Linear and Nonlinear Partial Differential Equations

This book provides a thorough introduction to scalar linear and nonlinear elliptic, parabolic, and hyperbolic partial differential equations. It presents both classical and modern methods for proving the existence and regularity of solutions to boundary and initial-boundary value problems for first- and second-order equations. The material is developed in a systematic and coherent way, progressing from elementary results for linear equations to advanced methods for nonlinear problems. A key feature is the careful derivation of the equations from physical principles. Part I covers first-order equations. Part II develops second-order theory, beginning with the wave equation and leading to elliptic and hyperbolic equations, including spectral methods, energy estimates, and the propagation of singularities. Part III focuses on nonlinear elliptic and parabolic equations, emphasizing monotonicity methods, variational approaches, regularity theory, and maximum principles. Modern techniques are based on Sobolev spaces and functional analysis, which are introduced with detailed proofs. The book also presents essential analytical tools needed for PDE theory. Intended for graduate students with a strong background in calculus and a basic knowledge of functional analysis, the wide range of topics covered in the book also makes it of interest to researchers.