Office Hours: WF 9-10Am and by Appointment;

MTH6230 PARTIAL DIFFERENTIAL EQUATIONS II; Fall16, TR 5-6:15; CRF230;

Office Hours: WF 9-10am and by appointment;

Professor Ugur G. Abdulla, Ph.D.

Catalog Description: (3 credits) Sobolev spaces and their properties, second order elliptic, parabolic and hyperbolic PDEs, weak solutions, Lax-Milgram theorem, energy estimates, regularity theory, Harnack inequalities, topics on nonlinear PDEs.

Prerequisites:

MTH5115-Functional Analysis & MTH5230-Partial Differential Equations

Textbook: L.C. Evans, Partial Differential Equations, American Mathematical Society, Graduate Studies in Mathematics, Volume 19, Second Edition, 2010, ISBN: 978-0-8218-4974-3;

Course Outcomes:

1. Understand and master methods of functional analysis to analyze linear and nonlinear PDEs arising in various applications such as mathematical physics, fluid mechanics, mathematical biology, economics.

2. Understand the concept of weak differentiability and the notion of generalized solution of the PDEs, which do not possess smooth solutions.

3. Understand the structure and intrinsic properties of Sobolev spaces – relevant set of weakly differentiable functions, where PDE problems are well-posed.

4. Understand and master advanced analysis methods to prove regularity of solutions to linear and nonlinear PDEs arising in applications.

Topics Covered:

1. Weak differentiability, Sobolev spaces and their properties: approximation,

extension, traces.

2. Gagliardo-Nirenberg-Sobolev inequality, Morrey inequality, compactness, Poincare’s inequality.

3. Second-order elliptic equations, weak solutions. Lax-Milgram theorem. Energy estimates. Fredholm alternative. Regularity properties. Maximum principles. Harnack inequality.

4. Second order parabolic and hyperbolic equations. Weak solutions. Energy estimates and well-posedness. Regularity of weak solutions.

5. 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.

Format and Teaching Methods:

Lectures, homework and project assignments, two midterm and final exams.

Total score of 45 (15%) will be available from homework; Each of the two midterm exams will be graded in 60’s (20%), project presentation in 45’s (15%) and final exam in 90’s (30%) (i.e. maximum score is 300). Final grade will be determined by curving all final scores.