Guide to Elliptic Boundary Value Problems for Dirac-Type Operators

Abstract

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contain local elliptic boundary conditions in the sense of Lopatinski and Shapiro as well as the Atiyah–Patodi–Singer boundary conditions. We discuss boundary regularity of solutions and also spectral and index theory. The emphasis is on providing the reader with a working knowledge.

Appendices

Appendix 1: Dirac Operators in the Sense of Gromov and Lawson

Here we discuss an important subclass of Dirac-type operators. Note that the connection in Corollary 2.4 is not metric, in general.

Definition A.1.

A formally self-adjoint operator D: C ^∞(M, E) → C ^∞(M, E) of Dirac type is called a Dirac operator in the sense of Gromov and Lawson if there exists a metric connection ∇ on E such that

• (1) $D =\sum \nolimits _{j}\sigma _{D}(e_{j}^{{\ast}}) \circ \nabla _{e_{j}}$, for any local orthonormal tangent frame (e ₁, …, e _n );

• (2) the principal symbol σ _D of D is parallel with respect to ∇ and to the Levi-Civita connection.

This is equivalent to the definition of generalized Dirac operators in [GL, Sect. 1] or to Dirac operators on Dirac bundles in [LM, Chap. II, § 5].

Remark A.2.

For a Dirac operator in the sense of Gromov and Lawson, the connection ∇ in Definition A.1 and the connection in the Weitzenböck formula (9) coincide and are uniquely determined by these properties. We will call ∇ the connection associated with the Dirac operator D. Moreover, the endomorphism field $\mathcal{K}$ in the Weitzenböck formula takes the form

$$\displaystyle{ \mathcal{K} = \frac{1} {2}\sum _{i,j}\sigma _{D}(e_{i}^{{\ast}}) \circ \sigma _{ D}(e_{j}^{{\ast}}) \circ R^{\nabla }(e_{ i},e_{j}) }$$

where R ^∇ is the curvature tensor of ∇. See [GL, Proposition 2.5] for a proof.

Next, we show how to explicitly construct an adapted operator on the boundary satisfying (13) for a Dirac operator in the sense of Gromov and Lawson. Let ∇ be the associated connection. Along the boundary we define

$$\displaystyle{ A_{0}:=\sigma _{D}(\nu ^{\flat })^{-1}D -\nabla _{\nu } =\sigma _{ D}(\nu ^{\flat })^{-1}\sum _{ j=2}^{n}\sigma _{ D}(e_{j}^{{\ast}})\nabla _{ e_{j}}. }$$

(23)

Here (e ₂, …, e _n ) is any local tangent frame for ∂ M. Then A ₀ is a first-order differential operator acting on section of E |  _{∂ M}  → ∂ M with principal symbol $\sigma _{A_{0}}(\xi ) =\sigma _{D}(\nu ^{\flat })^{-1}\sigma _{D}(\xi )$ as required for an adapted boundary operator. From the Weitzenböck formula (9) we get, using Proposition 2.1 twice, once for D and once for ∇, for all $\Phi,\Psi \in C_{c}^{\infty }(M,E)$:

$$\displaystyle\begin{array}{rcl} 0& =& \int _{M}\big(\langle D^{2}\Phi,\Psi \rangle -\langle \nabla ^{{\ast}}\nabla \Phi,\Psi \rangle -\langle \mathcal{K}\Phi,\Psi \rangle \big)\mathop{\mathrm{dV}}\nolimits \\ & =& \int _{M}\big(\langle D\Phi,D\Psi \rangle -\langle \nabla \Phi,\nabla \Psi \rangle -\langle \mathcal{K}\Phi,\Psi \rangle \big)\mathop{\mathrm{dV}}\nolimits \\ & & \quad +\int _{\partial M}\big(-\langle \sigma _{D}(\nu ^{\flat })D\Phi,\Psi \rangle +\langle \sigma _{ \nabla ^{{\ast}}}(\nu ^{\flat })\nabla \Phi,\Psi \rangle \big)\mathop{\mathrm{dS}}\nolimits.{}\end{array}$$

(24)

For the boundary contribution we have

$$\displaystyle\begin{array}{rcl} -\langle \sigma _{D}(\nu ^{\flat })D\Phi,\Psi \rangle +\langle \sigma _{ \nabla ^{{\ast}}}(\nu ^{\flat })\nabla \Phi,\Psi \rangle & =& \langle \sigma _{ D}(\nu ^{\flat })^{-1}D\Phi,\Psi \rangle -\langle \nabla \Phi,\sigma _{ \nabla }(\nu ^{\flat })\Psi \rangle \\ & =& \langle \sigma _{D}(\nu ^{\flat })^{-1}D\Phi,\Psi \rangle -\langle \nabla \Phi,\nu ^{\flat } \otimes \Psi \rangle \\ & =& \langle \sigma _{D}(\nu ^{\flat })^{-1}D\Phi,\Psi \rangle -\langle \nabla _{\nu }\Phi,\Psi \rangle \\ & =& \langle A_{0}\Phi,\Psi \rangle. {}\end{array}$$

(25)

Inserting (25) into (24) we get

$$\displaystyle{ \int _{M}\big(\langle D\Phi,D\Psi \rangle -\langle \nabla \Phi,\nabla \Psi \rangle -\langle \mathcal{K}\Phi,\Psi \rangle \big)\mathop{\mathrm{dV}}\nolimits = -\int _{\partial M}\langle A_{0}\phi,\psi \rangle \mathop{\mathrm{dS}}\nolimits }$$

(26)

where $\phi:= \Phi \vert _{\partial M}$ and $\psi:= \Psi \vert _{\partial M}$. Since the left-hand side of (26) is symmetric in $\Phi$ and $\Psi$, the right-hand side is symmetric as well, hence A ₀ is formally self-adjoint. This shows that A ₀ is an adapted boundary operator for D.

In general, A ₀ does not anticommute with σ _D (ν ^♭) however. We will rectify this by adding a suitable zero-order term. First, let us compute the anticommutator of A ₀ and σ _D (ν ^♭):

$$\displaystyle\begin{array}{rcl} \{\sigma _{D}(\nu ^{\flat }),A_{ 0}\}\phi & =& \sum _{j=2}^{n}\sigma _{ D}(e_{j}^{{\ast}})\nabla _{ e_{j}}\phi +\sigma _{D}(\nu ^{\flat })^{-1}\sum _{ j=2}^{n}\sigma _{ D}(e_{j}^{{\ast}})\nabla _{ e_{j}}(\sigma _{D}(\nu ^{\flat })\phi ) {}\\ & =& \sum _{j=2}^{n}\Big(\sigma _{ D}(e_{j}^{{\ast}})\nabla _{ e_{j}}\phi +\sigma _{D}(\nu ^{\flat })^{-1}\sigma _{ D}(e_{j}^{{\ast}})\sigma _{ D}(\nu ^{\flat })\nabla _{ e_{j}}\phi {}\\ & & \quad +\sigma _{D}(\nu ^{\flat })^{-1}\sigma _{ D}(e_{j}^{{\ast}})\sigma _{ D}(\nabla _{e_{j}}\nu ^{\flat })\phi \Big) {}\\ & =& \sigma _{D}(\nu ^{\flat })^{-1}\sum _{ j=2}^{n}\sigma _{ D}(e_{j}^{{\ast}})\sigma _{ D}(\nabla _{e_{j}}\nu ^{\flat })\phi. {}\\ \end{array}$$

Now ∇_⋅ ν is the negative of the Weingarten map of the boundary with respect to the normal field ν. We choose the orthonormal tangent frame (e ₂, …, e _n ) to consist of eigenvectors of the Weingarten map. The corresponding eigenvalues κ ₂, …, κ _n are the principal curvatures of ∂ M. We get

$$\displaystyle{ \sum _{j=2}^{n}\sigma _{ D}(e_{j}^{\flat })\sigma _{ D}(\nabla _{e_{j}}\nu ^{\flat }) = -\sum _{ j=2}^{n}\sigma _{ D}(e_{j}^{\flat })\sigma _{ D}(\kappa _{j}e_{j}^{\flat }) =\sum _{ j=2}^{n}\kappa _{ j} = (n - 1)H, }$$

where H is the mean curvature of ∂ M with respect to ν. Therefore,

$$\displaystyle{ \{\sigma _{D}(\nu ^{\flat }),A_{ 0}\} = (n - 1)H\sigma _{D}(\nu ^{\flat })^{-1} = -(n - 1)H\sigma _{ D}(\nu ^{\flat }). }$$

Since clearly

$$\displaystyle{\{\sigma _{D}(\nu ^{\flat }),(n - 1)H\} = 2(n - 1)H\sigma _{ D}(\nu ^{\flat }),}$$

the operator

$$\displaystyle{ A:= A_{0} + \frac{n - 1} {2} H =\sigma _{D}(\nu ^{\flat })^{-1}D -\nabla _{\nu } + \frac{n - 1} {2} H }$$

is an adapted boundary operator for D satisfying (13). From (26) we also have

$$\displaystyle{ \int _{M}\big(\langle D\Phi,D\Psi \rangle -\langle \nabla \Phi,\nabla \Psi \rangle -\langle \mathcal{K}\Phi,\Psi \rangle \big)\mathop{\mathrm{dV}}\nolimits =\int _{\partial M}\langle (\tfrac{n-1} {2} H - A)\phi,\psi \rangle \mathop{\mathrm{dS}}\nolimits. }$$

(27)

Definition A.3.

For a Dirac operator D in the sense of Gromov and Lawson as above, we call A the canonical boundary operator for D.

Remark A.4.

The canonical boundary operator A is again a Dirac operator in the sense of Gromov and Lawson. Namely, define a connection on E |  _{∂ M} by

$$\displaystyle{\nabla _{X}^{\partial }\phi:= \nabla _{ X}\phi + \frac{1} {2}\sigma _{D}(\nu ^{\flat })^{-1}\sigma _{ D}(\nabla _{X}\nu ^{\flat })\phi.}$$

The Clifford relations (6) show that the term σ _D (ν ^♭)⁻¹ σ _D (∇ _X ν ^♭) = σ _D (ν ^♭)^∗ σ _D (∇ _X ν ^♭) is skewhermitian, hence ∇^∂ is a metric connection. By (23), $A_{0} =\sum _{ j=2}^{n}\sigma _{A_{0}}(e_{j}^{{\ast}}) \circ \nabla _{e_{j}}$. This, $\sigma _{A_{0}} =\sigma _{A}$, and

$$\displaystyle{\sum _{j=2}^{n}\sigma _{ A_{0}}(e_{j}^{{\ast}})\sigma _{ D}(\nu ^{\flat })^{{\ast}}\sigma _{ D}(\nabla _{e_{j}}\nu ^{\flat }) = \frac{n - 1} {2} \,H}$$

show that

$$\displaystyle{A =\sum _{ j=2}^{n}\sigma _{ A}(e_{j}^{{\ast}}) \circ \nabla _{ e_{j}}^{\partial }.}$$

Moreover, a straightforward computation using the Gauss equation for the Levi-Civita connections ∇ _X ξ = ∇ _X ^∂ ξ −ξ(∇ _X ν)ν ^♭ shows that σ _A is parallel with respect to the boundary connections ∇^∂.

Remark A.5.

The triangle inequality and the Cauchy–Schwarz inequality show

$$\displaystyle\begin{array}{rcl} \vert D\Phi \vert ^{2}& =& \big\vert \sum _{ j=1}^{n}\sigma _{ D}(e_{j}^{\flat })\nabla _{ e_{j}}\Phi \big\vert ^{2} \leq \big (\sum _{ j=1}^{n}\vert \sigma _{ D}(e_{j}^{\flat })\nabla _{ e_{j}}\Phi \vert \big)^{2} \\ & \leq & n \cdot \sum _{j=1}^{n}\vert \sigma _{ D}(e_{j}^{\flat })\nabla _{ e_{j}}\Phi \vert ^{2} = n \cdot \sum _{ j=1}^{n}\langle \sigma _{ D}(e_{j}^{\flat })^{{\ast}}\sigma _{ D}(e_{j}^{\flat })\nabla _{ e_{j}}\Phi,\nabla _{e_{j}}\Phi \rangle \\ & =& n \cdot \sum _{j=1}^{n}\vert \nabla _{ e_{j}}\Phi \vert ^{2} = n \cdot \vert \nabla \Phi \vert ^{2}, {}\end{array}$$

(28)

for any orthonormal tangent frame (e ₁, …, e _n ) and all $\Phi \in C^{\infty }(M,E)$.

When does equality hold? Equality in the Cauchy–Schwarz inequality implies that all summands $\vert \sigma _{D}(e_{j}^{\flat })\nabla _{e_{j}}\Phi \vert$ are equal, i.e., $\vert \sigma _{D}(e_{j}^{\flat })\nabla _{e_{j}}\Phi \vert = \vert \sigma _{D}(e_{1}^{\flat })\nabla _{e_{1}}\Phi \vert$. Equality in the triangle inequality then implies $\sigma _{D}(e_{j}^{\flat })\nabla _{e_{j}}\Phi =\sigma _{D}(e_{1}^{\flat })\nabla _{e_{1}}\Phi$ for all j. Thus

$$\displaystyle{\sigma _{D}(e_{1}^{\flat })\nabla _{ e_{1}}\Phi = \frac{1} {n}\sum _{j=1}^{n}\sigma _{ D}(e_{j}^{\flat })\nabla _{ e_{j}}\Phi = \frac{1} {n}D\Phi,}$$

hence $\nabla _{e_{1}}\Phi = \frac{1} {n}\sigma _{D}(e_{1}^{\flat })^{{\ast}}D\Phi$. Since e ₁ is arbitrary, this shows the twistor equation

$$\displaystyle{ \nabla _{X}\Phi = \frac{1} {n}\sigma _{D}(X^{\flat })^{{\ast}}D\Phi, }$$

(29)

for all vector fields X on M. Conversely, if $\Phi$ solves the twistor equation, one sees directly that equality holds in (28).

Inserting (28) into (27) yields

$$\displaystyle{ \tfrac{n-1} {n} \int _{M}\vert D\Phi \vert ^{2}\mathop{\mathrm{dV}}\nolimits \geq \int _{ M}\langle \mathcal{K}\Phi,\Phi \rangle \mathop{\mathrm{dV}}\nolimits +\int _{\partial M}\langle (\tfrac{n-1} {2} H - A)\phi,\phi \rangle \mathop{\mathrm{dS}}\nolimits, }$$

for all $\Phi \in C^{\infty }_{c}(M,E)$, where $\phi:= \Phi \vert _{\partial M}$. Moreover, equality holds if and only if $\Phi$ solves the twistor equation (29).

Appendix 2: Proofs of Some Auxiliary Results

In this section we collect the proofs of some of the auxiliary results.

Proof of Proposition 2.3.

We start by choosing an arbitrary connection $\bar{\nabla }$ on E and define

$$\displaystyle{ \bar{D}: C^{\infty }(M,E) \rightarrow C^{\infty }(M,F),\quad \bar{D}\Phi:=\sum \nolimits _{ j}\sigma _{D}(e_{j}^{{\ast}})\bar{\nabla }_{ e_{j}}\Phi. }$$

Then $\bar{D}$ has the same principal symbol as D and, therefore, the difference $S:= D -\bar{ D}$ is of order 0. In other words, S is a field of homomorphisms from E to F.

Since $\mathcal{A}_{D}$ is onto, the restriction $\mathcal{A}$ of $\mathcal{A}_{D}$ to the orthogonal complement of the kernel of $\mathcal{A}_{D}$ is a fiberwise isomorphism. We put $V:= \mathcal{A}^{-1}(S)$ and define a new connection by

$$\displaystyle{\nabla:=\bar{ \nabla } + V.}$$

We compute

$$\displaystyle\begin{array}{rcl} \sum \nolimits _{j}\sigma _{D}(e_{j}^{{\ast}}) \circ \nabla _{ e_{j}}& =& \sum \nolimits _{j}\sigma _{D}(e_{j}^{{\ast}}) \circ \bar{\nabla }_{ e_{j}} +\sum \nolimits _{j}\sigma _{D}(e_{j}^{{\ast}}) \circ V (e_{ j}) {}\\ & =& \bar{D} + \mathcal{A}_{D}(V ) {}\\ & =& \bar{D} + S = D. {}\\ \end{array}$$

 □ 

Proof of Proposition 3.1.

Let $\widetilde{\nabla }$ be any metric connection on E. Then $F:= D^{{\ast}}D -\widetilde{\nabla }^{{\ast}}\widetilde{\nabla }$ is formally self-adjoint. Since both, D ^∗ D and $\widetilde{\nabla }^{{\ast}}\widetilde{\nabla }$, have the same principal symbol $-\vert \xi \vert ^{2} \cdot \mathop{\mathrm{id}}\nolimits$, the operator F is of order at most one. Any other metric connection ∇ on E is of the form $\nabla =\widetilde{ \nabla } + B$ where B is a 1-form with values in skewhermitian endomorphisms of E. Hence

$$\displaystyle{ D^{{\ast}}D = (\nabla - B)^{{\ast}}(\nabla - B) + F = \nabla ^{{\ast}}\nabla \mathop{\underbrace{-\nabla ^{{\ast}}B - B^{{\ast}}\nabla + B^{{\ast}}B + F}}\limits _{ =:\,\mathcal{K}}. }$$

In general, $\mathcal{K}$ is of first order and we need to show that there is a unique B such that $\mathcal{K}$ is of order zero. Since B ^∗ B is of order zero, $\mathcal{K}$ is of order zero if and only if F −∇^∗ B − B ^∗∇ is of order zero, i.e., if and only if $\sigma _{F}(\xi ) =\sigma _{\nabla ^{{\ast}}B+B^{{\ast}}\nabla }(\xi )$ for all ξ ∈ T ^∗ M. We compute, using a local tangent frame e ₁, …, e _n ,

$$\displaystyle\begin{array}{rcl} \left \langle \sigma _{\nabla ^{{\ast}}B+B^{{\ast}}\nabla }(\xi )\varphi,\psi \right \rangle & =& \left \langle \big(\sigma _{\nabla ^{{\ast}}}(\xi ) \circ B + B^{{\ast}}\circ \sigma _{ \nabla }(\xi )\big)\varphi,\psi \right \rangle {}\\ & =& -\left \langle B\varphi,\sigma _{\nabla }(\xi )\psi \right \rangle + \left \langle \sigma _{\nabla }(\xi )\varphi,B\psi \right \rangle {}\\ & =& -\left \langle B\varphi,\xi \otimes \psi \right \rangle + \left \langle \xi \otimes \varphi,B\psi \right \rangle {}\\ & =& -\big\langle \sum _{i}e_{i}^{{\ast}}\otimes B_{ e_{i}}\varphi,\xi \otimes \psi \big\rangle +\big\langle \xi \otimes \varphi,\sum _{i}e_{i}^{{\ast}}\otimes B_{ e_{i}}\psi \big\rangle {}\\ & =& -\sum _{i}\langle e_{i}^{{\ast}},\xi \rangle \langle B_{ e_{i}}\varphi,\psi \rangle +\sum _{i}\langle e_{i}^{{\ast}},\xi \rangle \left \langle \varphi,B_{ e_{i}}\psi \right \rangle {}\\ & =& -\left \langle B_{\xi ^{\sharp }}\varphi,\psi \right \rangle + \left \langle \varphi,B_{\xi ^{\sharp }}\psi \right \rangle {}\\ & =& -2\left \langle B_{\xi ^{\sharp }}\varphi,\psi \right \rangle. {}\\ \end{array}$$

Hence, $\sigma _{\nabla ^{{\ast}}B+B^{{\ast}}\nabla }(\xi ) = -2B_{\xi ^{\sharp }}$. Thus, $\mathcal{K}$ is of order 0 if and only if

$$\displaystyle{B_{X} = -\tfrac{1} {2}\,\sigma _{F}(X^{b})}$$

for all X ∈ TM. Note that σ _F (ξ) is indeed skewhermitian because F is formally self-adjoint. □ 

Proof of Lemma 3.2.

Since D is formally self-adjoint and of Dirac type,

$$\displaystyle{ -\sigma _{D}(\nu ^{\flat }) =\sigma _{ D}(\nu ^{\flat })^{{\ast}} =\sigma _{ D}(\nu ^{\flat })^{-1}, }$$

(30)

by (1) and (8). Let A ₀ be adapted to D along ∂ M and ξ ∈ T _x ^∗ ∂ M, as usual extended to T _x ^∗ M by ξ(ν(x)) = 0. Then, again using (6) and (11),

$$\displaystyle\begin{array}{rcl} \sigma _{A_{0}}(\xi ) +\sigma _{D}(\nu (x)^{\flat })& \sigma _{ A_{0}}& (\xi )\sigma _{D}(\nu (x)^{\flat })^{{\ast}} {}\\ & =& \sigma _{D}(\nu (x)^{\flat })^{-1}\sigma _{ D}(\xi ) +\sigma _{D}(\xi )\sigma _{D}(\nu (x)^{\flat })^{{\ast}} {}\\ & =& \sigma _{D}(\nu (x)^{\flat })^{{\ast}}\sigma _{ D}(\xi ) +\sigma _{D}(\xi )^{{\ast}}\sigma _{ D}(\nu (x)^{\flat }) {}\\ & =& 2\langle \nu (x)^{\flat },\xi \rangle \cdot \mathop{\mathrm{id}}\nolimits _{ E} {}\\ & =& 0. {}\\ \end{array}$$

Hence 2S: = A ₀ +σ _D (ν ^♭)A ₀ σ _D (ν ^♭)^∗ is of order 0, that is, S is a field of endomorphisms of E along ∂ M. Since A ₀ is formally self-adjoint so is S and, by (30),

$$\displaystyle{ \sigma _{D}(\nu ^{\flat })2S =\sigma _{ D}(\nu ^{\flat })A_{ 0} + A_{0}\sigma _{D}(\nu ^{\flat }) = 2S\sigma _{ D}(\nu ^{\flat }). }$$

Hence A: = A ₀ − S is adapted to D along ∂ M and

$$\displaystyle\begin{array}{rcl} \sigma _{D}(\nu ^{\flat })A + A\sigma _{ D}(\nu ^{\flat })& =& \sigma _{ D}(\nu ^{\flat })A_{ 0} + A_{0}\sigma _{D}(\nu ^{\flat }) -\sigma _{ D}(\nu ^{\flat })S - S\sigma _{ D}(\nu ^{\flat }) {}\\ & =& \sigma _{D}(\nu ^{\flat })\big(A_{ 0} -\sigma _{D}(\nu ^{\flat })A_{ 0}\sigma _{D}(\nu ^{\flat }) - 2S\big) {}\\ & =& \sigma _{D}(\nu ^{\flat })\big(A_{ 0} +\sigma _{D}(\nu ^{\flat })A_{ 0}\sigma _{D}(\nu ^{\flat })^{{\ast}}- 2S\big) {}\\ & =& 0. {}\\ \end{array}$$

 □