Diff. Geom, Math. Phys., PDE: Mostafa Fazly

  • Date: 01/08/2013
  • Time: 15:30
Mostafa Fazly, UBC

University of British Columbia


m-Liouville theorems for elliptic PDEs

De Giorgi in 1978 conjectured that bounded and monotone solutions of the Allen-Cahn equation must be one-dimensional up to dimension eight. This conjecture is known to be true for N=<3 and with extra (natural) assumptions for 4=<N=<8. We state a counterpart of the above conjecture for gradient systems introducing the concept of monotonicity for systems. Then, we prove this conjecture for dimensions up to three and applying a geometric Poincare inequality for stable solutions we show that the gradients of various components of the solutions are parallel.

On the other hand, we ask under what conditions we can prove solutions of a PDE are m-dimensional for 0=<m=<N-1. This leads us to define the concept of “m-Liouville theorem” for PDEs. The motivation to this definition is the Liouville theorem (or 0-Liouville theorem) that we have seen in elementary analysis stating that bounded harmonic functions on the whole space must be constant (0-dimensional).

This is the main part of my dissertation under the supervision of Nassif Ghoussoub.
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Location: ESB 2012