Analysis for Nonlinear Partial Differential Equations

This book offers a rigorous and self-contained introduction to the analytical methods used in nonlinear partial differential equations, with a special emphasis on dispersive equations.

The book addresses a central difficulty of the field: the prerequisite tools required for ****applying modern PDE theory ****are often scattered across different sources, either developed with incompatible notation or presented without considering dispersive analysis. In this book, they are organized into a coherent framework, offering rigorous, self-contained proofs and a unified point of view. Readers are guided from foundational topics such as Lebesgue and Lorentz spaces, convolution, weak convergence, and Fourier transforms to more advanced techniques including oscillatory integrals, maximal functions, singular integrals, dyadic methods, and spaces of regular functions. These foundational techniques can help readers in studying regularity, decay, dispersive estimates, and the behavior of solutions to evolution equations. The final part of the book shows how this analytical toolbox is applied to the Schrödinger and wave equations, highlighting the core ideas of Euclidean dispersive analysis in a concrete and mathematically rigorous way.

By combining breadth, detail, and an explicit focus on applications, the book serves both as a reference work and as a pathway into a demanding area of research. It is intended for graduate students and early career researchers in analysis, partial differential equations, harmonic analysis, and mathematical physics.

Presents the core tools of nonlinear and dispersive PDEs in a coherent framework Develops advanced methods, from Lebesgue spaces to oscillatory integrals, with complete proofs Illustrates the powerful theory with modern results on the Schrödinger and wave equations

Autorentext

Piero D'Ancona has been a full professor of mathematical analysis at Sapienza Università di Roma since 1997. He earned his PhD from the Scuola Normale Superiore di Pisa and has held extended visiting positions at several prestigious international institutions, including various institutes in Paris, the Courant Institute in New York, and Imperial College in London. Prior to his current role, he served as an assistant professor at the University Tor Vergata (1989-1991), an associate professor at the University of Pisa (1992-1993), and a full professor at the University of L'Aquila (1994-1997).

Inhalt

Part I. Real Analysis.- Chapter 1. Lebesgue spaces.- Chapter 2. Sequences in 𝑳𝒑.- Chapter 3. The fundamental inequalities.- Chapter 4. Density, separability and dual spaces.- Chapter 5. Lorentz spaces and rearrangements.- Chapter 6. Symmetry and duality.- Chapter 7. Convolution.- Chapter 8. Weak convergence.- Chapter 9. BochnerLebesgue spaces.- Chapter 10. Baby interpolation.- Chapter 11. Sobolev inequalities.- Chapter 12. Probability for Analysts.- Part II. Fourier Analysis.- Chapter 13. The Fourier transform on 𝑳1.- Chapter 14. The Fourier transform on S.- Chapter 15. The Fourier transform on 𝑳2.- Chapter 16. A review of distribution theory.- Chapter 17. The Fourier transform on S.- Chapter 18. A gallery of transforms.- Chapter 19. Fourier multipliers.- Chapter 20. Sobolev spaces 𝑯𝒔.- Chapter 21. Some applications.- Chapter 22. Oscillatory integrals.- Chapter 23. Fourier series.- Part III. Interpolation Theory.- Chapter 24. Bare necessities.- Chapter 25. Complex interpolation.- Chapter 26. Real interpolation.- Chapter 27. Technical lemmas.- Part IV. Harmonic Analysis.- Chapter 28. Covering lemmas and dyadic cubes.- Chapter 29. Maximal functions.- Chapter 30. The spaces 𝑩𝑴𝑶 and 𝕳1.- Chapter 31. Singular integrals.- Chapter 32. Dyadic analysis.- Part V. Spaces of Regular Functions.- Chapter 33. Functions of a real variable.- Chapter 34. Sobolev spaces.- Chapter 35. Sobolev spaces on domains.- Chapter 36. Other spaces.- Part VI. Linear Flows.- Chapter 37. The Schrodinger equation.- Chapter 38. The wave equation.