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Fundamental solutions and Liouville type theorems for fractional operators

Patricio Felmer (U. de Chile) - martes 22 febrero 2011, 12:00 horas
Ponente: Patricio Felmer (U. de Chile)
Cuándo 22/02/2011
de 12:00 a 13:00
Dónde Salón "Graciela Salicrup"
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Resumen:

In this talk we present basic properties of a class of nonlinear integral operators deeply related to the existence of fundamental solutions for the operator. Our goal is to establish Liouville type Theorems: non-existence Theorems are for positive entire solutions for Iu \leq 0 and for Iu + u^p \leq 0, p > 1.
We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamards Three Sphere Theorem, but we still obtain some of its consequences that are sufficient for the arguments.

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