Advanced Partial Differential Equations II
The course uses functional analytic techniques to further develop various aspects of the modern framework of linear and nonlinear partial differential equations. Sobolev spaces, distributions and operator theory are used in the treatment of linear second-order elliptic, parabolic, and hyperbolic equations. In particular we discuss the kind of potential, diffusion and wave equations that arise in inhomogeneous media, with an emphasis on the solvability of equations with different initial/boundary conditions. Then, we will survey the theory of semigroup of operators, which is one of the main tools in the study of the long-time behavior of solutions to nonlinear PDE. The theories and applications encountered in this course will create a strong foundation for studying nonlinear equations and nonlinear science in general.
MATH 950 or permission from the instructor.
Topics to be covered:
Sobolev spaces on bounded domains.
Second order linear elliptic PDE: Lax-Mailgram theorem, existence of weak solutions, regularity, maximum principle, spectral theory of self-adjoint operators.
Second order linear parabolic PDE: existence of weak solutions, regularity, maximum principle.
Second order linear hyperbolic PDE: existence of weak solutions, propagation of waves, causality.
Nonlinear operator theory techniques in nonlinear PDE: fixed point methods, monotone operators, bifurcation theory, nonlinear elliptic boundary value problems, nonlinear wave equations.
Variational methods in nonlinear PDE: criteria for existence of constrained and unconstrained minima of functions, concentration compactness, and applications to the existence and stability of solutions of nonlinear PDE.
Linear semigroup theory for evolution equations: infinitesimal generators, Hille-Yosida and Lumer-Philiips theorems, applications to existence, well-posedness, and dynamics of linear and nonlinear PDE.
McOwen, Partial Differential Equations, Methods and Applications, 2nd Edition, Pearson
Evans, Partial Differential Equations, 2nd Edition, Springer
(Johnson 2015 )