Fourier series in several variables with applications to partial differential equations

✍ Scribed by Shapiro V.

• Publisher: CRC
• Year: 2011
• Tongue: English
• Leaves: 348
• Series: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents......Page 13
Preface......Page 7
1. Introduction......Page 15
2. Iterated Fejer Summability of Fourier Series......Page 16
3. Bochner-Riesz Summability of Fourier Series......Page 22
4. Abel Summability of Fourier Series......Page 30
5. Gauss-Weierstrass Summability of Fourier Series......Page 43
6. Further Results and Comments......Page 50
1. Introduction......Page 53
2. Abel Summability of Conjugate Series......Page 62
3. Spherical Convergence of Conjugate Series......Page 70
4. The Cα-Condition......Page 80
5. An Application of the Cα-Condition......Page 84
6. An Application of the Lp-Condition......Page 88
7. Further Results and Comments......Page 92
1. Uniqueness for Abel Summability......Page 93
2. Uniqueness for Circular Convergence......Page 110
3. Uniqueness, Number Theory, and Fractals......Page 116
4. Further Results and Comments......Page 150
1. Positive Definite Functions on SN–1......Page 153
2. Positive Definite Functions on TN......Page 157
3. Positive Definite Functions on SN1–1× TN......Page 162
4. Further Results and Comments......Page 170
1. Reaction-Diffusion Equations on the N-Torus......Page 173
2. Quasilinear Ellipticity on the N-Torus......Page 200
3. Further Results and Comments......Page 229
1. Distribution Solutions......Page 233
2. Classical Solutions......Page 273
3. Further Results and Comments......Page 285
1. Integral Identities......Page 291
2. Estimates for Bessel Functions......Page 295
3. Surface Spherical Harmonics......Page 298
1. Convergence and Summability......Page 313
2. Tauberian Limit Theorems......Page 315
3. Distributions on the N-Torus......Page 319
4. Hj (x) and the Cα-Condition......Page 324
1. Harmonic Functions......Page 329
2. Subharmonic Functions......Page 340
Bibliography......Page 345