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Nonlinear Maxwell Equations (Summer Semester 2017)

The lecture of 24 May has to be cancelled. It is shifted to a later date which will be discussed during the lectures.

Schedule
Lecture: Wednesday 11:30-13:00 SR 3.69
Lecturers
Lecturer Prof. Dr. Roland Schnaubelt
Office hours: Wednesday, 11:30 - 13:00, and by appointment
Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30)
Email: schnaubelt@kit.edu

The Maxwell equations are the fundamental laws governing electro-magnetic phenomena. We will study the time-depending case and nonlinear instantaneous material laws. In the semilinear case where only the conductivity is nonlinear, one can treat the system successfully in L^2 within the framework of nonlinear contraction semigroups and monotone operators. We will sketch this theory in Hilbert spaces (which resembles the linear case) and then establish (global) wellposedness of semilinear Maxwell equations on domains under suitable monotonicity assumptions. Possibly we also look at decay properties. In the quasilinear case the polarization or magnetization depend on the fields in a nonlinear way. This case is much more demanding. Here we will focus on full space problems where one can establish local wellposedness for initial values in H^3 by energy methods. We will further discuss blow-up phenomena and sketch the available results on domains.

Examination

There is an oral exam of about 20 min.

References

  • S. Benzoni-Gavage and D. Serre: Multidimensional Hyperbolic Partial Differential Equations.
  • H. Brezis: Opérateurs maximaux monotones.
  • R. Showalter: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations.