Partial Differential Equations Preliminary Examination Topics

First-order PDEs:

• Cauchy problem, Local existence and uniqueness of solution;
• Method of characteristics for linear, quasilinear and fully nonlinear equations;
• IVP’s for conservation laws: integral solutions, shock formation, and Rankine-Hugoniot condition.

Second-order PDEs:

• Classification of 2nd order PDEs
• Formulation and interpretation of classical initial/boundary value problems; concepts of solution, well-posedness and stability,
• separation of variables, Fourier series, similarity solutions, Fourier transform.

Elliptic PDEs:

• Laplace and Poisson Equation: classical boundary value problems, Green’s identities, Green’s function, Poisson integral formula; harmonic and subharmonic functions: mean value property, maximum principle; energy methods.

Parabolic PDEs:

• Heat equation: fundamental solution and integral formula for solution of the heat equation; maximum principle for solution and subsolution to the heat equation; mean value property, regularity of solution; energy integral and estimates; Duhamel’s principle

Hyperbolic PDEs:

• Wave Equation: wave propagation, D’Alembert formula; energy integral; energy estimates; spherical means, Kirchoff solution, method of descent, global properties of solutions

Variational formulation and weak solutions

Text/References/Resources

• Evans, L.C. Partial Differential Equations, American Mathematical Society
• Han, Q. A basic course in partial differential equations, American Mathematical Society
• Zachmanoglou, E. C. and Thoe, D.W. Introduction to partial differential equations with applications
• W. Strauss, Partial Differential Equations: An Introduction, Wiley.