Instructor: Marius Mitrea

Description:

The aim is to study issues such as regularity, boundary behavior,
estimates, and integral representation formulas for solutions
of various classes of PDE's which include the Laplace operator,
the Lame system (of elastostatics) and the Stokes operator (of
hydrodynamics). The emphasis is on the tools, typically from
harmonic and functional analysis, which allow us to obtain
optimal results.

Contents:

(0) Hardy-Littlewood Maximal Function
(1) The Cauchy Integral Operator on Lipschitz Curves
(2) Singular Integrals, Maximal Operators, Cotlar's inequality
(3) Existence of pointwise values for singular integrals
(4) Fundamental solutions and layer potentials
(5) Jump relations
(6) Reduction of a boundary-value problem to a boundary integral equation
(7) Well-posedness in various smoothness spaces of basic problems
from mathematical physics

Prerequisites: Math 8420