Chapter III

Boundary Value Problems

1 Introduction

We shall recall two classical boundary value problems and show that an appropriate generalized or abstract formulation of each of these is a wellposed problem. This provides a weak global solution to each problem and motivates much of our latter discussion.

1.1

Suppose we are given a subset G of ℝ^{n} and a function F : G → 𝕂. We consider two boundary value problems for the partial differential equation

−Δ_{n}u(x) + u(x) = F (x) ,  x ∈ G .  (1.1)

The Dirichlet problem is to find a solution of (1.1) for which u = 0 on ∂G. The Neumann problem is to find a solution of (1.1) for which (∂u/∂ν) = 0 on ∂G. In order to formulate these problems in a meaningful way, we recall the first formula of Green

∫_{G} ((Δ_{n}u)v + ∇u · ∇v) = ∫_{∂G} \frac{∂u}{∂ν}v = ∫_{∂G} γ_{1}u · γ_{0}v  (1.2)

which holds if ∂G is sufficiently smooth and if u ∈ H^{2}(G), v ∈ H^{1}(G). Thus, if u is a solution of the Dirichlet problem and if u ∈ H^{2}(G), then we have u ∈ H_{0}^{1}(G) (since γ_{0}u = 0) and (from (1.1) and (1.2))

(u, v)_{H^{1}(G)} = (F, v)_{L^{2}(G)} ,  v ∈ H_{0}^{1}(G) .  (1.3)

Note that the identity (1.3) holds in general only for those v ∈ H^{1}(G) for which γ_{0}v = 0. If we drop the requirement that v vanish on ∂G, then there would be a contribution from (1.2) in the form of a boundary integral. Similarly, if u is a solution of the Neumann problem and u ∈ H^{2}(G), then (since γ_{1}u = 0) we obtain from (1.1) and (1.2) the identity (1.3) for all v ∈ H^{1}(G). That is, u ∈ H^{2}(G) and (1.3) holds for all v ∈ H^{1}(G).

Conversely, suppose u ∈ H^{2}(G) ∩ H_{0}^{1}(G) and (1.3) holds for all v ∈ H_{0}^{1}(G). Then (1.3) holds for all v ∈ C_{0}^{∞}(G), so (1.1) is satisfied in the sense of distributions on G, and γ_{0}u = 0 is a boundary condition. Thus, u is a solution of a Dirichlet problem. Similarly, if u ∈ H^{2}(G) and (1.3) holds for all v ∈ H^{1}(G), then C_{0}^{∞}(G) ⊂ H^{1}(G) shows (1.1) is satisfied as before, and substituting (1.1) into (1.3) gives us

∫_{∂G} γ_{1}u · γ_{0}v = 0 , v ∈ H^{1}(G) .

Since the range of γ_{0} is dense in L^{2}(∂G), this implies that γ_{1}u = 0, so u is a solution of a Neumann problem.

1.2

The preceding remarks suggest a weak formulation of the Dirichlet problem as follows:

Given F ∈ L^{2}(G), find u ∈ H_{0}^{1}(G) such that (1.3) holds for all
u ∈ H_{0}^{1}(G).

In particular, the condition that u ∈ H^{2}(G) is not necessary for this formulation to make sense. A similar formulation of the Neumann problem would be the following:

Given F ∈ L^{2}(G), find u ∈ H^{1}(G) such that (1.3) holds for all
v ∈ H^{1}(G).

This formulation does not require that u ∈ H^{2}(G), so we do not necessarily have γ_{1}u ∈ L^{2}(∂G). However, we can either extend the operator γ_{1} so (1.2) holds on a larger class of functions, or we may prove a regularity result to the effect that a solution of the Neumann problem is necessarily in H^{2}(G). We shall achieve both of these in the following, but for the present we consider the following abstract problem:

Given a Hilbert space V and f ∈ V′, find u ∈ V such that for all v ∈ V

(u, v)_{V} = f(v) .

By taking V = H_{0}^{1}(G) or V = H^{1}(G) and defining f to be the functional f(v) = (F, v)_{L^{2}(G)} of V′, we recover the weak formulations of the Dirichlet or Neumann problems, respectively. But Theorem I.4.5 shows that this problem is well-posed.

Theorem 1.1 For each f ∈ V′, there exists exactly one u ∈ V such that (u, v)_{V} = f(v) for all v ∈ V, and we have ∥u∥_{V} = ∥f∥_{V′}.

Corollary If u_{1} and u_{2} are the solutions corresponding to f_{1} and f_{2}, then

∥u_{1} − u_{2}∥_{V} = ∥f_{1} − f_{2}∥_{V′} .

Finally, we note that if V = H_{0}^{1}(G) or H^{1}(G), and if F ∈ L^{2}(G) then ∥f∥_{V′} ≤ ∥F ∥_{L^{2}(G)} where we identify L^{2}(G) ⊂ V′ as indicated.

2 Forms, Operators and Green’s Formula

2.1

We begin with a generalization of the weak Dirichlet problem and of the weak Neumann problem of Section 1:

Given a Hilbert space V, a continuous sesquilinear form a(·, ·) on V, and f ∈ V′, find u ∈ V such that

a(u, v) = f(v) ,  v ∈ V .  (2.1)

The sesquilinear form a(·, ·) determines a pair of operators α, β ∈ L(V) by the identities

a(u, v) = (α(u), v)_{V} = (u, β(v))_{V} ,  u, v ∈ V .  (2.2)

Theorem I.4.5 is used to construct α and β from a(·, ·), and a(·, ·) is clearly determined by either of α or β through (2.2). Theorem I.4.5 also defines the bijection J ∈ L(V′, V) for which

f(v) = (J(f), v)_{V} , f ∈ V′ , v ∈ V .

In fact, J is just the inverse of R_{V}. It is clear that u is a solution of the “weak” problem associated with (2.1) if and only if α(u) = J(f). Since J is a bijection, the solvability of this functional equation in V depends on the invertibility of the operator α. A useful sufficient condition for α to be a bijection is given in the following.

Definition. The sesquilinear form a(·, ·) on the Hilbert space V is V - coercive if there is a c > 0 such that

|a(v, v)| ≥ c∥v∥_{V}^{2} ,  v ∈ V .  (2.3)

We show that the weak problem associated with a V -coercive form is well-posed.

Theorem 2.1 Let a(·, ·) be a V -coercive continuous sesquilinear form. Then, for every f ∈ V′, there is a unique u ∈ V for which (2.1) is satisfied. Furthermore, ∥u∥_{V} ≤ (1/c)∥f∥_{V′}.

Proof : The estimate (2.3) implies that both α and β are injective, and we also obtain

∥α(v)∥_{V} ≥ c∥v∥_{V} , v ∈ V .

This estimate implies that the range of α is closed. But β is the adjoint of α in V, so the range of α, Rg(α), satisfies the orthogonality condition Rg(α)^{⊥} = K(β) = {0}. Hence, Rg(α) is dense in V, and this shows Rg(α) = V. Since J is norm-preserving the stated results follow easily.

2.2

We proceed now to construct some operators which characterize solutions of problem (2.1) as solutions of boundary value problems for certain choices of a(·, ·) and V. First, define A ∈ L(V, V′) by

a(u, v) = Au(v) ,  u, v ∈ V .  (2.4)

There is a one-to-one correspondence between continuous sesquilinear forms on V and linear operators from V to V′, and it is given by the identity (2.4). In particular, u is a solution of the weak problem (2.1) if and only if u ∈ V and Au = f, so the problem is determined by A when f ∈ V′ is regarded as data. We would like to know that the identity Au = f implies that u satisfies a partial differential equation. It will not be possible in all of our examples to identify V′ with a space of distributions on a domain G in ℝ^{n}. (For example, we are thinking of V = H^{1}(G) in a Neumann problem as in (1.1). The difficulty is that the space C_{0}^{∞}(G) is not dense in V.)

There are two “natural” ways around this difficulty. First, we assume there is a Hilbert space H such that V is dense and continuously imbedded in H (hence, we may identify H′ ⊂ V′) and such that H is identified with H′ through the Riesz map. Thus we have the inclusions

V ↪ H = H′ ↪ V′

and the identity

f(v) = (f, v)_{H} ,  f ∈ H , v ∈ V .  (2.5)

We call H the pivot space when we identify H = H′ as above. (For example, in the Neumann problem of Section 1, we choose H = L^{2}(G), and for this choice of H, the Riesz map is the identification of functions with functionals which is compatible with the identification of L^{2}(G) as a space of distributions on G; cf., Section I.5.3.) We define D = {u ∈ V : Au ∈ H}. In the examples, Au = f, u ∈ D, will imply that a partial differential equation is satisfied, since C_{0}^{∞}(G) will be dense in H. Note that u ∈ D if and only if u ∈ V and there is a K > 0 such that

|a(u, v)| ≤ K∥v∥_{H} , v ∈ V .

(This follows from Theorem I.4.5.) Finally, we obtain the following result.

Theorem 2.2 If a(·, ·) is V -coercive, then D is dense in V, hence, dense in H.

Proof : Let w ∈ V with (u, w)_{V} = 0 for all u ∈ D. Then the operator β from (2.2) being surjective implies w = β(v) for some v ∈ V. Hence, we obtain 0 = (u, β(v))_{V} = Au(v) = (Au, v)_{H} by (2.5), since u ∈ D. But A maps D onto H, so v = 0, hence, w = 0.

A second means of obtaining a partial differential equation from the continuous sesquilinear form a(·, ·) on V is to consider a closed subspace V_{0} of V, let i : V_{0} ↪ V denote the identity and ρ = i′ : V′ → V′_{0} the restriction to V_{0} of functionals on V, and define A = ρA : V → V′_{0}. The operator A ∈ L(V, V′_{0}) defined by

a(u, v) = Au(v) , u ∈ V , v ∈ V_{0}

is called the formal operator determined by a(·, ·), V and V_{0}. In examples, V_{0} will be the closure in V of C_{0}^{∞}(G), so V′_{0} is a space of distributions on G. Thus, Au = f ∈ V′_{0} will imply that a partial differential equation is satisfied.

2.3

We shall compare the operators A and A. Assume V_{0} is a closed subspace of V, H is a Hilbert space identified with its dual, the injection V ↪ H is continuous, and V_{0} is dense in H. Let D be given as above and define D_{0} = {u ∈ V : Au ∈ H}, where we identify H ⊂ V′_{0}. Note that u ∈ D_{0} if and only if u ∈ V and there is a K > 0 such that

|a(u, v)| ≤ K∥v∥_{H} , v ∈ V_{0} ,

so D ⊂ D_{0}. It is on D_{0} that we compare A and A. So, let u ∈ D_{0} be fixed in the following and consider the functional

φ(v) = Au(v) − (Au, v)_{H} ,  v ∈ V .  (2.6)

Then we have φ ∈ V′ and φ|_{V_{0}} = 0. But these are precisely the conditions that characterize those φ ∈ V′ which are in the range of q′ : (V/V_{0})′ → V′, the dual of the quotient map q : V → V/V_{0}. That is, there is a unique F ∈ (V/V_{0})′ such that q′(F) = F ∘ q = φ. Thus, (2.6) determines an F ∈ (V/V_{0})′ such that F (q(v)) = φ(v), v ∈ V. In order to characterize (V/V_{0}), let V_{0} be the kernel of a linear surjection γ : V → B and denote by γ̂ the quotient map which is a bijection of V/V_{0} onto B. Define a norm on B by ∥γ̂(x̂)∥_{B} = ∥x̂∥_{V/V_{0}} so γ̂ is bicontinuous. Then the dual operator γ̂′ : B′ → (V/V_{0})′ is a bijection. Given the functional F above, there is a unique ∂ ∈ B′ such that F = γ̂′(∂). That is, F = ∂ ∘ γ̂. We summarize the preceding discussion in the following result.

Theorem 2.3 Let V and H be Hilbert spaces with V dense and continuously imbedded in H. Let H be identified with its dual H′ so (2.5) holds. Suppose γ is a linear surjection of V onto a Hilbert space B such that the quotient map γ̂ : V/V_{0} → B is norm-preserving, where V_{0}, the kernel of γ, is dense in H.

Thus, we have V_{0} ↪ H ↪ V′_{0}. Let A ∈ L(V, V′) and define A ∈ L(V, V′_{0}) by A = ρA, where ρ : V′ → V′_{0} is restriction to V_{0}, the dual of the injection V_{0} ↪ V. Let D_{0} = {u ∈ V : Au ∈ H}. Then, for every u ∈ D_{0}, there is a unique ∂(u) ∈ B′ such that

Au(v) − (Au, v)_{H} = ∂(u)(γ(v)) ,  v ∈ V .  (2.7)

The mapping ∂ : D_{0} → B′ is linear.

When V′_{0} is a space of distributions, it is the formal operator A that determines a partial differential equation. When γ is a trace function and V_{0} consists of those elements of V which vanish on a boundary, the quotient V/V_{0} represents boundary values of elements of V. Thus B is a realization of these abstract boundary values as a function space and (2.7) is an abstract Green’s formula. We shall call ∂ the abstract Green’s operator.

Example. Let V = H^{1}(G) and γ : H^{1}(G) → L^{2}(∂G) be the trace map constructed in Theorem II.3.1. Then H_{0}^{1}(G) = V_{0} is the kernel of γ and we denote by B the range of γ. Since γ̂ is norm-preserving, the injection B ↪ L^{2}(∂G) is continuous and, by duality, L^{2}(∂G) ⊂ B′, where we identify L^{2}(∂G) with its dual space. In particular, B consists of functions on ∂G and L^{2}(∂G) is a subspace of B′. Continuing this example, we choose H = L^{2}(G) and a(u, v) = (u, v)_{H^{1}(G)}, so Au = −Δ_{n}u + u and D_{0} = {u ∈ H^{1}(G) : Δ_{n}u ∈ L^{2}(G)}. By comparing (2.7) with (1.2) we find that when ∂G is smooth ∂ : D_{0} → B′ is an extension of ∂/∂ν = γ_{1} : H^{2}(G) → L^{2}(∂G).

3 Abstract Boundary Value Problems

3.1

We begin by considering an abstract “weak” problem (2.1) motivated by certain carefully chosen formulations of the Dirichlet and Neumann problems for the Laplace differential operator. The sesquilinear form a(·, ·) led to two operators: A, which is equivalent to a(·, ·), and the formal operator A, which is determined by the action of A restricted to a subspace V_{0} of V. It is A that will be a partial differential operator in our applications, and its domain will be determined by the space V and the difference of A and A as characterized by the Green’s operator ∂ in Theorem 2.3. If V is prescribed by boundary conditions, then these same boundary conditions will be forced on a solution u of (2.1). Such boundary conditions are called stable or forced boundary conditions. A second set of constraints may arise from Theorem 2.3 and these are called unstable or variational boundary conditions. The complete set of both stable and unstable boundary conditions will be part of the characterization of the domain of the operator A.

We shall elaborate on these remarks by using Theorem 2.3 to characterize solutions of (2.1) in a setting with more structure than assumed before. This additional structure consists essentially of splitting the form a(·, ·) into the sum of a spatial part which determines the partial differential equation in the region and a second part which contributes only boundary terms. The functional f is split similarly into a spatial part and a boundary part.

3.2

We assume that we have a Hilbert space V and a linear surjection γ : V → B with kernel V_{0} and that B is a Hilbert space isomorphic to V/V_{0}. Let V be continuously imbedded in a Hilbert space H which is the pivot space identified with its dual, and let V_{0} be dense in H. Thus we have the continuous injections V_{0} ↪ H ↪ V′_{0} and V ↪ H ↪ V′ and the identity (2.5). Let a_{1} : V × V → 𝕂 and a_{2} : B × B → 𝕂 be continuous sesquilinear forms and define

a(u, v) = a_{1}(u, v) + a_{2}(γu, γv) , u, v ∈ V .

Similarly, let F ∈ H, g ∈ B′, and define

f(v) = (F, v)_{H} + g(γv) , v ∈ V .

The problem (2.1) is the following: find u ∈ V such that

a_{1}(u, v) + a_{2}(γu, γv) = (F, v)_{H} + g(γv) ,  v ∈ V .  (3.1)

We shall use Theorem 2.3 to show that (3.1) is equivalent to an abstract boundary value problem.

Theorem 3.1 Assume we are given the Hilbert spaces, sesquilinear forms and functionals as above. Let A_{2} : B → B′ be given by

A_{2}φ(ψ) = a_{1}(φ, ψ) , φ, ψ ∈ B ,

and A : V → V′_{0} by

Au(v) = a_{1}(u, v) , u ∈ V , v ∈ V_{0} .

Let D_{0} = {u ∈ V : Au ∈ H} and ∂_{1} ∈ L(D_{0}, B′) be given by (Theorem 2.3)

a_{1}(u, v) − (Au, v)_{H} = ∂_{1}u(γv) ,  u ∈ D_{0} , v ∈ V .  (3.2)

Then, u is a solution of (3.1) if and only if

u ∈ V , Au = F , ∂_{1}u + A_{2}(γu) = g .  (3.3)

Proof : Since a_{2}(γu, γv) = 0 for all v ∈ V_{0}, it follows that the formal operator A and space D_{0} (determined above by a_{1}(·, ·)) are equal, respectively, to the operator and domain determined by a(·, ·) in Section 2.3. Suppose u is a solution of (3.1). Then u ∈ V, and the identity (3.1) for v ∈ V_{0} and V_{0} being dense in H imply that Au = F ∈ H. This shows u ∈ D_{0} and using (3.2) in (3.1) gives

∂_{1}u(γv) + a_{2}(γu, γv) = g(γv) , v ∈ V .

Since γ is a surjection, this implies the remaining equation in (3.3). Similarly, (3.3) implies (3.1).

Corollary 3.2 Let D be the space of those u ∈ V such that for some F ∈ H

a(u, v) = (F, v)_{H} , v ∈ V .

Then u ∈ D if and only if u is a solution of (3.3) with g = 0.

Proof : Since V_{0} is dense in H, the functional f ∈ V′ defined above is in H if and only if g = 0.

4 Examples

We shall illustrate some applications of our preceding results in a variety of examples of boundary value problems. Our intention is to indicate the types of problems which can be described by Theorem 3.1.

4.1

Let there be given a set of (coefficient) functions

a_{ij} ∈ L^{∞}(G) , 1 ≤ i , j ≤ n ; a_{j} ∈ L^{∞}(G) , 0 ≤ j ≤ n ,

where G is open and connected in ℝ^{n}, and define

a(u, v) = ∫_{G}\{ ∑_{i,j=1}^{n} a_{ij}(x)∂_{i}u(x)∂_{j}\overline{v(x)} + ∑_{j=0}^{n} a_{j}(x)∂_{j}u(x)\overline{v(x)} \} dx ,
u, v ∈ H^{1}(G) ,  (4.1)

where ∂_{0}u = u. Let F ∈ L^{2}(G) ≡ H be given and define f(v) = (F, v)_{H}. Let Γ be a closed subset of ∂G and define

V = {v ∈ H^{1}(G) : γ_{0}(v)(s) = 0 , a.e. s ∈ Γ} .

V is a closed subspace of H^{1}(G), hence a Hilbert space. We let V_{0} = H_{0}^{1}(G) so the formal operator A : V → V′_{0} ⊂ D^{∗}(G) is given by

Au = − ∑_{i,j=1}^{n} ∂_{j}(a_{ij}∂_{i}u) + ∑_{j=0}^{n} a_{j}∂_{j}u .

Let γ be the restriction to V of the trace map H^{1}(G) → L^{2}(∂G), where we assume ∂G is appropriately smooth, and let B be the range of γ, hence B ↪ L^{2}(∂G ∼ Γ) ↪ B′. If all the a_{ij} ∈ C^{1}(Ḡ), then we have from the classical Green’s theorem

a(u, v) − (Au, v)_{H} = ∫_{∂G∼Γ} \frac{∂u}{∂ν_{A}} · γ_{0}(v) ds , u ∈ H^{2}(G) , v ∈ V

where

\frac{∂u}{∂ν_{A}} = ∑_{i=1}^{n} ∂_{i}u(s) ∑_{j=1}^{n} a_{ij}(s)ν_{j}(s)

denotes the (weighted) normal derivative on ∂G ∼ Γ. Thus, the operator ∂ is an extension of ∂/∂ν_{A} from H^{2}(G) to the domain D_{0} = {u ∈ V : Au ∈ L^{2}(G)}. Theorem 3.1 now asserts that u is a solution of the problem (2.1) if and only if u ∈ H^{1}(G), γ_{0}u = 0 on Γ, ∂u = 0 on ∂G ∼ Γ, and Au = F.

That is, u is a generalized solution of the mixed Dirichlet-Neumann boundary value problem

\begin{matrix}Au(x) = F (x) , & x ∈ G , \\ u(s) = 0 , & s ∈ Γ , \\ \frac{∂u(s)}{∂ν_{A}} = 0 , & s ∈ ∂G ∼ Γ . \end{matrix} \}  (4.2)

If Γ = ∂G, this is called the Dirichlet problem or the boundary value problem of first type. If Γ = ∅, it is called the Neumann problem or boundary value problem of second type.

4.2

We shall simplify the partial differential equation but introduce boundary integrals. Define H = L^{2}(G), V_{0} = H_{0}^{1}(G), and

a_{1}(u, v) = ∫_{G} ∇u · ∇v̄ u, v ∈ V  (4.3)

where V is a subspace of H^{1}(G) to be chosen below. The corresponding distribution-valued operator is given by A = −Δ_{n} and ∂_{1} is an extension of the standard normal derivative given by

\frac{∂u}{∂ν} = ∇u · ν .

Suppose we are given F ∈ L^{2}(G), g ∈ L^{2}(∂G), and α ∈ L^{∞}(∂G). We define

a_{2}(φ, ψ) = ∫_{∂G} α(s)φ(s)ψ̄(s) ds ,  φ, ψ ∈ L^{2}(∂G)
f(v) = (F, v)_{H} + (g, γ_{0}v)_{L^{2}(∂G)} ,  v ∈ V ,

and then use Theorem 3.1 to characterize a solution u of (2.1) for different choices of V.

If V = H^{1}(G), then u is a generalized solution of the boundary value problem

\begin{matrix}−Δ_{n}u(x) = F (x) , & x ∈ G , \\ \frac{∂u(s)}{∂ν} + α(s)u(s) = g(x) , & s ∈ ∂G . \end{matrix} \}  (4.4)

The boundary condition is said to be of third type at those points s ∈ ∂G where α(s) ≠ 0.

For an example of non-local boundary conditions, choose V = {v ∈ H^{1}(G) : γ_{0}(v) is constant}. Let g(s) = g_{0} and α(s) = α_{0} be constants, and define a_{2}(·, ·) and f as above. Then u is a solution of the boundary value problem of fourth type

\begin{matrix}−Δ_{n}u(x) = F (x) , & x ∈ G , \\ u(s) = u_{0} (constant) , & s ∈ ∂G , \\ (∫_{∂G} \frac{∂u(s)}{∂ν} ds/ ∫_{∂G} ds) + α_{0} · u_{0} = g_{0} . \end{matrix} \}  (4.5)

Note that B = 𝕂 in this example and u_{0} is not prescribed as data. Also, periodic boundary conditions are obtained when G is an interval.

4.3

We consider a problem with a prescribed derivative on the boundary in a direction which is not necessarily normal. For simplicity we assume n = 2, let c ∈ ℝ, and define

a(u, v) = ∫_{G}{∂_{1}u(∂_{1}v̄ + c∂_{2}v̄) + ∂_{2}u(∂_{2}v̄ − c∂_{1}v̄)}  (4.6)

for u, v ∈ V = H^{1}(G). Taking V_{0} = H_{0}^{1}(G) gives A = −Δ_{2} and the classical Green’s theorem shows that for u ∈ H^{2}(G) and v ∈ H^{1}(G) we have

a(u, v) − (Au, v)_{L^{2}(G)} = ∫_{∂G} (\frac{∂u}{∂ν} + c\frac{∂u}{∂τ}) v̄ ds

where

\frac{∂u}{∂τ} = ∇u · τ

is the derivative in the direction of the tangent vector τ = (ν_{2}, −ν_{1}) on ∂G. Thus ∂ is an extension of the oblique derivative in the direction ν +cτ on the boundary. If f is chosen as in (4.2), then Theorem 3.1 shows that problem (2.1) is equivalent to a weak form of the boundary value problem

−Δ_{2}u(x) = F (x) ,  x ∈ G ,
\frac{∂u}{∂ν} + c\frac{∂u}{∂τ} = g(s) ,  s ∈ ∂G .

4.4

Let G_{1} and G_{2} be disjoint open connected sets with smooth boundaries ∂G_{1} and ∂G_{2} which intersect in a C^{1} manifold Σ of dimension n − 1. If ν_{1} and ν_{2} denote the unit outward normals on ∂G_{1} and ∂G_{2}, then ν_{1}(s) = −ν_{2}(s) for s ∈ Σ. Let G be the interior of the closure of G_{1} ∪ G_{2}, so that

∂G = ∂G_{1} ∪ ∂G_{2} ∼ (Σ ∼ ∂Σ) .

For k = 1, 2, let γ_{0}^{k} be the trace map H^{1}(G_{k}) → L^{2}(∂G_{k}). Define V = H^{1}(G) and note that γ_{0}^{1}u_{1}(s) = γ_{0}^{2}u_{2}(s) for a.e. s ∈ Σ when u ∈ H^{1}(G) and u_{k} is the restriction of u to G_{k}, k = 1, 2. Thus we have a natural trace map

γ : H^{1}(G) ⟶ L^{2}(∂G) × L^{2}(Σ)
u ⟼ (γ_{0}u, γ_{0}^{1}u_{1}|_{Σ}) ,

where γ_{0}u(s) = γ_{0}^{k}u_{k}(s) for s ∈ ∂G_{k} ∼ Σ, k = 1, 2, and its kernel is given by V_{0} = H_{0}^{1}(G_{1}) × H_{0}^{1}(G_{2}).

Let a_{1} ∈ C^{1}(Ḡ_{1}), a_{2} ∈ C^{1}(Ḡ_{2}) and define

a(u, v) = ∫_{G_{1}} a_{1}∇u · ∇v̄ + ∫_{G_{2}} a_{2}∇u · ∇v̄ , u, v ∈ V .

The operator A takes values in D^{∗}(G_{1} ∪ G_{2}) and is given by

Au(x) = \{ \begin{matrix}− ∑_{j=1}^{n} ∂_{j}(a_{1}(x)∂_{j}u(x)) , & x ∈ G_{1} , \\ − ∑_{j=1}^{n} ∂_{j}(a_{2}(x)∂_{j}u(x)) , & x ∈ G_{2} . \end{matrix}

The classical Green’s formula applied to G_{1} and G_{2} gives

a(u, v) − (Au, v)_{L^{2}(G)} = ∫_{∂G_{1}} a_{1} \frac{∂u_{1}}{∂ν_{1}} v̄_{1} + ∫_{∂G_{2}} a_{2} \frac{∂u_{2}}{∂ν_{2}} v̄_{2}

for u ∈ H^{2}(G) and v ∈ H^{1}(G). It follows that the restriction of the operator ∂ to the space H^{2}(G) is given by ∂u = (∂_{0}u, ∂_{1}u) ∈ L^{2}(∂G) × L^{2}(Σ), where

∂_{0}u(s) = a_{k}(s)\frac{∂u_{k}(s)}{∂ν_{k}} , a.e. s ∈ ∂G_{k} ∼ Σ , k = 1, 2 ,
∂_{1}u(s) = a_{1}(s)\frac{∂u_{1}(s)}{∂ν_{1}} + a_{2}(s)\frac{∂u_{2}(s)}{∂ν_{2}} ,  s ∈ Σ .

Let f be given as in Section 4.2. Then a solution of u of (2.1) is characterized by Theorem 3.1 as a weak solution of the boundary value problem

\{ \begin{matrix}u_{1} ∈ H^{1}(G_{1}) , − ∑_{j=1}^{n} ∂_{j}a_{1}(x)∂_{j}u_{1}(x) = F (x) , & x ∈ G_{1} , \\ u_{2} ∈ H^{1}(G_{2}) , − ∑_{j=1}^{n} ∂_{j}a_{2}(x)∂_{j}u_{2}(x) = F (x) , & x ∈ G_{2} , \\ a_{1}(s)\frac{∂u_{1}(s)}{∂ν_{1}} = g(s) , & s ∈ ∂G_{1} ∼ Σ , \\ a_{2}(s)\frac{∂u_{2}(s)}{∂ν_{2}} = g(s) , & s ∈ ∂G_{2} ∼ Σ , \\ u_{1}(s) = u_{2}(s) , \\ a_{1}(s)\frac{∂u_{1}(s)}{∂ν_{1}} + a_{2}(s)\frac{∂u_{2}(s)}{∂ν_{2}} = 0 , & s ∈ Σ . \end{matrix}

Since ν_{1} = −ν_{2} on Σ, this last condition implies that the normal derivative has a prescribed jump on Σ which is determined by the ratio of a_{1}(s) to a_{2}(s). The pair of equations on the interface Σ are known as transition conditions.

4.5

Let the sets G_{1}, G_{2} and G be given as in Section 4.4. Suppose Σ_{0} is an open subset of the interface Σ which is also contained in the hyperplane {x = (x′, x_{n}) : x_{n} = 0} and define V = {v ∈ H_{0}^{1}(G) : γ_{0}^{1}u_{1}|_{Σ_{0}} ∈ H^{1}(Σ_{0})}. With the scalar product

(u, v)_{V} ≡ (u, v)_{H_{0}^{1}(G)} + (γ_{0}^{1}u, γ_{0}^{1}v)_{H^{1}(Σ_{0})} , u, v ∈ V ,

V is a Hilbert space. Let γ(u) = γ_{0}^{1}(u)|_{Σ} be the corresponding trace operator V → L^{2}(Σ), so K(γ) = H_{0}^{1}(G_{1}) × H_{0}^{1}(G_{2}) contains C_{0}^{∞}(G_{1} ∪ G_{2}) as a dense subspace. Let α ∈ L^{∞}(Σ_{0}) and define the sesquilinear form

a(u, v) = ∫_{G} ∇u · ∇v̄ + ∫_{Σ_{0}} α∇′(γu) · ∇′\overline{(γv)} ,  u, v ∈ V .  (4.7)

Where ∇′ denotes the gradient in the first n−1 coordinates. Then A = −Δ_{n} in D^{∗}(G_{1} ∪ G_{2}) and the classical Green’s formula shows that ∂u is given by

∂u(v) = ∫_{Σ} (\frac{∂u_{1}}{∂ν_{1}} v̄ + \frac{∂u_{2}}{∂ν_{2}} v̄) + ∫_{Σ_{0}} α∇′(γ(u))∇′v̄

for u ∈ H^{2}(G) and v ∈ B. Since the range of γ is dense in L^{2}(Σ ∼ Σ_{0}), it follows that if ∂u = 0 then

\frac{∂u_{1}(s)}{∂ν_{1}} + \frac{∂u_{2}(s)}{∂ν_{2}} = 0 , s ∈ Σ ∼ Σ_{0} .

But ν_{1} = −ν_{2} on Σ, so the normal derivative of u is continuous across Σ ∼ Σ_{0}. Since the range of γ contains C_{0}^{∞}(Σ_{0}), it follows that if ∂u = 0 then we obtain the identity

∫_{Σ_{0}} α∇′(γu)∇′\overline{(γv)} + ∫_{Σ_{0}} \frac{∂u_{1}}{∂ν_{1}} \overline{(γv)} + \frac{∂u_{2}}{∂ν_{2}} \overline{(γv)} = 0 , v ∈ V ,

and this shows that γu|_{Σ_{0}} satisfies the abstract boundary value

\begin{matrix}−Δ_{n−1}(γu)(s) = \frac{∂u_{2}(s)}{∂ν_{1}} − \frac{∂u_{1}(s)}{∂ν_{1}} , & s ∈ Σ_{0} , \\ (γu)(s) = 0 , & s ∈ ∂Σ_{0} ∩ ∂G , \\ \frac{∂(γu)(s)}{∂ν_{0}} = 0 , & s ∈ ∂Σ_{0} ∼ ∂G , \end{matrix}

where ν_{0} is the unit normal on ∂Σ_{0}, the (n − 2)-dimensional boundary of Σ_{0}.

Let F ∈ L^{2}(G) and f(v) = (F, v)_{L^{2}(G)} for v ∈ V. Then from Corollary 3.2 it follows that (3.3) is a generalized boundary value problem given by

\begin{matrix}−Δ_{n}u(x) = F (x) , & x ∈ G_{1} ∪ G_{2} , \\ u(s) = 0 , & s ∈ ∂G , \\ u_{1}(s) = u_{2}(s) , \frac{∂u_{1}(s)}{∂ν_{1}} = \frac{∂u_{2}(s)}{∂ν_{1}} , & s ∈ Σ ∼ Σ_{0} , \\ −Δ_{n−1}u(s) = \frac{∂u_{2}(s)}{∂ν_{1}} − \frac{∂u_{1}(s)}{∂ν_{1}} , & s ∈ Σ_{0} , \\ \frac{∂u(s)}{∂ν_{0}} = 0 , & s ∈ ∂Σ_{0} ∼ ∂G_{0} . \end{matrix} \}  (4.8)

Nonhomogeneous terms could be added as in previous examples and similar problems could be solved on interfaces which are not necessarily flat.

5 Coercivity; Elliptic Forms

5.1

Let G be an open set in ℝ^{n} and suppose we are given a collection of functions a_{ij}, 1 ≤ i, j ≤ n; a_{j}, 0 ≤ j ≤ n, in L^{∞}(G). Define the sesquilinear form

a(u, v) = ∫_{G}\{ ∑_{i,j=1}^{n} a_{ij}(x)∂_{i}u(x)∂_{j}v̄(x) + ∑_{j=0}^{n} a_{j}(x)∂_{j}u(x) · \overline{v(x)}\} dx (5.1)

on H^{1}(G). We saw in Section 4.1 that such forms lead to partial differential equations of second order on G.

Definition. The sesquilinear form (5.1) is called strongly elliptic if there is a constant c_{0} > 0 such that

Re ∑_{i,j=1}^{n} a_{ij}(x)ξ_{i}ξ̄_{j} ≥ c_{0} ∑_{j=1}^{n} |ξ_{j}|^{2} ,  ξ = (ξ_{1}, …, ξ_{n}) ∈ 𝕂^{n} , x ∈ G . (5.2)

We shall show that a strongly elliptic form can be made coercive over (any subspace of) H^{1}(G) by adding a sufficiently large multiple of the identity to it.

Theorem 5.1 Let (5.1) be strongly elliptic. Then there is a λ_{0} ∈ ℝ such that for every λ > λ_{0}, the form

a(u, v) + λ ∫_{G} u(x)v̄(x) dx

is H^{1}(G)-coercive.

Proof : Let K_{1} = max{∥a_{j}∥_{L^{∞}(G)} : 1 ≤ j ≤ n} and K_{0} = ess inf{Re a_{0}(x) : x ∈ G}. Then, for 1 ≤ j ≤ n and each ε > 0 we have

|(a_{j}∂_{j}u, u)_{L^{2}(G)}| ≤ K_{1}∥∂_{j}u∥_{L^{2}(G)} · ∥u∥_{L^{2}(G)}
≤ (K_{1}/2) (ε∥∂_{j}u∥_{L^{2}(G)}^{2} + (1/ε)∥u∥_{L^{2}(G)}^{2}) .

We also have Re(a_{0}u, u)_{L^{2}(G)} ≥ K_{0}∥u∥_{L^{2}(G)}^{2}, so using these with (5.2) in (5.1) gives

\begin{aligned}Re a(u, u) & ≥ (c_{0} − εK_{1}/2)∥∇u∥_{L^{2}(G)}^{2} \\ & +(K_{0} − nK_{1}/2ε)∥u∥_{L^{2}(G)}^{2} , u ∈ H^{1}(G) . \end{aligned}  (5.3)

We choose ε > 0 so that K_{1}ε = c_{0}. This gives us the desired result with λ_{0} = (nK_{1}^{2}/2c_{0}) − K_{0}.

Corollary 5.2 For every λ > λ_{0}, the boundary value problem (4.2) is wellposed, where

Au = − ∑_{i,j=1}^{n} ∂_{j}(a_{ij}∂_{i}u) + ∑_{j=1}^{n} a_{j}∂_{j}u + (a_{0} + λ)u .

Thus, for every F ∈ L^{2}(G), there is a unique u ∈ D such that (4.2) holds, and we have the estimate

∥(λ − λ_{0})u∥_{L^{2}(G)} ≤ ∥F ∥_{L^{2}(G)} .  (5.4)

Proof : The space D was defined in Section 2.2 and Corollary 3.2, so we need only to verify (5.4). For u ∈ D and λ > λ_{0} we have from (5.3)

(λ − λ_{0})∥u∥_{L^{2}(G)}^{2} ≤ a(u, u) + λ(u, u)_{L^{2}(G)} = (Au, u)_{L^{2}(G)}
≤ ∥Au∥_{L^{2}(G)} · ∥u∥_{L^{2}(G)}

and the estimate (5.4) now follows.

5.2

We indicate how coercivity may be obtained from the addition of boundary integrals to strongly elliptic forms.

Theorem 5.3 Let G be open in ℝ^{n} and suppose 0 ≤ x_{n} ≤ K for all x = (x′, x_{n}) ∈ G. Let ∂G be a C^{1}-manifold with G on one side of ∂G. Let ν(s) = (ν_{1}(s), …, ν_{n}(s)) be the unit outward normal on ∂G and define

Σ = {s ∈ ∂G : ν_{n}(s) > 0} .

Then for all u ∈ H^{1}(G) we have

∫_{G} |u|^{2} ≤ 2K ∫_{Σ} |γ_{0}u(s)|^{2} ds + 4K^{2} ∫_{G} |∂_{n}u|^{2} .

Proof : For u ∈ C^{1}(Ḡ), the Gauss Theorem gives

∫_{∂G} ν_{n}(s)s_{n}|u(s)|^{2} ds = ∫_{G} D_{n}(x_{n}|u(x)|^{2}) dx
= ∫_{G} |u|^{2} + ∫_{G} x_{n}D_{n}(|u(x)|^{2}) dx .

Thus, we obtain from the inequality

2|a| |b| ≤ \frac{|a|^{2}}{2K} + 2K|b|^{2} , a, b ∈ ℂ ,

the estimate

∫_{G} |u|^{2} ≤ ∫_{∂G} ν_{n}s_{n}|u(s)|^{2} ds + (1/2) ∫_{G} |u|^{2} + 2K^{2} ∫_{G} |D_{n}u|^{2} .

Since ν_{n}(s)s_{n} ≤ 0 for s ∈ ∂G ∼ Σ, the desired result follows.

Corollary 5.4 If (5.1) is strongly elliptic, a_{j} ≡ 0 for 1 ≤ j ≤ n, Re a_{0}(x) ≥ 0, x ∈ G, and if Σ ⊂ Γ, then the mixed Dirichlet-Neumann problem (4.2) is well-posed.

Corollary 5.5 If α ∈ L^{∞}(∂G) satisfies

Re α(x) ≥ 0 , x ∈ ∂G , Re α(x) ≥ c > 0 , x ∈ Σ ,

then the third boundary value problem (4.4) is well-posed. The fourth boundary value problem (4.5) is well-posed if Re(α_{0}) > 0.

Similar results can be obtained for the example of Section 4.3. Note that the form (4.6) satisfies

Re a(u, u) = ∫_{G}{|∂_{1}u|^{2} + |∂_{2}u|^{2}} , u ∈ H^{1}(G) ,

so coercivity can be obtained over appropriate subspaces of H^{1}(G) (as in Corollary 5.4) or by adding a positive multiple of the identity on G or boundary integrals (as in Corollary 5.5). Modification of (4.6) by restricting V, e.g., to consist of functions which vanish on a sufficiently large part of ∂G, or by adding forms, e.g., that are coercive over L^{2}(G) or L^{2}(∂G), will result in a well-posed problem.

Finally, we note that the first term in the form (4.7) is coercive over H_{0}^{1}(G) and, hence, over L^{2}(Σ). Thus, if Re α(x) ≥ c > 0, x ∈ Σ_{0}, then (4.7) is V -coercive and the problem (4.8) is well-posed.

5.3

In order to verify that the sesquilinear forms above were coercive over certain subspaces of H^{1}(G), we found it convenient to verify that they satisfied the following stronger condition.

Definition. The sesquilinear form a(·, ·) on the Hilbert space V is V -elliptic if there is a c > 0 such that

Re a(v, v) ≥ c∥v∥_{V}^{2} ,  v ∈ V .  (5.5)

Such forms will occur frequently in our following discussions.

6 Regularity

We begin this section with a consideration of the Dirichlet and Neumann problems for a simple elliptic equation. The original problems were to find solutions in H^{2}(G) but we found that it was appropriate to seek weak solutions in H^{1}(G). Our objective here is to show that those weak solutions are in H^{2}(G) when the domain G and data in the equation are sufficiently smooth. In particular, this shows that the solution of the Neumann problem satisfies the boundary condition in L^{2}(∂G) and not just in the sense of the abstract Green’s operator constructed in Theorem 2.3, i.e., in B′. (See the Example in Section 2.3.)

6.1

We begin with the Neumann problem; other cases will follow similarly.

Theorem 6.1 Let G be bounded and open in ℝ^{n} and suppose its boundary is a C^{2}-manifold of dimension n − 1. Let a_{ij} ∈ C^{1}(G), 1 ≤ i, j ≤ n, and a_{j} ∈ C^{1}(G), 0 ≤ j ≤ n, all have bounded derivatives and assume that the sesquilinear form defined by

a(φ, ψ) ≡ ∫_{G}\{ ∑_{i,j=1}^{n} a_{ij}∂_{i}φ\overline{∂_{j}ψ} + ∑_{j=0}^{n} a_{j}∂_{j}φψ̄\} dx ,  φ, ψ ∈ H^{1}(G) (6.1)

is strongly elliptic. Let F ∈ L^{2}(G) and suppose u ∈ H^{1}(G) satisfies

a(u, v) = ∫_{G} F v̄ dx ,  v ∈ H^{1}(G) .  (6.2)

Then u ∈ H^{2}(G).

Proof : Let {(φ_{k}, G_{k}) : 1 ≤ k ≤ N} be coordinate patches on ∂G and {β_{k} : 0 ≤ k ≤ N} the partition-of-unity construction in Section II.2.3. Let B_{k} denote the support of β_{k}, 0 ≤ k ≤ N. Since u = ∑(β_{k}u) in G and each B_{k} is compact in ℝ^{n}, it is sufficient to show the following:

(a) u|_{B_{k}∩G} ∈ H^{2}(B_{k} ∩ G), 1 ≤ k ≤ N, and

(b) β_{0}u ∈ H^{2}(B_{0}).

The first case (a) will be proved below, and the second case (b) will follow from a straightforward modification of the first.

6.2

We fix k, 1 ≤ k ≤ N, and note that the coordinate map φ_{k} : Q → G_{k} induces an isomorphism φ_{k}^{∗} : H^{m}(G_{k} ∩ G) → H^{m}(Q_{+}) for m = 0, 1, 2 by φ_{k}^{∗}(v) = v ∘ φ_{k}. Thus we define a continuous sesquilinear form on H^{1}(Q_{+}) by

a^{k} (φ_{k}^{∗}(w), φ_{k}^{∗}(v)) ≡ ∫_{G_{k}∩G}\{ ∑_{i,j=1}^{n} a_{ij}∂_{j}w\overline{∂_{j}v} + ∑_{j=0}^{n} a_{j}∂_{j}wv̄\} dx .  (6.3)

By making the appropriate change-of-variable in (6.3) and setting w_{k} = φ_{k}^{∗}(w), v_{k} = φ_{k}^{∗}(v), we obtain

a^{k}(w_{k}, v_{k}) = ∫_{Q_{+}} { ∑_{i,j=1}^{n} a_{ij}^{k}∂_{i}(w_{k})∂_{j}(v_{k}) + ∑_{j=0}^{n} a_{j}^{k}∂_{j}(w_{k})v_{k}} dy . (6.4)

The resulting form (6.4) is strongly-elliptic on Q_{+} (exercise).

Let u be the solution of (6.2) and let v ∈ H^{1}(G ∩ G_{k}) vanish in a neighborhood of ∂G_{k}. (That is, the support of v is contained in G_{k}.) Then the extension of v to all of G as zero on G ∼ G_{k} belongs to H^{1}(G) and we obtain from (6.4) and (6.2)

a^{k}(φ_{k}^{∗}(u), φ_{k}^{∗}(v)) = a(u, v) = ∫_{Q_{+}} F_{k}φ_{k}^{∗}(v) dy ,

where F_{k} ≡ φ_{k}^{∗}(F) · J(φ_{k}) ∈ L^{2}(Q_{+}). Letting V denote the space of those v ∈ H^{1}(Q_{+}) which vanish in a neighborhood of ∂Q, and u_{k} ≡ φ_{k}^{∗}(u), we have shown that u_{k} ∈ H^{1}(Q_{+}) satisfies

a^{k}(u_{k}, v_{k}) = ∫_{Q_{+}} F_{k}v_{k} dy ,  v_{k} ∈ V  (6.5)

where a^{k}(·, ·) is strongly elliptic with continuously differentiable coefficients with bounded derivatives and F_{k} ∈ L^{2}(Q_{+}). We shall show that the restriction of u_{k} to the compact subset K ≡ φ_{k}^{−1}(B_{k}) of Q belongs to H^{2}(Q_{+} ∩K). The first case (a) above will then follow.

6.3

Hereafter we drop the subscript “k” in (6.5). Thus, we have u ∈ H^{1}(Q_{+}), F ∈ L^{2}(Q_{+}) and

a(u, v) = ∫_{Q_{+}} F v ,  v ∈ V .  (6.6)

Since K ⊂⊂ Q, there is by Lemma II.1.1 a φ ∈ C_{0}^{∞}(Q) such that 0 ≤ φ(x) ≤ 1 for x ∈ Q and φ(x) = 1 for x ∈ K. We shall first consider φ · u.

Let w be a function defined on the half-space ℝ_{+}^{n}. For each h ∈ ℝ we define a translate of w by

(τ_{h}w)(x_{1}, x_{2}, …, x_{n}) = w(x_{1} + h, x_{2}, …, x_{n})

and a difference of w by

∇_{h}w = (τ_{h}w − w)/h

if h ≠ 0.

Lemma 6.2 If w, v ∈ L^{2}(Q_{+}) and the distance δ of the support of w to ∂Q is positive, then

(τ_{h}w, v)_{L^{2}(Q_{+})} = (w, τ_{−h}v)_{L^{2}(Q_{+})}

for all h ∈ ℝ with |h| < δ.

Proof : This follows by the obvious change of variable and the observation that each of the above integrands is non-zero only on a compact subset of Q_{+}.

Corollary ∥τ_{h}w∥_{L^{2}(Q_{+})} = ∥w∥_{L^{2}(Q_{+})}.

Lemma 6.3 If w ∈ V, then

∥∇_{h}w∥_{L^{2}(Q_{+})} ≤ ∥∂_{1}w∥_{L^{2}(Q_{+})} , 0 < |h| < δ .

Proof : It follows from the preceding Corollary that it is sufficient to consider the case where w ∈ C^{1}(Ḡ) ∩ V. Assuming this, and denoting the support of w by supp(w), we have

∇_{h}w(x) = h^{−1} ∫_{x_{1}}^{x_{1}+h} ∂_{1}w(t, x_{2}, …, x_{n}) dt , w ∈ supp(w) .

The Cauchy-Schwartz inequality gives

|∇_{h}w(x)| ≤ h^{−1/2}(∫_{x_{1}}^{x_{1}+h} |∂_{1}w(t, x_{2}, …, x_{n})|^{2} dt)^{1/2} , x ∈ supp(w) ,

and this leads to

∥∇_{h}w∥_{L^{2}(Q_{+})}^{2} ≤ h^{−1} ∫_{supp(w)} ∫_{x_{1}}^{x_{1}+h} |∂_{1}w(t, x_{2}, …, x_{n})|^{2} dt dx
= h^{−1} ∫_{Q_{+}} ∫_{0}^{h} |∂_{1}w(t + x_{1}, x_{2}, …, x_{n})|^{2} dt dx
= h^{−1} ∫_{0}^{h} ∫_{Q_{+}} |∂_{1}w(t + x_{1}, …, x_{n})|^{2} dx dt
= h^{−1} ∫_{0}^{h} ∫_{Q_{+}} |∂_{1}w(x_{1}, …, x_{n})|^{2} dx dt
= ∥∂_{1}w∥_{L^{2}(Q_{+})} .

Corollary lim_{h→0}(∇_{h}w) = ∂_{1}w in L^{2}(Q_{+}).

Proof : {∇_{h} : 0 < |h| < δ} is a family of uniformly bounded operators on L^{2}(Q_{+}), so it suffices to show the result holds on a dense subset.

We shall consider the forms

a_{h}(w, v) ≡ ∫_{Q_{+}}\{ ∑_{i,j=1}^{n}(∇_{h}a_{ij})∂_{i}w\overline{∂_{j}v} + ∑_{j=0}^{n}(∇_{h}a_{j})∂_{j}wv̄ \}

for w, v ∈ V and |h| < δ, δ being given as in Lemma 6.2. Since the coefficients in (6.5) have bounded derivatives, the mean-value theorem shows

|a_{h}(w, v)| ≤ C∥w∥_{H^{1}(Q_{+})} · ∥v∥_{H^{1}(Q_{+})}  (6.7)

where the constant is independent of w, v and h. Finally, we note that for w, v and h as above

a(∇_{h}w, v) + a(w, ∇_{−h}v) = −a_{−h}(τ_{−h}w, v) .  (6.8)

This follows from a computation starting with the first term above and Lemma 6.2.

After this lengthy preparation we continue with the proof of Theorem 6.1. From (6.6) we have the identity

a(∇_{h}(φu), v) = {a(∇_{h}(φu), v) + a(φu, ∇_{−h}v)}  (6.9)

+ {a(u, φ∇_{−h}v) − a(φu, ∇_{−h}v)} − (F, φ∇_{−h}v)_{L^{2}(Q_{+})}

for v ∈ V and 0 < |h| < δ, δ being the distance from K to ∂Q. The first term can be bounded appropriately by using (6.7) and (6.8). The third is similarly bounded and so we consider the second term in (6.9). An easy computation gives

a(u, φ∇_{−h}v) − a(φu, ∇_{−h}v)

= ∫_{Q_{+}}{ ∑_{i,j=1}^{n} a_{ij}(∂_{i}u∂_{j}φ∇_{−h}v − ∂_{i}φu∇_{−h}(∂_{j}v)) − ∑_{j=1}^{n} a_{j}∂_{j}φu(∇_{−h}v)} .

Thus, we obtain the estimate

|a(∇_{h}(φu), v)| ≤ C∥v∥_{H^{1}(Q_{+})} ,  v ∈ V , 0 < |h| < δ ,  (6.10)

in which the constant C is independent of h and v. Since a(·, ·) is stronglyelliptic we may assume it is coercive (Exercise 6.2), so setting v = ∇_{h}(φu) in (6.10) gives

c∥∇_{h}(φu)∥_{H^{1}(Q_{+})}^{2} ≤ C∥∇_{h}(φu)∥_{H^{1}(Q_{+})} ,  0 < |h| < δ ,  (6.11)

hence, {∇_{h}(φu) : |h| < δ} is bounded in the Hilbert space H^{1}(Q_{+}). By Theorem I.6.2 there is a sequence h_{n} → 0 for which ∇_{h_{n}}(φu) converges weakly to some w ∈ H^{1}(Q_{+}). But ∇_{h_{n}}(φu) converges weakly in L^{1}(Q_{+}) to ∂_{1}(φu), so the uniqueness of weak limits implies that ∂_{1}(φu) = w ∈ H^{1}(Q_{+}). It follows that ∂_{1}^{2}(φu) ∈ L^{2}(Q_{+}), and the same argument shows that each of the tangential derivatives ∂_{1}^{2}u, ∂_{2}^{2}u, …, ∂_{n−1}^{2}u belongs to L^{2}(K). (Recall φ = 1 on K.) This information together with the partial differential equation resulting from (6.6) implies that a_{nn} · ∂_{n}^{2}(u) ∈ L^{2}(K). The strong ellipticity implies a_{nn} has a positive lower bound on K, so ∂_{n}^{2}u ∈ L^{2}(K). Since n and all of its derivatives through second order are in L^{2}(K), it follows from Theorem II.5.5 that u ∈ H^{2}(K).

The preceding proves the case (a) above. The case (b) follows by using the differencing technique directly on β_{0}u. In particular, we can compute differences on β_{0}u in any direction. The details are an easy modification of those of this section and we leave them as an exercise.

6.4

We discuss some extensions of Theorem 6.1. First, we note that the result and proof of Theorem 6.1 also hold if we replace H^{1}(G) by H_{0}^{1}(G). This results from the observation that the subspace H_{0}^{1}(G) is invariant under multiplication by smooth functions and translations and differences in tangential directions along the boundary of G. Thus we obtain a regularity result for the Dirichlet problem.

Theorem 6.4 Let u ∈ H_{0}^{1}(G) satisfy

a(u, v) = ∫_{G} F v̄ , v ∈ H_{0}^{1}(G)

where the set G ⊂ ℝ^{n} and sesquilinear form a(·, ·) are given as in Theorem 6.1, and F ∈ L^{2}(G). Then u ∈ H^{2}(G).

When the data in the problem is smoother yet, one expects the same to be true of the solution. The following describes the situation which is typical of second-order elliptic boundary value problems.

Definition. Let V be a closed subspace of H^{1}(G) with H_{0}^{1}(G) ≤ V, and let a(·, ·) be a continuous sesquilinear form on V. Then a(·, ·) is called k-regular on V if for every F ∈ H^{s}(G) with 0 ≤ s ≤ k and every solution u ∈ V of

a(u, v) = (F, v)_{L^{2}(G)} , v ∈ V

7. CLOSED OPERATORS, ADJOINTS AND EIGENFUNCTION EXPANSIONS83

we have u ∈ H^{2+s}(G).

Theorems 6.1 and 6.4 give sufficient conditions for the form a(·, ·) given by (6.1) to be 0-regular over H^{1}(G) and H_{0}^{1}(G), respectively. Moreover, we have the following.

Theorem 6.5 The form a(·, ·) given by (6.1) is k-regular over H^{1}(G) and H_{0}^{1}(G) if ∂G is a C^{2+k}-manifold and the coefficients {a_{ij}, a_{j}} all belong to C^{1+k}(Ḡ).

7 Closed operators, adjoints and eigenfunction expansions

7.1

We were led in Section 2 to consider a linear map A : D → H whose domain D is a subspace of the Hilbert space H. We shall call such a map an (unbounded) operator on the Hilbert space H. Although an operator is frequently not continuous (with respect to the H-norm on D) it may have the property we now consider. The graph of A is the subspace

G(A) = {[x, Ax] : x ∈ D}

of the product H × H. (This product is a Hilbert space with the scalar product

([x_{1}, x_{2}], [y_{1}, y_{2}])_{H×H} = (x_{1}, y_{1})_{H} + (x_{2}, y_{2})_{H} .

The addition and scalar multiplication are defined componentwise.) The operator A on H is called closed if G(A) is a closed subset of H × H. That is, A is closed if for any sequence x_{n} ∈ D such that x_{n} → x and Ax_{n} → y in H, we have x ∈ D and Ax = y.

Lemma 7.1 If A is closed and continuous (i.e., ∥Ax∥_{H} ≤ K∥x∥_{H}, x ∈ H) then D is closed.

Proof : If x_{n} ∈ D and x_{n} → x ∈ H, then {x_{n}} and, hence, {Ax_{n}} are Cauchy sequences. H is complete, so Ax_{n} → y ∈ H and G(A) being closed implies x ∈ D.

When D is dense in H we define the adjoint of A as follows. The domain of the operator A^{∗} is the subspace D^{∗} of all y ∈ H such that the map x ↦ (Ax, y)_{H} : D → 𝕂 is continuous. Since D is dense in H, Theorem I.4.5 asserts that for each such y ∈ D^{∗} there is a unique A^{∗}y ∈ H such that

(Ax, y) = (x, A^{∗}y) ,  x ∈ D , y ∈ D^{∗} .  (7.1)

Then the function A^{∗} : D^{∗} → H is clearly linear and is called the adjoint of A. The following is immediate from (7.1).

Lemma 7.2 A^{∗} is closed.

Lemma 7.3 If D = H, then A^{∗} is continuous, hence, D^{∗} is closed.

Proof : If A^{∗} is not continuous there is a sequence x_{n} ∈ D^{∗} such that ∥x_{n}∥ = 1 and ∥A^{∗}x_{n}∥ → ∞. From (7.1) it follows that for each x ∈ H,

|(x, A^{∗}x_{n})_{H}| = |(Ax, x_{n})_{H}| ≤ ∥Ax∥_{H} ,

so the sequence {A^{∗}x_{n}} is weakly bounded. But Theorem I.6.1 implies that it is bounded, a contradiction.

Lemma 7.4 If A is closed, then D^{∗} is dense in H.

Proof : Let y ∈ H, y ≠ 0. Then [0, y] ∉ G(A) and G(A) closed in H × H imply there is an f ∈ (H × H)′ such that f[G(A)] = {0} and f(0, y) ≠ 0. In particular, let P : H × H → G(A)^{⊥} be the projection onto the orthogonal complement of G(A) in H × H, define [u, v] = P [0, y], and set

f(x_{1}, x_{2}) = (u, x_{1})_{H} + (v, x_{2})_{H} , x_{1}, x_{2} ∈ H .

Then we have

0 = f(x, Ax) = (u, x)_{H} + (v, Ax)_{H} , x ∈ D

so v ∈ D^{∗}, and 0 ≠ f(0, y) = (v, y)_{H}. The above shows (D^{∗})^{⊥} = {0}, so D^{∗} is dense in H.

The following result is known as the closed-graph theorem.

Theorem 7.5 Let A be an operator on H with domain D. Then A is closed and D = H if and only if A ∈ L(H).

Proof : If A is closed and D = H, then Lemma 7.3 and Lemma 7.4 imply A^{∗} ∈ L(H). Then Theorem I.5.2 shows (A^{∗})^{∗} ∈ L(H). But (7.1) shows A = (A^{∗})^{∗}, so A ∈ L(H). The converse in immediate.

The operators with which we are most often concerned are adjoints of another operator. The preceding discussion shows that the domain of such an operator, i.e., an adjoint, is all of H if and only if the operator is continuous. Thus, we shall most often encounter unbounded operators which are closed and densely defined.

We give some examples in H = L^{2}(G), G = (0, 1).

7.2

Let D = H_{0}^{1}(G) and A = i∂. If [u_{n}, Au_{n}] ∈ G(A) converges to [u, v] in H × H, then in the identity

∫_{0}^{1} Au_{n}φ dx = −i ∫_{0}^{1} u_{n} Dφ dx , φ ∈ C_{0}^{∞}(G) ,

we let n → ∞ and thereby obtain

∫_{0}^{1} vφ dx = −i ∫_{0}^{1} u Dφ dx , φ ∈ C_{0}^{∞}(G) .

This means v = i∂u = Au and u_{n} → u in H^{1}(G). Hence u ∈ H_{0}^{1}(G), and we have shown A is closed.

To compute the adjoint, we note that

∫_{0}^{1} Auv̄ dx = ∫_{0}^{1} uf̄dx , u ∈ H_{0}^{1}(G)

for some pair v, f ∈ L^{2}(G) if and only if v ∈ H^{1}(G) and f = i∂v. Thus D^{∗} = H^{1}(G) and A^{∗} = i∂ is a proper extension of A.

7.3

We consider the operator A^{∗} above: on its domain D^{∗} = H^{1}(G) it is given by A^{∗} = i∂. Since A^{∗} is an adjoint it is closed. We shall compute A^{∗∗} = (A^{∗})^{∗}, the second adjoint of A. We first note that the pair [u, f] ∈ H × H is in the graph of A^{∗∗} if and only if

∫_{0}^{1} A^{∗}vū dx = ∫_{0}^{1} vf̄dx , v ∈ H^{1}(G) .

This holds for all v ∈ C_{0}^{∞}(G), so we obtain i∂u = f. Substituting this into the above and using Theorem II.1.6, we obtain

i ∫_{0}^{1} ∂(vū) dx = ∫_{0}^{1}[(i∂v)ū − v\overline{(i∂u)} ] dx = 0 ,

hence, v(1)ū(1) − v(0)ū(0) = 0 for all v ∈ H^{1}(G). But this implies u(0) = u(1) = 0, hence, u ∈ H_{0}^{1}(G). From the above it follows that A^{∗∗} = A.

7.4

Consider the operator B = i∂ on L^{2}(G) with domain D(B) = {u ∈ H^{1}(G) : u(0) = cu(1)} where c ∈ ℂ is given. If v, f ∈ L^{2}(G), then B^{∗}v = f if and only if

∫_{0}^{1} i∂u · v̄ dx = ∫_{0}^{1} uf̄dx , u ∈ D .

But C_{0}^{∞}(G) ≤ D implies v ∈ H^{1}(G) and i∂v = f. We substitute this identity in the above and obtain

0 = i ∫_{0}^{1} ∂(uv̄) dx = iu(1)[v̄(1) − cv̄(0)] , u ∈ D .

The preceding shows that v ∈ D(B^{∗}) only if v ∈ H^{1}(0, 1) and v(1) = c̄v(0). It is easy to show that every such v belongs to D(B^{∗}), so we have shown that D(B^{∗}) = {v ∈ H^{1}(G) : v(1) = c̄v(0)} and B^{∗} = i∂.

7.5

We return to the situation of Section 2.2. Let a(·, ·) be a continuous sesquilinear form on the Hilbert space V which is dense and continuously imbedded in the Hilbert space H. We let D be the set of all u ∈ V such that the map v ↦ a(u, v) is continuous on V with the norm of H. For such a u ∈ D, there is a unique Au ∈ H such that

a(u, v) = (Au, v)_{H} ,  u ∈ D , v ∈ V .  (7.2)

This defines a linear operator A on H with domain D.

Consider the (adjoint) sesquilinear form on V defined by b(u, v) = \overline{a(v, u)}, u, v ∈ V. This gives another operator B on H with domain D(B) determined as before by

b(u, v) = (Bu, v)_{H} , u ∈ D(B) , v ∈ V .

Theorem 7.6 Assume there is a λ > 0 and c > 0 such that

Re a(u, u) + λ|u|_{H}^{2} ≥ c∥u∥_{V}^{2} ,  u ∈ V .  (7.3)

Then D is dense in H, A is closed, and A^{∗} = B, hence, D^{∗} = D(B).

Proof : Theorem 2.2 shows D is dense in H. If we prove A^{∗} = B, then by symmetry we obtain B^{∗} = A, hence A is closed by Lemma 7.2.

Suppose v ∈ D(B). Then for all u ∈ D(A) we have (Au, v)_{H} = a(u, v) = \overline{b(v, u)} = \overline{(Bv, u)}_{H}, hence, (Au, v)_{H} = (u, Bv)_{H}. This shows D(B) ≤ D^{∗} and A^{∗}|_{D(B)} = B. We need only to verify that D(B) = D^{∗}. Let u ∈ D^{∗}. Since B + λ is surjective, there is a u_{0} ∈ D(B) such that (B + λ)u_{0} = (A^{∗} + λ)u. Then for all v ∈ D we have

((A + λ)v, u)_{H} = (v, (B + λ)u_{0})_{H} = a(v, u_{0}) + λ(v, u_{0})_{H}
= ((A + λ)v, u_{0})_{H} .

But A+λ is a surjection, so this implies u = u_{0} ∈ D(B). Hence, D^{∗} = D(B).

For those operators as above which arise from a symmetric sesquilinear form on a space V which is compactly imbedded in H, we can apply the eigenfunction expansion theory for self-adjoint compact operators.

Theorem 7.7 Let V and H be Hilbert spaces with V dense in H and assume the injection V ↪ H is compact. Let A : D → H be the linear operator determined as above by a continuous sesquilinear form a(·, ·) on V which we assume is V -elliptic and symmetric:

a(u, v) = \overline{a(v, u)} , u, v ∈ V .

Then there is a sequence {v_{j}} of eigenfunctions of A with

\begin{matrix}Av_{j} = λ_{j}v_{j} ,  |v_{j}|_{H} = 1 , \\ (v_{i}, v_{j})_{H} = 0 ,  i ≠ j , \\ 0 < λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{n} → +∞ as n → +∞ , \end{matrix} \}  (7.4)

and {v_{j}} is a basis for H.

Proof : From Theorem 7.6 it follows that A = A^{∗} and, hence, A^{−1} ∈ L(H) is self-adjoint. The V -elliptic condition (5.5) shows that A^{−1} ∈ L(H, V).

Since the injection V ↪ H is compact, it follows that A^{−1} : H → V → H is compact. We apply Theorem I.7.5 to obtain a sequence {v_{j}} of eigenfunctions of A^{−1} which are orthonormal in H and form a basis for D = Rg(A^{−1}). If their corresponding eigenvalues are denoted by {μ_{j}}, then the symmetry of a(·, ·) and (5.5) shows that each μ_{j} is positive. We obtain (7.4) by setting λ_{j} = 1/μ_{j} for j ≥ 1 and noting that lim_{j→∞} μ_{j} = 0.

It remains to show {v_{j}} is a basis for H. (We only know that it is a basis for D.) Let f ∈ H and u ∈ D with Au = f. Let ∑ b_{j}v_{j} be the Fourier series for f, ∑ c_{j}v_{j} the Fourier series for u, and denote their respective partial sums by

u_{n} = ∑_{j=1}^{n} c_{j}v_{j} , f_{n} = ∑_{j=1}^{n} b_{j}v_{j} .

We know lim_{n→∞} u_{n} = u and lim_{n→∞} f_{n} = f_{∞} exists in H (cf. Exercise I.7.2). For each j ≥ 1 we have

b_{j} = (Au, v_{j})_{H} = (u, Av_{j})_{H} = λ_{j}c_{j} ,

so Au_{n} = f_{n} for all n ≥ 1. Since A is closed, it follows Au = f_{∞}, hence, f = lim_{n→∞} f_{n} as was desired.

If we replace A by A + λ in the proof of Theorem 7.7, we observe that ellipticity of a(·, ·) is not necessary but only that a(·, ·)+λ(·, ·)_{H} be V -elliptic for some λ ∈ ℝ.

Corollary 7.8 Let V and H be given as in Theorem 7.7, let a(·, ·) be continuous, sesquilinear, and symmetric. Assume also that

a(v, v) + λ|v|_{H}^{2} ≥ c∥v∥_{V}^{2} , v ∈ V

for some λ ∈ ℝ and c > 0. Then there is an orthonormal sequence of eigenfunctions of A which is a basis for H and the corresponding eigenvalues satisfy −λ < λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{n} → +∞ as n → +∞.

We give some examples in H = L^{2}(G), G = (0, 1). These eigenvalue problems are known as Sturm-Liouville problems. Additional examples are described in the exercises.

7.6

Let V = H_{0}^{1}(G) and define a(u, v) = ∫_{0}^{1} ∂u\overline{∂v} dx. The compactness of V → H follows from Theorem II.5.7 and Theorem 5.3 shows a(·, ·) is H_{0}^{1}(G)- elliptic. Thus Theorem 7.7 holds; it is a straightforward exercise to compute the eigenfunctions and corresponding eigenvalues for the operator A = −∂^{2} with domain D(A) = H_{0}^{1}(G) ∩ H^{2}(G):

v_{j}(x) = 2 sin(jπx) , λ = (jπ)^{2} , j = 1, 2, 3, ….

Since {v_{j}} is a basis for L^{2}(G), each F ∈ L^{2}(G) has a Fourier sine-series expansion. Similar results hold in higher dimension for, e.g., the eigenvalue problem

\{ \begin{matrix}−Δ_{n}v(x) = λv(x) , & x ∈ G , \\ v(s) = 0 , & s ∈ ∂G , \end{matrix}

but the actual computation of the eigenfunctions and eigenvalues is difficult except for very special regions G ⊂ ℝ^{n}.

7.7

Let V = H^{1}(G) and choose a(·, ·) as above. The compactness follows from Theorem II.5.8 so Corollary 7.8 applies for any λ > 0 to give a basis of eigenfunctions for A = −∂^{2} with domain D(A) = {v ∈ H^{2}(G) : v′(0) = v′(1) = 0}:

\begin{aligned}v_{0}(x) & = & 1 , v_{j}(x) = 2 cos(jπx) , & j ≥ 1 , \\ λ_{j} & = & (jπ)^{2} , & j ≥ 0 . \end{aligned}

As before, similar results hold for the Laplacean with boundary conditions of second type in higher dimensions.

7.8

Let a(·, ·) be given as above but set V = {v ∈ H^{1}(G) : v(0) = v(1)}. Then we can apply Corollary 7.8 to the periodic eigenvalue problem (cf. (4.5))

\begin{aligned}−∂^{2}v(x) & = & λv(x) , 0 < x < 1 , \\ v(0) & = & v(1) , v′(0) = v′(1) . \end{aligned}

The eigenfunction expansion is just the standard Fourier series.

Exercises

1.1. Use Theorem 1.1 to show the problem −Δ_{n}u = F in G, u = 0 on ∂G is well-posed. Hint: Use Theorem II.2.4 to obtain an appropriate norm on H_{0}^{1}(G).

1.2. Use Theorem 1.1 to solve (1.1) with the boundary condition ∂u/∂ν+u = 0 on ∂G. Hint: Use (u, v)_{V} ≡ (u, v)_{H^{1}(G)} + (γu, γv)_{L^{2}(∂G)} on H^{1}(G).

2.1. Give the details of the construction of α, β in (2.2).

2.2. Verify the remark on H = L^{2}(G) following (2.5) (cf. Section I.5.3).

2.3. Use Theorem I.1.1 to construct the F which appears after (2.6). Check that it is continuous.

2.4. Show that a(u, v) = ∫_{0}^{1} ∂u(x)∂v̄(x) dx, V = {u ∈ H^{1}(0, 1) : u(0) = 0}, and f(v) ≡ v(1/2) are admissible data in Theorem 2.1. Find a formula for the unique solution of the problem.

2.5. In Theorem 2.1 the continuous dependence of the solution u on the data f follows from the estimate made in the theorem. Consider the two abstract boundary value problems A_{1}u_{1} = f and A_{2}u_{2} = f where f ∈ V′, and A_{1}, A_{2} ∈ L(V, V′) are coercive with constants c_{1}, c_{2}, respectively. Show that the following estimates holds:

∥u_{1} − u_{2}∥ ≤ (1/c_{1})∥(A_{2} − A_{1})u_{2}∥ ,
∥u_{1} − u_{2}∥ ≤ (1/c_{1}c_{2})∥A_{2} − A_{1}∥ ∥f∥ .

Explain how these estimates show that the solution of (2.1) depends continuously on the form a(·, ·) or operator A.

3.1. Show (3.3) implies (3.1) in Theorem 3.1.

3.2. (Non-homogeneous Boundary Conditions.) In the situation of Theorem 3.1, assume we have a closed subspace V_{1} with V_{0} ⊂ V_{1} ⊂ V and u_{0} ∈ V. Consider the problem to find

u ∈ V , u − u_{0} ∈ V_{1} , a(u, v) = f(v) , v ∈ V_{1} .

(a) Show this problem is well-posed if a(·, ·) is V_{1}-coercive.

(b) Characterize the solution by u − u_{0} ∈ V_{1}, u ∈ D_{0}, Au = F, and ∂u(v) + a_{2}(γu, γv) = g(γv), v ∈ V_{1}.

(c) Construct an example of the above with V_{0} = H_{0}^{1}(G), V = H^{1}(G), V_{1} = {v ∈ V : v|_{Γ} = 0}, where Γ ⊂ ∂G is given.

7. CLOSED OPERATORS, ADJOINTS ... 91

4.1. Verify that the formal operator and Green’s theorem are as indicated

in Section 4.1. 4.2. Characterize the boundary value problem resulting from the choice of

V = {v ∈ H^{1}(G) : v = const. on G_{0}} in Section 4.2, where G_{0} ⊂ G is given.

4.3. When G is a cube in ℝ^{n}, show (4.5) is related to a problem on ℝ^{n} with

periodic solutions. 4.4. Choose V in Section 4.2 so that the solution u : ℝ^{n} → 𝕂 is periodic in

each coordinate direction. 5.1. Formulate and solve the problem (4.8) with non-homogeneous data pre-

scribed on ∂G and Σ. 5.2. Find choices for V in Section 4.3 which lead to well-posed problems.

Characterize the solution by a boundary value problem. 5.3. Prove Corollary 5.4. 5.4. Discuss coercivity of the form (4.6). Hint: Re(∫_{∂G} \frac{∂u}{∂τ} ū ds) = 0. 6.1. Show (6.4) is strongly-elliptic on Q_{+}. 6.2. Show that the result of Theorem 6.1 holds for a(·, ·) if and only if it

holds for a(·, ·) + λ(·, ·)_{L^{2}(G)}. Hence, one may infer coercivity from strong ellipticity without loss of generality.

6.3. If u ∈ H^{1}(G), show ∇_{h}(u) converges weakly in L^{2}(G) to ∂_{1}(u). 6.4. Prove the case (b) in Theorem 6.1. 6.5. Prove Theorem 6.5. 6.6. Give sufficient conditions for the solution of (6.2) to be a classical solu-

tion in C_{u}^{2}(G). 7.1. Prove Lemma 7.2 of Section 7.1. 7.2. Compute the adjoint of ∂ : {v ∈ H^{1}(G) : v(0) = 0} → L^{2}(G), G =

(0, 1).

7.3. Let D ≤ H^{2}(G), G = (0, 1), a_{1}(·), a_{2}(·) ∈ C^{1}(Ḡ), and define L : D → L^{2}(G) by Lu = ∂^{2}u + a_{1}∂u + a_{2}u. The formal adjoint of L is defined by

L^{∗}v(φ) = ∫_{0}^{1} v(x)\overline{Lφ(x)} dx , v ∈ L^{2}(G) , φ ∈ C_{0}^{∞}(G) .

(a) Show L^{∗}v = ∂^{2}v − ∂(ā_{1}v) + ā_{2}v in D^{∗}(G).

(b) If u, v ∈ H^{2}(G), then ∫_{0}^{1}(Luv̄ − uL^{∗}v̄) dx = J(u, v)|_{x=0}^{x=1}, where J(u, v) = v̄∂u − u∂v̄ + a_{1}uv̄.

(c) D(L^{∗}) = {v ∈ H^{2}(G) : J(u, v)|_{x=0}^{x=1} = 0, all u ∈ D} determines the domain of the L^{2}(G)-adjoint.

(d) Compute D(L^{∗}) when L = ∂^{2} + 1 and each of the following:

(i) D = {u : u(0) = u′(0) = 0}, (ii) D = {u : u(0) = u(1) = 0}, (iii) D = {u : u(0) = u(1), u′(0) = u′(1)}.

7.4. Let A be determined by {a(·, ·), V, H} as in (7.2) and A_{λ} by {a(·, ·) + λ(·, ·)_{H}, V, H}. Show D(A_{λ}) = D(A) and A_{λ} = A + λI.

7.5. Let H_{j}, V_{j} be Hilbert spaces with V_{j} continuously embedded in H_{j} for j = 1, 2. Show that if T ∈ L(H_{1}, H_{2}) and if T_{1} ≡ T |_{V_{1}} ∈ L(V_{1}, V_{2}), then T_{1} ∈ L(V_{1}, V_{2}).

7.6. In the situation of Section 6.4, let a(·, ·) be 0-regular on V and assume a(·, ·) is also V -elliptic. Let A be determined by {a(·, ·), V, L^{2}(G)} as in (7.2).

(a) Show A^{−1} ∈ L(L^{2}(G), V).

(b) Show A^{−1} ∈ L(L^{2}(G), H^{2}(G)).

(c) If a(·, ·) is k-regular, show A^{−p} ∈ L(L^{2}(G), H^{2+k}(G)) if p is sufficiently large.

7.7. Let A be self-adjoint on the complex Hilbert space H. That is, A = A^{∗}.

(a) Show that if Im(λ) ≠ 0, then λ−A is invertible and | Im(λ)| ∥x∥_{H} ≤ ∥(λ − A)x∥_{H} for all x ∈ D(A).

(b) Rg(λ − A) is dense in H.

(c) Show (λ − A)^{−1} ∈ L(H) and ∥(λ − A)^{−1}∥ ≤ | Im(λ)|^{−1}.

7.8. Show Theorem 7.7 applies to the mixed Dirichlet-Neumann eigenvalue problem

−∂^{2}v = λv(x) , 0 < x < 1 , v(0) = v′(1) = 0 .

Compute the eigenfunctions.

7.9. Show Corollary 7.8 applies to the eigenvalue problem with boundary conditions of third type

−∂^{2}v(x) = λv(x) , 0 < x < 1 ,

∂v(0) − hv(0) = 0 , ∂v(1) + hv(1) = 0 ,

where h > 0. Compute the eigenfunctions.

7.10. Take cc̄ = 1 in Section 7.4 and discuss the eigenvalue problem Bv = λv.

7.11. In the proof of Theorem 7.7, deduce that {v_{j}} is a basis for H directly from the fact that D̄ = H.