Summary: Nonhomogeneous Linear and Quasilinear
Elliptic and Parabolic Boundary Value
Problems
Herbert Amann
It is the purpose of this paper to describe some of the recent developments in the mathe­
matical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous
boundary conditions. For illustration we use the relatively simple set­up of reaction­diffusion
systems which are --- on the one hand --- typical for the whole class of systems to which
the general theory applies and --- on the other hand --- still simple enough to be easily de­
scribed without too many technicalities. In addition, quasilinear reaction­diffusion equations
are of great importance in applications and of actual mathematical and physical interest, as
is witnessed by the examples we include.
In particular, we try to elucidate the r“oles which are played in the theory of quasilinear
parabolic systems by the modern theory of function spaces, interpolation and extrapolation
techniques, and semigroup theory. Many of the proofs will be sketched only and we will
be rather brief at times. However, we try to explain the basic underlying ideas and give
references to the research literature where proofs can be found. A complete, detailed, and
coherent presentation will be given in the forthcoming monograph [Ama94] which will also
contain many additional results and extensions of the theory described in this paper.
1 Model Problems

Source: Amann, Herbert - Institut für Mathematik, Universität Zürich

Collections: Mathematics