Fourier Series in Several Variables with Applications to Partial Differential Equations

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.

Victor L. Shapiro is a Distinguished Professor Emeritus in the Department of Mathematics at the University of California, Riverside, where he has taught for 46 years. He earned his Ph.D. from the University of Chicago and completed postdoctoral work at the Institute for Advanced Study, where he was an NSF fellow.