Topics in Mathematical Analysis
This volume contains a collection of 13 lectures given over the period 2000-2003 at the University of Padova. The topics of these lectures are from various areas of mathematical analysis: potential theory, differential equations and harmonic analysis. Integral representations in complex, hypercomplex and Clifford analysis are treated in the H. Begehr lectures. O. Martio describes two approaches to Sobolev type spaces on metric spaces and their applications to minimizers of variational integrals. "An introduction to mean curvature flow" is the title of G. Bellettini's five hour course. P. Drábek concentrates on three basic tools used in bifurcation theory: the implicit function theorem, the degree theory and the variational method. The lectures by P. Lindquist deal with the eigenvalue problem for p-Laplacian. The existence of solutions to nonlinear elliptic equations with critical and supercritical exponents is the subject of D. Passaseos' lectures. The lectures of G. Rosenblum are devoted to asymptotics of eigenvalues of singular elliptic operators. The purpose of the lectures of E. Vesentini is to show an approach to semigroups of non-linear operators along the same lines as is followed in the linear case.
The last five lectures deal with the harmonic analysis approach to analysis of geometric objects. The injectivity of Radon's transform is studied in the lectures by M. Agranovsky. Connections with overdetermined boundary problems are also shown. The interplay of Fourier analysis and geometric combinatorics like the Szemeredi-Trotter incidence theorem is described in A. Iosevichs' lectures. C. D. Sogge presents the restriction theorem for Fourier transforms and estimates for the Bochner-Riesz means on Riemann manifolds and he shows how these results can be applied to estimate the eigenfunctions of the Laplacian on a compact Riemann manifold with or without boundary. The contribution "Five lectures on harmonic analysis" written by F. Soria explains how properties of classical equations (heat, Laplace, wave and Schrödinger) can be studied through methods of harmonic analysis (e.g. maximal operators and multipliers). H. Triebel shows that fractals can be interpreted as quasi-metric spaces and therefore the theory of function spaces on them is a decisive instrument, e.g. for the investigation of spectrum of the Laplacian on a fractal structure. All contributions are clearly written, contain motivations and give a large number of references and therefore can be recommended to graduate students. Non-experts can also find a lot of useful information on various aspects of partial differential equations.