MATH/MAPL 674, Spring 97: PARTIAL DIFFERENTIAL EQUATIONS II

DESCRIPTION

This course is a second semester of a two semester sequence. We will first finish our treatment of conservation laws by examining systems. We will then discuss modern methods for PDEs: distributions, functional analysis, Sobolev spaces, bounded and compact operators in Hilbert spaces, and linear elliptic PDEs.

TEXTBOOKS

L.C. Evans, Partial Differential Equations, Berkeley Lectures Notes, University of California

M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, Texts Appl. Math. 13, Springer, 1993

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1983.

SYLLABUS

1. Systems of conservation laws
   • Examples, linearization
   • Weak solutions, Rankine-Hugoniot condition
   • Simple waves, genuine nonlinearity and linear degeneracy
   • Lax shock condition, Riemann problem, p-system.
2. Distributions
   • Test functions and distributions, localization
   • Convergence of distributions
   • Derivatives, convolution, Fourier transform.
3. Hilbert spaces
   • Norms, completeness, separability, space $L^2$
   • Orthogonal projection, duality, Riesz representation theorem, weak convergence
   • Sobolev spaces, approximation, extension, Sobolev inequalities, compactness.
4. Bounded operators in Hilbert spaces
   • Domain, range, inverse, norm, spectrum and resolvent
   • Self-adjoint operators, compact operators
   • Fredholm altrenative, spectral theorem.
5. Second order linear elliptic PDEs
   • Ellipticity, Lax-Milgram theorem
   • Existence of weak solutions
   • Regularity
   • Eigenfunction expansions.

EVALUATION

Homeworks (50 %), Midterm Exam (20 %), and Final Exam (30 %).