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\vglue.6in
\centerline{\heading PROJECTION OPERATORS AND}
\centerline{\heading PARTIAL DIFFERENTIAL EQUATIONS}
\vfill
\centerline{by}
\smallskip
\centerline{John McCarthy}
\vfill
\eject
\centerline{\bf ABSTRACT}
\bigskip
This paper develops a method for approximating solutions of
differential equations which is based on the fact that the space of
solutions of a differential equation can be regarded as the
intersection of spaces with simpler properties.

In section 1 the general method is outlined and a necessary theorem
about Hilbert space is proved.

In section 2 the method is applied to systems of first order linear
partial differential equations of the form
$$a↑1_{(f)}(x) {\partial f\over \partial x↑1} = 0\quad (j =
1,\ldots, k).$$
In this case, a the space of gradients of solutions may be regarded as
the intersection of the space $M_1$ of gradient vector fields
$\partial f\over \partial x↑1$ with the space $M_2$ of vector fields
satisfying the relation $a↑1_{(j)} (x) f_i(x) = 0$.

In section 3 the method is applied to the Dirichlet problem for
self-adjoint elliptic partial differential equations.

In an appendix we prove that for certain regions $G$ of n-dimensional
Euclidean space there exist constants $K_G$ such that
$$\int_{\partial G}(f - {\overline f})↑2 dS \leq K_G
\int_G(\bigtriangledown f)↑2 dv.$$
where $\overline f$ is the average of $f(x)$ over $\partial G$.
\vfill
\eject
\centerline {\bf ACKNOWLEDGEMENTS}
\bigskip
I wish to thank Professor D. C. Spencer for his aid and encouragement
in the preparation of this thesis.  My thanks are also due to
Professors S. Lefschetz and M. Schiffer for getting me started on
differential equations.

To the Faculty of Princeton University, and to the Procter Fellowship
Fund, I wish to express my gratitude for the fellowship which I have
held while this work was in progress.
\vfill
\eject
\centerline{CONTENTS}
\bigskip
1.  The Method of Successive Projection for Solving Differential
Equations

2.  Application of the Method to First Order Linear Partial
Differential Equations

3.  Application to the Dirichlet Problem

Appendix

References
\vfill
\eject
\chapter{1}{THE METHOD OF SUCCESSIVE PROJECTION FOR SOLVING
DIFFERENTIAL EQUATIONS}

The application of operator methods to the solution of partial
differential equations has heretofore been largely directed towards
inverting the differential operator in one manner or another.  In this
paper a different approach is used.  We consider the space of
solutions of our differential equation as the intersection of spaces
of a simpler nature.  This enables us to develop iterative procedures
for obtaining approximate solutions of the differential equation.

We first develop the method in general, and then give applications to
two specific cases; namely, to the solution of systems of first order
linear partial differential equations and to the solution of the
Dirichlet problem for certain elliptic second order partial
differential equations.

Suppose we have a metric space and are interested in finding elements
of a certain subspace $M$.  Suppose further that $M$ is the
intersection of two other subspaces $M_1$ and $M_2$ which are easier
to deal with.  One might hope to obtain elements of $M$ by the
following procedure.  Let $P_1$ and $P_2$ be operators which map the
whole space onto $M_1$ and $M_2$ respectively and which leave
elements of $M_1$ and $M_2$ respectively fixed.  We now consider the
sequence of elements $f$, $P_1 f$, $P_2P_1 f$, $P_1P_2P_1 f$, etc.
where $f$ is any element of the space.  Since $f$ is left fixed by
$P_1$ and $P_2$ only if $f$ is in $M$ we see that if the sequence in
question converges it must converge to an element of $M$.

As an example of such a procedure consider two lines in the plane
intersecting in a point, and let the operators $P_1$ and $P_2$ be the
orthogonal projections onto the lines.  It is clear that in this case
the sequence converges to the intersection of the lines.  See figure
1.1.
\vskip 1.5in
In the case of closed linear manifolds in Hilbert space we can give a
theorem.

 \proclaim Theorem 1:1.  Let $M = M_1 \cap\ldots\cap M_k$ be the
intersection of the closed linear manifolds $M_i (i = 1,\ldots k)$ of
a Hilbert space $H$.  Further, let $P$ be the orthogonal projection
operator onto $M$ and $P_i$ the orthogonal projection onto $M_i$.
Then, if $f$ is any element of $H$, $\parallel (P_1\ldots P_kP_{k-1}\ldots
P_1)↑n f - Pf\parallel\rightarrow 0$ as $n\rightarrow\infty$.

The proof uses the spectral resolution of the operator $Q = P_1\ldots
P_kP_{k-1}\ldots P_1$.  We have $Q = \int_{-1}↑1 \lambda dE(\lambda)$
since $Q$ has norm less than or equal to one.

Since, for any $P_i$, $\parallel P_if\parallel = \parallel f\parallel$ implies $P_if =
f$, $Q$ has the same property, namely $\parallel Qf\parallel = \parallel f\parallel$
implies $Qf = f$.  Hence -1 is not an element of the point spectrum
of $Q$.  Moreover, $Qf = f$ implies $P_if = f$ for each $i$ and hence
$f\epsilon M$ and $Pf = f$.  Also $PQ = QP = P$.  We have
$$\eqalign{\parallel Q↑nf - Pf\parallel↑2 & = \parallel Q↑nf\parallel↑2 - \parallel
Pf\parallel↑2\cr & = \int_{-1}↑1\lambda↑{2n} d\parallel E (\lambda) f\parallel↑2 - \parallel
Pf\parallel↑2\cr}$$
As $n\rightarrow\infty$ the integral is easily seen to converge to
$\parallel[E(1) - E(1-0)] f\parallel↑2$, but $P = [E(1) - E(1-0)]$, and the
proof is concluded.  

For the case $k = 2$ this reduces to the theorem that $\parallel
(P_1P_2)↑nf- Pf\parallel\rightarrow 0$ as $n\rightarrow\infty$ which was
proved by von Neumann on p. 55 of (1).

In order to apply this method to partial differential equations we
proceed as follows.  Let there be given a system of partial
differential equations
$$\Phi_i(x, f(x), f↑{(1)}(x),\ldots, f↑{(p)}(x)) = 0,\quad
(2 = 1,\ldots, K)\leqno(1.1)$$
in a region $G$ of n-dimensional Euclidean space.  Here $x$ denotes a
variable point of $G$ with co-ordinates $x↑1, x↑2,\ldots, x↑n$, and
$f↑{(r)}(x)$ denotes the tensor field $f,_i,\ldots, i_r(x) =
{\partial↑rf\over \partial x↑{i_{1}}\ldots\partial x↑{i_{r}}}$.

Let $f$ be required to satisfy the boundary conditions
$$\theta_s[{x, f(x), \ldots, f↑{(p-1)}(x)] = 0,\quad (s = 1,\ldots, t)}$$
on the subsets $D_sCG$.

Consider the space $H$ of multiplets
$$F(x) = [f_o(x), \{f_{i_{11}}(x)\},\{f_{i_{21} i_{22}}(x)\},
\ldots,\{f_{i_{pi}\ldots i_{pp}}(x)\}]$$
where each $f_{i_{r1}}\ldots _{i_{rr}}(x)$ is a covariant tensor field
defined in $G$.  The space $M$ of solutions of (1.1) (or rather the
space of multiplets $F = [f, f↑{(1)}, \ldots, f↑{(p)}]$ where $f(x)$
is a solution of (1.1)) may be regarded as the intersection of the
following subspaces of $H$:
\medskip
\item{(i.)}  The spaces $M_s$ consisting of all $F(x)$ for which
$\theta_s[F(x)] = 0$ for $x$ in $D_s, (s = 1, \ldots, t)$.
\item{(ii.)}  The spaces $N_i$ consisting of all $F(x)$ for which $\Phi_i
[F(x)] = 0$ for $x$ in $G, (i = 1, \ldots, k)$.
\item{(iii.)}  The space $M↑*$ of all $F(x) = [f(x), f↑{(1)}(x), \ldots,
f↑{(p)} (x)]$ for some scalar function $f(x)$.
\medskip
If the elements $F(x)$ are subjected to the sort of condition one
would like to impose such as uniform continuity of all components, we
do not get a space $H$ in which the required projection operators can
conveniently be determine, i.e. we do not get Hilbert spaces.
However, we can get a Hilbert space by requiring the components of
$F(x)$ to be of integrable square in $G$ and making the definition
$$\parallel F(x)\parallel↑2 = \sum↑p_{r=1} \lambda↑2_r \int_G [F_{i_{r1}} \ldots
_{i_rr} (x)]↑2 dv$$

Moreover, in order to apply theorem 1 we must require the manifolds
$M_s$, $N_i$, and $M↑*$ to be closed linear manifolds of $H$.
(Strictly speaking, in order that the projection operators exist and
theorem 1 apply it is only necessary that the manifolds be flat, i.e.
have the property that if $f$ and $g$ are in the manifold then so is
$\lambda f + (1-\lambda) g$.  Hence, they need not contain the
origin.)  This seems to restrict us to linear differential equations
with linear boundary conditions.  However, in certain non-linear cases
it may be possible to approximate the manifolds $M_sN_i$, and $M↑*$ by
linear manifolds in the neighborhood of a point of intersection and so
apply the method to the non-linear case.

Another difficulty which may arise is that the solutions obtained may
not have continuous derivatives since in order to get a Hilbert space
we had to admit all functions of integrable square.  However, it turns
out in the cases discussed in this paper that it is possible to get
solutions in the ordinary sense, i.e. with continuous derivatives.

Even if the solutions obtained are sufficiently differentiable the
convergence of the sequences of successive approximations obtained by
this method may not be uniform.  There are reasons for believing that
in the cases considered the convergence is uniform although it has not
been possible to prove this in any case.

\vfill\eject
\section{2}  {APPLICATION OF THE METHOD TO FIRST ORDER LINEAR PARTIAL
DIFFERENTIAL EQUATIONS}
\bigskip
In this section the projection procedure described in the previous
section is used to solve systems of first order linear partial
differential equations with one dependent variable.  We consider
systems of the form
$$a↑i_{(j)} (x) f,_i(x) = 0,\qquad (j = 1,\ldots, k).\leqno (2.1)$$
Here $x$ is assumed to range over a region $G$ of n-dimensional
Euclidean space, $f,\,_1(x)$ denotes $\partial f\over\partial x↑i$, and
the $a↑i_{(j)}(x)$ are functions of $x$ with uniformly continuous
partial derivatives of all orders.

In order to simplify the proofs some very restrictive assumptions are
made.  We believe that most of them are unnecessary.
\medskip
\item{1.}  The $k$ equations (2.1) are assumed to be independent and
compatible at every point of $G$.  By this we mean that in the
neighborhood of every point the solutions depend on $n-k$ independent
solutions.  This hypothesis is probably unnecessary since if, for
example, the equations are such that constants are the only common
solutions, then the iterative procedure will probably converge to a
constant.
\item{2.}  The boundary $\partial G$ and $G$ is assumed to be such
that the Neumann problem can be solved for the region $G$.
\item{3.}  There is assumed to exist a mapping $y↑i = u↑i (x↑1,
\ldots, x↑n)$ of $G$ onto a region $G''$ such that
\itemitem{(i)}  The mapping is 1-1 and the partial derivatives
$\partial y↑i \over \partial x↑j$ and $\partial x↑i \over \partial
y↑j$ are uniformly bounded by a constant $M$.
\itemitem{(ii)}  $y↑1, \ldots, y↑{n-k}$ form a complete set of solutions
of (2.1).
\itemitem{(iii)}  The k-volume of the sub-manifolds $y↑1, \ldots,
y↑{n-k} = \hbox {const.}$ depends differentiably on $y↑1, \ldots,
y↑{n-k}$, and the region $G'$ in n-k dimensional Euclidean space onto
which $G$ is mapped by $u$ is assumed to have a boundary such that the
Neumann problem can be solved for $G'$.
\medskip

These assumptions are redundant to some extent.  Since they are used
in different parts of the proof, it seemed advisable to state them
separately rather than to combine them, in order that it should be
clear how a refinement in a section of the proofs could lead to a
weakening of the hypotheses.

It is clear from the above assumptions that the method to be outlined
gives no existence theorems and practically no qualitative
information about the solutions.

We now proceed to define the subspaces $M_1$ and $M_2$.

We work in the Hilbert space $H$ of vector fields of Lebesgue
integrable square in $G$.  The inner product $(f_1, g_1)$ of two
vector fields defined in $G$ is given by
$$(f_i, g_i) = \int_G g↑{ij}f_i(x) g_i(x) dv$$
where $g↑{ij}$ is the contravariant metric tensor for the Euclidean
metric.

The space of gradients of solutions of (2.1) may be regarded as the
intersection $M$ of two subspaces $M_1$ and $M_2$ of $H$.  Here $M_1$
is the space of all vector fields $f_i(x)$ such that there exists a
continuously differentiable function $f(x)$ defined in $G$ such that
$f_i(x) = {\partial f\over \partial x↑i}$.  $M_2$ is the space of all
vector fields $f_i(x)$ in $H$ satisfying
$$a↑i_{(j)}(x) f_i(x) = 0, \quad (j = 1, \ldots,
k)\leqno(2.2)$$
almost everywhere in $G$.  Note that $M_2$ is a closed subspace of $H$
and $M_1$ is not.  It is clear from these definitions that the
elements of $M$ are the gradients of solutions of (2.1).

We now proceed to determine the projection operators $P_1$ and $P_2$
on the manifolds $M_1$ and $M_2$.  Let $f_i(x)$ be a vector field
with uniformly continuous first partial derivatives, and suppose that
there exists a twice differentiable scalar function $f(x)$ such that
$$\int_G g↑{ij}[f_i(x) - f,i(x)] h,i(x)dv\leqno(2.3)$$
for all continuously differentiable $h(x)$.  Then $f,i(x) = P_1f_i
(x)$.

In fact, since the $h,_i(x)$ are dense in $M_1$, $f_i(x) -
f,_i(x)$ is orthogonal to all the elements of $\overline M_1$ and
$f,_i(x)$ is in $\overline M_1$.  But this is just the condition
that $f,_i(x)$ be the projection of $f_i(x)$ on ${\overline M}_1$.

Applying Gauss' theorem to (2.3) gives
$$\int_{\partial G} g↑{ij} [f_i(x) - f,_i(x)] h(x) n_j(x)
dS -\int_G g↑{ij} [f_{i,j} - f,_{ij}] hdv \leqno(2.4)$$
where $n_j(x)$ is the unit normal vector to $\partial G$ at the point
$x$.  Since $h(x)$ is arbitrary we have
$$g↑{ij}[f_i(x) - f,_i(x)]n_j(x) = 0 \quad {\rm on}\quad \partial
G\leqno (2.5)$$
and
$$g↑{ij}[f_{i,j} - f,_{ij}] = 0\quad {\rm in} \quad G,\leqno (2.6)$$
or, in another notation,
$${\partial f\over \partial n} = f_in↑i\quad {\rm on}\quad \partial G\leqno
(2.5')$$
and
$$\Delta f = f,↑i_i \quad {\rm in}\quad G, \leqno  (2.6')$$
where the raising of indices has its usual significance.

Hence, in order to obtain $f(x)$ we must solve Neumann's problem for
the region $G$.  We have
$$f(x) = - \int_G f,↑i_i(y) N(x, y) dv_y + h(x)\leqno (2.7)$$
where $h(x)$ is a harmonic function, and $N(x, y)$ is Neumann's
function for the region $G$.  See Kellogg (2) for the properties of
the Neumann function.  Taking the normal derivative of (2.7) gives
$$\leqalignno{{\partial f \over \partial n}& = - \int_G f,↑i_i (y)
{\partial \over \partial n_x} N(x, y) dv_y + {\partial h \over
\partial n}\cr
&= - K = {\partial h\over\partial n}&(2.8)\cr}$$
since ${\partial\over\partial n_x} N(x, y)$ is a constant independent
of $x$ and $y$.  This gives
$$\leqalignno{h(x) & = \int_{\partial G} {\partial h(y)\over\partial
n_y} N(x, y) dSy\cr
&= \int_{\partial G} f_i(y) n↑i(y) N(x, y) dS_y &(2.9)\cr}$$
since $\int_G N(x, y) d S_y = 0$

Hence, we have
$$f(x) = - \int_G f,↑i_i(y) N(x, y)dv_y + \int_{\partial G} f↑i(y) N(x,
y) n_i(y) dS_y$$
or, by Gauss' formula
$$f(x) \int_G f↑i(y) {\partial\over\partial y↑i} N(x, y) dv_y\leqno (2.10)$$
and hence
$$P_1f_i(x) = {\partial\over\partial x↑i}\int_G f↑j(y) {\partial \over
\partial y_j} N(x, y) dv_y\leqno (2.11)$$

We now determine $P_2$.  If $k_i(x) = P_2f_i(x)$ we must have $k_i(x)
- f_i(x)$ orthogonal to every $h_i(x)$ in $M_2$.  Indeed, $P_2f_i(x)$
is characterized by this condition together with the fact that it is
in $M_2$.  Hence, if for each $x$ we choose $k_i(x)$ to be the
projection in Euclidean n-space on the set of all $k_i(x)$ such that
$a↑i_{(j)}(x) h_i(x) = 0$ we will then have $g_{ij}[f_i(x) -
k_i(x)]h_j(x) = 0$ for all $h_i$ in $M_2$, and hence
$$\int _G g↑{ij}[f_i - k_i] h_j dv = 0$$
i.e. $k_i(x) = P_2f_i(x)$.  Thus, the projection onto $M_2$ is
obtained by projecting in Euclidean space at each point of $G$.
Therefore, finally, $k_i(x) = C↑j_i(x) f↑j(x)$ where $C↑j_i(x)$ is a
certain tensor field in $G$.

We then consider the following iterative procedure.  Let $f↑{(0)}(x)$
be a function of $x$ such that $f,↑{(0)}_i(x)$ is in $H$.  Given
$f↑{(n)}(x)$ we define $f↑{(n+1)}$ by
$$f↑{(n+1)}(x) = \int_G g↑{ij}(y) C↑l_j(y)f,_l(y)
{\partial\over\partial y_i} N(x, y) dV_y\leqno (2.12)$$
By theorem 1.1 the sequence $f↑{(n)}(x)$ converges to an element of
${\overline M_i} \cap M_2$.  Before establishing that this element is
actually the gradient of a solution of (2.1) we give a few
applications of the formal results.

Consider a system of ordinary differential equations
$${dx↑i\over dt} = u↑i(x)\leqno (2.13)$$
defined in $G$ and with no singular points in $G$.  Its solution
curves are the characteristic curves of the partial differential
equation
$$u↑i(x) {\partial f\over \partial x↑i} = 0\leqno (2.14)$$
The solutions of (2.14) are the integrals of (2.13), i.e. the
functions which are constant along the curves defined by (2.13).

In this case $P_2f_i(x)$ is given
$$k_i(x) = f_i(x) - {1\over\sqrt {g_{kl}u↑ku↑l}} f_m(x) u↑m(x) g_{ij}
u↑j(x)\leqno (2.15)$$
that is, $k_i(x)$ is the projection of $f_i(x)$ onto the set of vector
fields orthogonal to $u↑i(x)$ and hence is the difference of $f↑i(x)$
and the projection of $f↑i(x)$ on $u↑i(x)$.  The formulas are
simplified if we assume that $g _{kl} u↑ku↑l = 1$.  Then (2.12)
becomes
$$f↑{(n+1)}(x) = f↑{(n)}(x) - \int_G {\partial f↑{(n)}(y)\over
\partial y↑i} u↑i(y) u↑j(y){\partial\over\partial y↑j} N(x,y)dv_y\leqno
(2.16)$$

This procedure is related to the idea of an integrating factor in the
following way.  Let $k_i(x)$ be a vector field.  A function $\mu (x)$
is called an integrating factor of $k_i(x)$ if there exists a scalar
function $f(x)$ such that $f,_i(x) = \mu (x) k_i(x)$.  For $n = 2$
any $k_i(x)$ has an integrating factor, but for $n > 2$ an
integrating factor need not exist.  However, we can generalize the
notion of integrating factor in the following way.

Let $k_i↑{(1)}(x), \ldots, k_i↑{(r)}(x)$ be vector fields.  Then a
matrix function $\mu↑{(s, t)} (s, t = 1, \ldots, r)$ is called a
generalized integrating factor of $k_i↑{(1)}+i, \ldots, k↑{(r)}_i$ if
there exist scalar functions $f↑{(1)}(x), \ldots, f↑{(r)}(x)$ such
that
$$f,↑{(s)}_i = \sum↑r_{t=1} \mu↑{(s,t)} k↑{(t)}_i\leqno (2.17)$$
and, moreover, the $f,↑{(s)}_i$ are linearly independent for each $x$.

If $r = n-1$ and the $k↑{(t)}_i$ are linearly independent for each $x$
then a generalized integrating factor always exists, at least locally.
In fact, since there are $n-1$ linearly independent $k↑{(t)}_i$ there
is a vector field $u↑i(x)$ which is orthogonal to all of them at each
point $x$, i.e. $u↑i(x) k↑{(t)}_i (x) = 0$ for each $t$ and $x$.  If
$f↑{(1)}, \ldots, f↑{(n-1)}$ are $n-1$ independent solutions of
$u↑i(x) {\partial f\over \partial x↑i} = 0$ then each $f,↑{(t)}_i$ can
be written in the form $f,↑{(t)}_i = \sum↑{n-1}_{s=1} \mu↑{(s,t)}
k↑{(s)}_i$ since the $f,↑{(t)}_i$ are in the linear manifold generated
by the $k_i$.

Our iterative procedure may be regarded as an iterative procedure for
finding an integrating factor.  Suppose we are given $n-1$ vector
fields $k↑{(t)}_i$ which are linearly independent for each ??? and an
``approximate'' generalized integrating factor $\mu↑{(s,t)}$ (the
$k↑{(t)}_i$ may be, for example, a set of $n-1$ vector fields chosen
to be orthogonal to a particular vector field $u↑i$).  Since
$\mu↑{(s,t)}$ is not exactly an integrating factor the quantities
$\sum↑{n-1}_{s=1} \mu↑{(s,t)} k↑{(s)}_i$ are not exactly gradients.  So
we pick the closest gradients to them.  By this we mean to choose
$f↑{(1)}(x), \ldots, f↑{(n-1)}(x)$ so as to minimize $\parallel
f,↑{(t)}_i - \sum↑{n-1}_{s=1} \mu↑{(s,t)}k_1↑{(s)}\parallel↑2$.

This amounts to projecting $\sum↑{n-1}_{s=1} \mu↑{(s,t)} k_i↑{(s)}$ onto
the space of gradients, and hence we get
$$f↑{(t)}(x) = \int_G \sum↑{n-1}_{s=1} \mu↑{(s,t)}(y) k_i↑{(s)}(y)
g↑{ij}(y) {\partial\over\partial y↑j} N(x,y) dv_y\leqno (2.18)$$
or
$$f,↑{(t)}_i(x) = P_1 [\sum↑{n-1}_{s=1} \mu↑{(s,t)}(x) k_i(x)]\leqno
(2.19)$$
The $f,↑{(t)}_i$ obtained by this procedure, however, are no longer
exactly linear combinations of the $k_i↑{(s)}$.  So we now reproject
them onto the space of linear combinations of the $k_i↑{(s)}$ getting
elements of the form $\sum↑{n-1}_{s=1} \mu'↑{(s,t)} k_i↑{(s)}$.  Thus we
have calculated a new approximate integrating factor $\mu'↑{(s,t)}$.
Hence our iterated projection procedure may be regarded as a process
of successive approximations for getting an integrating factor.

The projection operators which we have been using are defined in the
$L↑2$ space $H$ of vector fields.  However, when one speaks of solving
a differential equation one usually wants continuously differentiable
functions.  In this connection, one would like a theorem of the
following sort if we start with a function which has a certain number
of derivatives and apply the iterative procedure, the sequence of
approximations should converge in these derivatives at each point of
$G$ to a function which is a solution of the differential equations.

We have not been able to prove any such theorem.  It seems likely that
there is such a theorem although the precise form stated above is
false.  However, we can prove the following theorem.

\proclaim Theorem 2:1.  If we start with a vector field $f_1$ having
uniformly continuous first derivatives, then the iterative converges
in the norm of $H$ to a uniformly continuous vector field.  Hence the
procedure gives an actual solution of the system of differential
equations.

The proof of this theorem uses all of the restrictive assumptions on
the region and on the differential equation.  It should be possible to
prove the theorem without these assumptions, but the proof would have
to be on quite different lines from the one presented here.

The proof is divided into two parts.  The iterative procedure has been
shown to give the projection of the initial vector field $f_i(x)$ onto
the intersection $\overline {M_1}\cap M_2$.  We first show that
$\overline {M_1} \cap M_2 = \overline {M_1\cap M_2}$, and then we show
that the projection onto $\overline M$ is an element of $M$.  Hence
the projection procedure converges to the gradient of a solution of
(2.1).

Let $G$ be the region and $Y↑1,\ldots, Y↑n$ the functions the
existence of which is stated in assumption 5, p.5.  We form the
Hilbert space $H↑1$ of vector fields defined on $G↑1$ with the norm
$$\parallel f↑1(y)\parallel↑2 = \int_G \sum↑n_{?=1} \mid f↑1_2(y)\mid↑2
dv_y\leqno (2.20)$$
There are natural correspondences between the functions and vector
fields on $G$ and those on $G↑1$; namely, $f(x) \leftrightarrow f↑1(y)
= f(x(y))$ and $f_i(x) \leftrightarrow f↑1_i(y) = {\partial x↑j\over
\partial y↑i} f_j(x(y))$.  On account of the boundedness of the
derivatives of the mapping of $G$ onto $G↑1$ the mapping $f_i(x)
\leftrightarrow f↑1_i(y)$ is a homeomorphism of $H$ and $H↑1$.

Under this mapping $M_1$ goes into the subspace $M'_1$, of gradients
of functions of $y$ and $M_2$ goes into the subspace $M'_2$ of vector
fields $f_i(y)$ such that $f_{n-k+1},\ldots, f_n = 0$.  In order to
prove that $\overline {M_1\cap M_2} = \overline {M_1}\cap M_2$ we
need only prove that $\overline {M'_1\cap M'_2} = \overline {M'_1}\cap
M'_2$ which we do as follows. 

Consider an element $f_i(y)$ of $\overline {M'_1}\cap M'_2$.  Since
each of its components is of integrable square each component may be
considered as a distribution $F_i$ in the sense of L. Schwartz (3).
Since $f_i(y)$ is in $\overline M'_1$ there exists a sequence of
differentiable functions $f↑{(n))}(y)$ such that $f,↑{(n)}_i(y)$
converges to $f_2(y)$ in the norm of $H'$ and hence converges to
$F_i$ as a sequence of distributions.  Therefore, there exists a
distribution $F$ such that $F_i = {\partial F\over \partial y↑i}$.
Since $f_i$ is in $M↑1_2$ we have ${\partial F\over \partial y↑j} = 0$ for
$j = n-k+1, \ldots, n$.  We then consider the functions $h↑{(n)}(y) =
F[\rho↑{(n)}0 (y↑1- y)]$.  (See (3) for interpretation of the notation)
where
$$\rho↑{(n)}(y) = K_n\, exp[- {r↑2\over {1\over n↑2} - r↑2}]\, {\rm for}\, r
< {1\over n}, \rho↑n(y) = 0\quad {\rm for}\, r > {1\over n}\leqno (2.21)$$
where $r↑2 = (y↑1)↑2 +,\ldots, +(y↑n)↑2$ and $K_n$ is such that $\int
\rho_n(y) dv = 1$.

The $h↑{(n)}(y)$ have the properties that they depend only on
$y↑1,\ldots, y↑{n-k)}$ if the distance of $y$ from the boundary is
greater than $1\over n$.  Furthermore, the sequence $h,↑{(n)}_i(y)$
converges to $f_i(y)$ in the norm of $H↑1$.  It is clear that we can
define new functions $h↑{(n)*}(y)$ such that $h↑{(n)*}(y) =
h↑{(n)}(y)$ when the distance of $y$ from the boundary is greater than
$1\over n$, such that $h↑{(n)*}(y)$, depends only on $y↑1,\ldots,
y↑{n-k}$, and such that $h,↑{(n)*}_i(y)$ converges to $f_i(y)$ in the
norm of $H↑1$.  Hence ${\overline M'_1}\cap M'_2 = {\overline {M'_1\cap
M'_2}}$, and by what was shown previously ${\overline M_1} \cap M_2 =
{\overline {M_1\cap M_2}}$.

Therefore, our iterative procedure converges to the projection of the
initial vector field onto $M = {\overline {M_1 \cap M_2}}$.  We shall
calculate this operator.

Let $f_i(x)$ have continuous first partial derivatives.  We are
looking for a vector field $k_i(x)$ in $M$ such that if $h_i(x)$ is
any element of $M$ then
$$\int_G g↑{ij} [f_i - k_i] h_j dv = 0\leqno (2.22)$$
Let $y↑1(x), \ldots, y↑{n-k}(x)$ be a complete set of solutions of
(2.1) such that the subsets of $G$ of the form $y↑\alpha(x) = C↑\alpha
= {\rm const.}\, \alpha = 1, \ldots, n-k$, are k-dimensional, of finite
k-volume, and such that this volume is a differentiable function of
$y↑1, \ldots, y↑{n-k}$.

The region $G$ is mapped onto a certain region $G↑1$ in the n-k
dimensional space of $y's$.  This region will be provided a metric and
regarded as a Riemannian space.

We have
$$\leqalignno{\int_G g↑{ij}(x) f,_i(y(x)) h,_i(y(x)) dv &=
\int_Gg↑{ij}(x) {\partial y↑\alpha\over \partial x↑i} {\partial
y↑\beta\over \partial x↑j} f,\alpha(y) h,\beta(y) dv\cr
&=\int_{G'} dy↑1,\ldots, dy↑{n-k} f,\alpha(y) h,\beta(y) \int_{G(y)}
g↑{ij} {\partial y↑\alpha\over\partial x↑i}{\partial
y↑\beta\over\partial x↑j}{\sqrt g_y(x)} dv'&(2.23)\cr}$$
where $f,\alpha (y) = {\partial f(y)\over\partial y↑\alpha}$ and Greek
indices run from 1 to $n-k$.  $g_y(x)$ denotes the determinant of the
metric tensor of the normal manifold to the manifold $y = {\rm
const}$ at the point $x$, and $dv'$ is the volume element of the
maniform $y = {\rm const}.\, (G_y)$ under the induced metric.  We can now write
$$\int_G g↑{ij} f,_j(y)h,_j(y) dv = \int_{G'} \gamma↑{\alpha\beta}(y) \psi
(y) f,\alpha(y) h,\beta(y) dv_y\leqno(2.24)$$
where $\gamma↑{\alpha\beta}$ is a metric tensor for $G↑1$ defined by
$$\gamma↑{\alpha\beta}(y)\psi(y)\sqrt \alpha = \int_{G(y)} g↑{ij}
{\partial y↑\alpha\over\partial x↑i}{\partial y↑\beta\over\partial x↑j} \sqrt {g_y} dv↑1\leqno(2.25)$$
where $\gamma = {\rm det}\, \left| \gamma_{\alpha\beta}\right|$ and $\psi
(y)$ is a positive function of $y$.  (If $n-k \not= 2$ we can choose
$\gamma↑{\alpha\beta}$ so as to make $\psi(y) = 1)$.

We can define a mapping of contravariant tensor fields on $G$ into
tensor fields of the same rank on $G'$ by
$$T↑{*\alpha\beta\ldots\gamma} = \int_{G(y)} T↑{ij\ldots K}{\partial
y↑\alpha\over\partial x↑j}{\partial y↑\beta\over\partial x↑j}\ldots
{\partial y↑\gamma\over\partial x↑k}\sqrt {g_y} dv↑1\leqno(2.26)$$
We define a similar mapping for covariant tensor fields by first
raising indices, then applying $↑*$ and finally lowering indices
again.

Since a solution of (2.1) is just a function of the $y's$ we see that
we are looking for a function $k(y)$ such that
$$\int g↑{ij} [f_i(x) - k,_i(y)] h,_j(y) dv = 0\leqno (2.27)$$ 
for all $h(y)$.  This equation becomes
$$\int_{G'} \psi (y) \gamma↑{\alpha\beta}(y) [f↑*_\alpha (y) -
k,_\alpha (y)] h,_\beta(y) dv_y\leqno(2.28)$$
for all $h (y)$.  In other words, we find outselves projecting the
vector field $f↑*_\alpha (y)$ onto the space of gradients.

Integration by parts gives
$$0 = \int_{\alpha G'}\Psi(y)h(y)[f↑*_\alpha(y) -
k,_\alpha(y)n↑\alpha dS'_y - \int_{G'}\gamma↑{\alpha\beta}[\Psi(y)
[f↑*_\alpha (y) - k,_\alpha(y)]],_\beta h(y)
dv_y\leqno(2.29)$$
Since this is to be true for all $h(y)$ we must have
$$k,_\alpha (y) n↑\alpha(y) = f↑*_\alpha(y) n↑\alpha (y)\, {\rm
 on}\, \partial G',\quad {\rm and}\leqno(1.)$$
$$\gamma↑{\alpha\beta}[\Psi(y) k,{\alpha}(y)],_\beta =
\gamma↑{\alpha \beta}[\Psi(y) f↑*_\alpha(y)],_\beta\quad {\rm\, in\,} 
G.\leqno(2.)$$
This is an elliptic partial differential equation with the normal
derivative of the unknown function prescribed on the boundary.  It is
easily seen that the integral of the prescribed values of the normal
derivative has the right value for a solution to exist.  Applying the
existence theorem for Neumann's problem we get a function $k(y)$ such
that
$$\int_G g↑{ij} [f_i(x) - k,_i(y)]h,_j(y) dv = 0$$
for any solution $h(y)$.  Hence $k,_i(x)$ is the vector field to
which our iterative procedure converges, and by the existence
theorems for elliptic differential equations we see that is actually a
differentiable function of $x$.
\vfill\eject
\section{3}  {APPLICATION TO THE DIRICHLET PROBLEM}
\bigskip
Let
$$a↑{ij}(x)f,_{ij}(x) + a,↑{ij}_j(x) f,_i(x) = 0\leqno (3.1)$$
be a self-adjoint second order partial differential equation of
elliptic type defined in a region $G$ Euclidean n-space which has a
normal vector at almost every point of its boundary and is such that
the Dirichlet problem can be solved for the equation (3.1).  We assume
that the $a↑{ij}(x)$ have uniformly continuous partial derivatives of
all orders.  We are looking for a solution of (3.1) taking on given
boundary values $k(x)$.

We operate in the Hilbert space $H$ of pairs $F = [f_i, f_{ij}]$ where
$f_i$ and $f_{ij}$ are covariant tensor fields of ranks one and two
respectively.  We define
$$(F,H) = \int_G g↑{ij} f_ih_jdv + \int_G g↑{ik}g↑{jl}
f_{ij}h_{kl}dv\leqno (3.2)$$
We assume that the boundary values $k(x)$ are such that (3.1) has a
solution $h(x)$ with these boundary values such that
$\int_Gg↑{ik}g↑{jl}h,_{ij}h,_{kl}dv$ exists.  It is not clear how much
of a restriction this is.  However, if the solution with boundary
values $k$ has uniformly continuous second partial derivatives, then
the above condition is satisfied.

We define $M_1$ to be the set of all elements $F$ for which $f_{ij} =
f,_{ij}$ and $f_i = f,_i$ for a scalar function $f$ with uniformly
continuous second partial derivatives which takes on the boundary
values $k$.

$M_2$ is defined to be the set of all elements $F$ for which $A[F] =
a↑{ij} f_{ij} + a,↑{ij}_j f_i = 0$ almost everywhere in $G$.  The
projection operator $P_2$ onto $M_2$ is an operator of the form
$$P_2F = [c↑{kl}_{ij} f_{kl} + c↑k_{ij} f_k, c↑{kl}_i f_{kl} +
c↑kf_k]\leqno (3.3)$$
where the $c$'s are functions of $x$ with uniformly continuous
derivatives of all orders.

The projection operator $P_1$ onto $M_1$ is determined as follows:
Let $f$ be a function such that $P_1F = [f,_{ij}f,_i]$.  Then
$$0 = \int_G g↑{ik} g↑{jl} [f_{ij} - f,_{ij}]h,_{kl} dv + \int_G
g↑{ij} [f_i - f,_i]h, _jdv\leqno (3.4)$$
for all sufficiently differentiable $h$ with zero boundary values.

Applying Gauss' theorem twice gives
$$\leqalignno{0 & = \int_{\partial G} (g↑{ik} g↑{jl} [f_{ij} -
f,_{ij}] h,_kn_1 - g↑{ik}g↑{jl} [f_{ij,1} - f,_{ijl}] hn_k \cr
& + g↑{ij} [f_i - f,_i] hn_j) dS \cr
& +\int_G(g↑{ik}g↑{jl} [f_{ij,lk} - f,_{ijlk}] - g↑{ij} [f_{i,j} -
f,_{ij}]) hdv&(3.5)\cr}$$
where $n_j$ denotes as usual the unit normal vector to the boundary of
$G$.

Remembering that $h = 0$ on $\partial G$ and hence $h,_k = {\partial
h\over \partial n} n_k$ we obtain
$$\leqalignno{\Delta\Delta f - \Delta f & = g↑{ik} g↑{jl} f_{ij,lk} -
g↑{ij} f_{i,j} {\rm\, in}\, G, {\rm\, and} &(3.6)\cr
{\partial↑2f\over \partial n↑2} & = f_{ij} n↑in↑j {\rm on} \partial G.
& (3.6') \cr}$$
There does not seem to be an existence theorem in the literature for
this boundary value problem in which the function and its second
normal derivative are given on the boundary.  Presumably, an
existence theorem can be proved by the method of integral equations.
However, this will not be done in this paper.  If the existence
theorem can be proved the projection operator $P_1$ will presumably be
expressable as a singular integral operator.

Applying the iterative procedure to an element $F↑{(0)} = [f,↑0_i$,
$f,↑0_{ij}]$ we get a sequence $F↑{(n)}$ of elements of $H$ of which
alternate members are in $M_1$ and $M_2$.  On account of our
assumption that the solution of (3.1) with the boundary values $k(x)$
is in $H$ the manifold $M = {\overline M}_1\cap M_2$ is non-empty.
Hence the iterative procedure converges to an element of this
manifold.  It remains to be shown that this element is the solution of
(3.1) with the boundary values $k$.

Let the limit element be $F$.  We shall regard its components as
distributions in the sense of L. Schwartz (3).  Since $F$ is the limit
in $H$ of elements of $M_1$, and since any $L↑2$ limit is a
distribution limit also, and since differentiation is a continuous
operation distribution-wise, $F = [f,_i,f,_{ij}]$ where $f$ is some
distribution.  Since $F$ is in $M_2 A[F] = 0$ and hence $f$ is a
distribution solution of (3.1).  By a theorem of L. Schwartz any
distribution solution of (3.1) is an infinitely differentiable
function.  If $F↑{(2n)} = [f,↑{(n)}_i , f,↑{(n)}_{ij}]$, then $f$ is
the limit in the Dirichlet norm of the sequence $f↑{(n)}$.

The Dirichlet norm $(\parallel f\parallel↑2 = \int_G g↑{ij}f,_i f,_jdv)$ is
equivalent to the norm $(\parallel f\parallel↑2_l = \int_G a↑{ij}f,_if,_jdv)$
association with equation (3.1).

In the space of functions with finite$\parallel\,\parallel_1$ consider the
projection operator $Q$ onto the set functions with boundary values
zero.  By a well known theorem $1 - Q$ is the projection on the
solutions of (3.1).  Since each $f↑{(n)}$ has the boundary values
$k(x), (1 - Q)f↑{(n)}$ is the solution of (3.1) with the boundary
values $k(x)$.  Since $f$ is a solution of (3.1) we have $f = (1 - Q)
f = \lim (1 - Q) f↑{(n)}$.  Hence $f$ is the solution of (3.1) with the
boundary values $k$.

In the application of the methods of this and the preceding we propose
to restrict ourselves to solid spheres.  Here the Neumann function
$N(x,y)$ is known, and, although it has not been proved, it seems
likely that the operator $P_1$ of this seciton will have a simple
expression.

This is not so much of a restriction as it may seem, because if we are
interested in other regions which can be mapped by a piecewise
differentiable homomorphism onto a solid sphere we can transform the
problem into one where the region is a sphere.  Under such a mapping
an equation of the form (2.1) or (3.1) goes into another equation of
the same kind.

This paper suggests many more questions than it solves.  The whole
relation between the Hilbert space properties of the projection
operators used and their local properties is obscure.  The proofs
that the result obtained is a solution in the ordinary sense are
consequently artificial and involve unnecessarily restrictive
assumptions.
\vfill\eject
\centerline {APPENDIX}
\bigskip
In this appendix we prove an inequality which at present has little
relation to the subject matter of the rest of this paper.  It is
included because it is a first step in carrying out the program of
connecting the local properties of the operators used in the paper
with their Hilbert space properties.

In potential theory the following inequality is much used:
$$\mid f(x_2) - f(x_1)\mid \leq\mid b-a\mid\int↑b_a[f'(x)]↑2dx\leqno
(A.1)$$
where $f(x)$ is a function defined on the interval $[a,b]$ with a
uniformly continuous first derivative.  It is proved by applying
Schwarz's inequality to the expression $\int↑{x_{2}}_{x_{1}}f'(x) dx$.

We shall give a generalization of this inequality to $n$ dimensions,
but first we need a couple of definitions.  

{\bf Definition 1.}  A closed
cell $G$ in Euclidean n-space is said to be a type $A$ if it can be
mapped homeomorphically onto the unit sphere with boundary by a
mapping which is piecewise differentiable in both directions inside
the region and on the boundary and which is such that the first
partial derivatives of the functions defining the mapping and its
inverse are uniformly bounded.

{\bf Definition 2.}  A closed region is of type $A$ if by introducing
partitions it can be made into a cell of type $A$.

Note that polyhedra are of type $A$.

{\bf Theorem 1.}  If $G$ is of type $A$ there exists a constant $K_G$
such that for any function $f(x)$ defined in $G$ with uniformly
continuous first partial derivatives we have
$$\int_{\partial G}[f(x) - {\overline f}]↑2 dS \leq
K_G\int_G[\bigtriangledown f(x)]↑2 dV\leqno (A.2)$$
where $\overline f$ is the average value of $f(x)$ on $\partial G$.

We shall first prove the theorem for the sphere $S_n$ of radius 1.  We
use the fact that any piecewise differentiable function on the surface
of $S_n$ can be expanded in a series of surface harmonics.  See
Kellogg (2).

Let $g(x)$ be the harmonic function with the same boundary values as
$f(x)$ and write
$$g(x) = \sum↑\infty_{n=0} r↑nT_n(\Omega)$$
where $\Omega$ takes values on the unit sphere and the $T_n(\Omega)$
are surface harmonics.

Then
$$\int_{\partial S_{n}}[f(x) - {\overline f}]↑2 dS = \sum↑\infty_{n=1}
\int_{\partial S_{n}} [T_n(\Omega)]↑2dS,$$
whereas
$$\eqalign {\int_{S_{n}} [\bigtriangledown f(x)]↑2dv & \geq
\int_{S_{n}} [\bigtriangledown g(x)]↑2 dv\cr
&\geq \int_{\partial S_{n}} g {\partial g\over \partial n} dS\cr
&\geq \sum↑\infty_{n=1} \int_{\partial S_{n}} n[T_n(\Omega)]↑2
dS.\cr}$$
Hence
$$\int_{\partial S_{n}} [f(x) - {\overline f}]↑2 dS \leq \int_{S_{n}}
[\bigtriangledown f(x)]↑2 dv.$$
Thus theorem 1 is true for the unit sphere with $K = 1$.

Suppose now that $G$ is any cell of type $A$ and that $K$ is the bound
for all the partial derivatives of the mapping onto $S_n$ as well as
the bound for the partial derivatives of the inverse mapping.  We then
have
$$\eqalign{\int_{\partial G} [f(x) - {\overline f}]↑2 dS &\leq
\int_{\partial G}[f(x) - {\overline f}']↑2dS\cr
&\leq (nK)↑{n-1} \int_{\partial S_{n}} [f(x(y)) - {\overline f}']↑2
dS\cr
\int_{\partial G}[f(x) - {\overline f}]↑2 dS &\leq (nK)↑{n-1}
\int_{S_{n}} [\bigtriangledown _y f(x(y))]↑2 dv_y\cr
&\leq (nK)↑{2n+1} \int_G[\bigtriangledown f(x)]↑2dv\cr}$$
where ${\overline f}'$ is the average of $f(x(y))$ over $\partial
S_n$.  Hence $K_G\leq (nK)↑{2n+1}$.  In $G$ is of type $A$ and not a
cell the theorem still holds.  In fact, $\int_G[\bigtriangledown
f(x)]↑2dv$ is unchanged by the introduction of the partitions
necessary to make $G$ a cell, while $\int_{\partial G}[f(x) -
{\overline f}]↑2dS$ is increased by the introduction of the partitions
since
$$\int_{\partial G}[f(x) - {\overline f}]↑2dS \leq\int_{\partial
G}[f(x) - {\overline f}'']↑2 dS\leq\int_{\partial G\cup P}[f(x) -
{\overline f}'']↑2 dS.$$
Here $P$ denotes the partitions and $\overline f''$ is the average
value of $f$ over $\partial G\cup P$.

Note that the proof of theorem 1 only involved the solution of the
Dirichlet problem for the sphere.

One consequence of theorem 1 is that if the functions with finite
Dirichlet norm over a region and average boundary value zero are
completed to a Hilbert space, then every element of the Hilbert space
may be regarded as having a function in $L↑2$ as boundary values.  The
mapping from a function to its boundary values is continuous.

The classical theorem of Poincar$\acute e$ which states that
$$\int_G [f(x)]↑2 dv \leq M_G\int_G [\bigtriangledown f(x)]↑2 dv$$
where $M_G$ is a constant and $f$ is assumed to have average boundary
value zero, can conveniently be proved by the same method as theorem
1.
\vfill\eject
\centerline {REFERENCES}
\bigskip
(1)  J.v. Neumann, Functional Operators, v. 2, Annals of Mathematics
Studies, no. 21, Princeton, 1950

(2)  O. D. Kellogg, Foundations of Potential Theory, Berlin, 1929.

(3)  L. Schwartz, Th$\acute e$orie des Distributions, v. 1, Paris,
1950.
\end