Modern Theory of Partial Differential Equations (PDEs) (MA 534, Spring 2019)

by Prof.Mayukh Mukherjee at IITB.

Review (2 page pdf):

\(\textbf{Motivation}\): I had been introduced to distributions by Prof.Anshuman Kumar in his course on Physics of Nanostructures. Dirac delta and green’s functions are truly understood only in distribution theory. He further told me about some modeling problems in COMSOL that need to use weak solutions of Maxwell’s equations. Prof. S. Bhaskar and Prof. Harsha Hutridurga also suggested me to take this course to build a good base in PDEs.

\(\textbf{Prerequisites}\): Multivariable Calculus, Linear Algebra, Real Analysis, some Functional Analysis (Hahn Banach, Riesz Representation and existence of Mollifiers, etc.) and Measure Theory. Prof. Mukherjee is heavy on differential geometry so concepts from manifold theory are helpful.

\(\textbf{Content Overview}\): Notes(161 pages pdf)

  • Review of Elliptic PDEs: Laplacian, Harmonic functions- variational characterization, Mean Value property of harmonic functions, Mollification, Liouville’s theorem, Maximum Modulus Principle, uniqueness of solution, pointwise gradient estimates, Harnack, Poincare \& Caccioppoli inequalities, Dirichlet Problem for unit ball, Riemann Mapping, Hopf boundary lemma. Fourier transforms, Parseval \& Plancheral theorem, Hahn-Banach theorem, Young’s \& Holder’s inequalities, Riesz Representation. \textit{Heat Equation}: Kernel and its properties, heat semigroup, Parabolic maximum principle.

  • Distribution theory: Tempered Distributions, Frechet space, semi-norms, subbase, Weak Star topology, Dirac-delta, derivatives of tempered distributions, Fourier Transform of tempered distributions, Marcinkiewicz \& Riesz-Thorin interpolation theorems, Fundamental Solutions, computing fundamental solutions for Laplace, Wave and Heat equations, Boundary Value Problem for upper half plane, Taylor Subordinate identity, Sharp Huygen’s Principle, Spectral theorem for unbounded self-adjoint operators.

  • Sobolev Spaces: Fourier characterization, Duality of \(H^s \& H^{-s}\), Sobolev & Compact Sobolev embedding, Trace theorem, Meyers-Serrin theorem. \textit{Elliptic Regularity}: \(L^2\) regularity, hypoellipticity, existence & uniqueness, eigenfunctions, Dirichlet form, strongly elliptic, coercivity, Garding’s Inequality, Lax-Milgram Lemma, Neumann-Poincare Inequality.

  • References: Folland, Evans, Jeffrey Rauch, Taylor, Jost.

  • Course Project Hypoellipticity of Elliptic Operators. References: Prof. Richard Melrose’s notes on Differential Analysis.

  • \(\textbf{Comments}\): Extra problem sessions on Saturday were held where students were supposed to present solutions to exercises. I spent maximum time on this course as I had to pick up a lot of material from diverse resources in very short time. I can’t describe how vast the study of PDEs has become. The number of hours spent in the library with Folland, Evans, notes was more than for any other course and yet it was just scratching the surface of thiis field.

  • Grading: Grade was based on Midsem, Endsem, Project, Homework and two quizzes. Project was individual presentation of a topic on blackboard.