28.02.2018

Inverse problems reading group (Autumn 2014)

The inverse problems reading group in Fall 2014 will meet on Wednesdays at 10.15–11.15 in room MaD245. This fall we will start by going through some basic results related to unique continuation for elliptic PDEs. The idea is to try to understand the ideas of the methods in simple cases. The reading group also provides a place to discuss the current affairs of the inverse problems group.

Notes on unique continuation (23.10.) Warning: rough draft, may contain mistakes, will be updated gradually.

Possible topics for the fall include:

  • PDE with real-analytic coefficients (Holmgren's theorem)
  • L^2 Carleman inequalities
  • Doubling/three spheres inequalities
  • Frequency function method
  • L^p Carleman estimates
  • The 2D case
  • Counterexamples to unique continuation
  • Pseudoconvexity for general operators
  • Nonlinear equations

The programme is as follows; it will be specified gradually during the fall.

References for topics

Holmgren's theorem:

  • F. John, Partial differential equations (Section 3.5), 4th edition, Springer-Verlag, 1982.
  • L. Hörmander, The analysis of linear partial differential operators, vol. 1 (Section 8.6).
  • F. Treves, Basic linear partial differential equations (Section II.21), Academic Press, 1975.

L^2 Carleman inequalities:

  • L. Hörmander, The analysis of linear partial differential operators, vol. 3 (Section 17.2).
  • L. Hörmander, The analysis of linear partial differential operators, vol. 4 (Chapter 28).
  • C. Kenig, CNA summer school lecture notes, https://www.math.cmu.edu/cna/LectureNotesFiles/Keniglecture1.pdf.
  • N. Lerner, Carleman inequalities (lecture notes), https://www.imj-prg.fr/~nicolas.lerner/m2carl.pdf.
  • J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators (lecture notes), http://hal.archives-ouvertes.fr/docs/00/55/95/68/PDF/carleman-notes.pdf.
  • D. Tataru, Carleman estimates (unfinished lecture notes), http://math.berkeley.edu/~tataru/papers/ucpnotes.ps.
  • D. Tataru, Unique continuation for PDEs (short expository note), http://math.berkeley.edu/~tataru/papers/shortucp.ps.

Doubling/three spheres inequalities, frequency function method:

  • G. Alessandrini, L. Rondi, E. Rosset, S. Vessella, The stability for the Cauchy problem for elliptic equations, http://arxiv.org/pdf/0907.2882.pdf.
  • N. Garofalo, F. Lin, Monotonicity properties of variational integrals, A_p weights and unique continuation, Indiana U Math J, 1986.
  • N. Garofalo, F. Lin, Unique continuation for elliptic operators: A geometric-variational approach, CPAM, 1987.
  • N. Marola, S. Granlund, On a frequency function approach to the unique continuation principle, http://arxiv.org/abs/1110.0945.

L^p Carleman inequalities:

  • C. Sogge, Fourier integrals in classical analysis (Section 5.1), Cambridge University Press, 1993.

Nonlinear equations:

  • S. Armstrong, L. Silvestre, Unique continuation for fully nonlinear elliptic equations, http://arxiv.org/abs/1102.1673.