Polyharmonic Boundary Value Problems - Filippo Gazzola

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Résumé :
Linear elliptic equations arise in several models describing various phenomena in the applied sciences, the most famous being the second order stationary heat eq- tion or,equivalently,the membraneequation. Forthis intensivelywell-studiedlinear problem there are two main lines of results. The ?rst line consists of existence and regularity results. Usually the solution exists and ?gains two orders of differen- ation? with respect to the source term. The second line contains comparison type results, namely the property that a positive source term implies that the solution is positive under suitable side constraints such as homogeneous Dirichlet bou- ary conditions. This property is often also called positivity preserving or, simply, maximum principle. These kinds of results hold for general second order elliptic problems, see the books by Gilbarg-Trudinger [198] and Protter-Weinberger [347]. For linear higher order elliptic problems the existence and regularitytype results - main, as one may say, in their full generality whereas comparison type results may fail. Here and in the sequel ?higher order? means order at least four. Most interesting models, however, are nonlinear. By now, the theory of second order elliptic problems is quite well developed for semilinear, quasilinear and even for some fully nonlinear problems. If one looks closely at the tools being used in the proofs, then one ?nds that many results bene?t in some way from the positivity preserving property. Techniques based on Harnack?s inequality, De Giorgi-Nash- Moser?s iteration, viscosity solutions etc.

Biographie:
Maurizio Garrione is Assistant Professor of Mathematical Analysis in the Department of Mathematics of the Politecnico di Milano, Italy. His research focus is in ordinary and partial differential equations, differential models and applications. Filippo Gazzola is Professor of Mathematical Analysis in the Department of Mathematics of the Politecnico di Milano, Italy. His research focus is in partial differential equations in a broad sense, in calculus of variations and, in particular, in models for suspension bridges....

Sommaire:
Prof. Filippo Gazzola, Department of Mathematics, Politecnico di Milano, Italy....

From the reviews:

This is an excellent book, full of well-explained ideas and techniques on the subject, and can be used as a textbook in an advanced course dealing with higher-order elliptic problems. The proofs of almost all of the theorems directly related to the higher-order elliptic problems are complete and well written. In general, the book itself is written in a very clear, pleasurable style, including a wealth of useful diagrams and figures, some of them in color. (Rodney Josu? Biezuner, Mathematical Reviews, Issue 2011 h)

The main tasks of the present Lecture Notes in Mathematics volume are nonlinear problems and positivity statements for higher order elliptic equations involving polyharmonic operators. In particular the biharmonic operator and semilinear operators related to it are investigated. ... That the authors are experienced researchers on the topic of the volume becomes evident from the excellent, clear and well motivated presentation. ... Whoever is interested in an exciting subject of the modern theory of higher order elliptic equations is recommended to study this exposition. (Heinrich Begehr, Zentralblatt MATH, Vol. 1239, 2012)