Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains

✍ Scribed by Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2010
• Tongue: English
• Leaves: 440
• Series: Lecture Notes in Mathematics 1991
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xviii
Models of Higher Order....Pages 1-25
Linear Problems....Pages 27-60
Eigenvalue Problems....Pages 61-98
Kernel Estimates....Pages 99-146
Positivity and Lower Order Perturbations....Pages 147-185
Dominance of Positivity in Linear Equations....Pages 187-226
Semilinear Problems....Pages 227-370
Willmore Surfaces of Revolution....Pages 371-392
Back Matter....Pages 393-429

✦ Subjects

Mathematics, general; Functional Analysis; Differential Geometry; Continuum Mechanics and Mechanics of Materials