Linear Elliptic Equations

Abstract

The first major topic of this chapter is the Dirichlet problem for the Laplace operator on a compact domain with boundary:

$$\Delta u = 0\text{ on }\Omega,\quad {u\bigr |}_{\partial\Omega } = f.$$

(0.1)

We also consider the nonhomogeneous problem Δu = g and allow for lower-order terms. As in Chap. 2, Δ is the Laplace operator determined by a Riemannian metric. In §1 we establish some basic results on existence and regularity of solutions, using the theory of Sobolev spaces. In §2 we establish maximum principles, which are useful for uniqueness theorems and for treating (0.1) for f continuous, among other things.

Appendices

A Spaces of generalized functions on manifolds with boundary

Let $\overline{M}$ be a compact manifold with smooth boundary. We will define a one-parameter family of spaces of functions and “generalized functions” on M, analogous to the Sobolev spaces defined when $\partial M =\varnothing$. The spaces will be defined in terms of a Laplace operator Δ on M, and a boundary condition for the Laplace operator. We will explicitly discuss only the Dirichlet boundary condition, though the results given work equally well for other coercive boundary conditions yielding self-adjoint operators, such as the Neumann boundary condition.

Fixing on the Dirichlet boundary condition, let us recall from (1.7) the map

$$T : {H}^{-1}(M)\rightarrow {H}^{1}(M), $$

(A.1)

inverting the Laplace operator

$$\Delta : {H}_{0}^{1}(M)\rightarrow {H}^{-1}(M). $$

(A.2)

The restriction of T to L²(M) is compact and self-adjoint, and we have an orthonormal basis of L²(M) consisting of eigenfunctions:

$${u}_{j}\in{H}_{0}^{1}(M)\cap{C}^{\infty }(\overline{M}),\quad T{u}_{ j} = -{\mu }_{j}{u}_{j},\quad\Delta {u}_{j} = -{\lambda }_{j}{u}_{j}, $$

(A.3)

where μ_j ↘ 0, 0 < λ_j ↗ ∞.

For a given v ∈ L²(M), set

$$v =\sum\limits_{j}\hat{v}(j)\ {u}_{j},\quad\hat{v}(j) = (v,{u}_{j}). $$

(A.4)

Now, for s ≥ 0, we define

$$\begin{array}{rcl}{\mathcal{D}}_{s }& =& {\Bigl\{v\in{L}^{2}(M) :\sum\limits_{j\geq 0}\vert\hat{v}(j){\vert }^{2}{\lambda }_{ j}^{s} <\infty\Bigr\}}\\ & =& {\Bigl\{v\in{L}^{2}(M) :\sum\limits_{j\geq 0}\hat{v}(j){\lambda }_{j}^{s/2}{u}_{ j}\in{L}^{2}(M)\Bigr\}}.\end{array}$$

(A.5)

In view of (A.3), an equivalent characterization is

$${\mathcal{D}}_{s } = {(-\mathcal{T} )}^{s /2}{\mathcal{L}}^{2}(M). $$

(A.6)

Clearly, we have

$${\mathcal{D}}_{0} = {\mathcal{L}}^{2}(M). $$

(A.7)

Also, ${\mathcal{D}}_{2} =\mathcal{T}\ {\mathcal{L}}^{2}(M)$, and by Theorem 1.3 we have

$${\mathcal{D}}_{2} = {\mathcal{H}}^{2}(M)\cap {\mathcal{H}}_{0}^{1}(M). $$

(A.8)

Generally, ${\mathcal{D}}_{s + 2 } =\mathcal{T}\ {\mathcal{D}}_{s }$, so Theorem 1.3 also gives, inductively,

$$\mathcal{D}_{2k} \subset H^{2k} (M), k = 1, 2, 3, \ldots .$$

(A.9)

A result perhaps slightly less obvious than (A.7)–(A.9) is that

$${\mathcal{D}}_{1 } = {\mathcal{H}}_{0}^{\infty }(M).$$

(A.10)

To see this, note that ${\mathcal{D}}_{s }$ is the completion of the space $\mathcal{F}$ of finite linear combinations of the eigenfunctions {u_j}, with respect to the ${\mathcal{D}}_{s }$-norm, defined by

$$\|{v\|}_{{\mathcal{D}}_{s }}^{2} =\sum\limits_{j}\vert\hat{v}(j){\vert }^{2}{\lambda }_{ j}^{s}. $$

(A.11)

Now, if $v\in\mathcal{F}$, then

$$(dv,dv) = (v,-\Delta v) =\sum \nolimits (v,{u}_{j})({u}_{j},-\Delta v) =\sum \nolimits\vert\hat{v}(j){\vert }^{2}{\lambda }_{ j}, $$

(A.12)

so

$$\|{v\|}_{{\mathcal{D}}_{1 }}^{2} =\| d{v\|}_{{ L}^{2}(M)}^{2}, $$

(A.13)

for $v\in\mathcal{F}$. In fact, ${\mathcal{D}}_{s }$ is the completion of ${\mathcal{D}}_{\sigma }$ in the ${\mathcal{D}}_{s }$-norm for any σ > s. We see that (A.13) holds for all $v\in {\mathcal{D}}_{2}$, and, with ${\mathcal{D}}_{2}$ characterized by (A.8), it is clear that the completion in the norm (A.13) is described by (A.10).

If the Neumann boundary condition were considered, we would replace λ_j by ⟨λ_j⟩ to take care of λ₀ = 0. In such a case, we would have

$${\mathcal{D}}_{2} ={\bigl\{u \in {\mathcal{H}}^{2}(M) :\frac{\partial u } {\partial\nu } = 0\text{ on }\partial M \Bigr\}},\quad {\mathcal{D}}_{1 } = {\mathcal{H}}^{1}(M).$$

Now, for s < 0, we define ${\mathcal{D}}_{s }$ to be the dual of ${\mathcal{D}}_{-s}$:

$${\mathcal{D}}_{s } = {\mathcal{D}}_{-s}^{{\ast}}. $$

(A.14)

In particular, for any $v\in {\mathcal{D}}_{s }$, and any $s\in \mathbb{R}$, $(v,{u}_{j}) =\hat{ v}(j)$ is defined, and we see that the characterizations involving the sums in (A.5) continue to hold for all $s\in \mathbb{R}$. Also the norm (A.11) provides a Hilbert space structure on ${\mathcal{D}}_{s }$ for all $s\in \mathbb{R}$. By (A.10) we have (for Dirichlet boundary conditions)

$${\mathcal{D}}_{-1} = {\mathcal{H}}^{-1 }(M). $$

(A.15)

Also, we have the interpolation identity

$${[{\mathcal{D}}_{s },{\mathcal{D}}_{\sigma }]}_{\theta } = {\mathcal{D}}_{\theta\sigma + (1 -\theta )s}, $$

(A.16)

for all $s,\sigma \in \mathbb{R},\theta \in [0,1]$, where the interpolation spaces are as defined in Chap. 4.

The isomorphism

$$\Delta : {\mathcal{D}}_{s + 2}\rightarrow {\mathcal{D}}_{s },\text{ with inverse }\mathcal{T} : {\mathcal{D}}_{s }\rightarrow {\mathcal{D}}_{s + 2}, $$

(A.17)

obviously valid for s ≥ 0, extends by duality to an isomorphism $\Delta : {\mathcal{D}}_{-s}\rightarrow {\mathcal{D}}_{- s -2 }$ for s ≥ 0, so (A.17) also holds for s ≤ − 2. By interpolation, it holds for all real s.

By interpolation, (A.9) implies

$${\mathcal{D}}_{s }\subset {\mathcal{H}}^{s }(M),\text{ for } s\ \geq 0. $$

(A.18)

The natural map ${\mathcal{D}}_{s }\hookrightarrow {\mathcal{H}}^{s}(M)$ is injective, for s ≥ 0, but it is not generally onto, and the transpose ${H}^{-s}(M)\rightarrow {\mathcal{D}}_{-s}$ is not generally injective. However, the natural map

$${H}_{\text{ comp}}^{-s}(M)\rightarrow {\mathcal{D}}_{ -s} $$

(A.19)

is injective, where ${H}_{\text{ comp}}^{-s}(M)$ denotes the space of elements of H^{− s}(N) (N being the double of M) with support in the interior of M. In particular, for any interior point p ∈ M,

$${\delta }_{p}\in {\mathcal{D}}_{-s},\text{ for } s > \frac{n}{2}\quad (n =\text{ dim } M ). $$

(A.20)

Note that as p → ∂M, δ_p → 0 in any of these spaces. From the isomorphism in (A.17), we have

$${G}_{p} = {\Delta }^{-1}{\delta }_{ p} = T{\delta }_{p} $$

(A.21)

well defined, and

$${G}_{p}\in {\mathcal{D}}_{-n /2 +2 -\epsilon },\text{ for all }\epsilon > 0. $$

(A.22)

This object is equivalent to the Green function studied in this chapter.

We can write any $v\in {\mathcal{D}}_{s }$, even for s < 0, as a Fourier series with respect to the eigenfunctions u_j. In fact, defining $\hat{v}(j) = (v,{u}_{j})$, as before, the series ${\sum}_{j}\hat{v}(j){u}_{j}$ is convergent in the space ${\mathcal{D}}_{s }$ to v, provided $v\in {\mathcal{D}}_{s }$, so we are justified in writing

$$v =\sum\limits_{j}\hat{v}(j){u}_{j},\quad v\in {\mathcal{D}}_{s },\text{ for any } s \in \mathbb{R}. $$

(A.23)

Note that $-\Delta : {\mathcal{D}}_{s }\rightarrow {\mathcal{D}}_{s -\in }$ is given by

$$-\Delta v =\sum\limits_{j}{\lambda }_{j}\hat{v}(j){u}_{j}, $$

(A.24)

for any $s\in \mathbb{R}$. We can define

$${(-\Delta )}^{(\sigma +i\tau )} : {\mathcal{D}}_{s }\rightarrow {\mathcal{D}}_{s -2 \sigma }, $$

(A.25)

for any $\sigma,\tau \in \mathbb{R}$, by

$${(-\Delta )}^{(\sigma +i\tau )}v =\sum \nolimits {\lambda }_{j}^{(\sigma +i\tau )}\hat{v}(j){u}_{ j}, $$

(A.26)

where $v\in {\mathcal{D}}_{s }$ is given by (A.23). The maps (A.25) are all isomorphisms. Note that we can write the ${\mathcal{D}}_{s }$-inner product coming from (A.11) as

$${(v,w)}_{{\mathcal{D}}_{s }} ={\bigl ( v,{(-\Delta )}^{s}w\bigr )}, $$

(A.27)

where on the right side the pairing arises from the natural ${\mathcal{D}}_{s } : {\mathcal{D}}_{-s}$ duality.

B The Mayer–Vietoris sequence in deRham cohomology

Here we establish a useful complement to the long exact sequence (9.67) and illustrate some of its implications. Let X be a smooth manifold, and suppose X is the union of two open sets, M₁ and M₂. Let U = M₁ ∩ M₂. The Mayer–Vietoris sequence has the form

$$\cdots \rightarrow {\mathcal{H}}^{k-1}(\overline{U})\mathop{\longrightarrow}\limits_{}^{\delta }{\mathcal{H}}^{k}(X)\mathop{\longrightarrow}\limits_{}^{\rho }{\mathcal{H}}^{k}({\overline{M}}_{ 1}) \oplus {\mathcal{H}}^{k}({\overline{M}}_{ 2})\mathop{\longrightarrow}\limits_{}^{\gamma }{\mathcal{H}}^{k}(\overline{U}) \rightarrow \cdots.$$

(B.1)

These maps are defined as follows. A closed form α ∈ Λ^k(X) restricts to a pair of closed forms on M₁ and M₂, yielding ρ in a natural fashion. The map γ also comes from restriction; if ι_ν : U ↪ M_ν, a pair of closed forms α_ν ∈ Λ^k(M_ν) goes to ${\iota }_{1}^{{\ast}}{\alpha }_{1} - {\iota }_{2}^{{\ast}}{\alpha }_{2}$, defining γ. Clearly, ${\iota }_{1}^{{\ast}}(\alpha {\vert }_{{M}_{1}}) = {\iota }_{2}^{{\ast}}(\alpha {\vert }_{{M}_{2}})$ if α ∈ Λ^k(X), so γ ∘ ρ = 0.

To define the “coboundary map” δ on a class [α], with α ∈ Λ^k(U) closed, pick β_ν ∈ Λ^k(M_ν) such that $\alpha = {\beta }_{1} - {\beta }_{2}$. Thus dβ₁ = dβ₂ on U. Set

$$\delta [\alpha ] = [\sigma ]\text{ with }\sigma = d{\beta }_{\nu }\text{ on }{M}_{\nu }. $$

(B.2)

To show that (B.2) is well defined, suppose β_ν ∈ Λ^k(M_ν) and ${\beta }_{1} - {\beta }_{2} = d\omega$ on U. Let {φ_ν} be a smooth partition of unity supported on {M_ν}, and consider $\psi = {\varphi }_{1}{\beta }_{1} + {\varphi }_{2}{\beta }_{2}$, where φ_νβ_ν is extended by 0 off M_ν. We have $d\psi = {\varphi }_{1}d{\beta }_{1} + {\varphi }_{2}d{\beta }_{2} + d{\varphi }_{1}\wedge ({\beta }_{1} - {\beta }_{2}) =\sigma + d{\varphi }_{1}\wedge ({\beta }_{1} - {\beta }_{2}).$ Since dφ₁ is supported on U, we can write

$$\sigma = d\psi - d(d{\varphi }_{1}\wedge \omega ),$$

an exact form on X, so (B.2) makes δ well defined. Obviously, the restriction of σ to each M_ν is always exact, so ρ ∘ δ = 0. Also, if $\alpha = {\iota }_{1}^{{\ast}}{\alpha }_{1} - {\iota }_{2}^{{\ast}}{\alpha }_{2}$ on U, we can pick β_ν = α_ν to define δ[α]. Then $d{\beta }_{\nu } = d{\alpha }_{\nu } = 0$, so δ ∘ γ = 0.

In fact, the sequence (B.1) is exact, that is,

$$\text{ im }\delta =\text{ ker }\rho,\quad\text{ im }\rho =\text{ ker }\gamma,\quad\text{ im }\gamma =\text{ ker }\delta. $$

(B.3)

We leave the verification of this as an exercise, which can be done with arguments similar to those sketched in Exercises 11–13 in the exercises on cohomology after §9.

If M_ν are the interiors of compact manifolds with smooth boundary, and $\overline{U} =\overline{{M}_{1}}\cap\overline{{M}_{2}}$ has smooth boundary, the argument above extends directly to produce an exact sequence

$$\cdots \rightarrow \mathcal{H}^{k-1} (\overline{U}) \mathop{\rightarrow}\limits^{\delta}\mathcal{H}^{k} (X) \mathop{\rightarrow}\limits^{\rho} \mathcal{H}^{k} (\overline{M}_1) \oplus \mathcal{H}^{k} (\overline{M}_2) \mathop{\rightarrow}\limits^{\gamma} \mathcal{H}^{k} (\overline{U}) \rightarrow \cdots .$$

(B.4)

Furthermore, suppose that instead $X ={\overline{M}}_{1}\cup {\overline{M}}_{2}$ and ${\overline{M}}_{1}\cap {\overline{M}}_{2} = Y$ is a smooth hypersurface in X. One also has an exact sequence

$$\cdots \rightarrow \mathcal{H}^{k-1} (\overline{Y}) \mathop{\rightarrow}\limits^{\delta}\mathcal{H}^{k} (X) \mathop{\rightarrow}\limits^{\rho} \mathcal{H}^{k} (\overline{M}_1) \oplus \mathcal{H}^{k} (\overline{M}_2) \mathop{\rightarrow}\limits^{\gamma} \mathcal{H}^{k} (Y) \rightarrow \cdots .$$

(B.5)

To relate (B.4) and (B.5), let U be a collar neighborhood of Y, and form (B.4) with ${\overline{M}}_{\nu }$ replaced by ${\overline{M}}_{\nu }\cup\overline{U}$. There is a map $\pi :\overline{U}\rightarrow Y$, collapsing orbits of a vector field transversal to Y, and π^∗ induces an isomorphism of cohomology groups, ${\pi }^{{\ast}} : {\mathcal{H}}^{k }(\overline{U})\approx {\mathcal{H}}^{k }(\mathcal{Y})$.

To illustrate the use of (B.5), suppose $X = {S}^{n},Y = {S}^{n-1}$ is the equator, and ${\overline{M}}_{\nu }$ are the upper and lower hemispheres, each diffeomorphic to the ball $\overline{{B}^{n}}$. Then we have an exact sequence

$$\begin{array}{l} \cdots \rightarrow \mathcal{H}^{k-1} (\overline{B^n}) \oplus \mathcal{H}^{k-1} (\overline{B^n}) \mathop{\rightarrow}\limits^{\gamma} \mathcal{H}^{k-1} (S^{n-1}) \mathop{\rightarrow}\limits^{\delta} \mathcal{H}^{k} (S^n) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mathop{\rightarrow}\limits^{\rho} \mathcal{H}^k (\overline{B^n}) \oplus \mathcal{H}^k (\overline{B^n}) \rightarrow \cdots . \end{array}$$

(B.6)

As in (9.70), ${\mathcal{H}}^{k }(\overline{{B}^{n}}) = 0$ except for k = 0, when you get $\mathbb{R}$. Thus

$$\delta : {\mathcal{H}}^{k - 1}({\mathcal{S}}^{n - 1})\mathop{\longrightarrow}\limits_{}^{\approx }{\mathcal{H}}^{k}({\mathcal{S}}^{n}),\text{ for } k > 1. $$

(B.7)

Granted that the computation ${\mathcal{H}}^{1}({\mathcal{S}}^{1})\approx \mathbb{R}$ is elementary, this implies ${\mathcal{H}}^{n }({\mathcal{S}}^{n })\approx \mathbb{R}$, for n ≥ 1. Looking at the segment

$$0\rightarrow {\mathcal{H}}^{0}({\mathcal{S}}^{n })\mathop{\longrightarrow}\limits_{}^{\rho }{\mathcal{H}}^{0}(\overline{{B}^{n}})\oplus {\mathcal{H}}^{0}(\overline{{B}^{n}})\mathop{\longrightarrow}\limits_{}^{\gamma }{\mathcal{H}}^{0}({\mathcal{S}}^{n - 1 })\mathop{\longrightarrow}\limits_{}^{\delta }{\mathcal{H}}^{1}({\mathcal{S}}^{n })\rightarrow 0,$$

we see that if n ≥ 2, then ker $\gamma \approx \mathbb{R}$, so γ is surjective, hence δ = 0, so ${\mathcal{H}}^{1}({\mathcal{S}}^{n }) = 0$, for n ≥ 2. Also, if 0 < k < n, we see by iterating (B.7) that ${\mathcal{H}}^{k}({\mathcal{S}}^{n})\approx {\mathcal{H}}^{1}({\mathcal{S}}^{n - k + 1})$, so ${\mathcal{H}}^{k}({\mathcal{S}}^{n}) = 0$, for 0 < k < n. Since obviously ${\mathcal{H}}^{0}({\mathcal{S}}^{n }) =\mathbb{R}$ for n ≥ 1, we have a fourth computation of ${\mathcal{H}}^{k}({\mathcal{S}}^{n})$, distinct from those sketched in Exercise 10 of §8 and in Exercises 10 and 14 of the set of exercises on cohomology after §9.

We note an application of (B.5) to the computation of Euler characteristics, namely

$$\chi ({\overline{M}}_{1}) +\chi ({\overline{M}}_{2}) =\chi (X) +\chi (Y ). $$

(B.8)

Note that this result contains some of the implications of Exercises 17 and 18 in the exercises on cohomology, in §9.

Using this, it is an exercise to show that if one two-dimensional surface X₁ is obtained from another X₀ by adding a handle, then $\chi ({X}_{1}) =\chi ({X}_{0}) - 2$. In particular, if M^g is obtained from S² by adding g handles, then $\chi ({M}^{g}) = 2 - 2g$. Thus, if M^g is orientable, since ${\mathcal{H}}^{0}({ M }^{g})\approx {\mathcal{H}}^{2}({ M }^{g})\approx \mathbb{R}$, we have

$${\mathcal{H}}^{1}({ M }^{g})\approx{\mathbb{R}}^{2g}. $$

(B.9)

It is useful to examine the beginning of the sequence (B.5):

$$0 \rightarrow \mathcal{H}^0 (X) \mathop{\rightarrow}\limits^{\rho} \mathcal{H}^0 (\overline{M}_1) \oplus \mathcal{H}^0 (\overline{M}_2) \mathop{\rightarrow}\limits^{\gamma} \mathcal{H}^0 (Y) \mathop{\rightarrow}\limits^{\delta} \mathcal{H}^1 (X) \rightarrow \cdots ,$$

(B.10)

Suppose C is a smooth, closed curve in S². Apply (B.10) with ${M}_{1} =\mathcal{C}$, a collar neighborhood of C, and ${\overline{M}}_{2} =\overline{\Omega }$, the complement of $\mathcal{C}$. Since $\partial\mathcal{C}$ is diffeomorphic to two copies of C, and since ${\mathcal{H}}^{1}({\mathcal{S}}^{2}) = 0$, (B.10) becomes

$$0\rightarrow \mathbb{R}\mathop{\longrightarrow}\limits_{}^{\rho }\mathbb{R}\oplus {\mathcal{H}}^{0}(\overline{\Omega })\mathop{\longrightarrow}\limits_{}^{\gamma }\mathbb{R}\oplus \mathbb{R}\mathop{\longrightarrow}\limits_{}^{\delta } 0. $$

(B.11)

Thus γ is surjective while ker γ = im $\rho \approx \mathbb{R}$. This forces

$${\mathcal{H}}^{0}(\overline{\Omega })\approx \mathbb{R}\oplus \mathbb{R}. $$

(B.12)

In other words, $\overline{\Omega }$ has exactly two connected components. This is the smooth case of the Jordan curve theorem. Jordan’s theorem holds when C is a homeomorphic image of S¹, but the trick of putting a collar about C does not extend to this case.

More generally, if X is a compact, connected, smooth, oriented manifold such that ${\mathcal{H}}^{1}(\mathcal{X}) = 0$, and if Y is a smooth, compact, connected, oriented hypersurface, then letting $\mathcal{C}$ be a collar neighborhood of Y and $\overline{\Omega } = X\setminus\mathcal{C}$, we again obtain the sequence (B.11) and hence the conclusion (B.12). The orientability ensures that $\partial\mathcal{C}$ is diffeomorphic to two copies of Y. This produces the following variant of (the smooth case of) the Jordan–Brouwer separation theorem.

Theorem B.1.

If X is a smooth manifold, Y is a smooth submanifold of codimension 1, both are

$$compact,\ connected,\ and\ oriented,$$

and

$${\mathcal{H}}^{1}(\mathcal{X}) = 0,$$

then X ∖ Y has precisely two connected components.

If all these conditions hold, except that Y is not orientable, then we replace $\mathbb{R}\oplus \mathbb{R}$ by $\mathbb{R}$ in (B.11) and conclude that X ∖ Y is connected, in that case. As an example, the real projective space $\mathbb{R}{\mathbb{P}}^{2}$ sits in $\mathbb{R}{\mathbb{P}}^{3}$ in such a fashion.

Recall from §19 of Chap. 1 the elementary proof of Theorem B.1 when $X = {\mathbb{R}}^{n+1}$, in particular the argument using degree theory that if Y is a compact, oriented surface in ${\mathbb{R}}^{n+1}$ (hence, in S^{n + 1}), then its complement has at least two connected components. One can extend the degree-theory argument to the nonorientable case, as follows.

There is a notion of degree mod 2 of a map F: Y → Sⁿ, which is well defined whether or not Y is orientable. For one approach, see [Mil]. This is also invariant under homotopy. Now, if in the proof of Theorem 19.11 of Chap. 1, one drops the hypothesis that the hypersurface Y (denoted X there) is orientable, it still follows that the mod 2 degree of F_p must jump by±1 when p crosses Y, so ${\mathbb{R}}^{n+1}\setminus Y$ still must have at least two connected components. In view of the result noted after Theorem B.1, this situation cannot arise. This establishes the following.

Proposition B.2.

If Y is a compact hypersurface of ${\mathbb{R}}^{n+1}$ (or S ⁿ⁺¹ ), then Y is orientable.