Theory of Partial Differential Equations

Code: MA752 | L-T-P-C: 4-0-0-8

MA752 Theory of Partial Differential Equations L-T-P-C [4-0-0-8]

Review of Sobolev spaces. weak solutions, eigenvalues and eigenfunctions of symmetric and non-symmetric elliptic operators. evolution equations, existence of weak solutions, maximum principle, interior and boundary regularities. Nonlinear elliptic equations: Nonlinear variational problems. first and second variations, existence of minimizers, nonlinear eigenvalue problems. Nonvariational techniques: monotonicity methods, fixed point methods, Nemytskii and pseudo-nRinotone operators. geometric properties of solutions. radial symmetry. Hamilton Jacobi equations: viscosity solutions, uniqueness, control theory, Hamilton-Jacobi-Bellman equations. Semigroup methods: Strongly continuous semigroups, infinitesimal generator, Hille-Yosida theorem, applications to wave and Schrodinger equations, analytic semigroups and their generators. Energy methods for evolution problems. System of conservation laws: Riemann's problem: simple waves, rarefraction waves, shock waves, contact discontinuities, local solution of Riemann's problem, vanishing viscosity, traveling waves, entropy/entropy-flux pairs.

Texts/References:

1. Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathmatics, Vol.19. American Mathematical Society. Providence, 1998.
2. D. Gilberg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York. 1983.
3. A. Pazy, Sentigroup,s of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
4. M. Renardy, B.C. Rogers. An Introduction to Partial Differential Equations, Springer, New York, 1993.
5. O.A. Ladyzhenskaya. N.N. Uraltseva, Linear and Quasilinear' Elliptic Equations, Academic Press, 1968.
6. P.-L. Lions. Generalized Solutions of Hamilton-Jacobi Equations, Research Notes in Mathematics 69, Pitman, 1982.
7. P. Lax, Hyperbolic Systems of Conservation Laws and Mathematical Theory of Shock Waves, SIAM, 1973.