geometric measures and conservation...
TRANSCRIPT
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Lecture Room C1 Weeks 1-4, Hilary Term 2020
Wednesdays 14:00-16:00
By Professor Gui-Qiang G. Chen
Geometric Measuresand
Conservation Laws
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Geometric Measures
have contributed greatly to the development of Conservation Laws, Partial Differential Equations, Calculus of Variations, Geometric Analysis, …
have a wide range of applications toDifferential Geometry/Topology,Continuum Physics, Fluid Mechanics,Stochastic Analysis, Dynamical Systems, ……
An introduction to some facets of Geometric Measures and Conservation Laws, and related applications.
Basic Analysis & PDE - the only essential prerequisites. However, some familiarity with basic measure theory,
functional analysis, nonlinear PDEs, and differentialgeometry is desirable.
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Topics:
1. Connections: Geometric Measures and Conservation Laws
2. Review: Basic Measure Theory
3. Hausdorff Measures
4. Area/Co-Area Formulas
5. BV Functions and Sets of Finite Perimeter
6. Theory of Divergence-Measure Fields and Connections with
Conservation Laws
7. *Differentiability and Approximation
8. *Varifolds and Currents
9. *Further Connections with Nonlinear PDEs
The topics with * are optional, depending on the course
development.
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References:::
1. H. Federer: Geometric Measure Theory, Springer-Verlag: Berlin,1996.
2. L. C. Evans & R. F. Gariepy: Measure Theory and Fine Properties of
Functions , CRC Press: Boca Raton, Florida, 1992.
3. R. Hardt & L. Simon: Seminar on Geometric Measure Theory,
Birkhauser, 1986.
4. L. Ambrosio, N. Fusco & D. Pallara: Functions of Bounded Variation
and Free Discontinuity Problems, Oxford Univ. Press, 2000.
5. W. P. Ziemer: Weakly Differentiable Functions, Springer: NY, 1989.
6. C. M. Dafermos: Hyperbolic Conservation Laws in Continuum Physics,
4th Edition, Springer-Verlag: Berlin, 2016.
7. F. Morgan: Geometric Measure Theory: A Beginners Guide,
Academic Press: Boston, 1988.
8. H. Whitney: Geometric Integration Theory, Princeton Univ. Press, 1957
9. L. C. Evans: Partial Differential Equations, 2nd ed., AMS: Providence,
RI, 2010.
10. G.-Q. Chen: Some Lecture Notes
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Geometric Measure Theory
could be described as differential geometry, generalised through measure theory to deal with
MapsSurfaces
that are not necessarily smooth.
DeGiorgi (1961), H. Federer (1969)
Almgren, Schoen-Simon, Bomberi, …..
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