Partial Differential Equations II
Analyze linear and nonlinear PDEs arising in various applications such as mathematical physics, fluid mechanics, mathematical biology, economics
Learn modern theory of partial differential equations (PDEs) with emphasis on linear and nonlinear PDEs arising in various applications such as mathematical physics, fluid mechanics, mathematical biology and economics. Explore topics including Sobolev spaces and their properties, second order elliptic, parabolic and hyperbolic PDEs, concept of weak differentiability, weak solutions, Lax-Milgram theorem, energy estimates, regularity theory, Harnack inequalities, topics on nonlinear PDEs.
Weak differentiability of functions
Sobolev spaces and their properties: approximation, extension, traces
Gagliardo-Nirenberg-Sobolev inequality
Morrey inequality, compactness, Poincare inequality
Second-order elliptic equations, weak solutions.
Lax-Milgram theorem. Energy estimates
Fredholm alternative
Regularity properties
Maximum principle, Harnack inequality
Second-order parabolic and hyperbolic equations, weak solutions
Energy estimates and well-posedness.
Regularity of weak solutions.
Nonlinear diffusion and reaction-diffusion type equations. Space-time scaling and instantaneous point source solution.
Weak solutions, energy estimates, existence and uniqueness. Holder continuity of weak solutions.
Homework 3 x 15 points each (15%), Midterm Exams 2 x 60 points each (40%), Project Presentation 45 points (15%), Final exam 90 points (30%), Total 300 points = 100%
A114 Functional Analysis and A108 Partial Differential Equations, or equivalent.
L.C. Evans, Partial Differential Equations, American Mathematical Society, Graduate Studies in Mathematics, Volume 19, Second Edition, 2010, ISBN: 978-0-8218-4974-3