Advanced Partial Differential Equations

Syllabus Details

• Unit I:(Distribution Theory) Test functions; distributions; order of the distribution; support of distributions; derivative of distributions; convolutions of distributions; fundamental solutions; the Schwartz space; Fourier transforms; tempered distributions.
• Unit II:(Laplace Equation in Higher Dimensions) Fundamental solution; Green’s functions; Poisson integral formula; mean value properties; weak and strong maximum principle; Hopf’s lemma; gradient estimates; removable singularity; Perron’s method.
• Unit III:(Heat Equation in Higher Dimensions) Fourier transforms; fundamental solution; regularity of solutions; weak and strong maximum principle; Harnack’s inequalities; gradient estimates.
• Unit IV:(Wave Equation in Higher Dimensions) Solution of the wave equation in higher dimensions using spherical means; Huygens’ principle; Duhamel’s principle; energy estimates.

Course Learning Objective

1. Describe the notion of the distributional derivative, and multiplication and convolution of a distribution with a test function.
2. Explain that the Fourier transform is an isometric isomorphism on L²(Rⁿ), and describe the notion of the Fourier transform of distributions.
3. Distinguish the behavior of the solution of the wave equation in odd and even dimensions.
4. Apply the maximum principle for elliptic and parabolic linear partial differential equations (PDEs).
5. Prove the energy estimates for linear PDEs.

Recommended Books

1. Kesavan, S. Topics in Functional Analysis and Applications. New Age International, 2003
2. Han, Qing. A Basic Course in Partial Differential Equations. Graduate Studies in Mathematics, Vol. 120, AMS, 2013
3. Strichartz, R. S. A Guide to Distribution Theory and Fourier Transforms. World Scientific, 2003
4. Stakgold, I. Green’s Functions and Boundary Value Problems. John Wiley & Sons, 1979
5. Hörmander, L. The Analysis of Linear Partial Differential Operators. Vol. 1. Springer, 1983