Chennai Mathematical Institute


3.30 pm, Seminar Hall
A test for harmonicity

E.K. Narayanan
Indian Institute of Science, Bangalore.


Morera's theorem in complex analysis says that a continuous function, whose integrals over closed curves in a disc vanish, is holomorphic. Questions regarding the family of curves required to test holomorphicty led to a lot of research in the recent past. Considering holomorphic functions as solutions to a PDE, lead to similar questions for harmonicity. Consider the unit disc in the complex plane and its translates by points in [0, 1]. Let F be a smooth enough function defined in a neighborhood of the union of these discs. If we assume that F extends to a harmonic function from the boundary of these discs to the interior (which always happens via the Poisson integral, but notice that F need not agree with this extension inside the disc) together with a condition on the normal derivatives, then we show that F is actually harmonic. If time permits, generalizations to higher order PDE will also be discussed.