A Weighted Approach on Elliptic Equations with Singularities

Elliptic equations in a two- or three-dimensional
bounded domain
may have singular solutions from the non-smoothness
of the
domain, changes of boundary conditions, and
discontinuities,
singularities of the coefficients. These
singularities give rise to
various difficulties in the theoretical analysis and
in the
development of numerical algorithms for these
equations. On the
other hand, most of the problems arising from physics,
engineering, and other applications have
singularities of this form.
In addition, the study on these elliptic equations
leads to good
understandings of other types of PDEs and systems of
PDEs. This
research, therefore, is not only of theoretical
interest, but also of
practical importance.
This book includes a-priori estimates
(well-posedness, regularity,
and Fredholm property) for these singular solutions
of general
elliptic equations in weighted Sobolev spaces, as
well as effective
finite element schemes and corresponding multigrid
estimates.
Graduate students and researchers will find it a very
useful
introductory reference.

Autorentext
Department of Mathematics, Syracuse University

Klappentext
Elliptic equations in a two- or three-dimensional bounded domain may have singular solutions from the non-smoothness of the domain, changes of boundary conditions, and discontinuities, singularities of the coefficients. These singularities give rise to various difficulties in the theoretical analysis and in the development of numerical algorithms for these equations. On the other hand, most of the problems arising from physics, engineering, and other applications have singularities of this form. In addition, the study on these elliptic equations leads to good understandings of other types of PDEs and systems of PDEs. This research, therefore, is not only of theoretical interest, but also of practical importance. This book includes a-priori estimates (well-posedness, regularity, and Fredholm property) for these singular solutions of general elliptic equations in weighted Sobolev spaces, as well as effective finite element schemes and corresponding multigrid estimates. Graduate students and researchers will find it a very useful introductory reference.