Topics in Partial Differential Equations - Math 825 - Fall 2006

Lecturer:

Irina Mitrea, 225 Kerchof Hall, im3p@virginia.edu, Phone: 928-2787.

General description:

The scope of the course is to present an up-to-date, rigorous, and to a large extent self-contained, treatment of some of the most basic partial differential equations of mathematical physics, via the modern tools of Harmonic Analysis. Examples include the Laplace equation, the Lamé system of elastostatics, the Stokes system of hydrostatics and the Maxwell system of electromagnetism. The topics presented in this class belong to an area that undergoes active developments. Along the way we will list open problems of research level.

Prerequisite: The course is appropriate for any student who has finished the graduate analysis sequence.

References:

B. Dahlberg, C.Kenig and G. Verchota, Boundary value problems for the system of elastostatics on Lipschitz domains, Duke Math. J., 57 (1988), 795--818.
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, 1998.
E. Fabes, Layer potential methods for boundary value problems on Lipschitz domains, Potential Theory, Surveys and Problems, J. Král et al eds., Springer-Verlag Lecture Notes in Math., No. 1344, 1988, 55--80.
E. Fabes, M. Jodeit and N. Riviére, Potential techniques for boundary value problems on $C^1$ domains, Acta Math. 141 (1978), 165--186.
E. B. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769--793.
C. E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, No. 83, AMS, Providence, RI, 1994.
V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israel Program for Scientific Translations, Jerusalem, 1965.
O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, 1963.
I. S. Sokolnikoff, Mathematical Theory of Elasticity, R. E. Krieger Publishing Comp., Florida, 1983.
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.
M. E. Taylor, Partial Differential Equations II, Qualitative Studies of Linear Equations, Texts in Applied Mathematics, Vol. 116, Springer-Verlag New York, 1996.
M. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, Vol. 81, AMS, 2000.
G. C. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, Journal of Functional Analysis, 69 (1984), 572--611.
W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, Springer Verlag, 1989.

Course web page:

http://www.people.virginia.edu/~im3p/mathpage825.html

Projects:

First Category (basic)

Fourier Transform: the Fourier transform of functions of rapid descent; the Fourier transform of distributions of slow growth; Plancherel's Theorem.
The Hardy Littlewood maximal function: Vitali's covering lemma; weak type 1-1 estimates, Lp boundedness for p∈(1,∞].
The Marcinkiewicz interpolation theorem: weak type (p,q), non-increasing rearrangements of measurable functions; integral inequalities for functions on (0,∞)
Decomposition in cubes of open sets in the Euclidean space: Whitney decompositions; the fundamental lemma of Calderon and Zygmund
Function Spaces in the Euclidean Space: Riesz and Bessel potentials, Sobolev spaces, Besov spaces, Hölder spaces.

Second Category (challenging)

The geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs by M. Melnikov and J. Verdera.
Boundary Value Problems for Second Order Elliptic Systems.

Lectures:
MWF 11:00- 11:50 a.m., Kerchof 317

Office Hours:

Irina Mitrea, 225 Kerchof Hall, MW 12:00 - 12:50 p.m. (or by appointment).

Grading:

Two projects per student (related to the covered material) will be assigned during the semester. Each project counts 45%, and the remaining 10% will be given for class participation.