Renormalized solutions of the fractional Laplace equation

We define renormalized solutions for the problems of the kind beta(u) + (-Delta)(s/2)u sic f in R(n), f is an element of L(1)(R(n)). Here beta is a maximal monotone graph in R, and (-Delta)(s/2), s is an element of (0,2), is the fractional Laplace operator which is a particular case of Levy 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. (C) 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.