Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains: Lecture Notes in Mathematics, cartea 1991

Models of Higher Order.- Linear Problems.- Eigenvalue Problems.- Kernel Estimates.- Positivity and Lower Order Perturbations.- Dominance of Positivity in Linear Equations.- Semilinear Problems.- Willmore Surfaces of Revolution.

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 Josué 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)

This monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. Underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity or - since, in contrast to second order equations, a general form of a comparison principle does not exist - on “near positivity.” The required kernel estimates are also presented in detail. As for nonlinear problems, several techniques well-known from second order equations cannot be utilized and have to be replaced by new and different methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the first part is emphasized and, moreover, a far-reaching Gidas-Ni-Nirenberg-type symmetry result is included. Finally, some recent progress on the Dirichlet problem for Willmore surfaces under symmetry assumptions is discussed.