Synopses & Reviews
Fourier Series in Several Variables with Applications to Partial Differential Equations illustrates the value of Fourier series methods in solving difficult nonlinear partial differential equations (PDEs). Using these methods, the author presents results for stationary Navier-Stokes equations, nonlinear reaction-diffusion systems, and quasilinear elliptic PDEs and resonance theory. He also establishes the connection between multiple Fourier series and number theory.
The book first presents four summability methods used in studying multiple Fourier series: iterated Fejer, Bochner-Riesz, Abel, and Gauss-Weierstrass. It then covers conjugate multiple Fourier series, the analogue of Cantor 's uniqueness theorem in two dimensions, surface spherical harmonics, and Schoenberg 's theorem. After describing five theorems on periodic solutions of nonlinear PDEs, the text concludes with solutions of stationary Navier-Stokes equations.
Discussing many results and studies from the literature, this book demonstrates the robust power of Fourier analysis in solving seemingly impenetrable nonlinear problems.
Synopsis
Presenting Fourier series from the aspect of several variables, this book covers connections between Fourier analysis and partial differential equations. It discusses newly emerging topics, such as the fundamental results on Calderona "Zygmund kernels in multi-dimensions and their relation to Fourier series and integrals, along with Fourier integrals and their applications to uniqueness for two-dimensional trigonometric series. The author explores important applications for the first time through reaction-diffusion equations and Naviera "Stokes equations. The text also covers spherical harmonics and includes problems to aid with application.