Maximum Principles are central to the theory and applications of second-order partial differential equations and systems. This self-contained text establishes the fundamental principles and provides a variety of applications.
1. The One-Dimensional Maximum Principle.- 1. The maximum principle.- 2. The generalized maximum principle.- 3. The initial value problem.- 4. Boundary value problems.- 5. Approximation in boundary value problems.- 6. Approximation in the initial value problem.- 7. The eigenvalue problem.- 8. Oscillation and comparison theorems.- 9. Nonlinear operators.- Bibliographical notes.- 2. Elliptic Equations.- 1. The Laplace operator.- 2. Second-order elliptic operators. Transformations.- 3. The maximum principle of E. Hopf.- 4. Uniqueness theorems for boundary value problems.- 5. The generalized maximum principle.- 6. Approximation in boundary value problems.- 7. Green’s identities and Green’s function.- 8. Eigenvalues.- 9. The Phragmèn-Lindelöf principle.- 10. The Harnack inequalities.- 11. Capacity.- 12. The Hadamard three-circles theorem.- 13. Derivatives of harmonic functions.- 14. Boundary estimates for the derivatives.- 15. Applications of bounds for derivatives.- 16. Nonlinear operators.- Bibliographical notes.- 3. Parabolic Equations.- 1. The heat equation.- 2. The one-dimensional parabolic operator.- 3. The general parabolic operator.- 4. Uniqueness theorems for boundary value problems.- 5. A three-curves theorem.- 6. The Phragmèn-Lindelöf principle.- 7. Nonlinear operators.- 8. Weakly coupled parabolic systems.- Bibliographical notes.- 4. Hyperbolic Equations.- 1. The wave equation.- 2. The wave operator with lower order terms.- 3. The two-dimensional hyperbolic operator.- 4. Bounds and uniqueness in the initial value problem.- 5. Riemann’s function.- 6. Initial-boundary value problems.- 7. Estimates for series solutions.- 8. The two-characteristic problem.- 9. The Goursat problem.- 10. Comparison theorems.- 11. The wave equation in higher dimensions.- Bibliographical notes.
