simply choose a neighbourhood U such that a ≠ 0 on Ū. Since ∂Ω is compact, there is finite collection U_{1}, ..., U_{k} of neighbourhoods as above covering ∂Ω such that each U_{j} either has the property (*) or a ≠ 0 on Ū_{j} . Extend U_{i} to be an open set in R^{n+2} such that U_{j} = I^{n} × I × I_{ϵ}, I_{ϵ} = (−ϵ, ϵ), U_{j} ∩ ∂Ω = I^{n} × I × {0}, and U_{j} ∩ Ω = {r > 0}. Let U_{0} = {r > δ }, so that if δ is small enough, then

∪_{j=0}^{k} U_{j} ⊃ Ω̄.

On those U (we drop the index j for the convenience) with property (*), equations (7) and (8) now become

\{ \begin{matrix}P v & = & 0 & in & r > 0 \\ ℓ^{→} v & = & g̃− T^{→} F & on & r = 0 \end{matrix}  (9)

and

\{ \begin{matrix}P v & = & 0 & in & r > 0 \\ ℓ^{→} v & = & g̃− T^{→} F & on & r = 0 \\ v & = & h − F & on & t = r = 0. \end{matrix}  (10)

where

\begin{matrix}P = & \frac{∂^{2}}{∂r^{2}} + n+1 ; ∑_{k=1} b_{k} \frac{∂}{∂y_{k}} \frac{∂}{∂r} + n+1 ; ∑_{i,j=1} g_{ij} \frac{∂}{∂y_{i}} \frac{∂}{∂y_{j}} + lower order terms, \\ ℓ^{→}= & \frac{∂}{∂t} − a\frac{∂}{∂r} . \end{matrix}

Now, if w_{0} = w |_{r=0} and F ≠ f |_{∂Ω}, then w is uniquely determined by w_{0} via the Poisson operator Ẽ of P, namely w = Ẽw_{0}. Therefore, to solve (9) and (10), we need only calculate ℓ^{→} Ẽw_{0} and then solve for w_{0} on the boundary. The calculation is standard if the coefficients of P are smooth (see e.g. [BFG]), but otherwise some care must be exercised. Here, Ẽ is defined by the theory of the Dirichlet problem for elliptic operators (see e.g. [ADN]), but we will instead use an approximate version E defined as follows.

Write

\begin{aligned}P & = (\frac{∂^{2}}{∂r^{2}} + 2B(x, D_{x})\frac{∂}{∂r} + P_{1}(x, D_{x})) \\ & +B′(r, x, D_{x})\frac{∂}{∂r} + P′_{1}(r, x, D_{x}) + lower order terms, \end{aligned}

where B′(0, x, D_{x}) = P′_{1}(0, x, D_{x}) = 0. Now set δ = max \{\frac{1}{λ+1}, \frac{1}{2}\}. Using the symbol splitting (e.g., see Proposition 4 in [GSII]), we define an approximate Poisson operator by

(Eu_{0}) (r, x) = ∫ e^{ix·ξ}e^{−r[\sqrt{−P_{1}^{♯}(x,iξ)+B^{♯}(x,iξ)^{2}}+B^{♯}(x,iξ)]}û_{0}(ξ)dξ  (11)

where ξ = (ξ_{1}, ..., ξ_{n+1}) is the dual variable of x = (x_{1}, ..., x_{n+1}), and

B = B^{♯} + B^{♭}, P_{1} = P_{1}^{♯} + P^{♭},  (12)

with

B^{♯} = i ∑_{k=1}^{n+1} b_{k}^{♯} (x, ξ) ξ_{k}, P_{1}^{♯} = − ∑_{i,j=1}^{n+1} g_{i,j}^{♯} (x, ξ) ξ_{i}ξ_{j},  (13)

where

\begin{matrix}b_{k}^{♯}, g_{i,j}^{♯} ∈ S_{1,δ}^{0}, \\ b_{k}^{♭} = b_{k} − b_{k}^{♯}, g_{i,j}^{♭} = g_{i,j} − g_{i,j}^{♯} ∈ C^{λ}S_{1,δ}^{−λδ}, \end{matrix}

and where

g̃_{i,j}^{♯} = g_{i,j}^{♯} − b_{i}^{♯}b_{j}^{♯}  (14)

is positive definite for |ξ| large.

Since E is a Poisson integral, we have that E : Λ^{s}(∂Ω) → Λ^{s}(Ω) for 0 < s ≤ λ. A straightfoward computation shows

P E = B_{1},  (15)

where

B_{1} : Λ^{s+1+δ}(∂Ω) → Λ^{s}(Ω), for 0 ≤ s ≤ λ,

As for the boundary, we have

ℓ^{→} E |_{∂Ω}= L_{1}

where

\begin{aligned}L_{1} & =T^{→} +a\sqrt{P_{1}^{♯}(x, D_{x}) − B^{♯}(x, D_{x})^{2}} + aB^{♯}(x, D_{x}) \\ & =T^{→} +a\sqrt{∑_{i,j=1}^{n+1} g̃_{i,j}^{♯} (x, ξ) ξ_{i}ξ_{j}} + ia ∑_{k=1}^{n+1} b_{k}^{♯} (x, ξ) ξ_{k}. \end{aligned}  (16)

Thus we have reduced the solving of (9) and (10) to the solution of the pseudodifferential equations

L_{1}u = g̃− T^{→} F  (17)

and

\{ \begin{matrix}L_{1}u = g̃− T^{→} F \\ u |_{t=0}= h − F. \end{matrix} .  (18)

Note that the pseudodifferential operator L_{1} is elliptic on the cone {| ξ_{n+1} |> λ′ | ξ′ |} for any λ′ > 0, and thus we can easily invert L_{1} in this cone. So it remains to investigate L_{1} on a cone {| ξ_{n+1} |< λ | ξ′ |}. As in [GS], our goal now is to eliminate ξ_{n+1} from underneath the root sign in (16). Since the functions are not smooth now, we can’t use the Malgrange Preparation Theorem directly. Here we again use the technique of symbol spitting for pseudodifferential operators with limited smoothness.

We singal out x_{n+1} = t and ξ_{n+1} = τ, and still denote x and ξ for the rest of variables. Now decompose a (x, t, ξ, τ) = a^{♯} (x, t, ξ, τ) + a^{♭} (x, t, ξ, τ) with a^{♯} ∈ S_{1,δ}^{0} and a^{♭} ∈ C^{λ+2}S_{1,δ}^{−δ(λ+2)} where δ = max \{\frac{1}{λ+1}, \frac{1}{2}\}, and let

L̃_{1}(x, t, ξ, τ) = iτ
+a^{♯} (x, t, ξ, τ) \sqrt{g̃_{n+1,n+1}^{♯}τ^{2} + ∑_{j=1}^{n} g̃_{n+1,j}^{♯} (x, t, ξ, τ) ξ_{i}ξ_{j} + ∑_{i,j=1}^{n+1} g̃_{i,j}^{♯} (x, t, ξ, τ) ξ_{i}ξ_{j}}
+ia^{♯} (x, t, ξ, τ) ∑_{k=1}^{n+1} b_{k}^{♯} (x, t, ξ, τ) ξ_{k} + ia^{♯} (x, t, ξ, τ) b_{n+1}^{♯} (x, t, ξ, τ) τ.

Since a is small in U, we may assume that a^{♯} is also small there. Lemma 4 L̃_{1}(x, t, ξ, τ) = L̃_{0}(x, t, ξ, τ)L̃ (x, t, ξ, τ) where

L̃ (x, t, ξ, τ) = iτ + ia^{♯}Q (x, t, ξ, a^{♯}) ,

with Q, (\frac{∂}{∂z} )^{k} Q (x, t, ξ, z) ∈ S_{1,δ}^{1}, c |ξ| ≤ ReQ ≤ c^{−1} |ξ| for some c > 0; and

L̃_{0} (x, t, ξ, τ) = 1 + a^{♯}Q_{0} (x, t, ξ, a^{♯}) ,

with Q_{0} ∈ S_{1,δ}^{1}.

Proof. Let z = a^{♯}and write

\begin{aligned}L̃_{1}(x, t, ξ, τ) & = iτ + z (\sqrt{g̃_{n+1,n+1}^{♯}τ^{2} + ∑_{j=1}^{n} g̃_{n+1,j}^{♯}ξ_{i}ξ_{j} + ∑_{i,j=1}^{n+1} g̃_{i,j}^{♯}ξ_{i}ξ_{j}} + iz ∑_{k=1}^{n+1} b_{k}^{♯}ξ_{k} + ib_{n+1}^{♯}τ) \\ & = f (x, t, ξ̃, τ̃, z) |ξ| , \end{aligned}

where ξ̃ = \frac{ξ}{|ξ|}, τ̃ = \frac{τ}{|ξ|} and |\frac{∂}{∂_{τ}}f (x, t, ξ̃, τ̃, z) |_{_{τ}=0}| ≥ 1 − c |z| for some constant c > 0. Since

z is small, \frac{∂}{∂_{τ}}f |_{_{τ}=0} and so by the Malgrange preparation theorem, f (x, t, ξ̃, τ̃, z) = f_{0} (x, t, ξ̃, τ̃, z) (iτ̃ + g_{0} (x, t, ξ̃, z)) , f_{0} ≠ 0.

But f (x, t, ξ̃, τ̃, z) = 0 and so g_{0} (x, t, ξ̃, z) = zg_{1} (x, t, ξ̃, z) and |(\frac{∂}{∂z} )^{k} g| ≤ C_{k}. Now let

L̃ (x, t, ξ, τ) = |ξ| (iτ̃ + g_{0} (x, t, ξ̃, a^{♯})) = iτ + a^{♯} |ξ| g_{1} (x, t, ξ̃, a^{♯}) = iτ + a^{♯}Q

and

L̃_{0} (x, t, ξ, τ) = f_{0} (x, t, ξ̃, τ̃, a^{♯}) .

A simple computation shows that \frac{ReQ}{|ξ|} is bounded above and below by two positive constants and that f_{0} (x, t, ξ̃, τ̃, 0) = 1. This completes the proof of Lemma 4.

To simplify the notation for mapping property of the operators. We give Definition 5 An operator B belongs to \overline{O}_{I}^{m} if it is bounded from Λ^{t+m} to Λ^{t} for every t in the closed interval \overline{I}.

We now give the corresponding operator decomposition. Lemma 6 Let L and L_{0} denote the operators with symbols iτ + aQ (x, t, ξ, a) and 1 + a^{♯}Q_{0} (x, t, ξ, a^{♯}) where Q and Q_{0} are as in Lemma 4 respectively. We have

L_{1} = L_{0} ∘ L + aB_{δ} + B_{−δ}  (19)

where B_{δ} ∈ \overline{O}_{(0,λ+2)}^{δ} and B_{−δ} ∈ \overline{O}_{(0,λ+2)}^{−δ}.

Proof. Since a^{♭} ∈ C^{λ+2}S_{1,δ}^{−δ−1}, L_{1} − L̃_{1} ∈ \overline{O}_{(0,λ+2)}^{−δ}. By Lemma 4 and Propostion 5 in [GSII],

\begin{matrix}L̃_{0} ∘ L̃ − L̃_{1} = (a^{♯}Q_{0}) ∘ (a^{♯}Q) − a^{♯}Q_{0}a^{♯}Q = (aQ_{0}) ∘ (a^{♯}Q) − aQ_{0}a^{♯}Q + B_{−δ} \\ = a \{Q_{0} ∘ (a^{♯}Q) − Q_{0}a^{♯}Q\} + B_{−δ} = aB_{δ} + B_{−δ}. \end{matrix}

On the other hand,

L̃ − L = a^{♯}Q (x, t, ξ, a^{♯}) − aQ (x, t, ξ, a) = a^{♭}B_{1} = B_{−δ}.

If we now write

\begin{matrix}L_{1} = L̃_{1} + (L_{1} − L̃_{1}) = L̃_{0} ∘ L̃ + (L̃_{1} − L̃_{0} ∘ L̃) + (L_{1} − L̃_{1}) \\ = L̃_{0} ∘ L + L̃_{0} ∘ (L̃ − L) + (L̃_{1} − L̃_{0} ∘ L̃) + (L_{1} − L̃_{1}) , \end{matrix}

the lemma follows.

2.2 Parametrix and Estimates

In our local setting, the conditions in cases (I), and (II) in §1 can be restated as follows: (I) a(x, t) ≥ 0 for (x, t)ϵ I^{n} × I or a(x, t) ≤ 0 for (x, t)ϵ I^{n} × I; a(x, ±1) ≠ 0 for x ∈ I^{n}. (II) a(x, t) ≥ 0 for t ≥ 0, x ϵ I^{n} and a(x, t) ≤ 0 for t ≤ 0, x ϵ I^{n}; a(x, ±1) ≠ 0 for

x ∈ I^{n}.

First, we will construct a left parametrix K for L microlocally in the cone {|τ| < λ |ξ|} in cases (I) and (II). Then we use this to obtain a parametrix for L_{1}, and finally to obtain a parametrix and a priori estimates for the original problem (5) and (6).

By the choice of the coordinate charts, there exist c > 0, 1 > c′ > 0, such that |a(x, t)| ≥ c for |t| ≥ c′. Since ρ = ρ_{j} in the previous section is compactly supported in I^{n} × I, we may pick ρ^{∗} and ρ̃ ∈ C_{c}^{∞}(I^{n} × I), ρ^{∗} = 1 on supp ρ, and ρ̃ = 1 on supp ρ^{∗}. By making the change of variable t → −t if necessary, we may assume that a(x, t) ≥ 0 in case (I). Recall that L in Lemma 4 is given by

L(t, x, τ, ξ) = iτ + a(t, x)Q(x, t, ξ, a),

where c |ξ| ≤ |ReQ| ≤ c^{−1} |ξ|. Now, let

Ã(x, t, t′, ξ) = ∫_{t′}^{t} a(x, θ)Q(x, θ, ξ, a)dθ,
A(x, t, t′) = ∫_{t′}^{t} a(x, θ)dθ,
K(x, t, t′, ξ) = ρ̃(x, t)e^{−Ã(x,t,t′,ξ)}.

Define

Ku(x, t) = (2π)^{− \frac{n}{2}} ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ}K (x, t, t′, ξ) ũ^{∗} (ξ, t′) dξdt′,

T u(x, t) = (2π)^{− \frac{n}{2}} ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ}K (x, t, t′, ξ) a (x, t′) Q (x, t′, ξ, a (x, t′)) ũ^{∗} (ξ, t′) dξdt′, Ku(x, t) = −(2π)^{− \frac{n}{2}} ∫_{|ξ|≥1} e^{ix·ξ}ρ̃(x, t_{0})e^{−Ã(x,t,t_{0},ξ)}ũ^{∗} (ξ, t′) dξ,

(20) along with the commutator

R = K ∘ ãQ − KãQ,  (21)

where u^{∗} = ρ^{∗}u , ũ^{∗}(ξ, t′) = ∫_{R^{n}} e^{−ix·ξ}u^{∗}(x, t′)dx, u_{0}^{∗} (x) = ρ^{∗} (x, 0) u_{0} (x) , ũ_{0}^{∗}(ξ) = ∫_{R^{n}} e^{−ix·ξ}u_{0}^{∗}(x)dx, and t_{0} = −1 if a satisfies case (I), while t_{0} = 0 if a satisfies case (II).

Remark 1 The choice of t_{0} is crucial. With the above choice, a(x, θ) keeps the same sign in (t′, t) for t′ ∈ (t_{0}, t), and Re (Ã(x, t, t′, ξ)) ≥ 0 and A(x, t, t′) ≥ 0 for t′ ∈ (t_{0}, t).

Remark 2 The operators K, T and K are pseudodifferential operators with parameters (see [GSII]).

We can treat above operators in a similar way as in [Gu] and [GS], and obtainning following mapping properties of the corresponding operators (see [GSII]).

Proposition 7

K,K ∘ φ ∈ \overline{O}_{(−1,λ+2)}^{0},
T ∘ φ : Λ^{s} → Λ^{s}, 0 < s ≤ λ + 2,
K ∘ a ∘ φ : Λ^{s} → Λ^{s+1}, 0 < s ≤ λ + 1,
R ∘ φ ∈ \overline{O}_{(−1,λ+2)}^{−μ},

Claim The operator K is a left parametrix for L in Case I, and a left parametrix for L in Case II for functions u with u vanishing at t = 0 (i.e., KLu = u + Su if u |_{t=0}= 0).

Proof. We have

KL = K ∘ ∂_{t} + K ∘ aQ.

Performing integration by parts in K ∘ ∂_{t} we get

KLu = u − Ru − Ku_{0} + O_{(−1,λ+2)}^{−∞},

where u_{0} = u (t_{0}, x). By the previous Proposition, we have that R is in \overline{O}_{(−1,λ+2)}^{−μ} . We get

KLu = u + S′u − K ∘ u_{0},

where S′ is in O_{(−1,λ+2)}^{−μ}. In Case I, u_{0} = u (x, −1) = 0 since supp u is compact. This completes the proof of the claim.

Now we turn to constructing a left parametrix for L_{1}. Since L_{0} ∈ C^{λ+2}S_{1,δ}^{0} is elliptic, there is L_{0}^{−1} ∈ C^{λ+2}S_{1,δ}^{0} such that L_{0}^{−1} ∘ L_{0} = I + J where J is in \overline{O}_{(0,λ+2)}^{−2}. Now set K̃ = K ∘ L_{0}^{−1}. By the rough ψdo calculus and Lemma 6 (since a ∈ C^{λ+2}), we have

K̃ ∘ L_{1}u = K ∘ Lu + K ∘ aB_{δ}u + \overline{O}_{(0,λ+2)}^{−δ}u
= K ∘ Lu + (K ∘ a) B_{δ}u + \overline{O}_{(0,λ+2)}^{−δ}u
= u − Ru − Ku_{0} + \overline{O}_{(0,λ+2)}^{−μ}u,

where u_{0} = u |_{t=t_{0}}, since K ∘ a ∈ \overline{O}_{(0,λ+2)}^{−1} . Therefore,

K̃ ∘ L_{1}u = u − Ku_{0} + S̃u,

with S̃ ∈ \overline{O}_{(0,λ+2)}^{−μ}. Note that in case (I), u_{0} is smooth and so K̃ ∘ L_{1} = I + S̃.

Up to this point we have only worked on the cone {|τ| < |ξ′|} for L_{1} microlocally. We can patch the microlocal solutions together to obtain a left parametrix for the model problem in the coordinate chart:

L_{1}u = g − ∂_{t}F + aw  (22)

or

\{ \begin{matrix}L_{1}u & = g − ∂_{t}F + aw \\ u |_{t=0} & = h − F \end{matrix} .  (23)

Proposition 8 If u satisfies either (22) or (23) above with the assumption that g, F, u and w are smooth at t = −1 and t = 1, then

\begin{aligned}u & = K̃(φg) + T ∘ L_{0}^{−1} (F) + K̃ ∘ aw \\ & +\overline{O}_{(0,λ+2)}^{0} (F) + \overline{O}_{(0,λ+2)}^{−μ} (u) + \overline{O}_{(0,λ+2)}^{−1} (w) + \overline{O}_{(0,λ+2)}^{−1} (g) , \end{aligned}

in case (I), and

\begin{aligned}ρu_{b} & = K̃(φg) + T ∘ L_{0}^{−1} (F) + K̃ ∘ aw + K \{h + B (g) |_{t=0}\} \\ & +\overline{O}_{(0,λ+2)}^{0} (F) + \overline{O}_{(0,λ+2)}^{−μ} (u) + \overline{O}_{(0,λ+2)}^{−1} (w) + \overline{O}_{(0,λ+2)}^{−1} (g) , \end{aligned}

in case (II), where B ∈ \overline{O}_{(0,λ)}^{−1} (∂Ω).

Now we return to the original equations (5) and (6) and obtain a left parametrix for them. Here we will use elliptic theory for the Dirichlet problem in a crucial way. As in the beginning of this section, \overline{Ω} ⊂ ∪_{j=0}^{k}U_{j} and ∂Ω ⊂ ∪_{j=1}^{k}U_{j}. Let ρ ∈ C^{λ+2} (Ω) ∩C_{c}^{∞} (U_{j})for some j ≥ 1 (elliptic theory handles the case j = 0) . Let U = U_{j}.

Proposition 9 Suppose that u ∈ Λ^{s} (Ω) for some s > 2. Let ρ, ρ̃ ∈ C_{c}^{∞} (U) with ρ̃ = 1 in a rectangular (relative to the coordinate system) subdomain Ũ of U with Ũ ⊃ supp ρ. If u satisfies (5), then

\begin{aligned}ρu_{b} & = ρK̃(ρ̃g) ρT ∘ L_{0}^{−1} (ρ̃Nf |_{∂Ω}) \\ & +\overline{O}_{(0,λ+2)}^{0} (ρ̃Nf |_{∂Ω}) + \overline{O}_{(0,λ+2)}^{−μ} (ρ̃u_{b}, ρ^{∗}u_{b}) + \overline{O}_{(0,λ+2)}^{−1} (ρ̃g) , \end{aligned}  (24)

while if u satisfies (6), then

\begin{aligned}ρu_{b} & = ρK̃(ρ̃g) + ρT ∘ L_{0}^{−1} (ρ̃Nf |_{∂Ω}) − ρK \{ρ̃h + \overline{B} (ρ̃g) |_{Γ}\} \\ & +\overline{O}_{(0,λ+2)}^{0} (ρ̃Nf |_{∂Ω}) + \overline{O}_{(0,λ+2)}^{−μ} (ρ̃u_{b}, ρ^{∗}u_{b}) + \overline{O}_{(0,λ+2)}^{−1} (ρ̃g) , \end{aligned}  (25)

where u_{b} = u |_{∂Ω}, \overline{B} in \overline{O}_{(0,λ+2)}^{−1}.

Proof: We prove (25), the proof for (24) being similar but easier. Let v = u − Nf, where N denotes the Neumann operator. We have

\{ \begin{matrix}P v = & 0 & on Ω \\ ℓ^{→} v = & g− T^{→} Nf |_{∂Ω} & on ∂Ω \\ v = & h − Nf & on Γ \end{matrix} .

We must prove (25) for v. Choose ρ^{∗} ∈ C_{c}^{∞} (U) with ρ^{∗} = 1 on a rectangular subdomain U^{∗} where U^{∗} ⊃ supp ρ, a ≠ 0 on the top and bottom of U^{∗}, and ρ̃ = 1 on supp ρ^{∗}. Then ρ^{∗}v satisfies

\{ \begin{matrix}P (ρ^{∗}v) = & 0 & on Ω \\ ℓ^{→} (ρ^{∗}v) = & ρ^{∗}g − ρ^{∗} T^{→} Nf |_{∂Ω} − ( ℓ^{→} ρ^{∗}) v & on ∂Ω \\ ρ^{∗}v = & ρ^{∗}h − ρ^{∗}Nf & on Γ \end{matrix} .

Now set ṽ = Eρ^{∗}v_{b} so that ℓ^{→} ṽ |_{∂Ω}= L_{1} (ρ^{∗}v_{b}) by the construction of E.

Meanwhile,

\begin{matrix}P (ρ^{∗}v − ṽ) = & [ρ^{∗}, P] v − B_{1} (ρ^{∗}v_{b}) & on Ω \\ ρ^{∗}v − ṽ = & 0 & on ∂Ω \end{matrix} ,  (26)

where B_{1} is as in (15). Solving the Dirichlet problem (26), we obtain

ρ^{∗}v − ṽ = G {[ρ^{∗}, P] v − B_{1} (ρ^{∗}v_{b})} ,

where G ∈ \overline{O}_{(1,λ+2)}^{−2} (Ω), P G = I, and G |_{∂Ω}= 0. Since T^{→} (ρ^{∗}v − ṽ) = 0 on ∂Ω, we conclude that

ℓ^{→} (ρ^{∗}v − ṽ) |_{∂Ω}= a\overrightarrow{n}G [ρ^{∗}, P] v − aB′_{0} (ρ^{∗}v_{b}) ,

where B′_{0} = (\overrightarrow{n}GB_{1}) |_{∂Ω}. So B′_{0} : Λ^{s+1−δλ} (∂Ω) → Λ^{s} (∂Ω) for 0 < s ≤ λ + 1. But, for any ρ̃ ∈ C_{c}^{∞} (Ω) with ρ̃ = 1 on the support of ρ^{∗} we get

\begin{matrix}ρ^{∗}g − ρ^{∗} T^{→} Nf |_{∂Ω} − ( ℓ^{→} ρ^{∗}) v_{b} − L_{1}v_{b} \\ = −aB′_{0} (ρ^{∗}v_{b}) + aB̃_{0} (ρ̃v_{b}) , \end{matrix}

and

\overrightarrow{n}G [ρ^{∗}, P] ρ̃v |_{∂Ω}= B̃_{0} (ρ̃v_{b}) ,

with B̃_{0} ∈ \overline{O}_{(0,λ+1)}^{0} (∂Ω). By Proposition 8, with w = B′_{0} (ρ^{∗}v_{b}) + B̃_{0} (ρ^{∗}v_{b}),

ρ^{∗}v_{b} = K̃(ρ^{∗}g) + T ∘ L_{0}^{−1} (ρ^{∗}Nf |_{∂Ω}) − K̃( ℓ^{→} ρ^{∗}) v_{b} + (K̃ ∘ a) B̃_{0}ρ̃v_{b}

+K̃ ∘ a ∘ B′_{0} (ρ^{∗}v_{b}) − K \{ρ^{∗}h + \overline{B}_{1} (ρ^{∗}g) |_{Γ} +\overline{B}_{1} (( ℓ^{→} ρ^{∗}) v_{b}) |_{Γ}\}

+\overline{O}_{(0,λ+2)}^{0} (ρ̃Nf |_{∂Ω}) + \overline{O}_{(0,λ+2)}^{−μ} (ρ^{∗}v_{b}) + \overline{O}_{(0,λ+2)}^{−1} (ρ̃g) .

If we now multiply the above identity by ρ, commute a with L_{0}^{−1}, and note that K̃ ∘ a ∈

\overline{O}_{(0,λ+2)}^{−1+ \frac{1}{p(λ+3)}} (∂Ω). So we have K̃ ∘ a ∘ B̃_{0}, K̃ ∘ a ∘ B′_{0} ∈ \overline{O}_{(0,λ+2)}^{−μ_{p}} (∂Ω) and

\begin{matrix}|x − y| ≥ δ_{0} > 0, \\ ∫_{t′}^{t} a (x, θ) dθ ≥ δ_{0} > 0, \end{matrix}

for (x, t) ∈ supp ρ, (y, t′) ∈ supp ( ℓ^{→} ρ^{∗}) ∪ supp (ρ̃ − ρ^{∗}), we finally conclude

\begin{aligned}ρK̃(ρ̃ − ρ^{∗}) , ρ\overline{K}∘ ℓ^{→} ρ^{∗}, ρK ∘ (B_{1} ℓ^{→} ρ^{∗}) |_{Γ}, ρK ∘ (ρ̃ − ρ^{∗}) & ∈ \overline{O}_{(0,λ+2)}^{−1} (∂Ω) \\ ρT ∘ L_{0}^{−1} (ρ̃ − ρ^{∗}) & ∈ \overline{O}_{(0,λ+2)}^{0} (∂Ω) \end{aligned} ,

and (25) now follows easily.

We now consider a priori estimates for the original equations (5) and (6). Suppose that P, Ω, and \overrightarrow{ℓ}=\overrightarrow{T}+a\overrightarrow{n} are as at the beginning of section 1 with λ > 0. Consider the following problems:

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \end{matrix} ,  (27)

and

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \\ u & = & h & on Γ \end{matrix} .  (28)

Proposition 10 Suppose that P, Ω, and ℓ⃗ are as at the beginning of section 1 with λ > 0, let 0 < μ < min \{1 − δ, \frac{λ+2}{2(λ+3)}\} = min \{\frac{λ}{λ+1}, \frac{λ+2}{2(λ+3)}\} (by our choice of δ) .

(i) Let 2 < s ≤ λ + 2. If ℓ⃗ satisfies case (I) and if u ∈ Λ^{s′}(Ω) for some 2 ≤ s′ satisfies (27), then u ∈ Λ^{s}(Ω), and there is C_{s} such that

∥u∥_{Λ^{s}(Ω)} ≤ C_{s} (∥u∥_{Λ^{s−μ}(Ω)} + ∥f∥_{Λ^{s−2}(Ω)} + ∥g∥_{Λ^{s}(∂Ω)}) .  (29)

(ii) Let 2 < s ≤ λ + 2. If ℓ⃗ satisfies case (II) and if u ∈ Λ^{s′}(Ω) for some 2 ≤ s′ satisfies (28), then u ∈ Λ^{s}(Ω), and there is C_{s} such that

∥u∥_{Λ^{s}(Ω)} ≤ C_{s} (∥u∥_{Λ^{s−μ}(Ω)} + ∥f∥_{Λ^{s−2}(Ω)} + ∥g∥_{Λ^{s}(∂Ω)} + ∥h∥_{Λ^{s}(Γ)}) .  (30)

Proof: Note that

K̃a = K ∘ L_{0}^{−1} ∘ a = K ∘ a + K ∘ [L_{0}^{−1}, a] ,
[L_{0}^{−1}, a] = L_{0}^{−1} ∘ a − aL_{0}^{−1} = L_{0}^{−1} ∘ a^{♯} − a^{♯}L_{0}^{−1} + L_{0}^{−1} ∘ a^{♭} − a^{♭}L_{0}^{−1}
= (L_{0}^{−1})_{ξ} ∘ (a^{♯})_{x} + \overline{O}_{(0,λ+2)}^{−1−δ} ∈ \overline{O}_{(0,λ+2)}^{−1},

by the improved estimates in Propostion 4 in [GSII]. The Proposition now follows from Proposition 9, Proposition 25 in [GSII].

Proposition 11 If the zero order term c (x) in the differential operator P is negative (respectively nonnegative) and μ > 0, then the solution u to (27) (respectively (28)) is unique in the spaces Λ^{s} for s > 1.

Proof: Suppose u_{1} and u_{2} are solutions, set u = u_{1} − u_{2}. Then u satisfies (27) or (28) with f = g = h = 0. By the Proposition, u is in C^{1}, the standard argument using the Hopf lemma as in [GS] shows that u ≡ 0, so u_{1} = u_{2}.

Proposition 12 Suppose s > 2 and that P = ∑_{i,j}^{n+2} a_{ij}∂_{i}∂_{j} + ∑_{k}^{n+2} b_{k}∂_{k} + c where c (x) ≤ c_{0} < 0 for some constant c_{0}. Let ∥P ∥_{Λ^{λ}} ≡ ∑_{i,j}^{n+2} ∥a_{ij}∥_{Λ^{λ}} +∑_{k}^{n+2} ∥b_{k}∥_{Λ^{λ}} +∥c∥_{Λ^{λ}} and suppose that

∑_{i,j=1}^{n+2} a_{ij}ξ_{i}ξ_{j} ≥ c_{1} |ξ|^{2} for all ξ ∈ R^{n+2}.

If u satisfies (27) or (28), then the terms ∥u∥_{Λ^{s−μ}(Ω)} can be omitted from the inequalities

(29) to (30) in Proposition 10 if we replace C_{s} by C (c_{0}, c_{1}, ∥P ∥_{Λ^{λ}} , Ω, \overrightarrow{ℓ}).

Proof: If for example, if the inequality (30) fails, then there are sequences {P_{k}}, {u_{k}}, {f_{k}}, {g_{k}}, and {h_{k}} such that

∥P_{k}∥_{Λ^{λ}(Ω)} ≤ C′ and P_{k} is uniformly elliptic for all k,
∥u_{k}∥_{Λ^{λ}(Ω)} = 1 for all k,
∥f_{k}∥_{Λ^{s−2+α}(Ω)} + ∥g_{k}∥_{Λ^{s}(∂Ω)} + ∥h_{k}∥_{Λ^{s}(Γ)} → 0 as k → ∞.

¿From (30) we obtain that ∥u_{k}∥_{Λ^{s}(Ω)} ≤ C for all k, and so by compactness there is a subsequence, which we continue to denote by {u_{k}}, such that u_{k} converges in Λ^{s−ϵ}(Ω) to some function u in Λ^{s−ϵ}(Ω). Also, there is an elliptic operator P such that P_{k} → P in Λ^{λ−ϵ}. However since P_{k} ∈ Λ^{λ}, a straightforward computation with a difference operator shows that P ∈ Λ^{λ}. Now P_{k}u_{k} → P u in Λ^{s−ϵ−2}(Ω), ℓ⃗u_{k} |_{∂Ω}→ ℓ⃗u |_{∂Ω} in Λ^{s−ϵ−1}(∂Ω) and u_{k} |_{Γ}→ u |_{Γ} in Λ^{s−ϵ}(Γ). Thus u satisfies (4) with f = g = h ≡ 0 and ∥u∥_{Λ^{s−μ}(Ω)} = 1, contradicting Proposition 11.

Now we can give the proof of Theorem 1.

Proof (of Theorem 1 ). We may assume that P (x, 0) < 0, the result for general P will follow from this special case and standard functional analysis by looking at the operator P − C for some large constant C.

Now suppose 2 < s ≤ λ + 2 and f ∈ Λ^{s−2}(Ω), g ∈ Λ^{s}(∂Ω) and h ∈ Λ^{s}(Γ) (again, we consider only case (II)). We want to show that there is a solution u in Λ^{s}. Choose f_{k} ∈ C^{∞}(Ω), g_{k} ∈ C^{∞}(∂Ω), h_{k} ∈ C^{∞}(Γ) and P_{k} ∈ C^{∞} (Ω) such that

∥f − f_{k}∥_{Λ^{s−2}(Ω)} + ∥g − g_{k}∥_{Λ^{s}(∂Ω)} + ∥h − h_{k}∥_{Λ^{s}(Γ)} → 0 as k → ∞,
∥P − P_{k}∥_{Λ^{λ}(Ω)} → 0 as k → ∞.

Then by a theorem in [Win], there is u_{k} ∈ Λ^{λ+2} (Ω) satisfying

\{ \begin{matrix}P_{k}u_{k} & = & f_{k} & in Ω \\ ℓ^{→} u_{k} & = & g_{k} & on ∂Ω \\ u_{k} & = & h_{k} & on Γ \end{matrix} .

But the two earlier Propositions, together with the apriori estimates in Proposition 10, show that

∥u_{k}∥_{Λ^{s}(Ω)} ≤ C, for all k.  (31)

Thus there is a subsequence, which we continue to denote by {u_{k}}, that converges to u in Λ^{r} (Ω) for any r < s. It follows upon letting k → ∞ that

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \\ u & = & h & on Γ \end{matrix} .

By (31), we obtain ∥u∥_{Λ^{s}(Ω)} ≤ C also.

Finally, if v in Λ^{s′} (Ω) for some s′ > 1 satisfies the equation, then v = u by Proposition 11, where u is the solution we just constructed in above. By Proposition 10, we obtain apriori estimates for v. This completes the proof.

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