Introductin to PDE

Winter/Spring.
I. Various ways of representing solutions to various PDEs: Fundamental solutions. Mean value formula (for Laplace and heat equations). Maximal principle. Liouville theorem. Analyticity. Self-similar solutions. Hopf-Cole transformations. Potential functions. Hodograph, Legendre, and Lewy transformations. Power series, Cauchy-Kovalevskaya theorem. Lewy's counterexample.

II. Qualitative theory for (linear elliptic and parabolic) PDEs: Sobolev spaces. Existence of weak solutions. Regularity. Maximum principle. Harnack inequality. Eigenvalues and eigenfunctions.

III. Variational and non-variational techniques for nonlinear PDEs: Existence of minimizers. Regularity. Constraints. Critical points. Monotonicity method. Fixed point methods. Sub/super solutions. Nonexistence. Moving plane method. Viscosity solutions.

Textbook: Partial differential equations, L. C. Evans. [E]

Reserved reference: PDE, F. John [J]

Prerequisite: Advanced calculus.

HW1 1/10 [E] Sec1.5 2, 4 Sec2.5 1, 3, 4 [J] Sec4.1 Problem 5
HW2 1/24 [E] Sec2.5 5, 6, 7, 8, 9
HW3 2/21 [E] Sec2.5 10, 11, 12, 13, 14 [J] p. 73 #3
HW4 3/9 [E] Sec4.7 1, 2, 3, 4, 9

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