%% R.E. Showalter: Chapter 3
\chapter{Boundary Value Problems}
\section{Introduction}
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We shall recall two classical boundary value problems and show that an
appropriate generalized or abstract formulation of each of these is a
well-posed problem.
This provides a weak global solution to each problem and motivates much
of our latter discussion.
\subsection{} %1.1
Suppose we are given a subset $G$ of $\RR^n$ and a function $F:G\to \KK$.
We consider two boundary value problems for the partial differential
equation
\begin{equation}\label{eq311}
-\Delta_n u(x) + u(x) = F(x)\ ,\qquad x\in G\ .
\end{equation}
The {\it Dirichlet problem\/} is to find a solution of \eqn{311} for
which $u=0$ on $\partial G$.
The {\it Neumann problem\/} is to find a solution of \eqn{311} for
which $(\partial u/\partial\nu)=0$ on $\partial G$.
In order to formulate these problems in a meaningful way, we recall the
first formula of Green
\begin{equation}\label{eq312}
\int_G \left( (\Delta_n u) v +\nabla u \cdot \nabla v\right)
=\int_{\partial G} {\partial u\over\partial \nu} v
= \int_{\partial G} \gamma_1 u\cdot\gamma_0 v
\end{equation}
which holds if $\partial G$ is sufficiently smooth and if $u\in H^2(G)$,
$v\in H^1(G)$.
Thus, if $u$ is a solution of the Dirichlet problem and if $u\in H^2(G)$,
then we have $u\in H_0^1 (G)$ (since $\gamma_0 u=0$) and
(from \eqn{311} and \eqn{312})
\begin{equation}\label{eq313}
(u,v)_{H^1(G)} = (F,v)_{L^2(G)}\ ,\qquad v\in H_0^1 (G)\ .
\end{equation}
Note that the identity \eqn{313} holds in general only for those
$v\in H^1(G)$ for which $\gamma_0v=0$.
If we drop the requirement that $v$ vanish on $\partial G$, then there would
be a contribution from \eqn{312} in the form of a boundary integral.
Similarly, if $u$ is a solution of the Neumann problem and $u\in H^2(G)$,
then (since $\gamma_1u=0$) we obtain from \eqn{311} and \eqn{312} the
identity \eqn{313} for all $v\in H^1(G)$.
That is, $u\in H^2(G)$ and \eqn{313} holds for all $v\in H^1(G)$.
Conversely, suppose $u\in H^2(G)\cap H_0^1(G)$ and \eqn{313} holds for
all $v\in H_0^1(G)$.
Then \eqn{313} holds for all $v\in C_0^\infty (G)$, so \eqn{311}
is satisfied in the sense of distributions on $G$, and $\gamma_0u=0$ is
a boundary condition.
Thus, $u$ is a solution of a Dirichlet problem.
Similarly, if $u\in H^2(G)$ and \eqn{313} holds for all $v\in H^1(G)$,
then $C_0^\infty (G) \subset H^1(G)$ shows \eqn{311} is satisfied
as before, and substituting \eqn{311} into \eqn{313} gives us
$$\int_{\partial G} \gamma_1 u\cdot\gamma_0 v=0\ ,\qquad v\in H^1(G)\ .$$
Since the range of $\gamma_0$ is dense in $L^2(\partial G)$, this implies
that $\gamma_1u=0$, so $u$ is a solution of a Neumann problem.
\subsection{} %1.2
The preceding remarks suggest a weak formulation of the Dirichlet problem
as follows:
\begin{quotation}
\noindent
Given $F\in L^2(G)$, find $u\in H_0^1(G)$ such that \eqn{313} holds for
all $u\in H_0^1(G)$.
\end{quotation}
\noindent In particular, the condition that $u\in H^2(G)$ is not necessary
for this formulation to make sense.
A similar formulation of the Neumann problem would be the following:
\begin{quotation}
\noindent
Given $F\in L^2(G)$, find $u\in H^1(G)$ such that \eqn{313} holds for
all $v\in H^1(G)$.
\end{quotation}
\noindent This formulation does not require that $u\in H^2(G)$, so we do
not necessarily have $\gamma_1u\in L^2(\partial G)$.
However, we can either extend the operator $\gamma_1$ so \eqn{312} 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:
\begin{quotation}
\noindent Given a Hilbert space $V$ and $f\in V'$, find $u\in V$ such that
for all $v\in V$
$$(u,v)_V = f(v)\ .$$
\end{quotation}
\noindent 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.\ref{thm1-4E} shows that this problem is well-posed.
\begin{theorem}\label{thm3-1A}
For each $f\in V'$, there exists exactly one $u\in V$ such that
$(u,v)_V = f(v)$ for all $v\in V$, and we have $\|u\|_V = \|f\|_{V'}$.
\end{theorem}
\plaincor
{If $u_1$ and $u_2$ are the solutions corresponding to $f_1$ and $f_2$, then
$$\|u_1-u_2\|_V = \|f_1-f_2\|_{V'}\ .$$}
\noindent Finally, we note that if $V= H_0^1(G)$ or $H^1(G)$, and
if $F\in L^2(G)$ then $\|f\|_{V'} \le \|F\|_{L^2(G)}$ where we identify
$L^2(G)\subset V'$ as indicated.
\section{Forms, Operators and Green's Formula} %2
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\subsection{} %2.1
We begin with a generalization of the weak Dirichlet problem and of the
weak Neumann problem of Section 1:
\begin{quotation}
\noindent
Given a Hilbert space $V$, a continuous sesquilinear form $a(\cdot,\cdot)$
on $V$, and $f\in V'$, find $u\in V$ such that
\begin{equation}\label{eq321}
a(u,v) =f(v)\ ,\qquad v\in V\ .
\end{equation}
\end{quotation}
The sesquilinear form $a(\cdot,\cdot)$ determines a pair of operators
$\alpha,\beta \in \L(V)$ by the identities
\begin{equation}\label{eq322}
a(u,v) = ( \alpha (u),v)_V = (u,\beta (v))_V\ ,\qquad u,v\in V\ .
\end{equation}
Theorem I.\ref{thm1-4E} is used to construct $\alpha$ and $\beta$ from
$a(\cdot,\cdot)$, and $a(\cdot,\cdot)$ is clearly determined by either
of $\alpha$ or $\beta$ through \eqn{322}.
Theorem I.\ref{thm1-4E} also defines the bijection $J\in \L(V',V)$ for which
$$f(v) = (J(f),v)_V\ ,\qquad f\in V'\ ,\ v\in V\ .$$
In fact, $J$ is just the inverse of $R_V$.
It is clear that $u$ is a solution of the ``weak'' problem associated
with \eqn{321} if and only if $\alpha (u)=J(f)$.
Since $J$ is a bijection, the solvability of this functional equation in $V$
depends on the invertibility of the operator $\alpha$.
A useful sufficient condition for $\alpha$ to be a bijection is given in the
following.
\definition
The sesquilinear form $a(\cdot,\cdot)$ on the Hilbert space $V$ is
{\it $V$-coercive\/} if there is a $c>0$ such that
\begin{equation}\label{eq323}
|a(v,v)| \ge c\|v\|_V^2\ ,\qquad v\in V\ .
\end{equation}
\qed
We show that the weak problem associated with a $V$-coercive form is
well-posed.
\begin{theorem}\label{thm3-2A}
Let $a(\cdot,\cdot)$ be a $V$-coercive continuous sesquilinear form.
Then, for every $f\in V'$, there is a unique $u\in V$ for which \eqn{321}
is satisfied. Furthermore, $\|u\|_V \le (1/c)\|f\|_{V'}$.
\end{theorem}
\proof
The estimate \eqn{323} implies that both $\alpha$ and $\beta$ are
injective, and we also obtain
$$\|\alpha (v)\|_V \ge c\|v\|_V\ ,\qquad v\in V\ .$$
This estimate implies that the range of $\alpha$ is closed.
But $\beta$ is the adjoint of $\alpha$ in $V$, so the range of
$\alpha$, $\Rg (\alpha)$, satisfies the orthogonality condition
$\Rg (\alpha)^\bot = K(\beta) = \{0\}$.
Hence, $\Rg (\alpha)$ is dense in $V$, and this shows $\Rg (\alpha)=V$.
Since $J$ is norm-preserving the stated results follow easily.
\subsection{} %2.2
We proceed now to construct some operators which characterize solutions
of problem \eqn{321} as solutions of boundary value problems for certain
choices of $a(\cdot,\cdot)$ and $V$.
First, define $\A \in \L(V,V')$ by
\begin{equation}\label{eq324}
a(u,v) = \A u(v)\ ,\qquad u,v\in V\ .
\end{equation}
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 \eqn{324}.
In particular, $u$ is a solution of the weak problem \eqn{321} if and
only if $u\in V$ and $\A u=f$, so the problem is determined by $\A$ when
$f\in V'$ is regarded as data.
We would like to know that the identity $\A u=f$ 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 $\RR^n$.
(For example, we are thinking of $V=H^1(G)$ in a Neumann problem as in
\eqn{311}.
The difficulty is that the space $C_0^\infty (G)$ is {\it 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 $H'\subset V'$)
and such that $H$ is identified with $H'$ through the Riesz map.
Thus we have the inclusions
$$V\hookrightarrow H = H' \hookrightarrow V'$$
and the identity
\begin{equation}\label{eq325}
f(v) = (f,v)_H\ ,\qquad f\in H\ ,\ v\in V\ .
\end{equation}
We call $H$ the {\it 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 $D= \{ u\in V: \A u\in H\}$.
In the examples, $\A u =f$, $u\in D$, will imply that a partial differential
equation is satisfied, since $C_0^\infty (G)$ will be dense in $H$.
Note that $u\in D$ if and only if $u\in V$ and there is a $K>0$ such that
$$|a(u,v)| \le K\|v\|_H \ ,\qquad v\in V\ .$$
(This follows from Theorem I.\ref{thm1-4E}.)
Finally, we obtain the following result.
\begin{theorem}\label{thm3-2B}
If $a(\cdot,\cdot)$ is $V$-coercive, then $D$ is dense in $V$,
hence, dense in $H$.
\end{theorem}
\proof
Let $w\in V$ with $(u,w)_V=0$ for all $u\in D$.
Then the operator $\beta$ from \eqn{322}
being surjective implies $w=\beta (v)$ for some $v\in V$.
Hence, we obtain $0=(u,\beta (v))_V = \A u(v) = (\A u,v)_H$ by
\eqn{325}, since $u\in D$.
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 $a(\cdot,\cdot)$ on $V$ is to consider a
closed subspace $V_0$ of $V$, let $i:V_0\hookrightarrow V$ denote the
identity and $\rho = i' :V'\to V'_0$ the restriction to $V_0$ of functionals
on $V$, and define $A= \rho \A :V\to V'_0$.
The operator $A\in \L(V,V'_0)$ defined by
$$a(u,v) = Au (v)\ ,\qquad u\in V\ ,\ v\in V_0$$
is called the {\it formal operator\/} determined by $a(\cdot,\cdot)$,
$V$ and $V_0$.
In examples, $V_0$ will be the closure in $V$ of $C_0^\infty (G)$, so
$V'_0$ is a space of distributions on $G$.
Thus, $Au = f\in V'_0$ will imply that a partial differential equation
is satisfied.
\subsection{} %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 $V\hookrightarrow H$ is continuous, and $V_0$
is dense in $H$.
Let $D$ be given as above and define $D_0=\{u\in V:Au\in H\}$, where we
identify $H\subset V'_0$.
Note that $u\in D_0$ if and only if $u\in V$ and there is a $K>0$ such that
$$|a(u,v)| \le K\|v\|_H\ ,\qquad v\in V_0\ ,$$
so $D\subset D_0$.
It is on $D_0$ that we compare $\A$ and $A$.
So, let $u\in D_0$ be fixed in the following and consider the functional
\begin{equation}\label{eq326}
\varphi (v) = \A u (v) - (Au,v)_H\ ,\qquad v\in V\ .
\end{equation}
Then we have $\varphi\in V'$ and $\varphi|_{V_0} =0$.
But these are precisely the conditions that characterize those
$\varphi \in V'$ which are in the range of $q':(V/V_0)' \to V'$,
the dual of the quotient map $q:V\to V/V_0$.
That is, there is a unique $F\in (V/V_0)'$ such that $q'(F)=F\circ q=\varphi$.
Thus, \eqn{326} determines an $F\in (V/V_0)'$ such that
$F(q(v))= \varphi (v)$, $v\in V$.
In order to characterize $(V/V_0)$, let $V_0$ be the kernel of a linear
surjection $\gamma :V\to B$ and denote by $\hat\gamma$ the quotient map
which is a bijection of $V/V_0$ onto $B$.
Define a norm on $B$ by $\|\hat\gamma (\hat x)\|_B = \|\hat x\|_{V/V_0}$
so $\hat \gamma$ is bicontinuous.
Then the dual operator $\hat\gamma{}' :B'\to (V/V_0)'$ is a bijection.
Given the functional $F$ above, there is a unique $\partial \in B'$ such
that $F=\hat\gamma{}'(\partial)$.
That is, $F=\partial \circ\hat\gamma$.
We summarize the preceding discussion in the following result.
\begin{theorem}\label{thm3-2C}
Let $V$ and $H$ be Hilbert spaces with $V$ dense and continuously imbedded
in $H$.
Let $H$ be identified with its dual $H'$ so \eqn{325} holds.
Suppose $\gamma$ is a linear surjection of $V$ onto a Hilbert space $B$
such that the quotient map $\hat\gamma :V/V_0\to B$ is norm-preserving,
where $V_0$, the kernel of $\gamma$, is dense in $H$.
Thus, we have $V_0\hookrightarrow H\hookrightarrow V'_0$.
Let $\A \in \L(V,V')$ and define $A\in \L(V,V'_0)$ by $A= \rho\A$, where
$\rho :V'\to V'_0$ is restriction to $V_0$, the dual of the injection
$V_0\hookrightarrow V$.
Let $D_0=\{u\in V: Au\in H\}$.
Then, for every $u\in D_0$, there is a unique $\partial (u)\in B'$ such that
\begin{equation}\label{eq327}
\A u (v) - (Au,v)_H = \partial (u) (\gamma (v))\ ,\qquad v\in V\ .
\end{equation}
The mapping $\partial :D_0\to B'$ is linear.
\end{theorem}
When $V'_0$ is a space of distributions, it is the formal operator $A$
that determines a partial differential equation.
When $\gamma$ 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 \eqn{327} is an abstract Green's formula.
We shall call $\partial$ the abstract {\it Green's operator\/}.
\example
Let $V= H^1(G)$ and $\gamma :H^1(G)\to L^2(\partial G)$ be the trace map
constructed in Theorem II.\ref{thm2-3A}.
Then $H_0^1(G)=V_0$ is the kernel of $\gamma$ and we denote by $B$ the
range of $\gamma$. Since $\hat\gamma$ is norm-preserving, the injection
$B\hookrightarrow L^2(\partial G)$ is continuous and, by duality,
$L^2(\partial G)\subset B'$, where we identify $L^2(\partial G)$ with
its dual space.
In particular, $B$ consists of functions on $\partial G$ and $L^2(\partial G)$
is a subspace of $B'$.
Continuing this example, we choose $H=L^2(G)$ and $a(u,v)=(u,v)_{H^1(G)}$,
so $Au = -\Delta_n u+u$ and $D_0 = \{u\in H^1(G): \Delta_n u\in L^2(G)\}$.
By comparing \eqn{327} with \eqn{312} we find that when $\partial G$
is smooth $\partial :D_0\to B'$ is an extension of
$\partial/\partial\nu = \gamma_1:H^2 (G)\to L^2(\partial G)$.
\section{Abstract Boundary Value Problems} %3
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\subsection{} %3.1
We begin by considering an abstract ``weak'' problem \eqn{321} motivated by
certain carefully chosen formulations of the Dirichlet and Neumann problems
for the Laplace differential operator.
The sesquilinear form $a(\cdot,\cdot)$ led to two operators:
$\A$, which is equivalent to $a(\cdot,\cdot)$, 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 $\partial$ in
Theorem \ref{thm3-2C}.
If $V$ is prescribed by boundary conditions, then these same boundary
conditions will be forced on a solution $u$ of \eqn{321}.
Such boundary conditions are called {\it stable\/} or {\it forced\/}
boundary conditions.
A second set of constraints may arise from Theorem \ref{thm3-2C} and
these are called {\it unstable\/} or {\it 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 \ref{thm3-2C} to
characterize solutions of \eqn{321} in a setting with more structure than
assumed before.
This additional structure consists essentially of splitting the form
$a(\cdot,\cdot)$ 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.
\subsection{} %3.2
We assume that we have a Hilbert space $V$ and a linear surjection
$\gamma : V\to B$ 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 $V_0\hookrightarrow H\hookrightarrow
V'_0$ and $V\hookrightarrow H\hookrightarrow V'$ and the identity \eqn{325}.
Let $a_1 :V\times V\to \KK$ and $a_2 :B\times B\to \KK$ be continuous
sesquilinear forms and define
$$a(u,v) = a_1(u,v) + a_2 (\gamma u,\gamma v)\ ,\qquad u,v\in V\ .$$
Similarly, let $F\in H$, $g\in B'$, and define
$$f(v) = (F,v)_H + g(\gamma v)\ ,\qquad v\in V\ .$$
The problem \eqn{321} is the following: find $u\in V$ such that
\begin{equation}\label{eq331}
a_1(u,v) + a_2 (\gamma u,\gamma v) = (F,v)_H + g(\gamma v)\ ,\qquad v\in V\ .
\end{equation}
We shall use Theorem \ref{thm3-2C} to show that \eqn{331} is equivalent
to an abstract boundary value problem.
\begin{theorem}\label{thm3-3A}
Assume we are given the Hilbert spaces, sesquilinear forms and functionals
as above. Let $\A_2 :B\to B'$ be given by
$$\A_2 \varphi (\psi) = a_1 (\varphi,\psi)\ ,\qquad \varphi,\psi\in B\ ,$$
and $A:V\to V'_0$ by
$$Au (v) = a_1(u,v)\ ,\qquad u\in V\ ,\ v\in V_0\ .$$
Let $D_0 = \{u\in V:Au\in H\}$ and $\partial_1 \in L(D_0,B')$ be given
by (Theorem \ref{thm3-2C})
\begin{equation}\label{eq332}
a_1(u,v) - (Au,v)_H = \partial_1 u(\gamma v)\ ,\qquad u\in D_0\ ,\ v\in V\ .
\end{equation}
Then, $u$ is a solution of \eqn{331} if and only if
\begin{equation}\label{eq333}
u\in V\ ,\quad Au =F\ ,\quad \partial_1 u+\A_2 (\gamma u) = g\ .
\end{equation}
\end{theorem}
\proof
Since $a_2(\gamma u,\gamma v)=0$ for all $v\in V_0$, it follows that the
formal operator $A$ and space $D_0$ (determined above by $a_1(\cdot,\cdot)$)
are equal, respectively, to the operator and domain determined by
$a(\cdot,\cdot)$ in Section 2.3.
Suppose $u$ is a solution of \eqn{331}.
Then $u\in V$, and the identity \eqn{331} for $v\in V_0$ and $V_0$
being dense in $H$ imply that $Au=F\in H$.
This shows $u\in D_0$ and using \eqn{332} in \eqn{331} gives
$$\partial_1 u(\gamma v) + a_2 (\gamma u,\gamma v) = g(\gamma v)\ ,\qquad
v\in V\ .$$
Since $\gamma$ is a surjection, this implies the remaining equation
in \eqn{333}. Similarly, \eqn{333} implies \eqn{331}.
\begin{corollary}\label{cor3-3B}
Let $D$ be the space of those $u\in V$ such that for some $F\in H$
$$a(u,v) = (F,v)_H\ ,\qquad v\in V\ .$$
Then $u\in D$ if and only if $u$ is a solution of \eqn{333} with $g=0$.
\end{corollary}
\proof
Since $V_0$ is dense in $H$, the functional $f\in V'$ defined above
is in $H$ if and only if $g=0$.
\section{Examples} %4
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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 \ref{thm3-3A}.
\subsection{} %4.1
Let there be given a set of (coefficient) functions
$$a_{ij} \in L^\infty (G)\ ,\quad 1\le i\ ,\ j\le n\ ;\qquad
a_j\in L^\infty (G)\ ,\quad 0\le j\le n\ ,$$
where $G$ is open and connected in $\RR^n$, and define
\begin{eqnarray}\label{eq341}
a(u,v) = \int_G \biggl\{ \sum_{i,j=1}^n a_{ij} (x)\partial_i u(x)
\partial_j \overline{v(x)}
&+& \sum_{j=0}^n a_j (x) \partial_j u(x)
\overline{v(x)}\,\biggr\}\,dx\ ,\nonumber\\
&&\qquad u,v\in H^1(G)\ ,
\end{eqnarray}
where $\partial_0 u=u$.
Let $F\in L^2(G)\equiv H$ be given and define $f(v) = (F,v)_H$.
Let $\Gamma$ be a closed subset of $\partial G$ and define
$$V= \{ v\in H^1(G) :\gamma_0 (v) (s)=0\ ,\ \mbox{ a.e. }\ s\in\Gamma\}\ .$$
$V$ is a closed subspace of $H^1(G)$, hence a Hilbert space.
We let $V_0 = H_0^1(G)$ so the formal operator $A:V\to V'_0 \subset \D^*(G)$
is given by
$$Au =- \sum_{i,j=1}^n \partial_j (a_{ij}\partial_i u) + \sum_{j=0}^n
a_j\partial_j u\ .$$
Let $\gamma$ be the restriction to $V$ of the trace map $H^1(G)\to
L^2(\partial G)$, where we assume $\partial G$ is appropriately smooth,
and let $B$ be the range of $\gamma$, hence $B\hookrightarrow L^2(\partial G
\sim \Gamma) \hookrightarrow B'$.
If all the $a_{ij}\in C^1(\bar G)$, then we have from the classical
Green's theorem
$$a(u,v) - (Au,v)_H = \int_{\partial G\sim\Gamma}
{\partial u\over\partial\nu_A} \cdot \gamma_0 (v)\,ds\ ,\qquad
u\in H^2(G)\ ,\ v\in V$$
where
$${\partial u\over\partial \nu_A} = \sum_{i=1}^n \partial_i u(s)
\sum_{j=1}^n a_{ij} (s)\nu_j (s)$$
denotes the (weighted) normal derivative on $\partial G\sim\Gamma$.
Thus, the operator $\partial$ is an extension of $\partial /\partial \nu_A$
from $H^2(G)$ to the domain $D_0=\{u\in V :Au\in L^2(G)\}$.
Theorem \ref{thm3-3A} now asserts that $u$ is a solution of the problem
\eqn{321} if and only if $u\in H^1(G)$, $\gamma_0 u =0$ on $\Gamma$,
$\partial u=0$ on $\partial G\sim \Gamma$, and $Au=F$.
That is, $u$ is a generalized solution of the {\it mixed Dirichlet-Neumann
boundary value problem\/}
\begin{equation}\label{eq342}
\left. \begin{array}{ll}
Au(x)= F(x)\ ,&x\in G\ ,\\
\noalign{\vskip4pt}
u(s)=0\ ,&s\in \Gamma\ ,\\
\noalign{\vskip4pt}
\displaystyle {\partial u(s)\over\partial\nu_A}=0\ ,&s\in \partial G\sim
\Gamma\ .\\
\end{array} \right\}
\end{equation}
If $\Gamma=\partial G$, this is called the Dirichlet problem or the boundary
value problem of {\it first type\/}.
If $\Gamma=\emptyset$, it is called the Neumann problem or boundary value
problem of {\it second type\/}.
\subsection{} %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
\begin{equation}\label{eq343}
a_1(u,v) = \int_G \nabla u\cdot \nabla \bar v\qquad u,v\in V
\end{equation}
where $V$ is a subspace of $H^1(G)$ to be chosen below.
The corresponding distribution-valued operator is given by $A=-\Delta_n$
and $\partial_1$ is an extension of the standard normal derivative given by
$${\partial u\over\partial \nu} = \nabla u \cdot \nu\ .$$
Suppose we are given $F\in L^2(G)$, $g\in L^2(\partial G)$, and
$\alpha \in L^\infty (\partial G)$.
We define
\begin{eqnarray*}
a_2(\varphi,\psi) &=& \int_{\partial G} \alpha (s) \varphi (s) \bar{\psi}(s)
\, ds\ ,\qquad \varphi,\psi\in L^2(\partial G)\\
f(v) &=&(F,v)_H + (g,\gamma_0 v)_{L^2(\partial G)}\ ,\qquad v\in V\ ,
\end{eqnarray*}
and then use Theorem \ref{thm3-3A} to characterize a solution $u$ of
\eqn{321} for different choices of $V$.
If $V=H^1(G)$, then $u$ is a generalized solution of the boundary value
problem
\begin{equation}\label{eq344}
\left. \begin{array}{ll}
-\Delta_n u(x) = F(x)\ ,&x\in G\ ,\cr
\noalign{\vskip6pt}
\displaystyle{\partial u(s)\over\partial\nu} + \alpha (s)u(s)
= g(x)\ ,&s\in \partial G\ .\cr
\end{array} \right\}
\end{equation}
The boundary condition is said to be of {\it third type\/} at those points
$s\in\partial G$ where $\alpha (s)\ne0$.
For an example of {\it non-local\/} boundary conditions, choose
$V= \{v\in H^1(G):\gamma_0(v)$ is constant$\}$.
Let $g(s)= g_0$ and $\alpha (s)=\alpha_0$ be constants, and define
$a_2(\cdot,\cdot)$ and $f$ as above.
Then $u$ is a solution of the boundary value problem of {\it fourth type\/}
\begin{equation}\label{eq345}
\left. \begin{array}{ll}
-\Delta_n u(x)=F(x)\ ,\qquad&x\in G\ ,\cr
\noalign{\vskip6pt}
u(s) = u_0\ \mbox{ (constant)}\ ,&s\in \partial G\ ,\cr
\noalign{\vskip6pt}
\displaystyle \biggl( \int_{\partial G} {\partial u(s)\over\partial \nu} \,ds
\Big/ \int_{\partial G} \,ds\biggr) +\alpha_0 \cdot u_0=g_0\ .&
\end{array} \right\}
\end{equation}
Note that $B=\KK$ in this example and $u_0$ is not prescribed as data.
Also, {\it periodic\/} boundary conditions are obtained when $G$ is an
interval.
\subsection{} %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 $c\in\RR$, and define
\begin{equation}\label{eq346}
a(u,v) = \int_G \{ \partial_1 u(\partial_1\bar v + c\partial_2\bar v\,)
+ \partial_2 u (\partial_2 \bar v- c\partial_1 \bar v\,)\}
\end{equation}
for $u$, $v\in V=H^1(G)$.
Taking $V_0 =H_0^1(G)$ gives $A=-\Delta_2$ and the classical Green's
theorem shows that for $u\in H^2(G)$ and $v\in H^1(G)$ we have
$$a(u,v) - (Au,v)_{L^2(G)} = \int_{\partial G}
\left( {\partial u\over\partial \nu} + c{\partial u\over\partial\tau}\right)
\,\bar v\, ds$$
where
$${\partial u\over\partial\tau} = \nabla u\cdot\tau$$
is the derivative in the direction of the tangent vector $\tau =
(\nu_2,-\nu_1)$ on $\partial G$.
Thus $\partial$ is an extension of the {\it oblique derivative\/} in the
direction $\nu +c\tau$ on the boundary.
If $f$ is chosen as in \eqn{342}, then Theorem \ref{thm3-3A} shows that
problem \eqn{321} is equivalent to a weak form of the boundary value problem
\begin{eqnarray*}
-\Delta_2 u(x) = F(x)\ ,&\qquad& x\in G\ ,\\
{\partial u\over\partial \nu} + c{\partial u\over\partial\tau}
= g(s)\ ,& & s\in\partial G\ .\end{eqnarray*}
\subsection{} %4.4
Let $G_1$ and $G_2$ be disjoint open connected sets with smooth boundaries
$\partial G_1$ and $\partial G_2$ which intersect in a $C^1$ manifold
$\Sigma$ of dimension $n-1$.
If $\nu_1$ and $\nu_2$ denote the unit outward normals on $\partial G_1$
and $\partial G_2$, then $\nu_1 (s)=-\nu_2(s)$ for $s\in\Sigma$.
Let $G$ be the interior of the closure of $G_1\cup G_2$, so that
$$\partial G= \partial G_1 \cup \partial G_2 \sim (\Sigma\sim \partial
\Sigma)\ .$$
For $k=1,2$, let $\gamma_0^k$ be the trace map $H^1(G_k)\to L^2(\partial G_k)$.
Define $V=H^1(G)$ and note that $\gamma_0^1u_1(s) = \gamma_0^2 u_2(s)$ for
a.e.\ $s\in\Sigma$ when $u\in H^1(G)$ and $u_k$ is the restriction of $u$
to $G_k$, $k=1,2$.
Thus we have a natural trace map
\begin{eqnarray*}
\gamma :H^1(G) &\longrightarrow&L^2(\partial G)\times L^2(\Sigma)\\
u&\longmapsto&(\gamma_0 u,\gamma_0^1 u_1|_{\Sigma})\ ,\end{eqnarray*}
where $\gamma_0 u(s) = \gamma_0^k u_k(s)$ for $s\in\partial G_k\sim\Sigma$,
$k=1,2$, and its kernel is given by $V_0=H_0^1(G_1)\times H_0^1(G_2)$.
Let $a_1\in C^1(\bar G_1)$, $a_2 \in C^1(\bar G_2)$ and define
$$a(u,v) = \int_{G_1} a_1\nabla u\cdot \nabla \bar v
+ \int_{G_2} a_2\nabla u\cdot \nabla \bar v\ ,\qquad u,v\in V\ .$$
The operator $A$ takes values in $\D^* (G_1\cup G_2)$ and is given by
$$Au(x) = \cases{ \displaystyle -\sum_{j=1}^n \partial_j
(a_1(x)\partial_ju(x))\ ,&$x\in G_1\ ,$\cr
\noalign{\vskip6pt}
\displaystyle -\sum_{j=1}^n \partial_j (a_2(x)\partial_ju(x))\ ,
&$x\in G_2\ .$\cr}$$
The classical Green's formula applied to $G_1$ and $G_2$ gives
$$a(u,v) -(Au,v)_{L^2(G)} = \int_{\partial G_1}
a_1 {\partial u_1\over\partial \nu_1} \ \bar v_1
+ \int_{\partial G_2} a_2 {\partial u_2\over\partial \nu_2}\ \bar v_2$$
for $u\in H^2(G)$ and $v\in H^1(G)$.
It follows that the restriction of the operator $\partial$ to the space
$H^2(G)$ is given by $\partial u = (\partial_0u,\partial_1u)\in L^2(\partial
G)\times L^2(\Sigma)$, where
\begin{eqnarray*}
\partial_0 u(s) &=& a_k(s) {\partial u_k(s)\over\partial \nu_k}\ ,
\mbox{ a.e. }\ s\in \partial G_k\sim\Sigma\ ,\quad k=1,2\ ,\\
\noalign{\vskip6pt}
\partial_1 u(s) &=& a_1(s) {\partial u_1(s)\over\partial\nu_1}
+ a_2 (s) {\partial u_2(s)\over\partial \nu_2}\ ,\qquad s\in\Sigma\ .
\end{eqnarray*}
Let $f$ be given as in Section 4.2.
Then a solution of $u$ of \eqn{321} is characterized by
Theorem \ref{thm3-3A} as a weak solution of the boundary value problem
$$\cases{
\displaystyle u_1\in H^1(G_1)\ ,\ -\sum_{j=1}^n \partial_j a_1 (x)
\partial_j u_1(x)=F(x)\ ,&$x\in G_1\ ,$\cr
\noalign{\vskip6pt}
\displaystyle u_2\in H^1(G_2)\ ,\ -\sum_{j=1}^n \partial_j a_2 (x)
\partial_j u_2(x)=F(x)\ ,&$x\in G_2\ ,$\cr
\noalign{\vskip6pt}
\displaystyle a_1(s) {\partial u_1(s)\over\partial\nu_1} = g(s)\ ,
&$s\in\partial G_1 \sim\Sigma\ ,$\cr
\noalign{\vskip6pt}
\displaystyle a_2(s) {\partial u_2(s)\over\partial\nu_2} = g(s)\ ,
&$s\in\partial G_2\sim\Sigma\ ,$\cr
\noalign{\vskip6pt}
\displaystyle u_1(s)= u_2(s)\ ,&\cr
\noalign{\vskip6pt}
\displaystyle a_1(s) {\partial u_1(s)\over\partial\nu_1} +
a_2(s) {\partial u_2(s)\over\partial\nu_2} =0\ ,&$s\in\Sigma\ .$} $$
Since $\nu_1= -\nu_2$ on $\Sigma$, this last condition implies that the normal
derivative has a prescribed jump on $\Sigma$ which is determined by the ratio
of $a_1(s)$ to $a_2(s)$.
The pair of equations on the interface $\Sigma$ are known as
{\it transition conditions\/}.
\subsection{} %4.5
Let the sets $G_1$, $G_2$ and $G$ be given as in Section 4.4.
Suppose $\Sigma_0$ is an open subset of the interface $\Sigma$ which is also
contained in the hyperplane $\{x=(x',x_n): x_n=0\}$ and define
$V= \{v\in H_0^1(G):\gamma_0^1 u_1|_{\Sigma_0} \in H^1(\Sigma_0)\}$.
With the scalar product
$$(u,v)_V \equiv (u,v)_{H_0^1(G)} + (\gamma_0^1u,\gamma_0^1v)_{H^1(\Sigma_0)}
\ ,\qquad u,v\in V\ ,$$
$V$ is a Hilbert space.
Let $\gamma (u) = \gamma_0^1(u)|_{\Sigma}$ be the corresponding trace operator
$V\to L^2(\Sigma)$, so $K(\gamma) = H_0^1(G_1)\times H_0^1(G_2)$ contains
$C_0^\infty (G_1\cup G_2)$ as a dense subspace.
Let $\alpha \in L^\infty (\Sigma_0)$ and define the sesquilinear form
\begin{equation}\label{eq347}
a(u,v) = \int_G \nabla u\cdot \nabla \bar v +\int_{\Sigma_0}
\alpha\nabla' (\gamma u)\cdot\nabla' \overline{(\gamma v)}\ ,\qquad
u,v\in V\ .
\end{equation}
Where $\nabla'$ denotes the gradient in the first $n-1$ coordinates.
Then $A= -\Delta_n$ in $\D^* (G_1\cup G_2)$ and the classical Green's
formula shows that $\partial u$ is given by
$$\partial u(v) = \int_\Sigma \left( {\partial u_1\over\partial\nu_1}\, \bar v
+{\partial u_2\over\partial\nu_2}\,\bar v\right)
+ \int_{\Sigma_0} \alpha\nabla' \bigl(\gamma(u)\bigr) \nabla' \bar v$$
for $u\in H^2(G)$ and $v\in B$.
Since the range of $\gamma$ is dense in $L^2(\Sigma\sim\Sigma_0)$,
it follows that if $\partial u=0$ then
$${\partial u_1(s)\over\partial\nu_1} +
{\partial u_2(s)\over\partial\nu_2}=0\ ,\qquad s\in \Sigma\sim\Sigma_0\ .$$
But $\nu_1= -\nu_2$ on $\Sigma$, so the normal derivative of $u$ is
continuous across $\Sigma\sim\Sigma_0$.
Since the range of $\gamma$ contains $C_0^\infty (\Sigma_0)$, it follows
that if $\partial u=0$ then we obtain the identity
$$\int_{\Sigma_0} \alpha \nabla' (\gamma u) \nabla'\overline{(\gamma v)}
+ \int_{\Sigma_0} {\partial u_1\over\partial \nu_1} \overline{(\gamma v)}
+ {\partial u_2\over\partial\nu_2} \overline{(\gamma v)} =0\ ,\qquad
v\in V\ ,$$
and this shows that $\gamma u|_{\Sigma_0}$ satisfies the abstract
boundary value
$$\begin{array}{ll}
-\Delta_{n-1}(\gamma u)(s) = \displaystyle{\partial u_2(s)\over\partial\nu_1}
- \displaystyle{\partial u_1(s)\over \partial\nu_1}\ ,\qquad &s\in\Sigma_0\ ,\\
\noalign{\vskip6pt}
(\gamma u)(s)=0\ ,&s\in \partial \Sigma_0\cap \partial G\ ,\\
\noalign{\vskip6pt}
\displaystyle{\partial (\gamma u)(s)\over\partial\nu_0}=0\ ,
&s\in \partial\Sigma_0
\sim \partial G\ ,\end{array}$$
where $\nu_0$ is the unit normal on $\partial \Sigma_0$, the
$(n-2)$-dimensional boundary of $\Sigma_0$.
Let $F\in L^2(G)$ and $f(v) =(F,v)_{L^2(G)}$ for $v\in V$.
Then from Corollary \ref{cor3-3B} it follows that \eqn{333}
is a generalized boundary value problem given by
\begin{equation}\label{eq348}
\left. \begin{array}{ll}
-\Delta_nu(x) = F(x)\ ,\qquad&x\in G_1\cup G_2\ ,\\
\noalign{\vskip6pt}
u(s)= 0\ ,&s\in \partial G\ ,\\
\noalign{\vskip6pt}
\displaystyle u_1(s) = u_2(s)\ ,\quad
{\partial u_1(s)\over\partial\nu_1} = {\partial u_2(s)\over\partial\nu_1}\ ,
&s\in \Sigma\sim\Sigma_0\ ,\\
\noalign{\vskip6pt}
\displaystyle -\Delta_{n-1}u(s) = {\partial u_2(s)\over\partial\nu_1}
- {\partial u_1(s)\over\partial\nu_1}\ ,&s\in\Sigma_0\ ,\\
\noalign{\vskip6pt}
\displaystyle {\partial u(s)\over\partial\nu_0}=0\ ,&s\in\partial\Sigma_0
\sim\partial G_0\ .
\end{array} \right\}
\end{equation}
Nonhomogeneous terms could be added as in previous examples and similar
problems could be solved on interfaces which are not necessarily flat.
\section{Coercivity; Elliptic Forms} % 5
\setcounter{equation}{0}
\subsection{} %5.1
Let $G$ be an open set in $\RR^n$ and suppose we are given a collection
of functions $a_{ij}$, $1\le i$, $j\le n$; $a_j$, $0\le j\le n$, in
$L^\infty (G)$.
Define the sesquilinear form
\begin{equation}\label{eq351}
a(u,v) = \int_G \biggl\{ \sum_{i,j=1}^n a_{ij} (x)\partial_i u(x)
\partial_j \bar v (x) + \sum_{j=0}^n a_j(x)\partial_j u(x)\cdot
\overline{v(x)}\biggr\}\,dx
\end{equation}
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 \eqn{351} is called {\it strongly elliptic\/} if
there is a constant $c_0>0$ such that
\begin{equation}\label{eq352}
\Re \sum_{i,j=1}^n a_{ij} (x) \xi_i \bar \xi_j \ge c_0 \sum_{j=1}^n
|\xi_j|^2\ ,\qquad \xi = (\xi_1,\ldots,\xi_n)\in \KK^n\ ,\ x\in G\ .
\end{equation}
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.
\begin{theorem}\label{thm3-5A}
Let \eqn{351} be strongly elliptic.
Then there is a $\lambda_0\in\RR$ such that for every $\lambda>\lambda_0$,
the form
$$a(u,v)+\lambda \int_G u(x)\bar v(x)\,dx $$
is $H^1(G)$-coercive.
\end{theorem}
\proof
Let $K_1=\max \{\|a_j\|_{L^\infty (G)} :1\le j\le n\}$ and
$K_0 = \ess \inf \{\Re a_0(x): x\in G\}$.
Then, for $1\le j\le n$ and each $\varep>0$ we have
\begin{eqnarray*}
|(a_j\partial_j u,u)_{L^2(G)} | &\le & K_1\|\partial_j u\|_{L^2(G)}
\cdot \|u\|_{L^2(G)} \\
&\le & (K_1/2) \left(\varep\|\partial_j u\|_{L^2(G)}^2
+ (1/\varep) \|u\|_{L^2(G)}^2 \right)\ .
\end{eqnarray*}
We also have $\Re (a_0u,u)_{L^2(G)} \ge K_0\|u\|_{L^2(G)}^2$,
so using these with \eqn{352} in \eqn{351} gives
\begin{equation}\label{eq353}
\begin{array}{rl}
\Re a (u,u)&\ge (c_0-\varep K_1/2)\|\nabla u\|_{L^2(G)}^2\\
\noalign{\vskip6pt}
&\qquad + (K_0-nK_1/2\varep) \|u\|_{L^2(G)}^2\ ,\quad u\in H^1(G)\ .
\end{array}
\end{equation}
We choose $\varep>0$ so that $K_1\varep = c_0$.
This gives us the desired result with $\lambda_0 = (nK_1^2/2c_0)-K_0$.
\begin{corollary}\label{cor3-5B}
For every $\lambda >\lambda_0$, the boundary value problem \eqn{342}
is well-posed, where
$$Au =- \sum_{i,j=1}^n \partial_j (a_{ij}\partial_iu)
+ \sum_{j=1}^n a_j\partial_j u+ (a_0 +\lambda) u\ .$$
Thus, for every $F\in L^2(G)$, there is a unique $u\in D$ such that
\eqn{342} holds, and we have the estimate
\begin{equation}\label{eq354}
\|(\lambda -\lambda_0)u\|_{L^2(G)} \le \|F\|_{L^2(G)}\ .
\end{equation}
\end{corollary}
\proof
The space $D$ was defined in Section 2.2 and Corollary \ref{cor3-3B},
so we need only to verify \eqn{354}.
For $u\in D$ and $\lambda>\lambda_0$ we have from \eqn{353}
\begin{eqnarray*}
(\lambda -\lambda_0) \|u\|_{L^2(G)}^2
&\le & a(u,u) + \lambda (u,u)_{L^2(G)} = (Au,u)_{L^2(G)}\\
\noalign{\vskip6pt}
&\le & \|Au\|_{L^2(G)} \cdot \|u\|_{L^2(G)}
\end{eqnarray*}
and the estimate \eqn{354} now follows.
\qed
\subsection{} %5.2
We indicate how coercivity may be obtained from the addition of boundary
integrals to strongly elliptic forms.
\begin{theorem}\label{thm3-5C}
Let $G$ be open in $\RR^n$ and suppose $0\le x_n\le K$ for all $x=(x',x_n)\in
G$. Let $\partial G$ be a $C^1$-manifold with $G$ on one side of
$\partial G$. Let $\nu (s) = (\nu_1(s),\ldots,\nu_n(s))$ be the unit
outward normal on $\partial G$ and define
$$\Sigma = \bigl\{ s\in \partial G: \nu_n (s)>0\bigr\}\ .$$
Then for all $u\in H^1(G)$ we have
$$\int_G |u|^2 \le 2K\int_\Sigma |\gamma_0u(s)|^2\,ds
+ 4K^2 \int_G |\partial_n u|^2\ .$$
\end{theorem}
\proof
For $u\in C^1(\bar G)$, the Gauss Theorem gives
\begin{eqnarray*}
\int_{\partial G} \nu_n (s) s_n|u(s)|^2\,ds
&=& \int_G D_n (x_n|u(x)|^2)\,dx\\
\noalign{\vskip6pt}
&=& \int_G |u|^2 + \int_G x_nD_n (|u(x)|^2)\,dx \ .
\end{eqnarray*}
Thus, we obtain from the inequality
$$2|a|\, |b| \le {|a|^2 \over 2K} + 2K|b|^2\ ,\qquad a,b\in \CC\ ,$$
the estimate
$$\int_G |u|^2 \le \int_{\partial G} \nu_n s_n |u(s)|^2\,ds
+ (1/2) \int_G |u|^2 + 2K^2 \int_G |D_nu|^2\ .$$
Since $\nu_n (s) s_n\le 0$ for $s\in\partial G\sim \Sigma$,
the desired result follows.
\begin{corollary}\label{cor3-5D}
If \eqn{351} is strongly elliptic, $a_j\equiv 0$ for $1\le j\le n$,
$\Re a_0(x) \ge 0$, $x\in G$, and if $\Sigma \subset\Gamma$, then the mixed
Dirichlet-Neumann problem \eqn{342} is well-posed.
\end{corollary}
\begin{corollary}\label{cor3-5E}
If $\alpha\in L^\infty (\partial G)$ satisfies
$$\Re \alpha (x)\ge0\ ,\quad x\in\partial G\ ,\quad
\Re \alpha (x)\ge c>0\ ,\quad x\in\Sigma\ ,$$
then the third boundary value problem \eqn{344} is well-posed.
The fourth boundary value problem \eqn{345} is well-posed if
$\Re (\alpha_0)>0$.
\end{corollary}
\noindent
Similar results can be obtained for the example of Section 4.3.
Note that the form \eqn{346} satisfies
$$\Re a(u,u) = \int_G \Bigl\{ |\partial_1u|^2 +|\partial_2u|^2\Bigr\}\ ,
\qquad u\in H^1(G)\ ,$$
so coercivity can be obtained over appropriate subspaces of $H^1(G)$
(as in Corollary \ref{cor3-5D}) or by adding a positive multiple of the
identity on $G$ or boundary integrals (as in Corollary \ref{cor3-5E}).
Modification of \eqn{346} by restricting $V$, e.g., to consist of
functions which vanish on a sufficiently large part of $\partial G$,
or by adding forms, e.g., that are coercive over $L^2(G)$ or $L^2(\partial G)$,
will result in a well-posed problem.
Finally, we note that the first term in the form \eqn{347} is coercive
over $H_0^1(G)$ and, hence, over $L^2(\Sigma)$.
Thus, if $\Re \alpha (x) \ge c>0$, $x\in\Sigma_0$, then \eqn{347} is
$V$-coercive and the problem \eqn{348} is well-posed.
\subsection{} %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 $a(\cdot,\cdot)$ on the Hilbert space $V$ is
{\it $V$-elliptic\/} if there is a $c>0$ such that
\begin{equation}\label{eq355}
\Re a(v,v) \ge c\|v\|_V^2\ ,\qquad v\in V\ .
\end{equation}
Such forms will occur frequently in our following discussions.
\section{Regularity} %6
\setcounter{equation}{0}
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 $L^2(\partial G)$ and not just in the sense of
the abstract Green's operator constructed in Theorem \ref{thm3-2C}, i.e.,
in $B'$. (See the Example in Section 2.3.)
\subsection{} %6.1
We begin with the Neumann problem; other cases will follow similarly.
\begin{theorem}\label{thm3-6A}
Let $G$ be bounded and open in $\RR^n$ and suppose its boundary is a
$C^2$-manifold of dimension ${n-1}$.
Let $a_{ij} \in C^1(G)$, $1\le i$, $j\le n$, and $a_j\in C^1(G)$,
$0\le j\le n$, all have bounded derivatives and assume that the
sesquilinear form defined by
\begin{equation}\label{eq361}
a(\varphi,\psi) \equiv \int_G \biggl\{ \sum_{i,j=1}^n a_{ij}\partial_i
\varphi \overline{\partial_j\psi}
+ \sum_{j=0}^n a_j\partial_j \varphi \bar\psi \biggr\}\,dx\ ,\qquad
\varphi,\psi \in H^1(G)
\end{equation}
is strongly elliptic.
Let $F\in L^2(G)$ and suppose $u\in H^1(G)$ satisfies
\begin{equation}\label{eq362}
a(u,v) = \int_G F\bar v\,dx\ ,\qquad v\in H^1(G)\ .
\end{equation}
Then $u\in H^2(G)$.
\end{theorem}
\proof
Let $\{(\varphi_k,G_k):1\le k\le N\}$ be coordinate patches on $\partial G$
and $\{\beta_k :0\le k\le N\}$ the partition-of-unity construction in
Section~II.2.3.
Let $B_k$ denote the support of $\beta_k$, $0\le k\le N$.
Since $u=\sum (\beta_k u)$ in $G$ and each $B_k$ is compact in $\RR^n$,
it is sufficient to show the following:
\begin{description}
\item[(a)] $u|_{B_k\cap G} \in H^2(B_k\cap G)$, $1\le k\le N$, and
\item[(b)] $\beta_0 u\in H^2(B_0)$.
\end{description}
The first case (a) will be proved below, and the second case (b) will
follow from a straightforward modification of the first.
\subsection{} %6.2
We fix $k$, $1\le k\le N$, and note that the coordinate map
$\varphi_k :Q\to G_k$ induces an isomorphism $\varphi_k^* :H^m(G_k\cap G)
\to H^m(Q_+)$ for $m=0,1,2$ by $\varphi_k^*(v) =v\circ \varphi_k$.
Thus we define a continuous sesquilinear form on $H^1(Q_+)$ by
\begin{equation}\label{eq363}
a^k\left( \varphi_k^* (w),\varphi_k^*(v)\right)
\equiv \int_{G_k\cap G}\biggl\{\sum_{i,j=1}^n a_{ij}\partial_j w
\overline{\partial_j v} + \sum_{j=0}^n a_j\partial_j w\bar v\biggr\} \,dx\ .
\end{equation}
By making the appropriate change-of-variable in \eqn{363} and setting
$w_k=\varphi_k^*(w)$, $v_k=\varphi_k^*(v)$, we obtain
\begin{equation}\label{eq364}
a^k(w_k,v_k) = \int_{Q_+} \bigg\{\sum_{i,j=1}^n a_{ij}^k \partial_i (w_k)
\partial_j (v_k) + \sum_{j=0}^n a_j^k\partial_j(w_k)v_k\biggr\}\,dy\ .
\end{equation}
The resulting form \eqn{364} is strongly-elliptic on $Q_+$ (exercise).
\vskip.2in
{\epsfxsize=4truein\epsfysize=2truein
\centerline{\epsfbox{figure3-6.eps}}}
\medskip
Let $u$ be the solution of \eqn{362} and let $v\in H^1(G\cap G_k)$ vanish
in a neighborhood of $\partial G_k$.
(That is, the support of $v$ is contained in $G_k$.)
Then the extension of $v$ to all of $G$ as zero on $G\sim G_k$ belongs
to $H^1(G)$ and we obtain from \eqn{364} and \eqn{362}
$$a^k(\varphi_k^*(u),\varphi_k^*(v)) = a(u,v) = \int_{Q_+} F_k
\varphi_k^* (v)\,dy\ ,$$
where $F_k\equiv \varphi_k^* (F)\cdot J(\varphi_k)\in L^2(Q_+)$.
Letting $\V$ denote the space of those $v\in H^1(Q_+)$ which vanish
in a neighborhood of $\partial Q$, and $u_k\equiv \varphi_k^*(u)$, we
have shown that $u_k\in H^1(Q_+)$ satisfies
\begin{equation}\label{eq365}
a^k(u_k,v_k) = \int_{Q_+} F_kv_k\,dy\ ,\qquad v_k\in\V
\end{equation}
where $a^k(\cdot,\cdot)$ is strongly elliptic with continuously
differentiable coefficients with bounded derivatives and $F_k\in L^2(Q_+)$.
We shall show that the restriction of $u_k$ to the compact subset
$K\equiv \varphi_k^{-1}(B_k)$ of $Q$ belongs to $H^2(Q_+\cap K)$.
The first case (a) above will then follow.
\subsection{} % 6.3
Hereafter we drop the subscript ``$k$'' in \eqn{365}.
Thus, we have $u\in H^1(Q_+)$, $F\in L^2(Q_+)$ and
\begin{equation}\label{eq366}
a(u,v) = \int_{Q_+} Fv\ ,\qquad v\in \V\ .
\end{equation}
Since $K\subset\subset Q$, there is by Lemma II.\ref{lem2-1-1} a
$\varphi\in C_0^\infty (Q)$ such that $0\le \varphi(x)\le1$ for $x\in Q$
and $\varphi (x)=1$ for $x\in K$.
We shall first consider $\varphi\cdot u$.
Let $w$ be a function defined on the half-space $\RR_+^n$.
For each $h\in\RR$ we define a {\it translate\/} of $w$ by
$$(\tau_h w) (x_1,x_2,\ldots,x_n) = w(x_1+h, x_2,\ldots,x_n)$$
and a {\it difference\/} of $w$ by
$$\nabla_h w = (\tau_h w-w)/h$$
if $h\ne0$.
\begin{lemma}\label{lem3-6-1}
If $w,v\in L^2(Q_+)$ and the distance $\delta$ of the support of $w$
to $\partial Q$ is positive, then
$$(\tau_h w,v)_{L^2(Q_+)} = (w,\tau_{-h} v)_{L^2(Q_+)}$$
for all $h\in \RR$ with $|h| <\delta$.
\end{lemma}
\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 $Q_+$.
\plaincor
{$\|\tau_hw\|_{L^2(Q_+)} = \|w\|_{L^2(Q_+)}$.}
\begin{lemma}\label{lem3-6-2}
If $w\in \V$, then
$$\|\nabla_h w\|_{L^2(Q_+)} \le \|\partial_1 w\|_{L^2(Q_+)} \ ,\qquad
0<|h| <\delta\ .$$
\end{lemma}
\proof
It follows from the preceding Corollary that it is sufficient to consider
the case where $w\in C^1(\bar G)\cap \V$.
Assuming this, and denoting the support of $w$ by $\supp (w)$, we have
$$\nabla_h w(x) = h^{-1} \int_{x_1}^{x_1+h} \partial_1 w(t,x_2,\ldots,x_n)\,
dt\ ,\qquad w\in \supp(w)\ .$$
The Cauchy-Schwartz inequality gives
$$|\nabla_h w(x)| \le h^{-1/2} \biggl( \int_{x_1}^{x_1+h}
|\partial_1 w(t,x_2,\ldots, x_n)|^2\,dt\biggr)^{1/2}\ ,
\qquad x\in\supp (w)\ ,$$
and this leads to
\begin{eqnarray*}
\|\nabla_h w\|_{L^2(Q_+)}^2
&\le & h^{-1} \int_{\supp (w)} \int_{x_1}^{x_1+h}
|\partial_1 w(t,x_2,\ldots,x_n)|^2\,dt\,dx \cr
\noalign{\vskip6pt}
&= & h^{-1} \int_{Q_+}\int_0^h
|\partial_1 w(t+x_1,x_2,\ldots,x_n)|^2\,dt\,dx\cr
\noalign{\vskip6pt}
&= & h^{-1} \int_0^h \int_{Q_+}
|\partial_1 w(t+x_1,\ldots,x_n)|^2\,dx\, dt\cr
\noalign{\vskip6pt}
&= & h^{-1} \int_0^h \int_{Q_+}
|\partial_1 w(x_1,\ldots,x_n)|^2\,dx\,dt\cr
\noalign{\vskip6pt}
&= &\|\partial_1w\|_{L^2(Q_+)}\ .
\end{eqnarray*}
\plaincor
{$\lim_{h\to0} (\nabla_h w) = \partial_1 w$ in $L^2(Q_+)$.}
\proof
$\{\nabla_h :0 <|h|<\delta\}$ 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
$$a_h(w,v) \equiv \int_{Q_+} \biggl\{ \sum_{i,j=1}^n (\nabla_h a_{ij})
\partial_i w\overline{\partial_j v} + \sum_{j=0}^n (\nabla_h a_j)
\partial_j w\bar v\,\biggr\}$$
for $w,v\in \V$ and $|h|<\delta$, $\delta$ being given as in
Lemma \ref{lem3-6-1}.
Since the coefficients in \eqn{365} have bounded derivatives, the
mean-value theorem shows
\begin{equation}\label{eq367}
|a_h (w,v)| \le C\|w\|_{H^1(Q_+)} \cdot \|v\|_{H^1(Q_+)}
\end{equation}
where the constant is independent of $w,v$ and $h$.
Finally, we note that for $w,v$ and $h$ as above
\begin{equation}\label{eq368}
a(\nabla_h w,v) + a(w,\nabla_{-h}v) = -a_{-h} (\tau_{-h} w,v)\ .
\end{equation}
This follows from a computation starting with the first term above and
Lemma \ref{lem3-6-1}.
After this lengthy preparation we continue with the proof of
Theorem \ref{thm3-6A}. From \eqn{366} we have the identity
\begin{eqnarray}\label{eq369}
a(\nabla_h (\varphi u),v)
&=& \left\{ a(\nabla_h (\varphi u),v) + a(\varphi u,\nabla_{-h}v)\right\}\\
&&\quad + \left\{ a(u,\varphi \nabla_{-h} v)
- a(\varphi u,\nabla_{-h}v)\right\}
- (F,\varphi \nabla_{-h} v)_{L^2(Q_+)} \nonumber
\end{eqnarray}
for $v\in \V$ and $0<|h|<\delta$, $\delta$ being the distance from $K$
to $\partial Q$.
The first term can be bounded appropriately by using \eqn{367} and \eqn{368}.
The third is similarly bounded and so we consider the second term in \eqn{369}.
An easy computation gives
\begin{eqnarray*}
&&a(u,\varphi\nabla_{-h}v) -a(\varphi u,\nabla_{-h}v)\\
&&\quad =\int_{Q_+} \biggl\{ \sum_{i,j=1}^n a_{ij} (\partial_i u\partial_j
\varphi\nabla_{-h} v-\partial_i \varphi u\nabla_{-h} (\partial_j v))
- \sum_{j=1}^n a_j \partial_j \varphi u(\nabla_{-h} v)\biggr\}\ .
\end{eqnarray*}
Thus, we obtain the estimate
\begin{equation}\label{eq3610}
|a(\nabla_h (\varphi u),v)| \le C\|v\|_{H^1(Q_+)}\ ,\qquad
v\in \V\ ,\ 0<|h| <\delta\ ,
\end{equation}
in which the constant $C$ is independent of $h$ and $v$.
Since $a(\cdot,\cdot)$ is strongly-elliptic we may assume it is coercive
(Exercise 6.2), so setting $v=\nabla_h (\varphi u)$ in \eqn{3610} gives
\begin{equation}\label{eq3611}
c\|\nabla_h (\varphi u)\|_{H^1(Q_+)}^2 \le C\|\nabla_h (\varphi u)\|_{H^1(Q_+)}
\ ,\qquad 0<|h|<\delta\ ,
\end{equation}
hence, $\{\nabla_h (\varphi u): |h| <\delta\}$ is bounded in the Hilbert
space $H^1(Q_+)$.
By Theorem I.\ref{thm1-6B} there is a sequence $h_n\to0$
for which $\nabla_{h_n}
(\varphi u)$ converges weakly to some $w\in H^1(Q_+)$.
But $\nabla_{h_n}(\varphi u)$ converges weakly in $L^1(Q_+)$ to
$\partial_1 (\varphi u)$, so the uniqueness of weak limits implies that
$\partial_1(\varphi u) = w\in H^1(Q_+)$.
It follows that $\partial_1^2 (\varphi u) \in L^2(Q_+)$, and the same
argument shows that each of the tangential derivatives $\partial_1^2u,
\partial_2^2 u,\ldots,\partial_{n-1}^2 u$ belongs to $L^2(K)$.
(Recall $\varphi=1$ on $K$.)
This information together with the partial differential equation resulting
from \eqn{366} implies that $a_{nn}\cdot \partial_n^2(u)\in L^2(K)$.
The strong ellipticity implies $a_{nn}$ has a positive lower bound on $K$,
so $\partial_n^2 u\in L^2(K)$.
Since $n$ and all of its derivatives through second order are in $L^2(K)$,
it follows from Theorem II.\ref{thm2-5B} that $u\in H^2(K)$.
The preceding proves the case (a) above.
The case (b) follows by using the differencing technique directly
on $\beta_0u$.
In particular, we can compute differences on $\beta_0u$ in {\it any\/}
direction. The details are an easy modification of those of this section
and we leave them as an exercise.
\subsection{} % 6.4
We discuss some extensions of Theorem \ref{thm3-6A}.
First, we note that the result and proof of Theorem \ref{thm3-6A} 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.
\begin{theorem}\label{thm3-6B}
Let $u\in H_0^1(G)$ satisfy
$$a(u,v) = \int_G F\bar v\ ,\qquad v\in H_0^1(G)$$
where the set $G\subset \RR^n$ and sesquilinear form $a(\cdot,\cdot)$
are given as in Theorem \ref{thm3-6A}, and $F\in L^2(G)$.
Then $u\in H^2(G)$.
\end{theorem}
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 $H_0^1(G)\le V$,
and let $a(\cdot,\cdot)$ be a continuous sesquilinear form on $V$.
Then $a(\cdot,\cdot)$ is called {\it $k$-regular on\/} $V$ if for every
$F\in H^s(G)$ with $0\le s\le k$ and every solution $u\in V$ of
$$a(u,v) = (F,v)_{L^2(G)}\ ,\qquad v\in V$$
we have $u\in H^{2+s}(G)$.
\qed
Theorems \ref{thm3-6A} and \ref{thm3-6B} give sufficient conditions for
the form $a(\cdot,\cdot)$ given by \eqn{361} to be $0$-regular over $H^1(G)$
and $H_0^1(G)$, respectively.
Moreover, we have the following.
\begin{theorem}\label{thm3-6C}
The form $a(\cdot,\cdot)$ given by \eqn{361} is $k$-regular over $H^1(G)$
and $H_0^1(G)$ if $\partial G$ is a $C^{2+k}$-manifold and the coefficients
$\{a_{ij},a_j\}$ all belong to $C^{1+k}(\bar G)$.
\end{theorem}
\section{Closed operators, adjoints and eigenfunction expansions} %7
\markright{7. CLOSED OPERATORS, ADJOINTS ...}
\setcounter{equation}{0}
\subsection{} % 7.1
We were led in Section 2 to consider a linear map $A:D\to H$ whose domain
$D$ is a subspace of the Hilbert space $H$.
We shall call such a map an ({\it unbounded\/}) {\it 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 {\it graph\/} of $A$ is the subspace
$$G(A) = \{ [x,Ax] : x\in D\}$$
of the product $H\times H$.
(This product is a Hilbert space with the scalar product
$$([x_1,x_2], [y_1,y_2])_{H\times H}
= (x_1,y_1)_H + (x_2,y_2)_H\ .$$
The addition and scalar multiplication are defined componentwise.)
The operator $A$ on $H$ is called {\it closed\/} if $G(A)$ is a
closed subset of $H\times H$.
That is, $A$ is closed if for any sequence $x_n\in D$ such that $x_n\to x$
and $Ax_n\to y$ in $H$, we have $x\in D$ and $Ax=y$.
\begin{lemma}\label{lem3-7-1}
If $A$ is closed and continuous (i.e., $\|Ax\|_H \le K\|x\|_H$, $x\in H$)
then $D$ is closed.
\end{lemma}
\proof
If $x_n\in D$ and $x_n\to x\in H$, then $\{x_n\}$ and, hence, $\{Ax_n\}$
are Cauchy sequences.
$H$ is complete, so $Ax_n\to y\in H$ and $G(A)$ being closed implies $x\in D$.
When $D$ is dense in $H$ we define the {\it adjoint\/} of $A$ as follows.
The domain of the operator $A^*$ is the subspace $D^*$ of all $y\in H$
such that the map $x\mapsto (Ax,y)_H:D\to \KK$ is continuous.
Since $D$ is dense in $H$, Theorem I.\ref{thm1-4E} asserts that for each
such $y\in D^*$ there is a unique $A^*y\in H$ such that
\begin{equation}\label{eq371}
(Ax,y) = (x,A^* y)\ ,\qquad x\in D\ ,\ y\in D^*\ .
\end{equation}
Then the function $A^*:D^*\to H$ is clearly linear and is called the
{\it adjoint\/} of $A$.
The following is immediate from \eqn{371}.
\begin{lemma}\label{lem3-7-2}
$A^*$ is closed.
\end{lemma}
\begin{lemma}\label{lem3-7-3}
If $D=H$, then $A^*$ is continuous, hence, $D^*$ is closed.
\end{lemma}
\proof
If $A^*$ is not continuous there is a sequence $x_n\in D^*$ such that
$\|x_n\|=1$ and $\|A^* x_n\|\to \infty$.
From \eqn{371} it follows that for each $x\in H$,
$$|(x,A^* x_n)_H| = |(Ax,x_n)_H| \le \|Ax\|_H\ ,$$
so the sequence $\{A^* x_n\}$ is weakly bounded.
But Theorem I.\ref{thm1-6A} implies that it is bounded, a contradiction.
\begin{lemma}\label{lem3-7-4}
If $A$ is closed, then $D^*$ is dense in $H$.
\end{lemma}
\proof
Let $y\in H$, $y\ne0$.
Then $[0,y] \notin G(A)$ and $G(A)$ closed in $H\times H$ imply there
is an $f\in (H\times H)'$ such that $f[G(A)] = \{0\}$ and $f(0,y)\ne 0$.
In particular, let $P: H\times H\to G(A)^\bot$ be the projection onto
the orthogonal complement of $G(A)$ in $H\times H$, define $[u,v]=P[0,y]$,
and set
$$f(x_1,x_2) = (u,x_1)_H + (v,x_2)_H \ ,\qquad x_1,x_2\in H\ .$$
Then we have
$$0= f(x,Ax) = (u,x)_H + (v,Ax)_H\ ,\qquad x\in D$$
so $v\in D^*$, and $0\ne f(0,y) = (v,y)_H$.
The above shows $(D^*)^\bot = \{0\}$, so $D^*$ is dense in $H$.
\qed
The following result is known as the {\it closed-graph theorem\/}.
\begin{theorem}\label{thm3-7A}
Let $A$ be an operator on $H$ with domain $D$.
Then $A$ is closed and $D=H$ if and only if $A\in \L(H)$.
\end{theorem}
\proof
If $A$ is closed and $D=H$, then Lemma \ref{lem3-7-3} and
Lemma \ref{lem3-7-4}
imply $A^*\in \L(H)$.
Then Theorem I.\ref{thm1-5B} shows $(A^*)^* \in \L(H)$.
But \eqn{371} shows $A= (A^*)^*$, so $A\in \L(H)$.
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.
\qed
We give some examples in $H=L^2(G)$, $G= (0,1)$.
\subsection{} % 7.2
Let $D=H_0^1(G)$ and $A= i\partial$.
If $[u_n,Au_n]\in G(A)$ converges to $[u,v]$ in $H\times H$, then in the
identity
$$\int_0^1 Au_n \varphi\,dx = -i\int_0^1 u_n\,D\varphi\,dx\ ,\qquad
\varphi\in C_0^\infty (G)\ ,$$
we let $n\to\infty$ and thereby obtain
$$\int_0^1 v\varphi\,dx = -i\int_0^1 u\,D\varphi\,dx\ ,\qquad
\varphi \in C_0^\infty (G)\ .$$
This means $v=i\partial u= Au$ and $u_n\to u$ in $H^1(G)$.
Hence $u\in H_0^1(G)$, and we have shown $A$ is closed.
To compute the adjoint, we note that
$$\int_0^1 Au\bar v\,dx = \int_0^1 u\bar f\,dx\ ,\qquad
u\in H_0^1(G)$$
for some pair $v,f\in L^2(G)$ if and only if $v\in H^1(G)$ and $f=i\partial v$.
Thus $D^*=H^1(G)$ and $A^* = i\partial$ is a proper extension of $A$.
\subsection{} % 7.3
We consider the operator $A^*$ above:
on its domain $D^* =H^1(G)$ it is given by $A^*=i\partial$.
Since $A^*$ is an adjoint it is closed.
We shall compute $A^{**} = (A^*)^*$, the second adjoint of $A$.
We first note that the pair $[u,f]\in H\times H$ is in the graph of
$A^{**}$ if and only if
$$\int_0^1 A^* v\bar u\,dx = \int_0^1 v\bar f\,dx\ ,\qquad v\in H^1(G)\ .$$
This holds for all $v\in C_0^\infty (G)$, so we obtain $i\partial u=f$.
Substituting this into the above and using Theorem II.\ref{thm2-1D}, we obtain
$$i\int_0^1 \partial (v\bar u)\,dx = \int_0^1 \Bigl[ (i\partial v)\bar u
- v\overline{(i\partial u)}\,\Bigr]\,dx = 0\ ,$$
hence, $v(1) \bar u(1) -v(0)\bar u(0)=0$ for all $v\in H^1(G)$.
But this implies $u(0)=u(1)=0$, hence, $u\in H_0^1(G)$.
From the above it follows that $A^{**}=A$.
\subsection{} % 7.4
Consider the operator $B=i\partial$ on $L^2(G)$ with domain $D(B)=\{u\in H^1
(G) :u(0)= cu(1)\}$ where $c\in \CC$ is given.
If $v,f\in L^2(G)$, then $B^*v=f$ if and only if
$$\int_0^1 i\partial u\cdot \bar v\, dx = \int_0^1 u\bar f\,dx\ ,
\qquad u\in D\ .$$
But $C_0^\infty (G)\le D$ implies $v\in H^1 (G)$ and $i\partial v=f$.
We substitute this identity in the above and obtain
$$0 = i\int_0^1 \partial (u\bar v\,) \, dx= iu(1) \bigl[ \bar v(1) - c\bar v
(0)\bigr] \ ,\qquad u\in D\ .$$
The preceding shows that $v\in D(B^*)$ only if $v\in H^1(0,1)$ and
$v(1) = \bar cv (0)$.
It is easy to show that every such $v$ belongs to $D(B^*)$, so we have
shown that $D(B^*) = \{v\in H^1(G) : v(1) = \bar c v(0)\}$ and
$B^* = i\partial$.
\subsection{} % 7.5
We return to the situation of Section 2.2.
Let $a(\cdot,\cdot)$ 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 $u\in V$ such that the map $v\mapsto a(u,v)$
is continuous on $V$ with the norm of $H$.
For such a $u\in D$, there is a unique $Au\in H$ such that
\begin{equation}\label{eq372}
a(u,v) = (Au,v)_H\ ,\qquad u\in D\ ,\ v\in V\ .
\end{equation}
This defines a linear operator $A$ on $H$ with domain $D$.
Consider the (adjoint) sesquilinear form on $V$ defined by $b(u,v)=
\overline{a(v,u)}$, $u,v\in V$.
This gives another operator $B$ on $H$ with domain $D(B)$ determined
as before by
$$b(u,v) = (Bu,v)_H\ ,\qquad u\in D(B)\ ,\ v\in V\ .$$
\begin{theorem}\label{thm3-7B}
Assume there is a $\lambda >0$ and $c>0$ such that
\begin{equation}\label{eq373}
\Re\, a(u,u) + \lambda |u|_H^2 \ge c\|u\|_V^2\ ,\qquad u\in V\ .
\end{equation}
Then $D$ is dense in $H$, $A$ is closed, and $A^*=B$, hence,
$D^* = D(B)$.
\end{theorem}
\proof
Theorem \ref{thm3-2B} shows $D$ is dense in $H$.
If we prove $A^*=B$, then by symmetry we obtain $B^*=A$, hence $A$
is closed by Lemma \ref{lem3-7-2}.
Suppose $v\in D(B)$.
Then for all $u\in D(A)$ 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 $D(B) \le D^*$ and $A^*|_{D(B)} =B$.
We need only to verify that $D(B) = D^*$.
Let $u\in D^*$.
Since $B+\lambda$ is surjective, there is a $u_0\in D(B)$ such that
$(B+\lambda)u_0 = (A^*+\lambda)u$.
Then for all $v\in D$ we have
\begin{eqnarray*}
((A+\lambda)v,u)_H &=& (v,(B+\lambda)u_0)_H = a(v,u_0)+\lambda(v,u_0)_H\\
&=& ((A+\lambda )v,u_0)_H\ .
\end{eqnarray*}
But $A+\lambda$ is a surjection, so this implies $u=u_0\in D(B)$.
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.
\begin{theorem}\label{thm3-7C}
Let $V$ and $H$ be Hilbert spaces with $V$ dense in $H$ and assume the
injection $V\hookrightarrow H$ is compact.
Let $A:D\to H$ be the linear operator determined as above by a
continuous sesquilinear form $a(\cdot,\cdot)$ on $V$ which we assume
is $V$-elliptic and symmetric:
$$a(u,v) = \overline{a(v,u)}\ ,\qquad u,v\in V\ .$$
Then there is a sequence $\{v_j\}$ of eigenfunctions of $A$ with
\begin{equation}\label{eq374}
\left. \begin{array}{c}
Av_j = \lambda_j v_j\ ,\qquad |v_j|_H=1\ ,\\
\noalign{\vskip6pt}
(v_i,v_j)_H=0\ ,\qquad i\ne j\ ,\\
\noalign{\vskip6pt}
0<\lambda_1\le \lambda_2\le\cdots\le \lambda_n \to +\infty\ \mbox{ as }
n\to +\infty\ ,
\end{array} \right\}
\end{equation}
and $\{v_j\}$ is a basis for $H$.
\end{theorem}
\proof
From Theorem \ref{thm3-7B} it follows that $A=A^*$ and, hence, $A^{-1}\in
\L(H)$ is self-adjoint.
The $V$-elliptic condition \eqn{355} shows that $A^{-1}\in \L(H,V)$.
Since the injection $V\hookrightarrow H$ is compact, it follows that
$A^{-1}:H\to V\to H$ is compact.
We apply Theorem I.\ref{thm1-7C} 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 $\{\mu_j\}$, then the
symmetry of $a(\cdot,\cdot)$ and \eqn{355} shows that each $\mu_j$
is positive.
We obtain \eqn{374} by setting $\lambda_j = 1/\mu_j$ for $j\ge1$ and
noting that $\lim_{j\to\infty} \mu_j=0$.
It remains to show $\{v_j\}$ is a basis for $H$.
(We only know that it is a basis for $D$.)
Let $f\in H$ and $u\in D$ with $Au=f$.
Let $\sum b_jv_j$ be the Fourier series for $f$, $\sum c_jv_j$ the
Fourier series for $u$, and denote their respective partial sums by
$$u_n = \sum_{j=1}^n c_j v_j\quad ,\quad
f_n = \sum_{j=1}^n b_j v_j\ .$$
We know $\lim_{n\to\infty} u_n = u$ and $\lim_{n\to\infty} f_n=f_\infty$
exists in $H$ (cf.\ Exercise I.7.2).
For each $j\ge1$ we have
$$b_j = (Au,v_j)_H = (u,Av_j)_H = \lambda_j c_j\ ,$$
so $Au_n =f_n$ for all $n\ge1$.
Since $A$ is closed, it follows $Au=f_\infty$, hence,
$f= \lim_{n\to\infty} f_n$ as was desired.
If we replace $A$ by $A+\lambda$ in the proof of Theorem \ref{thm3-7C},
we observe that ellipticity of $a(\cdot,\cdot)$ is not necessary but only
that $a(\cdot,\cdot)+ \lambda(\cdot,\cdot)_H$ be $V$-elliptic for some
$\lambda \in\RR$.
\begin{corollary}\label{cor3-7D}
Let $V$ and $H$ be given as in Theorem \ref{thm3-7C}, let $a(\cdot,\cdot)$
be continuous, sesquilinear, and symmetric.
Assume also that
$$a(v,v) + \lambda |v|_H^2 \ge c\|v\|_V^2\ ,\qquad v\in V$$
for some $\lambda\in\RR$ and $c>0$.
Then there is an orthonormal sequence of eigenfunctions of $A$ which
is a basis for $H$ and the corresponding eigenvalues satisfy
$-\lambda <\lambda_1 \le \lambda_2\le \cdots\le\lambda_n\to+\infty$
as $n\to +\infty$.
\end{corollary}
We give some examples in $H= L^2(G)$, $G= (0,1)$.
These eigenvalue problems are known as {\it Sturm-Liouville problems\/}.
Additional examples are described in the exercises.
\subsection{} % 7.6
Let $V= H_0^1(G)$ and define
$a(u,v) = \int_0^1 \partial u\overline{\partial v}\,dx$.
The compactness of $V\to H$ follows from Theorem II.\ref{thm2-5C} and
Theorem \ref{thm3-5C} shows $a(\cdot,\cdot)$ is $H_0^1(G)$-elliptic.
Thus Theorem \ref{thm3-7C} holds; it is a straightforward exercise
to compute the eigenfunctions and corresponding eigenvalues for the
operator $A= -\partial^2$ with domain $D(A)= H_0^1(G)\cap H^2(G)$:
$$v_j(x) = 2\sin (j\pi x)\ ,\quad \lambda = (j\pi)^2\ ,\qquad
j=1,2,3,\ldots\ .$$
Since $\{v_j\}$ is a basis for $L^2(G)$, each $F\in L^2(G)$ has a Fourier
sine-series expansion.
Similar results hold in higher dimension for, e.g., the eigenvalue problem
$$\cases{-\Delta_n v(x) = \lambda v(x)\ ,&$x\in G$\ ,\cr
\noalign{\vskip6pt}
v(s) =0\ ,&$s\in\partial G$\ ,\cr}$$
but the actual computation of the eigenfunctions and eigenvalues is
difficult except for very special regions $G\subset \RR^n$.
\subsection{} % 7.7
Let $V= H^1(G)$ and choose $a(\cdot,\cdot)$ as above.
The compactness follows from Theorem II.\ref{thm2-5D} so
Corollary \ref{cor3-7D} applies for any $\lambda >0$ to give a basis
of eigenfunctions for $A= -\partial^2$ with domain $D(A)=\{v\in H^2(G):
v'(0)= v'(1)=0\}$:
$$\begin{array}{rcll}
v_0(x)&=&1\ ,\quad v_j(x)= 2\cos (j\pi x)\ ,&j\ge1\ ,\\
\noalign{\vskip6pt}
\lambda_j&=&(j\pi)^2\ ,&j\ge0\ .
\end{array}$$
As before, similar results hold for the Laplacean with boundary conditions
of second type in higher dimensions.
\subsection{} % 7.8
Let $a(\cdot,\cdot)$ be given as above but set $V= \{v\in H^1(G): v(0)=v(1)\}$.
Then we can apply Corollary \ref{cor3-7D} to the {\it periodic
eigenvalue problem\/} (cf.\ \eqn{345})
$$\begin{array}{rcl}
-\partial^2 v(x)&=& \lambda v(x)\ ,\qquad 00$. Compute the eigenfunctions.
\item[7.10.]
Take $c\bar c=1$ in Section 7.4 and discuss the eigenvalue problem $Bv=
\lambda v$.
\item[7.11.]
In the proof of Theorem \ref{thm3-7C}, deduce that $\{v_j\}$ is a basis for
$H$ directly from the fact that $\bar D=H$.
\end{description}