Analysis of Heat Equations on Domains
Although the title suggests that heat equations would be the only topic of the book, the reader soon observes that the book is devoted to a more general study of the Lp theory of evolution equations associated with non-self-adjoint operators in divergence form. The author uses the technique of sesquilinear forms and semigroup theory, which avoids using Sobolev embeddings and therefore does not need smoothness properties of the boundary. On the other hand, this means that the question of regularity is not addressed by this approach.
One of the aims of the author was to write a self-contained contribution, which will be useful for a majority of readers. First we learn some necessary background material from functional analysis, semigroup theory, sesquilinear forms and the theory of evolutionary partial differential equations needed to understand the topic. The author proceeds to study semigroups associated with sesquilinear forms, both for uniformly elliptic and degenerate elliptic operators. Finally, an even more general approach is presented. The Lp estimates for the Schrödinger and wave-type equations are given in the setting of abstract operators on domains of metric spaces. This framework includes operators on general Riemannian manifolds, sub-Laplacians on Lie groups or Laplacians on fractals. The book is intended not only for specialists in partial differential equations but also for graduate students who want to learn about the sesquilinear form technique and the semigroup approach to partial differential equations containing second-order elliptic operators in divergence form.