Renormalized solutions of the fractional Laplace equation [Solutions renormalisées de l'équation de Laplace fractionnaire]

摘要

We define renormalized solutions for the problems of the kind β(u)+(-Δ)~(s/2)u?f in R~n, f?L~1(Rn). Here β is a maximal monotone graph in R, and (-Δ)~(s/2), s?(0,2), is the fractional Laplace operator which is a particular case of Lévy diffusions. We prove well-posedness in the framework of renormalized solutions. Then the Cauchy problem for the associated evolution equations can be solved using the Crandall-Liggett semigroup technique.