Elliptic Equations: An Introductory Course

Description

The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and complicated refinements. Apart from the basic theory of equations in divergence form, it includes subjects as singular perturbations, homogenization, computations, asymptotic behavior of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes systems, p-Laplace type operators, large solutions, and mountain pass techniques. Just a minimum on Sobolev spaces has been introduced and work on integration on the boundary has been carefully avoided to keep the reader attention focused on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original, and have not been published elsewhere. The book will be of interest to graduate students and researchers specializing in partial differential equations. This new edition contains two more chapters devoted to interesting techniques in partial differential equations. Namely, Chap. 20 is concerned with the so-called large solutions, i.e., with solutions of elliptic problems which blow up at the boundary of the domain; Chap. 21 investigates mountain pass techniques which allow the finding of critical points which are not local minima of some functionals. In addition, Chaps. 6, 14, and 17 have been completely revisited. In particular, Chap. 14 contains now a very simple proof of existence of a solution to the stationary Navier–Stokes problem including non-homogeneous boundary conditions and an elegant and simple technique to prove the uniqueness of the Poiseuille flow. This is completed by a new appendix devoted to the divergence equation. Chapter 6 benefits of the new developments regarding existence and uniqueness of solutions in some unbounded domains, and Chap. 17 introduces new techniques for p-Laplacian type problems.