Sourdis, Christos (2014) Uniform estimates for positive solutions of a class of semilinear elliptic equations and related Liouville and onedimensional symmetry results. (Submitted)
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Abstract
We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a nondegeneracy condition that was assumed in the latter reference. Furthermore, we can generalize an old result, concerning semilinear elliptic nonlinear eigenvalue problems. Moreover, we study the boundary layer of global minimizers of the corresponding singular perturbation problem. For the above applications, our approach is based on a refinement of a useful result, concerning the behavior of global minimizers of the associated energy over large balls, subject to Dirichlet conditions. Combining this refinement with global bifurcation theory and the celebrated sliding method, we can prove uniform estimates for solutions away from their nodal set. Combining our approach with apriori estimates that we obtain by blowup, a doubling lemma, and known Liouville type theorems, we can give a new proof of a known Liouville type theorem without using boundary blowup solutions. We can also provide an alternative proof, and a useful extension, of a Liouville theorem, involving the presence of an obstacle. Making use of the latter extension, we consider the singular perturbation problem with mixed boundary conditions. Moreover, we prove some new onedimensional symmetry and rigidity properties of certain entire solutions to AllenCahn type equations, as well as in half spaces, convex cylindrical domains. In particular, we provide a new proof of Gibbons' conjecture.
Item Type:  Article 

Divisions:  Faculty of Engineering, Science and Mathematics > School of Mathematics > Department of Applied Mathematics 
Depositing User:  Dr. Christos Sourdis 
Date Deposited:  05 Mar 2014 16:22 
Last Modified:  11 Dec 2014 16:03 
URI:  http://preprints.acmac.uoc.gr/id/eprint/299 
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Uniform estimates for positive solutions of a class of semilinear elliptic equations and related Liouville and onedimensional symmetry results. (deposited 17 Mar 2013 11:31)
 Uniform estimates for positive solutions of a class of semilinear elliptic equations and related Liouville and onedimensional symmetry results. (deposited 05 Mar 2014 16:22) [Currently Displayed]
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