# Elliptic Partial Differential Operators and Symplectic Algebra

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*W. N. Everitt; L. Markus*

This investigation introduces a new description and classification for the set
of all self-adjoint operators (not just those defined by differential
boundary conditions) which are generated by a linear elliptic partial
differential expression
\[A(\mathbf{x},D)=\sum_{0\,\leq\,\left| s\right| \,\leq\,2m}a_{s}
(\mathbf{x})D^{s}\text{ for all }\mathbf{x}\in\Omega\]
in a region \(\Omega\), with compact closure
\(\overline{\Omega}\) and \(C^{\infty }\)-smooth boundary
\(\partial\Omega\), in Euclidean space \(\mathbb{E}^{r}\)
\((r\geq2).\) The order \(2m\geq2\) and the spatial dimension
\(r\geq2\) are arbitrary. We assume that the coefficients \(a_{s}\in
C^{\infty}(\overline {\Omega})\) are complex-valued, except real for the
highest order terms (where \(\left| s\right| =2m\)) which satisfy the
uniform ellipticity condition in \(\overline{\Omega}\). In addition,
\(A(\cdot,D)\) is Lagrange symmetric so that the corresponding linear
operator \(A\), on its classical domain
\(D(A):=C_{0}^{\infty}(\Omega)\subset L_{2}(\Omega)\), is symmetric; for
example the familiar Laplacian \(\Delta\) and the higher order
polyharmonic operators \(\Delta^{m}\).

Through the methods of complex symplectic algebra, which the authors have
previously developed for ordinary differential operators, the Stone-von Neumann
theory of symmetric linear operators in Hilbert space is reformulated and
adapted to the determination of all self-adjoint extensions of \(A\) on
\(D(A)\), by means of an abstract generalization of the
Glazman-Krein-Naimark (GKN) Theorem. In particular the authors construct a
natural bijective correspondence between the set \(\{T\}\) of all such
self-adjoint operators on domains \(D(T)\supset D(A)\), and the set
\(\{\mathsf{L}\}\) of all complete Lagrangian subspaces of the boundary
complex symplectic space \(\mathsf{S}=D(T_{1})\,/\,D(T_{0})\), where
\(T_{0}\) on \(D(T_{0})\) and \(T_{1}\) on
\(D(T_{1})\) are the minimal and maximal operators, respectively,
determined by \(A\) on \(D(A)\subset L_{2}(\Omega)\). In the case
of the elliptic partial differential operator \(A\), we verify
\(D(T_{0})=\overset{\text{o}}{W}{}^{2m}(\Omega)\) and provide a novel
definition and structural analysis for
\(D(T_{1})=\overset{A}{W}{}^{2m}(\Omega)\), which extends the GKN-theory
from ordinary differential operators to a certain class of elliptic partial
differential operators. Thus the boundary complex symplectic space
\(\mathsf{S}=
\overset{A}{W}{}^{2m}(\Omega)\,/\,\overset{\text{o}}{W}{}^{2m}(\Omega)\)
effects a classification of all self-adjoint extensions of \(A\) on
\(D(A)\), including those operators that are not specified by
differential boundary conditions, but instead by global (i.e.
non-local) generalized boundary conditions. The scope of the theory is
illustrated by several familiar, and other quite unusual, self-adjoint
operators described in special examples.

An Appendix is attached to present the basic definitions and concepts of
differential topology and functional analysis on differentiable manifolds. In
this Appendix care is taken to list and explain all special mathematical terms
and symbols - in particular, the notations for Sobolev Hilbert spaces and the
appropriate trace theorems.

An Acknowledgment and subject Index complete this Memoir.