m-Liouville theorems and regularity results for elliptic PDEs

• Main Author: Fazly, Mostafa
• Language: English
• Published: University of British Columbia 2012
• Online Access: http://hdl.handle.net/2429/43751

This thesis which is a compendium of seven papers, focuses on the study of the semilinear elliptic equations and systems, on both bounded and unbounded domains of dimension n, most importantly the Allen-Cahn equation and the De Giorgi’s conjecture (1978). This conjecture brings together two groups of mathematicians: one specializing in nonlinear partial differential equations and another in differential geometry, more specifically on minimal surfaces and constant mean curvature surfaces. De Giorgi conjectured that the monotone and bounded solutions of the Allen-Cahn equation on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This is known to be true for n ≤ 3 and with extra (natural) assumptions for 4 ≤ n ≤ 8. Motivated by this conjecture, I have introduced two main concepts: The first concept is the “H-monotone solutions” that allows us to formulate a counterpart of the De Giorgi’s conjecture for system of equations stating that the H-monotone and bounded solutions of the gradient systems on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This seems to be in the right track to extend the De Giorgi's conjecture to systems. The second concept is the “m-Liouville theorem” for m = 0, · · · , n − 1 that allows us to formulate a counterpart of the De Giorgi’s conjecture for equations but this time for higher-dimensional solutions as opposed to 1-dimensional solutions. We use the induction idea that is to use 0-Liouville theorem (0- dimensional solutions) to prove 1-Liouville theorem (1-dimensional solutions) and then to prove (n − 1)- Liouville theorem ((n − 1)-dimensional solutions). The reason that we call this “m-Liouville theorem” is because of the great mathematician Joseph Liouville (1809-1882) who proved a classical theorem in complex analysis stating that "bounded harmonic functions on the whole space must be constant" and constants are 0-dimensional objects. 0-Liouville theorem is at the heart of this thesis and it includes various 0-Liouville theorems for various equations and system. In particular, we give a positive answer to the Henon-Lane-Emden conjecture in dimension three under an extra boundedness assumption. On the other hand, it is well known that there is a close relationship between the regularity of solutions on bounded domains and 0-Liouville theorem for related “limiting equations” on the whole space, via rescaling and blow up procedures. In this direction, we present regularity of solutions for gradient and twisted-gradient systems as well as the uniqueness results for nonlocal eigenvalue problems. The novelty here is a stability inequality for both gradient and twisted-gradient systems that gives us the chance to adjust the known techniques and ideas (for equations) to systems.