Klaus, Boehmer, Professor, University of Marburg
 
Title: Linear and Nonlinear Partial Elliptic Differential Equations and Systems, A Survey
 
Abstract: Guided by the numerical needs, we present the key results for
Linear and nonlinear partial elliptic differential equations and systems:
We discuss results for existence and uniqueness, regularity of solutions
(smooth data yield smooth solutions), linearizations + satisfy the
Fredholm alternative. A preprint with 90 pages is available with all the
detailed results. \\ We formulate the different cases for the Laplacian
$\Delta u$ and only now and then indicate, in a footnote, the general form
for specialists. Classical, strong and weak solutions,
 relation to coercive and elliptic bilinear forms,  $(\Delta)^m u$, different types of nonlinearity, regularity, linear and
nonlinear systems and  the Navier-Stokes equations are discussed. All linearizations in this
huge class of operators, are, under slightly stronger conditions, compact
perturbations of boundedly invertible, often coercive, operators with
stable discretizations. They satisfy the Fredholm alternative and, in
case of invertability, inherit the stability.