MATH742: Geometric Analysis (Fall 2015)
This will be an advanced course in elliptic and parabolic PDE with a slant
towards applications in geometry and geometric (as opposed to frequency or
Fourier domain) methods such as ball coverings, the coarea formula, the
isoperimetric inequality, and Hausdorff measure.
Prerequisites: MATH673 and MATH674, or permission of the instructor.
MATH740 is not required. If needed I will review some basic differential
geometry of submanifolds such as the first and second fundamental form of a hypersurface in R^n.
1) Scalar elliptic equations
- Linear theory in Holder spaces (Riesz potentials, Simon's blowup method)
- Linear theory in W^{k,p} spaces, p not equal to 2 (Calderon-Zygmund theory)
- Linear theory with measurable coefficients (De Giorgi-Moser iteration)
2) Scalar parabolic equations
- Basic properties of the heat kernel on R^n (uniqueness, growth, analyticity)
- Heat kernel estimates and the logarithmic Sobolev inequality
- Nash's entropy method and heat kernel regularity under measurable coefficients
3) Elliptic and parabolic systems
- C^{0,alpha} and higher regularity of bounded weak solutions
- unbounded and pathological weak solutions, regularity off of negligible sets
- detailed analysis of a specific problem such as minimal surfaces, Yang-Mills, or Navier-Stokes
In 1) we will often be using the book by Gilbarg and Trudinger, which has plenty of
exercises. There will also be options for student projects.